Solução_Calculo_Stewart_6e

F.
TX.10
Solução detalhada de todos os exercícios ímpares do livro: Cálculo, James Stewart - VOLUME I e VOLUME II
"O que sabemos é uma gota, o que ignoramos é um oceano." Isaac Newton
23/11/2010.TX

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TX.10
1 FUNCTIONS AND MODELS
1.1 Four Ways to Represent a Function
In exercises requiring estimations or approximations, your answers may vary slightly from the answers given here.
1. (a) The point (−1, −2) is on the graph of f,sof(−1) = −2.
(b) When x =2, y is about 2.8,sof(2) ≈ 2.8.
(c) f(x) =2is equivalent to y =2.Wheny =2,wehavex = −3 and x =1.
(d) Reasonable estimates for x when y =0are x = −2.5 and x =0.3.
(e) The domain of f consists of all x-values on the graph of f. For this function, the domain is −3 ≤ x ≤ 3,or[−3, 3].
The range of f consists of all y-values on the graph of f. For this function, the range is −2 ≤ y ≤ 3,or[−2, 3].
(f ) As x increases from −1 to 3, y increases from −2 to 3. Thus,f is increasing on the interval [−1, 3].
3. From Figure 1 in the text, the lowest point occurs at about (t, a) =(12, −85). The highest point occurs at about (17, 115).
Thus, the range of the vertical ground acceleration is −85 ≤ a ≤ 115. Written in interval notation, we get [−85, 115].
5. No, the curve is not the graph of a function because a vertical line intersects the curve more than once. Hence, the curve fails
the Vertical Line Test.
7. Yes, the curve is the graph of a function because it passes the Vertical Line Test. The domain is [−3, 2] and the range
is [−3, −2) ∪ [−1, 3].
9. The person’s weight increased to about 160 pounds at age 20 and stayed fairly steady for 10 years. The person’s weight
dropped to about 120 pounds for the next 5 years, then increased rapidly to about 170 pounds. The next 30 years saw a gradual
increase to 190 pounds. Possible reasons for the drop in weight at 30 years of age: diet, exercise, health problems.
11. The water will cool down almost to freezing as the ice
melts. Then, when the ice has melted, the water will
slowly warm up to room temperature.
13. Of course, this graph depends strongly on the
geographical location!
15. As the price increases, the amount sold decreases. 17.
9

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10 ¤ CHAPTER 1 FUNCTIONS AND MODELS
19. (a) (b) From the graph, we estimate the number of
cell-phone subscribers worldwide to be about
92 million in 1995 and 485 million in 1999.
21. f(x) =3x 2 − x +2.
f(2) = 3(2) 2 − 2+2=12− 2+2=12.
f(−2) = 3(−2) 2 − (−2)+2=12+2+2=16.
f(a) =3a 2 − a +2.
f(−a) =3(−a) 2 − (−a)+2=3a 2 + a +2.
f(a +1)=3(a +1) 2 − (a +1)+2=3(a 2 +2a +1)− a − 1+2=3a 2 +6a +3− a +1=3a 2 +5a +4.
2f(a) =2· f(a) =2(3a 2 − a +2)=6a 2 − 2a +4.
f(2a) =3(2a) 2 − (2a)+2=3(4a 2 ) − 2a +2=12a 2 − 2a +2.
f(a 2 )=3(a 2 ) 2 − (a 2 )+2=3(a 4 ) − a 2 +2=3a 4 − a 2 +2.
[f(a)] 2 = 3a 2 − a +2 2 = 3a 2 − a +2 3a 2 − a +2
=9a 4 − 3a 3 +6a 2 − 3a 3 + a 2 − 2a +6a 2 − 2a +4=9a 4 − 6a 3 +13a 2 − 4a +4.
f(a + h) =3(a + h) 2 − (a + h)+2=3(a 2 +2ah + h 2 ) − a − h +2=3a 2 +6ah +3h 2 − a − h +2.
23. f(x) =4+3x − x 2 ,sof(3 + h) =4+3(3+h) − (3 + h) 2 =4+9+3h − (9 + 6h + h 2 )=4− 3h − h 2 ,
and
f(3 + h) − f(3)
h
= (4 − 3h − h2 ) − 4
h
=
h(−3 − h)
h
= −3 − h.
25.
f(x) − f(a)
x − a
=
1
x − 1 a − x
a
x − a = xa
x − a =
a − x −1(x − a)
=
xa(x − a) xa(x − a) = − 1
ax
27. f(x) =x/(3x − 1) is defined for all x except when 0=3x − 1 ⇔ x = 1 , so the domain
3
is
x ∈ R | x 6= 1 3 = −∞,
1
3 ∪ 1 , ∞ 3
.
29. f(t) = √ t + 3√ t is defined when t ≥ 0. These values of t give real number results for √ t, whereas any value of t gives a real
number result for 3√ t.Thedomainis[0, ∞).
31. h(x) =1 √ 4
x 2 − 5x is defined when x 2 − 5x >0 ⇔ x(x − 5) > 0. Note that x 2 − 5x 6= 0since that would result in
division by zero. The expression x(x − 5) is positive if x5. (See Appendix A for methods for solving
inequalities.) Thus, the domain is (−∞, 0) ∪ (5, ∞).

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SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ¤ 11
33. f(x) =5is defined for all real numbers, so the domain is R,or(−∞, ∞).
The graph of f is a horizontal line with y-intercept 5.
35. f(t) =t 2 − 6t is defined for all real numbers, so the domain is R,or
(−∞, ∞). The graph of f is a parabola opening upward since the coefficient
of t 2 is positive. To find the t-intercepts, let y =0and solve for t.
0=t 2 − 6t = t(t − 6) ⇒ t =0and t =6.Thet-coordinate of the
vertex is halfway between the t-intercepts, that is, at t =3.Since
f(3) = 3 2 − 6 · 3=−9,thevertexis(3, −9).
37. g(x) = √ x − 5 is defined when x − 5 ≥ 0 or x ≥ 5,sothedomainis[5, ∞).
Since y = √ x − 5 ⇒ y 2 = x − 5 ⇒ x = y 2 +5,weseethatg is the
tophalfofaparabola.
39. G(x) =
3x + |x|
x if x ≥ 0
.Since|x| =
x
−x if x0
if x0
if x0
2 if x

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12 ¤ CHAPTER 1 FUNCTIONS AND MODELS
47. We need to solve the given equation for y. x +(y − 1) 2 =0 ⇔ (y − 1) 2 = −x ⇔ y − 1=± √ −x ⇔
y =1± √ −x. The expression with the positive radical represents the top half of the parabola, and the one with the negative
radical represents the bottom half. Hence, we want f(x) =1− √ −x. Note that the domain is x ≤ 0.
49. For 0 ≤ x ≤ 3, the graph is the line with slope −1 and y-intercept 3,thatis,y = −x +3.For3

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SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ¤ 13
61. f is an odd function because its graph is symmetric about the origin. g is an even function because its graph is symmetric with
respect to the y-axis.
63. (a) Because an even function is symmetric with respect to the y-axis, and the point (5, 3) is on the graph of this even function,
the point (−5, 3) must also be on its graph.
(b) Because an odd function is symmetric with respect to the origin, and the point (5, 3) is on the graph of this odd function,
the point (−5, −3) must also be on its graph.
65. f(x) = x
x 2 +1 .
f(−x) =
−x
(−x) 2 +1 =
So f is an odd function.
−x
x 2 +1 = − x
x 2 +1 = −f(x).
67. f(x) = x
x +1 ,sof(−x) =
−x
−x +1 =
x
x − 1 .
Sincethisisneitherf(x) nor −f(x), the function f is
neither even nor odd.
69. f(x) =1+3x 2 − x 4 .
f(−x) =1+3(−x) 2 − (−x) 4 =1+3x 2 − x 4 = f(x).
So f is an even function.
1.2 Mathematical Models: A Catalog of Essential Functions
1. (a) f(x) = 5√ x is a root function with n =5.
(b) g(x) = √ 1 − x 2 is an algebraic function because it is a root of a polynomial.
(c) h(x) =x 9 + x 4 is a polynomial of degree 9.
(d) r(x) = x2 +1
is a rational function because it is a ratio of polynomials.
x 3 + x
(e) s(x) =tan2x is a trigonometric function.
(f ) t(x) =log 10 x is a logarithmic function.
3. We notice from the figure that g and h are even functions (symmetric with respect to the y-axis) and that f is an odd function
(symmetric with respect to the origin). So (b) y = x 5 must be f. Sinceg is flatter than h near the origin, we must have
(c) y = x 8 matched with g and (a) y = x 2 matched with h.

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14 ¤ CHAPTER 1 FUNCTIONS AND MODELS
5. (a) An equation for the family of linear functions with slope 2
is y = f(x) =2x + b,whereb is the y-intercept.
(b) f(2) = 1 means that the point (2, 1) is on the graph of f. We can use the
point-slope form of a line to obtain an equation for the family of linear
functions through the point (2, 1). y − 1=m(x − 2), which is equivalent
to y = mx +(1− 2m) in slope-intercept form.
(c) To belong to both families, an equation must have slope m =2, so the equation in part (b), y = mx +(1− 2m),
becomes y =2x − 3. Itistheonly function that belongs to both families.
7. All members of the family of linear functions f(x) =c − x have graphs
that are lines with slope −1. They-intercept is c.
9. Since f(−1) = f(0) = f(2) = 0, f has zeros of −1, 0,and2,soanequationforf is f(x) =a[x − (−1)](x − 0)(x − 2),
or f(x) =ax(x +1)(x − 2). Because f(1) = 6, we’ll substitute 1 for x and 6 for f(x).
6=a(1)(2)(−1) ⇒ −2a =6 ⇒ a = −3, so an equation for f is f(x) =−3x(x +1)(x − 2).
11. (a) D =200,soc =0.0417D(a +1)=0.0417(200)(a +1)=8.34a +8.34. The slope is 8.34, which represents the
change in mg of the dosage for a child for each change of 1 year in age.
(b) For a newborn, a =0,soc =8.34 mg.
13. (a) (b) The slope of 9 means that F increases 9 degrees for each increase
5 5
of 1 ◦ C. (Equivalently, F increases by 9 when C increases by 5
and F decreases by 9 when C decreases by 5.) The F -intercept of
32 is the Fahrenheit temperature corresponding to a Celsius
temperature of 0.

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SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ¤ 15
15. (a) Using N in place of x and T in place of y,wefind the slope to be T 2 − T 1 80 − 70
=
N 2 − N 1 173 − 113 = 10
equation is T − 80 = 1 (N − 173) ⇔ T − 80 = 1 N − 173 ⇔ T = 1 N + 307
6 6 6 6 6
60 = 1 . So a linear
6
307
6
=51.16 .
(b) The slope of 1 means that the temperature in Fahrenheit degrees increases one-sixth as rapidly as the number of cricket
6
chirps per minute. Said differently, each increase of 6 cricket chirps per minute corresponds to an increase of 1 ◦ F.
(c) When N =150, the temperature is given approximately by T = 1 307
(150) + =76.16 ◦ F ≈ 76 ◦ F.
6 6
17. (a) We are given
change in pressure
10 feet change in depth = 4.34 =0.434. UsingP for pressure and d for depth with the point
10
(d, P )=(0, 15), we have the slope-intercept form of the line, P =0.434d +15.
(b) When P = 100,then100 = 0.434d +15 ⇔ 0.434d =85 ⇔ d = 85
0.434
≈ 195.85 feet. Thus, the pressure is
100 lb/in 2 at a depth of approximately 196 feet.
19. (a) The data appear to be periodic and a sine or cosine function would make the best model. A model of the form
f(x) =a cos(bx)+c seems appropriate.
(b) The data appear to be decreasing in a linear fashion. A model of the form f(x) =mx + b seems appropriate.
Some values are given to many decimal places. These are the results given by several computer algebra systems — rounding is left to the reader.
21. (a)
(b) Using the points (4000, 14.1) and (60,000, 8.2),weobtain
8.2 − 14.1
y − 14.1 = (x − 4000) or, equivalently,
60,000 − 4000
y ≈−0.000105357x +14.521429.
A linear model does seem appropriate.
(c) Using a computing device, we obtain the least squares regression line y = −0.0000997855x +13.950764.
The following commands and screens illustrate how to find the least squares regression line on a TI-83 Plus.
Enter the data into list one (L1) and list two (L2). Press
to enter the editor.
Find the regession line and store it in Y 1.Press .

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16 ¤ CHAPTER 1 FUNCTIONS AND MODELS
Note from the last figure that the regression line has been stored in Y 1 andthatPlot1hasbeenturnedon(Plot1is
highlighted). You can turn on Plot1 from the Y= menu by placing the cursor on Plot1 and pressing
or by
pressing .
Now press
to produce a graph of the data and the regression
line. Note that choice 9 of the ZOOM menu automatically selects a window
that displays all of the data.
(d) When x =25,000, y ≈ 11.456; or about 11.5 per 100 population.
(e) When x =80,000, y ≈ 5.968;orabouta6% chance.
(f ) When x = 200,000, y is negative, so the model does not apply.
23. (a)
(b)
A linear model does seem appropriate.
Using a computing device, we obtain the least squares
regression line y =0.089119747x − 158.2403249,
where x is the year and y is the height in feet.
(c) When x = 2000,themodelgivesy ≈ 20.00 ft. Note that the actual winning height for the 2000 Olympics is less than the
winning height for 1996—so much for that prediction.
(d) When x =2100, y ≈ 28.91 ft. This would be an increase of 9.49 ft from 1996 to 2100. Even though there was an increase
of 8.59 ft from 1900 to 1996, it is unlikely that a similar increase will occur over the next 100 years.
25. Using a computing device, we obtain the cubic
function y = ax 3 + bx 2 + cx + d with
a =0.0012937, b = −7.06142, c =12,823,
and d = −7,743,770. Whenx = 1925,
y ≈ 1914 (million).

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SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ¤ 17
1.3 New Functions from Old Functions
1. (a) If the graph of f is shifted 3 units upward, its equation becomes y = f(x)+3.
(b) If the graph of f is shifted 3 units downward, its equation becomes y = f(x) − 3.
(c) If the graph of f is shifted 3 units to the right, its equation becomes y = f(x − 3).
(d) If the graph of f is shifted 3 units to the left, its equation becomes y = f(x +3).
(e) If the graph of f is reflected about the x-axis, its equation becomes y = −f(x).
(f ) If the graph of f is reflected about the y-axis, its equation becomes y = f(−x).
(g) If the graph of f isstretchedverticallybyafactorof3, its equation becomes y =3f(x).
(h) If the graph of f is shrunk vertically by a factor of 3, its equation becomes y = 1 f(x). 3
3. (a) (graph 3) The graph of f is shifted 4 units to the right and has equation y = f(x − 4).
(b) (graph 1) The graph of f is shifted 3 units upward and has equation y = f(x)+3.
(c) (graph 4) The graph of f is shrunk vertically by a factor of 3 and has equation y = 1 f(x). 3
(d) (graph 5) The graph of f is shifted 4 units to the left and reflected about the x-axis. Its equation is y = −f(x +4).
(e) (graph 2) The graph of f is shifted 6 units to the left and stretched vertically by a factor of 2. Its equation is
y =2f(x +6).
5. (a) To graph y = f(2x) we shrink the graph of f
horizontally by a factor of 2.
(b) To graph y = f 1
2 x westretchthegraphoff
horizontally by a factor of 2.
The point (4, −1) on the graph of f corresponds to the
point 1
2 · 4, −1 =(2, −1).
(c) To graph y = f(−x) we reflect the graph of f about
the y-axis.
The point (4, −1) on the graph of f corresponds to the
point (2 · 4, −1) = (8, −1).
(d) To graph y = −f(−x) we reflect the graph of f about
the y-axis, then about the x-axis.
The point (4, −1) on the graph of f corresponds to the
point (−1 · 4, −1) = (−4, −1).
The point (4, −1) on the graph of f corresponds to the
point (−1 · 4, −1 · −1) = (−4, 1).

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18 ¤ CHAPTER 1 FUNCTIONS AND MODELS
7. The graph of y = f(x) = √ 3x − x 2 has been shifted 4 units to the left, reflected about the x-axis, and shifted downward
1 unit. Thus, a function describing the graph is
y = −1 ·
reflect
about x-axis
f (x +4)
shift
4 units left
− 1
shift
1 unit left
This function can be written as
y = −f(x +4)− 1=− 3(x +4)− (x +4) 2 − 1=− 3x +12− (x 2 +8x +16)− 1=− √ −x 2 − 5x − 4 − 1
9. y = −x 3 : Start with the graph of y = x 3 and reflect
about the x-axis. Note: Reflecting about the y-axis
gives the same result since substituting −x for x gives
us y =(−x) 3 = −x 3 .
11. y =(x +1) 2 : Start with the graph of y = x 2
and shift 1 unit to the left.
13. y =1+2cosx: Start with the graph of y =cosx, stretch vertically by a factor of 2,andthenshift1 unit upward.
15. y =sin(x/2): Start with the graph of y =sinx and stretch horizontally by a factor of 2.

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SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ¤ 19
17. y = √ x +3: Start with the graph of
y = √ x and shift 3 units to the left.
19. y = 1 2 (x2 +8x) = 1 2 (x2 +8x +16)− 8= 1 2 (x +4)2 − 8: Start with the graph of y = x 2 , compress vertically by a
factor of 2,shift4 units to the left, and then shift 8 units downward.
0 0 0 0
21. y =2/(x +1): Start with the graph of y =1/x,shift1 unit to the left, and then stretch vertically by a factor of 2.
23. y = |sin x|: Start with the graph of y =sinx and reflect all the parts of the graph below the x-axis about the x-axis.
25. This is just like the solution to Example 4 except the amplitude of the curve (the 30 ◦ NcurveinFigure9onJune21)is
14 − 12 = 2. So the function is L(t) =12+2sin 2π
365 (t − 80) . March 31 is the 90th day of the year, so the model gives
L(90) ≈ 12.34 h. The daylight time (5:51 AM to 6:18 PM)is12 hours and 27 minutes, or 12.45 h. The model value differs
from the actual value by 12.45−12.34
12.45
≈ 0.009,lessthan1%.
27. (a) To obtain y = f(|x|), the portion of the graph of y = f(x) to the right of the y-axisisreflected about the y-axis.
(b) y =sin|x|
(c) y = |x|

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20 ¤ CHAPTER 1 FUNCTIONS AND MODELS
29. f(x) =x 3 +2x 2 ; g(x) =3x 2 − 1. D = R for both f and g.
(f + g)(x) =(x 3 +2x 2 )+(3x 2 − 1) = x 3 +5x 2 − 1, D = R.
(f − g)(x) =(x 3 +2x 2 ) − (3x 2 − 1) = x 3 − x 2 +1, D = R.
(fg)(x) =(x 3 +2x 2 )(3x 2 − 1) = 3x 5 +6x 4 − x 3 − 2x 2 , D = R.
f
(x) = x3 +2x 2
g 3x 2 − 1 , D = x | x 6=± √ 1
since 3x 2 − 1 6= 0.
3
31. f(x) =x 2 − 1, D = R; g(x) =2x +1, D = R.
(a) (f ◦ g)(x) =f(g(x)) = f(2x +1)=(2x +1) 2 − 1=(4x 2 +4x +1)− 1=4x 2 +4x, D = R.
(b) (g ◦ f)(x) =g(f(x)) = g(x 2 − 1) = 2(x 2 − 1) + 1 = (2x 2 − 2) + 1 = 2x 2 − 1, D = R.
(c) (f ◦ f)(x) =f(f(x)) = f(x 2 − 1) = (x 2 − 1) 2 − 1=(x 4 − 2x 2 +1)− 1=x 4 − 2x 2 , D = R.
(d) (g ◦ g)(x) =g(g(x)) = g(2x +1)=2(2x +1)+1=(4x +2)+1=4x +3, D = R.
33. f(x) =1− 3x; g(x) =cosx. D = R for both f and g, and hence for their composites.
(a) (f ◦ g)(x) =f(g(x)) = f(cos x) =1− 3cosx.
(b) (g ◦ f)(x) =g(f(x)) = g(1 − 3x) =cos(1− 3x).
(c) (f ◦ f)(x) =f(f(x)) = f(1 − 3x) =1− 3(1 − 3x) =1− 3+9x =9x − 2.
(d) (g ◦ g)(x) =g(g(x)) = g(cos x) =cos(cosx)
[Note that this is not cos x · cos x.]
35. f(x) =x + 1 +1
, D = {x | x 6= 0}; g(x) =x , D = {x | x 6=−2}
x x +2
x +1
(a) (f ◦ g)(x) =f(g(x)) = f = x +1
x +2 x +2 + 1 = x +1
x +1 x +2 + x +2
x +1
x +2
=
(x +1)(x +1)+(x +2)(x +2)
(x +2)(x +1)
=
x 2 +2x +1 + x 2 +4x +4
(x +2)(x +1)
Since g(x) is not defined for x = −2 and f(g(x)) is not defined for x = −2 and x = −1,
the domain of (f ◦ g)(x) is D = {x | x 6=−2, −1}.
(b) (g ◦ f)(x) =g(f(x)) = g x + 1
=
x
x + 1
+1
x
x + 1 =
+2
x
x 2 +1+x
x
x 2 +1+2x
x
Since f(x) is not defined for x =0and g(f(x)) is not defined for x = −1,
the domain of (g ◦ f)(x) is D = {x | x 6=−1, 0}.
(c) (f ◦ f)(x)=f(f(x)) = f x + 1
= x + 1
+ 1
x x x + 1 x
= x(x) x 2 +1 +1 x 2 +1 + x(x)
x(x 2 +1)
= x4 +3x 2 +1
, D = {x | x 6= 0}
x(x 2 +1)
= x + 1 x + 1
x 2 +1
x
= x4 + x 2 + x 2 +1+x 2
x(x 2 +1)
= x2 + x +1
x 2 +2x +1 = x2 + x +1
(x +1) 2
= x + 1 x + x
x 2 +1
= 2x2 +6x +5
(x +2)(x +1)

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TX.10
x +1
(d) (g ◦ g)(x) =g(g(x)) = g =
x +2
x +1
x +2 +1
=
x +1
x +2 +2
SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ¤ 21
x +1+1(x +2)
x +2
x +1+2(x +2)
x +2
Since g(x) is not defined for x = −2 and g(g(x)) is not defined for x = − 5 3 ,
the domain of (g ◦ g)(x) is D = x | x 6=−2, − 5 3
.
37. (f ◦ g ◦ h)(x) =f(g(h(x))) = f(g(x − 1)) = f(2(x − 1)) = 2(x − 1) + 1 = 2x − 1
39. (f ◦ g ◦ h)(x) =f(g(h(x))) = f(g(x 3 +2))=f[(x 3 +2) 2 ]
= f(x 6 +4x 3 +4)= (x 6 +4x 3 +4)− 3= √ x 6 +4x 3 +1
=
x +1+x +2 2x +3
=
x +1+2x +4 3x +5
41. Let g(x) =x 2 +1and f(x) =x 10 .Then(f ◦ g)(x) =f(g(x)) = f(x 2 +1)=(x 2 +1) 10 = F (x).
43. Let g(x) = 3√ x and f(x) = x
1+x .Then(f ◦ g)(x) =f(g(x)) = f( 3√ x )= 3√ x
1+ 3√ x = F (x).
45. Let g(t) =cost and f(t) = √ t.Then(f ◦ g)(t) =f(g(t)) = f(cos t) = √ cos t = u(t).
47. Let h(x) =x 2 , g(x) =3 x ,andf(x) =1− x. Then
(f ◦ g ◦ h)(x) =f(g(h(x))) = f(g(x 2 )) = f 3 x2 =1− 3 x2 = H(x).
49. Let h(x) = √ x, g(x) =secx, andf(x) =x 4 .Then
(f ◦ g ◦ h)(x) =f(g(h(x))) = f(g( √ x )) = f(sec √ x )=(sec √ x ) 4 =sec 4 ( √ x )=H(x).
51. (a) g(2) = 5, because the point (2, 5) is on the graph of g. Thus,f(g(2)) = f(5) = 4, because the point (5, 4) is on the
graph of f.
(b) g(f(0)) = g(0) = 3
(c) (f ◦ g)(0) = f(g(0)) = f(3) = 0
(d) (g ◦ f)(6) = g(f(6)) = g(6). This value is not defined, because there is no point on the graph of g that has
x-coordinate 6.
(e) (g ◦ g)(−2) = g(g(−2)) = g(1) = 4
(f ) (f ◦ f)(4) = f(f(4)) = f(2) = −2
53. (a) Using the relationship distance = rate · time with the radius r as the distance, we have r(t) =60t.
(b) A = πr 2 ⇒ (A ◦ r)(t) =A(r(t)) = π(60t) 2 = 3600πt 2 . This formula gives us the extent of the rippled area
(in cm 2 )atanytimet.
55. (a) From the figure, we have a right triangle with legs 6 and d, and hypotenuse s.
By the Pythagorean Theorem, d 2 +6 2 = s 2 ⇒ s = f(d) = √ d 2 +36.
(b) Using d = rt,wegetd =(30km/hr)(t hr) =30t (in km). Thus,
d = g(t) =30t.
(c) (f ◦ g)(t) =f(g(t)) = f(30t) = (30t) 2 +36= √ 900t 2 +36. This function represents the distance between the
lighthouse and the ship as a function of the time elapsed since noon.

F.
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22 ¤ CHAPTER 1 FUNCTIONS AND MODELS
57. (a)
(b)
H(t) =
0 if t

F.
TX.10
1.4 Graphing Calculators and Computers
SECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS ¤ 23
1. f(x) = √ x 3 − 5x 2
(a) [−5, 5] by [−5, 5]
(There is no graph shown.)
(b) [0, 10] by [0, 2] (c) [0, 10] by [0, 10]
The most appropriate graph is produced in viewing rectangle (c).
3. Since the graph of f(x) =5+20x − x 2 is a
parabola opening downward, an appropriate viewing
rectangle should include the maximum point.
5. f(x) = 4√ 81 − x 4 is defined when 81 − x 4 ≥ 0 ⇔
x 4 ≤ 81 ⇔ |x| ≤ 3, so the domain of f is [−3, 3]. Also
0 ≤ 4√ 81 − x 4 ≤ 4√ 81 = 3,sotherangeis[0, 3].
7. The graph of f(x) =x 3 − 225x is symmetric with respect to the origin.
Since f(x) =x 3 − 225x = x(x 2 − 225) = x(x + 15)(x − 15),there
are x-intercepts at 0, −15,and15. f(20) = 3500.
9. The period of g(x) = sin(1000x) is 2π ≈ 0.0063 and its range is
1000
[−1, 1]. Sincef(x) =sin 2 (1000x) is the square of g,itsrangeis
[0, 1] and a viewing rectangle of [−0.01, 0.01] by [0, 1.1] seems
appropriate.
11. The domain of y = √ x is x ≥ 0,sothedomainoff(x) =sin √ x is [0, ∞)
and the range is [−1, 1]. With a little trial-and-error experimentation, we find
that an Xmax of 100 illustrates the general shape of f,soanappropriate
viewing rectangle is [0, 100] by [−1.5, 1.5].

F.
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24 ¤ CHAPTER 1 FUNCTIONS AND MODELS
13. The first term, 10 sin x, has period 2π and range [−10, 10]. It will be the dominant term in any “large” graph of
y =10sinx +sin100x, as shown in the first figure. The second term, sin 100x,hasperiod 2π = π and range [−1, 1].
100 50
Itcausesthebumpsinthefirst figure and will be the dominant term in any “small” graph, as shown in the view near the
origin in the second figure.
15. We must solve the given equation for y to obtain equations for the upper and
lower halves of the ellipse.
4x 2 +2y 2 =1 ⇔ 2y 2 =1− 4x 2 ⇔ y 2 = 1 − 4x2
2
1 − 4x
2
y = ±
2
⇔
17. From the graph of y =3x 2 − 6x +1
and y =0.23x − 2.25 in the viewing
rectangle [−1, 3] by [−2.5, 1.5],itis
difficult to see if the graphs intersect.
If we zoom in on the fourth quadrant,
we see the graphs do not intersect.
19. From the graph of f(x) =x 3 − 9x 2 − 4, we see that there is one solution
of the equation f(x) =0and it is slightly larger than 9. By zooming in or
using a root or zero feature, we obtain x ≈ 9.05.
21. We see that the graphs of f(x) =x 2 and g(x) =sinx intersect twice. One
solution is x =0. The other solution of f = g is the x-coordinate of the
point of intersection in the first quadrant. Using an intersect feature or
zooming in, we find this value to be approximately 0.88. Alternatively, we
could find that value by finding the positive zero of h(x) =x 2 − sin x.
Note: After producing the graph on a TI-83 Plus, we can find the approximate value 0.88 by using the following keystrokes:
. The “1” is just a guess for 0.88.

F.
TX.10
SECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS ¤ 25
23. g(x) =x 3 /10 is larger than f(x) =10x 2 whenever x>100.
25. We see from the graphs of y = |sin x − x| and y =0.1 that there are
two solutions to the equation |sin x − x| =0.1: x ≈−0.85 and
x ≈ 0.85. The condition |sin x − x| < 0.1 holds for any x lying
between these two values, that is, −0.85

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26 ¤ CHAPTER 1 FUNCTIONS AND MODELS
33. y 2 = cx 3 + x 2 .Ifc0 (b) R (c) (0, ∞) (d) See Figures 4(c), 4(b), and 4(a), respectively.
3. All of these graphs approach 0 as x →−∞, all of them pass through the point
(0, 1), and all of them are increasing and approach ∞ as x →∞.Thelargerthe
base, the faster the function increases for x>0, and the faster it approaches 0 as
x →−∞.
Note: The notation “x →∞” can be thought of as “x becomes large” at this point.
More details on this notation are given in Chapter 2.
5. The functions with bases greater than 1 (3 x and 10 x ) are increasing, while those
with bases less than 1
1 x
and
1 x
3
10 are decreasing. The graph of 1
x
is the
3
reflection of that of 3 x about the y-axis, and the graph of 1
10
x
is the reflection of
that of 10 x about the y-axis. The graph of 10 x increases more quickly than that of
3 x for x>0, and approaches 0 faster as x →−∞.
7. We start with the graph of y =4 x (Figure 3) and then
shift 3 units downward. This shift doesn’t affect the
domain, but the range of y =4 x − 3 is (−3, ∞) .
There is a horizontal asymptote of y = −3.
y =4 x y =4 x − 3

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SECTION 1.5 EXPONENTIAL FUNCTIONS ¤ 27
9. We start with the graph of y =2 x (Figure 3),
reflectitaboutthey-axis, and then about the
x-axis (or just rotate 180 ◦ to handle both
reflections) to obtain the graph of y = −2 −x .
In each graph, y =0is the horizontal
asymptote.
y =2 x y =2 −x y = −2 −x
11. We start with the graph of y = e x (Figure 13) and reflect about the y-axis to get the graph of y = e −x . Then we compress the
graphverticallybyafactorof2 to obtain the graph of y = 1 2 e−x andthenreflect about the x-axis to get the graph of
y = − 1 2 e−x . Finally, we shift the graph upward one unit to get the graph of y =1− 1 2 e−x .
13. (a) To find the equation of the graph that results from shifting the graph of y = e x 2 units downward, we subtract 2 from the
original function to get y = e x − 2.
(b) To find the equation of the graph that results from shifting the graph of y = e x 2 units to the right, we replace x with x − 2
in the original function to get y = e (x−2) .
(c) To find the equation of the graph that results from reflecting the graph of y = e x about the x-axis, we multiply the original
function by −1 to get y = −e x .
(d) To find the equation of the graph that results from reflecting the graph of y = e x about the y-axis, we replace x with −x in
the original function to get y = e −x .
(e) To find the equation of the graph that results from reflecting the graph of y = e x about the x-axis and then about the
y-axis, we first multiply the original function by −1 (to get y = −e x ) and then replace x with −x in this equation to
get y = −e −x .
15. (a) The denominator 1+e x is never equal to zero because e x > 0, so the domain of f(x) =1/(1 + e x ) is R.
(b) 1 − e x =0 ⇔ e x =1 ⇔ x =0, so the domain of f(x) =1/(1 − e x ) is (−∞, 0) ∪ (0, ∞).
17. Use y = Ca x with the points (1, 6) and (3, 24). 6=Ca 1 C =
6
a
6
and 24 = Ca 3 ⇒ 24 = a 3
a
⇒
4=a 2 ⇒ a =2 [since a>0] andC = 6 2 =3. The function is f(x) =3· 2x .
19. If f(x) =5 x ,then
f(x + h) − f(x)
h
= 5x+h − 5 x
h
= 5x 5 h − 5 x
h
= 5x 5 h − 1
h
5
=5 x h − 1
.
h

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28 ¤ CHAPTER 1 FUNCTIONS AND MODELS
21. 2 ft =24in, f(24) = 24 2 in = 576 in =48ft. g(24) = 2 24 in =2 24 /(12 · 5280) mi ≈ 265 mi
23. The graph of g finally surpasses that of f at x ≈ 35.8.
25. (a) Fifteen hours represents 5 doubling periods (one doubling period is three hours). 100 · 2 5 = 3200
(b) In t hours, there will be t/3 doubling periods. The initial population is 100,
so the population y at time t is y = 100 · 2 t/3 .
(c) t =20 ⇒ y = 100 · 2 20/3 ≈ 10,159
(d) We graph y 1 = 100 · 2 x/3 and y 2 =50,000. The two curves intersect at
x ≈ 26.9, so the population reaches 50,000 in about 26.9 hours.
27. An exponential model is y = ab t ,wherea =3.154832569 × 10 −12
and b =1.017764706. This model gives y(1993) ≈ 5498 million and
y(2010) ≈ 7417 million.
29. From the graph, it appears that f is an odd function (f is undefined for x =0).
To prove this, we must show that f(−x) =−f(x).
f(−x) = 1 − e1/(−x) 1 −
1 − 1
e(−1/x)
=
1+e1/(−x) 1+e = e 1/x
(−1/x)
1+ 1
e 1/x
= − 1 − e1/x
= −f(x)
1+e1/x so f is an odd function.
· e1/x
e = e1/x − 1
1/x e 1/x +1
1.6 Inverse Functions and Logarithms
1. (a) See Definition 1.
(b) It must pass the Horizontal Line Test.
3. f is not one-to-one because 2 6= 6,butf(2) = 2.0 =f(6).
5. No horizontal line intersects the graph of f more than once. Thus, by the Horizontal Line Test, f is one-to-one.

F.
TX.10
SECTION 1.6 INVERSE FUNCTIONS AND LOGARITHMS ¤ 29
7. The horizontal line y =0(the x-axis) intersects the graph of f in more than one point. Thus, by the Horizontal Line Test,
f is not one-to-one.
9. The graph of f(x) =x 2 − 2x is a parabola with axis of symmetry x = − b
2a = − −2 =1.Pickanyx-values equidistant
2(1)
from 1 to find two equal function values. For example, f(0) = 0 and f(2) = 0,sof is not one-to-one.
11. g(x) =1/x. x 1 6=x 2 ⇒ 1/x 1 6=1/x 2 ⇒ g (x 1 ) 6=g (x 2 ),sog is one-to-one.
Geometric solution: The graph of g is the hyperbola shown in Figure 14 in Section 1.2. It passes the Horizontal Line Test,
so g is one-to-one.
13. A football will attain every height h up to its maximum height twice: once on the way up, and again on the way down. Thus,
even if t 1 does not equal t 2 , f(t 1 ) may equal f(t 2 ),sof is not 1-1.
15. Since f(2) = 9 and f is 1-1, we know that f −1 (9) = 2. Remember, if the point (2, 9) is on the graph of f, then the point
(9, 2) is on the graph of f −1 .
17. First, we must determine x such that g(x) =4. By inspection, we see that if x =0,theng(x) =4.Sinceg is 1-1 (g is an
increasing function), it has an inverse, and g −1 (4) = 0.
19. We solve C = 5 (F − 32) for F : 9 C = F − 32 ⇒ F = 9 9 5 5
C +32. This gives us a formula for the inverse function, that
is, the Fahrenheit temperature F as a function of the Celsius temperature C. F ≥−459.67 ⇒ 9 5 C +32≥−459.67 ⇒
9
5
C ≥−491.67 ⇒ C ≥−273.15, the domain of the inverse function.
21. f(x) = √ 10 − 3x ⇒ y = √ 10 − 3x (y ≥ 0) ⇒ y 2 =10− 3x ⇒ 3x =10− y 2 ⇒ x = − 1 3 y2 + 10
3 .
Interchange x and y: y = − 1 3 x2 + 10
3 .Sof −1 (x) =− 1 3 x2 + 10
3 . Note that the domain of f −1 is x ≥ 0.
23. y = f(x) =e x3 ⇒ ln y = x 3 ⇒ x = 3√ ln y. Interchange x and y: y = 3√ ln x. Sof −1 (x) = 3√ ln x.
25. y = f(x) =ln(x +3) ⇒ x +3=e y ⇒ x = e y − 3. Interchange x and y: y = e x − 3. Sof −1 (x) =e x − 3.
27. y = f(x) =x 4 +1 ⇒ y − 1=x 4 ⇒ x = 4√ y − 1 (not ± since
x ≥ 0). Interchange x and y: y = 4√ x − 1. Sof −1 (x) = 4√ x − 1. The
graph of y = 4√ x − 1 is just the graph of y = 4√ x shifted right one unit.
From the graph, we see that f and f −1 are reflections about the line y = x.
29. Reflect the graph of f about the line y = x. The points (−1, −2), (1, −1),
(2, 2),and(3, 3) on f are reflected to (−2, −1), (−1, 1), (2, 2),and(3, 3)
on f −1 .

F.
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30 ¤ CHAPTER 1 FUNCTIONS AND MODELS
31. (a) It is defined as the inverse of the exponential function with base a,thatis,log a x = y ⇔ a y = x.
(b) (0, ∞) (c) R (d) See Figure 11.
33. (a) log 5 125 = 3 since 5 3 =125. (b) log 3
1
27 = −3 since 3−3 = 1 3 3 = 1 27 .
35. (a) log 2 6 − log 2 15 + log 2 20 = log 2 ( 6
15 )+log 2 20 [by Law 2]
=log 2 ( 6 · 20) [by Law 1]
15
=log 2 8,and log 2 8=3since 2 3 =8.
(b) log 3 100 − log 3 18 − log 3 50 = log 100
3 18 − log3 50 = log 100
3 18·50
37. ln5+5ln3=ln5+ln3 5 [by Law 3]
=ln(5· 3 5 ) [by Law 1]
=ln1215
=log 3 ( 1 9 ),and log 3 1
9
= −2 since 3 −2 = 1 9 .
39. ln(1 + x 2 )+ 1 2 ln x − ln sin x =ln(1+x2 )+lnx 1/2 − ln sin x =ln[(1+x 2 ) √ x ] − ln sin x =ln (1 + x2 ) √ x
sin x
41. To graph these functions, we use log 1.5 x = ln x
ln 1.5 and log 50 x = ln x
ln 50 .
These graphs all approach −∞ as x → 0 + , and they all pass through the
point (1, 0). Also, they are all increasing, and all approach ∞ as x →∞.
The functions with larger bases increase extremely slowly, and the ones with
smaller bases do so somewhat more quickly. The functions with large bases
approach the y-axis more closely as x → 0 + .
43. 3 ft =36in, so we need x such that log 2 x =36 ⇔ x =2 36 =68,719,476,736. Inmiles,thisis
68,719,476,736 in · 1 ft
12 in · 1 mi
≈ 1,084,587.7 mi.
5280 ft
45. (a) Shift the graph of y =log 10 x five units to the left to
obtain the graph of y =log 10 (x +5).Notethevertical
asymptote of x = −5.
(b) Reflect the graph of y =lnx about the x-axis to obtain
the graph of y = − ln x.
y =log 10 x y =log 10 (x +5)
47. (a) 2lnx =1 ⇒ ln x = 1 2
⇒ x = e 1/2 = √ e
(b) e −x =5 ⇒ −x =ln5 ⇒ x = − ln 5
y =lnx
y = − ln x
49. (a) 2 x−5 =3 ⇔ log 2 3=x − 5 ⇔ x =5+log 2 3.
Or: 2 x−5 =3 ⇔ ln 2 x−5 =ln3 ⇔ (x − 5) ln 2 = ln 3 ⇔ x − 5= ln 3
ln 2
⇔ x =5+ ln 3
ln 2

F.
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SECTION 1.6 INVERSE FUNCTIONS AND LOGARITHMS ¤ 31
(b) ln x +ln(x − 1) = ln(x(x − 1)) = 1 ⇔ x(x − 1) = e 1 ⇔ x 2 − x − e =0. The quadratic formula (with a =1,
b = −1,andc = −e)givesx = 1 2
1 ±
√ 1+4e
, but we reject the negative root since the natural logarithm is not
defined for xe −1 ⇒ x>e −1 ⇒ x ∈ (1/e, ∞)
53. (a) For f(x) = √ 3 − e 2x ,wemusthave3 − e 2x ≥ 0 ⇒ e 2x ≤ 3 ⇒ 2x ≤ ln 3 ⇒ x ≤ 1 2
ln 3. Thus, the domain
of f is (−∞, 1 2
ln 3].
(b) y = f(x) = √ 3 − e 2x [note that y ≥ 0] ⇒ y 2 =3− e 2x ⇒ e 2x =3− y 2 ⇒ 2x =ln(3− y 2 ) ⇒
x = 1 2 ln(3 − y2 ). Interchange x and y: y = 1 2 ln(3 − x2 ).Sof −1 (x) = 1 2 ln(3 − x2 ). For the domain of f −1 ,wemust
have 3 − x 2 > 0 ⇒ x 2 < 3 ⇒ |x| < √ 3 ⇒ − √ 3

F.
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32 ¤ CHAPTER 1 FUNCTIONS AND MODELS
63. (a) In general, tan(arctan x) =x for any real number x. Thus,tan(arctan 10) = 10.
(b) sin −1 sin 7π 3
=sin
−1 sin π 3
=sin
−1 √ 3
2 = π 3 since sin π 3 = √ 3
2 and π 3 is in − π 2 , π 2
.
[Recall that 7π 3
= π 3
+2π and the sine function is periodic with period 2π.]
65. Let y =sin −1 x.Then− π 2 ≤ y ≤ π 2
⇒ cos y ≥ 0,socos(sin −1 x)=cosy = 1 − sin 2 y = √ 1 − x 2 .
67. Let y =tan −1 x.Thentan y = x, so from the triangle we see that
sin(tan −1 x)=siny =
x
√
1+x
2 .
69. The graph of sin −1 x is the reflection of the graph of
sin x about the line y = x.
71. g(x) =sin −1 (3x +1).
Domain (g) ={x | −1 ≤ 3x +1≤ 1} = {x | −2 ≤ 3x ≤ 0} = x | − 2 3 ≤ x ≤ 0 = − 2 3 , 0 .
Range (g) = y | − π 2 ≤ y ≤ π 2
=
−
π
2 , π 2
.
73. (a) If the point (x, y) is on the graph of y = f(x), then the point (x − c, y) is that point shifted c units to the left. Since f is
1-1, the point (y, x) is on the graph of y = f −1 (x) and the point corresponding to (x − c, y) on the graph of f is
(y, x − c) on the graph of f −1 . Thus, the curve’s reflection is shifted down the same number of units as the curve itself is
shiftedtotheleft.Soanexpressionfortheinversefunctionisg −1 (x) =f −1 (x) − c.
(b) If we compress (or stretch) a curve horizontally, the curve’s reflection in the line y = x is compressed (or stretched)
vertically by the same factor. Using this geometric principle, we see that the inverse of h(x) =f(cx) canbeexpressedas
h −1 (x) =(1/c) f −1 (x).

F.
TX.10
CHAPTER 1 REVIEW ¤ 33
1 Review
1. (a) A function f is a rule that assigns to each element x in a set A exactly one element, called f(x),inasetB. ThesetA is
called the domain of the function. The range of f is the set of all possible values of f(x) as x varies throughout the
domain.
(b) If f is a function with domain A,thenitsgraph is the set of ordered pairs {(x, f(x)) | x ∈ A}.
(c) Use the Vertical Line Test on page 16.
2. The four ways to represent a function are: verbally, numerically, visually, and algebraically. An example of each is given
below.
Verbally: An assignment of students to chairs in a classroom (a description in words)
Numerically: A tax table that assigns an amount of tax to an income (a table of values)
Visually: A graphical history of the Dow Jones average (a graph)
Algebraically: A relationship between distance, rate, and time: d = rt (an explicit formula)
3. (a) An even function f satisfies f(−x) =f(x) for every number x in its domain. It is symmetric with respect to the y-axis.
(b) An odd function g satisfies g(−x) =−g(x) for every number x in its domain. It is symmetric with respect to the origin.
4. A function f is called increasing on an interval I if f(x 1)

F.
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34 ¤ CHAPTER 1 FUNCTIONS AND MODELS
(c) (d) (e)
(f ) (g) (h)
9. (a) The domain of f + g is the intersection of the domain of f and the domain of g;thatis,A ∩ B.
(b) The domain of fg is also A ∩ B.
(c) The domain of f/g must exclude values of x that make g equal to 0;thatis,{x ∈ A ∩ B | g(x) 6= 0}.
10. Given two functions f and g,thecomposite function f ◦ g is defined by (f ◦ g)(x) =f(g (x)). The domain of f ◦ g is the
set of all x in the domain of g such that g(x) is in the domain of f.
11. (a) If the graph of f is shifted 2 units upward, its equation becomes y = f(x)+2.
(b) If the graph of f is shifted 2 units downward, its equation becomes y = f(x) − 2.
(c) If the graph of f is shifted 2 units to the right, its equation becomes y = f(x − 2).
(d) If the graph of f is shifted 2 units to the left, its equation becomes y = f(x +2).
(e) If the graph of f is reflected about the x-axis, its equation becomes y = −f(x).
(f ) If the graph of f is reflected about the y-axis, its equation becomes y = f(−x).
(g) If the graph of f isstretchedverticallybyafactorof2, its equation becomes y =2f(x).
(h) If the graph of f is shrunk vertically by a factor of 2, its equation becomes y = 1 2 f(x).
(i) If the graph of f is stretched horizontally by a factor of 2, its equation becomes y = f 1
2 x .
(j) If the graph of f is shrunk horizontally by a factor of 2, its equation becomes y = f(2x).
12. (a) A function f is called a one-to-one function if it never takes on the same value twice; that is, if f(x 1 ) 6=f(x 2 ) whenever
x 1 6=x 2 .(Or,f is 1-1 if each output corresponds to only one input.)
once.
Use the Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than
(b) If f is a one-to-one function with domain A and range B,thenitsinverse function f −1 has domain B and range A and is
defined by
f −1 (y) =x ⇔ f(x) =y
for any y in B. The graph of f −1 is obtained by reflecting the graph of f about the line y = x.

F.
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CHAPTER 1 REVIEW ¤ 35
13. (a)Theinversesinefunctionf(x) =sin −1 x is defined as follows:
sin −1 x = y ⇔ sin y = x and − π 2 ≤ y ≤ π 2
Its domain is −1 ≤ x ≤ 1 and its range is − π 2 ≤ y ≤ π 2 .
(b) The inverse cosine function f(x) =cos −1 x is defined as follows:
cos −1 x = y ⇔ cos y = x and 0 ≤ y ≤ π
Its domain is −1 ≤ x ≤ 1 and its range is 0 ≤ y ≤ π.
(c) The inverse tangent function f(x) =tan −1 x is defined as follows:
tan −1 x = y ⇔ tan y = x and − π 2

F.
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36 ¤ CHAPTER 1 FUNCTIONS AND MODELS
3. f(x) =x 2 − 2x +3,sof(a + h) =(a + h) 2 − 2(a + h)+3=a 2 +2ah + h 2 − 2a − 2h +3,and
f(a + h) − f(a)
h
= (a2 +2ah + h 2 − 2a − 2h +3)− (a 2 − 2a +3)
h
=
h(2a + h − 2)
h
=2a + h − 2.
5. f(x) =2/(3x − 1). Domain: 3x − 1 6= 0 ⇒ 3x 6= 1 ⇒ x 6= 1 . D =
−∞, 1 3 3 ∪ 1 , ∞ 3
Range: all reals except 0 (y =0is the horizontal asymptote for f.) R =(−∞, 0) ∪ (0, ∞)
7. h(x) =ln(x +6). Domain: x +6> 0 ⇒ x>−6. D =(−6, ∞)
Range: x +6> 0,soln(x +6)takes on all real numbers and, hence, the range is R.
R =(−∞, ∞)
9. (a)Toobtainthegraphofy = f(x)+8,weshiftthegraphofy = f(x) up 8 units.
(b)Toobtainthegraphofy = f(x +8),weshiftthegraphofy = f(x) left 8 units.
(c) To obtain the graph of y =1+2f(x), we stretch the graph of y = f(x) vertically by a factor of 2, and then shift the
resulting graph 1 unit upward.
(d)Toobtainthegraphofy = f(x − 2) − 2, we shift the graph of y = f(x) right 2 units (for the “−2” insidethe
parentheses), and then shift the resulting graph 2 units downward.
(e)Toobtainthegraphofy = −f(x),wereflect the graph of y = f(x) about the x-axis.
(f)Toobtainthegraphofy = f −1 (x),wereflect the graph of y = f(x) about the line y = x (assuming f is one–to-one).
11. y = − sin 2x: Start with the graph of y =sinx, compress horizontally by a factor of 2,andreflect about the x-axis.
13. y = 1 (1 + 2 ex ):
Startwiththegraphofy = e x ,
shift 1 unit upward, and compress
vertically by a factor of 2.
15. f(x) = 1
x +2 :
Startwiththegraphoff(x) =1/x
and shift 2 units to the left.

F.
TX.10
CHAPTER 1 REVIEW ¤ 37
17. (a) The terms of f are a mixture of odd and even powers of x,sof is neither even nor odd.
(b) The terms of f are all odd powers of x,sof is odd.
(c) f(−x) =e −(−x)2 = e −x2 = f(x),sof is even.
(d) f(−x) =1+sin(−x) =1− sin x. Nowf(−x) 6=f(x) and f(−x) 6=−f(x),sof is neither even nor odd.
19. f(x) =lnx, D =(0, ∞); g(x) =x 2 − 9, D = R.
(a) (f ◦ g)(x) =f(g(x)) = f(x 2 − 9) = ln(x 2 − 9).
Domain: x 2 − 9 > 0 ⇒ x 2 > 9 ⇒ |x| > 3 ⇒ x ∈ (−∞, −3) ∪ (3, ∞)
(b) (g ◦ f)(x) =g(f(x)) = g(ln x) =(lnx) 2 − 9. Domain: x>0,or(0, ∞)
(c) (f ◦ f)(x) =f(f(x)) = f(ln x) =ln(lnx). Domain: ln x>0 ⇒ x>e 0 =1,or(1, ∞)
(d) (g ◦ g)(x) =g(g(x)) = g(x 2 − 9) = (x 2 − 9) 2 − 9. Domain: x ∈ R,or(−∞, ∞)
21. Many models appear to be plausible. Your choice depends on whether you
think medical advances will keep increasing life expectancy, or if there is
bound to be a natural leveling-off of life expectancy. A linear model,
y =0.2493x − 423.4818, gives us an estimate of 77.6 years for the
year 2010.
23. We need to know the value of x such that f(x) =2x +lnx =2.Sincex =1gives us y =2, f −1 (2) = 1.
25. (a) e 2ln3 = e ln 3 2
=3 2 =9
(b) log 10 25 + log 10 4=log 10 (25 · 4) = log 10 100 = log 10 10 2 =2
(c) tan
arcsin 1 2 =tan
π
= √ 1 6 3
(d) Let θ =cos −1 4 5 ,socos θ = 4 5 .Thensin cos −1 4 5
=sinθ =
√
1 − cos2 θ =
1 −
4 2 9
5
= = 3 . 25 5
27. (a) The population would reach 900 in about 4.4 years.
100,000
(b) P =
100 + 900e ⇒ 100P −t +900Pe−t = 100,000 ⇒ 900Pe −t = 100,000 − 100P ⇒
e −t 100,000 − 100P
1000 − P
1000 − P
= ⇒ −t =ln
⇒ t = − ln
,orln
900P
9P
9P
required for the population to reach a given number P .
9 · 900
(c) P =900 ⇒ t =ln
=ln81≈ 4.4 years, as in part (a).
1000 − 900
9P
1000 − P
;thisisthetime

F.
TX.10

F.
TX.10
PRINCIPLES OF PROBLEM SOLVING
1. By using the area formula for a triangle, 1 (base)(height), in two ways, we see that
2
1
(4) (y) = 1 (h)(a),soa = 4y 2 2
h .Since42 + y 2 = h 2 , y = √ h 2 − 16,and
a = 4√ h 2 − 16
.
h
3. |2x − 1| =
2x − 1 if x ≥
1
2
1 − 2x if x< 1 2
and |x +5| =
x +5
−x − 5
if x ≥−5
if x

F.
TX.10
40 ¤ CHAPTER 1 PRINCIPLES OF PROBLEM SOLVING
9. |x| + |y| ≤ 1. The boundary of the region has equation |x| + |y| =1. In quadrants
I,II,III,andIV,thisbecomesthelinesx + y =1, −x + y =1, −x − y =1,and
x − y =1respectively.
11. (log 2 3)(log 3 4)(log 4 5) ···(log 31 32) =
ln 3
ln 2
ln 4
ln 3
ln 5
···
ln 4
ln 32 ln 32 ln 25
= =
ln 31 ln 2 ln 2 = 5ln2
ln 2 =5
13. ln x 2 − 2x − 2 ≤ 0 ⇒ x 2 − 2x − 2 ≤ e 0 =1 ⇒ x 2 − 2x − 3 ≤ 0 ⇒ (x − 3)(x +1)≤ 0 ⇒ x ∈ [−1, 3].
Since the argument must be positive, x 2 − 2x − 2 > 0 ⇒ x − 1 − √ 3 x − 1+ √ 3 > 0 ⇒
x ∈ −∞, 1 − √ 3 ∪ 1+ √ 3, ∞ . The intersection of these intervals is −1, 1 − √ 3 ∪ 1+ √ 3, 3 .
15. Let d be the distance traveled on each half of the trip. Let t 1 and t 2 be the times taken for the first and second halves of the trip.
For the first half of the trip we have t 1 = d/30 and for the second half we have t 2 = d/60. Thus, the average speed for the
entire trip is
is 40 mi/h.
total distance
total time
= 2d
t 1 + t 2
=
2d
d
30 + d 60
17. Let S n be the statement that 7 n − 1 is divisible by 6.
• S 1 is true because 7 1 − 1=6is divisible by 6.
· 60
60 = 120d
2d + d = 120d =40. The average speed for the entire trip
3d
• Assume S k is true, that is, 7 k − 1 is divisible by 6. Inotherwords,7 k − 1=6m for some positive integer m. Then
7 k+1 − 1=7 k · 7 − 1=(6m +1)· 7 − 1=42m +6=6(7m +1), which is divisible by 6,soS k+1 is true.
• Therefore, by mathematical induction, 7 n − 1 is divisible by 6 for every positive integer n.
19. f 0 (x) =x 2 and f n+1 (x) =f 0 (f n (x)) for n =0, 1, 2,....
f 1 (x) =f 0 (f 0 (x)) = f 0
x
2 = x 22 = x 4 , f 2 (x) =f 0 (f 1 (x)) = f 0 (x 4 )=(x 4 ) 2 = x 8 ,
f 3(x) =f 0(f 2(x)) = f 0(x 8 )=(x 8 ) 2 = x 16 , .... Thus, a general formula is f n(x) =x 2n+1 .

F.
TX.10
2 LIMITS AND DERIVATIVES
2.1 The Tangent and Velocity Problems
(b) Using the values of t that correspond to the points
1. (a) Using P (15, 250), we construct the following table:
30 (30, 0)
0−250
30−15 15
closest to P (t =10and t =20), we have
t Q slope = m PQ
5 (5, 694)
694−250
= − 444 −38.8+(−27.8)
5−15 10
= −33.3
2
444−250
10 (10, 444)
= − 194 = −38.8
10−15 5
111−250
20 (20, 111)
20−15
= − 139
5
= −27.8
28−250
25 (25, 28)
25−15 10
(c) From the graph, we can estimate the slope of the
tangent line at P to be −300
9
= −33.3.
3. (a)
x Q m PQ
(i) 0.5 (0.5, 0.333333) 0.333333
(ii) 0.9 (0.9, 0.473684) 0.263158
(iii) 0.99 (0.99, 0.497487) 0.251256
(iv) 0.999 (0.999, 0.499750) 0.250125
(v) 1.5 (1.5, 0.6) 0.2
(vi) 1.1 (1.1, 0.523810) 0.238095
(vii) 1.01 (1.01, 0.502488) 0.248756
(viii) 1.001 (1.001, 0.500250) 0.249875
(b) The slope appears to be 1 4 .
(c) y − 1 2 = 1 4 (x − 1) or y = 1 4 x + 1 4 .
5. (a) y = y(t) =40t − 16t 2 .Att =2, y =40(2)− 16(2) 2 =16. The average velocity between times 2 and 2+h is
y(2 + h) − y(2)
40(2 + h) − 16(2 + h)
2
− 16 −24h − 16h2
v ave = =
= = −24 − 16h, ifh 6= 0.
(2 + h) − 2
h
h
(i) [2, 2.5]: h =0.5, v ave = −32 ft/s
(ii) [2, 2.1]: h =0.1, v ave = −25.6 ft/s
(iii) [2, 2.05]: h =0.05, v ave = −24.8 ft/s
(iv) [2, 2.01]: h =0.01, v ave = −24.16 ft/s
(b) The instantaneous velocity when t =2(h approaches 0)is−24 ft/s.
41

F.
42 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
7. (a) (i) On the interval [1, 3], v ave =
(ii) On the interval [2, 3], v ave =
(iii) On the interval [3, 5], v ave =
(iv) On the interval [3, 4], v ave =
s(3) − s(1)
3 − 1
s(3) − s(2)
3 − 2
s(5) − s(3)
5 − 3
s(4) − s(3)
4 − 3
=
=
=
=
10.7 − 1.4
2
10.7 − 5.1
1
25.8 − 10.7
2
17.7 − 10.7
1
= 9.3
2
=5.6 m/s.
= 15.1
2
=7m/s.
=4.65 m/s.
=7.55 m/s.
(b)
Using the points (2, 4) and (5, 23) from the approximate tangent
line, the instantaneous velocity at t =3is about 23 − 4
5 − 2
≈ 6.3 m/s.
9. (a) For the curve y =sin(10π/x) and the point P (1, 0):
x Q m PQ
2 (2, 0) 0
1.5 (1.5, 0.8660) 1.7321
1.4 (1.4, −0.4339) −1.0847
1.3 (1.3, −0.8230) −2.7433
1.2 (1.2, 0.8660) 4.3301
1.1 (1.1, −0.2817) −2.8173
x Q m PQ
0.5 (0.5, 0) 0
0.6 (0.6, 0.8660) −2.1651
0.7 (0.7, 0.7818) −2.6061
0.8 (0.8, 1) −5
0.9 (0.9, −0.3420) 3.4202
As x approaches 1, the slopes do not appear to be approaching any particular value.
(b)
We see that problems with estimation are caused by the frequent
oscillations of the graph. The tangent is so steep at P that we need to
take x-values much closer to 1 in order to get accurate estimates of
its slope.
(c) If we choose x =1.001, then the point Q is (1.001, −0.0314) and m PQ ≈−31.3794. Ifx =0.999, thenQ is
(0.999, 0.0314) and m PQ = −31.4422. The average of these slopes is −31.4108. So we estimate that the slope of the
tangent line at P is about −31.4.

F.
TX.10
SECTION 2.2 THE LIMIT OF A FUNCTION ¤ 43
2.2 The Limit of a Function
1. As x approaches 2, f(x) approaches 5. [Or,thevaluesoff(x) can be made as close to 5 as we like by taking x sufficiently
close to 2 (but x 6= 2).] Yes, the graph could have a hole at (2, 5) and be definedsuchthatf(2) = 3.
3. (a) lim f(x) =∞ means that the values of f(x) canbemadearbitrarilylarge(aslargeasweplease)bytakingx
x→−3
(b)
sufficiently close to −3 (but not equal to −3).
lim f(x) =−∞ means that the values of f(x) can be made arbitrarily large negative by taking x sufficiently close to 4
x→4 +
through values larger than 4.
5. (a) f(x) approaches 2 as x approaches 1 from the left, so lim f(x) =2.
x→1− (b) f(x) approaches 3 as x approaches 1 from the right, so lim f(x) =3.
x→1 +
(c) lim
x→1
f(x) does not exist because the limits in part (a) and part (b) are not equal.
(d) f(x) approaches 4 as x approaches 5 from the left and from the right, so lim
x→5
f(x) =4.
(e) f(5) is not defined, so it doesn’t exist.
7. (a) lim g(t) =−1
t→0− (b) lim g(t) =−2
+
(c) lim
t→0
g(t) does not exist because the limits in part (a) and part (b) are not equal.
(d) lim g(t) =2
t→2− t→0
(e) lim g(t) =0
+
(f ) lim
t→2
g(t) does not exist because the limits in part (d) and part (e) are not equal.
(g) g(2) = 1
t→2
(h) lim
t→4
g(t) =3
9. (a) lim f(x) =−∞ (b) lim f(x) =∞
x→−7 x→−3
(d)
lim f(x) =−∞
x→6− (e) lim f(x) =∞
+
x→6
(f ) The equations of the vertical asymptotes are x = −7, x = −3, x =0,andx =6.
(c) lim f(x) =∞
x→0
11. (a) lim f(x) =1
x→0− (b)
lim f(x) =0
x→0 +
(c) lim
x→0
f(x) does not exist because the limits
in part (a) and part (b) are not equal.
13. lim f(x) =2, lim
x→1− x→1
f(x) =−2, f(1) = 2 15. lim f(x) =4, lim
+ +
x→3
f(3) = 3, f(−2) = 1
f(x) =2, lim f(x) =2,
x→3− x→−2

F.
44 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
17. For f(x) = x2 − 2x
x 2 − x − 2 :
x
f(x)
2.5 0.714286
2.1 0.677419
2.05 0.672131
2.01 0.667774
2.005 0.667221
2.001 0.666778
x
f(x)
1.9 0.655172
1.95 0.661017
1.99 0.665552
1.995 0.666110
1.999 0.666556
x 2 − 2x
It appears that lim
x→2 x 2 − x − 2 =0.¯6 = 2 . 3
TX.10
19. For f(x) = ex − 1 − x
x 2 :
x
f(x)
1 0.718282
0.5 0.594885
0.1 0.517092
0.05 0.508439
0.01 0.501671
x
f(x)
−1 0.367879
−0.5 0.426123
−0.1 0.483742
−0.05 0.491770
−0.01 0.498337
It appears that lim
x→0
e x − 1 − x
x 2 =0.5 = 1 2 .
√ x +4− 2
21. For f(x) =
:
x
x
f(x)
1 0.236068
0.5 0.242641
0.1 0.248457
0.05 0.249224
0.01 0.249844
It appears that lim
x→0
√ x +4− 2
x
23. For f(x) = x6 − 1
x 10 − 1 :
x f(x)
x f(x)
−1 0.267949
0.5 0.985337
−0.5 0.258343
0.9 0.719397
−0.1 0.251582
0.95 0.660186
−0.05 0.250786
0.99 0.612018
−0.01 0.250156
0.999 0.601200
It appears that lim
x→1
x 6 − 1
x 10 − 1 =0.6 = 3 5 .
x
f(x)
1.5 0.183369
1.1 0.484119
1.05 0.540783
1.01 0.588022
1.001 0.598800
25. lim
x→−3 + x +2
x +3 = −∞ since the numerator is negative and the denominator approaches 0 from the positive side as x →−3+ .
2 − x
27. lim = ∞ since the numerator is positive and the denominator approaches 0 through positive values as x → 1.
x→1 (x − 1)
2
29. Let t = x 2 − 9. Thenasx → 3 + , t → 0 + ,and lim ln(x2 − 9) = lim ln t = −∞ by (3).
x→3 + t→0 +
31. lim
x→2π − x csc x =
lim
x→2π −
x
= −∞ since the numerator is positive and the denominator approaches 0 through negative
sin x
values as x → 2π − .
33. (a) f(x) = 1
x 3 − 1 .
lim f(x) =−∞ and lim x→1− x→1 + 0.99 −33.7
x f(x)
0.5 −1.14
From these calculations, it seems that
0.9 −3.69
0.999 −333.7
0.9999 −3333.7
0.99999 −33,333.7
x f(x)
1.5 0.42
1.1 3.02
1.01 33.0
1.001 333.0
1.0001 3333.0
1.00001 33,333.3

F.
TX.10
SECTION 2.2 THE LIMIT OF A FUNCTION ¤ 45
(b) If x is slightly smaller than 1,thenx 3 − 1 will be a negative number close to 0,andthereciprocalofx 3 − 1,thatis,f(x),
will be a negative number with large absolute value. So lim f(x) =−∞.
x→1− If x is slightly larger than 1,thenx 3 − 1 will be a small positive number, and its reciprocal, f(x), will be a large positive
number. So lim f(x) =∞.
x→1 +
(c) It appears from the graph of f that
lim
x→1
f(x) =−∞ and lim f(x) =∞.
− +
x→1
35. (a) Let h(x) =(1+x) 1/x .
(b)
x
h(x)
−0.001 2.71964
−0.0001 2.71842
−0.00001 2.71830
−0.000001 2.71828
0.000001 2.71828
0.00001 2.71827
0.0001 2.71815
0.001 2.71692
It appears that lim
x→0
(1 + x) 1/x ≈ 2.71828, which is approximately e.
In Section 3.6 we will see that the value of the limit is exactly e.
37. For f(x) =x 2 − (2 x /1000):
(a)
x f(x)
1 0.998000
0.8 0.638259
0.6 0.358484
0.4 0.158680
0.2 0.038851
0.1 0.008928
0.05 0.001465
It appears that lim
x→0
f(x) =0.
(b)
x
f(x)
0.04 0.000572
0.02 −0.000614
0.01 −0.000907
0.005 −0.000978
0.003 −0.000993
0.001 −0.001000
It appears that lim
x→0
f(x) =−0.001.

F.
46 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
39. No matter how many times we zoom in toward the origin, the graphs of f(x) =sin(π/x) appear to consist of almost-vertical
lines. This indicates more and more frequent oscillations as x → 0.
41. There appear to be vertical asymptotes of the curve y =tan(2sinx) at x ≈ ±0.90
and x ≈ ±2.24. Tofind the exact equations of these asymptotes, we note that the
graph of the tangent function has vertical asymptotes at x = π + πn. Thus,we
2
must have 2sinx = π + πn, or equivalently, sin x = π + π n.Since
2 4 2
−1 ≤ sin x ≤ 1,wemusthavesin x = ± π and so x = ± 4 sin−1 π (corresponding
4
to x ≈ ±0.90). Just as 150 ◦ is the reference angle for 30 ◦ , π − sin −1 π 4
is the
reference angle for sin −1 π .Sox = ±
4
π − sin −1 π 4 are also equations of
vertical asymptotes (corresponding to x ≈ ±2.24).
2.3 Calculating Limits Using the Limit Laws
1. (a) lim
x→2
[f(x)+5g(x)] = lim
x→2
f(x) + lim
x→2
[5g(x)] [Limit Law 1]
=lim
x→2
f(x)+5lim
x→2
g(x) [Limit Law 3]
=4+5(−2) = −6
3
(b) lim [g(x)] 3 = lim g(x) [Limit Law 6]
x→2 x→2
=(−2) 3 = −8
(c) lim
x→2
f(x)=
lim
x→2
f(x) [Limit Law 11]
= √ 4=2

F.
TX.10
SECTION 2.3 CALCULATING LIMITS USING THE LIMIT LAWS ¤ 47
3f(x)
lim [3f(x)]
(d) lim
x→2 g(x) = x→2
lim g(x) [Limit Law 5]
x→2
=
3lim
x→2
f(x)
lim
x→2 g(x) [Limit Law 3]
= 3(4)
−2 = −6
g(x)
(e) Because the limit of the denominator is 0, we can’t use Limit Law 5. The given limit, lim , does not exist because the
x→2 h(x)
denominator approaches 0 while the numerator approaches a nonzero number.
g(x) h(x)
lim [g(x) h(x)]
x→2
(f ) lim =
x→2 f(x) lim f(x) [Limit Law 5]
x→2
=
lim g(x) · lim h(x)
x→2 x→2
lim f(x) [Limit Law 4]
x→2
= −2 · 0
4
=0
3. lim
x→−2 (3x4 +2x 2 − x +1)= lim
x→−2 3x4 + lim
x→−2 2x2 − lim x + lim 1 [Limit Laws 1 and 2]
x→−2 x→−2
= 3 lim
x→−2 x4 +2 lim
x→−2 x2 − lim
x→−2 x + lim
x→−2 1 [3]
=3(−2) 4 +2(−2) 2 − (−2) + (1)
[9,8,and7]
=48+8+2+1=59
5. lim (1 + 3√ x )(2− 6x 2 + x 3 ) = lim (1 + 3√ x ) · lim(2 − 6x 2 + x 3 ) [Limit Law 4]
x→8 x→8 x→8
= lim 1 + lim 3√ x · lim 2 − 6lim
x→8 x→8
x→8 x→8 x2 +limx 3 [1,2,and3]
x→8
7. lim
x→1
3
1+3x
=
1+4x 2 +3x 4
=
lim
x→1
= 1+ 3√ 8 · 2
− 6 · 8 2 +8 3 [7, 10, 9]
= (3)(130) = 390
3
1+3x
[6]
1+4x 2 +3x 4 3
[5]
lim (1 + 3x)
x→1
lim (1 +
x→1 4x2 +3x 4 )
lim 1+3limx
3
x→1 x→1
=
lim 1+4lim
[2, 1, and 3]
x→1 x→1 x2 +3limx 4
x→1
3 3 3
1 + 3(1)
4 1
=
= = = 1 [7, 8, and 9]
1 + 4(1) 2 +3(1) 4 8 2 8
9. lim
x→4 − √
16 − x2 = lim
x→4 − (16 − x2 ) [11]
= lim 16 − lim x2 [2]
x→4− x→4 −
= 16 − (4) 2 =0 [7 and 9]

F.
48 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
x 2 + x − 6 (x +3)(x − 2)
11. lim
=lim
=lim(x +3)=2+3=5
x→2 x − 2 x→2 x − 2
x→2
x 2 − x +6
13. lim
does not exist since x − 2 → 0 but x 2 − x +6→ 8 as x → 2.
x→2 x − 2
15. lim
t→−3
t 2 − 9
2t 2 +7t +3 = lim
t→−3
(t +3)(t − 3)
(2t +1)(t +3) = lim
t→−3
t − 3
2t +1 = −3 − 3
2(−3) + 1 = −6
−5 = 6 5
(4 + h) 2 − 16 (16 + 8h + h 2 ) − 16 8h + h 2 h(8 + h)
17. lim
=lim
=lim =lim =lim(8 + h) =8+0=8
h→0 h
h→0 h
h→0 h h→0 h h→0
19. Bytheformulaforthesumofcubes,wehave
lim
x→−2
x +2
x 3 +8 = lim
x→−2
x +2
(x +2)(x 2 − 2x +4) = lim
x→−2
1
x 2 − 2x +4 = 1
4+4+4 = 1
12 .
√ √
9 − t 3+ t 3 − t
21. lim
t→9 3 − √ t =lim
t→9 3 − √ √ √
=lim 3+ t =3+ 9=6
t
t→9
√ √ √ x +2− 3 x +2− 3 x +2+3
(x +2)− 9
23. lim
=lim
· √ = lim
x→7 x − 7 x→7 x − 7 x +2+3 x→7 (x − 7) √ x +2+3
25. lim
x→−4
27. lim
x→16
29. lim
t→0
=lim
x→7
x − 7
(x − 7) √ x +2+3 = lim
x→7
1
4 + 1 x +4
x
4+x = lim 4x
x→−4 4+x = lim
x→−4
4 − √ x
16x − x 2 = lim
x→16
1
t √ 1+t − 1
t
x +4
4x(4 + x) = lim
x→−4
(4 − √ x )(4 + √ x )
(16x − x 2 )(4 + √ x ) = lim
x→16
1
√ x +2+3
=
1
√
9+3
= 1 6
1
4x = 1
4(−4) = − 1 16
16 − x
x(16 − x)(4 + √ x )
1
= lim
x→16 x(4 + √ x ) = 1
16 4+ √ 16 = 1
16(8) = 1
128
1 − √ √ √
1+t 1 − 1+t 1+ 1+t
=lim
t→0 t √ = lim
1+t t→0 t √ t +1 1+ √ 1+t =lim
t→0
=lim
t→0
−1
√ 1+t
1+
√ 1+t
=
−1
√ 1+0
1+
√ 1+0
= − 1 2
−t
t √ 1+t 1+ √ 1+t
31. (a)
lim
x→0
x
√ 1+3x − 1
≈ 2 3
(b)
x
f(x)
−0.001 0.6661663
−0.0001 0.6666167
−0.00001 0.6666617
−0.000001 0.6666662
0.000001 0.6666672
0.00001 0.6666717
0.0001 0.6667167
0.001 0.6671663
The limit appears to be 2 3 .

F.
(c) lim
x→0
TX.10 SECTION 2.3 CALCULATING LIMITS USING THE LIMIT LAWS ¤ 49
√
x 1+3x +1 x √ 1+3x +1
x √ 1+3x +1
√ · √ =lim
=lim
1+3x − 1 1+3x +1 x→0 (1 + 3x) − 1 x→0 3x
= 1 3 lim √
1+3x +1
x→0
= 1 3
lim
= 1 3
lim
x→0
(1 + 3x)+lim
x→0
1
x→0
1+3lim
x→0
x +1
= 1 √
1+3· 0+1
3
[Limit Law 3]
[1 and 11]
[1,3,and7]
[7 and 8]
= 1 3 (1 + 1) = 2 3
33. Let f(x) =−x 2 , g(x) =x 2 cos 20πx and h(x) =x 2 .Then
−1 ≤ cos 20πx ≤ 1 ⇒ −x 2 ≤ x 2 cos 20πx ≤ x 2 ⇒ f(x) ≤ g(x) ≤ h(x).
So since lim
x→0
f(x) = lim
x→0
h(x) =0, by the Squeeze Theorem we have
lim g(x) =0.
x→0
35. We have lim
x→4
(4x − 9) = 4(4) − 9=7and lim
x→4
x 2 − 4x +7 =4 2 − 4(4) + 7 = 7. Since4x − 9 ≤ f(x) ≤ x 2 − 4x +7
for x ≥ 0, lim
x→4
f(x) =7by the Squeeze Theorem.
37. −1 ≤ cos(2/x) ≤ 1 ⇒ −x 4 ≤ x 4 cos(2/x) ≤ x 4 .Sincelim
−x
4
=0and lim x 4 =0,wehave
x→0 x→0
lim x 4 cos(2/x) =0by the Squeeze Theorem.
x→0
39. |x − 3| =
41.
x − 3 if x − 3 ≥ 0
−(x − 3) if x − 3 < 0 =
x − 3 if x ≥ 3
3 − x if x

F.
50 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
1
43. Since |x| = −x for x0, lim sgn x = lim 1=1.
x→0 + x→0 +
(ii) Since sgn x = −1 for x

F.
TX.10
SECTION 2.4 THE PRECISE DEFINITION OF A LIMIT ¤ 51
f(x) − 8
f(x) − 8
55. lim [f(x) − 8] = lim
· (x − 1) =lim · lim (x − 1) = 10 · 0=0.
x→1 x→1 x − 1
x→1 x − 1 x→1
Thus, lim
x→1
f(x) = lim
x→1
{[f(x) − 8] + 8} =lim
x→1
[f(x) − 8] + lim
x→1
8=0+8=8.
f(x) − 8
f(x) − 8
Note: The value of lim does not affect the answer since it’s multiplied by 0. What’s important is that lim
x→1 x − 1
x→1 x − 1
exists.
57. Observe that 0 ≤ f(x) ≤ x 2 for all x,and lim
x→0
0=0= lim
x→0
x 2 . So, by the Squeeze Theorem, lim
x→0
f(x) =0.
59. Let f(x) =H(x) and g(x) =1− H(x),whereH is the Heaviside function definedinExercise1.3.57.
Thus, either f or g is 0 for any value of x. Thenlim
x→0
f(x) and lim
x→0
g(x) do not exist, but lim
x→0
[f(x)g(x)] = lim
x→0
0=0.
61. Since the denominator approaches 0 as x →−2, the limit will exist only if the numerator also approaches
0 as x →−2. In order for this to happen, we need lim 3x 2 + ax + a +3 =0
x→−2
⇔
3(−2) 2 + a(−2) + a +3=0 ⇔ 12 − 2a + a +3=0 ⇔ a =15.Witha =15, the limit becomes
3x 2 +15x +18 3(x +2)(x +3)
lim
= lim
x→−2 x 2 + x − 2 x→−2 (x − 1)(x +2) = lim 3(x +3)
= 3(−2+3) = 3
x→−2 x − 1 −2 − 1 −3 = −1.
2.4 The Precise Definition of a Limit
1. On the left side of x =2,weneed|x − 2| < 10
− 2 = 4 . On the right side, we need |x − 2| < 10
− 2 = 4 . For both of
7 7 3 3
these conditions to be satisfied at once, we need the more restrictive of the two to hold, that is, |x − 2| < 4 . So we can choose
7
δ = 4 7
, or any smaller positive number.
3. The leftmost question mark is the solution of √ x =1.6 and the rightmost, √ x =2.4. Sothevaluesare1.6 2 =2.56 and
2.4 2 =5.76. On the left side, we need |x − 4| < |2.56 − 4| =1.44. On the right side, we need |x − 4| < |5.76 − 4| =1.76.
To satisfy both conditions, we need the more restrictive condition to hold — namely, |x − 4| < 1.44. Thus, we can choose
δ =1.44, or any smaller positive number.
5. From the graph, we find that tan x =0.8 when x ≈ 0.675,so
π
− 4 δ1 ≈ 0.675 ⇒ δ1 ≈ π 4
− 0.675 ≈ 0.1106. Also,tan x =1.2
when x ≈ 0.876,so π 4 + δ 2 ≈ 0.876 ⇒ δ 2 =0.876 − π 4 ≈ 0.0906.
Thus, we choose δ =0.0906 (or any smaller positive number) since this is
the smaller of δ 1 and δ 2 .

F.
52 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
7. For ε =1,thedefinition of a limit requires that we find δ such that 4+x − 3x 3 − 2 < 1 ⇔ 1 < 4+x − 3x 3 < 3
whenever 0 < |x − 1|

F.
TX.10
SECTION 2.4 THE PRECISE DEFINITION OF A LIMIT ¤ 53
15. Given ε>0, we need δ>0 such that if 0 < |x − 1|

F.
54 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
29. Given ε>0,weneedδ>0 such that if 0 < |x − 2|

F.
TX.10
SECTION 2.5 CONTINUITY ¤ 55
39. Suppose that lim f(x) =L. Givenε = 1 ,thereexistsδ>0 such that 0 < |x|

F.
56 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
4
11. lim f(x) = lim x +2x
3 4 = lim x +2 lim
x→−1 x→−1
x→−1 x→−1 x3 = −1+2(−1) 34 =(−3) 4 =81=f(−1).
By the definition of continuity, f is continuous at a = −1.
13. For a>2,wehave
2x +3
lim (2x +3)
lim f(x)= lim
x→a x→a x − 2 = x→a
lim (x − 2) [Limit Law 5]
x→a
2 lim x + lim 3
x→a x→a
=
lim x − lim 2
x→a x→a
=
2a +3
a − 2
= f(a)
TX.10
[1,2,and3]
[7 and 8]
Thus, f is continuous at x = a for every a in (2, ∞);thatis,f is continuous on (2, ∞).
15. f(x) =ln|x − 2| is discontinuous at 2 since f(2) = ln 0 is not defined.
17. f(x) =
e
x
if x

F.
TX.10
SECTION 2.5 CONTINUITY ¤ 57
25. By Theorem 7, the exponential function e −5t and the trigonometric function cos 2πt are continuous on (−∞, ∞).
By part 4 of Theorem 4, L(t) =e −5t cos 2πt is continuous on (−∞, ∞).
27. By Theorem 5, the polynomial t 4 − 1 is continuous on (−∞, ∞). By Theorem 7, ln x is continuous on its domain, (0, ∞).
By Theorem 9, ln t 4 − 1 is continuous on its domain, which is
t | t 4 − 1 > 0 = t | t 4 > 1 = {t ||t| > 1} =(−∞, −1) ∪ (1, ∞)
29. The function y =
1
is discontinuous at x =0because the
1+e1/x left- and right-hand limits at x =0are different.
31. Because we are dealing with root functions, 5+ √ x is continuous on [0, ∞), √ x +5is continuous on [−5, ∞), sothe
quotient f(x) = 5+√ x
√ 5+x
is continuous on [0, ∞). Sincef is continuous at x =4, lim
x→4
f(x) =f(4) = 7 3 .
33. Because x 2 − x is continuous on R, the composite function f(x) =e x2 −x is continuous on R, so
lim
x→1 f(x) =f(1) = e1 − 1 = e 0 =1.
x
2
if x

F.
58 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
⎧
x +2 ⎪⎨
if x1
TX.10
f is continuous on (−∞, 0) and (1, ∞) sinceoneachoftheseintervals
it is a polynomial; it is continuous on (0, 1) since it is an exponential.
Now lim f(x) = lim +2)=2and lim f(x) = lim
x→0− x→0−(x ex =1,sof is discontinuous at 0. Sincef(0) = 1, f is
x→0 + x→0 +
continuous from the right at 0. Also lim f(x) = lim ex = e and lim f(x) = lim − x) =1,sof is discontinuous
x→1− x→1 − x→1 + x→1 +(2
at 1. Sincef(1) = e, f is continuous from the left at 1.
41. f(x) =
cx 2 +2x if x

F.
TX.10
SECTION 2.5 CONTINUITY ¤ 59
51. (a) f(x) =cosx − x 3 is continuous on the interval [0, 1], f(0) = 1 > 0,andf(1) = cos 1 − 1 ≈−0.46 < 0. Since
1 > 0 > −0.46, there is a number c in (0, 1) such that f(c) =0by the Intermediate Value Theorem. Thus, there is a root
of the equation cos x − x 3 =0,orcos x = x 3 , in the interval (0, 1).
(b) f(0.86) ≈ 0.016 > 0 and f(0.87) ≈−0.014 < 0, so there is a root between 0.86 and 0.87, that is, in the interval
(0.86, 0.87).
53. (a) Let f(x) =100e −x/100 − 0.01x 2 . Then f(0) = 100 > 0 and
f(100) = 100e −1 − 100 ≈−63.2 < 0. So by the Intermediate
Value Theorem, there is a number c in (0, 100) such that f(c) =0.
This implies that 100e −c/100 =0.01c 2 .
(b) Using the intersect feature of the graphing device, we find that the
root of the equation is x =70.347, correct to three decimal places.
55. (⇒)Iff is continuous at a,thenbyTheorem8withg(h) =a + h,wehave
lim f(a + h) =f lim (a + h) = f(a).
h→0 h→0
(⇐)Letε>0. Sincelim
h→0
f(a + h) =f(a),thereexistsδ>0 such that 0 < |h|

F.
60 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
lim
x→0 (−x4 )=0and lim x 4 =0, the Squeeze Theorem gives us lim(x 4 sin(1/x)) = 0,whichequalsf(0). Thus,f is
x→0 x→0
continuous at 0 and, hence, on (−∞, ∞).
65. Define u(t) to be the monk’s distance from the monastery, as a function of time, on the first day, and define d(t) to be his
distance from the monastery, as a function of time, on the second day. Let D be the distance from the monastery to the top of
the mountain. From the given information we know that u(0) = 0, u(12) = D, d(0) = D and d(12) = 0. Now consider the
function u − d, which is clearly continuous. We calculate that (u − d)(0) = −D and (u − d)(12) = D.Sobythe
Intermediate Value Theorem, there must be some time t 0 between 0 and 12 such that (u − d)(t 0)=0 ⇔ u(t 0)=d(t 0).
So at time t 0 after 7:00 AM, the monk will be at the same place on both days.
2.6 Limits at Infinity; Horizontal Asymptotes
1. (a) As x becomes large, the values of f(x) approach 5.
(b) As x becomes large negative, the values of f(x) approach 3.
3. (a) lim f(x) =∞ (b) lim f(x) =∞ (c) lim f(x) =−∞
x→2 − +
x→−1
(d) lim f(x) =1 (e) lim f(x) =2 (f ) Vertical: x = −1, x =2; Horizontal: y =1, y =2
x→∞ x→−∞
x→−1
5. f(0) = 0, f(1) = 1,
lim f(x) =0,
x→∞
f is odd
7. lim f(x) =−∞, lim f(x) =∞, 9. f(0) = 3, lim f(x) =4,
x→2 x→∞ x→0− lim f(x) =0, lim f(x) =∞, lim f(x) =2,
x→−∞ x→0 +
x→0 +
lim f(x) =−∞ lim f(x) =−∞, lim f(x) =−∞,
x→0 − x→−∞ x→4− lim f(x) =∞, lim
x→4 +
f(x) =3
x→∞
11. If f(x) =x 2 /2 x , then a calculator gives f(0) = 0, f(1) = 0.5, f(2) = 1, f(3) = 1.125, f(4) = 1, f(5) = 0.78125,
f(6) = 0.5625, f(7) = 0.3828125, f(8) = 0.25, f(9) = 0.158203125, f(10) = 0.09765625, f(20) ≈ 0.00038147,
f(50) ≈ 2.2204 × 10 −12 , f(100) ≈ 7.8886 × 10 −27 .
It appears that lim x 2 /2 x =0.
x→∞

F.
TX.10
SECTION 2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES ¤ 61
13. lim
x→∞
3x 2 − x +4
2x 2 +5x − 8 = lim
x→∞
(3x 2 − x +4)/x 2
(2x 2 +5x − 8)/x 2 [divide both the numerator and denominator by x 2
(the highest power of x thatappears in the denominator)]
=
lim (3 − 1/x
x→∞ +4/x2 )
lim (2 + 5/x −
x→∞ 8/x2 )
lim 3 − lim (1/x)+ lim
x→∞ x→∞ x→∞ (4/x2 )
=
lim 2 + lim (5/x) − lim
x→∞ x→∞ x→∞ (8/x2 )
=
3 − lim (1/x)+4 lim
x→∞ x→∞ (1/x2 )
2+5 lim(1/x) − 8 lim
x→∞ x→∞ (1/x2 )
= 3 − 0+4(0)
2+5(0)− 8(0)
[Limit Law 5]
[Limit Laws 1 and 2]
[Limit Laws 7 and 3]
[Theorem 5 of Section 2.5]
= 3 2
1
15. lim
x→∞ 2x +3 = lim 1/x
lim (1/x)
lim
x→∞ (2x +3)/x = x→∞
=
lim (2 + 3/x)
1 − x − x 2
17. lim
x→−∞ 2x 2 − 7
x→∞
x→∞ (1/x)
lim 2 + 3 lim (1/x) = 0
x→∞ x→∞
(1 − x − x 2 )/x 2 lim
x→−∞ (1/x2 − 1/x − 1)
= lim
=
x→−∞ (2x 2 − 7)/x 2 lim (2 −
x→−∞ 7/x2 )
lim
x→−∞ (1/x2 ) − lim (1/x) − lim
x→−∞
=
lim 2 − 7 lim
x→−∞ x→−∞ (1/x2 )
x→−∞ 1
= 0 − 0 − 1
2 − 7(0) = −1 2
2 + 3(0) = 0 2 =0
19. Divide both the numerator and denominator by x 3 (the highest power of x that occurs in the denominator).
lim
x→∞
x 3 +5x
2x 3 − x 2 +4 = lim
x→∞
=
x 3 +5x
x 3
2x 3 − x 2 +4
x 3
lim 1+5 lim
x→∞ x→∞
lim 2 − lim
x→∞ x→∞
= lim
x→∞
1
x 2
1
+4 lim
x x→∞
1+ 5 x 2
2 − 1 x + 4 x 3 =
lim
1+ 5
x
2 2 − 1 x + 4
x 3
x→∞
lim
x→∞
= 1 + 5(0)
1 2 − 0+4(0) = 1 2
x 3
21. First, multiply the factors in the denominator. Then divide both the numerator and denominator by u 4 .
lim
u→∞
4u 4 +5
(u 2 − 2)(2u 2 − 1) = lim
u→∞
=
lim
u→∞
4u 4 +5
2u 4 − 5u 2 +2 = lim
u→∞
lim
4+ 5
u
2 4 − 5 u + 2 =
2 u 4
u→∞
4u 4 +5
u 4
2u 4 − 5u 2 +2
u 4
= lim
u→∞
lim 4+5 lim
u→∞ u→∞
lim 2 − 5 lim
u→∞ u→∞
1
u 4
1
+2 lim
u2 u→∞
4+ 5 u 4
2 − 5 u 2 + 2 u 4
=
1
u 4
4+5(0)
2 − 5(0) + 2(0) = 4 2 =2

F.
62 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
√ √
9x6 − x 9x6 − x/x 3
23. lim
= lim
=
x→∞ x 3 +1 x→∞ (x 3 +1)/x 3
=
lim 9 − 1/x
5
x→∞
lim
x→∞
= √ 9 − 0=3
lim
1+ lim
x→∞ (1/x3 ) =
lim (9x6 − x)/x 6
x→∞
lim (1 +
x→∞ 1/x3 )
x→∞ 9 − lim
x→∞ (1/x5 )
1+0
[since x 3 = √ x 6 for x>0]
√
25. lim 9x2 + x − 3x = lim
x→∞
x→∞
= lim
x→∞
= lim
x→∞
√
27. lim x2 + ax − √ x 2 + bx = lim
x→∞
x→∞
√
9x2 + x − 3x √ 9x 2 + x +3x
√
9x2 + x +3x
9x 2 + x − 9x 2
√
9x2 + x +3x = lim
x→∞
x/x
9x2 /x 2 + x/x 2 +3x/x = lim
x→∞
= lim
x→∞
= lim
x→∞
= lim
x→∞
x
√
9x2 + x +3x · 1/x
1/x
1
9+1/x +3
=
√
9x2 + x 2
− (3x)
2
√
9x2 + x +3x
√
x2 + ax − √ x 2 + bx √ x 2 + ax + √ x 2 + bx
√
x2 + ax + √ x 2 + bx
(x 2 + ax) − (x 2 + bx)
√
x2 + ax + √ x 2 + bx = lim
x→∞
a − b
1+a/x +
1+b/x
=
1
√
9+3
= 1
3+3 = 1 6
[(a − b)x]/x
√
x2 + ax + √ x 2 + bx / √ x 2
a − b
√ 1+0+
√ 1+0
= a − b
2
x + x 3 + x 5
29. lim
x→∞ 1 − x 2 + x = lim (x + x 3 + x 5 )/x 4
[divide by the highest power of x in the denominator]
4 x→∞ (1 − x 2 + x 4 )/x 4
= lim
x→∞
1/x 3 +1/x + x
1/x 4 − 1/x 2 +1 = ∞
because (1/x 3 +1/x + x) →∞and (1/x 4 − 1/x 2 +1)→ 1 as x →∞.
31. lim
x→−∞ (x4 + x 5 )= lim
x→−∞ x5 ( 1 +1) [factor out the largest power of x] = −∞ because x x5 →−∞and 1/x +1→ 1
as x →−∞.
Or: lim x 4 + x 5 = lim
x→−∞
x→−∞ x4 (1 + x) =−∞.
33. lim
x→∞
1 − e x
1+2e x = lim
x→∞
(1 − e x )/e x
(1 + 2e x )/e = lim 1/e x − 1
x x→∞ 1/e x +2 = 0 − 1
0+2 = −1 2
35. Since −1 ≤ cos x ≤ 1 and e −2x > 0, wehave−e −2x ≤ e −2x cos x ≤ e −2x .Weknowthat lim
lim
e
−2x
=0,sobytheSqueezeTheorem, lim
x→∞
x→∞ (e−2x cos x) =0.
x→∞ (−e−2x )=0and
37. (a)
x
f(x)
−10,000 −0.4999625
−100,000 −0.4999962
−1,000,000 −0.4999996
From the graph of f(x) = √ x 2 + x +1+x,weestimate
the value of
lim
x→−∞
f(x) to be −0.5.
(b)
Fromthetable,weestimatethelimit
to be −0.5.

F.
(c)
lim
x→−∞
√
x2 + x +1+x =
lim
x→−∞
= lim
x→−∞
=
√
x2 + x +1+x √
x 2 + x +1− x
√ = lim
x2 + x +1− x x→−∞
(x +1)(1/x)
√
x2 + x +1− x (1/x) =
1+0
− √ 1+0+0− 1 = − 1 2
lim
x→−∞
1+(1/x)
− 1+(1/x)+(1/x 2 ) − 1
Note that for x

F.
64 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
45. From the graph, it appears y =1is a horizontal asymptote.
3x 3 +500x 2
3x 3 +500x 2
lim
x→∞ x 3 + 500x 2 +100x +2000 = lim
x 3
3+(500/x)
= lim
x→∞ x 3 +500x 2 + 100x + 2000 x→∞ 1+(500/x) + (100/x 2 )+(2000/x 3 )
x 3
3+0
=
=3, soy =3is a horizontal asymptote.
1+0+0+0
The discrepancy can be explained by the choice of the viewing window. Try
[−100,000, 100,000] by [−1, 4] to get a graph that lends credibility to our
calculation that y =3is a horizontal asymptote.
47. Let’s look for a rational function.
(1) lim f(x) =0 ⇒ degree of numerator < degree of denominator
x→±∞
(2) lim f(x) =−∞ ⇒ there is a factor of x 2 in the denominator (not just x, since that would produce a sign
x→0
change at x =0), and the function is negative near x =0.
(3) lim f(x) =∞ and lim f(x) =−∞ ⇒ vertical asymptote at x =3;thereisafactorof(x − 3) in the
x→3− x→3 +
denominator.
(4) f(2) = 0 ⇒ 2 is an x-intercept; there is at least one factor of (x − 2) in the numerator.
Combining all of this information and putting in a negative sign to give us the desired left- and right-hand limits gives us
f(x) =
2 − x as one possibility.
x 2 (x − 3)
49. y = f(x) =x 4 − x 6 = x 4 (1 − x 2 )=x 4 (1 + x)(1 − x). They-intercept is
f(0) = 0. Thex-intercepts are 0, −1,and1 [found by solving f(x) =0for x].
Since x 4 > 0 for x 6= 0, f doesn’t change sign at x =0. The function does change
sign at x = −1 and x =1.Asx → ±∞, f(x) =x 4 (1 − x 2 ) approaches −∞
because x 4 →∞and (1 − x 2 ) →−∞.
51. y = f(x) =(3− x)(1 + x) 2 (1 − x) 4 .They-intercept is f(0) = 3(1) 2 (1) 4 =3.
The x-intercepts are 3, −1,and1. There is a sign change at 3,butnotat−1 and 1.
When x is large positive, 3 − x is negative and the other factors are positive, so
lim f(x) =−∞. Whenx is large negative, 3 − x is positive, so
x→∞
lim f(x) =∞.
x→−∞
53. (a) Since −1 ≤ sin x ≤ 1 for all x, − 1 x ≤ sin x
x
≤ 1 for x>0. Asx →∞, −1/x → 0 and 1/x → 0, so by the Squeeze
x
sin x
Theorem, (sin x)/x → 0. Thus, lim
x→∞ x =0.

F.
TX.10
SECTION 2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES ¤ 65
(b) From part (a), the horizontal asymptote is y =0. The function
y =(sinx)/x crosses the horizontal asymptote whenever sin x =0;
that is, at x = πn for every integer n. Thus, the graph crosses the
asymptote an infinite number of times.
55. Divide the numerator and the denominator by the highest power of x in Q(x).
(a) If deg Pdeg Q, then the numerator → ±∞ but the denominator doesn’t, so lim [P (x)/Q(x)] = ±∞
x→∞
57. lim
x→∞
(depending on the ratio of the leading coefficients of P and Q).
5 √ x
√ · 1/√ x
x − 1 1/ √ x = lim
x→∞
5
1 − (1/x)
=
5
√ 1 − 0
=5and
10e x − 21
lim
· 1/ex
x→∞ 2e x 1/e = lim 10 − (21/e x )
= 10 − 0 =5.Since 10ex − 21

F.
66 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
65. (a) 1/x 2 < 0.0001 ⇔ x 2 > 1/0.0001 = 10 000 ⇔ x>100 (x >0)
(b) If ε>0 is given, then 1/x 2 1/ε ⇔ x>1/ √ ε.LetN =1/ √ ε.
Then x>N ⇒ x> √ 1 ⇒
ε 1
x − 0 = 1 1
M.Nowe x >M ⇔ x>ln M, sotake
N =max(1, ln M). (ThisensuresthatN>0.) Then x>N=max(1, ln M) ⇒ e x > max(e, M) ≥ M,
so lim
x→∞ ex = ∞.
71. Suppose that lim f(x) =L. Then for every ε>0 there is a corresponding positive number N such that |f(x) − L| N.Ift =1/x,thenx>N ⇔ 0 < 1/x < 1/N ⇔ 0 0 (namely 1/N ) such that |f(1/t) − L|

F.
TX.10
SECTION 2.7 DERIVATIVES AND RATES OF CHANGE ¤ 67
(c)
The graph of y =2x +1is tangent to the graph of y =4x − x 2 at the
point (1, 3). Now zoom in toward the point (1, 3) until the parabola and
the tangent line are indistiguishable.
5. Using (1) with f(x) = x − 1 and P (3, 2),
x − 2
x − 1
f(x) − f(a)
m = lim
=lim
x − 2 − 2
x→a x − a x→3 x − 3
=lim
x→3
3 − x
(x − 2)(x − 3) =lim
x→3
x − 1 − 2(x − 2)
=lim
x − 2
x→3 x − 3
−1
x − 2 = −1
1 = −1
Tangent line: y − 2=−1(x − 3) ⇔ y − 2=−x +3 ⇔ y = −x +5
√ √
x − 1 ( √ x − 1)( √ x +1)
7. Using (1), m =lim =lim
x→1 x − 1 x→1 (x − 1)( √ x − 1
= lim
x +1) x→1 (x − 1)( √ x +1) =lim 1
√ = 1
x→1 x +1 2 .
Tangent line: y − 1= 1 2 (x − 1) ⇔ y = 1 2 x + 1 2
9. (a) Using (2) with y = f(x) =3+4x 2 − 2x 3 ,
f(a + h) − f(a) 3+4(a + h) 2 − 2(a + h) 3 − (3 + 4a 2 − 2a 3 )
m =lim
= lim
h→0 h
h→0 h
=lim
h→0
3+4(a 2 +2ah + h 2 ) − 2(a 3 +3a 2 h +3ah 2 + h 3 ) − 3 − 4a 2 +2a 3
h
=lim
h→0
3+4a 2 +8ah +4h 2 − 2a 3 − 6a 2 h − 6ah 2 − 2h 3 − 3 − 4a 2 +2a 3
h
8ah +4h 2 − 6a 2 h − 6ah 2 − 2h 3 h(8a +4h − 6a 2 − 6ah − 2h 2 )
=lim
=lim
h→0 h
h→0 h
=lim
h→0
(8a +4h − 6a 2 − 6ah − 2h 2 )=8a − 6a 2
(b) At (1, 5): m =8(1)− 6(1) 2 =2,soanequationofthetangentline
is y − 5=2(x − 1) ⇔ y =2x +3.
(c)
At (2, 3): m =8(2)− 6(2) 2 = −8, so an equation of the tangent
line is y − 3=−8(x − 2) ⇔ y = −8x +19.
11. (a) The particle is moving to the right when s is increasing; that is, on the intervals (0, 1) and (4, 6). The particle is moving to
the left when s is decreasing; that is, on the interval (2, 3). The particle is standing still when s is constant; that is, on the
intervals (1, 2) and (3, 4).

F.
68 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
(b) The velocity of the particle is equal to the slope of the tangent line of the
graph. Note that there is no slope at the corner points on the graph. On the
interval (0, 1), the slope is 3 − 0 =3.Ontheinterval(2, 3), the slope is
1 − 0
1 − 3
3 − 1
= −2. On the interval (4, 6), the slope is
3 − 2 6 − 4 =1.
13. Let s(t) =40t − 16t 2 .
s(t) − s(2)
40t − 16t
2
− 16 −16t 2 +40t − 16 −8 2t 2 − 5t +2
v(2) = lim
=lim
=lim
=lim
t→2 t − 2 t→2 t − 2
t→2 t − 2
t→2 t − 2
−8(t − 2)(2t − 1)
=lim
= −8lim(2t − 1) = −8(3) = −24
t→2 t − 2
t→2
Thus, the instantaneous velocity when t =2is −24 ft/s.
1
s(a + h) − s(a) (a + h) − 1 15. v(a)= lim
= lim
2 a 2
h→0 h
h→0 h
a 2 − (a + h) 2
a
=lim
2 (a + h) 2
h→0 h
= lim
h→0
a 2 − (a 2 +2ah + h 2 )
ha 2 (a + h) 2
−(2ah + h 2 )
=lim
h→0 ha 2 (a + h) =lim −h(2a + h)
2 h→0 ha 2 (a + h) = lim −(2a + h)
2 h→0 a 2 (a + h) = −2a
2 a 2 · a = −2
2 a m/s 3
So v (1) = −2
−2
= −2 m/s, v(2) =
13 2 = − 1 −2
m/s, and v(3) = 3 4 3 = − 2 3 27 m/s.
17. g 0 (0) istheonlynegativevalue.Theslopeatx =4is smaller than the slope at x =2and both are smaller than the slope at
x = −2. Thus, g 0 (0) < 0

F.
TX.10
SECTION 2.7 DERIVATIVES AND RATES OF CHANGE ¤ 69
23. (a) Using Definition 2 with F (x) =5x/(1 + x 2 ) and the point (2, 2),wehave (b)
5(2 + h)
F 0 F (2 + h) − F (2) 1+(2+h) − 2
(2) = lim
=lim
2
h→0 h
h→0 h
5h +10
= lim
h 2 +4h +5 − 2 5h +10− 2(h 2 +4h +5)
=lim
h 2 +4h +5
h→0 h
h→0 h
−2h 2 − 3h
= lim
h→0 h(h 2 +4h +5) =lim h(−2h − 3)
h→0 h(h 2 +4h +5) =lim −2h − 3
h→0 h 2 +4h +5 = −3
5
So an equation of the tangent line at (2, 2) is y − 2=− 3 (x − 2) or y = − 3 x + 16 . 5 5 5
25. Use Definition 2 with f(x) =3− 2x +4x 2 .
f 0 f(a + h) − f(a) [3 − 2(a + h)+4(a + h) 2 ] − (3 − 2a +4a 2 )
(a) = lim
=lim
h→0 h
h→0 h
= lim
h→0
(3 − 2a − 2h +4a 2 +8ah +4h 2 ) − (3 − 2a +4a 2 )
h
−2h +8ah +4h 2 h(−2+8a +4h)
= lim
=lim
=lim(−2+8a +4h) =−2+8a
h→0 h
h→0 h
h→0
27. Use Definition 2 with f(t) =(2t +1)/(t +3).
2(a + h)+1
f 0 f(a + h) − f(a) (a + h)+3
(a) = lim
=lim
h→0 h
h→0 h
−
2a +1
a +3
= lim
h→0
(2a 2 +6a +2ah +6h + a +3)− (2a 2 +2ah +6a + a + h +3)
h(a + h +3)(a +3)
= lim
h→0
5h
h(a + h +3)(a +3) =lim
h→0
29. Use Definition 2 with f(x) =1/ √ x +2.
f 0 f(a + h) − f(a)
(a) = lim
=lim
h→0 h
√ √ a +2− a + h +2
= lim
h→0 h √ a + h +2 √ a +2 ·
= lim
h→0
5
(a + h +3)(a +3) = 5
(a +3) 2
= lim
h→0
(2a +2h +1)(a +3)− (2a +1)(a + h +3)
h(a + h +3)(a +3)
√ √
1
1
− √ a +2− a + h +2
√ √ (a + h)+2 a +2
a + h +2 a +2
= lim
h→0 h
h→0 h
√ √ a +2+ a + h +2
(a +2)− (a + h +2)
√ √ = lim
a +2+ a + h +2 h→0 h √ a + h +2 √ a +2 √ a +2+ √ a + h +2
−h
h √ a + h +2 √ a +2 √ a +2+ √ a + h +2 =lim
h→0
−1
1
= √ 2 √ = −
a +2 2 a +2 2(a +2) 3/2
Note that the answers to Exercises 31 – 36 are not unique.
(1 + h) 10 − 1
31. By Definition 2, lim
= f 0 (1),wheref(x) =x 10 and a =1.
h→0 h
(1 + h) 10 − 1
Or: By Definition 2, lim
= f 0 (0),wheref(x) =(1+x) 10 and a =0.
h→0 h
−1
√ a + h +2
√ a +2
√ a +2+
√ a + h +2

F.
70 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
33. By Equation 3, lim
x→5
2 x − 32
x − 5 = f 0 (5),wheref(x) =2 x and a =5.
cos(π + h)+1
35. By Definition 2, lim
= f 0 (π),wheref(x) =cosx and a = π.
h→0 h
cos(π + h)+1
Or: ByDefinition 2, lim
= f 0 (0),wheref(x) =cos(π + x) and a =0.
h→0 h
37. v(5) = f 0 f(5 + h) − f(5) [100 + 50(5 + h) − 4.9(5 + h) 2 ] − [100 + 50(5) − 4.9(5) 2 ]
(5) = lim
=lim
h→0 h
h→0 h
(100 + 250 + 50h − 4.9h 2 − 49h − 122.5) − (100 + 250 − 122.5) −4.9h 2 + h
=lim
= lim
h→0 h
h→0 h
h(−4.9h +1)
=lim
=lim(−4.9h +1)=1m/s
h→0 h
h→0
The speed when t =5is |1| =1m/s.
39. Thesketchshowsthegraphforaroomtemperatureof72 ◦ and a refrigerator
temperature of 38 ◦ . The initial rate of change is greater in magnitude than the
rate of change after an hour.
41. (a) (i) [2000, 2002]:
(ii) [2000, 2001]:
(iii) [1999, 2000]:
P (2002) − P (2000)
2002 − 2000
P (2001) − P (2000)
2001 − 2000
P (2000) − P (1999)
2000 − 1999
(b) Using the values from (ii) and (iii), we have
=
=
=
77 − 55
2
68 − 55
1
55 − 39
1
13 + 16
2
= 22
2 =11percent/year
=13percent/year
=16percent/year
=14.5 percent/year.
(c) Estimating A as (1999, 40) and B as (2001, 70), the slope at 2000 is
70 − 40
2001 − 1999 = 30
2 =15percent/year.
43. (a) (i) ∆C
∆x
(ii) ∆C
∆x
(b)
=
C(105) − C(100)
105 − 100
=
6601.25 − 6500
5
6520.05 − 6500
1
=$20.25/unit.
C(101) − C(100)
= = =$20.05/unit.
101 − 100
5000 + 10(100 + h)+0.05(100 + h)
2
− 6500
=
=
h
=20+0.05h, h 6= 0
C(100 + h) − C(100)
h
20h +0.05h2
h
C(100 + h) − C(100)
So the instantaneous rate of change is lim
= lim (20 + 0.05h) =$20/unit.
h→0 h
h→0
45. (a) f 0 (x) is the rate of change of the production cost with respect to the number of ounces of gold produced. Its units are
dollars per ounce.

F.
TX.10
SECTION 2.8 THEDERIVATIVEASAFUNCTION ¤ 71
(b) After 800 ounces of gold have been produced, the rate at which the production cost is increasing is $17/ounce. So the cost
of producing the 800th (or 801st) ounce is about $17.
(c) In the short term, the values of f 0 (x) will decrease because more efficient use is made of start-up costs as x increases. But
eventually f 0 (x) might increase due to large-scale operations.
47. T 0 (10) is the rate at which the temperature is changing at 10:00 AM.ToestimatethevalueofT 0 (10), we will average the
difference quotients obtained using the times t =8and t =12.LetA =
B =
T (12) − T (10)
12 − 10
=
88 − 81
2
T (8) − T (10)
8 − 10
=
72 − 81
−2
=4.5 and
=3.5. ThenT 0 T (t) − T (10)
(10) = lim
≈ A + B = 4.5+3.5 =4 ◦ F/h.
t→10 t − 10 2 2
49. (a) S 0 (T ) is the rate at which the oxygen solubility changes with respect to the water temperature. Its units are (mg/L)/ ◦ C.
(b) For T =16 ◦ C, it appears that the tangent line to the curve goes through the points (0, 14) and (32, 6). So
S 0 (16) ≈ 6 − 14
32 − 0 = − 8
32 = −0.25 (mg/L)/◦ C. This means that as the temperature increases past 16 ◦ C, the oxygen
solubility is decreasing at a rate of 0.25 (mg/L)/ ◦ C.
51. Since f(x) =x sin(1/x) when x 6= 0and f(0) = 0, wehave
f 0 f(0 + h) − f(0) h sin(1/h) − 0
(0) = lim
=lim
=limsin(1/h). This limit does not exist since sin(1/h) takes the
h→0 h
h→0 h
h→0
values −1 and 1 on any interval containing 0. (Compare with Example 4 in Section 2.2.)
2.8 The Derivative as a Function
1. It appears that f is an odd function, so f 0 will be an even
function—that is, f 0 (−a) =f 0 (a).
(a) f 0 (−3) ≈ 1.5
(b) f 0 (−2) ≈ 1
(c) f 0 (−1) ≈ 0
(d) f 0 (0) ≈−4
(e) f 0 (1) ≈ 0
(f ) f 0 (2) ≈ 1
(g) f 0 (3) ≈ 1.5
3. (a) 0 = II, since from left to right, the slopes of the tangents to graph (a) start out negative, become 0, then positive, then 0,then
negative again. The actual function values in graph II follow the same pattern.
(b) 0 = IV, since from left to right, the slopes of the tangents to graph (b) start out at a fixed positive quantity, then suddenly
become negative, then positive again. The discontinuities in graph IV indicate sudden changes in the slopes of the tangents.
(c) 0 = I, since the slopes of the tangents to graph (c) are negative for x0,asarethefunctionvaluesof
graph I.
(d) 0 = III, since from left to right, the slopes of the tangents to graph (d) are positive, then 0, then negative, then 0, then
positive, then 0, then negative again, and the function values in graph III follow the same pattern.

F.
72 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
Hints for Exercises 4 –11: First plot x-intercepts on the graph of f 0 for any horizontal tangents on the graph of f . Look for any corners on the graph
of f— there will be a discontinuity on the graph of f 0 . On any interval where f has a tangent with positive (or negative) slope, the graph of f 0 will be
positive (or negative). If the graph of the function is linear, the graph of f 0 will be a horizontal line.
5. 7.
9. 11.
13. It appears that there are horizontal tangents on the graph of M for t = 1963
and t = 1971. Thus, there are zeros for those values of t on the graph of
M 0 . The derivative is negative for the years 1963 to 1971.
15.
The slope at 0 appears to be 1 and the slope at 1
appears to be 2.7. Asx decreases, the slope gets
closer to 0. Since the graphs are so similar, we might
guess that f 0 (x) =e x .

F.
TX.10
SECTION 2.8 THEDERIVATIVEASAFUNCTION ¤ 73
17. (a) By zooming in, we estimate that f 0 (0) = 0, f 0 1
2
=1, f 0 (1) = 2,
and f 0 (2) = 4.
(b) By symmetry, f 0 (−x) =−f 0 (x). Sof 0 − 1 2
= −1, f 0 (−1) = −2,
and f 0 (−2) = −4.
(c) It appears that f 0 (x) is twice the value of x, so we guess that f 0 (x) =2x.
(d) f 0 f(x + h) − f(x) (x + h) 2 − x 2
(x) =lim
=lim
h→0 h
h→0 h
x 2 +2hx + h 2 − x 2 2hx + h 2 h(2x + h)
=lim
=lim =lim
=lim(2x + h) =2x
h→0 h
h→0 h h→0 h
h→0
1
19. f 0 f(x + h) − f(x)
(x)= lim
= lim
(x + h) − 1
2 3 − 1 x −
1
2 3
h→0 h
h→0 h
=lim
1
h 2
h→0
h =lim
h→0
1
2 = 1 2
Domain of f = domain of f 0 = R.
21. f 0 f(t + h) − f(t)
5(t + h) − 9(t + h)
2
− (5t − 9t 2 )
(t) = lim
=lim
h→0 h
h→0 h
1
= lim
x + 1 h − 1 − 1 x + 1 2 2 3 2 3
h→0 h
5t +5h − 9(t 2 +2th + h 2 ) − 5t +9t 2 5t +5h − 9t 2 − 18th − 9h 2 − 5t +9t 2
= lim
= lim
h→0 h
h→0 h
5h − 18th − 9h 2 h(5 − 18t − 9h)
= lim
= lim
=lim(5 − 18t − 9h) =5− 18t
h→0 h
h→0 h
h→0
Domain of f = domain of f 0 = R.
23. f 0 f(x + h) − f(x) (x + h) 3 − 3(x + h)+5 − (x 3 − 3x +5)
(x) = lim
=lim
h→0 h
h→0 h
x 3 +3x 2 h +3xh 2 + h 3 − 3x − 3h +5 − x 3 − 3x +5
3x 2 h +3xh 2 + h 3 − 3h
= lim
= lim
h→0 h
h→0 h
h 3x 2 +3xh + h 2 − 3
= lim
=lim 3x 2 +3xh + h 2 − 3 =3x 2 − 3
h→0 h
h→0
Domain of f = domain of f 0 = R.
√ √
25. g 0 g(x + h) − g(x) 1+2(x + h) − 1+2x 1+2(x + h)+ 1+2x
(x) =lim
= lim
√
h→0 h
h→0 h
1+2(x + h)+ 1+2x
(1 + 2x +2h) − (1 + 2x)
2
2
=lim
h→0
1+2(x √ =lim√ √ =
h
+ h)+ 1+2x h→0 1+2x +2h + 1+2x 2 √ 1+2x = 1
√ 1+2x
Domain of g = − 1 2 , ∞ , domain of g 0 = − 1 2 , ∞ .

F.
74 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
4(t + h)
27. G 0 G(t + h) − G(t) (t + h)+1 − 4t 4(t + h)(t +1)− 4t(t + h +1)
t +1
(t + h +1)(t +1)
(t) =lim
=lim
=lim
h→0 h
h→0 h
h→0 h
4t 2 +4ht +4t +4h − 4t 2 +4ht +4t
4h
=lim
= lim
h→0 h(t + h +1)(t +1)
h→0 h(t + h +1)(t +1)
=lim
h→0
4
(t + h +1)(t +1) = 4
(t +1) 2
Domain of G = domain of G 0 =(−∞, −1) ∪ (−1, ∞).
29. f 0 f(x + h) − f(x) (x + h) 4 − x 4 x 4 +4x 3 h +6x 2 h 2 +4xh 3 + h 4 − x 4
(x) =lim
=lim
=lim
h→0 h
h→0 h
h→0 h
4x 3 h +6x 2 h 2 +4xh 3 + h 4
=lim
=lim 4x 3 +6x 2 h +4xh 2 + h 3 =4x 3
h→0 h
h→0
Domain of f = domain of f 0 = R.
31. (a) f 0 f(x + h) − f(x) [(x + h) 4 +2(x + h)] − (x 4 +2x)
(x)= lim
=lim
h→0 h
h→0 h
=lim
h→0
x 4 +4x 3 h +6x 2 h 2 +4xh 3 + h 4 +2x +2h − x 4 − 2x
h
4x 3 h +6x 2 h 2 +4xh 3 + h 4 +2h h(4x 3 +6x 2 h +4xh 2 + h 3 +2)
=lim
=lim
h→0 h
h→0 h
=lim
h→0
(4x 3 +6x 2 h +4xh 2 + h 3 +2)=4x 3 +2
(b) Notice that f 0 (x) =0when f has a horizontal tangent, f 0 (x) is
positive when the tangents have positive slope, and f 0 (x) is
negative when the tangents have negative slope.
33. (a) U 0 (t) is the rate at which the unemployment rate is changing with respect to time. Its units are percent per year.
(b) To find U 0 U(t + h) − U(t) U(t + h) − U(t)
(t),weuse lim
≈ for small values of h.
h→0 h
h
For 1993: U 0 (1993) ≈
U(1994) − U(1993)
1994 − 1993
=
6.1 − 6.9
1
= −0.80
For 1994: We estimate U 0 (1994) by using h = −1 and h =1, and then average the two results to obtain a final estimate.
h = −1 ⇒ U 0 (1994) ≈
h =1 ⇒ U 0 (1994) ≈
U(1993) − U(1994)
1993 − 1994
U(1995) − U(1994)
1995 − 1994
=
=
6.9 − 6.1
−1
5.6 − 6.1
1
So we estimate that U 0 (1994) ≈ 1 [(−0.80) + (−0.50)] = −0.65.
2
= −0.80;
= −0.50.
t 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
U 0 (t) −0.80 −0.65 −0.35 −0.35 −0.45 −0.35 −0.25 0.25 0.90 1.10

F.
TX.10
SECTION 2.8 THEDERIVATIVEASAFUNCTION ¤ 75
35. f is not differentiable at x = −4, because the graph has a corner there, and at x =0, because there is a discontinuity there.
37. f is not differentiable at x = −1, because the graph has a vertical tangent there, and at x =4, because the graph has a corner
there.
39. As we zoom in toward (−1, 0), the curve appears more and more like a
straight line, so f(x) =x + |x| is differentiable at x = −1. But no
matter how much we zoom in toward the origin, the curve doesn’t straighten
out—we can’t eliminate the sharp point (a cusp). So f is not differentiable
at x =0.
41. a = f, b = f 0 , c = f 00 . We can see this because where a has a horizontal tangent, b =0,andwhereb has a horizontal tangent,
c =0. We can immediately see that c can be neither f nor f 0 , since at the points where c has a horizontal tangent, neither a
nor b is equal to 0.
43. We can immediately see that a is the graph of the acceleration function, since at the points where a has a horizontal tangent,
neither c nor b is equal to 0. Next, we note that a =0at the point where b has a horizontal tangent, so b must be the graph of
the velocity function, and hence, b 0 = a. We conclude that c is the graph of the position function.
45. f 0 f(x + h) − f(x)
1+4(x + h) − (x + h)
2
− (1 + 4x − x 2 )
(x) = lim
= lim
h→0 h
h→0 h
(1 + 4x +4h − x 2 − 2xh − h 2 ) − (1 + 4x − x 2 ) 4h − 2xh − h 2
=lim
= lim
= lim (4 − 2x − h) =4− 2x
h→0 h
h→0 h
h→0
f 00 f 0 (x + h) − f 0 (x) [4 − 2(x + h)] − (4 − 2x) −2h
(x) = lim
= lim
=lim
h→0 h
h→0 h
h→0 h
= lim (−2) = −2
h→0
We see from the graph that our answers are reasonable because the graph of
f 0 is that of a linear function and the graph of f 00 is that of a constant
function.

F.
76 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
47. f 0 f(x + h) − f(x) 2(x + h) 2 − (x + h) 3 − (2x 2 − x 3 )
(x) = lim
= lim
h→0 h
h→0 h
h(4x +2h − 3x 2 − 3xh − h 2 )
=lim
=lim(4x +2h − 3x 2 − 3xh − h 2 )=4x − 3x 2
h→0 h
h→0
f 00 f 0 (x + h) − f 0 (x)
4(x + h) − 3(x + h)
2
− (4x − 3x 2 ) h(4 − 6x − 3h)
(x) = lim
= lim
= lim
h→0 h
h→0 h
h→0 h
=lim(4 − 6x − 3h) =4− 6x
h→0
f 000 f 00 (x + h) − f 00 (x) [4 − 6(x + h)] − (4 − 6x) −6h
(x) = lim
=lim
=lim
h→0 h
h→0 h
h→0 h
=lim (−6) = −6
h→0
f (4) f 000 (x + h) − f 000 (x) −6 − (−6) 0
(x) = lim
=lim
=lim
h→0 h
h→0 h
h→0 h =lim(0)
= 0
h→0
The graphs are consistent with the geometric interpretations of the
derivatives because f 0 has zeros where f has a local minimum and a local
maximum, f 00 has a zero where f 0 has a local maximum, and f 000 is a
constant function equal to the slope of f 00 .
49. (a)Notethatwehavefactoredx − a as the difference of two cubes in the third step.
f 0 f(x) − f(a) x 1/3 − a 1/3
x 1/3 − a 1/3
(a) =lim
=lim
= lim
x→a x − a x→a x − a x→a (x 1/3 − a 1/3 )(x 2/3 + x 1/3 a 1/3 + a 2/3 )
1
=lim
x→a x 2/3 + x 1/3 a 1/3 + a = 1
2/3 3a or 1 2/3 3 a−2/3
(b) f 0 f(0 + h) − f(0)
√ 3
h − 0
(0) = lim
= lim = lim
h→0 h
h→0 h
exist, and therefore f 0 (0) does not exist.
(c) lim
x→0
|f 0 (x)| =lim
51. f(x) =|x − 6| =
x→0
1
TX.10
1
h→0 h
x − 6 = lim
x→6
|x − 6| .
2/3
. This function increases without bound, so the limit does not
= ∞ and f is continuous at x =0(root function), so f has a vertical tangent at x =0.
3x2/3
x − 6 if x − 6 ≥ 6 x − 6 if x ≥ 6
−(x − 6) if x − 6 < 0 = 6 − x if x6
However, a formula for f 0 is f 0 (x) =
−1 if x

F.
TX.10
SECTION 2.8 THEDERIVATIVEASAFUNCTION ¤ 77
53. (a) f(x) =x |x| =
x
2
if x ≥ 0
−x 2 if x0.
[See Exercise 2.8.17(d).] Similarly, since f(x) =−x 2 for
x

F.
78 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
2 Review
1. (a) lim
x→a
f(x) =L: SeeDefinition2.2.1andFigures1and2inSection2.2.
(b)
(c)
lim f(x) =L: See the paragraph after Definition 2.2.2 and Figure 9(b) in Section 2.2.
x→a +
lim f(x) =L: SeeDefinition 2.2.2 and Figure 9(a) in Section 2.2.
x→a− (d) lim
x→a
f(x) =∞: SeeDefinition 2.2.4 and Figure 12 in Section 2.2.
(e) lim f(x) =L: SeeDefinition 2.6.1 and Figure 2 in Section 2.6.
x→∞
2. In general, the limit of a function fails to exist when the function does not approach a fixed number. For each of the following
functions, the limit fails to exist at x =2.
The left- and right-hand
limits are not equal.
There is an
infinite discontinuity.
There are an infinite
number of oscillations.
3. (a) – (g) See the statements of Limit Laws 1– 6 and 11 in Section 2.3.
4. See Theorem 3 in Section 2.3.
5. (a) See Definition 2.2.6 and Figures 12–14 in Section 2.2.
(b) See Definition 2.6.3 and Figures 3 and 4 in Section 2.6.
6. (a) y = x 4 :Noasymptote (b)y =sinx: No asymptote
(c) y =tanx: Vertical asymptotes x = π + πn, n an integer (d) y 2 =tan−1 x: Horizontal asymptotes y = ± π 2
(e) y = e x : Horizontal asymptote y =0
lim
x→−∞ ex =0
(g) y =1/x: Vertical asymptote x =0,
horizontal asymptote y =0
(f ) y =lnx: Vertical asymptote x =0
lim
x→0 + ln x = −∞
(h) y = √ x: No asymptote
7. (a) A function f is continuous at a number a if f(x) approaches f(a) as x approaches a;thatis, lim
x→a
f(x) =f(a).
(b) A function f is continuous on the interval (−∞, ∞) if f is continuous at every real number a. The graph of such a
function has no breaks and every vertical line crosses it.

F.
TX.10
CHAPTER 2 REVIEW ¤ 79
8. See Theorem 2.5.10.
9. See Definition 2.7.1.
10. See the paragraph containing Formula 3 in Section 2.7.
11. (a) The average rate of change of y with respect to x over the interval [x 1 ,x 2 ] is
f(x2) − f(x1)
x 2 − x 1
.
f(x 2) − f(x 1)
(b) The instantaneous rate of change of y with respect to x at x = x 1 is lim
.
x 2 →x 1 x 2 − x 1
12. See Definition 2.7.2. The pages following the definition discuss interpretations of f 0 (a) as the slope of a tangent line to the
graph of f at x = a and as an instantaneous rate of change of f(x) with respect to x when x = a.
13. See the paragraphs before and after Example 6 in Section 2.8.
14. (a) A function f is differentiable at a number a if its derivative f 0 exists
(c)
at x = a;thatis,iff 0 (a) exists.
(b) See Theorem 2.8.4. This theorem also tells us that if f is not
continuous at a,thenf is not differentiable at a.
15. See the discussion and Figure 7 on page 159.
1. False. Limit Law 2 applies only if the individual limits exist (these don’t).
3. True. Limit Law 5 applies.
x(x − 5) sin(x − 5)
5. False. Consider lim or lim .Thefirst limit exists and is equal to 5. By Example 3 in Section 2.2,
x→5 x − 5 x→5 x − 5
we know that the latter limit exists (and it is equal to 1).
7. True. A polynomial is continuous everywhere, so lim
x→b
p(x) exists and is equal to p(b).
9. True. See Figure 8 in Section 2.6.
11. False. Consider f(x) =
1/(x − 1) if x 6= 1
2 if x =1
13. True. Use Theorem 2.5.8 with a =2, b =5,andg(x) =4x 2 − 11. Notethatf(4) = 3 is not needed.
15. True, by the definition of a limit with ε =1.
17. False. See the note after Theorem 4 in Section 2.8.
19. False.
d 2 2
y
dy
dx is the second derivative while is the first derivative squared. For example, if y = x,
2 dx
then d 2 2
y dy
dx =0,but =1.
2 dx

F.
80 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
1. (a) (i) lim f(x) =3 (ii) lim f(x) =0
x→2 + x→−3 +
(iii)
lim f(x) does not exist since the left and right limits are not equal. (The left limit is −2.)
x→−3
(iv) lim
x→4
f(x) =2
(v) lim f(x) =∞ (vi) lim f(x) =−∞
x→0 −
(vii) lim f(x) =4 (viii) lim f(x) =−1
x→∞ x→−∞
(b) The equations of the horizontal asymptotes are y = −1 and y =4.
(c) The equations of the vertical asymptotes are x =0and x =2.
(d) f is discontinuous at x = −3, 0, 2,and4. The discontinuities are jump, infinite, infinite, and removable, respectively.
x→2
3. Since the exponential function is continuous, lim
x→1
e x3 −x = e 1−1 = e 0 =1.
x 2 − 9
5. lim
x→−3 x 2 +2x − 3 = lim (x +3)(x − 3)
x→−3 (x +3)(x − 1) = lim x − 3
x→−3 x − 1 = −3 − 3
−3 − 1 = −6
−4 = 3 2
(h − 1) 3 +1 h 3 − 3h 2 +3h − 1 +1 h 3 − 3h 2 +3h
7. lim
= lim
= lim
= lim h 2 − 3h +3 =3
h→0 h
h→0 h
h→0 h
h→0
Another solution: Factor the numerator as a sum of two cubes and then simplify.
(h − 1) 3 +1 (h − 1) 3 +1 3 [(h − 1) + 1] (h − 1) 2 − 1(h − 1) + 1 2
lim
= lim
=lim
h→0 h
h→0 h
h→0 h
9. lim
r→9
= lim
h→0
(h − 1) 2 − h +2 =1− 0+2=3
√ r
(r − 9) 4 = ∞ since (r − 9)4 → 0 as r → 9 and
√ r
> 0 for r 6= 9.
(r − 9)
4
u 4 − 1
11. lim
u→1 u 3 +5u 2 − 6u = lim (u 2 +1)(u 2 − 1)
u→1 u(u 2 +5u − 6) =lim (u 2 +1)(u +1)(u − 1) (u 2 +1)(u +1)
= lim
= 2(2)
u→1 u(u +6)(u − 1) u→1 u(u +6) 1(7) = 4 7
13. Since x is positive, √ x 2 = |x| = x. Thus,
√ √
x2 − 9
lim
x→∞ 2x − 6
= lim
x2 − 9/ √ √
x 2 1 − 9/x
x→∞ (2x − 6)/x = lim
2 1 − 0
x→∞ 2 − 6/x = 2 − 0 = 1 2
15. Let t =sinx. Thenasx → π − , sin x → 0 + ,sot → 0 + . Thus, lim ln(sin x) = lim ln t = −∞.
x→π− t→0 +
√
17. lim x2 +4x +1− x √ √
x2 +4x +1− x x2 +4x +1+x (x 2 +4x +1)− x 2
= lim
· √ = lim √
x→∞
x→∞ 1
x2 +4x +1+x x→∞ x2 +4x +1+x
(4x +1)/x
= lim
x→∞ ( √ x 2 +4x +1+x)/x
divide by x = √
x 2 for x>0
4+1/x
= lim
x→∞ 1+4/x +1/x2 +1 = 4+0
√ = 4 1+0+0+1 2 =2

F.
TX.10
CHAPTER 2 REVIEW ¤ 81
19. Let t =1/x. Thenasx → 0 + , t →∞,and lim
x→0 + tan−1 (1/x) = lim
t→∞
tan −1 t = π 2 .
21. From the graph of y = cos 2 x /x 2 , it appears that y =0is the horizontal
asymptote and x =0is the vertical asymptote. Now 0 ≤ (cos x) 2 ≤ 1
0
x ≤ cos2 x
≤ 1 ⇒ 0 ≤ cos2 x
≤ 1 .But lim
2 x 2 x 2 x 2 x 0=0and
2 x→±∞
lim
x→±∞
1
x
2
=0, so by the Squeeze Theorem, lim
x→±∞
cos 2 x
x 2 =0.
⇒
cos 2 x
Thus, y =0is the horizontal asymptote. lim = ∞ because cos 2 x → 1 and x 2 → 0 as x → 0, sox =0is the
x→0 x 2
vertical asymptote.
23. Since 2x − 1 ≤ f(x) ≤ x 2 for 0 0 such that if 0 < |x − 2|

F.
82 ¤ CHAPTER 2 LIMITS AND DERIVATIVES
TX.10
35. (a) The slope of the tangent line at (2, 1) is
f(x) − f(2) 9 − 2x 2 − 1 8 − 2x 2
lim
=lim
= lim
x→2 x − 2 x→2 x − 2 x→2 x − 2
= lim −2(x 2 − 4) −2(x − 2)(x +2)
= lim
x→2 x − 2 x→2 x − 2
=lim
x→2
[−2(x +2)]=−2 · 4=−8
(b) An equation of this tangent line is y − 1=−8(x − 2) or y = −8x +17.
37. (a) s = s(t) =1+2t + t 2 /4. The average velocity over the time interval [1, 1+h] is
v ave =
s(1 + h) − s(1)
(1 + h) − 1
= 1+2(1+h)+(1+h)2 4 − 13/4
h
=
10h + h2
4h
= 10 + h
4
So for the following intervals the average velocities are:
(i) [1, 3]: h =2, v ave =(10+2)/4 =3m/s
(iii) [1, 1.5]: h =0.5, v ave =(10+0.5)/4 =2.625 m/s
(ii) [1, 2]: h =1, v ave =(10+1)/4 =2.75 m/s
(iv) [1, 1.1]: h =0.1, v ave =(10+0.1)/4 =2.525 m/s
s(1 + h) − s(1) 10 + h
(b) When t =1, the instantaneous velocity is lim
= lim = 10
h→0 h
h→0 4 4
=2.5 m/s.
39. (a) f 0 f(x) − f(2) x 3 − 2x − 4
(2) = lim
=lim
x→2 x − 2 x→2 x − 2
(x − 2) x 2 +2x +2
= lim
x→2 x − 2
= lim
x→2
x 2 +2x +2 =10
(c)
(b) y − 4=10(x − 2) or y =10x − 16
41. (a) f 0 (r) is the rate at which the total cost changes with respect to the interest rate. Its units are dollars/(percent per year).
(b) The total cost of paying off the loan is increasing by $1200/(percent per year) as the interest rate reaches 10%. Soifthe
interest rate goes up from 10% to 11%, the cost goes up approximately $1200.
(c) As r increases, C increases. So f 0 (r) will always be positive.
43.

F.
TX.10
CHAPTER 2 REVIEW ¤ 83
√
45. (a) f 0 f(x + h) − f(x) 3 − 5(x + h) − 3 − 5x
(x) =lim
=lim
h→0 h
h→0 h
[3 − 5(x + h)] − (3 − 5x)
=lim
h→0
3 √ = lim
h − 5(x + h)+ 3 − 5x h→0
3 − 5(x + h)+
√ 3 − 5x
3 − 5(x + h)+
√ 3 − 5x
−5
3 − 5(x + h)+
√ 3 − 5x
=
−5
2 √ 3 − 5x
(b) Domain of f: (the radicand must be nonnegative) 3 − 5x ≥ 0
5x ≤ 3 ⇒ x ∈
−∞, 3 5
⇒
Domain of f 0 :exclude 3 because it makes the denominator zero;
5
x ∈
−∞, 3 5
(c) Our answer to part (a) is reasonable because f 0 (x) is always negative and f
is always decreasing.
47. f is not differentiable: at x = −4 because f is not continuous, at x = −1 because f has a corner, at x =2because f is not
continuous, and at x =5because f has a vertical tangent.
49. C 0 (1990) is the rate at which the total value of US currency in circulation is changing in billions of dollars per year. To
estimate the value of C 0 (1990), we will average the difference quotients obtained using the times t =1985and t = 1995.
Let A =
B =
C(1985) − C(1990)
1985 − 1990
C(1995) − C(1990)
1995 − 1990
C 0 (1990) =
=
=
187.3 − 271.9
−5
409.3 − 271.9
5
= −84.6
−5
= 137.4
5
=16.92 and
=27.48. Then
C(t) − C(1990)
lim
≈ A + B 16.92 + 27.48
= = 44.4 =22.2 billion dollars/year.
t→1990 t − 1990 2
2 2
51. |f(x)| ≤ g(x) ⇔ −g(x) ≤ f(x) ≤ g(x) and lim
x→a
g(x) = 0 = lim
x→a
−g(x).
Thus, by the Squeeze Theorem, lim
x→a
f(x) =0.

F.
TX.10

F.
TX.10
PROBLEMS PLUS
1. Let t = 6√ x,sox = t 6 .Thent → 1 as x → 1,so
lim
x→1
3√ x − 1
√ x − 1
= lim
t→1
t 2 − 1
t 3 − 1 =lim
t→1
(t − 1)(t +1)
(t − 1) (t 2 + t +1) =lim
t→1
t +1
t 2 + t +1 = 1+1
1 2 +1+1 = 2 3 .
Another method: Multiply both the numerator and the denominator by ( √
x +1)
3√
x2 + 3√
x +1 .
3. For − 1 0,so|2x − 1| = −(2x − 1) and |2x +1| =2x +1.
2 2
|2x − 1| − |2x +1| −(2x − 1) − (2x +1) −4x
Therefore, lim
=lim
=lim
x→0 x
x→0 x
x→0 x
= lim (−4) = −4.
x→0
5. Since [x] ≤ x 0,thenG(a +180 ◦ )=T (a +360 ◦ ) − T (a + 180 ◦ )=T (a) − T (a +180 ◦ )=−G(a) < 0.
Also, G is continuous since temperature varies continuously. So, by the Intermediate Value Theorem, G has a zero on the
interval [a, a +180 ◦ ].IfG(a) < 0, then a similar argument applies.
85

F.
86 ¤ CHAPTER 2 PROBLEMS PLUS
TX.10
(b) Yes. The same argument applies.
(c) The same argument applies for quantities that vary continuously, such as barometric pressure. But one could argue that
altitude above sea level is sometimes discontinuous, so the result might not always hold for that quantity.
13. (a) Put x =0and y =0in the equation: f(0 + 0) = f(0) + f(0) + 0 2 · 0+0· 0 2 ⇒ f(0) = 2f(0).
Subtracting f(0) from each side of this equation gives f(0) = 0.
(b) f 0 f(0 + h) − f(0) f(0) + f(h)+0 2 h +0h 2 − f(0) f(h)
(0) = lim
=lim
=lim
h→0 h
h→0 h
h→0 h
= lim f(x)
x→0 x =1
(c) f 0 f(x + h) − f(x) f(x)+f(h)+x 2 h + xh 2 − f(x)
(x) =lim
=lim
h→0 h
h→0 h
f(h)
=lim
h→0 h
+ x2 + xh =1+x 2
=lim
h→0
f(h)+x 2 h + xh 2
h

F.
TX.10
3 DIFFERENTIATION RULES
3.1 Derivatives of Polynomials and Exponential Functions
e h − 1
1. (a) e is the number such that lim =1.
h→0 h
(b)
x
2.7 x − 1
x
−0.001 0.9928
−0.0001 0.9932
0.001 0.9937
0.0001 0.9933
x
2.8 x − 1
x
−0.001 1.0291
−0.0001 1.0296
0.001 1.0301
0.0001 1.0297
From the tables (to two decimal places),
2.7 h − 1
2.8 h − 1
lim =0.99 and lim =1.03.
h→0 h
h→0 h
Since 0.99 < 1 < 1.03, 2.7

F.
88 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
27. We first expand using the Binomial Theorem (see Reference Page 1).
H(x) =(x + x −1 ) 3 = x 3 +3x 2 x −1 +3x(x −1 ) 2 +(x −1 ) 3 = x 3 +3x +3x −1 + x −3
⇒
H 0 (x) =3x 2 +3+3(−1x −2 )+(−3x −4 )=3x 2 +3− 3x −2 − 3x −4
29. u = 5√ t +4 √
t 5 = t 1/5 +4t 5/2 ⇒ u 0 = 1 5 t−4/5 5
+4
2 t3/2 = 1 5 t−4/5 +10t 3/2 or 1/
5 5√
t 4 +10 √ t 3
31. z = A
y + 10 Bey = Ay −10 + Be y ⇒ z 0 = −10Ay −11 + Be y = − 10A
y + 11 Bey
33. y = 4√ x = x 1/4 ⇒ y 0 = 1 4 x−3/4 = 1
4 4√ x . At(1, 1), 3 y0 = 1 4
and an equation of the tangent line is
y − 1= 1 4 (x − 1) or y = 1 4 x + 3 4 .
35. y = x 4 +2e x ⇒ y 0 =4x 3 +2e x . At (0, 2), y 0 =2and an equation of the tangent line is y − 2=2(x − 0)
or y =2x +2. The slope of the normal line is − 1 (the negative reciprocal of 2) and an equation of the normal line is
2
y − 2=− 1 (x − 0) or y = − 1 x +2.
2 2
37. y =3x 2 − x 3 ⇒ y 0 =6x − 3x 2 .
At (1, 2), y 0 =6− 3=3, so an equation of the tangent line is
y − 2=3(x − 1) or y =3x − 1.
39. f(x) =e x − 5x ⇒ f 0 (x) =e x − 5.
Notice that f 0 (x) =0when f has a horizontal tangent, f 0 is positive
when f is increasing, and f 0 is negative when f is decreasing.
41. f(x) =3x 15 − 5x 3 +3 ⇒ f 0 (x) =45x 14 − 15x 2 .
Notice that f 0 (x) =0when f has a horizontal tangent, f 0 is positive
when f is increasing, and f 0 is negative when f is decreasing.

F.
SECTION TX.10 3.1 DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS ¤ 89
43. (a) (b) From the graph in part (a), it appears that f 0 is zero at x 1 ≈−1.25, x 2 ≈ 0.5,
and x 3 ≈ 3. The slopes are negative (so f 0 is negative) on (−∞,x 1) and
(x 2,x 3). The slopes are positive (so f 0 is positive) on (x 1,x 2) and (x 3, ∞).
(c) f(x) =x 4 − 3x 3 − 6x 2 +7x +30
⇒
f 0 (x) =4x 3 − 9x 2 − 12x +7
45. f(x) =x 4 − 3x 3 +16x ⇒ f 0 (x) =4x 3 − 9x 2 +16 ⇒ f 00 (x) =12x 2 − 18x
47. f(x) =2x − 5x 3/4 ⇒ f 0 (x) =2− 15
4 x−1/4 ⇒ f 00 (x) = 15
16 x−5/4
Note that f 0 is negative when f is decreasing and positive when f is
increasing. f 00 is always positive since f 0 is always increasing.
49. (a) s = t 3 − 3t ⇒ v(t) =s 0 (t) =3t 2 − 3 ⇒ a(t) =v 0 (t) =6t
(b) a(2) = 6(2) = 12 m/s 2
(c) v(t) =3t 2 − 3=0when t 2 =1,thatis,t =1and a(1) = 6 m/s 2 .
51. The curve y =2x 3 +3x 2 − 12x +1has a horizontal tangent when y 0 =6x 2 +6x − 12 = 0 ⇔ 6(x 2 + x − 2) = 0 ⇔
6(x +2)(x − 1) = 0 ⇔ x = −2 or x =1. The points on the curve are (−2, 21) and (1, −6).
53. y =6x 3 +5x − 3 ⇒ m = y 0 =18x 2 +5,butx 2 ≥ 0 for all x,som ≥ 5 for all x.
55. The slope of the line 12x − y =1(or y =12x − 1)is12, so the slope of both lines tangent to the curve is 12.
y =1+x 3 ⇒ y 0 =3x 2 . Thus, 3x 2 =12 ⇒ x 2 =4 ⇒ x = ±2, which are the x-coordinates at which the tangent
lines have slope 12. The points on the curve are (2, 9) and (−2, −7), so the tangent line equations are y − 9=12(x − 2)
or y =12x − 15 and y + 7 = 12(x +2)or y =12x +17.

F.
90 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
57. The slope of y = x 2 − 5x +4is given by m = y 0 =2x − 5. The slope of x − 3y =5 ⇔ y = 1 3 x − 5 3 is 1 3 ,
so the desired normal line must have slope 1 , and hence, the tangent line to the parabola must have slope −3. This occurs if
3
2x − 5=−3 ⇒ 2x =2 ⇒ x =1.Whenx =1, y =1 2 − 5(1) + 4 = 0, and an equation of the normal line is
y − 0= 1 3 (x − 1) or y = 1 3 x − 1 3 .
59. Let a, a 2 be a point on the parabola at which the tangent line passes through the
point (0, −4). The tangent line has slope 2a and equation y − (−4) = 2a(x − 0)
⇔
y =2ax − 4. Since a, a 2 also lies on the line, a 2 =2a(a) − 4,ora 2 =4.So
a = ±2 and the points are (2, 4) and (−2, 4).
1
61. f 0 f(x + h) − f(x)
(x) = lim
= lim
x + h − 1 x
h→0 h
h→0 h
= lim
h→0
x − (x + h)
hx(x + h) =lim
h→0
−h
hx(x + h) =lim
h→0
63. Let P (x) =ax 2 + bx + c. ThenP 0 (x) =2ax + b and P 00 (x) =2a. P 00 (2) = 2 ⇒ 2a =2 ⇒ a =1.
P 0 (2) = 3 ⇒ 2(1)(2) + b =3 ⇒ 4+b =3 ⇒ b = −1.
P (2) = 5 ⇒ 1(2) 2 +(−1)(2) + c =5 ⇒ 2+c =5 ⇒ c =3.SoP (x) =x 2 − x +3.
−1
x(x + h) = − 1 x 2
65. y = f(x) =ax 3 + bx 2 + cx + d ⇒ f 0 (x) =3ax 2 +2bx + c. Thepoint(−2, 6) is on f, sof(−2) = 6 ⇒
−8a +4b − 2c + d =6 (1). The point (2, 0) is on f, sof(2) = 0 ⇒ 8a +4b +2c + d =0 (2). Since there are
horizontal tangents at (−2, 6) and (2, 0), f 0 (±2) = 0. f 0 (−2) = 0 ⇒ 12a − 4b + c =0 (3) and f 0 (2) = 0 ⇒
12a +4b + c =0 (4). Subtracting equation (3) from (4) gives 8b =0 ⇒ b =0. Adding (1) and (2) gives 8b +2d =6,
so d =3since b =0.From(3) we have c = −12a,so(2) becomes 8a +4(0)+2(−12a)+3=0 ⇒ 3=16a ⇒
a = 3
3
16 16 = −
9
3
and the desired cubic function is y =
4 16 x3 − 9 x +3. 4
67. f(x) =2− x if x ≤ 1 and f(x) =x 2 − 2x +2if x>1. Now we compute the right- and left-hand derivatives defined in
Exercise 2.8.54:
f 0 −(1) =
f(1 + h) − f(1) 2 − (1 + h) − 1 −h
lim
= lim
= lim
h→0 − h
h→0 − h
h→0 − h = lim −1 =−1 and
h→0− f+(1) 0 f(1 + h) − f(1) (1 + h) 2 − 2(1 + h)+2− 1 h 2
= lim
= lim
= lim
h→0 + h
h→0 + h
h→0 + h = lim h =0.
h→0 +
Thus, f 0 (1) does not exist since f−(1) 0 6=f+(1),sof
0
is not differentiable at 1. Butf 0 (x) =−1 for x1.

F.
SECTION TX.10 3.1 DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS ¤ 91
69. (a) Note that x 2 − 9 < 0 for x 2 < 9 ⇔ |x| < 3 ⇔ −3

F.
92 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
79. y = x 2 ⇒ y 0 =2x, so the slope of a tangent line at the point (a, a 2 ) is y 0 =2a and the slope of a normal line is −1/(2a),
for a 6= 0. The slope of the normal line through the points (a, a 2 ) and (0,c) is a2 − c − c
a − 0 ,soa2 = − 1
a 2a
⇒
a 2 − c = − 1 ⇒ a 2 = c − 1 . The last equation has two solutions if c> 1 , one solution if c = 1 , and no solution if
2 2 2 2
c< 1 2 . Since the y-axis is normal to y = x2 regardless of the value of c (thisisthecasefora =0), we have three normal lines
if c> 1 2 and one normal line if c ≤ 1 2 .
3.2 The Product and Quotient Rules
1. Product Rule: y =(x 2 +1)(x 3 +1) ⇒
y 0 =(x 2 +1)(3x 2 )+(x 3 + 1)(2x) =3x 4 +3x 2 +2x 4 +2x =5x 4 +3x 2 +2x.
Multiplying first: y =(x 2 +1)(x 3 +1)=x 5 + x 3 + x 2 +1 ⇒ y 0 =5x 4 +3x 2 +2x (equivalent).
3. By the Product Rule, f(x) =(x 3 +2x)e x ⇒
f 0 (x)=(x 3 +2x)(e x ) 0 + e x (x 3 +2x) 0 =(x 3 +2x)e x + e x (3x 2 +2)
= e x [(x 3 +2x)+(3x 2 +2)]=e x (x 3 +3x 2 +2x +2)
x 2 d
5. By the Quotient Rule, y = ex
⇒ y 0 = dx (ex ) − e x d
dx (x2 )
= x2 (e x ) − e x (2x)
= xex (x − 2)
= ex (x − 2)
.
x 2 (x 2 ) 2 x 4 x 4 x 3
The notations
PR
⇒
and
QR
⇒
indicate the use of the Product and Quotient Rules, respectively.
7. g(x) = 3x − 1
2x +1
QR
⇒ g 0 (2x + 1)(3) − (3x − 1)(2) 6x +3− 6x +2 5
(x) = = =
(2x +1) 2 (2x +1) 2 (2x +1) 2
9. V (x) =(2x 3 +3)(x 4 − 2x)
PR
⇒
V 0 (x) =(2x 3 + 3)(4x 3 − 2) + (x 4 − 2x)(6x 2 )=(8x 6 +8x 3 − 6) + (6x 6 − 12x 3 )=14x 6 − 4x 3 − 6
1
11. F (y) =
y − 3
(y +5y 3 )= y −2 − 3y −4 y +5y 3 PR
⇒
2 y 4
F 0 (y) =(y −2 − 3y −4 )(1 + 15y 2 )+(y +5y 3 )(−2y −3 +12y −5 )
=(y −2 +15− 3y −4 − 45y −2 )+(−2y −2 +12y −4 − 10 + 60y −2 )
=5+14y −2 +9y −4 or 5+14/y 2 +9/y 4
13. y = x3
1 − x 2 QR
⇒
y 0 = (1 − x2 )(3x 2 ) − x 3 (−2x)
= x2 (3 − 3x 2 +2x 2 )
= x2 (3 − x 2 )
(1 − x 2 ) 2 (1 − x 2 ) 2 (1 − x 2 ) 2

F.
TX.10
SECTION 3.2 THE PRODUCT AND QUOTIENT RULES ¤ 93
15. y =
t 2 +2
t 4 − 3t 2 +1
QR
⇒
y 0 = (t4 − 3t 2 +1)(2t) − (t 2 +2)(4t 3 − 6t)
= 2t[(t4 − 3t 2 +1)− (t 2 + 2)(2t 2 − 3)]
(t 4 − 3t 2 +1) 2 (t 4 − 3t 2 +1) 2
= 2t(t4 − 3t 2 +1− 2t 4 − 4t 2 +3t 2 +6)
= 2t(−t4 − 4t 2 +7)
(t 4 − 3t 2 +1) 2 (t 4 − 3t 2 +1) 2
17. y =(r 2 − 2r)e r PR
⇒ y 0 =(r 2 − 2r)(e r )+e r (2r − 2) = e r (r 2 − 2r +2r − 2) = e r (r 2 − 2)
19. y = v3 − 2v √ v
v
= v 2 − 2 √ v = v 2 − 2v 1/2 ⇒ y 0 =2v − 2 1
2
v −1/2 =2v − v −1/2 .
We can change the form of the answer as follows: 2v − v −1/2 =2v − 1 √ v
= 2v √ v − 1
√ v
= 2v3/2 − 1
√ v
21. f(t) = 2t
2+ √ t
QR
⇒
(2 + t 1/2 1
)(2) − 2t
f 0 (t) =
(2 + √ t ) 2 = 4+2t1/2 − t 1/2
(2 + √ t ) 2 = 4+t1/2
(2 + √ t ) 2 or 4+ √ t
(2 + √ t ) 2
2 t−1/2
23. f(x) =
A
B + Ce x
QR
⇒ f 0 (x) = (B + Cex ) · 0 − A(Ce x )
= − ACex
(B + Ce x ) 2 (B + Ce x ) 2
x
25. f(x) =
x + c/x ⇒ f 0 (x) = (x + c/x)(1) − x(1 − c/x2 ) x + c/x − x + c/x
x + c 2
= x 2 2
= 2c/x
+ c (x 2 + c) 2
x
x
x 2
· x2
x = 2cx
2 (x 2 + c) 2
27. f(x) =x 4 e x ⇒ f 0 (x) =x 4 e x + e x · 4x 3 = x 4 +4x 3 e x or x 3 e x (x +4) ⇒
f 00 (x)=(x 4 +4x 3 )e x + e x (4x 3 +12x 2 )=(x 4 +4x 3 +4x 3 +12x 2 )e x
=(x 4 +8x 3 +12x 2 )e x
or x 2 e x (x +2)(x +6)
29. f(x) = x2
1+2x ⇒ f 0 (x) = (1 + 2x)(2x) − x2 (2)
(1 + 2x) 2 = 2x +4x2 − 2x 2
(1 + 2x) 2 = 2x2 +2x
(1 + 2x) 2 ⇒
f 00 (x)= (1 + 2x)2 (4x +2)− (2x 2 +2x)(1 + 4x +4x 2 ) 0
= 2(1 + 2x)2 (2x +1)− 2x(x +1)(4+8x)
[(1 + 2x) 2 ] 2 (1 + 2x) 4
= 2(1 + 2x)[(1 + 2x)2 − 4x(x +1)]
= 2(1 + 4x +4x2 − 4x 2 − 4x) 2
=
(1 + 2x) 4 (1 + 2x) 3 (1 + 2x) 3
31. y = 2x
x +1
⇒
y 0 =
(x +1)(2)− (2x)(1) 2
=
(x +1) 2 (x +1) . 2
At (1, 1), y 0 = 1 2 , and an equation of the tangent line is y − 1= 1 2 (x − 1),ory = 1 2 x + 1 2 .
33. y =2xe x ⇒ y 0 =2(x · e x + e x · 1) = 2e x (x +1).
At (0, 0), y 0 =2e 0 (0 + 1) = 2 · 1 · 1=2, and an equation of the tangent line is y − 0=2(x − 0),ory =2x. Theslopeof
the normal line is − 1 2 , so an equation of the normal line is y − 0=− 1 2 (x − 0),ory = − 1 2 x.

F.
94 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
35. (a) y = f(x) = 1
1+x 2 ⇒
f 0 (x) = (1 + x2 )(0) − 1(2x)
= −2x . So the slope of the
(1 + x 2 ) 2 (1 + x 2 )
2
(b)
tangent line at the point
−1, 1 2 is f 0 (−1) = 2 2 = 1 2 2
and its
equation is y − 1 2 = 1 2 (x +1)or y = 1 2 x +1.
37. (a) f(x) = ex
⇒ f 0 (x) = x3 (e x ) − e x (3x 2 )
= x2 e x (x − 3)
= ex (x − 3)
x 3 (x 3 ) 2 x 6 x 4
(b)
f 0 =0when f has a horizontal tangent line, f 0 is negative when
f is decreasing, and f 0 is positive when f is increasing.
39. (a) f(x) =(x − 1)e x ⇒ f 0 (x) =(x − 1)e x + e x (1) = e x (x − 1+1)=xe x .
f 00 (x) =x(e x )+e x (1) = e x (x +1)
(b)
f 0 =0when f has a horizontal tangent and f 00 =0when f 0 has a
horizontal tangent. f 0 is negative when f is decreasing and positive when f
is increasing. f 00 is negative when f 0 is decreasing and positive when f 0 is
increasing. f 00 is negative when f is concave down and positive when f is
concave up.
41. f(x) = x2
1+x ⇒ f 0 (x) = (1 + x)(2x) − x2 (1)
(1 + x) 2 = 2x +2x2 − x 2
(1 + x) 2 = x2 +2x
x 2 +2x +1
⇒
so f 00 (1) =
f 00 (x) = (x2 +2x +1)(2x +2)− (x 2 +2x)(2x +2)
(x 2 +2x +1) 2 = (2x +2)(x2 +2x +1− x 2 − 2x)
[(x +1) 2 ] 2
=
2
(1 + 1) 3 = 2 8 = 1 4 .
2(x +1)(1) 2
=
(x +1) 4 (x +1) 3 ,
43. We are given that f(5) = 1, f 0 (5) = 6, g(5) = −3,andg 0 (5) = 2.
(a) (fg) 0 (5) = f(5)g 0 (5) + g(5)f 0 (5) = (1)(2) + (−3)(6) = 2 − 18 = −16
0 f
(b) (5) = g(5)f 0 (5) − f(5)g 0 (5) (−3)(6) − (1)(2)
= = − 20 g
[g(5)] 2 (−3) 2 9
0 g
(c) (5) = f(5)g0 (5) − g(5)f 0 (5)
=
f
[f(5)] 2
(1)(2) − (−3)(6)
(1) 2 =20
45. f(x) =e x g(x) ⇒ f 0 (x) =e x g 0 (x)+g(x)e x = e x [g 0 (x)+g(x)]. f 0 (0) = e 0 [g 0 (0) + g(0)] = 1(5 + 2) = 7

F.
TX.10
SECTION 3.2 THE PRODUCT AND QUOTIENT RULES ¤ 95
47. (a) From the graphs of f and g, we obtain the following values: f(1) = 2 since the point (1, 2) is on the graph of f;
g(1) = 1 since the point (1, 1) is on the graph of g; f 0 (1) = 2 since the slope of the line segment between (0, 0) and (2, 4)
is 4 − 0
2 − 0 =2; 0 − 4
g0 (1) = −1 since the slope of the line segment between (−2, 4) and (2, 0) is
2 − (−2) = −1.
Now u(x) =f(x)g(x),sou 0 (1) = f(1)g 0 (1) + g(1) f 0 (1) = 2 · (−1) + 1 · 2=0.
(b) v(x) =f(x)/g(x),sov 0 (5) = g(5)f 0 (5) − f(5)g 0 (5)
= 2
− 1 3 − 3 · 2
3
= − 8 3
[g(5)] 2 2 2 4 = − 2 3
49. (a) y = xg(x) ⇒ y 0 = xg 0 (x)+g(x) · 1=xg 0 (x)+g(x)
(b) y =
x
g(x)
⇒
y 0 = g(x) · 1 − xg0 (x)
[g(x)] 2
= g(x) − xg0 (x)
[g(x)] 2
(c) y = g(x)
x
⇒ y 0 = xg0 (x) − g(x) · 1
(x) 2
= xg0 (x) − g(x)
x 2
51. If y = f(x) = x
x +1 ,thenf 0 (x +1)(1)− x(1) 1
(x) = = .Whenx = a, the equation of the tangent line is
(x +1) 2 (x +1)
2
y −
a
a +1 = 1
a
(x − a). This line passes through (1, 2) when 2 −
(a +1)
2
a +1 = 1
(1 − a) ⇔
(a +1)
2
2(a +1) 2 − a(a +1)=1− a ⇔ 2a 2 +4a +2− a 2 − a − 1+a =0 ⇔ a 2 +4a +1=0.
The quadratic formula gives the roots of this equation as a = −4 ± 4 2 − 4(1)(1)
2(1)
so there are two such tangent lines. Since
f −2 ± √ 3 = −2 ± √ 3
−2 ± √ 3+1 = −2 ± √ 3
−1 ± √ 3 · −1 ∓ √ 3
−1 ∓ √ 3
the lines touch the curve at A
= 2 ± 2 √ 3 ∓ √ 3 − 3
1 − 3
−2+ √ 3, 1 − √ 3
2
and B −2 − √
3, 1+√ 3
≈ (−3.73, 1.37).
2
= −1 ± √ 3
−2
≈ (−0.27, −0.37)
= 1 ∓ √ 3
,
2
= −4 ± √ 12
2
= −2 ± √ 3,
53. If P (t) denotes the population at time t and A(t) the average annual income, then T (t) =P (t)A(t) is the total personal
income. The rate at which T (t) is rising is given by T 0 (t) =P (t)A 0 (t)+A(t)P 0 (t)
⇒
T 0 (1999) = P (1999)A 0 (1999) + A(1999)P 0 (1999) = (961,400)($1400/yr)+($30,593)(9200/yr)
=$1,345,960,000/yr +$281,455,600/yr =$1,627,415,600/yr
So the total personal income was rising by about $1.627 billion per year in 1999.
The term P (t)A 0 (t) ≈ $1.346 billion represents the portion of the rate of change of total income due to the existing
population’s increasing income. The term A(t)P 0 (t) ≈ $281 million represents the portion of the rate of change of total
income due to increasing population.

F.
96 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
We will sometimes use the form f 0 g + fg 0 rather than the form fg 0 + gf 0 for the Product Rule.
55. (a) (fgh) 0 =[(fg)h] 0 =(fg) 0 h +(fg)h 0 =(f 0 g + fg 0 )h +(fg)h 0 = f 0 gh + fg 0 h + fgh 0
(b) Putting f = g = h in part (a), we have d
dx [f(x)]3 =(fff) 0 = f 0 ff + ff 0 f + fff 0 =3fff 0 =3[f(x)] 2 f 0 (x).
(c)
d
dx (e3x )= d
dx (ex ) 3 =3(e x ) 2 e x =3e 2x e x =3e 3x
57. For f(x) =x 2 e x , f 0 (x) =x 2 e x + e x (2x) =e x (x 2 +2x). Similarly, we have
f 00 (x) =e x (x 2 +4x +2)
f 000 (x) =e x (x 2 +6x +6)
f (4) (x) =e x (x 2 +8x + 12)
f (5) (x) =e x (x 2 +10x +20)
It appears that the coefficient of x in the quadratic term increases by 2 with each differentiation. The pattern for the
constant terms seems to be 0=1· 0, 2=2· 1, 6=3· 2, 12 = 4 · 3, 20 = 5 · 4. So a reasonable guess is that
f (n) (x) =e x [x 2 +2nx + n(n − 1)].
Proof: Let S n be the statement that f (n) (x) =e x [x 2 +2nx + n(n − 1)].
1. S 1 is true because f 0 (x) =e x (x 2 +2x).
2. Assume that S k is true; that is, f (k) (x) =e x [x 2 +2kx + k(k − 1)]. Then
f (k+1) (x) = d
f (k) (x) = e x (2x +2k)+[x 2 +2kx + k(k − 1)]e x
dx
= e x [x 2 +(2k +2)x +(k 2 + k)] = e x [x 2 +2(k +1)x +(k +1)k]
This shows that S k+1 is true.
3. Therefore, by mathematical induction, S n is true for all n;thatis,f (n) (x) =e x [x 2 +2nx + n(n − 1)] for every
positive integer n.
3.3 Derivatives of Trigonometric Functions
1. f(x) =3x 2 − 2cosx ⇒ f 0 (x) =6x − 2(− sin x) =6x +2sinx
3. f(x) =sinx + 1 2 cot x ⇒ f 0 (x) =cosx − 1 2 csc2 x
5. g(t) =t 3 cos t ⇒ g 0 (t) =t 3 (− sin t)+(cost) · 3t 2 =3t 2 cos t − t 3 sin t or t 2 (3 cos t − t sin t)
7. h(θ) =cscθ + e θ cot θ ⇒ h 0 (θ) =− csc θ cot θ + e θ (− csc 2 θ)+(cotθ)e θ = − csc θ cot θ + e θ (cot θ − csc 2 θ)
9. y =
x
2 − tan x ⇒ y0 = (2 − tan x)(1) − x(− sec2 x)
= 2 − tan x + x sec2 x
(2 − tan x) 2 (2 − tan x) 2
11. f(θ) = sec θ
1+secθ
⇒
f 0 (θ) =
(1 + sec θ)(sec θ tan θ) − (sec θ)(sec θ tan θ) (sec θ tan θ) [(1+secθ) − sec θ] sec θ tan θ
= =
(1 + sec θ) 2 (1 + sec θ) 2 (1 + sec θ) 2

F.
TX.10
SECTION 3.3 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS ¤ 97
13. y = sin x ⇒ y 0 = x2 cos x − (sin x)(2x) x(x cos x − 2sinx) x cos x − 2sinx
x 2 (x 2 ) 2 = =
x 4
x 3
15. Using Exercise 3.2.55(a), f(x) =xe x csc x ⇒
17.
19.
f 0 (x)=(x) 0 e x csc x + x(e x ) 0 csc x + xe x (csc x) 0 =1e x csc x + xe x csc x + xe x (− cot x csc x)
= e x csc x (1 + x − x cot x)
d
d 1 (sin x)(0) − 1(cos x)
(csc x) = =
dx dx sin x
sin 2 = − cos x
x
sin 2 x = − 1
sin x · cos x = − csc x cot x
sin x
d
d
(cot x) =
dx dx
cos x
(sin x)(− sin x) − (cos x)(cos x)
=
sin x
sin 2 = − sin2 x +cos 2 x
x
sin 2 = − 1
x sin 2 x = − csc2 x
21. y =secx ⇒ y 0 =secx tan x,soy 0 ( π 3 )=secπ 3 tan π 3 =2√ 3. An equation of the tangent line to the curve y =secx
at the point π
, 2 is y − 2=2 √ 3 √ √
x − π 3 3 or y =2 3 x +2−
2
3 3 π.
23. y = x +cosx ⇒ y 0 =1− sin x. At(0, 1), y 0 =1, and an equation of the tangent line is y − 1=1(x − 0),ory = x +1.
25. (a) y =2x sin x ⇒ y 0 =2(x cos x +sinx · 1). At π
2 ,π ,
y 0 =2 π
cos π
2 2 +sinπ 2 =2(0+1)=2, and an equation of the
tangent line is y − π =2
x − π 2 ,ory =2x.
(b)
27. (a) f(x) =secx − x ⇒ f 0 (x) =secx tan x − 1
(b)
Note that f 0 =0where f has a minimum. Also note that f 0 is negative
when f is decreasing and f 0 is positive when f is increasing.
29. H(θ) =θ sin θ ⇒ H 0 (θ) =θ (cos θ) +(sinθ) · 1=θ cos θ +sinθ ⇒
H 00 (θ) =θ (− sin θ)+(cosθ) · 1+cosθ = −θ sin θ +2cosθ
31. (a) f(x) = tan x − 1
sec x
⇒
f 0 (x) = sec x(sec2 x) − (tan x − 1)(sec x tan x)
(sec x) 2
(b) f(x) = tan x − 1
sec x
=
sin x
cos x − 1
=
1
cos x
(c) From part (a), f 0 (x) = 1+tanx
sec x
sin x − cos x
cos x
1
cos x
= sec x(sec2 x − tan 2 x +tanx)
sec 2 x
= 1+tanx
sec x
=sinx − cos x ⇒ f 0 (x) =cosx − (− sin x) =cosx +sinx
= 1
sec x + tan x
sec x =cosx +sinx, which is the expression for f 0 (x) in part (b).

F.
98 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
33. f(x) =x +2sinx has a horizontal tangent when f 0 (x) =0 ⇔ 1+2cosx =0 ⇔ cos x = − 1 2
⇔
x = 2π +2πn or 4π +2πn,wheren is an integer. Note that 4π and 2π are ± π units from π.Thisallowsustowritethe
3 3 3 3 3
solutions in the more compact equivalent form (2n +1)π ± π 3
, n an integer.
35. (a) x(t) =8sint ⇒ v(t) =x 0 (t) =8cost ⇒ a(t) =x 00 (t) =−8sint
(b) The mass at time t = 2π 3
has position x √3
2π
3 =8sin
2π
=8 3 2
and acceleration a √3
2π
3 = −8sin
2π
= −8 3 2
=4 √ 3,velocityv
2π
3 =8cos
2π
=8
3
− 1 2 = −4,
= −4 √ 3.Sincev
2π
3 < 0, the particle is moving to the left.
37. From the diagram we can see that sin θ = x/10 ⇔ x =10sinθ. Wewanttofind the rate
of change of x with respect to θ,thatis,dx/dθ. Taking the derivative of x =10sinθ,weget
dx/dθ = 10(cos θ). Sowhenθ = π 3 , dx
dθ =10cosπ 3 =10 1
2
=5ft/rad.
39. lim
x→0
sin 3x
x
3sin3x
=lim
x→0 3x
=3 lim
3x→0
sin 3x
3x
[multiply numerator and denominator by 3]
[as x → 0, 3x → 0]
sin θ
=3lim
θ→0 θ
[let θ =3x]
=3(1) [Equation 2]
=3
tan 6t sin 6t
41. lim
t→0 sin 2t =lim ·
t→0 t
sin 6t
=6lim
t→0 6t
1
cos 6t ·
t
6sin6t 1
=lim · lim
sin 2t t→0 6t t→0 cos 6t · lim
t→0
1
· lim
t→0 cos 6t · 1
2 lim
t→0
sin(cos θ)
sin lim cos θ
θ→0
43. lim =
θ→0 sec θ lim sec θ = sin 1 =sin1
1
θ→0
45. Divide numerator and denominator by θ. (sin θ also works.)
lim
θ→0
47. lim
x→π/4
49. (a)
(b)
sin θ
θ +tanθ =lim
θ→0
1 − tan x
sin x − cos x =
d
dx tan x = d
dx
d
dx sec x = d
dx
sin θ
θ
1+ sin θ
θ
lim
x→π/4
·
1
cos θ
sin x
cos x ⇒ sec2 x =
1
cos x
=
2t
sin 2t =6(1)· 1
1 · 1
2 (1) = 3
sin θ
lim
θ→0 θ
sin θ
1+lim lim
θ→0 θ θ→0
1 − sin x
· cos x
cos x
(sin x − cos x) · cos x =
lim
x→π/4
1
cos θ
cos x cos x − sin x (− sin x)
cos 2 x
=
2t
2sin2t
1
1+1· 1 = 1 2
cos x − sin x
(sin x − cos x)cosx =
lim
x→π/4
−1
cos x = −1
1/ √ 2 = −√ 2
= cos2 x +sin 2 x
. Sosec 2 x = 1
cos 2 x
cos 2 x .
(cos x)(0) − 1(− sin x)
⇒ sec x tan x = . Sosec x tan x = sin x
cos 2 x
cos 2 x .

F.
TX.10
SECTION 3.4 THE CHAIN RULE ¤ 99
(c)
d
d 1+cotx
(sin x +cosx) =
dx dx csc x
⇒
cos x − sin x = csc x (− csc2 x) − (1 + cot x)(− csc x cot x)
csc 2 x
= − csc2 x +cot 2 x +cotx
csc x
So cos x − sin x = cot x − 1
csc x .
= −1+cotx
csc x
= csc x [− csc2 x +(1+cotx) cotx]
csc 2 x
51. By the definition of radian measure, s = rθ,wherer is the radius of the circle. By drawing the bisector of the angle θ,wecan
see that sin θ 2 = d/2
r
⇒ d =2r sin θ s
.So lim
2 θ→0 + d = lim
θ→0 +
rθ
2r sin(θ/2) = lim
θ→0 +
2 · (θ/2)
2sin(θ/2) =lim
θ→0
θ/2
sin(θ/2) =1.
sin x
[This is just the reciprocal of the limit lim =1combined with the fact that as θ → 0, θ → 0 also.]
x→0 x
2
3.4 The Chain Rule
1. Let u = g(x) =4x and y = f(u) =sinu. Then dy
dx = dy du
=(cosu)(4) = 4 cos 4x.
du dx
3. Let u = g(x) =1− x 2 and y = f(u) =u 10 . Then dy
dx = dy du
du dx =(10u9 )(−2x) =−20x(1 − x 2 ) 9 .
5. Let u = g(x) = √ x and y = f(u) =e u .Then dy
dx = dy du
du dx =(eu )
7. F (x) =(x 4 +3x 2 − 2) 5 ⇒ F 0 (x) =5(x 4 +3x 2 − 2) 4 ·
or 10x(x 4 +3x 2 − 2) 4 (2x 2 +3)
1
2 x−1/2
= e √x ·
1
2 √ x = e√ x
2 √ x .
d
x 4 +3x 2 − 2 =5(x 4 +3x 2 − 2) 4 (4x 3 +6x)
dx
9. F (x) = 4√ 1+2x + x 3 =(1+2x + x 3 ) 1/4 ⇒
F 0 (x) = 1 4 (1 + 2x + x3 ) −3/4 ·
11. g(t) =
2+3x 2
=
4 4 (1 + 2x + x 3 ) 3
d
dx (1 + 2x + x3 )=
1
4(1 + 2x + x 3 ) 3/4 · (2 + 3x2 )=
2+3x 2
4(1 + 2x + x 3 ) 3/4
1
(t 4 +1) 3 =(t4 +1) −3 ⇒ g 0 (t) =−3(t 4 +1) −4 (4t 3 )=−12t 3 (t 4 +1) −4 = −12t3
(t 4 +1) 4
13. y =cos(a 3 + x 3 ) ⇒ y 0 = − sin(a 3 + x 3 ) · 3x 2 [a 3 is just a constant] = −3x 2 sin(a 3 + x 3 )
15. y = xe −kx ⇒ y 0 = x e −kx (−k) + e −kx · 1=e −kx (−kx +1)
or (1 − kx)e
−kx
17. g(x) =(1+4x) 5 (3 + x − x 2 ) 8 ⇒
g 0 (x) =(1+4x) 5 · 8(3 + x − x 2 ) 7 (1 − 2x)+(3+x − x 2 ) 8 · 5(1 + 4x) 4 · 4
=4(1+4x) 4 (3 + x − x 2 ) 7 2(1 + 4x)(1 − 2x)+5(3+x − x 2 )
=4(1+4x) 4 (3 + x − x 2 ) 7 (2 + 4x − 16x 2 )+(15+5x − 5x 2 ) =4(1+4x) 4 (3 + x − x 2 ) 7 (17 + 9x − 21x 2 )

F.
100 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
19. y =(2x − 5) 4 (8x 2 − 5) −3 ⇒
y 0 =4(2x − 5) 3 (2)(8x 2 − 5) −3 +(2x − 5) 4 (−3)(8x 2 − 5) −4 (16x)
=8(2x − 5) 3 (8x 2 − 5) −3 − 48x(2x − 5) 4 (8x 2 − 5) −4
[This simplifies to 8(2x − 5) 3 (8x 2 − 5) −4 (−4x 2 +30x − 5).]
21. y =
x 2 3
+1
x 2 − 1
⇒
x
y 0 2 2
+1
=3
·
x 2 − 1
x 2 +1
=3
x 2 − 1
d
dx
x 2 +1
=3
x 2 − 1
23. y = e x cos x ⇒ y 0 = e x cos x ·
25. F (z) =
2
· 2x[x2 − 1 − (x 2 +1)]
(x 2 − 1) 2 =3
1/2
z − 1 z − 1
z +1 = z +1
F 0 (z) = 1 −1/2 z − 1
·
2 z +1
d
dz
x 2 2
+1
· (x2 − 1)(2x) − (x 2 +1)(2x)
x 2 − 1
(x 2 − 1) 2
x 2 2
+1
·
x 2 − 1
2x(−2)
(x 2 − 1) = −12x(x2 +1) 2
2 (x 2 − 1) 4
d
dx (x cos x) =ex cos x [x(− sin x)+(cosx) · 1] = e x cos x (cos x − x sin x)
⇒
z − 1
= 1 z +1 2
1/2 z +1
·
z − 1
(z +1)(1)− (z − 1)(1)
(z +1) 2
= 1 (z +1) 1/2 z +1− z +1
· = 1 (z +1) 1/2
2 (z − 1)
1/2
(z +1) 2 2 (z − 1) · 2
1/2 (z +1) = 1
2 (z − 1) 1/2 (z +1) 3/2
27. y =
r
√
r2 +1
⇒
y 0 =
√
√
r 2
r2 +1(1)− r · 1
2 (r2 +1) −1/2 r2 +1− √
(2r)
√
r2 +1 r2 +1
2
= √
r2 +1 2
=
√
r2 +1 √ r 2 +1− r 2
√
r2 +1
√
r2 +1 2
=
r 2 +1 − r 2
√
r2 +1 1
3
=
(r 2 +1) or 3/2 (r2 +1) −3/2
Another solution: Write y as a product and make use of the Product Rule. y = r(r 2 +1) −1/2
⇒
y 0 = r · − 1 2 (r2 +1) −3/2 (2r)+(r 2 +1) −1/2 · 1=(r 2 +1) −3/2 [−r 2 +(r 2 +1) 1 ]=(r 2 +1) −3/2 (1) = (r 2 +1) −3/2 .
The step that students usually have trouble with is factoring out (r 2 +1) −3/2 . But this is no different than factoring out x 2
from x 2 + x 5 ; that is, we are just factoring out a factor with the smallest exponent that appears on it. In this case, − 3 2 is
smaller than − 1 2 .
29. y = sin(tan 2x) ⇒ y 0 =cos(tan2x) ·
31. Using Formula 5 and the Chain Rule, y =2 sin πx ⇒
y 0 =2 sin πx (ln 2) ·
d
dx (tan 2x) = cos(tan 2x) · d
sec2 (2x) ·
dx (2x) = 2 cos(tan 2x)sec2 (2x)
d
dx (sin πx) =2sin πx (ln 2) · cos πx · π =2 sin πx (π ln 2) cos πx

F.
TX.10
SECTION 3.4 THE CHAIN RULE ¤ 101
33. y =sec 2 x +tan 2 x =(secx) 2 +(tanx) 2 ⇒
y 0 =2(secx)(sec x tan x) + 2(tan x)(sec 2 x)=2sec 2 x tan x +2sec 2 x tan x =4sec 2 x tan x
1 − e
2x
35. y =cos
1+e 2x
⇒
1 − e
y 0 2x
= − sin
·
1+e 2x
d 1 − e
2x
1 − e
2x
= − sin
· (1 + e2x )(−2e 2x ) − (1 − e 2x )(2e 2x )
dx 1+e 2x 1+e 2x (1 + e 2x ) 2
1 − e
2x
1 − e
2x
= − sin
· −2e2x (1 + e 2x )+(1− e 2x )
= − sin
1+e 2x (1 + e 2x ) 2
1+e 2x
· −2e2x (2)
(1 + e 2x ) = 4e 2x 1 − e
2x
2 (1 + e 2x ) · sin 2 1+e 2x
37. y =cot 2 (sin θ) =[cot(sinθ)] 2 ⇒
y 0 = 2[cot(sin θ)] ·
d
dθ [cot(sin θ)] = 2 cot(sin θ) · [− csc2 (sin θ) · cos θ] =−2cosθ cot(sin θ) csc 2 (sin θ)
39. f(t) =tan(e t )+e tan t ⇒ f 0 (t) =sec 2 (e t ) · d
dt (et )+e tan t · d
dt (tan t) =sec2 (e t ) · e t + e tan t · sec 2 t
41. f(t) =sin 2 e sin2 t
= sin e
t 2 sin2 ⇒
f 0 (t)=2 sin e sin2 t
· d
=2sin e sin2 t
cos
dt sin
e sin2 t
e sin2 t
=4sin e sin2 t
cos e sin2 t
e sin2t sin t cos t
=2sin e sin2 t
· cos e sin2 t
· d
· e sin2t · d
dt sin2 t =2sin e sin2 t
cos
dt esin2 t
e sin2 t
e sin2t · 2sint cos t
43. g(x) =(2ra rx + n) p ⇒
g 0 (x) =p(2ra rx + n) p−1 ·
45. y =cos sin(tan πx) = cos(sin(tan πx)) 1/2 ⇒
y 0 = − sin(sin(tan πx)) 1/2 ·
= − sin sin(tan πx)
2 sin(tan πx)
d
dx (2rarx + n) =p(2ra rx + n) p−1 · 2ra rx (ln a) · r =2r 2 p(ln a)(2ra rx + n) p−1 a rx
d
dx (sin(tan πx))1/2 = − sin(sin(tan πx)) 1/2 · 1 (sin(tan d
2 πx))−1/2 · (sin(tan πx))
dx
· cos(tan πx) ·
= −π cos(tan πx)sec2 (πx)sin sin(tan πx)
2 sin(tan πx)
d
dx tan πx = − sin sin(tan πx)
2 · cos(tan πx) · sec 2 (πx) · π
sin(tan πx)
47. h(x) = √ x 2 +1 ⇒ h 0 (x) = 1 2 (x2 +1) −1/2 (2x) =
x
√
x2 +1
⇒
h 00 (x) =
√
1
x2 +1· 1 − x
2 (x2 +1) −1/2 (2x)
√
x2 +1 2
=
x 2 +1 −1/2 (x 2 +1)− x 2
(x 2 +1) 1 =
1
(x 2 +1) 3/2

F.
102 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
49. y = e αx sin βx ⇒ y 0 = e αx · β cos βx +sinβx · αe αx = e αx (β cos βx + α sin βx) ⇒
y 00 = e αx (−β 2 sin βx + αβ cos βx)+(β cos βx + α sin βx) · αe αx
= e αx (−β 2 sin βx + αβ cos βx + αβ cos βx + α 2 sin βx) =e αx (α 2 sin βx − β 2 sin βx +2αβ cos βx)
= e αx (α 2 − β 2 )sinβx +2αβ cos βx
51. y =(1+2x) 10 ⇒ y 0 =10(1+2x) 9 · 2=20(1+2x) 9 .
At (0, 1), y 0 = 20(1 + 0) 9 =20, and an equation of the tangent line is y − 1=20(x − 0),ory =20x +1.
53. y =sin(sinx) ⇒ y 0 =cos(sinx) · cos x. At (π, 0), y 0 =cos(sinπ) · cos π =cos(0)· (−1) = 1(−1) = −1,andan
equation of the tangent line is y − 0=−1(x − π),ory = −x + π.
55. (a) y =
2
⇒ y 0 = (1 + e−x )(0) − 2(−e −x ) 2e −x
=
1+e −x (1 + e −x ) 2 (1 + e −x ) . 2
(b)
At (0, 1), y 0 =
2e 0
(1 + e 0 ) 2 = 2(1)
(1 + 1) 2 = 2 2 2 = 1 2 .Soanequationofthe
tangent line is y − 1= 1 2 (x − 0) or y = 1 2 x +1.
57. (a) f(x) =x √ 2 − x 2 = x(2 − x 2 ) 1/2 ⇒
f 0 (x) =x · 1
2 (2 − x2 ) −1/2 (−2x)+(2− x 2 ) 1/2 · 1=(2− x 2 ) −1/2 −x 2 +(2− x 2 ) = 2 − 2x2 √
2 − x
2
(b)
f 0 =0when f has a horizontal tangent line, f 0 is negative when f is
decreasing, and f 0 is positive when f is increasing.
59. For the tangent line to be horizontal, f 0 (x) =0. f(x) =2sinx +sin 2 x ⇒ f 0 (x) =2cosx +2sinx cos x =0 ⇔
2cosx(1 + sin x) =0 ⇔ cos x =0or sin x = −1, sox = π 2 +2nπ or 3π 2
+2nπ, wheren is any integer. Now
f π
2
=3and f
3π
2
where n is any integer.
= −1, so the points on the curve with a horizontal tangent are
π
2 +2nπ, 3 and 3π
2 +2nπ, −1 ,
61. F (x) =f(g(x)) ⇒ F 0 (x) =f 0 (g(x)) · g 0 (x),soF 0 (5) = f 0 (g(5)) · g 0 (5) = f 0 (−2) · 6=4· 6=24
63. (a) h(x) =f(g(x)) ⇒ h 0 (x) =f 0 (g(x)) · g 0 (x),soh 0 (1) = f 0 (g(1)) · g 0 (1) = f 0 (2) · 6=5· 6=30.
(b) H(x) =g(f(x)) ⇒ H 0 (x) =g 0 (f(x)) · f 0 (x),soH 0 (1) = g 0 (f(1)) · f 0 (1) = g 0 (3) · 4=9· 4=36.
65. (a) u(x) =f(g(x)) ⇒ u 0 (x) =f 0 (g(x))g 0 (x). Sou 0 (1) = f 0 (g(1))g 0 (1) = f 0 (3)g 0 (1). Tofind f 0 (3), note that f is
linear from (2, 4) to (6, 3), so its slope is 3 − 4
6 − 2 = − 1 4 .Tofind g0 (1), note that g is linear from (0, 6) to (2, 0), so its slope
is 0 − 6
2 − 0 = −3. Thus, f 0 (3)g 0 (1) =
− 1 4 (−3) =
3
. 4
(b) v(x) =g(f(x)) ⇒ v 0 (x) =g 0 (f(x))f 0 (x). Sov 0 (1) = g 0 (f(1))f 0 (1) = g 0 (2)f 0 (1), which does not exist since
g 0 (2) does not exist.

F.
TX.10
SECTION 3.4 THE CHAIN RULE ¤ 103
(c) w(x) =g(g(x)) ⇒ w 0 (x) =g 0 (g(x))g 0 (x). Sow 0 (1) = g 0 (g(1))g 0 (1) = g 0 (3)g 0 (1). Tofind g 0 (3), note that g is
linear from (2, 0) to (5, 2),soitsslopeis 2 − 0
5 − 2 = 2 3 . Thus, g0 (3)g 0 (1) = 2
3
(−3) = −2.
67. (a) F (x) =f(e x ) ⇒ F 0 (x) =f 0 (e x ) d
dx (ex )=f 0 (e x )e x
(b) G(x) =e f(x) ⇒ G 0 (x) =e f(x) d
dx f(x) =ef(x) f 0 (x)
69. r(x) =f(g(h(x))) ⇒ r 0 (x) =f 0 (g(h(x))) · g 0 (h(x)) · h 0 (x),so
r 0 (1) = f 0 (g(h(1))) · g 0 (h(1)) · h 0 (1) = f 0 (g(2)) · g 0 (2) · 4=f 0 (3) · 5 · 4=6· 5 · 4=120
71. F (x) =f(3f(4f(x))) ⇒
F 0 (x)=f 0 (3f(4f(x))) ·
d
dx (3f(4f(x))) = f 0 (3f(4f(x))) · 3f 0 d
(4f(x)) ·
dx (4f(x))
= f 0 (3f(4f(x))) · 3f 0 (4f(x)) · 4f 0 (x), so
F 0 (0) = f 0 (3f(4f(0))) · 3f 0 (4f(0)) · 4f 0 (0) = f 0 (3f(4 · 0)) · 3f 0 (4 · 0) · 4 · 2=f 0 (3 · 0) · 3 · 2 · 4 · 2=2· 3 · 2 · 4 · 2=96.
73. y = Ae −x + Bxe −x ⇒
y 0 = A(−e −x )+B[x(−e −x )+e −x · 1] = −Ae −x + Be −x − Bxe −x =(B − A)e −x − Bxe −x
⇒
y 00 =(B − A)(−e −x ) − B[x(−e −x )+e −x · 1] = (A − B)e −x − Be −x + Bxe −x =(A − 2B)e −x + Bxe −x ,
so
y 00 +2y 0 + y =(A − 2B)e −x + Bxe −x +2[(B − A)e −x − Bxe −x ]+Ae −x + Bxe −x
=[(A − 2B)+2(B − A)+A]e −x +[B − 2B + B]xe −x =0.
75. The use of D, D 2 , ..., D n is just a derivative notation (see text page 157). In general, Df(2x) =2f 0 (2x),
D 2 f(2x) =4f 00 (2x), ..., D n f(2x) =2 n f (n) (2x). Sincef(x) =cosx and 50 = 4(12) + 2, wehave
f (50) (x) =f (2) (x) =− cos x,soD 50 cos 2x = −2 50 cos 2x.
77. s(t) =10+ 1 sin(10πt) ⇒ the velocity after t seconds is v(t) 4 =s0 (t) = 1 cos(10πt)(10π) = 5π cos(10πt) cm/s.
4 2
79. (a) B(t) =4.0+0.35 sin 2πt ⇒
dB
5.4 dt = 0.35 cos 2πt 2π
= 0.7π 2πt
cos
5.4 5.4 5.4 5.4 = 7π 2πt
cos
54 5.4
(b) At t =1, dB
dt = 7π 2π
cos
54 5.4 ≈ 0.16.
81. s(t) =2e −1.5t sin 2πt ⇒
v(t) =s 0 (t) =2[e −1.5t (cos 2πt)(2π)+(sin2πt)e −1.5t (−1.5)] = 2e −1.5t (2π cos 2πt − 1.5sin2πt)

F.
104 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
83. By the Chain Rule, a(t) = dv
dt = dv ds
ds dt = dv
dv
v(t) =v(t)
ds ds .
Thederivativedv/dt is the rate of change of the velocity
with respect to time (in other words, the acceleration) whereas the derivative dv/ds is the rate of change of the velocity with
respect to the displacement.
85. (a) Using a calculator or CAS, we obtain the model Q = ab t with a ≈ 100.0124369 and b ≈ 0.000045145933.
(b) Use Q 0 (t) =ab t ln b (from Formula 5) with the values of a and b from part (a) to get Q 0 (0.04) ≈−670.63 μA.
The result of Example 2 in Section 2.1 was −670 μA.
87. (a) Derive gives g 0 (t) =
45(t − 2)8
without simplifying. With either Maple or Mathematica, we first get
(2t +1)
10
g 0 (t − 2)8 (t − 2)9
(t) =9 − 18 , and the simplification command results in the expression given by Derive.
(2t +1)
9
(2t +1)
10
(b) Derive gives y 0 =2(x 3 − x +1) 3 (2x +1) 4 (17x 3 +6x 2 − 9x +3)without simplifying. With either Maple or
Mathematica, we first get y 0 =10(2x +1) 4 (x 3 − x +1) 4 +4(2x +1) 5 (x 3 − x +1) 3 (3x 2 − 1). Ifweuse
Mathematica’s Factor or Simplify,orMaple’sfactor, we get the above expression, but Maple’s simplify gives
the polynomial expansion instead. For locating horizontal tangents, the factored form is the most helpful.
89. (a) If f is even, then f(x) =f(−x). Using the Chain Rule to differentiate this equation, we get
f 0 (x) =f 0 (−x) d
dx (−x) =−f 0 (−x). Thus,f 0 (−x) =−f 0 (x),sof 0 is odd.
(b) If f is odd, then f(x) =−f(−x).
even.
91. (a)
(b)
d
dx (sinn x cos nx) =n sin n−1 x cos x cos nx +sin n x (−n sin nx)
Differentiating this equation, we get f 0 (x) =−f 0 (−x)(−1) = f 0 (−x),sof 0 is
[Product Rule]
= n sin n−1 x (cos nx cos x − sin nx sin x) [factor out n sin n−1 x]
= n sin n−1 x cos(nx + x) [Addition Formula for cosine]
= n sin n−1 x cos[(n +1)x] [factor out x]
d
dx (cosn x cos nx) =n cos n−1 x (− sin x)cosnx +cos n x (−n sin nx)
[Product Rule]
= −n cos n−1 x (cos nx sin x +sinnx cos x) [factor out −n cos n−1 x]
= −n cos n−1 x sin(nx + x) [Addition Formula for sine]
= −n cos n−1 x sin[(n +1)x] [factor out x]
93. Since θ ◦ =
π
d
180 θ rad, we have
dθ (sin θ◦ )= d
sin
π
180
dθ
θ = π cos π θ = π cos 180 180 180 θ◦ .
95. The Chain Rule says that dy
dx = dy
du
dy
d 2 y
dx = d
2 dx
dx
du
dx ,so
= d
dx
dy du
du dx
d
=
dx
dy du
du
dx + dy d
du dx
du
dx
[Product Rule]
d
=
du
dy
du
du du
dx dx + dy
d 2 u
du dx = d2 y du
2 du 2 dx
2
+ dy d 2 u
du dx 2

F.
TX.10
SECTION 3.5 IMPLICIT DIFFERENTIATION ¤ 105
3.5 Implicit Differentiation
d
1. (a)
dx (xy +2x +3x2 )= d
dx (4) ⇒ (x · y0 + y · 1)+2+6x =0 ⇒ xy 0 = −y − 2 − 6x ⇒
y 0 −y − 2 − 6x
= or y 0 = −6 − y +2
x
x .
(b) xy +2x +3x 2 =4 ⇒ xy =4− 2x − 3x 2 ⇒ y =
3. (a)
5.
(c) From part (a), y 0 =
−y − 2 − 6x
x
=
−(4/x − 2 − 3x) − 2 − 6x
x
4 − 2x − 3x2
x
=
= 4 x − 2 − 3x,soy0 = − 4 x 2 − 3.
−4/x − 3x
x
= − 4 x 2 − 3.
d 1
dx x + 1
= d
y dx (1) ⇒ − 1 x − 1
2 y 2 y0 =0 ⇒ − 1 y 2 y0 = 1 ⇒ y 0 = − y2
x 2 x 2
(b) 1 x + 1 y =1 ⇒ 1 y =1− 1 x = x − 1
x
(c) y 0 = − y2
2
− 1)]
= −[x/(x
x2 x 2
x 2
⇒ y = x (x − 1)(1) − (x)(1)
x − 1 ,soy0 = = −1
(x − 1) 2 (x − 1) . 2
= −
x 2 (x − 1) = − 1
2 (x − 1) 2
d
x 3 + y 3 = d
dx
dx (1) ⇒ 3x2 +3y 2 · y 0 =0 ⇒ 3y 2 y 0 = −3x 2 ⇒ y 0 = − x2
y 2
7.
d
dx (x2 + xy − y 2 )= d
dx (4) ⇒ 2x + x · y0 + y · 1 − 2yy 0 =0 ⇒
xy 0 − 2yy 0 = −2x − y ⇒ (x − 2y) y 0 = −2x − y ⇒ y 0 = −2x − y
x − 2y
= 2x + y
2y − x
9.
11.
13.
15.
d
x 4 (x + y) = d
y 2 (3x − y) ⇒ x 4 (1 + y 0 )+(x + y) · 4x 3 = y 2 (3 − y 0 )+(3x − y) · 2yy 0 ⇒
dx
dx
x 4 + x 4 y 0 +4x 4 +4x 3 y =3y 2 − y 2 y 0 +6xy y 0 − 2y 2 y 0 ⇒ x 4 y 0 +3y 2 y 0 − 6xy y 0 =3y 2 − 5x 4 − 4x 3 y ⇒
(x 4 +3y 2 − 6xy) y 0 =3y 2 − 5x 4 − 4x 3 y ⇒ y 0 = 3y2 − 5x 4 − 4x 3 y
x 4 +3y 2 − 6xy
d
dx (x2 y 2 + x sin y) = d
dx (4) ⇒ x2 · 2yy 0 + y 2 · 2x + x cos y · y 0 +siny · 1=0 ⇒
2x 2 yy 0 + x cos y · y 0 = −2xy 2 − sin y ⇒ (2x 2 y + x cos y)y 0 = −2xy 2 − sin y ⇒ y 0 = −2xy2 − sin y
2x 2 y + x cos y
d
(4 cos x sin y) =
d
dx
dx (1) ⇒ 4[cosx · cos y · y0 +siny · (− sin x)] = 0 ⇒
y 0 (4 cos x cos y) =4sinx sin y ⇒ y 0 =
d
dx (ex/y )=
d
dx (x − y) ⇒ ex/y ·
4sinx sin y
=tanx tan y
4cosx cos y
d x
=1− y 0 ⇒
dx y
e x/y · y · 1 − x · y0
=1− y 0 ⇒ e x/y · 1
y 2 y − xex/y
· y 0 =1− y 0 ⇒ y 0 − xex/y
· y 0 =1− ex/y
y 2 y 2
y
⇒
y 0 1 − xex/y
= y − ex/y
y 2 y
⇒ y 0 =
y − e x/y
y
= y(y − ex/y )
y 2 − xe x/y y 2 − xe x/y
y 2

F.
106 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
17.
xy =1+x 2 y ⇒ 1 2 (xy)−1/2 (xy 0 + y · 1) = 0 + x 2 y 0 + y · 2x ⇒
19.
21.
23.
y 0 x
2 xy − x2 =2xy −
y
2 xy
⇒
y 0 x − 2x 2
xy
2 xy
= 4xy xy − y
2 xy
x
2 xy y0 + y
2 xy = x2 y 0 +2xy ⇒
⇒
y 0 = 4xy xy − y
x − 2x 2 xy
d
dx (ey cos x) = d
dx [1 + sin(xy)] ⇒ ey (− sin x)+cosx · e y · y 0 =cos(xy) · (xy 0 + y · 1) ⇒
−e y sin x + e y cos x · y 0 = x cos(xy) · y 0 + y cos(xy) ⇒ e y cos x · y 0 − x cos(xy) · y 0 = e y sin x + y cos(xy) ⇒
[e y cos x − x cos(xy)] y 0 = e y sin x + y cos(xy) ⇒ y 0 = ey sin x + y cos(xy)
e y cos x − x cos(xy)
d
f(x)+x 2 [f(x)] 3 = d
dx
dx (10) ⇒ f 0 (x)+x 2 · 3[f(x)] 2 · f 0 (x)+[f(x)] 3 · 2x =0. Ifx =1,wehave
f 0 (1) + 1 2 · 3[f(1)] 2 · f 0 (1) + [f(1)] 3 · 2(1) = 0 ⇒ f 0 (1) + 1 · 3 · 2 2 · f 0 (1) + 2 3 · 2=0 ⇒
f 0 (1) + 12f 0 (1) = −16 ⇒ 13f 0 (1) = −16 ⇒ f 0 (1) = − 16
13 .
d
dy (x4 y 2 − x 3 y +2xy 3 )= d
dy (0) ⇒ x4 · 2y + y 2 · 4x 3 x 0 − (x 3 · 1+y · 3x 2 x 0 )+2(x · 3y 2 + y 3 · x 0 )=0 ⇒
4x 3 y 2 x 0 − 3x 2 yx 0 +2y 3 x 0 = −2x 4 y + x 3 − 6xy 2 ⇒ (4x 3 y 2 − 3x 2 y +2y 3 ) x 0 = −2x 4 y + x 3 − 6xy 2 ⇒
x 0 = dx
dy = −2x4 y + x 3 − 6xy 2
4x 3 y 2 − 3x 2 y +2y 3
25. x 2 + xy + y 2 =3 ⇒ 2x + xy 0 + y · 1+2yy 0 =0 ⇒ xy 0 +2yy 0 = −2x − y ⇒ y 0 (x +2y) =−2x − y ⇒
y 0 = −2x − y
x +2y . Whenx =1and y =1,wehavey0 = −2 − 1
1+2· 1 = −3 = −1, so an equation of the tangent line is
3
y − 1=−1(x − 1) or y = −x +2.
27. x 2 + y 2 =(2x 2 +2y 2 − x) 2 ⇒ 2x +2yy 0 =2(2x 2 +2y 2 − x)(4x +4yy 0 − 1). Whenx =0and y = 1 ,wehave
2
0+y 0 =2( 1 2 )(2y0 − 1) ⇒ y 0 =2y 0 − 1 ⇒ y 0 =1, so an equation of the tangent line is y − 1 =1(x − 0)
2
or y = x + 1 . 2
29. 2(x 2 + y 2 ) 2 =25(x 2 − y 2 ) ⇒ 4(x 2 + y 2 )(2x +2yy 0 ) = 25(2x − 2yy 0 ) ⇒
4(x + yy 0 )(x 2 + y 2 )=25(x − yy 0 ) ⇒ 4yy 0 (x 2 + y 2 )+25yy 0 =25x − 4x(x 2 + y 2 ) ⇒
y 0 = 25x − 4x(x2 + y 2 )
25y +4y(x 2 + y 2 ) . Whenx =3and y =1,wehavey0 75 − 120
= = − 45 = − 9 ,
25 + 40 65 13
so an equation of the tangent line is y − 1=− 9
13 (x − 3) or y = − 9 13 x + 40
13 .
31. (a) y 2 =5x 4 − x 2 ⇒ 2yy 0 =5(4x 3 ) − 2x ⇒ y 0 = 10x3 − x
.
y
(b)
So at the point (1, 2) we have y 0 = 10(1)3 − 1
2
of the tangent line is y − 2= 9 2 (x − 1) or y = 9 2 x − 5 2 .
= 9 , and an equation
2

F.
TX.10
SECTION 3.5 IMPLICIT DIFFERENTIATION ¤ 107
33. 9x 2 + y 2 =9 ⇒ 18x +2yy 0 =0 ⇒ 2yy 0 = −18x ⇒ y 0 = −9x/y ⇒
y · 1 − x · y
y 00 0 y − x(−9x/y)
= −9
= −9
= −9 · y2 +9x 2 9
= −9 · [since x and y must satisfy the original
y 2
y 2
y 3 y 3
equation, 9x 2 + y 2 =9].Thus,y 00 = −81/y 3 .
35. x 3 + y 3 =1 ⇒ 3x 2 +3y 2 y 0 =0 ⇒ y 0 = − x2
y 2
y 00 = − y2 (2x) − x 2 · 2yy 0
= − 2xy2 − 2x 2 y(−x 2 /y 2 )
= − 2xy4 +2x 4 y
= − 2xy(y3 + x 3 )
= − 2x
(y 2 ) 2 y 4
y 6 y 6 y , 5
since x and y must satisfy the original equation, x 3 + y 3 =1.
37. (a) There are eight points with horizontal tangents: four at x ≈ 1.57735 and
four at x ≈ 0.42265.
(b) y 0 =
3x 2 − 6x +2
2(2y 3 − 3y 2 − y +1)
⇒
⇒ y 0 = −1 at (0, 1) and y 0 = 1 at (0, 2).
3
Equations of the tangent lines are y = −x +1and y = 1 3 x +2.
(c) y 0 =0 ⇒ 3x 2 − 6x +2=0 ⇒ x =1± 1 3
√
3
(d) By multiplying the right side of the equation by x − 3, we obtain the first
graph. By modifying the equation in other ways, we can generate the other
graphs.
y(y 2 − 1)(y − 2)
= x(x − 1)(x − 2)(x − 3)
y(y 2 − 4)(y − 2)
= x(x − 1)(x − 2)
y(y +1)(y 2 − 1)(y − 2)
= x(x − 1)(x − 2)
(y +1)(y 2 − 1)(y − 2)
=(x − 1)(x − 2)
x(y +1)(y 2 − 1)(y − 2)
= y(x − 1)(x − 2)
y(y 2 +1)(y − 2)
= x(x 2 − 1)(x − 2)
y(y +1)(y 2 − 2)
= x(x − 1)(x 2 − 2)

F.
108 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
39. From Exercise 29, a tangent to the lemniscate will be horizontal if y 0 =0 ⇒ 25x − 4x(x 2 + y 2 )=0 ⇒
x[25 − 4(x 2 + y 2 )] = 0 ⇒ x 2 + y 2 = 25 4
(1). (Note that when x is 0, y is also 0, and there is no horizontal tangent
at the origin.) Substituting 25 for 4
x2 + y 2 in the equation of the lemniscate, 2(x 2 + y 2 ) 2 =25(x 2 − y 2 ),weget
x 2 − y 2 = 25 (2). Solving (1)and(2), we have x 2 = 75 and 8 16 y2 = 25 , so the four points are ± 5√ 3
, ± 5 .
16 4 4
41.
x 2
a − y2
2x
=1 ⇒ 2 b2 a − 2yy0 =0 ⇒ y 0 = b2 x
2 b 2 a 2 y
⇒
an equation of the tangent line at (x 0 ,y 0 ) is
y − y 0 = b2 x 0
(x − x
a 2 0 ). Multiplying both sides by y0 y0y
gives
y 0 b2 b − y2 0
2 b 2
= x0x
a 2 − x2 0
a 2 .Since(x 0,y 0 ) lies on the hyperbola,
we have x0x
a 2
− y0y
b 2 = x2 0
a 2 − y2 0
b 2 =1.
43. If the circle has radius r, its equation is x 2 + y 2 = r 2 ⇒ 2x +2yy 0 =0 ⇒ y 0 = − x , so the slope of the tangent line
y
at P (x 0 ,y 0 ) is − x 0
−1
. The negative reciprocal of that slope is = y 0
, which is the slope of OP, so the tangent line at
y 0 −x 0 /y 0 x 0
P is perpendicular to the radius OP.
45. y =tan √ −1 x ⇒ y 0 1
= √ 2 ·
1+ x
d
√
x = 1
dx 1+x
1
2 x−1/2
=
1
2 √ x (1 + x)
47. y =sin −1 (2x +1) ⇒
y 0 =
1
· d
1 − (2x +1)
2 dx (2x +1)= 1
1 − (4x2 +4x +1) · 2= 2
√
−4x2 − 4x = 1
√
−x2 − x
49. G(x) = √ 1 − x 2 arccos x ⇒ G 0 (x) = √ 1 − x 2 ·
51. h(t) =cot −1 (t) +cot −1 (1/t) ⇒
h 0 (t) =− 1
1+t − 1
2 1+(1/t) · d 1
2 dt t = − 1
1+t − t2
2 t 2 +1 ·
−1
√ +arccosx · 1
1 − x
2 2 (1 − x2 ) −1/2 (−2x) =−1 − x √ arccos x
1 − x
2
− 1 t 2
= − 1
1+t 2 + 1
t 2 +1 =0.
Note that this makes sense because h(t) = π 2 for t>0 and h(t) =3π 2
for t

F.
TX.10
SECTION 3.5 IMPLICIT DIFFERENTIATION ¤ 109
57. Let y =cos −1 x.Thencos y = x and 0 ≤ y ≤ π ⇒ −sin y dy
dx =1
dy
dx = − 1
sin y = − 1
1 − cos2 y = − 1
√
1 − x
2 .
⇒
[Notethatsin y ≥ 0 for 0 ≤ y ≤ π.]
59. x 2 + y 2 = r 2 is a circle with center O and ax + by =0is a line through O [assume a
and b are not both zero]. x 2 + y 2 = r 2 ⇒ 2x +2yy 0 =0 ⇒ y 0 = −x/y,sothe
slope of the tangent line at P 0 (x 0,y 0) is −x 0/y 0. The slope of the line OP 0 is y 0/x 0,
which is the negative reciprocal of −x 0/y 0. Hence, the curves are orthogonal, and the
families of curves are orthogonal trajectories of each other.
61. y = cx 2 ⇒ y 0 =2cx and x 2 +2y 2 = k [assume k>0] ⇒ 2x +4yy 0 =0 ⇒
2yy 0 = −x ⇒ y 0 = − x
2(y) = − x
2(cx 2 ) = − 1 , so the curves are orthogonal if
2cx
c 6= 0.Ifc =0, then the horizontal line y = cx 2 =0intersects x 2 +2y 2 = k orthogonally
at ± √
k, 0 , since the ellipse x 2 +2y 2 = k has vertical tangents at those two points.
63. To find the points at which the ellipse x 2 − xy + y 2 =3crosses the x-axis, let y =0and solve for x.
y =0 ⇒ x 2 − x(0) + 0 2 =3 ⇔ x = ± √ 3. So the graph of the ellipse crosses the x-axis at the points ± √ 3, 0 .
Using implicit differentiation to find y 0 ,weget2x − xy 0 − y +2yy 0 =0 ⇒ y 0 (2y − x) =y − 2x ⇔ y 0 = y − 2x
2y − x .
So y 0 at √ 3, 0 is 0 − 2 √ 3
2(0) − √ 3 =2and y0 at − √ 3, 0 is 0+2√ 3
2(0) + √ =2. Thus, the tangent lines at these points are parallel.
3
65. x 2 y 2 + xy =2 ⇒ x 2 · 2yy 0 + y 2 · 2x + x · y 0 + y · 1=0 ⇔ y 0 (2x 2 y + x) =−2xy 2 − y ⇔
y 0 = − 2xy2 + y
2x 2 y + x .So− 2xy2 + y
2x 2 y + x = −1 ⇔ 2xy2 + y =2x 2 y + x ⇔ y(2xy +1)=x(2xy +1) ⇔
y(2xy +1)− x(2xy +1)=0 ⇔ (2xy +1)(y − x) =0 ⇔ xy = − 1 2 or y = x.Butxy = − 1 2
⇒
x 2 y 2 + xy = 1 4 − 1 2 6=2,sowemusthavex = y. Then x2 y 2 + xy =2 ⇒ x 4 + x 2 =2 ⇔ x 4 + x 2 − 2=0 ⇔
(x 2 +2)(x 2 − 1) = 0. Sox 2 = −2, which is impossible, or x 2 =1 ⇔ x = ±1. Sincex = y, the points on the curve
where the tangent line has a slope of −1 are (−1, −1) and (1, 1).
67. (a) If y = f −1 (x),thenf(y) =x. Differentiating implicitly with respect to x and remembering that y is a function of x,
we get f 0 (y) dy dy
=1,so
dx dx = 1
f 0 (y)
⇒ f −10 (x) =
1
f 0 (f −1 (x)) .
(b) f(4) = 5 ⇒ f −1 (5) = 4. Bypart(a), f −10 (5) =
1
f 0 (f −1 (5)) = 1
f 0 (4) =1
2
3 =
3
. 2

F.
110 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
69. x 2 +4y 2 =5 ⇒ 2x +4(2yy 0 )=0 ⇒ y 0 = − x .Nowleth be the height of the lamp, and let (a, b) be the point of
4y
tangency of the line passing through the points (3,h) and (−5, 0). Thislinehasslope(h − 0)/[3 − (−5)] = 1 8 h.Butthe
slope of the tangent line through the point (a, b) can be expressed as y 0 = − a 4b ,oras b − 0
a − (−5) = b [since the line
a +5
passes through (−5, 0) and (a, b)], so − a 4b =
b
a +5
⇔ 4b 2 = −a 2 − 5a ⇔ a 2 +4b 2 = −5a. Buta 2 +4b 2 =5
[since (a, b) is on the ellipse], so 5=−5a ⇔ a = −1. Then4b 2 = −a 2 − 5a = −1 − 5(−1) = 4 ⇒ b =1, since the
point is on the top half of the ellipse. So h 8 =
x-axis.
b
a +5 = 1
−1+5 = 1 4
⇒
h =2. So the lamp is located 2 units above the
3.6 Derivatives of Logarithmic Functions
1. The differentiation formula for logarithmic functions,
3. f(x) =sin(lnx) ⇒ f 0 (x) =cos(lnx) ·
5. f(x) =log 2 (1 − 3x) ⇒ f 0 (x) =
d
dx (log a x) = 1 ,issimplestwhena = e because ln e =1.
x ln a
d
1 cos(ln x)
ln x =cos(lnx) · =
dx x x
1
(1 − 3x)ln2
d
−3
(1 − 3x) =
dx
(1 − 3x) ln2 or 3
(3x − 1) ln 2
7. f(x) = 5√ ln x =(lnx) 1/5 ⇒ f 0 (x) = 1 5 (ln x)−4/5 d
dx (ln x) = 1
5(ln x) 4/5 · 1
x = 1
5x 5 (ln x) 4
9. f(x) =sinx ln(5x) ⇒ f 0 (x) =sinx ·
11. F (t) =ln
F 0 (t) =3·
1
5x · d
sin x · 5
(5x)+ln(5x) · cos x = +cosx ln(5x) = sin x +cosx ln(5x)
dx 5x
x
(2t +1)3
(3t − 1) 4 =ln(2t +1)3 − ln(3t − 1) 4 =3ln(2t +1)− 4ln(3t − 1) ⇒
1
2t +1 · 2 − 4 · 1
3t − 1 · 3= 6
2t +1 − 12
3t − 1 , or combined, −6(t +3)
(2t +1)(3t − 1) .
13. g(x) =ln x √ x 2 − 1 =lnx +ln(x 2 − 1) 1/2 =lnx + 1 2 ln(x2 − 1) ⇒
g 0 (x) = 1 x + 1 2 ·
1
x 2 − 1 · 2x = 1 x + x
x 2 − 1 = x2 − 1+x · x
= 2x2 − 1
x(x 2 − 1) x(x 2 − 1)
15. f(u) =
ln u
1+ln(2u)
⇒
f 0 (u) =
[1 + ln(2u)] · 1
u − ln u · 1
2u · 2
[1 + ln(2u)] 2 =
17. y =ln 2 − x − 5x 2 ⇒ y 0 =
1
u
[1 + ln(2u) − ln u] 1+(ln2+lnu) − ln u
= =
[1 + ln(2u)] 2 u[1 + ln(2u)] 2
1
−10x − 1
· (−1 − 10x) =
2 − x − 5x2 2 − x − 5x or 10x +1
2 5x 2 + x − 2
19. y =ln(e −x + xe −x )=ln(e −x (1 + x)) = ln(e −x )+ln(1+x) =−x +ln(1+x) ⇒
y 0 = −1+ 1 −1 − x +1
= = − x
1+x 1+x 1+x
1+ln2
u[1 + ln(2u)] 2

F.
TX.10
SECTION 3.6 DERIVATIVES OF LOGARITHMIC FUNCTIONS ¤ 111
√
21. y =2x log 10 x =2x log10 x 1/2 =2x · 1 log 2 10 x = x log 10 x ⇒ y 0 1
= x ·
x ln 10 +log 10 x · 1= 1
ln 10 +log 10 x
Note:
1
ln 10 = ln e
ln 10 =log 10 e, so the answer could be written as 1
ln 10 +log 10 x =log 10 e +log 10 x =log 10 ex.
23. y = x 2 ln(2x) ⇒ y 0 = x 2 ·
y 00 =1+2x ·
1 · 2+ln(2x) · (2x) =x +2x ln(2x)
2x ⇒
1 · 2+ln(2x) · 2=1+2+2ln(2x) =3+2ln(2x)
2x
25. y =ln x + √ 1+x 2 ⇒
y 0 1
=
x + √ 1+x 2
=
1
x + √ 1+
1+x 2
d √
x + 1+x
2
=
dx
x
√ = 1+x
2
1
x + √ 1+ 1
1+x (1 + 2 2 x2 ) −1/2 (2x)
1
x + √ 1+x 2 ·
√
1+x2 + x
√
1+x
2
=
1
√
1+x
2
⇒
y 00 = − 1 2 (1 + x2 ) −3/2 (2x) =
−x
(1 + x 2 ) 3/2
27. f(x) =
x
1 − ln(x − 1)
⇒
−1
[1 − ln(x − 1)] · 1 − x ·
f 0 (x) =
x − 1
[1 − ln(x − 1)] 2 =
=
2x − 1 − (x − 1) ln(x − 1)
(x − 1)[1 − ln(x − 1)] 2
(x − 1)[1 − ln(x − 1)] + x
x − 1
x − 1 − (x − 1) ln(x − 1) + x
=
[1 − ln(x − 1)] 2 (x − 1)[1 − ln(x − 1)] 2
Dom(f) ={x | x − 1 > 0 and 1 − ln(x − 1) 6= 0} = {x | x>1 and ln(x − 1) 6= 1}
= x | x>1 and x − 1 6=e 1 = {x | x>1 and x 6= 1+e} =(1, 1+e) ∪ (1 + e, ∞)
29. f(x) =ln(x 2 − 2x) ⇒ f 0 (x) =
1
2(x − 1)
(2x − 2) =
x 2 − 2x x(x − 2) .
Dom(f) ={x | x(x − 2) > 0} =(−∞, 0) ∪ (2, ∞).
31. f(x) = ln x ⇒ f 0 (x) = x2 (1/x) − (ln x)(2x) x − 2x ln x x(1 − 2lnx)
= = = 1 − 2lnx ,
x 2 (x 2 ) 2 x 4
x 4
x 3
so f 0 (1) = 1 − 2ln1 = 1 − 2 · 0 =1.
1 3 1
33. y =ln xe x2 =lnx +lne x2 =lnx + x 2 ⇒ y 0 = 1 +2x. At(1, 1), the slope of the tangent line is
x
y 0 (1) = 1 + 2 = 3, and an equation of the tangent line is y − 1=3(x − 1),ory =3x − 2.
35. f(x) =sinx +lnx ⇒ f 0 (x) =cosx +1/x.
This is reasonable, because the graph shows that f increases when f 0 is
positive, and f 0 (x) =0when f has a horizontal tangent.

F.
112 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
37. y =(2x +1) 5 (x 4 − 3) 6 ⇒ ln y =ln (2x +1) 5 (x 4 − 3) 6 ⇒ ln y =5ln(2x +1)+6ln(x 4 − 3) ⇒
1
1
y y0 =5·
2x +1 · 2+6· 1
x 4 − 3 · 4x3
10
y 0 = y
2x +1 + 24x3
x 4 − 3
⇒
10
=(2x +1) 5 (x 4 − 3) 6 2x +1 + 24x3 .
x 4 − 3
[The answer could be simplified to y 0 =2(2x +1) 4 (x 4 − 3) 5 (29x 4 +12x 3 − 15), but this is unnecessary.]
39. y = sin2 x tan 4 x
(x 2 +1) 2 ⇒ ln y =ln(sin 2 x tan 4 x) − ln(x 2 +1) 2 ⇒
ln y =ln(sinx) 2 +ln(tanx) 4 − ln(x 2 +1) 2 ⇒ ln y =2ln|sin x| +4ln|tan x| − 2ln(x 2 +1) ⇒
1 1
y y0 =2·
sin x · cos x +4· 1
tan x · 1
sec2 x − 2 ·
x 2 +1 · 2x ⇒ y0 = sin2 x tan 4 x
2cotx + 4sec2 x
(x 2 +1) 2 tan x − 4x
x 2 +1
41. y = x x ⇒ ln y =lnx x ⇒ ln y = x ln x ⇒ y 0 /y = x(1/x)+(lnx) · 1 ⇒ y 0 = y(1 + ln x) ⇒
y 0 = x x (1 + ln x)
43. y = x sin x ⇒ ln y =lnx sin x ⇒ ln y =sinx ln x ⇒ y0
y =(sinx) · 1 +(lnx)(cos x)
x ⇒
sin x
sin x
y 0 = y
x
+lnx cos x ⇒ y 0 = x sin x x
+lnx cos x
45. y =(cosx) x ⇒ ln y =ln(cosx) x ⇒ ln y = x ln cos x ⇒ 1 y y0 = x ·
y 0 = y ln cos x − x sin x
cos x
⇒ y 0 =(cosx) x (ln cos x − x tan x)
1 · (− sin x)+lncosx · 1
cos x ⇒
47. y =(tanx) 1/x ⇒ ln y =ln(tanx) 1/x ⇒ ln y = 1 ln tan x ⇒
x
1
y y0 = 1 x · 1
tan x · sec2 x +lntanx ·
− 1
sec
⇒ y 0 2 x ln tan x
= y −
x 2 x tan x x 2
y 0 =(tanx) 1/x sec 2 x
x tan x
49. y =ln(x 2 + y 2 ) ⇒ y 0 =
ln tan x
−
x 2
1
x 2 + y 2
or y 0 = (tan x) 1/x · 1
x
x 2 y 0 + y 2 y 0 − 2yy 0 =2x ⇒ (x 2 + y 2 − 2y)y 0 =2x ⇒ y 0 =
51. f(x) =ln(x − 1) ⇒ f 0 (x) =
csc x sec x −
⇒
ln tan x
x
d
dx (x2 + y 2 ) ⇒ y 0 2x +2yy0
= ⇒ x 2 y 0 + y 2 y 0 =2x +2yy 0 ⇒
x 2 + y 2
2x
x 2 + y 2 − 2y
1
(x − 1) =(x − 1)−1 ⇒ f 00 (x) =−(x − 1) −2 ⇒ f 000 (x) =2(x − 1) −3 ⇒
f (4) (x) =−2 · 3(x − 1) −4 ⇒ ··· ⇒ f (n) (x) =(−1) n−1 · 2 · 3 · 4 ·····(n − 1)(x − 1) −n =(−1) n−1 (n − 1)!
(x − 1) n
53. If f(x) =ln(1+x), thenf 0 (x) = 1
1+x ,sof 0 (0) = 1.
ln(1 + x) f(x)
Thus, lim =lim
x→0 x x→0 x
= lim f(x) − f(0)
= f 0 (0) = 1.
x→0 x − 0

F.
SECTION TX.10 3.7 RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES ¤ 113
3.7 Rates of Change in the Natural and Social Sciences
1. (a) s = f(t) =t 3 − 12t 2 +36t ⇒ v(t) =f 0 (t) =3t 2 − 24t +36
(b) v(3) = 27 − 72 + 36 = −9 ft/s
(c) The particle is at rest when v(t) =0. 3t 2 − 24t +36=0 ⇔ 3(t − 2)(t − 6) = 0 ⇔ t =2sor6 s.
(d) The particle is moving in the positive direction when v(t) > 0. 3(t − 2)(t − 6) > 0 ⇔ 0 ≤ t6.
(e) Since the particle is moving in the positive direction and in the (f )
negative direction, we need to calculate the distance traveled in the
intervals [0, 2], [2, 6],and[6, 8] separately.
|f(2) − f(0)| = |32 − 0| =32.
|f(6) − f(2)| = |0 − 32| =32.
|f(8) − f(6)| = |32 − 0| =32.
The total distance is 32 + 32 + 32 = 96 ft.
(g) v(t) =3t 2 − 24t +36
a(t) =v 0 (t) =6t − 24.
⇒
(h )
a(3) = 6(3) − 24 = −6(ft/s)/s orft/s 2 .
(i) The particle is speeding up when v and a havethesamesign.Thisoccurswhen2

F.
114 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
(i) The particle is speeding up when v and a have the same sign. This occurs when 0

F.
SECTION TX.10 3.7 RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES ¤ 115
(c) The circumference is C(r) =2πr = A 0 (r). Thefigure suggests that if ∆r is small,
then the change in the area of the circle (a ring around the outside) is approximately equal
to its circumference times ∆r. Straightening out this ring gives us a shape that is approximately
rectangular with length 2πr and width ∆r, so∆A ≈ 2πr(∆r). Algebraically,
∆A = A(r + ∆r) − A(r) =π(r + ∆r) 2 − πr 2 =2πr(∆r)+π(∆r) 2 .
So we see that if ∆r is small, then ∆A ≈ 2πr(∆r) and therefore, ∆A/∆r ≈ 2πr.
15. S(r) =4πr 2 ⇒ S 0 (r) =8πr ⇒
(a) S 0 (1) = 8π ft 2 /ft (b) S 0 (2) = 16π ft 2 /ft (c) S 0 (3) = 24π ft 2 /ft
As the radius increases, the surface area grows at an increasing rate. In fact, the rate of change is linear with respect to the
radius.
17. The mass is f(x) =3x 2 , so the linear density at x is ρ(x) =f 0 (x) =6x.
(a) ρ(1) = 6 kg/m (b) ρ(2) = 12kg/m (c) ρ(3) = 18 kg/m
Since ρ is an increasing function, the density will be the highest at the right end of the rod and lowest at the left end.
19. The quantity of charge is Q(t) =t 3 − 2t 2 +6t +2,sothecurrentisQ 0 (t) =3t 2 − 4t +6.
(a) Q 0 (0.5) = 3(0.5) 2 − 4(0.5) + 6 = 4.75 A
(b) Q 0 (1) = 3(1) 2 − 4(1) + 6 = 5 A
The current is lowest when Q 0 has a minimum. Q 00 (t) =6t − 4 < 0 when t< 2 3 . So the current decreases when t< 2 3 and
increases when t> 2 3 . Thus, the current is lowest at t = 2 3 s.
21. (a) To find the rate of change of volume with respect to pressure, we first solve for V in terms of P .
PV = C ⇒ V = C P ⇒ dV
dP = − C P 2 .
(b) From the formula for dV/dP in part (a), we see that as P increases, the absolute value of dV/dP decreases.
Thus, the volume is decreasing more rapidly at the beginning.
(c) β = − 1 V
dV
dP = − 1 − C
=
V P 2
C
(PV)P =
C
CP = 1 P
23. In Example 6, the population function was n =2 t n 0. Since we are tripling instead of doubling and the initial population is
400, the population function is n(t) =400· 3 t . The rate of growth is n 0 (t) =400· 3 t · ln 3, so the rate of growth after
2.5 hours is n 0 (2.5) = 400 · 3 2.5 · ln 3 ≈ 6850 bacteria/hour.
1860 − 1750
25. (a) 1920: m 1 =
1920 − 1910 = 110
2070 − 1860
=11, m2 =
10 1930 − 1920 = 210
10 =21,
(m 1 + m 2 )/ 2 = (11 + 21)/2 =16million/year
4450 − 3710
1980: m 1 =
1980 − 1970 = 740
10 =74, m 5280 − 4450
2 =
1990 − 1980 = 830
10 =83,
(m 1 + m 2)/ 2 = (74 + 83)/2 =78.5 million/year
(b) P (t) =at 3 + bt 2 + ct + d (in millions of people), where a ≈ 0.0012937063, b ≈−7.061421911, c ≈ 12,822.97902,
and d ≈−7,743,770.396.
(c) P (t) =at 3 + bt 2 + ct + d ⇒ P 0 (t) =3at 2 +2bt + c (in millions of people per year)

F.
116 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
(d) P 0 (1920) = 3(0.0012937063)(1920) 2 +2(−7.061421911)(1920) + 12,822.97902
≈ 14.48 million/year [smaller than the answer in part (a), but close to it]
P 0 (1980) ≈ 75.29 million/year (smaller, but close)
(e) P 0 (1985) ≈ 81.62 million/year, so the rate of growth in 1985 was about 81.62 million/year.
27. (a) Using v = P
4ηl (R2 − r 2 ) with R =0.01, l =3, P =3000,andη =0.027,wehavev as a function of r:
v(r) = 3000
4(0.027)3 (0.012 − r 2 ). v(0) = 0.925 cm/s, v(0.005) = 0.694 cm/s, v(0.01) = 0.
(b) v(r) = P
4ηl (R2 − r 2 ) ⇒ v 0 (r) = P
Pr
(−2r) =−
4ηl 2ηl .
Whenl =3, P =3000,andη =0.027,wehave
v 0 (r) =−
3000r
2(0.027)3 . v0 (0) = 0, v 0 (0.005) = −92.592 (cm/s)/cm, and v 0 (0.01) = −185.185 (cm/s)/cm.
(c) The velocity is greatest where r =0(at the center) and the velocity is changing most where r = R =0.01 cm
(at the edge).
29. (a) C(x) = 1200 + 12x − 0.1x 2 +0.0005x 3 ⇒ C 0 (x) =12− 0.2x +0.0015x 2 $/yard, which is the marginal cost
function.
(b) C 0 (200) = 12 − 0.2(200) + 0.0015(200) 2 =$32/yard, and this is the rate at which costs are increasing with respect to
the production level when x =200.
C 0 (200) predicts the cost of producing the 201st yard.
(c) The cost of manufacturing the 201st yard of fabric is C(201) − C(200) = 3632.2005 − 3600 ≈ $32.20,whichis
approximately C 0 (200).
31. (a) A(x) = p(x)
x
⇒ A 0 (x) = xp0 (x) − p(x) · 1
x 2
= xp0 (x) − p(x)
x 2 .
A 0 (x) > 0 ⇒ A(x) is increasing; that is, the average productivity increases as the size of the workforce increases.
(b) p 0 (x) is greater than the average productivity ⇒ p 0 (x) >A(x) ⇒ p 0 (x) > p(x)
x
xp 0 (x) − p(x) > 0 ⇒ xp0 (x) − p(x)
x 2 > 0 ⇒ A 0 (x) > 0.
⇒ xp 0 (x) >p(x) ⇒
33. PV = nRT ⇒ T = PV
nR = PV
(10)(0.0821) = 1 (PV). Using the Product Rule, we have
0.821
dT
dt = 1
0.821 [P (t)V 0 (t)+V (t)P 0 (t)] = 1 [(8)(−0.15) + (10)(0.10)] ≈−0.2436 K/min.
0.821
35. (a) If the populations are stable, then the growth rates are neither positive nor negative; that is, dC
dt =0and dW dt
(b) “The caribou go extinct” means that the population is zero, or mathematically, C =0.
=0.
(c)Wehavetheequations dC
dt = aC − bCW and dW dt
= −cW + dCW .LetdC/dt = dW/dt =0, a =0.05, b =0.001,
c =0.05,andd =0.0001 to obtain 0.05C − 0.001CW =0 (1) and −0.05W +0.0001CW =0 (2). Adding10 times
(2) to (1) eliminates the CW-terms and gives us 0.05C − 0.5W =0 ⇒ C =10W . Substituting C =10W into (1)

F.
TX.10
SECTION 3.8 EXPONENTIAL GROWTH AND DECAY ¤ 117
results in 0.05(10W ) − 0.001(10W )W =0 ⇔ 0.5W − 0.01W 2 =0 ⇔ 50W − W 2 =0 ⇔
W (50 − W )=0 ⇔ W =0or 50. SinceC =10W , C =0or 500. Thus, the population pairs (C, W ) that lead to
stable populations are (0, 0) and (500, 50).Soitispossibleforthetwospeciestoliveinharmony.
3.8 Exponential Growth and Decay
1. The relative growth rate is 1 P
dP
dt
Thus, P (6) = 2e 0.7944(6) ≈ 234.99 or about 235 members.
=0.7944,so
dP
dt =0.7944P and, by Theorem 2, P (t) =P (0)e0.7944t =2e 0.7944t .
3. (a) By Theorem 2, P (t) =P (0)e kt = 100e kt . NowP (1) = 100e k(1) = 420 ⇒ e k = 420
100
⇒ k =ln4.2.
So P (t) =100e (ln 4.2)t =100(4.2) t .
(b) P (3) = 100(4.2) 3 =7408.8 ≈ 7409 bacteria
(c) dP/dt = kP ⇒ P 0 (3) = k · P (3) = (ln 4.2) 100(4.2) 3 [from part (a)] ≈ 10,632 bacteria/hour
(d) P (t) = 100(4.2) t =10,000 ⇒ (4.2) t =100 ⇒ t =(ln100)/(ln 4.2) ≈ 3.2 hours
5. (a) Let the population (in millions) in the year t be P (t). Since the initial time is the year 1750, we substitute t − 1750 for t in
Theorem 2, so the exponential model gives P (t) =P (1750)e k(t−1750) .ThenP (1800) = 980 = 790e k(1800−1750)
⇒
980
= 790 ek(50) ⇒ ln 980 =50k ⇒ k = 1 980
ln ≈ 0.0043104. Sowiththismodel,wehave
790 50 790
P (1900) = 790e k(1900−1750) ≈ 1508 million, and P (1950) = 790e k(1950−1750) ≈ 1871 million. Both of these
estimates are much too low.
(b) In this case, the exponential model gives P (t) =P (1850)e k(t−1850) ⇒ P (1900) = 1650 = 1260e k(1900−1850) ⇒
ln 1650 = k(50) ⇒ k = 1 1650
ln ≈ 0.005393. Sowiththismodel,weestimate
1260 50 1260
P (1950) = 1260e k(1950−1850) ≈ 2161 million. This is still too low, but closer than the estimate of P (1950) in part (a).
(c) The exponential model gives P (t) =P (1900)e k(t−1900) ⇒ P (1950) = 2560 = 1650e k(1950−1900) ⇒
ln 2560 = k(50) ⇒ k = 1 2560
1650 50
ln
1650
≈ 0.008785. With this model, we estimate
P (2000) = 1650e k(2000−1900) ≈ 3972 million. This is much too low. The discrepancy is explained by the fact that the
world birth rate (average yearly number of births per person) is about the same as always, whereas the mortality rate
(especially the infant mortality rate) is much lower, owing mostly to advances in medical science and to the wars in the first
part of the twentieth century. The exponential model assumes, among other things, that the birth and mortality rates will
remain constant.
7. (a) If y =[N 2O 5] then by Theorem 2, dy
dt = −0.0005y ⇒ y(t) =y(0)e−0.0005t = Ce −0.0005t .
(b) y(t) =Ce −0.0005t =0.9C ⇒ e −0.0005t =0.9 ⇒ −0.0005t =ln0.9 ⇒ t = −2000 ln 0.9 ≈ 211 s

F.
118 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
9. (a) If y(t) is the mass (in mg) remaining after t years, then y(t) =y(0)e kt =100e kt .
y(30) = 100e 30k = 1 (100) ⇒ 2 e30k = 1 2
⇒ k = −(ln 2)/30 ⇒ y(t) = 100e −(ln 2)t/30 = 100 · 2 −t/30
(b) y(100) = 100 · 2 −100/30 ≈ 9.92 mg
(c) 100e −(ln 2)t/30 =1 ⇒ −(ln 2)t/30 = ln 1
ln 0.01
100
⇒ t = −30
ln 2
≈ 199.3 years
11. Let y(t) be the level of radioactivity. Thus, y(t) =y(0)e −kt and k is determined by using the half-life:
y(5730) = 1 y(0) ⇒ 2 y(0)e−k(5730) = 1 y(0) ⇒ 2 e−5730k = 1 ⇒ −5730k =ln 1 ⇒ k = − ln 1 2
2 2
5730 = ln 2
5730 .
If 74% of the 14 C remains, then we know that y(t) =0.74y(0) ⇒ 0.74 = e −t(ln 2)/5730 ⇒ ln 0.74 = − t ln 2 ⇒
5730
5730(ln 0.74)
t = − ≈ 2489 ≈ 2500 years.
ln 2
13. (a) Using Newton’s Law of Cooling, dT
dt = k(T − T s), wehave dT = k(T − 75). Nowlety = T − 75, so
dt
y(0) = T (0) − 75 = 185 − 75 = 110,soy is a solution of the initial-value problem dy/dt = ky with y(0) = 110 and by
Theorem 2 we have y(t) =y(0)e kt =110e kt .
y(30) = 110e 30k =150− 75 ⇒ e 30k = 75 = 15
110 22
⇒ k = 1 15
30
ln ,soy(t) = 110e 30 1 t ln( 15
22 )
22 and
y(45) = 110e 45
30
ln( 15
22 ) ≈ 62 ◦ F.Thus,T (45) ≈ 62 + 75 = 137 ◦ F.
(b) T (t) =100 ⇒ y(t) =25. y(t) = 110e 30 1 t ln( 15
1
22 ) =25 ⇒ e 30
t ln( 15
22 ) =
25
110
⇒ 1 15 25
30
t ln
22
=ln110 ⇒
25
30 ln
110
t =
ln 15 ≈ 116 min.
22
dT
15. = k(T − 20). Lettingy = T − 20, wegetdy
dt dt = ky,soy(t) =y(0)ekt . y(0) = T (0) − 20 = 5 − 20 = −15, so
y(25) = y(0)e 25k = −15e 25k ,andy(25) = T (25) − 20 = 10 − 20 = −10,so−15e 25k = −10 ⇒ e 25k = 2 . Thus,
3
25k =ln
2
3 and k =
1
ln
2
25 3 ,soy(t) =y(0)e kt = −15e (1/25) ln(2/3)t .Moresimply,e 25k = 2 ⇒ e k =
2 1/25
⇒
3 3
e kt =
2 t/25
⇒ y(t) =−15 ·
2 t/25
.
3
3
(a) T (50) = 20 + y(50) = 20 − 15 ·
2 50/25
=20− 15 ·
2 2
=20− 20 =13.¯3 ◦ C
3
3
3
(b) 15 = T (t) =20+y(t) =20− 15 ·
2 t/25
3
⇒ 15 ·
2 t/25
3
=5 ⇒
2 t/25
3
= 1 3
⇒
(t/25) ln
2
3 =ln 1
⇒ t =25ln
1
3
3 ln 2
3 ≈ 67.74 min.
17. (a) Let P (h) be the pressure at altitude h. ThendP/dh = kP ⇒ P (h) =P (0)e kh =101.3e kh .
P (1000) = 101.3e 1000k =87.14 ⇒ 1000k =ln
87.14
⇒ k = 1 ln
87.14
⇒
101.3
1000 101.3
P (h) =101.3 e 1000 1 h ln( 87.14
101.3) ,soP (3000) = 101.3e
3ln( 87.14
101.3 ) ≈ 64.5 kPa.
(b) P (6187) = 101.3 e 6187 ln( 87.14
1000 101.3 ) ≈ 39.9 kPa

F.
TX.10
19. (a) Using A = A 0
1+ r n
nt
with A0 = 3000, r =0.05,andt =5,wehave:
SECTION 3.9 RELATED RATES ¤ 119
(i) Annually: n =1; A =3000
1+ 0.05 1·5
1
= $3828.84
(ii) Semiannually: n =2; A =3000
1+ 0.05 2·5
2
= $3840.25
(iii) Monthly: n =12; A =3000 1+ 0.05
12
(iv) Weekly: n =52; A =3000 1+ 0.05
52
(v) Daily: n =365; A =3000 1+ 0.05
365
12·5
= $3850.08
52·5
= $3851.61
365·5
= $3852.01
(vi) Continuously: A =3000e (0.05)5 = $3852.08
(b) dA/dt =0.05A and A(0) = 3000.
3.9 Related Rates
1. V = x 3 ⇒ dV
dt = dV dx dx
dx dt =3x2 dt
3. Let s denote the side of a square. The square’s area A is given by A = s 2 . Differentiating with respect to t gives us
dA
dt
ds
ds dA
=2s .WhenA =16, s =4. Substitution 4 for s and 6 for gives us
dt dt dt = 2(4)(6) = 48 cm2 /s.
5. V = πr 2 h = π(5) 2 h =25πh ⇒ dV
dt
=25π
dh
dt
⇒
3=25π dh
dt
⇒
dh
dt = 3
25π m/min.
7. y = x 3 +2x ⇒ dy
dt = dy dx
dx dt =(3x2 + 2)(5) = 5(3x 2 +2).Whenx =2, dy
dt =5(14)=70.
9. z 2 = x 2 + y 2 ⇒ 2z dz dx dy
=2x +2y
dt dt dt
z 2 =5 2 +12 2 ⇒ z 2 =169 ⇒ z = ±13. For dx
dt
⇒
dz
dt = 1
x dx
z dt + y dy
.Whenx =5and y =12,
dt
dy dz
=2and =3,
dt dt = 1 (5 · 2+12· 3) = ±46
±13 13 .
11. (a) Given: a plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station.
If we let t be time (in hours) and x be the horizontal distance traveled by the plane (in mi), then we are given
that dx/dt = 500 mi/h.
(b) Unknown: the rate at which the distance from the plane to the station is increasing
(c)
when it is 2 mi from the station. If we let y be the distance from the plane to the station,
then we want to find dy/dt when y =2mi.
(d) By the Pythagorean Theorem, y 2 = x 2 +1 ⇒ 2y (dy/dt) =2x (dx/dt).
(e) dy
dt = x dx
y dt = x y (500). Sincey2 = x 2 +1,wheny =2, x = √ 3,so dy
√
3
dt = 2 (500) = 250 √ 3 ≈ 433 mi/h.
13. (a) Given: a man 6 ft tall walks away from a street light mounted on a 15-ft-tall pole at a rate of 5 ft/s. If we let t be time (in s)
and x be the distance from the pole to the man (in ft), then we are given that dx/dt =5ft/s.

F.
120 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
(b) Unknown: the rate at which the tip of his shadow is moving when he is 40 ft
from the pole. If we let y be the distance from the man to the tip of his
(c)
shadow(inft),thenwewanttofind d (x + y) when x =40ft.
dt
(d) By similar triangles, 15
6 = x + y
y
⇒ 15y =6x +6y ⇒ 9y =6x ⇒ y = 2 3 x.
(e) The tip of the shadow moves at a rate of d dt (x + y) = d dt
15. We are given that dx
dt
2z dz dx dy
=2x +2y
dt dt dt
17. We are given that dx
dt
x + 2
3 x = 5 dx
3 dt = 5 (5) = 25 3 3
ft/s.
=60mi/hand
dy
dt =25mi/h. z2 = x 2 + y 2
⇒
z dz
dt = x dx
dt + y dy
dt
⇒
⇒
dz
dt = 1
x dx
z dt + y dy
.
dt
After 2 hours, x = 2 (60) = 120 and y =2(25)=50 ⇒ z = √ 120 2 +50 2 = 130,
so dz
dt = 1
x dx
z dt + y dy
120(60) + 50(25)
= =65mi/h.
dt
130
2z dz
dx
dt =2(x + y) dt + dy
dt
dy
=4ft/sand
dt =5ft/s. z2 =(x + y) 2 + 500 2 ⇒
. 15 minutes after the woman starts, we have
x =(4ft/s)(20 min)(60 s/min) = 4800 ft and y =5· 15 · 60 = 4500
⇒
z = (4800 + 4500) 2 +500 2 = √ 86,740,000,so
dz
dt = x + y dx
z dt + dy
4800 + 4500
= √ (4 + 5) = √ 837 ≈ 8.99 ft/s.
dt 86,740,000 8674
19. A = 1 dh
bh,whereb is the base and h is the altitude. We are given that
2
dt
Product Rule, we have dA
dt = 1 2
b =20,so2= 1
20 · 1+10 db
2
dt
b dh
dt + h db
dt
⇒
=1cm/min and
dA
dt =2cm2 /min. Using the
.Whenh =10and A = 100,wehave100 = 1 b(10) 2 ⇒ 1 b =10 2 ⇒
4=20+10 db
dt
⇒
db
dt = 4 − 20 = −1.6 cm/min.
10
21. We are given that dx
dt
2z dz
dx
dt =2(x + y) dt + dy
dt
dy
=35km/hand
dt =25km/h. z2 =(x + y) 2 + 100 2 ⇒
.At4:00PM, x =4(35)=140and y = 4(25) = 100
⇒
z = (140 + 100) 2 +100 2 = √ 67,600 = 260,so
dz
dt = x + y dx
z dt + dy
140 + 100
= (35 + 25) = 720 ≈ 55.4 km/h.
dt 260
13

F.
TX.10
SECTION 3.9 RELATED RATES ¤ 121
23. If C = the rate at which water is pumped in, then dV
dt = C − 10,000,where
V = 1 3 πr2 h is the volume at time t. By similar triangles, r 2 = h 6
⇒ r = 1 3 h ⇒
V = 1 π 1
3 3 h2 h = π 27 h3 ⇒ dV
dt = π dh
9 h2 .Whenh =200cm,
dt
dh
dt =20cm/min,soC − 10,000 = π 9 (200)2 (20) ⇒ C =10,000 + 800,000 π ≈ 289,253 cm 3 /min.
9
25. The figure is labeled in meters. The area A of a trapezoid is
1
(base1 + base2)(height), and the volume V of the 10-meter-long trough is 10A.
2
Thus, the volume of the trapezoid with height h is V =(10) 1 2 [0.3+(0.3+2a)]h.
By similar triangles, a h = 0.25
0.5 = 1 2 ,so2a = h ⇒ V =5(0.6+h)h =3h +5h2 .
Now dV
dt = dV dh
dh dt
⇒
0.2 =(3+10h) dh
dt
dh
dt = 0.2
3 + 10(0.3) = 0.2
6 m/min = 1 10
m/min or
30 3 cm/min.
⇒
dh
dt = 0.2 .Whenh =0.3,
3+10h
27. We are given that dV
dt =30ft3 /min. V = 1 3 πr2 h = 1 2 h
3 π h = πh3
2 12
⇒
dV
dt = dV dh
dh dt
⇒
30 = πh2
4
dh
dt
⇒
dh
dt = 120
πh . 2
When h =10ft, dh
dt = 120
10 2 π = 6 ≈ 0.38 ft/min.
5π
29. A = 1 2 bh,butb =5mandsin θ = h 4
⇒
h =4sinθ,soA = 1 (5)(4 sin θ) =10sinθ.
2
We are given dθ
dA
=0.06 rad/s, so
dt
When θ = π 3 , dA
dt =0.6 cos π 3
dθ
=(10cosθ)(0.06) = 0.6cosθ.
dt
dt = dA
dθ
=(0.6)
1
2 =0.3 m 2 /s.
31. Differentiating both sides of PV = C with respect to t and using the Product Rule gives us P dV
dt + V dP dt =0
dV
dt = −V P
dP
dP
.WhenV = 600, P = 150 and
dt dt
decreasing at a rate of 80 cm 3 /min.
33. With R 1 =80and R 2 = 100,
dV
=20,sowehave
dt = − 600 (20) = −80. Thus, the volume is
150
1
R = 1 + 1 = 1 R 1 R 2 80 + 1
100 = 180
8000 = 9 400
,soR =
400
with respect to t,wehave− 1 dR
R 2 dt = − 1 dR 1
− 1 dR 2
R1
2 dt R2
2 dt
R 2 = 100, dR
dt = 4002 1
9 2 80 (0.3) + 1
2 100 (0.2) = 107 ≈ 0.132 Ω/s.
2 810
⇒
dR 1
dt = dR 1
R2 + 1 R1
2 dt R2
2
dR 2
dt
⇒
9 . Differentiating 1 R = 1 + 1 R 1 R 2
.WhenR 1 =80and

F.
122 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
35. We are given dθ/dt =2 ◦ /min = π 90
rad/min. By the Law of Cosines,
x 2 =12 2 +15 2 − 2(12)(15) cos θ =369− 360 cos θ
2x dx
dt
=360sinθ
dθ
dt
⇒
dx
dt
=
180 sin θ
x
x = √ 369 − 360 cos 60 ◦ = √ 189 = 3 √ 21,so dx
dt
⇒
dθ
dt .Whenθ =60◦ ,
=
180 sin 60◦
3 √ 21
37. (a) By the Pythagorean Theorem, 4000 2 + y 2 = 2 . Differentiating with respect to t,
we obtain 2y dy
dt
d
dy
=2 . We know that =600ft/s, so when y =3000ft,
dt dt
= √ 4000 2 + 3000 2 = √ 25,000,000 = 5000 ft
and d
dt = y dy
(b) Here tan θ =
dt = 3000
5000
y
4000
⇒
(600) =
1800
5
=360ft/s.
d
dt (tan θ) = d y
dt 4000
⇒
π
90 = π √ √
3 7 π
3 √ 21 = ≈ 0.396 m/min.
21
sec 2 θ dθ
dt = 1 dy
4000 dt
⇒
dθ
dt = cos2 θ dy
4000 dt .When
y =3000ft, dy
4000
=600ft/s, =5000and cos θ = = 4000
dt 5000 = 4 dθ
,so
5 dt = (4/5)2 (600) = 0.096 rad/s.
4000
39. cot θ = x 5
dx
dt = 5π 6
⇒
−csc 2 θ dθ
dt = 1 dx
5 dt
⇒
2 2√3
= 10 π km/min [≈ 130 mi/h]
9
− csc π 2
− π
= 1 dx
3 6 5 dt
⇒
41. We are given that dx
dt
=300km/h. By the Law of Cosines,
y 2 = x 2 +1 2 − 2(1)(x) cos 120 ◦ = x 2 +1− 2x − 1 2
= x 2 + x +1,so
2y dy dx
=2x
dt dt + dx
dt
⇒
dy
dt
=
2x +1
2y
dx
300
.After1 minute, x =
60
dt =5km ⇒
y = √ 5 2 +5+1= √ 31 km ⇒ dy
dt = 2(5) + 1
2 √ 1650
(300) = √ ≈ 296 km/h.
31 31
43. Let the distance between the runner and the friend be .ThenbytheLawofCosines,
2 =200 2 +100 2 − 2 · 200 · 100 · cos θ =50,000 − 40,000 cos θ (). Differentiating
implicitly with respect to t,weobtain2 d
dt
dθ
= −40,000(− sin θ) .NowifD is the
dt
distance run when the angle is θ radians, then by the formula for the length of an arc
on a circle, s = rθ,wehaveD = 100θ,soθ = 1
100 D ⇒ dθ
dt = 1 dD
100 dt = 7 . To substitute into the expression for
100
d
dt , we must know sin θ atthetimewhen =200,whichwefind from (): 2002 =50,000 − 40,000 cos θ
cos θ = 1 ⇒ sin θ = 1 −
1 2
= √ 15
d
. Substituting, we get 2(200)
4 4 4
dt =40,000 √
15 7
⇒
4 100
d/dt = 7 √ 15
4
≈ 6.78 m/s. Whether the distance between them is increasing or decreasing depends on the direction in which
the runner is running.
⇔

F.
TX.10SECTION 3.10 LINEAR APPROXIMATIONS AND DIFFERENTIALS ¤ 123
3.10 Linear Approximations and Differentials
1. f(x) =x 4 +3x 2 ⇒ f 0 (x) =4x 3 +6x,sof(−1) = 4 and f 0 (−1) = −10.
Thus, L(x) =f(−1) + f 0 (−1)(x − (−1)) = 4 + (−10)(x +1)=−10x − 6.
3. f(x) =cosx ⇒ f 0 (x) =− sin x, sof π
2 =0and f
0 π
Thus, L(x) =f π
2
+ f
0 π
2
5. f(x) = √ 1 − x ⇒ f 0 (x) =
x −
π
2
=0− 1
x −
π
2
2
= −1.
= −x +
π
2 .
−1
2 √ 1 − x ,sof(0) = 1 and f 0 (0) = − 1 2 .
Therefore,
√ 1 − x = f(x) ≈ f(0) + f 0 (0)(x − 0) = 1 + − 1 2
(x − 0) = 1 −
1
2 x.
So √ 0.9 = √ 1 − 0.1 ≈ 1 − 1 (0.1) = 0.95
2
and √ 0.99 = √ 1 − 0.01 ≈ 1 − 1 (0.01) = 0.995.
2
7. f(x) = 3√ 1 − x =(1− x) 1/3 ⇒ f 0 (x) =− 1 3 (1 − x)−2/3 ,sof(0) = 1
and f 0 (0) = − 1 .Thus,f(x) ≈ f(0) + f 0 3
(0)(x − 0) = 1 − 1 x.Weneed
3
3√ 1 − x − 0.1 < 1 −
1
x< 3√ 3
1 − x +0.1, which is true when
−1.204

F.
124 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
17. (a) y =tanx ⇒ dy =sec 2 xdx
(b) When x = π/4 and dx = −0.1, dy =[sec(π/4)] 2 (−0.1) = √ 2 2
(−0.1) = −0.2.
19. y = f(x) =2x − x 2 , x =2, ∆x = −0.4 ⇒
∆y = f(1.6) − f(2) = 0.64 − 0=0.64
dy =(2− 2x) dx =(2− 4)(−0.4) = 0.8
21. y = f(x) =2/x, x =4, ∆x =1 ⇒
∆y = f(5) − f(4) = 2 5 − 2 4 = −0.1
dy = − 2 x dx = − 2 (1) = −0.125
2 42 23. To estimate (2.001) 5 ,we’llfind the linearization of f(x) =x 5 at a =2.Sincef 0 (x) =5x 4 , f(2) = 32,andf 0 (2) = 80,
we have L(x) = 32 + 80(x − 2) = 80x − 128. Thus, x 5 ≈ 80x − 128 when x is near 2 ,so
(2.001) 5 ≈ 80(2.001) − 128 = 160.08 − 128 = 32.08.
25. To estimate (8.06) 2/3 ,we’llfind the linearization of f(x) =x 2/3 at a =8.Sincef 0 (x) = 2 3 x−1/3 =2/ 3 3√
x ,
f(8) = 4, andf 0 (8) = 1 ,wehaveL(x) 3 =4+1 (x − 8) = 1 x + 4 . Thus, 3 3 3 x2/3 ≈ 1 x + 4 when x is near 8, so
3 3
(8.06) 2/3 ≈ 1 3 (8.06) + 4 3 = 12.06
3
=4.02.
27. y = f(x) =tanx ⇒ dy =sec 2 xdx.Whenx =45 ◦ and dx = −1 ◦ ,
dy =sec 2 45 ◦ (−π/180) = √ 2 2
(−π/180) = −π/90,sotan 44 ◦ = f(44 ◦ ) ≈ f(45 ◦ )+dy =1− π/90 ≈ 0.965.
29. y = f(x) =secx ⇒ f 0 (x) =secx tan x, sof(0) = 1 and f 0 (0) = 1 · 0=0. The linear approximation of f at 0 is
f(0) + f 0 (0)(x − 0) = 1 + 0(x) =1.Since0.08 is close to 0, approximating sec 0.08 with 1 is reasonable.
31. y = f(x) =lnx ⇒ f 0 (x) =1/x, sof(1) = 0 and f 0 (1) = 1. The linear approximation of f at 1 is
f(1) + f 0 (1)(x − 1) = 0 + 1(x − 1) = x − 1. Nowf(1.05) = ln 1.05 ≈ 1.05 − 1=0.05, so the approximation
is reasonable.
33. (a) If x is the edge length, then V = x 3 ⇒ dV =3x 2 dx. Whenx =30and dx =0.1, dV =3(30) 2 (0.1) = 270,sothe
maximum possible error in computing the volume of the cube is about 270 cm 3 . The relative error is calculated by dividing
the change in V , ∆V ,byV . We approximate ∆V with dV .
Relative error = ∆V
V ≈ dV V = 3x2 dx
=3 dx 0.1
x 3 x =3 =0.01.
30
Percentage error = relative error × 100% = 0.01 × 100% = 1%.

F.
TX.10SECTION 3.10 LINEAR APPROXIMATIONS AND DIFFERENTIALS ¤ 125
(b) S =6x 2 ⇒ dS =12xdx.Whenx =30and dx =0.1, dS = 12(30)(0.1) = 36, so the maximum possible error in
computing the surface area of the cube is about 36 cm 2 .
Relative error = ∆S
S
≈ dS S = 12xdx =2 dx 0.1
6x 2 x =2 30
=0.006.
Percentage error = relative error × 100% = 0.006 × 100% = 0.6%.
35. (a)Forasphereofradiusr, the circumference is C =2πr and the surface area is S =4πr 2 ,so
r = C 2 C
2π ⇒ S =4π = C2
⇒ dS = 2 2π π
π CdC.WhenC =84and dC =0.5, dS = 2 84
(84)(0.5) =
π π ,
so the maximum error is about 84
π ≈ 27 cm2 . Relative error ≈ dS S = 84/π
84 2 /π = 1
84 ≈ 0.012
(b) V = 4 3 πr3 = 4 3 π C
2π
3
= C3
6π 2 ⇒ dV = 1
2π 2 C2 dC. WhenC =84and dC =0.5,
dV = 1
2π 2 (84)2 (0.5) = 1764
1764
, so the maximum error is about ≈ 179 cm 3 .
π2 π 2
The relative error is approximately dV V = 1764/π2
(84) 3 /(6π 2 ) = 1 56 ≈ 0.018.
37. (a) V = πr 2 h ⇒ ∆V ≈ dV =2πrh dr =2πrh ∆r
(b) The error is
∆V − dV =[π(r + ∆r) 2 h − πr 2 h] − 2πrh ∆r = πr 2 h +2πrh ∆r + π(∆r) 2 h − πr 2 h − 2πrh ∆r = π(∆r) 2 h.
39. V = RI ⇒ I = V R ⇒ dI = − V ∆I
dR. The relative error in calculating I is ≈ dI
R2 I I = −(V/R2 ) dR
= − dR V/R R .
Hence, the relative error in calculating I is approximately the same (in magnitude) as the relative error in R.
41. (a) dc = dc dx =0dx =0
dx
(c) d(u + v) = d
du
dx (u + v) dx = dx + dv
dx
(d) d(uv) = d
dx (uv) dx = u dv
dx + v du
dx
u
(e) d = d u
v du
dx = dx − u dv
dx dx =
v dx v
v 2
(f ) d (x n )= d
dx (xn ) dx = nx n−1 dx
dx = du dv
dx + dx = du + dv
dx dx
dx = u dv du
dx + v dx = udv+ vdu
dx dx
v du dv
dx − u
dx dx dx vdu− udv
=
v 2
v 2
43. (a) The graph shows that f 0 (1) = 2,soL(x) =f(1) + f 0 (1)(x − 1) = 5 + 2(x − 1) = 2x +3.
f(0.9) ≈ L(0.9) = 4.8 and f(1.1) ≈ L(1.1) = 5.2.
d
du
(b) d(cu) = (cu) dx = c dx = cdu
dx dx
(b) From the graph, we see that f 0 (x) is positive and decreasing. This means that the slopes of the tangent lines are positive,
but the tangents are becoming less steep. So the tangent lines lie above the curve. Thus, the estimates in part (a) are too
large.

F.
126 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
3.11 Hyperbolic Functions
1. (a) sinh 0 = 1 2 (e0 − e 0 )=0 (b) cosh 0 = 1 2 (e0 + e 0 )= 1 2
(1 + 1) = 1
3. (a) sinh(ln 2) = eln 2 − e −ln 2
2
= eln 2 − (e ln 2 ) −1
2
= 2 − 2−1
2
= 2 − 1 2
2
= 3 4
(b) sinh 2 = 1 2 (e2 − e −2 ) ≈ 3.62686
5. (a) sech 0 = 1
cosh 0 = 1 1 =1 (b) cosh−1 1=0because cosh 0 = 1.
7. sinh(−x) = 1 2 [e−x − e −(−x) ]= 1 2 (e−x − e x )=− 1 2 (e−x − e x )=− sinh x
9. cosh x +sinhx = 1 2 (ex + e −x )+ 1 2 (ex − e −x )= 1 2 (2ex )=e x
11. sinh x cosh y +coshx sinh y = 1
2 (ex − e −x ) 1
2 (ey + e −y ) + 1
2 (ex + e −x ) 1
2 (ey − e −y )
= 1 4 [(ex+y + e x−y − e −x+y − e −x−y )+(e x+y − e x−y + e −x+y − e −x−y )]
= 1 4 (2ex+y − 2e −x−y )= 1 2 [ex+y − e −(x+y) ]=sinh(x + y)
13. Divide both sides of the identity cosh 2 x − sinh 2 x =1by sinh 2 x:
cosh 2 x
sinh 2 x − sinh2 x
sinh 2 x = 1
sinh 2 x ⇔ coth2 x − 1=csch 2 x.
15. Putting y = x in the result from Exercise 11, we have
sinh 2x =sinh(x + x) =sinhx cosh x +coshx sinh x =2sinhx cosh x.
17. tanh(ln x) =
sinh(ln x)
cosh(ln x) = (eln x − e − ln x )/2
(e ln x + e − ln x )/2 = x − (eln x ) −1 x − x−1 x − 1/x
= =
x +(e ln x )
−1
x + x−1 x +1/x = (x2 − 1)/x
(x 2 +1)/x = x2 − 1
x 2 +1
19. By Exercise 9, (cosh x +sinhx) n =(e x ) n = e nx =coshnx +sinhnx.
21. sech x = 1
cosh x ⇒ sech x = 1
5/3 = 3 5 .
cosh 2 x − sinh 2 x =1 ⇒ sinh 2 x =cosh 2 x − 1=
5 2
− 1= 16 ⇒ sinh x = 4 [because x>0].
3
9 3
csch x = 1
sinh x ⇒ csch x = 1
4/3 = 3 4 .
tanh x = sinh x
cosh x
⇒ tanh x =
4/3
5/3 = 4 5 .
coth x = 1
tanh x ⇒ coth x = 1
4/5 = 5 4 .
e x − e −x
23. (a) lim tanh x = lim
x→∞ x→∞ e x + e
(b) lim tanh x =
x→−∞
lim
x→−∞
−x ·
e−x
= lim
e−x e x − e −x ex
·
e x + e−x e = x
x→∞
lim
x→−∞
1 − e −2x
1+e −2x = 1 − 0
1+0 =1
e 2x − 1
e 2x +1 = 0 − 1
0+1 = −1

F.
TX.10
SECTION 3.11 HYPERBOLIC FUNCTIONS ¤ 127
e x − e −x
(c) lim sinh x = lim = ∞
x→∞ x→∞ 2
(d) lim sinh x =
x→−∞
lim
x→−∞
e x − e −x
2
= −∞
2
(e) lim sech x = lim =0
x→∞ x→∞ e x + e−x e x + e −x
(f ) lim coth x = lim
x→∞ x→∞ e x − e
(g)
(h)
−x ·
e−x
= lim
e−x x→∞
1+e −2x 1+0
= =1 [Or: Use part (a)]
1 − e−2x 1 − 0
cosh x
lim coth x = lim = ∞,sincesinh x → 0 through positive values and cosh x → 1.
x→0 + x→0 + sinh x
cosh x
lim coth x = lim = −∞,sincesinh x → 0 through negative values and cosh x → 1.
x→0− x→0 − sinh x
(i) lim csch x =
x→−∞
lim
x→−∞
2
=0
e x − e−x 25. Let y =sinh −1 x.Thensinh y = x and, by Example 1(a), cosh 2 y − sinh 2 y =1 ⇒ [with cosh y>0]
cosh y = 1+sinh 2 y = √ 1+x 2 .SobyExercise9,e y =sinhy +coshy = x + √ 1+x 2 ⇒ y =ln x + √ 1+x 2 .
27. (a) Let y =tanh −1 x.Thenx =tanhy = sinh y
cosh y = (ey − e −y )/2
(e y + e −y )/2 · ey
e = e2y − 1
y e 2y +1
1+x = e 2y − xe 2y ⇒ 1+x = e 2y (1 − x) ⇒ e 2y = 1+x
1+x
1 − x ⇒ 2y =ln 1 − x
(b) Let y =tanh −1 x.Thenx =tanhy, so from Exercise 18 we have
e 2y = 1+tanhy
1 − tanh y = 1+x
1+x
1 − x ⇒ 2y =ln 1 − x
⇒ y = 1 2 ln 1+x
1 − x
29. (a) Let y =cosh −1 x.Thencosh y = x and y ≥ 0 ⇒ sinh y dy
dx =1
dy
dx = 1
sinh y = 1
cosh 2 y − 1 = 1
√
x2 − 1
⇒
.
[since sinh y ≥ 0 for y ≥ 0]. Or: Use Formula 4.
⇒ xe 2y + x = e 2y − 1 ⇒
1+x
⇒ y = 1 ln .
2
1 − x
(b) Let y =tanh −1 x.Thentanh y = x ⇒ sech 2 y dy
dy
=1 ⇒
dx dx = 1
sech 2 y = 1
1 − tanh 2 y = 1
1 − x . 2
Or: Use Formula 5.
(c) Let y =csch −1 x.Thencsch y = x ⇒ −csch y coth y dy
dy
=1 ⇒
dx dx = − 1
. By Exercise 13,
csch y coth y
coth y = ± csch 2 y +1=± √ x 2 +1.Ifx>0,thencoth y>0,socoth y = √ x 2 +1.Ifx0.]
1 − x2 ⇒

F.
128 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
(e) Let y =coth −1 x.Thencoth y = x ⇒ −csch 2 y dy
dy
=1 ⇒
dx dx = − 1
csch 2 y = 1
1 − coth 2 y = 1
1 − x 2
by Exercise 13.
31. f(x) =x sinh x − cosh x ⇒ f 0 (x) =x (sinh x) 0 +sinhx · 1 − sinh x = x cosh x
33. h(x) =ln(coshx) ⇒ h 0 (x) = 1
cosh x (cosh x)0 = sinh x
cosh x =tanhx
35. y = e cosh 3x ⇒ y 0 = e cosh 3x · sinh 3x · 3=3e cosh 3x sinh 3x
37. f(t) =sech 2 (e t )=[sech(e t )] 2 ⇒
f 0 (t) =2[sech(e t )] [sech(e t )] 0 =2sech(e t ) − sech(e t )tanh(e t ) · e t = −2e t sech 2 (e t )tanh(e t )
39. y = arctan(tanh x) ⇒ y 0 =
1
1+(tanhx) (tanh 2 x)0 =
sech2 x
1+tanh 2 x
41. G(x) = 1 − cosh x
1+coshx
⇒
G 0 (x) =
(1 + cosh x)(− sinh x) − (1 − cosh x)(sinh x) − sinh x − sinh x cosh x − sinh x +sinhx cosh x
=
(1 + cosh x) 2 (1 + cosh x) 2
= −2sinhx
(1 + cosh x) 2
43. y =tanh −1√ x ⇒ y 0 1
= √ 2 · 1
1
2 x−1/2 =
1 − x 2 √ x (1 − x)
45. y = x sinh −1 (x/3) − √ 9+x 2 ⇒
x
y 0 =sinh −1 1/3
+ x
3
−
1+(x/3)
2
47. y =coth −1√ x 2 +1 ⇒ y 0 1
=
1 − (x 2 +1)
2x
x
2 √ 9+x 2 =sinh−1 +
3
49. As the depth d of the water gets large, the fraction 2πd
L
approaches 1. Thus, v =
x
√
9+x
2 −
2x
2 √ x 2 +1 = − 1
x √ x 2 +1
gL 2πd gL gL
2π tanh ≈
L 2π (1) = 2π .
x
x
√
9+x
2 =sinh−1 3
2πd
gets large, and from Figure 3 or Exercise 23(a), tanh L
51. (a) y =20cosh(x/20) − 15 ⇒ y 0 = 20 sinh(x/20) · 1 = sinh(x/20). Since the right pole is positioned at x =7,
20
we have y 0 (7) = sinh 7
20 ≈ 0.3572.
(b) If α is the angle between the tangent line and the x-axis, then tan α = slope of the line =sinh 7 20 ,so
α =tan −1 sinh 7
20
≈ 0.343 rad ≈ 19.66 ◦ .Thus,theanglebetweenthelineandthepoleisθ =90 ◦ − α ≈ 70.34 ◦ .
53. (a) y = A sinh mx + B cosh mx ⇒ y 0 = mA cosh mx + mB sinh mx ⇒
y 00 = m 2 A sinh mx + m 2 B cosh mx = m 2 (A sinh mx + B cosh mx) =m 2 y

F.
TX.10
CHAPTER 3 REVIEW ¤ 129
(b) From part (a), a solution of y 00 =9y is y(x) =A sinh 3x + B cosh 3x. So−4 =y(0) = A sinh 0 + B cosh 0 = B,so
B = −4. Nowy 0 (x) =3A cosh 3x − 12 sinh 3x ⇒ 6=y 0 (0) = 3A ⇒ A =2,soy =2sinh3x − 4cosh3x.
55. The tangent to y =coshx has slope 1 when y 0 =sinhx =1 ⇒ x =sinh −1 1=ln 1+ √ 2 ,byEquation3.
Since sinh x =1and y =coshx = 1+sinh 2 x,wehavecosh x = √ 2. The point is ln 1+ √ 2 , √ 2 .
57. If ae x + be −x = α cosh(x + β) [or α sinh(x + β)], then
ae x + be −x = α 2 e x+β ± e −x−β
= α 2 e x e β ± e −x e −β = α
eβ e x ± α
e−β e −x . Comparing coefficients of e x
2 2
and e −x ,wehavea = α 2 eβ (1) and b = ± α 2 e−β (2). Weneedtofind α and β. Dividing equation (1) by equation (2)
gives us a b = ±e2β ⇒ () 2β =ln ± a b
⇒ β = 1 2 ln ± a b
. Solving equations (1) and (2) for e β gives us
e β = 2a α and eβ = ± α 2b ,so 2a α = ± α 2b
⇒ α 2 = ±4ab ⇒ α =2 √ ±ab.
() If a b > 0, weusethe+ sign and obtain a cosh function, whereas if a b
< 0, weusethe− sign and obtain a sinh
function.
In summary, if a and b havethesamesign,wehaveae x + be −x =2 √ ab cosh x + 1 2 ln a b
, whereas, if a and b have the
opposite sign, then ae x + be −x =2 √ −ab sinh x + 1 2 ln − a b
.
3 Review
1. (a) The Power Rule: If n is any real number, then d
dx (xn )=nx n−1 . The derivative of a variable base raised to a constant
power is the power times the base raised to the power minus one.
(b) The Constant Multiple Rule: If c is a constant and f is a differentiable function, then d
dx [cf(x)] = c d
dx f(x).
The derivative of a constant times a function is the constant times the derivative of the function.
(c) The Sum Rule: If f and g are both differentiable, then d
dx [f(x)+g(x)] = d
dx f(x)+ d g(x). The derivative of a sum
dx
of functions is the sum of the derivatives.
(d) The Difference Rule: If f and g are both differentiable, then d
d
[f(x) − g(x)] =
dx dx f(x) − d g(x). The derivative of a
dx
difference of functions is the difference of the derivatives.
(e) The Product Rule: If f and g are both differentiable, then d
d
[f(x) g(x)] = f(x)
dx dx g(x)+g(x) d f(x). The
dx
derivative of a product of two functions is the first function times the derivative of the second function plus the second
function times the derivative of the first function.

F.
130 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
(f ) The Quotient Rule: If f and g are both differentiable, then d
dx
f(x)
g(x) d
d
f(x) − f(x)
= dx dx g(x)
g(x)
[ g(x)] 2 .
The derivative of a quotient of functions is the denominator times the derivative of the numerator minus the numerator
times the derivative of the denominator, all divided by the square of the denominator.
(g) The Chain Rule: If f and g are both differentiable and F = f ◦ g is the composite function defined by F (x) =f(g(x)),
then F is differentiable and F 0 is given by the product F 0 (x) =f 0 (g(x)) g 0 (x). The derivative of a composite function is
the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
2. (a) y = x n ⇒ y 0 = nx n−1 (b) y = e x ⇒ y 0 = e x
(c) y = a x ⇒ y 0 = a x ln a (d) y =lnx ⇒ y 0 =1/x
(e) y =log a x ⇒ y 0 =1/(x ln a) (f ) y =sinx ⇒ y 0 =cosx
(g) y =cosx ⇒ y 0 = − sin x (h) y =tanx ⇒ y 0 =sec 2 x
(i) y =cscx ⇒ y 0 = − csc x cot x (j) y =secx ⇒ y 0 =secx tan x
(k) y =cotx ⇒ y 0 = − csc 2 x (l) y =sin −1 x ⇒ y 0 =1/ √ 1 − x 2
(m) y =cos −1 x ⇒ y 0 = −1/ √ 1 − x 2 (n) y =tan −1 x ⇒ y 0 =1/(1 + x 2 )
(o) y =sinhx ⇒ y 0 =coshx (p) y =coshx ⇒ y 0 =sinhx
(q) y =tanhx ⇒ y 0 =sech 2 x (r) y =sinh −1 x ⇒ y 0 =1/ √ 1+x 2
(s) y =cosh −1 x ⇒ y 0 =1/ √ x 2 − 1 (t) y =tanh −1 x ⇒ y 0 =1/(1 − x 2 )
e h − 1
3. (a) e is the number such that lim =1.
h→0 h
(b) e =lim
x→0
(1 + x) 1/x
(c) The differentiation formula for y = a x [y 0 = a x ln a] is simplest when a = e because ln e =1.
(d) The differentiation formula for y =log a x [y 0 =1/(x ln a)] issimplestwhena = e because ln e =1.
4. (a) Implicit differentiation consists of differentiating both sides of an equation involving x and y with respect to x,andthen
solving the resulting equation for y 0 .
(b) Logarithmic differentiation consists of taking natural logarithms of both sides of an equation y = f(x), simplifying,
differentiating implicitly with respect to x, and then solving the resulting equation for y 0 .
5. (a) The linearization L of f at x = a is L(x) =f(a)+f 0 (a)(x − a).
(b) If y = f(x), then the differential dy is given by dy = f 0 (x) dx.
(c)SeeFigure5inSection3.10.

F.
TX.10
CHAPTER 3 REVIEW ¤ 131
1. True. This is the Sum Rule.
3. True. This is the Chain Rule.
5. False.
√
d
√ f 0 x
dx f x =
2 √ by the Chain Rule.
x
7. False.
9. True.
d
dx 10x =10 x ln 10
d
dx (tan2 x)=2tanx sec 2 x,and d
dx (sec2 x)=2secx (sec x tan x) =2tanx sec 2 x.
Or:
d
dx (sec2 x)= d
dx (1 + tan2 x)= d
dx (tan2 x).
11. True. g(x) =x 5 ⇒ g 0 (x) =5x 4 ⇒ g 0 (2) = 5(2) 4 =80,andbythedefinition of the derivative,
g(x) − g(2)
lim
= g 0 (2) = 80.
x→2 x − 2
1. y =(x 4 − 3x 2 +5) 3 ⇒
y 0 =3(x 4 − 3x 2 +5) 2
d
dx (x4 − 3x 2 +5)=3(x 4 − 3x 2 +5) 2 (4x 3 − 6x) =6x(x 4 − 3x 2 +5) 2 (2x 2 − 3)
3. y = √ x + 1
3√
x
4 = x1/2 + x −4/3 ⇒ y 0 = 1 2 x−1/2 − 4 3 x−7/3 = 1
2 √ x − 4
3 3√ x 7
5. y =2x √ x 2 +1 ⇒
y 0 =2x · 1
2 (x2 +1) −1/2 (2x)+ √ x 2 +1(2)= √ 2x2
x2 +1 +2√ x 2 +1= 2x2 +2(x 2 +1)
√ = 2(2x2 +1)
√
x2 +1 x2 +1
7. y = e sin 2θ ⇒ y 0 = e sin 2θ d
dθ (sin 2θ) =esin 2θ sin 2θ
(cos 2θ)(2) = 2 cos 2θe
9. y = t
1 − t 2 ⇒ y 0 = (1 − t2 )(1) − t(−2t)
(1 − t 2 ) 2 = 1 − t2 +2t 2
(1 − t 2 ) 2 = t2 +1
(1 − t 2 ) 2
11. y = √ x cos √ x ⇒
y 0 = √
x cos √ x
0
+cos
√
x
√
x
0
=
√
x
− sin √ x
=
− √ 1 2 x−1/2 x sin √ x +cos √
x = cos √ x − √ x sin √ x
2 √ x
1
2 x−1/2
+cos √ x
1
2 x−1/2
13. y = e1/x
⇒ y 0 = x2 (e 1/x ) 0 − e 1/x x 2 0
= x2 (e 1/x )(−1/x 2 ) − e 1/x (2x)
= −e1/x (1 + 2x)
x 2 (x 2 ) 2 x 4 x 4

F.
132 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
15.
d
dx (xy4 + x 2 y)= d
dx (x +3y) ⇒ x · 4y3 y 0 + y 4 · 1+x 2 · y 0 + y · 2x =1+3y 0 ⇒
y 0 (4xy 3 + x 2 − 3) = 1 − y 4 − 2xy ⇒ y 0 = 1 − y4 − 2xy
4xy 3 + x 2 − 3
17. y =
sec 2θ
1+tan2θ
⇒
y 0 = (1 + tan 2θ)(sec 2θ tan 2θ · 2) − (sec 2θ)(sec2 2θ · 2)
= 2sec2θ [(1 + tan 2θ)tan2θ − sec2 2θ]
(1 + tan 2θ) 2 (1 + tan 2θ) 2
= 2sec2θ (tan 2θ +tan2 2θ − sec 2 2θ) 2sec2θ (tan 2θ − 1)
= 1+tan 2 x =sec 2 x
(1 + tan 2θ) 2 (1 + tan 2θ) 2
19. y = e cx (c sin x − cos x) ⇒
y 0 = e cx (c cos x +sinx)+ce cx (c sin x − cos x) =e cx (c 2 sin x − c cos x + c cos x +sinx)
= e cx (c 2 sin x +sinx) =e cx sin x (c 2 +1)
21. y =3 x ln x ⇒ y 0 =3 x ln x · ln 3 ·
d
dx (x ln x) =3x ln x · ln 3 x · 1
x +lnx · 1 =3 x ln x · ln 3(1 + ln x)
23. y =(1− x −1 ) −1 ⇒
y 0 = −1(1 − x −1 ) −2 [−(−1x −2 )] = −(1 − 1/x) −2 x −2 = −((x − 1)/x) −2 x −2 = −(x − 1) −2
25. sin(xy) =x 2 − y ⇒ cos(xy)(xy 0 + y · 1) = 2x − y 0 ⇒ x cos(xy)y 0 + y 0 =2x − y cos(xy) ⇒
y 0 [x cos(xy)+1]=2x − y cos(xy) ⇒ y 0 =
2x − y cos(xy)
x cos(xy)+1
27. y =log 5 (1 + 2x) ⇒ y 0 =
1 d
(1 + 2x) ln5dx (1 + 2x) = 2
(1 + 2x) ln5
29. y =lnsinx − 1 2 sin2 x ⇒ y 0 = 1
sin x · cos x − 1 · 2sinx · cos x =cotx − sin x cos x
2
31. y = x tan −1 (4x) ⇒ y 0 = x ·
1
1+(4x) · 4x
2 4+tan−1 (4x) · 1=
1+16x 2 +tan−1 (4x)
33. y =ln|sec 5x +tan5x| ⇒
y 0 =
1
sec 5x +tan5x (sec 5x tan 5x · 5sec5x (tan 5x +sec5x)
5+sec2 5x · 5) = =5sec5x
sec 5x +tan5x
35. y =cot(3x 2 +5) ⇒ y 0 = − csc 2 (3x 2 + 5)(6x) =−6x csc 2 (3x 2 +5)
37. y =sin tan √ 1+x 3 ⇒ y 0 =cos tan √ 1+x 3 sec 2 √ 1+x 3 3x 2 2 √ 1+x 3
39. y =tan 2 (sin θ) =[tan(sinθ)] 2 ⇒ y 0 =2[tan(sinθ)] · sec 2 (sin θ) · cos θ

F.
TX.10
CHAPTER 3 REVIEW ¤ 133
√ x +1(2− x)
5
41. y =
⇒ ln y = 1
(x +3) 7 2
√ x +1(2− x)
y 0 5
1
=
(x +3) 7 2(x +1) − 5
2 − x − 7
x +3
ln(x +1)+5ln(2− x) − 7ln(x +3) ⇒
y0
y = 1
2(x +1) + −5
2 − x − 7
or y 0 = (2 − x)4 (3x 2 − 55x − 52)
2 √ x +1(x +3) 8 .
x +3
⇒
43. y = x sinh(x 2 ) ⇒ y 0 = x cosh(x 2 ) · 2x +sinh(x 2 ) · 1=2x 2 cosh(x 2 )+sinh(x 2 )
45. y =ln(cosh3x) ⇒ y 0 =(1/ cosh 3x)(sinh 3x)(3) = 3 tanh 3x
47. y =cosh −1 (sinh x) ⇒ y 0 =
49. y =cos e √
tan 3x
⇒
1
(sinh x)2 − 1 · cosh x = cosh x
sinh 2 x − 1
y 0 = − sin e √
tan 3x
· e √ 0
tan 3x = − sin e √
tan 3x
e √ tan 3x · 1 (tan 2 3x)−1/2 · sec 2 (3x) · 3
−3sin e √
tan 3x
e √ tan 3x sec 2 (3x)
=
2 √ tan 3x
51. f(t) = √ 4t +1 ⇒ f 0 (t) = 1 2 (4t +1)−1/2 · 4=2(4t +1) −1/2 ⇒
f 00 (t) =2(− 1 2 )(4t +1)−3/2 · 4=−4/(4t +1) 3/2 ,sof 00 (2) = −4/9 3/2 = − 4
27 .
53. x 6 + y 6 =1 ⇒ 6x 5 +6y 5 y 0 =0 ⇒ y 0 = −x 5 /y 5 ⇒
y 00 = − y5 (5x 4 ) − x 5 (5y 4 y 0 )
= − 5x4 y 4 y − x(−x 5 /y 5 ) = − 5x4 (y 6 + x 6 )/y 5
= − 5x4
(y 5 ) 2 y 10
y 6
y 11
55. We firstshowitistrueforn =1: f(x) =xe x ⇒ f 0 (x) =xe x + e x =(x +1)e x . We now assume it is true
for n = k: f (k) (x) =(x + k)e x . With this assumption, we must show it is true for n = k +1:
f (k+1) (x) = d
f (k) (x) = d
dx
dx [(x + k)ex ]=(x + k)e x + e x =[(x + k)+1]e x =[x +(k +1)]e x .
Therefore, f (n) (x) =(x + n)e x by mathematical induction.
57. y =4sin 2 x ⇒ y 0 =4· 2sinx cos x. At π
6 , 1 , y 0 =8· 1
2 · √3
2 =2√ 3, so an equation of the tangent line
is y − 1=2 √ 3 x − π 6
,ory =2
√
3 x +1− π
√
3/3.
59. y = √ 1+4sinx ⇒ y 0 = 1 2 (1 + 4 sin x)−1/2 · 4cosx =
2cosx
√ 1+4sinx
.
At (0, 1), y 0 = 2 √
1
=2, so an equation of the tangent line is y − 1=2(x − 0),ory =2x +1.
61. y =(2+x)e −x ⇒ y 0 =(2+x)(−e −x )+e −x · 1=e −x [−(2 + x)+1]=e −x (−x − 1).
At (0, 2), y 0 =1(−1) = −1, so an equation of the tangent line is y − 2=−1(x − 0),ory = −x +2.
The slope of the normal line is 1, so an equation of the normal line is y − 2=1(x − 0),ory = x +2.

F.
134 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
63. (a) f(x) =x √ 5 − x ⇒
1
f 0 (x) =x
2 (5 − x)−1/2 (−1) + √ −x
5 − x =
2 √ 5 − x + √ 5 − x · 2 √ 5 − x
2 √ 5 − x = −x
2 √ 2(5 − x)
+
5 − x 2 √ 5 − x
=
−x +10− 2x
2 √ 5 − x
=
10 − 3x
2 √ 5 − x
(b) At (1, 2): f 0 (1) = 7 4 .
(c)
So an equation of the tangent line is y − 2= 7 4 (x − 1) or y = 7 4 x + 1 4 .
At (4, 4): f 0 (4) = − 2 2 = −1.
So an equation of the tangent line is y − 4=−1(x − 4) or y = −x +8.
(d)
The graphs look reasonable, since f 0 is positive where f has tangents with
positive slope, and f 0 is negative where f has tangents with negative slope.
65. y =sinx +cosx ⇒ y 0 =cosx − sin x =0 ⇔ cos x =sinx and 0 ≤ x ≤ 2π ⇔ x = π or 5π , so the points
4 4
are π
4 , √ 2 and 5π
4 , −√ 2 .
67. f(x) =(x − a)(x − b)(x − c) ⇒ f 0 (x) =(x − b)(x − c)+(x − a)(x − c)+(x − a)(x − b).
So f 0 (x)
f(x)
=
(x − b)(x − c)+(x − a)(x − c)+(x − a)(x − b)
(x − a)(x − b)(x − c)
= 1
x − a + 1
x − b + 1
x − c .
Or: f(x) =(x − a)(x − b)(x − c) ⇒ ln |f(x)| =ln|x − a| +ln|x − b| +ln|x − c| ⇒
f 0 (x)
f(x) = 1
x − a + 1
x − b + 1
x − c
69. (a) h(x) =f(x) g(x) ⇒ h 0 (x) =f(x) g 0 (x)+g(x) f 0 (x) ⇒
h 0 (2) = f(2) g 0 (2) + g(2) f 0 (2) = (3)(4) + (5)(−2) = 12 − 10 = 2
(b) F (x) =f(g(x)) ⇒ F 0 (x) =f 0 (g(x)) g 0 (x) ⇒ F 0 (2) = f 0 (g(2)) g 0 (2) = f 0 (5)(4) = 11 · 4=44
71. f(x) =x 2 g(x) ⇒ f 0 (x) =x 2 g 0 (x)+g(x)(2x) =x[xg 0 (x)+2g(x)]
73. f(x) =[g(x)] 2 ⇒ f 0 (x) =2[g(x)] · g 0 (x) =2g(x) g 0 (x)
75. f(x) =g(e x ) ⇒ f 0 (x) =g 0 (e x ) e x
77. f(x) =ln|g(x)| ⇒ f 0 (x) = 1
g(x) g0 (x) = g0 (x)
g(x)

F.
TX.10
CHAPTER 3 REVIEW ¤ 135
79. h(x) =
f(x) g(x)
f(x)+g(x)
⇒
h 0 (x) = [f(x)+g(x)] [f(x) g0 (x)+g(x) f 0 (x)] − f(x) g(x)[f 0 (x)+g 0 (x)]
[f(x)+g(x)] 2
= [f(x)]2 g 0 (x)+f(x) g(x) f 0 (x)+f(x) g(x) g 0 (x)+[g(x)] 2 f 0 (x) − f(x) g(x) f 0 (x) − f(x) g(x) g 0 (x)
[f(x)+g(x)] 2
= f 0 (x)[g(x)] 2 + g 0 (x)[f(x)] 2
[f(x)+g(x)] 2
81. Using the Chain Rule repeatedly, h(x) =f(g(sin 4x)) ⇒
h 0 (x) =f 0 (g(sin 4x)) ·
83. y =[ln(x +4)] 2 ⇒ y 0 =2[ln(x +4)] 1 ·
d
dx (g(sin 4x)) = f 0 (g(sin 4x)) · g 0 d
(sin 4x) ·
dx (sin 4x) =f 0 (g(sin 4x))g 0 (sin 4x)(cos 4x)(4).
1
+4)
· 1=2ln(x and y 0 =0 ⇔ ln(x +4)=0 ⇔
x +4 x +4
x +4=e 0 ⇒ x +4=1 ⇔ x = −3, so the tangent is horizontal at the point (−3, 0).
85. y = f(x) =ax 2 + bx + c ⇒ f 0 (x) =2ax + b. We know that f 0 (−1) = 6 and f 0 (5) = −2, so−2a + b =6and
10a + b = −2. Subtracting the first equation from the second gives 12a = −8 ⇒ a = − 2 . Substituting − 2 for a in the
3 3
first equation gives b = 14 3 .Nowf(1) = 4 ⇒ 4=a + b + c,soc =4+2 3 − 14
3 =0and hence, f(x) =− 2 3 x2 + 14
3 x.
87. s(t) =Ae −ct cos(ωt + δ) ⇒
v(t) =s 0 (t) =A{e −ct [−ω sin(ωt + δ)] + cos(ωt + δ)(−ce −ct )} = −Ae −ct [ω sin(ωt + δ)+c cos(ωt + δ)]
⇒
a(t) =v 0 (t) =−A{e −ct [ω 2 cos(ωt + δ) − cω sin(ωt + δ)] + [ω sin(ωt + δ)+c cos(ωt + δ)](−ce −ct )}
= −Ae −ct [ω 2 cos(ωt + δ) − cω sin(ωt + δ) − cω sin(ωt + δ) − c 2 cos(ωt + δ)]
= −Ae −ct [(ω 2 − c 2 )cos(ωt + δ) − 2cω sin(ωt + δ)] = Ae −ct [(c 2 − ω 2 )cos(ωt + δ)+2cω sin(ωt + δ)]
89. (a) y = t 3 − 12t +3 ⇒ v(t) =y 0 =3t 2 − 12 ⇒ a(t) =v 0 (t) =6t
(b) v(t) =3(t 2 − 4) > 0 when t>2,soitmovesupwardwhent>2 anddownwardwhen0 ≤ t2. The particle is slowing down when v and a have opposite
signs; that is, when 0

F.
136 ¤ CHAPTER 3 DIFFERENTIATION RULES
TX.10
93. (a) y(t) =y(0)e kt =200e kt ⇒ y(0.5) = 200e 0.5k =360 ⇒ e 0.5k =1.8 ⇒ 0.5k =ln1.8 ⇒
k =2ln1.8 =ln(1.8) 2 =ln3.24 ⇒ y(t) = 200e (ln 3.24)t = 200(3.24) t
(b) y(4) = 200(3.24) 4 ≈ 22,040 bacteria
(c) y 0 (t) = 200(3.24) t · ln 3.24,soy 0 (4) = 200(3.24) 4 · ln 3.24 ≈ 25,910 bacteria per hour
(d) 200(3.24) t =10,000 ⇒ (3.24) t =50 ⇒ t ln 3.24 = ln 50 ⇒ t =ln50/ ln 3.24 ≈ 3.33 hours
95. (a) C 0 (t) =−kC(t) ⇒ C(t) =C(0)e −kt by Theorem 9.4.2. But C(0) = C 0 ,soC(t) =C 0 e −kt .
(b) C(30) = 1 2 C0 since the concentration is reduced by half. Thus, 1 2 C0 = C0e−30k ⇒ ln 1 2 = −30k ⇒
k = − 1 ln 1 = 1 ln 2. Since10% of the original concentration remains if 90% is eliminated, we want the value of t
30 2 30
such that C(t) = 1 10 C 0. Therefore,
1
C 10 0 = C 0 e −t(ln 2)/30 ⇒ ln 0.1 =−t(ln 2)/30 ⇒ t = − 30 ln 0.1 ≈ 100 h.
ln 2
97. If x = edge length, then V = x 3 ⇒ dV/dt =3x 2 dx/dt =10 ⇒ dx/dt =10/(3x 2 ) and S =6x 2 ⇒
dS/dt =(12x) dx/dt =12x[10/(3x 2 )] = 40/x. Whenx =30, dS/dt = 40
30 = 4 3 cm2 /min.
99. Given dh/dt =5and dx/dt =15, find dz/dt. z 2 = x 2 + h 2 ⇒
2z dz dx dh
=2x +2h
dt dt dt
⇒
dz
dt = 1 (15x +5h). Whent =3,
z
h =45+3(5)=60and x = 15(3) = 45 ⇒ z = √ 45 2 +60 2 =75,
so dz
dt = 1 [15(45) + 5(60)] = 13 ft/s.
75
101. We are given dθ/dt = −0.25 rad/h. tan θ =400/x ⇒
x =400cotθ ⇒ dx
dt = −400 csc2 θ dθ
dt .Whenθ = π , 6
dx
dt = −400(2)2 (−0.25) = 400 ft/h.
103. (a) f(x) = 3√ 1+3x =(1+3x) 1/3 ⇒ f 0 (x) =(1+3x) −2/3 , so the linearization of f at a =0is
L(x) =f(0) + f 0 (0)(x − 0) = 1 1/3 +1 −2/3 x =1+x. Thus,
3√
1.03 =
3 1+3(0.01) ≈ 1+(0.01) = 1.01.
3√ 1+3x ≈ 1+x
⇒
(b) The linear approximation is 3√ 1+3x ≈ 1+x,sofortherequiredaccuracy
we want 3√ 1+3x − 0.1 < 1+x< 3√ 1+3x +0.1. From the graph,
itappearsthatthisistruewhen−0.23

F.
TX.10
CHAPTER 3 REVIEW ¤ 137
105. A = x 2 + 1 π 1
2 2 x2 =
1+ π 8 x
2
⇒ dA = 2+ π 4
xdx.Whenx =60
and dx =0.1, dA = 2+ π 4
approximately 12 + 3π 2 ≈ 16.7 cm2 .
√
4
16 + h − 2 d
107. lim
=
h→0 h dx
60(0.1) = 12 +
3π
2
, so the maximum error is
4√
x
x =16
= 1 4 x−3/4 x
=16
=
1
4 √ 4
16 3 = 1
32
√ √ √ √ √ √
1+tanx − 1+sinx 1+tanx − 1+sinx 1+tanx + 1+sinx
109. lim
=lim
x→0 x 3
x→0 x √ 3 1+tanx + √ 1+sinx
(1 + tan x) − (1 + sin x)
=lim
x→0 x 3√ 1+tanx + √ 1+sinx =lim sin x (1/ cos x − 1)
x→0 x 3√ 1+tanx + √ 1+sinx · cos x
cos x
sin x (1 − cos x)
=lim
x→0 x 3√ 1+tanx + √ 1+sinx cos x · 1+cosx
1+cosx
sin x · sin 2 x
=lim
x→0 x 3√ 1+tanx + √ 1+sinx cos x (1 + cos x)
= lim
x→0
=1 3 ·
3
sin x
1
lim √ √
x x→0 1+tanx + 1+sinx cos x (1 + cos x)
1
√
1+
√
1
· 1 · (1 + 1)
= 1 4
111.
d
dx [f(2x)] = x2 ⇒ f 0 (2x) · 2=x 2 ⇒ f 0 (2x) = 1 2 x2 .Lett =2x. Thenf 0 (t) = 1 1
2 2 t2 = 1 8 t2 ,sof 0 (x) = 1 8 x2 .

F.
TX.10

F.
TX.10
PROBLEMS PLUS
1. Let a be the x-coordinate of Q. Since the derivative of y =1− x 2 is y 0 = −2x,theslopeatQ is −2a. But since the triangle
is equilateral, AO/OC = √ 3/1, so the slope at Q is − √ 3. Therefore, we must have that −2a = − √ 3 ⇒ a = √ 3
Thus, the point Q has coordinates
√3
√3
, 1 − 2 2
2 √3
= , 1 andbysymmetry,P has coordinates − √
3
, 1 .
2 4
2 4
2 .
3. We must show that r (in the figure) is halfway between p and q,thatis,
r =(p + q)/2. For the parabola y = ax 2 + bx + c, the slope of the tangent line is
given by y 0 =2ax + b. An equation of the tangent line at x = p is
y − (ap 2 + bp + c) =(2ap + b)(x − p). Solving for y gives us
y =(2ap + b)x − 2ap 2 − bp +(ap 2 + bp + c)
or y =(2ap + b)x + c − ap 2 (1)
Similarly, an equation of the tangent line at x = q is
y =(2aq + b)x + c − aq 2 (2)
We can eliminate y and solve for x by subtracting equation (1) from equation (2).
[(2aq + b) − (2ap + b)]x − aq 2 + ap 2 =0
(2aq − 2ap)x = aq 2 − ap 2
2a(q − p)x = a(q 2 − p 2 )
x =
a(q + p)(q − p)
2a(q − p)
= p + q
2
Thus, the x-coordinate of the point of intersection of the two tangent lines, namely r,is(p + q)/2.
5. Let y =tan −1 x.Thentan y = x, so from the triangle we see that
sin(tan −1 x)=siny =
sin(tan −1 (sinh x)) =
x
√ . Using this fact we have that
1+x
2
sinh x
1+sinh 2 x = sinh x
cosh x =tanhx.
Hence, sin −1 (tanh x) =sin −1 (sin(tan −1 (sinh x))) = tan −1 (sinh x).
7. We use mathematical induction. Let S n be the statement that
S 1 is true because
d n
dx n (sin4 x +cos 4 x)=4 n−1 cos(4x + nπ/2).
d
dx (sin4 x +cos 4 x)=4sin 3 x cos x − 4cos 3 x sin x =4sinx cos x sin 2 x − cos 2 x x
= −4sinx cos x cos 2x = −2sin2x cos 2 = − sin 4x =sin(−4x)
=cos π
− (−4x) 2
=cos π
+4x 2
=4 n−1 cos
4x + n π 2 when n =1
[continued]
139

F.
140 ¤ CHAPTER 3 PROBLEMS PLUS
TX.10
d k
Now assume S k is true, that is, sin 4 x +cos 4 x =4 k−1 cos
4x + k π
dx k 2 .Then
d k+1
dx k+1 (sin4 x +cos 4 x)= d
d
k
dx dx k (sin4 x +cos 4 x) = d 4 k−1 cos
4x + k π 2
dx
which shows that S k+1 is true.
Therefore,
Another proof: First write
= −4 k−1 sin
4x + k π d
2 ·
dx
=4 k sin −4x − k π 2
4x + k
π
2 = −4 k sin
4x + k π 2
=4 k cos π
2 − −4x − k π 2
=4 k cos 4x +(k +1) π 2
d n
dx n (sin4 x +cos 4 x)=4 n−1 cos
4x + n π 2 for every positive integer n, by mathematical induction.
sin 4 x +cos 4 x =(sin 2 x +cos 2 x) 2 − 2sin 2 x cos 2 x =1− 1 2 sin2 2x =1− 1 (1 − cos 4x) = 3 + 1 cos 4x
4 4 4
Then we have dn
dx n (sin4 x +cos 4 x)= dn 3
dx n 4 + 1
4 cos 4x = 1
4 · 4n cos 4x + n π
=4 n−1 cos 4x + n π
.
2
2
9. We must find a value x 0 such that the normal lines to the parabola y = x 2 at x = ±x 0 intersectatapointoneunitfromthe
points
±x 0 ,x 2 0 . The normals to y = x 2 at x = ±x 0 have slopes − 1 and pass through
±x 0 ,x 2 0 respectively, so the
±2x 0
normals have the equations y − x 2 0 = − 1
2x 0
(x − x 0) and y − x 2 0 = 1
2x 0
(x + x 0). The common y-intercept is x 2 0 + 1 2 .
We want to find the value of x 0 for which the distance from 0,x 2 0 + 1 2
to
x0,x 2 0
equals 1. The square of the distance is
(x 0 − 0) 2 + x 2 0 − x 2 0 + 1 2
the center of the circle is at 0, 5 4
.
2
= x 2 0 + 1 4 =1 ⇔ x 0 = ± √ 3
2 .Forthesevaluesofx 0,they-intercept is x 2 0 + 1 2 = 5 4 ,so
Another solution: Let the center of the circle be (0,a). Then the equation of the circle is x 2 +(y − a) 2 =1.
Solving with the equation of the parabola, y = x 2 ,wegetx 2 +(x 2 − a) 2 =1 ⇔ x 2 + x 4 − 2ax 2 + a 2 =1 ⇔
x 4 +(1− 2a)x 2 + a 2 − 1=0. The parabola and the circle will be tangent to each other when this quadratic equation in x 2
has equal roots; that is, when the discriminant is 0. Thus,(1 − 2a) 2 − 4(a 2 − 1) = 0
⇔
1 − 4a +4a 2 − 4a 2 +4=0 ⇔ 4a =5,soa = 5 4 . The center of the circle is 0, 5 4
.
11. We can assume without loss of generality that θ =0at time t =0,sothatθ =12πt rad. [The angular velocity of the wheel
is 360 rpm =360· (2π rad)/(60 s) =12π rad/s.] Then the position of A as a function of time is
A =(40cosθ, 40 sin θ) = (40 cos 12πt, 40 sin 12πt),sosin α =
y 40 sin θ
= = sin θ = 1 sin 12πt.
1.2 m 120 3 3
(a) Differentiating the expression for sin α,wegetcos α · dα
dt = 1 3 · 12π · cos 12πt =4π cos θ. Whenθ = π 3 ,wehave
sin α = 1 √
√ 2
3
3 11
3 sin θ = 6 ,socos α = dα
1 − = and
6 12 dt = 4π cos π 3
cos α = 2π
= 4π √ 3
√ ≈ 6.56 rad/s.
11/12 11

F.
TX.10
CHAPTER 3 PROBLEMS PLUS ¤ 141
(b) By the Law of Cosines, |AP | 2 = |OA| 2 + |OP| 2 − 2 |OA||OP| cos θ
⇒
120 2 =40 2 + |OP| 2 − 2 · 40 |OP| cos θ ⇒ |OP| 2 − (80 cos θ) |OP| − 12,800 = 0 ⇒
|OP| = 1 2
80 cos θ ±
√ 6400 cos2 θ +51,200 =40cosθ ± 40 √ cos 2 θ +8=40 cos θ + √ 8+cos 2 θ cm
[since |OP| > 0]. As a check, note that |OP| =160cm when θ =0and |OP| =80 √ 2 cm when θ = π 2 .
(c) By part (b), the x-coordinate of P is given by x =40 cos θ + √ 8+cos 2 θ ,so
dx
dt = dx
dθ
dθ
dt =40 − sin θ −
2cosθ sin θ
2 √ · 12π = −480π sin θ 1+
8+cos 2 θ
In particular, dx/dt =0cm/swhenθ =0and dx/dt = −480π cm/swhenθ = π 2 .
13. Consider the statement that
d
dx (eax sin bx) =ae ax sin bx + be ax cos bx,and
d n
dx n (eax sin bx) =r n e ax sin(bx + nθ). Forn =1,
cos θ
√ cm/s.
8+cos2 θ
re ax sin(bx + θ) =re ax [sin bx cos θ +cosbx sin θ] =re ax a
r sin bx + b r cos bx
= ae ax sin bx + be ax cos bx
since tan θ = b a
⇒ sin θ = b r and cos θ = a . So the statement is true for n =1.
r
But
Assume it is true for n = k. Then
d k+1
dx k+1 (eax sin bx)= d
r k e ax sin(bx + kθ) = r k ae ax sin(bx + kθ)+r k e ax b cos(bx + kθ)
dx
= r k e ax [a sin(bx + kθ)+b cos(bx + kθ)]
sin[bx +(k +1)θ] =sin[(bx + kθ)+θ] =sin(bx + kθ)cosθ +sinθ cos(bx + kθ) = a sin(bx + kθ)+ b cos(bx + kθ).
r r
Hence, a sin(bx + kθ)+b cos(bx + kθ) =r sin[bx +(k +1)θ].So
d k+1
dx k+1 (eax sin bx) =r k e ax [a sin(bx+kθ)+b cos(bx+kθ)] = r k e ax [r sin(bx+(k +1)θ)] = r k+1 e ax [sin(bx+(k +1)θ)].
Therefore, the statement is true for all n by mathematical induction.
15. It seems from the figure that as P approaches the point (0, 2) from the right, x T →∞and y T → 2 + .AsP approaches the
point (3, 0) from the left, it appears that x T → 3 + and y T →∞.Soweguessthatx T ∈ (3, ∞) and y T ∈ (2, ∞). Itis
more difficult to estimate the range of values for x N and y N. We might perhaps guess that x N ∈ (0, 3),
and y N ∈ (−∞, 0) or (−2, 0).
In order to actually solve the problem, we implicitly differentiate the equation of the ellipse to find the equation of the
tangent line: x2
9 + y2
4 =1 ⇒ 2x 9 + 2y 4 y0 =0,soy 0 = − 4 9
x
. So at the point (x0,y0) on the ellipse, an equation of the
y

F.
142 ¤ CHAPTER 3 PROBLEMS PLUS
TX.10
tangent line is y − y 0 = − 4 x 0
(x − x 0) or 4x 0x +9y 0y =4x 2 0 +9y
9 y
0. 2 This can be written as x 0x
0 9 + y 0y
4 = x2 0
9 + y2 0
4 =1,
because (x 0 ,y 0 ) lies on the ellipse. So an equation of the tangent line is x 0x
9 + y 0y
4 =1.
Therefore, the x-intercept x T for the tangent line is given by x 0x T
9
=1 ⇔ x T = 9 x 0
,andthey-intercept y T is given
by y0yT
4
=1 ⇔ y T = 4 y 0
.
So as x 0 takes on all values in (0, 3), x T takes on all values in (3, ∞),andasy 0 takes on all values in (0, 2), y T takes on
all values in (2, ∞).
1
At the point (x 0,y 0) on the ellipse, the slope of the normal line is −
y 0 (x = 9 ,andits
0,y 0) 4 x 0
y 0
y 0
equation is y − y 0 = 9 (x − x 0). Sothex-intercept x N for the normal line is given by 0 − y 0 = 9 (x N − x 0)
4 x 0 4 x 0
y 0
⇒
x N = − 4x0
9 + x 0 = 5x0
9 ,andthey-intercept y N is given by y N − y 0 = 9 4
y 0
(0 − x 0 ) ⇒ y N = − 9y0
x 0 4 + y 0 = − 5y0
4 .
So as x 0 takes on all values in (0, 3), x N takes on all values in 0, 5 3
,andasy0 takes on all values in (0, 2), y N takes on
all values in − 5 2 , 0 .
17. (a) If the two lines L 1 and L 2 have slopes m 1 and m 2 and angles of
inclination φ 1 and φ 2 ,thenm 1 =tanφ 1 and m 2 =tanφ 2 . The triangle
in the figure shows that φ 1 + α +(180 ◦ − φ 2 ) = 180 ◦ and so
α = φ 2 − φ 1 . Therefore, using the identity for tan(x − y),wehave
tan α =tan(φ 2 − φ 1 )= tan φ 2 − tan φ 1
m2 − m1
and so tan α = .
1+tanφ 2 tan φ 1 1+m 1m 2
(b) (i) The parabolas intersect when x 2 =(x − 2) 2 ⇒ x =1.Ify = x 2 ,theny 0 =2x, so the slope of the tangent
to y = x 2 at (1, 1) is m 1 =2(1)=2.Ify =(x − 2) 2 ,theny 0 =2(x − 2), so the slope of the tangent to
y =(x − 2) 2 at (1, 1) is m 2 =2(1− 2) = −2. Therefore, tan α =
so α =tan −1 4
3
≈ 53 ◦ [or 127 ◦ ].
m2 − m1
= −2 − 2
1+m 1m 2 1+2(−2) = 4 3 and
(ii) x 2 − y 2 =3and x 2 − 4x + y 2 +3=0intersect when x 2 − 4x +(x 2 − 3) + 3 = 0 ⇔ 2x(x − 2) = 0 ⇒
x =0or 2,but0 is extraneous. If x =2,theny = ±1.Ifx 2 − y 2 =3then 2x − 2yy 0 =0 ⇒ y 0 = x/y and
x 2 − 4x + y 2 +3=0 ⇒ 2x − 4+2yy 0 =0 ⇒ y 0 = 2 − x .At(2, 1) the slopes are m 1 =2and
y
m 2 =0,so tan α = 0 − 2
1+2· 0 = −2 ⇒ α ≈ 117◦ .At(2, −1) the slopes are m 1 = −2 and m 2 =0,
so tan α =
0 − (−2)
1+(−2) (0) =2 ⇒ α ≈ 63◦ [or 117 ◦ ].

F.
TX.10
CHAPTER 3 PROBLEMS PLUS ¤ 143
19. Since ∠ROQ = ∠OQP = θ, the triangle QOR is isosceles, so
|QR| = |RO| = x. By the Law of Cosines, x 2 = x 2 + r 2 − 2rx cos θ. Hence,
2rx cos θ = r 2 ,sox =
sin θ = y/r), and hence x →
r2
2r cos θ = r
2cosθ .Notethatasy → 0+ , θ → 0 + (since
r
2cos0 = r .Thus,asP is taken closer and closer
2
to the x-axis, the point R approaches the midpoint of the radius AO.
21. lim
x→0
sin(a +2x) − 2sin(a + x)+sina
x 2
= lim
x→0
sin a cos 2x +cosa sin 2x − 2sina cos x − 2cosa sin x +sina
x 2
= lim
x→0
sin a (cos 2x − 2cosx +1)+cosa (sin 2x − 2sinx)
x 2
= lim
x→0
sin a (2 cos 2 x − 1 − 2cosx +1)+cosa (2 sin x cos x − 2sinx)
x 2
= lim
x→0
sin a (2 cos x)(cos x − 1) + cos a (2 sin x)(cos x − 1)
x 2
2(cos x − 1)[sin a cos x +cosa sin x](cos x +1)
= lim
x→0 x 2 (cos x +1)
−2sin 2 2
x [sin(a + x)]
sin x sin(a + x)
= lim
= −2 lim ·
x→0 x 2 (cos x +1)
x→0 x cos x +1 = sin(a +0)
−2(1)2 cos 0 + 1 = − sin a
23. Let f(x) =e 2x and g(x) =k √ x [k >0]. From the graphs of f and g,
So we must have k √ a =
k
4 √ a
k =2e 1/2 =2 √ e ≈ 3.297.
25. y =
⇒
we see that f will intersect g exactly once when f and g share a tangent
line. Thus, we must have f = g and f 0 = g 0 at x = a.
f(a) =g(a) ⇒ e 2a = k √ a ()
and f 0 (a) =g 0 (a) ⇒ 2e 2a = k
2 √ ⇒ e 2a = k
a
4 √ a .
√ 2 k a = ⇒ a = 1
4k
.From(), 4 e2(1/4) = k 1/4 ⇒
x
√
a2 − 1 − 2
√
a2 − 1 arctan sin x
a + √ a 2 − 1+cosx .Letk = a + √ a 2 − 1. Then
y 0 =
=
1
√
a2 − 1 − 2
√
a2 − 1 ·
1
1+sin 2 x/(k +cosx) · cos x(k +cosx)+sin2 x
2 (k +cosx) 2
1
√
a2 − 1 − 2
√
a2 − 1 · k cos x +cos2 x +sin 2 x
(k +cosx) 2 +sin 2 x = 1
√
a2 − 1 − 2
√
a2 − 1 · k cos x +1
k 2 +2k cos x +1
= k2 +2k cos x +1− 2k cos x − 2
√
a2 − 1(k 2 +2k cos x +1)
=
k 2 − 1
√
a2 − 1(k 2 +2k cos x +1)
But k 2 =2a 2 +2a √ a 2 − 1 − 1=2a a + √ a 2 − 1 − 1=2ak − 1,sok 2 +1=2ak,andk 2 − 1=2(ak − 1).
[continued]

F.
144 ¤ CHAPTER 3 PROBLEMS PLUS
TX.10
So y 0 =
2(ak − 1)
√
a2 − 1(2ak +2k cos x) = ak − 1
√
a2 − 1k (a +cosx) .Butak − 1=a2 + a √ a 2 − 1 − 1=k √ a 2 − 1,
so y 0 =1/(a +cosx).
27. y = x 4 − 2x 2 − x ⇒ y 0 =4x 3 − 4x − 1. The equation of the tangent line at x = a is
y − (a 4 − 2a 2 − a) =(4a 3 − 4a − 1)(x − a) or y =(4a 3 − 4a − 1)x +(−3a 4 +2a 2 ) and similarly for x = b. Soifat
x = a and x = b we have the same tangent line, then 4a 3 − 4a − 1=4b 3 − 4b − 1 and −3a 4 +2a 2 = −3b 4 +2b 2 .Thefirst
equation gives a 3 − b 3 = a − b ⇒ (a − b)(a 2 + ab + b 2 )=(a − b). Assuming a 6=b, wehave1=a 2 + ab + b 2 .
The second equation gives 3(a 4 − b 4 )=2(a 2 − b 2 ) ⇒ 3(a 2 − b 2 )(a 2 + b 2 )=2(a 2 − b 2 ) which is true if a = −b.
Substituting into 1=a 2 + ab + b 2 gives 1=a 2 − a 2 + a 2 ⇒ a = ±1 so that a =1and b = −1 or vice versa. Thus,
the points (1, −2) and (−1, 0) have a common tangent line.
As long as there are only two such points, we are done. So we show that these are in fact the only two such points.
Suppose that a 2 − b 2 6=0.Then3(a 2 − b 2 )(a 2 + b 2 )=2(a 2 − b 2 ) gives 3(a 2 + b 2 )=2 or a 2 + b 2 = 2 3 .
Thus, ab =(a 2 + ab + b 2 ) − (a 2 + b 2 )=1− 2 3 = 1 3 ,sob = 1
3a . Hence, a2 + 1
9a 2 = 2 3 ,so9a4 +1=6a 2
0=9a 4 − 6a 2 +1=(3a 2 − 1) 2 .So3a 2 − 1=0 ⇒ a 2 = 1 3
that a 2 6=b 2 .
⇒ b 2 = 1
9a 2 = 1 3 = a2 , contradicting our assumption
29. Because of the periodic nature of the lattice points, it suffices to consider the points in the 5 × 2 grid shown. We can see that
the minimum value of r occurs when there is a line with slope 2 which touches the circle centered at (3, 1) and the circles
5
centered at (0, 0) and (5, 2).
⇒
To find P , the point at which the line is tangent to the circle at (0, 0), we simultaneously solve x 2 + y 2 = r 2 and
y = − 5 2 x ⇒ x2 + 25 4 x2 = r 2 ⇒ x 2 = 4 29 r2 ⇒ x = 2 √
29
r, y = − 5 √
29
r.Tofind Q, we either use symmetry or
solve (x − 3) 2 +(y − 1) 2 = r 2 and y − 1=− 5 2
2
(x − 3).Asabove,wegetx =3− √
29
r, y =1+ √ 5
29
r. Now the slope of
the line PQis 2 5 ,som PQ =
1+ √ 5
29
r − − √ 5
29
r
3 − 2 √
29
r − 2 √
29
r
=
10 √
1+ √
29
r 29 + 10r
3 − √ 4
29
r = 3 √ 29 − 4r = 2 5
5 √ 29 + 50r =6 √ 29 − 8r ⇔ 58r = √ 29 ⇔ r = √ 29
58 . So the minimum value of r for which any line with slope 2 5
intersects circles with radius r centered at the lattice points on the plane is r = √ 29
58 ≈ 0.093.
⇒

F.
TX.10
31. By similar triangles, r 5 = h 16
⇒
CHAPTER 3 PROBLEMS PLUS ¤ 145
r = 5h . The volume of the cone is
16
2 5h
V = 1 3 πr2 h = 1 π h = 25π
3
16 768 h3 ,so dV
dt = 25π dh
256 h2 dt .Nowtherateof
change of the volume is also equal to the difference of what is being added
(2 cm 3 /min) and what is oozing out (kπrl,whereπrl is the area of the cone and k
Equating the two expressions for dV
dt
is a proportionality constant). Thus, dV
dt
and substituting h =10,
dh
dt
=2− kπrl.
= −0.3, r =
5(10)
16
= 25 8 ,and l
√
281
= 10
16
⇔
l = 5 8
√ 25π
281,weget
256 (10)2 (−0.3) = 2 − kπ 25
8 · 5 √ 125kπ √ 281
281 ⇔
8
64
=2+ 750π . Solving for k gives us
256
256 + 375π
k =
250π √ . To maintain a certain height, the rate of oozing, kπrl, must equal the rate of the liquid being poured in;
281
that is, dV =0. Thus, the rate at which we should pour the liquid into the container is
dt
kπrl =
256 + 375π
250π √ 281 · π · 25 8 · 5 √ 281 256 + 375π
= ≈ 11.204 cm 3 /min
8 128

F.
TX.10

F.
TX.10
4 APPLICATIONS OF DIFFERENTIATION
4.1 Maximum and Minimum Values
1. Afunctionf has an absolute minimum at x = c if f(c) is the smallest function value on the entire domain of f, whereas
f has a local minimum at c if f(c) is the smallest function value when x is near c.
3. Absolute maximum at s, absolute minimum at r,localmaximumatc, local minima at b and r, neither a maximum nor a
minimum at a and d.
5. Absolute maximum value is f(4) = 5; there is no absolute minimum value; local maximum values are f(4) = 5 and
f(6) = 4; local minimum values are f(2) = 2 and f(1) = f(5) = 3.
7. Absolute minimum at 2, absolute maximum at 3,
local minimum at 4
9. Absolute maximum at 5, absolute minimum at 2,
local maximum at 3, local minima at 2 and 4
11. (a) (b) (c)
13. (a) Note: By the Extreme Value Theorem,
f must not be continuous; because if it
were, it would attain an absolute
minimum.
(b)
147

F.
148 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
15. f(x) =8− 3x, x ≥ 1. Absolute maximum f(1) = 5;no
local maximum. No absolute or local minimum.
TX.10
17. f(x) =x 2 , 0

F.
TX.10
SECTION 4.1 MAXIMUM AND MINIMUM VALUES ¤ 149
29. f(x) =5x 2 +4x ⇒ f 0 (x) =10x +4. f 0 (x) =0 ⇒ x = − 2 ,so− 2 5 5
is the only critical number.
31. f(x) =x 3 +3x 2 − 24x ⇒ f 0 (x) =3x 2 +6x − 24 = 3(x 2 +2x − 8).
f 0 (x) =0 ⇒ 3(x +4)(x − 2) = 0 ⇒ x = −4, 2. These are the only critical numbers.
33. s(t) =3t 4 +4t 3 − 6t 2 ⇒ s 0 (t) =12t 3 +12t 2 − 12t. s 0 (t) =0 ⇒ 12t(t 2 + t − 1) ⇒
t =0 or t 2 + t − 1=0. Using the quadratic formula to solve the latter equation gives us
t = −1 ± 1 2 − 4(1)(−1)
2(1)
= −1 ± √ 5
2
≈ 0.618, −1.618. The three critical numbers are 0, −1 ± √ 5
.
2
35. g(y) =
y − 1
y 2 − y +1
⇒
g 0 (y) = (y2 − y +1)(1)− (y − 1)(2y − 1)
= y2 − y +1− (2y 2 − 3y +1)
= −y2 +2y y(2 − y)
=
(y 2 − y +1) 2 (y 2 − y +1) 2 (y 2 − y +1)
2
(y 2 − y +1) . 2
g 0 (y) =0 ⇒ y =0, 2. The expression y 2 − y +1is never equal to 0,sog 0 (y) exists for all real numbers.
The critical numbers are 0 and 2.
37. h(t) =t 3/4 − 2t 1/4 ⇒ h 0 (t) = 3 4 t−1/4 − 2 4 t−3/4 = 1 4 t−3/4 (3t 1/2 − 2) = 3 √ t − 2
4 4√ t 3 .
h 0 (t) =0 ⇒ 3 √ t =2 ⇒ √ t = 2 3
⇒ t = 4 9 . h0 (t) does not exist at t =0, so the critical numbers are 0 and 4 9 .
39. F (x) =x 4/5 (x − 4) 2 ⇒
F 0 (x) =x 4/5 · 2(x − 4) + (x − 4) 2 · 4
5 x−1/5 = 1 5 x−1/5 (x − 4)[5 · x · 2+(x − 4) · 4]
=
(x − 4)(14x − 16) 2(x − 4)(7x − 8)
=
5x 1/5 5x 1/5
F 0 (x) =0 ⇒ x =4, 8 7 . F 0 (0) does not exist. Thus, the three critical numbers are 0, 8 7 ,and4.
41. f(θ) =2cosθ +sin 2 θ ⇒ f 0 (θ) =−2sinθ +2sinθ cos θ. f 0 (θ) =0 ⇒ 2sinθ (cos θ − 1) = 0 ⇒ sin θ =0
or cos θ =1 ⇒ θ = nπ [n an integer] or θ =2nπ. The solutions θ = nπ include the solutions θ =2nπ, so the critical
numbers are θ = nπ.
43. f(x) =x 2 e −3x ⇒ f 0 (x) =x 2 (−3e −3x )+e −3x (2x) =xe −3x (−3x +2). f 0 (x) =0 ⇒ x =0, 2 3
[e −3x is never equal to 0]. f 0 (x) always exists, so the critical numbers are 0 and 2 3 .
45. The graph of f 0 (x) =5e −0.1|x| sin x − 1 has 10 zeros and exists
everywhere, so f has 10 critical numbers.

F.
150 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
47. f(x) =3x 2 − 12x +5, [0, 3]. f 0 (x) =6x − 12 = 0 ⇔ x =2. Applying the Closed Interval Method, we find that
f(0) = 5, f(2) = −7,andf(3) = −4. Sof(0) = 5 is the absolute maximum value and f(2) = −7 is the absolute minimum
value.
49. f(x) =2x 3 − 3x 2 − 12x +1, [−2, 3]. f 0 (x) =6x 2 − 6x − 12 = 6(x 2 − x − 2) = 6(x − 2)(x +1)=0 ⇔
x =2, −1.
f(−2) = −3, f(−1) = 8, f(2) = −19,andf(3) = −8. Sof(−1) = 8 is the absolute maximum value and
f(2) = −19 is the absolute minimum value.
51. f(x) =x 4 − 2x 2 +3, [−2, 3]. f 0 (x) =4x 3 − 4x =4x(x 2 − 1) = 4x(x +1)(x − 1) = 0 ⇔ x = −1, 0, 1.
f(−2) = 11, f(−1) = 2, f(0) = 3, f(1) = 2, f(3) = 66. Sof(3) = 66 is the absolute maximum value and f(±1) = 2 is
the absolute minimum value.
53. f(x) = x , [0, 2].
x 2 +1 f
0 (x) = (x2 +1)− x(2x)
= 1 − x2
=0 ⇔ x = ±1,but−1 is not in [0, 2]. f(0) = 0,
(x 2 +1) 2 (x 2 +1)
2
f(1) = 1 , f(2) = 2 .Sof(1) = 1 2 5 2
is the absolute maximum value and f(0) = 0 is the absolute minimum value.
55. f(t) =t √ 4 − t 2 , [−1, 2].
f 0 (t) =t · 1 (4 − 2 t2 ) −1/2 (−2t)+(4− t 2 ) 1/2 · 1= √ −t2 + √ 4 − t 2 = −t2 +(4− t 2 )
√ = √ 4 − 2t2 .
4 − t
2 4 − t
2 4 − t
2
f 0 (t) =0 ⇒ 4 − 2t 2 =0 ⇒ t 2 =2 ⇒ t = ± √ 2,butt = − √ 2 is not in the given interval, [−1, 2].
f 0 (t) does not exist if 4 − t 2 =0 ⇒ t = ±2, but−2 is not in the given interval. f(−1) = − √ 3, f √ 2 =2,and
f(2) = 0. Sof √ 2 =2is the absolute maximum value and f(−1) = − √ 3 is the absolute minimum value.
57. f(t) =2cost +sin2t, [0, π/2].
f 0 (t) =−2sint +cos2t · 2=−2sint +2(1− 2sin 2 t)=−2(2 sin 2 t +sint − 1) = −2(2 sin t − 1)(sin t +1).
f 0 (t) =0 ⇒ sin t = 1 or sin t = −1 ⇒ t = π . f(0) = 2, f( π )=√ √ √
2 6 6
3+ 1 2 3=
3
2 3 ≈ 2.60,andf(
π
)=0. 2
So f( π )= √ 3
6 2 3 is the absolute maximum value and f(
π
2
)=0is the absolute minimum value.
59. f(x) =xe −x2 /8 , [−1, 4]. f 0 (x) =x · e −x2 /8 · (− x 4 )+e−x2 /8 · 1=e −x2 /8 (− x2
4 +1).Sincee−x2 /8 is never 0,
f 0 (x) =0 ⇒ −x 2 /4+1=0 ⇒ 1=x 2 /4 ⇒ x 2 =4 ⇒ x = ±2,but−2 is not in the given interval, [−1, 4].
f(−1) = −e −1/8 ≈−0.88, f(2) = 2e −1/2 ≈ 1.21,andf(4) = 4e −2 ≈ 0.54. Sof(2) = 2e −1/2 is the absolute maximum
value and f(−1) = −e −1/8 is the absolute minimum value.
61. f(x) =ln(x 2 + x +1), [−1, 1]. f 0 1
(x) =
x 2 + x +1 · (2x +1)=0 ⇔ x = − 1 2 .Sincex2 + x +1> 0 for all x,the
domain of f and f 0 is R. f(−1)=ln1=0, f
− 1 2 =ln
3
≈ −0.29,andf(1) = ln 3 ≈ 1.10. Sof(1) = ln 3 ≈ 1.10 is
4
the absolute maximum value and f
− 1 2 =ln
3
4
≈ −0.29 is the absolute minimum value.

F.
TX.10
SECTION 4.1 MAXIMUM AND MINIMUM VALUES ¤ 151
63. f(x) =x a (1 − x) b , 0 ≤ x ≤ 1, a>0, b>0.
f 0 (x) =x a · b(1 − x) b−1 (−1) + (1 − x) b · ax a−1 = x a−1 (1 − x) b−1 [x · b(−1) + (1 − x) · a]
= x a−1 (1 − x) b−1 (a − ax − bx)
At the endpoints, we have f(0) = f(1) = 0 [the minimum value of f ]. In the interval (0, 1), f 0 (x) =0 ⇔ x = a
a + b .
f
a
a + b
=
a
So f
a + b
a
a + b
a
1 − a b
=
a + b
a a
b a + b − a
=
(a + b) a a + b
a a b b
=
is the absolute maximum value.
a+b
(a + b)
a a
(a + b) a ·
b b
(a + b) = a a b b
b (a + b) . a+b
65. (a) From the graph, it appears that the absolute maximum value is about
f(−0.77) = 2.19, and the absolute minimum value is about
f(0.77) = 1.81.
(b) f(x) =x 5 − x 3 +2 ⇒ f 0 (x) =5x 4 − 3x 2 = x 2 (5x 2 − 3). Sof 0 3
(x) =0 ⇒ x =0, ± . 5
5 3
3
3
f − = −
5
5 −
3
−
5 +2=− 3
2 3
+ 3 3
+2= 3
−
9 3
+2= 6 3
+2(maximum)
5 5 5 5 5 25 5 25 5
and similarly, f
3
5
= − 6 25
3
5 +2(minimum).
67. (a) From the graph, it appears that the absolute maximum value is about
f(0.75) = 0.32, and the absolute minimum value is f(0) = f(1) = 0;
that is, at both endpoints.
(b) f(x) =x √ x − x 2 ⇒ f 0 1 − 2x
(x) =x ·
2 √ x − x + √ x − x 2 = (x − 2x2 )+(2x − 2x 2 )
2 2 √ 3x − 4x2
=
x − x 2 2 √ x − x . 2
So f 0 (x) =0 ⇒ 3x − 4x 2 =0 ⇒ x(3 − 4x) =0 ⇒ x =0or 3 . 4
f(0) = f(1) = 0 (minimum), and f
3
4 =
3
4
3
4 − 3
4
2
= 3 4
3
= 3 √ 3
(maximum).
16 16
69. The density is defined as ρ = mass
volume = 1000
V (T ) (in g/cm3 ). But a critical point of ρ will also be a critical point of V
[since dρ
dT = −1000V −2 dV
dT
and V is never 0],andV is easier to differentiate than ρ.
V (T )=999.87 − 0.06426T +0.0085043T 2 − 0.0000679T 3 ⇒ V 0 (T )=−0.06426 + 0.0170086T − 0.0002037T 2 .
Setting this equal to 0 and using the quadratic formula to find T ,weget
T = −0.0170086 ± √ 0.0170086 2 − 4 · 0.0002037 · 0.06426
2(−0.0002037)
≈ 3.9665 ◦ Cor79.5318 ◦ C. Since we are only interested

F.
152 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
in the region 0 ◦ C ≤ T ≤ 30 ◦ C, we check the density ρ at the endpoints and at 3.9665 ◦ C: ρ(0) ≈ 1000
999.87 ≈ 1.00013;
ρ(30) ≈ 1000
1000
≈ 0.99625; ρ(3.9665) ≈ ≈ 1.000255. So water has its maximum density at
1003.7628 999.7447
about 3.9665 ◦ C.
71. Let a = −0.000 032 37, b =0.000 903 7, c = −0.008 956, d =0.03629, e = −0.04458, andf =0.4074.
Then S(t) =at 5 + bt 4 + ct 3 + dt 2 + et + f and S 0 (t) =5at 4 +4bt 3 +3ct 2 +2dt + e.
We now apply the Closed Interval Method to the continuous function S on the interval 0 ≤ t ≤ 10. SinceS 0 exists for all t,
the only critical numbers of S occur when S 0 (t) =0.Weusearootfinder on a CAS (or a graphing device) to find that
S 0 (t) =0when t 1 ≈ 0.855, t 2 ≈ 4.618, t 3 ≈ 7.292, andt 4 ≈ 9.570. The values of S at these critical numbers are
S(t 1) ≈ 0.39, S(t 2) ≈ 0.43645, S(t 3) ≈ 0.427,andS(t 4) ≈ 0.43641. The values of S at the endpoints of the interval are
S(0) ≈ 0.41 and S(10) ≈ 0.435. Comparing the six numbers, we see that sugar was most expensive at t 2 ≈ 4.618
(corresponding roughly to March 1998) and cheapest at t 1 ≈ 0.855 (June 1994).
73. (a) v(r) =k(r 0 − r)r 2 = kr 0 r 2 − kr 3 ⇒ v 0 (r) =2kr 0 r − 3kr 2 . v 0 (r) =0 ⇒ kr(2r 0 − 3r) =0 ⇒
r =0or 2 r 3 0 (but 0 is not in the interval). Evaluating v at 1 r 2 0, 2 r 3 0,andr 0 ,wegetv 1
r
2 0 =
1
8 kr3 0, v 2
r
3 0 =
4
27 kr3 0,
and v(r 0 )=0.Since 4 27 > 1 8 , v attains its maximum value at r = 2 3 r 0. This supports the statement in the text.
(b) From part (a), the maximum value of v is 4 27 kr3 0.
(c)
75. f(x) =x 101 + x 51 + x +1 ⇒ f 0 (x) =101x 100 +51x 50 +1≥ 1 for all x,sof 0 (x) =0has no solution. Thus, f(x)
has no critical number, so f(x) can have no local maximum or minimum.
77. If f has a local minimum at c,theng(x) =−f(x) has a local maximum at c,sog 0 (c) =0by the case of Fermat’s Theorem
proved in the text. Thus, f 0 (c) =−g 0 (c) =0.
4.2 The Mean Value Theorem
1. f(x) =5− 12x +3x 2 , [1, 3]. Sincef is a polynomial, it is continuous and differentiable on R, so it is continuous on [1, 3]
and differentiable on (1, 3). Alsof(1) = −4 =f(3). f 0 (c) =0 ⇔ −12 + 6c =0 ⇔ c =2, which is in the open
interval (1, 3),soc =2satisfies the conclusion of Rolle’s Theorem.

F.
TX.10
SECTION 4.2 THE MEAN VALUE THEOREM ¤ 153
3. f(x) = √ x − 1 3
x, [0, 9]. f, being the difference of a root function and a polynomial, is continuous and differentiable
on [0, ∞), so it is continuous on [0, 9] and differentiable on (0, 9). Also,f(0) = 0 = f(9). f 0 (c) =0
⇔
1
2 √ c − 1 3 =0 ⇔ 2 √ c =3 ⇔ √ c = 3 2
conclusion of Rolle’s Theorem.
⇒
c = 9 4 , which is in the open interval (0, 9),soc = 9 satisfies the
4
5. f(x) =1− x 2/3 . f(−1) = 1 − (−1) 2/3 =1− 1=0=f(1). f 0 (x) =− 2 3 x−1/3 ,sof 0 (c) =0has no solution. This
does not contradict Rolle’s Theorem, since f 0 (0) does not exist, and so f is not differentiable on (−1, 1).
7.
f(8) − f(0)
8 − 0
= 6 − 4
8
= 1 4 . The values of c which satisfy f 0 (c) = 1 seem to be about c =0.8, 3.2, 4.4,and6.1.
4
9. (a), (b) The equation of the secant line is
(c) f(x) =x +4/x ⇒ f 0 (x) =1− 4/x 2 .
y − 5= 8.5 − 5
8 − 1 (x − 1) ⇔ y = 1 x + 9 . So f 0 (c) = 1 2
⇒ c 2 =8 ⇒ c =2 √ 2,and
2 2
f(c) =2 √ 2+ 4
2 √ =3√ 2. Thus, an equation of the
2
tangent line is y − 3 √ √
2= 1 2 x − 2 2 ⇔
y = 1 2 x +2√ 2.
11. f(x) =3x 2 +2x +5, [−1, 1]. f is continuous on [−1, 1] and differentiable on (−1, 1) since polynomials are continuous
and differentiable on R.
c =0, which is in (−1, 1).
f 0 (c) =
f(b) − f(a)
b − a
⇔
6c +2=
f(1) − f(−1)
1 − (−1)
= 10 − 6
2
=2 ⇔ 6c =0 ⇔
13. f(x) =e −2x , [0, 3]. f is continuous and differentiable on R, so it is continuous on [0, 3] and differentiable on (0, 3).
f 0 f(b) − f(a)
(c) = ⇔ −2e −2c = e−6 − e 0
⇔ e −2c = 1 − e−6
1 − e
−6
⇔ −2c =ln
⇔
b − a
3 − 0
6
6
c = − 1 1 − e
−6
2 ln ≈ 0.897, which is in (0, 3).
6

F.
154 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
15. f(x) =(x − 3) −2 ⇒ f 0 (x) =−2(x − 3) −3 . f(4) − f(1) = f 0 (c)(4 − 1) ⇒ 1 1 − 1
2 (−2) = −2
2 (c − 3) · 3 ⇒
3
3
4 = −6 ⇒ (c − 3) 3 = −8 ⇒ c − 3=−2 ⇒ c =1, which is not in the open interval (1, 4). This does not
(c − 3) 3
contradict the Mean Value Theorem since f is not continuous at x =3.
17. Let f(x) =1+2x + x 3 +4x 5 .Thenf(−1) = −6 < 0 and f(0) = 1 > 0. Since f is a polynomial, it is continuous, so the
Intermediate Value Theorem says that there is a number c between −1 and 0 such that f(c) =0. Thus, the given equation has
a real root. Suppose the equation has distinct real roots a and b with a

F.
TX.10 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH ¤ 155
27. We use Exercise 26 with f(x) = √ 1+x, g(x) =1+ 1 2
x,anda =0.Noticethatf(0) = 1 = g(0) and
f 0 (x) =
1
2 √ 1+x < 1 2 = g0 (x) for x>0.SobyExercise26,f(b)

F.
156 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
5. (a) Since f 0 (x) > 0 on (1, 5), f is increasing on this interval. Since f 0 (x) < 0 on (0, 1) and (5, 6), f is decreasing on these
intervals.
(b) Since f 0 (x) =0at x =1and f 0 changes from negative to positive there, f changes from decreasing to increasing and has
a local minimum at x =1.Sincef 0 (x) =0at x =5and f 0 changes from positive to negative there, f changes from
increasing to decreasing and has a local maximum at x =5.
7. There is an inflection point at x =1because f 00 (x) changes from negative to positive there, and so the graph of f changes
from concave downward to concave upward. There is an inflection point at x =7because f 00 (x) changes from positive to
negative there, and so the graph of f changes from concave upward to concave downward.
9. (a) f(x) =2x 3 +3x 2 − 36x ⇒ f 0 (x) =6x 2 +6x − 36 = 6(x 2 + x − 6) = 6(x +3)(x − 2).
We don’t need to include the “6”inthecharttodeterminethesignoff 0 (x).
Interval x +3 x − 2 f 0 (x) f
x− 1 2 ,and
f 00 (x) < 0 ⇔ x

F.
TX.10 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH ¤ 157
13. (a) f(x) =sinx +cosx, 0 ≤ x ≤ 2π. f 0 (x) =cosx − sin x =0 ⇒ cos x =sinx ⇒ 1= sin x
cos x
tan x =1 ⇒ x = π 4 or 5π 4 . Thus, f 0 (x) > 0 ⇔ cos x − sin x>0 ⇔ cos x>sin x ⇔ 0 ln 1 2
⇔
x> 1 (ln 1 − ln 2) ⇔ x>− 1 ln 2 [≈ −0.23] andf 0 3 3
(x) < 0 if x 0 [the sum of two positive terms].
point of inflection.
Thus, f is concave upward on (−∞, ∞) and there is no
17. (a) y = f(x) = ln x √
x
.(Notethatf is only defined for x>0.)
f 0 (x) =
√
x (1/x) − ln x
1
2 x−1/2
x
=
1
√ − ln x
x 2 √ x
· 2 √ x
x 2 √ x = 2 − ln x > 0 ⇔ 2 − ln x>0 ⇔
2x 3/2
ln xe 8/3 ,sof is concave upward on (e 8/3 , ∞) and
concave downward on (0,e 8/3 ).Thereisaninflection point at
e 8/3 , 8 3 e−4/3 ≈ (14.39, 0.70).

F.
158 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
19. f(x) =x 5 − 5x +3 ⇒ f 0 (x) =5x 4 − 5=5(x 2 +1)(x +1)(x − 1).
First Derivative Test: f 0 (x) < 0 ⇒ −1 1 or x 0 ⇒ f(1) = −1 is a local minimum value.
Preference: For this function, the two tests are equally easy.
21. f(x) =x + √ 1 − x ⇒ f 0 (x) =1+ 1 2 (1 − x)−1/2 (−1) = 1 −
1
2 √ .Notethatf is defined for 1 − x ≥ 0;thatis,
1 − x
for x ≤ 1. f 0 (x) =0 ⇒ 2 √ 1 − x =1 ⇒ √ 1 − x = 1 2
⇒ 1 − x = 1 4
⇒ x = 3 4 . f 0 does not exist at x =1,
but we can’t have a local maximum or minimum at an endpoint.
First Derivative Test: f 0 (x) > 0 ⇒ x< 3 and f 0 (x) < 0 ⇒ 3

F.
TX.10 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH ¤ 159
29. The function must be always decreasing (since the first derivative is always negative)
and concave downward (since the second derivative is always negative).
31. (a) f is increasing where f 0 is positive, that is, on (0, 2), (4, 6),and(8, ∞); and decreasing where f 0 is negative, that is,
on (2, 4) and (6, 8).
(b) f has local maxima where f 0 changes from positive to negative, at x =2and at x =6, and local minima where f 0 changes
from negative to positive, at x =4and at x =8.
(c) f is concave upward (CU) where f 0 is increasing, that is, on (3, 6) and (6, ∞),
(e)
and concave downward (CD) where f 0 is decreasing, that is, on (0, 3).
(d) There is a point of inflection where f changes from being CD to being CU, that
is, at x =3.
33. (a) f(x) =2x 3 − 3x 2 − 12x ⇒ f 0 (x) =6x 2 − 6x − 12 = 6(x 2 − x − 2) = 6(x − 2)(x +1).
f 0 (x) > 0 ⇔ x2 and f 0 (x) < 0 ⇔ −1 0 ⇔ x

F.
160 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
(c) f 00 (x) =4− 12x 2 =4(1− 3x 2 ). f 00 (x) =0 ⇔ 1 − 3x 2 =0 ⇔
(d)
x 2 = 1 3
⇔ x = ±1/ √ 3. f 00 (x) > 0 on −1/ √ 3, 1/ √ 3 and f 00 (x) < 0
on −∞, −1/ √ 3 and 1/ √ 3, ∞ .Sof is concave upward on
−1/
√
3, 1/
√
3
and f is concave downward on
−∞, −1/
√
3
and
√ √ 1/ 3, ∞ . f ±1/ 3 =2+
2
− 1 = 23 . There are points of inflection
3 9 9
at ±1/ √
3, 23
9 .
37. (a) h(x) =(x +1) 5 − 5x − 2 ⇒ h 0 (x) =5(x +1) 4 − 5. h 0 (x) =0 ⇔ 5(x +1) 4 =5 ⇔ (x +1) 4 =1 ⇒
(x +1) 2 =1 ⇒ x +1=1or x +1=−1 ⇒ x =0or x = −2. h 0 (x) > 0 ⇔ x0 and
h 0 (x) < 0 ⇔ −2 −1 and
h 00 (x) < 0 ⇔ x 0 for x>−2 and A 0 (x) < 0 for −3 −3,soA is concave upward on (−3, ∞). Thereisnoinflection point.
41. (a) C(x) =x 1/3 (x +4)=x 4/3 +4x 1/3 ⇒ C 0 (x) = 4 3 x1/3 + 4 3 x−2/3 = 4 3 x−2/3 (x +1)=
4(x +1)
3 3√ x 2 . C 0 (x) > 0 if
−1

F.
TX.10 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH ¤ 161
(b) C(−1) = −3 is a local minimum value.
(c) C 00 (x) = 4 9 x−2/3 − 8 9 x−5/3 = 4 9 x−5/3 (x − 2) =
4(x − 2)
9 3√ x 5 .
C 00 (x) < 0 for 0 0 ⇔ π

F.
162 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
(d) f 00 (x) = (x2 − 1) 2 (−2) − (−2x) · 2(x 2 − 1)(2x)
[(x 2 − 1) 2 ] 2
= 2(x2 − 1)[−(x 2 − 1) + 4x 2 ]
(x 2 − 1) 4 = 2(3x2 +1)
(x 2 − 1) 3 .
The sign of f 00 (x) is determined by the denominator; that is, f 00 (x) > 0 if
|x| > 1 and f 00 (x) < 0 if |x| < 1. Thus,f is CU on (−∞, −1) and (1, ∞),
and f is CD on (−1, 1). Therearenoinflection points.
(e)
√
47. (a) lim x2 +1− x = ∞ and
x→−∞
√
lim x2 +1− x √
= lim x2 +1− x √ x 2 +1+x
√
x→∞
x→∞
x2 +1+x = lim 1
√ =0,soy =0is a HA.
x→∞ x2 +1+x
(b) f(x) = √ x 2 +1− x ⇒ f 0 x
x
(x) = √ − 1. Since√ x2 +1 x2 +1 < 1 for all x, f 0 (x) < 0,sof is decreasing on R.
(c) No minimum or maximum
(d) f 00 (x) = (x2 +1) 1/2 (1) − x · 1
2 (x2 +1) −1/2 (2x)
√
x2 +1 2
(e)
(x 2 +1) 1/2 x 2
−
(x
=
2 +1) 1/2
x 2 +1
so f is CU on R. NoIP
= (x2 +1)− x 2
(x 2 +1) 3/2 =
1
> 0,
(x 2 +1)
3/2
49. f(x) =ln(1− ln x) is defined when x>0 (so that ln x is defined) and 1 − ln x>0 [so that ln(1 − ln x) is defined].
The second condition is equivalent to 1 > ln x ⇔ x 0 ⇔ ln x

F.
TX.10 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH ¤ 163
(d) f 00 (x) = (x +1)2 e −1/(x+1) 1/(x +1) 2 − e −1/(x+1) [2(x +1)]
[(x +1) 2 ] 2
= e−1/(x+1) [1 − (2x +2)]
(x +1) 4 = − e−1/(x+1) (2x +1)
(x +1) 4 ⇒
(e)
f 00 (x) > 0 ⇔ 2x +1< 0 ⇔ x 0 ⇒ (x − 3) 5 > 0 ⇒ x − 3 > 0 ⇒ x>3. Thus,f is increasing on the interval (3, ∞).
55. (a) From the graph, we get an estimate of f(1) ≈ 1.41 as a local maximum
value, and no local minimum value.
f(x) = x +1 √
x2 +1
⇒ f 0 (x) =
1 − x
(x 2 +1) 3/2 .
f 0 (x) =0 ⇔ x =1. f(1) = 2 √
2
= √ 2 is the exact value.
(b) From the graph in part (a), f increases most rapidly somewhere between x = − 1 and x = − 1 .Tofind the exact value,
2 4
we need to find the maximum value of f 0 , which we can do by finding the critical numbers of f 0 .
f 00 (x) = 2x2 − 3x − 1
=0 ⇔ x = 3 ± √ 17
. x = 3+√ 17
corresponds to the minimum value of f 0 .
(x 2 +1) 5/2 4
4
Themaximumvalueoff 0 3 −
is at
√
17 7
4
, − √ 17
≈ (−0.28, 0.69).
6 6
57. f(x) =cosx + 1 2 cos 2x ⇒ f 0 (x) =− sin x − sin 2x ⇒ f 00 (x) =− cos x − 2cos2x
(a)
From the graph of f,itseemsthatf is CD on (0, 1),CUon(1, 2.5),CDon
(2.5, 3.7),CUon(3.7, 5.3), and CD on (5.3, 2π). The points of inflection
appear to be at (1, 0.4), (2.5, −0.6), (3.7, −0.6),and(5.3, 0.4).
(b)
From the graph of f 00 (and zooming in near the zeros), it seems that f is CD
on (0, 0.94),CUon(0.94, 2.57),CDon(2.57, 3.71),CUon(3.71, 5.35),
and CD on (5.35, 2π). Refined estimates of the inflection points are
(0.94, 0.44), (2.57, −0.63), (3.71, −0.63),and(5.35, 0.44).
59. In Maple, we define f andthenusethecommand
plot(diff(diff(f,x),x),x=-2..2);. InMathematica,wedefine f
andthenusePlot[Dt[Dt[f,x],x],{x,-2,2}]. Weseethatf 00 > 0 for
x0.0 [≈ 0.03] andf 00 < 0 for −0.6

F.
164 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
61. (a) The rate of increase of the population is initially very small, then gets larger until it reaches a maximum at about
t =8hours, and decreases toward 0 as the population begins to level off.
(b) The rate of increase has its maximum value at t =8hours.
(c) The population function is concave upward on (0, 8) and concave downward on (8, 18).
(d) At t =8, the population is about 350,sotheinflection point is about (8, 350).
63. Most students learn more in the third hour of studying than in the eighth hour, so K(3) − K(2) is larger than K(8) − K(7).
In other words, as you begin studying for a test, the rate of knowledge gain is large and then starts to taper off, so K 0 (t)
decreases and the graph of K is concave downward.
65. S(t) =At p e −kt with A =0.01, p =4,andk =0.07. Wewillfind the
zeros of f 00 for f(t) =t p e −kt .
f 0 (t) =t p (−ke −kt )+e −kt (pt p−1 )=e −kt (−kt p + pt p−1 )
f 00 (t)=e −kt (−kpt p−1 + p(p − 1)t p−2 )+(−kt p + pt p−1 )(−ke −kt )
= t p−2 e −kt [−kpt + p(p − 1) + k 2 t 2 − kpt]
= t p−2 e −kt (k 2 t 2 − 2kpt + p 2 − p)
Using the given values of p and k gives us f 00 (t) =t 2 e −0.07t (0.0049t 2 − 0.56t +12).SoS 00 (t) =0.01f 00 (t) and its zeros
are t =0and the solutions of 0.0049t 2 − 0.56t +12=0, which are t 1 = 200
7
≈ 28.57 and t 2 = 600
7
≈ 85.71.
At t 1 minutes, the rate of increase of the level of medication in the bloodstream is at its greatest and at t 2 minutes, the rate of
decrease is the greatest.
67. f(x) =ax 3 + bx 2 + cx + d ⇒ f 0 (x) =3ax 2 +2bx + c.
We are given that f(1) = 0 and f(−2) = 3,sof(1) = a + b + c + d =0and
f(−2) = −8a +4b − 2c + d =3.Alsof 0 (1) = 3a +2b + c =0and
f 0 (−2) = 12a − 4b + c =0by Fermat’s Theorem. Solving these four equations, we get
a = 2 , b = 1 , c = − 4 , d = 7 , so the function is f(x) = 1
9 3 3 9 9 2x 3 +3x 2 − 12x +7 .
69. y = 1+x ⇒ y 0 = (1 + x2 )(1) − (1 + x)(2x) 1 − 2x − x2
= ⇒
1+x 2 (1 + x 2 ) 2 (1 + x 2 ) 2
y 00 = (1 + x2 ) 2 (−2 − 2x) − (1 − 2x − x 2 ) · 2(1 + x 2 )(2x)
= 2(1 + x2 )[(1 + x 2 )(−1 − x) − (1 − 2x − x 2 )(2x)]
[(1 + x 2 ) 2 ] 2 (1 + x 2 ) 4
= 2(−1 − x − x2 − x 3 − 2x +4x 2 +2x 3 )
= 2(x3 +3x 2 − 3x − 1)
= 2(x − 1)(x2 +4x +1)
(1 + x 2 ) 3 (1 + x 2 ) 3 (1 + x 2 ) 3
So y 00 =0 ⇒ x =1, −2 ± √ 3.Leta = −2 − √ 3, b = −2+ √ 3,andc =1. We can show that f(a) = 1 4
1 −
√
3
,

F.
TX.10 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH ¤ 165
f(b) = 1 4
1+
√
3
,andf(c) =1. To show that these three points of inflection lie on one straight line, we’ll show that the
slopes m ac and m bc are equal.
m ac =
m bc =
f(c) − f(a)
c − a
f(c) − f(b)
c − b
= 1 − √
1
4 1 − 3
3
1 − −2 − √ 3 = + √ 1
4 4 3
3+ √ 3 = 1 4
= 1 − √
1
4 1+ 3
3
1 − −2+ √ 3 = − √ 1
4 4 3
3 − √ 3 = 1 4
71. Suppose that f is differentiable on an interval I and f 0 (x) > 0 for all x in I except x = c. Toshowthatf is increasing on I,
let x 1, x 2 be two numbers in I with x 1

F.
166 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
77. Let the cubic function be f(x) =ax 3 + bx 2 + cx + d ⇒ f 0 (x) =3ax 2 +2bx + c ⇒ f 00 (x) =6ax +2b.
So f is CU when 6ax +2b>0 ⇔ x>−b/(3a),CDwhenx0,so(0, 0) is an inflection point. But g 00 (0) does not
exist.
83. (a) f(x) =x 4 sin 1 x ⇒ f 0 (x) =x 4 cos 1 − 1
+sin 1 x x 2 x (4x3 )=4x 3 sin 1 x − x2 cos 1 x .
g(x) =x 4 2+sin 1
=2x 4 + f(x) ⇒ g 0 (x) =8x 3 + f 0 (x).
x
h(x) =x 4 −2+sin 1
= −2x 4 + f(x) ⇒ h 0 (x) =−8x 3 + f 0 (x).
x
It is given that f(0) = 0, sof 0 f(x) − f(0)
x 4 sin 1
(0) = lim
=lim x − 0
=limx 3 sin 1
x→0 x − 0 x→0 x
x→0 x .Since
− x 3 ≤ x 3 sin 1 x ≤ x 3 and lim
x→0
x 3 =0,weseethatf 0 (0) = 0 by the Squeeze Theorem. Also,
g 0 (0) = 8(0) 3 + f 0 (0) = 0 and h 0 (0) = −8(0) 3 + f 0 (0) = 0,so0 is a critical number of f, g,andh.
For x 2n = 1
2nπ
For x 2n+1 =
1
1
[n a nonzero integer], sin =sin2nπ =0and cos =cos2nπ =1,sof 0 (x 2n )=−x 2 2n < 0.
x 2n x 2n
1
(2n +1)π , sin 1
1
=sin(2n +1)π =0and cos =cos(2n +1)π = −1, so
x 2n+1 x 2n+1
f 0 (x 2n+1 )=x 2 2n+1 > 0. Thus,f 0 changes sign infinitely often on both sides of 0.
Next, g 0 (x 2n) =8x 3 2n + f 0 (x 2n) =8x 3 2n − x 2 2n = x 2 2n(8x 2n − 1) < 0 for x 2n < 1 8 ,but
g 0 (x 2n+1 )=8x 3 2n+1 + x 2 2n+1 = x 2 2n+1(8x 2n+1 +1)> 0 for x 2n+1 > − 1 8 ,sog0 changes sign infinitely often on both
sides of 0.
Last, h 0 (x 2n) =−8x 3 2n + f 0 (x 2n) =−8x 3 2n − x 2 2n = −x 2 2n(8x 2n +1)< 0 for x 2n > − 1 8 and
h 0 (x 2n+1 )=−8x 3 2n+1 + x 2 2n+1 = x 2 2n+1(−8x 2n+1 +1)> 0 for x 2n+1 < 1 8 ,soh0 changes sign infinitely often on both
sides of 0.

F.
TX.10SECTION 4.4 INDETERMINATE FORMS AND L’HOSPITAL’S RULE ¤ 167
(b) f(0) = 0 and since sin 1 x and hence x4 sin 1 is both positive and negative inifinitely often on both sides of 0,and
x
arbitrarily close to 0, f has neither a local maximum nor a local minimum at 0.
Since 2+sin 1
x ≥ 1, g(x) =x4 2+sin 1
> 0 for x 6= 0,sog(0) = 0 is a local minimum.
x
Since −2+sin 1
x ≤−1, h(x) =x4 −2+sin 1
< 0 for x 6= 0,soh(0) = 0 is a local maximum.
x
4.4 Indeterminate Forms and L'Hospital's Rule
Note: The use of l’Hospital’s Rule is indicated by an H above the equal sign:
1. (a) lim
x→a
f(x)
g(x) is an indeterminate form of type 0 0 .
f(x)
(b) lim =0because the numerator approaches 0 while the denominator becomes large.
x→a p(x)
h(x)
(c) lim =0because the numerator approaches a finite number while the denominator becomes large.
x→a p(x)
H
=
p(x)
(d) If lim p(x) =∞ and f(x) → 0 through positive values, then lim
x→a x→a f(x) = ∞. [Forexample,takea =0, p(x) =1/x2 ,
and f(x) =x 2 p(x)
.] If f(x) → 0 through negative values, then lim
x→a f(x) = −∞. [Forexample,takea =0, p(x) =1/x2 ,
and f(x) =−x 2 .] If f(x) → 0 through both positive and negative values, then the limit might not exist. [For example,
take a =0, p(x) =1/x 2 ,andf(x) =x.]
p(x)
(e) lim
x→a q(x) is an indeterminate form of type ∞ ∞ .
3. (a) When x is near a, f(x) is near 0 and p(x) is large, so f(x) − p(x) is large negative. Thus, lim
x→a
[f(x) − p(x)] = −∞.
(b) lim
x→a
[ p(x) − q(x)] is an indeterminate form of type ∞−∞.
(c) When x is near a, p(x) and q(x) are both large, so p(x)+q(x) is large. Thus, lim
x→a
[ p(x)+q(x)] = ∞.
5. This limit has the form 0 0 .Wecansimplyfactorandsimplifytoevaluatethelimit.
x 2 − 1
lim
x→1 x 2 − x =lim (x +1)(x − 1) x +1
= lim = 1+1 =2
x→1 x(x − 1) x→1 x 1
7. This limit has the form 0 . lim x 9 − 1 H 9x 8
0
=lim
x→1 x 5 − 1 x→1 5x = 9 4 5 lim
x→1 x4 = 9 5 (1) = 9 5
9. This limit has the form 0 0 . lim
x→(π/2) +
11. This limit has the form 0 . lim e t − 1
0 t→0 t 3
cos x
1 − sin x
H
=lim
t→0
H − sin x
= lim
x→(π/2) + − cos x =
lim
x→(π/2)
+
tan x = −∞.
e t
3t 2 = ∞ since et → 1 and 3t 2 → 0 + as t → 0.

F.
168 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
13. This limit has the form 0 . lim tan px H p sec 2 px
=lim
0 x→0 tan qx x→0 q sec 2 qx = p(1)2
q(1) = p 2 q
15. This limit has the form ∞ . lim ln x
√
∞
= H lim
x→∞ x
x→∞
1
2
1/x
= lim
x−1/2 x→∞
2
√
x
=0
17. lim
x→0 + [(ln x)/x] =−∞ since ln x →−∞as x → 0+ anddividingbysmallvaluesofx just increases the magnitude of the
quotient (ln x)/x. L’Hospital’s Rule does not apply.
19. This limit has the form ∞ . lim e x
∞ x→∞ x 3
21. This limit has the form 0 . lim e x − 1 − x
0 x→0 x 2
= H e x
lim
x→∞ 3x 2
= H e x
lim
x→∞ 6x
H e x
= lim
x→∞ 6 = ∞
=lim
H e x − 1 H e x
=lim
x→0 2x x→0 2 = 1 2
23. This limit has the form 0 . lim tanh x H sech 2 x
=lim
0 x→0 tan x x→0 sec 2 x = sech2 0
sec 2 0 = 1 1 =1
25. This limit has the form 0 . lim 5 t − 3 t
0 t→0 t
H 5 t ln 5 − 3 t ln 3
=lim
=ln5− ln 3 = ln 5
t→0 3
1
27. This limit has the form 0 . lim sin −1 x H 1/ √ 1 − x
= lim
2
1
=lim√ 0 x→0 x x→0 1
= 1 x→0 1 − x
2 1 =1
29. This limit has the form 0 . lim 1 − cos x
0
x→0 x 2
=lim
H sin x H cos x
= lim = 1
x→0 2x x→0 2 2
x +sinx
31. lim
x→0 x +cosx = 0+0
0+1 = 0 =0. L’Hospital’s Rule does not apply.
1
33. This limit has the form 0 . lim 1 − x +lnx
0 x→1 1+cosπx
H
=lim
x→1
−1+1/x
−π sin πx
H −1/x 2
=lim
x→1 −π 2 cos πx =
−1
−π 2 (−1) = − 1 π 2
35. This limit has the form 0 . lim x a − ax + a − 1 H ax a−1 − a H a(a − 1)x a−2 a(a − 1)
=lim
= lim
=
0 x→1 (x − 1) 2 x→1 2(x − 1) x→1 2
2
37. This limit has the form 0 . lim cos x − 1+ 1 2 x2
0 x→0 x 4
39. This limit has the form ∞ · 0.
H =lim
x→0
− sin x + x
4x 3
H = lim
x→0
− cos x +1
12x 2
sin(π/x) H cos(π/x)(−π/x 2 )
lim x sin(π/x) = lim = lim
= π lim cos(π/x) =π(1) = π
x→∞ x→∞ 1/x x→∞ −1/x 2
x→∞
41. This limit has the form ∞ · 0. We’ll change it to the form 0 0 .
sin 6x H 6cos6x
lim cot 2x sin 6x =lim =lim
x→0 x→0 tan 2x x→0 2sec 2 2x = 6(1)
2(1) =3 2
=lim
H sin x H cos x
= lim
x→0 24x x→0 24 = 1
24
x 3
43. This limit has the form ∞ · 0. lim
x→∞ x3 e −x2 = lim
x→∞ e x2
= H 3x 2
lim
x→∞ 2xe x2
= lim
x→∞
3x
2e x2
= H 3
lim =0
x→∞ x2
4xe

F.
TX.10SECTION 4.4 INDETERMINATE FORMS AND L’HOSPITAL’S RULE ¤ 169
45. This limit has the form 0 · (−∞).
lim ln x tan(πx/2) = lim
x→1 + x→1 +
47. This limit has the form ∞−∞.
x
lim
x→1 x − 1 − 1
ln x
ln x
cot(πx/2)
=lim
x→1
x ln x − (x − 1)
(x − 1) ln x
H
= lim
x→1 + 1/x
(−π/2) csc 2 (πx/2) = 1
(−π/2)(1) 2 = − 2 π
H
= lim
x→1
x(1/x)+lnx − 1
(x − 1)(1/x)+lnx = lim
x→1
H 1/x
=lim
x→1 1/x 2 +1/x · x2
x = lim x
2 x→1 1+x = 1
1+1 = 1 2
ln x
1 − (1/x)+lnx
49. We will multiply and divide by the conjugate of the expression to change the form of the expression.
√
lim x2 + x − x √ √
x2 + x − x x2 + x + x
x 2 + x − x 2
= lim
· √ = lim √
x→∞
x→∞ 1 x2 + x + x x→∞ x2 + x + x
x
= lim √
x→∞ x2 + x + x = lim 1
1
= √ = 1
x→∞ 1+1/x +1 1+1 2
As an alternate solution, write √ x 2 + x − x as √ x 2 + x − √ x 2 ,factorout √ x 2 ,rewriteas( 1+1/x − 1)/(1/x),and
apply l’Hospital’s Rule.
51. The limit has the form ∞−∞and we will change the form to a product by factoring out x.
lim (x − ln x) = lim x 1 − ln x
ln x H 1/x
= ∞ since lim = lim
x→∞ x→∞ x
x→∞ x x→∞ 1 =0.
53. y = x x2 ⇒ ln y = x 2 ln x
ln x,so lim ln y = lim x2 ln x = lim
H 1/x
= lim
x→0 + x→0 + x→0 + 1/x 2 x→0 + −2/x = lim − 1 3 x→0 + 2 x2 =0 ⇒
lim
x→0 + xx2 = lim eln y = e 0 =1.
x→0 +
55. y =(1− 2x) 1/x ⇒ ln y = 1 x
lim
x→0 (1 − 2x)1/x =lim
x→0
e ln y = e −2 .
57. y =
1+ 3 x + 5 x
⇒ ln y = x ln
1+ 3 x 2 x + 5
x 2
ln
1+ 3 x + 5
x
lim ln y = lim
2
x→∞ x→∞ 1/x
ln(1 − 2x) H −2/(1 − 2x)
ln(1 − 2x), solim ln y =lim
= lim
= −2 ⇒
x→0 x→0 x
x→0 1
H
= lim
x→∞
so lim 1+ 3
x→∞ x + 5 x
= lim
x 2 x→∞ eln y = e 3 .
⇒
− 3 x 2 − 10
x 3
1+ 3 x + 5 x 2
−1/x 2
59. y = x 1/x ln x
⇒ ln y =(1/x) lnx ⇒ lim ln y = lim
x→∞ x→∞ x
lim
x→∞ x1/x = lim
x→∞ eln y = e 0 =1
= lim
x→∞
H 1/x
= lim
x→∞ 1 =0 ⇒
3+ 10 x
1+ 3 x + 5 x 2 =3,

F.
170 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
61. y =(4x +1) cot x ln(4x +1) H
⇒ ln y =cotx ln(4x +1),so lim ln y = lim
= lim
x→0 + x→0 + tan x
x→0 + 4
4x +1
sec 2 x =4
⇒
lim
x→0 +(4x +1)cot x = lim
x→0 + eln y = e 4 .
63. y =(cosx) 1/x2 ⇒ ln y = 1 ln cos x
ln cos x ⇒ lim ln y = lim
x2 x→0 + x→0 + x 2
= H − tan x H − sec 2 x
lim = lim = − 1
x→0 + 2x x→0 + 2 2
⇒
lim
x→0 +(cos x)1/x2 = lim
x→0 + eln y = e −1/2 =1/ √ e
65. From the graph, if x = 500, y ≈ 7.36. The limit has the form 1 ∞ .
Now y = 1+ 2 x
⇒ ln y = x ln 1+ 2
⇒
x x
1
− 2
ln(1 + 2/x) H 1+2/x x
lim ln y = lim
= lim
2
x→∞ x→∞ 1/x x→∞ −1/x 2
=2 lim
x→∞
1
1+2/x =2(1)=2
lim 1+ 2 x
= lim
x→∞ x x→∞ eln y = e 2 [≈ 7.39]
f(x)
67. From the graph, it appears that lim
x→0 g(x) =lim f 0 (x)
x→0 g 0 (x) =0.25.
We calculate lim
x→0
f(x)
g(x) = lim
x→0
e x − 1
x 3 +4x
⇒
H
= lim
x→0
e x
3x 2 +4 = 1 4 .
e x
69. lim
x→∞ x n
71. lim
x→∞
= H e x
lim
x→∞ nx n−1
x
√
x2 +1
H
= lim
x→∞
H = lim
x→∞
e x
n(n − 1)x n−2
1
1
2 (x2 +1) −1/2 (2x) = lim
x→∞
= H ··· H= e x
lim
x→∞ n! = ∞
√
x2 +1
. Repeated applications of l’Hospital’s Rule result in the
x
original limit or the limit of the reciprocal of the function. Another method is to try dividing the numerator and denominator
by x:
lim
x→∞
x
√
x2 +1 = lim
x→∞
x/x
x2 /x 2 +1/x 2 = lim
x→∞
1
1+1/x
2 = 1 1 =1
73. First we will find lim 1+ r nt,
which is of the form 1
n→∞ n ∞ . y = 1+ r nt
⇒ ln y = nt ln 1+ r
,so
n n
−r/n
2
lim ln y = lim
1+ nt ln r
ln(1 + r/n)
= t lim
n→∞ n→∞ n n→∞ 1/n
H
= t lim
n→∞
(1 + r/n)(−1/n 2 ) = t lim
n→∞
r
1+i/n = tr
⇒
lim y =
n→∞ ert .Thus,asn →∞, A = A 0
1+ r nt
→ A0 e rt .
n

F.
e E
75. lim P (E) = lim + e −E
E→0 + E→0 + e E − e − 1
−E E
TX.10SECTION 4.4 INDETERMINATE FORMS AND L’HOSPITAL’S RULE ¤ 171
E e E + e −E − 1 e E − e −E Ee E + Ee −E − e E + e −E
= lim
= lim
form is
0
E→0 + (e E − e −E ) E
E→0 + Ee E − Ee −E 0
H
= lim
E→0 + Ee E + e E · 1+E −e −E + e −E · 1 − e E + −e −E
Ee E + e E · 1 − [E(−e −E )+e −E · 1]
= lim
E→0 +
Ee E − Ee −E
= lim
Ee E + e E + Ee −E − e−E E→0 +
= 0
e E − e −E
, whereL = lim
2+L E→0 + E
Thus, lim P (E) = 0
E→0 + 2+2 =0.
e E − e −E
e E + eE E + e−E − e−E
E
form is
0
0
H = lim
E→0 + e E + e −E
1
77. We see that both numerator and denominator approach 0, so we can use l’Hospital’s Rule:
√
2a3 x − x
lim
4 − a 3√ aax
x→a a − 4√ ax 3
[divide by E]
= 1+1
1
=lim
H 1
2 (2a3 x − x 4 ) −1/2 (2a 3 − 4x 3 ) − a
1
3 (aax) −2/3 a 2
x→a − 1 4 (ax3 ) −3/4 (3ax 2 )
=
1
2 (2a3 a − a 4 ) −1/2 (2a 3 − 4a 3 ) − 1 3 a3 (a 2 a) −2/3
− 1 4 (aa3 ) −3/4 (3aa 2 )
=2
= (a4 ) −1/2 (−a 3 ) − 1 3 a3 (a 3 ) −2/3
− 3 4 a3(a4 )−3/4
= −a − 1 3 a
− 3 4
= 4 3
4
3 a = 16 9 a
79. Since f(2) = 0, the given limit has the form 0 0 .
f(2 + 3x)+f(2 + 5x) H f 0 (2 + 3x) · 3+f 0 (2 + 5x) · 5
lim
=lim
= f 0 (2) · 3+f 0 (2) · 5=8f 0 (2) = 8 · 7=56
x→0 x
x→0 1
81. Since lim
h→0
[f(x + h) − f(x − h)] = f(x) − f(x) =0 (f is differentiable and hence continuous) and lim
h→0
2h =0,weuse
l’Hospital’s Rule:
f(x + h) − f(x − h) H f 0 (x + h)(1) − f 0 (x − h)(−1)
lim
=lim
= f 0 (x)+f 0 (x)
= 2f 0 (x)
= f 0 (x)
h→0 2h
h→0 2
2
2
f(x + h) − f(x − h)
2h
is the slope of the secant line between
(x − h, f(x − h)) and (x + h, f(x + h)). Ash → 0,thislinegetscloser
to the tangent line and its slope approaches f 0 (x).
f(x)
83. (a) We show that lim
x→0 x =0for every integer n ≥ 0. Lety = 1 n
x .Then 2
f(x)
lim
x→0 x =lim e −1/x2 y n
2n x→0 (x 2 ) n = lim
y→∞ e y
= H ny n−1
lim
y→∞ e y
= H ··· H= n!
lim
y→∞ e =0 ⇒
y
f(x)
lim
x→0 x = lim f(x)
n x→0 xn x =lim f(x)
2n x→0 xn lim
x→0 x =0.Thus,f 0 f(x) − f(0) f(x)
(0) = lim
= lim
2n
x→0 x − 0 x→0 x =0.

F.
172 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
(b) Using the Chain Rule and the Quotient Rule we see that f (n) (x) exists for x 6= 0. In fact, we prove by induction that for
each n ≥ 0, there is a polynomial p n and a non-negative integer k n with f (n) (x) =p n (x)f(x)/x kn for x 6= 0.Thisis
true for n =0; suppose it is true for the nth derivative. Then f 0 (x) =f(x)(2/x 3 ),so
which has the desired form.
f (n+1) (x)= x k n
[p 0 n(x) f(x)+p n(x)f 0 (x)] − k nx k n−1 p n(x) f(x) x −2k n
= x k n
p 0 n(x)+p n(x) 2/x 3 − k nx k n−1 p n(x) f(x)x −2k n
= x kn+3 p 0 n(x)+2p n (x) − k n x kn+2 p n (x) f(x)x −(2kn+3)
Now we show by induction that f (n) (0) = 0 for all n.Bypart(a),f 0 (0) = 0. Suppose that f (n) (0) = 0. Then
f (n+1) f (n) (x) − f (n) (0) f (n) (x) p n(x) f(x)/x k n
(0) = lim
= lim =lim
x→0 x − 0
x→0 x x→0 x
f(x)
=limp n (x) lim
x→0 x→0 x = p n(0) · 0=0
kn+1
= lim
x→0
p n(x) f(x)
x k n+1
4.5 Summary of Curve Sketching
1. y = f(x) =x 3 + x = x(x 2 +1) A. f is a polynomial, so D = R.
H.
B. x-intercept =0, y-intercept = f(0) = 0 C. f(−x) =−f(x),sof is
odd; the curve is symmetric about the origin.
D. f is a polynomial, so there is
no asymptote.
E. f 0 (x) =3x 2 +1> 0,sof is increasing on (−∞, ∞).
F. There is no critical number and hence, no local maximum or minimum value.
G. f 00 (x) =6x>0 on (0, ∞) and f 00 (x) < 0 on (−∞, 0),sof is CU on
(0, ∞) and CD on (−∞, 0). Since the concavity changes at x =0,thereisan
inflection point at (0, 0).
3. y = f(x) =2− 15x +9x 2 − x 3 = −(x − 2) x 2 − 7x +1 A. D = R B. y-intercept: f(0) = 2; x-intercepts:
f(x) =0 ⇒ x =2or (by the quadratic formula) x = 7 ± √ 45
2
≈ 0.15, 6.85 C. No symmetry D. No asymptote
E. f 0 (x) =−15 + 18x − 3x 2 = −3(x 2 − 6x +5)
= −3(x − 1)(x − 5) > 0 ⇔ 1

F.
TX.10
SECTION 4.5 SUMMARY OF CURVE SKETCHING ¤ 173
5. y = f(x) =x 4 +4x 3 = x 3 (x +4) A. D = R B. y-intercept: f(0) = 0;
H.
x-intercepts: f(x) =0 ⇔ x = −4, 0 C. No symmetry
D. No asymptote E. f 0 (x) =4x 3 +12x 2 =4x 2 (x +3)> 0 ⇔
x>−3,sof is increasing on (−3, ∞) and decreasing on (−∞, −3).
F. Local minimum value f(−3) = −27, no local maximum
G. f 00 (x) =12x 2 +24x =12x(x +2)< 0 ⇔ −2 1,sof is CU on (1, ∞) and
(x − 1)
3
11. y = f(x) =1/(x 2 − 9) A. D = {x | x 6=±3} =(−∞, −3) ∪ (−3, 3) ∪ (3, ∞) B. y-intercept = f(0) = − 1 9 ,no
1
x-intercept C. f(−x) =f(x) ⇒ f is even; the curve is symmetric about the y-axis. D. lim =0,soy =0
x→±∞ x 2 − 9
is a HA.
1
1
lim = −∞, lim
x→3 − x 2 − 9 x→3 + x 2 − 9 = ∞,
lim 1
x→−3 − x 2 − 9 = ∞,
lim 1
= −∞, sox =3and x = −3
x→−3 + x 2 − 9
are VA. E. f 0 2x
(x) =−
(x 2 2
> 0 ⇔ x

F.
174 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
decreasing on (0, 3) and (3, ∞). F. Local maximum value f(0) = − 1 9 .
H.
G. y 00 = −2(x2 − 9) 2 +(2x)2(x 2 − 9)(2x)
(x 2 − 9) 4 = 6(x2 +3)
(x 2 − 9) 3 > 0 ⇔
x 2 > 9 ⇔ x>3 or x 0 ⇔ −3

F.
TX.10
SECTION 4.5 SUMMARY OF CURVE SKETCHING ¤ 175
17. y = f(x) = x2
x 2 +3 = (x2 +3)− 3
=1− 3
x 2 +3 x 2 +3
A. D = R B. y-intercept: f(0) = 0;
x-intercepts: f(x) =0 ⇔ x =0 C. f(−x) =f(x),sof is even; the graph is symmetric about the y-axis.
D. lim
x→±∞
x 2
x 2 +3 =1,soy =1is a HA. No VA. E. Using the Reciprocal Rule, f 0 (x) =−3 ·
f 0 (x) > 0 ⇔ x>0 and f 0 (x) < 0 ⇔ x 0 ⇒ x>− 1 and f 0 (x) < 0 ⇒ x

F.
176 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
23. y = f(x) =x/ √ x 2 +1 A. D = R B. y-intercept: f(0) = 0; x-intercepts: f(x) =0 ⇒ x =0
C. f(−x) =−f(x),sof is odd; the graph is symmetric about the origin.
D. lim f(x) = lim
x→∞ x→∞
and
lim f(x) =
x→−∞
=
lim
x→−∞
x
√
x2 +1 = lim
x→∞
x
√
x2 +1 =
lim
x→−∞
x/x
√
x2 +1/x = lim
x→∞
x/x
√
x2 +1/x =
1
− √ = −1 so y = ±1 are HA.
1+0
lim
x→−∞
x/x
√
x2 +1/ √ x 2 = lim
x→∞
x/x
√
x2 +1/− √ x 2 = lim
x→−∞
1
1+1/x
2 = 1
√ 1+0
=1
1
− 1+1/x 2
No VA.
√ 2x
x2 +1− x ·
E. f 0 2 √ x
(x) =
2 +1
= x2 +1− x 2
[(x 2 +1) 1/2 ] 2 (x 2 +1) = 1
> 0 for all x,sof is increasing on R.
3/2 (x 2 3/2
+1)
F. No extreme values
G. f 00 (x) =− 3 2 (x2 +1) −5/2 · 2x =
−3x
(x 2 +1) 5/2 ,sof 00 (x) > 0 for x0. Thus,f is CU on (−∞, 0) and CD on (0, ∞).
IP at (0, 0)
25. y = f(x) = √ 1 − x 2 /x A. D = {x ||x| ≤ 1, x 6= 0} =[−1, 0) ∪ (0, 1] B. x-intercepts ±1, noy-intercept
√
1 − x
2
C. f(−x) =−f(x), so the curve is symmetric about (0, 0) . D. lim
x→0 + x
−x 2 / √ 1 − x 2 − √ 1 − x 2
so x =0is a VA. E. f 0 (x) =
on (−1, 0) and (0, 1).
G. f 00 (x) =
f is CU on −1, −
2
IP at ± , ± √ 1
3 2
F. No extreme values
2 − 3x 2
x 3 (1 − x 2 ) > 0 3/2 ⇔
−1 1 and f 0 (x) < 0 when 0 < |x| < 1,sof is increasing on (−∞, −1) and (1, ∞),and
decreasing on (−1, 0) and (0, 1) [hence decreasing on (−1, 1) since f is
H.
H.
continuous on (−1, 1)].
F. Local maximum value f(−1) = 2, local minimum
value f(1) = −2 G. f 00 (x) = 2 3 x−5/3 < 0 when x 0
when x>0,sof is CD on (−∞, 0) and CU on (0, ∞). IPat(0, 0)

F.
TX.10
SECTION 4.5 SUMMARY OF CURVE SKETCHING ¤ 177
29. y = f(x) = 3√ x 2 − 1 A. D = R B. y-intercept: f(0) = −1; x-intercepts: f(x) =0 ⇔ x 2 − 1=0 ⇔
x = ±1 C. f(−x) =f(x), so the curve is symmetric about the y-axis. D. No asymptote
E. f 0 (x) = 1 3 (x2 − 1) −2/3 (2x) =
increasing on (0, ∞) and decreasing on (−∞, 0).
G. f 00 (x)= 2 3 · (x2 − 1) 2/3 (1) − x · 2
3 (x2 − 1) −1/3 (2x)
[(x 2 − 1) 2/3 ] 2
2x
3 3 (x 2 − 1) 2 . f 0 (x) > 0 ⇔ x>0 and f 0 (x) < 0 ⇔ x 0 ⇔ −1 0 ⇔ x ∈ 2nπ − π , 2nπ + π
2 2 for each integer n, andf 0 (x) < 0 ⇔ cos x0 and sin x 6=±1 ⇔ x ∈
2nπ, 2nπ + π 2 ∪ 2nπ +
π
, 2nπ + π for some integer n.
2
f 00 (x) > 0 ⇔ sin x

F.
178 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
35. y = f(x) = 1 2
x − sin x, 0 0 on (0,α) and (β,2π) [ f is CU].
The inflection points occur when x = α, β.
H.

F.
TX.10
SECTION 4.5 SUMMARY OF CURVE SKETCHING ¤ 179
41. y =1/(1 + e −x ) A. D = R B. No x-intercept; y-intercept = f(0) = 1 2
. C. No symmetry
D. lim 1/(1 +
x→∞ e−x )= 1 =1and lim 1/(1 + 1+0 x→−∞ e−x )=0 since
lim
x→−∞ e−x = ∞], so f has horizontal asymptotes
y =0and y =1. E. f 0 (x) =−(1 + e −x ) −2 (−e −x )=e −x /(1 + e −x ) 2 . This is positive for all x,sof is increasing on R.
F. No extreme values G. f 00 (x) = (1 + e−x ) 2 (−e −x ) − e −x (2)(1 + e −x )(−e −x )
= e−x (e −x − 1)
(1 + e −x ) 4 (1 + e −x ) 3
The second factor in the numerator is negative for x>0 and positive for x 0 ⇒ 1 > 1/x ⇒ x>1 and
H.
f 0 (x) < 0 ⇒ 0 0]
x→−∞
f(x) =1,soy =0and y =1are HA; no VA
E. f 0 (x) =−2(1 + e x ) −3 e x = −2ex < 0,sof is decreasing on R F. No local extrema
(1 + e x )
3
G. f 00 (x)=(1+e x ) −3 (−2e x )+(−2e x )(−3)(1 + e x ) −4 e x
H.
= −2e x (1 + e x ) −4 [(1 + e x ) − 3e x ]= −2ex (1 − 2e x )
(1 + e x ) 4 .
f 00 (x) > 0 ⇔ 1 − 2e x < 0 ⇔ e x > 1 2
⇔ x>ln 1 2 and
f 00 (x) < 0 ⇔ x0} = ∞
n=−∞
(2nπ, (2n +1)π) =···∪ (−4π, −3π) ∪ (−2π, −π) ∪ (0,π) ∪ (2π, 3π) ∪ ···
B. No y-intercept; x-intercepts: f(x) =0 ⇔ ln(sin x) =0 ⇔ sin x = e 0 =1 ⇔ x =2nπ + π for each
2
integer n. C. f is periodic with period 2π. D. lim f(x) =−∞ and lim
x→(2nπ) +
x = nπ are VAs for all integers n.
x→[(2n+1)π]
−
f(x) =−∞, so the lines
E. f 0 (x) = cos x
sin x =cotx,sof 0 (x) > 0 when 2nπ < x < 2nπ + π for each
2
integer n,andf 0 (x) < 0 when 2nπ + π 2

F.
180 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
decreasing on 2nπ + π 2 , (2n +1)π for each integer n.
TX.10
H.
F. Local maximum values f 2nπ + π 2
=0, no local minimum.
G. f 00 (x) =− csc 2 x 0 ⇔ x 2 < 1 2
⇔ |x| < 1 √
2
,sof is increasing on
and decreasing on −∞, − √ 1
2
and
√
1
1
2
, ∞ . F. Local maximum value f √
2
=1/ √ 2e, local minimum
value f − √ 1
2
= −1/ √ 2e G. f 00 (x) =−2xe −x2 (1 − 2x 2 ) − 4xe −x2 =2xe −x2 (2x 2 − 3) > 0 ⇔
3
x> or − 3
2e −2x [multiply by e 2x ] ⇔
H.
e 5x > 2 3
⇔ 5x >ln 2 3
⇔ x> 1 5 ln 2 3 ≈−0.081. Similarly, f 0 (x) < 0 ⇔
x< 1 5 ln 2 3 . f is decreasing on −∞, 1 5 ln 2 3
and increasing on
1
5 ln 2 3 , ∞ .
F. Local minimum value f 1
ln 2
5 3 = 2
3/5
+
2 −2/5
≈ 1.96; no local maximum.
3
3
G. f 00 (x) =9e 3x +4e −2x ,so f 00 (x) > 0 for all x,andf is CU on (−∞, ∞). NoIP
53. m = f(v) =
m 0
1 − v2 /c .Them-intercept is f(0) = m 0.Therearenov-intercepts. lim f(v) =∞,sov = c is a VA.
2 v→c− f 0 (v) =− 1 2 m0(1 − v2 /c 2 ) −3/2 (−2v/c 2 )=
increasing on (0,c). There are no local extreme values.
m 0 v
c 2 (1 − v 2 /c 2 ) 3/2 =
f 00 (v)= (c2 − v 2 ) 3/2 (m 0 c) − m 0 cv · 3
2 (c2 − v 2 ) 1/2 (−2v)
[(c 2 − v 2 ) 3/2 ] 2
= m 0c(c 2 − v 2 ) 1/2 [(c 2 − v 2 )+3v 2 ]
(c 2 − v 2 ) 3 = m 0c(c 2 +2v 2 )
(c 2 − v 2 ) 5/2 > 0,
so f is CU on (0,c). Therearenoinflection points.
m 0 v
c 2 (c 2 − v 2 ) 3/2
c 3 =
m 0 cv
> 0,sof is
(c 2 − v 2 )
3/2

F.
TX.10
SECTION 4.5 SUMMARY OF CURVE SKETCHING ¤ 181
55. y = − W
24EI x4 + WL
12EI x3 − WL2
24EI x2 = − W
24EI x2 x 2 − 2Lx + L 2
= −W
24EI x2 (x − L) 2 = cx 2 (x − L) 2
where c = − W isanegativeconstantand0 ≤ x ≤ L. Wesketch
24EI
f(x) =cx 2 (x − L) 2 for c = −1. f(0) = f(L) =0.
f 0 (x) =cx 2 [2(x − L)] + (x − L) 2 (2cx) =2cx(x − L)[x +(x − L)] = 2cx(x − L)(2x − L). Sofor0

F.
182 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
C. No symmetry D. lim f(x) =−∞ and lim
x→(1/2) −
lim
x→±∞
[f(x) − (−x +2)]= lim
E. f 0 (x) =−1 −
f(x) =∞,sox = 1 x→(1/2) + 2
is a VA.
1
=0,sotheliney = −x +2is a SA.
x→±∞ 2x − 1
2
(2x − 1) < 0 for x 6= 1 ,sof is decreasing on
−∞, 1 2 2 2
and 1
2 , ∞ . F. No extreme values G. f 0 (x) =−1 − 2(2x − 1) −2 ⇒
H.
f 00 (x) =−2(−2)(2x − 1) −3 8
(2) =
(2x − 1) 3 ,sof 00 (x) > 0 when x> 1 and 2
f 00 (x) < 0 when x< 1 . Thus, f is CU on 1
, ∞ and CD on
−∞, 1 2 2 2 .NoIP
63. y = f(x) =(x 2 +4)/x = x +4/x A. D = {x | x 6= 0} =(−∞, 0) ∪ (0, ∞) B. No intercept
C. f(−x) =−f(x) ⇒ symmetry about the origin D. lim (x +4/x) =∞ but f(x) − x =4/x → 0 as x → ±∞,
x→∞
so y = x is a slant asymptote.
lim (x +4/x) =∞ and
x→0 +
lim
x→0 − (x +4/x) =−∞,sox =0 is a VA. E. f 0 (x) =1− 4/x 2 > 0 ⇔
x 2 > 4 ⇔ x>2 or x 0 ⇔ x>0 so f is CU on
(0, ∞) and CD on (−∞, 0). NoIP
65. y = f(x) = 2x3 + x 2 +1
x 2 +1
=2x +1+ −2x
x 2 +1
A. D = R B. y-intercept: f(0) = 1; x-intercept: f(x) =0 ⇒
0=2x 3 + x 2 +1=(x + 1)(2x 2 − x +1) ⇒ x = −1 C. No symmetry D. No VA
lim
x→±∞
[f(x) − (2x + 1)] = lim
x→±∞
−2x
x 2 +1 =
lim
x→±∞
−2/x
=0, so the line y =2x +1is a slant asymptote.
1+1/x2 E. f 0 (x) =2+ (x2 +1)(−2) − (−2x)(2x)
= 2(x4 +2x 2 +1)− 2x 2 − 2+4x 2
= 2x4 +6x 2
(x 2 +1) 2 (x 2 +1) 2 (x 2 +1) = 2x2 (x 2 +3)
2 (x 2 +1) 2
so f 0 (x) > 0 if x 6= 0.Thus,f is increasing on (−∞, 0) and (0, ∞). Sincef is continuous at 0, f is increasing on R.
F. No extreme values
G. f 00 (x) = (x2 +1) 2 · (8x 3 +12x) − (2x 4 +6x 2 ) · 2(x 2 +1)(2x)
[(x 2 +1) 2 ] 2
= 4x(x2 +1)[(x 2 + 1)(2x 2 +3)− 2x 4 − 6x 2 ]
= 4x(−x2 +3)
(x 2 +1) 4 (x 2 +1) 3
so f 00 (x) > 0 for x

F.
TX.10
SECTION 4.5 SUMMARY OF CURVE SKETCHING ¤ 183
67. y = f(x) =x − tan −1 x, f 0 (x) =1− 1
1+x 2 = 1+x2 − 1
1+x 2 = x2
1+x 2 ,
f 00 (x) = (1 + x2 )(2x) − x 2 (2x)
= 2x(1 + x2 − x 2 ) 2x
=
(1 + x 2 ) 2 (1 + x 2 ) 2 (1 + x 2 ) . 2
lim f(x) − x −
π
x→∞
2 = lim π −
x→∞
2 tan−1 x = π − π =0,soy = x − π is a SA.
2 2 2
Also,
lim
x→−∞
f(x) −
x +
π
2
= lim
x→−∞
−
π
2 − tan−1 x = − π 2 − − π 2
=0,
so y = x + π 2 is also a SA. f 0 (x) ≥ 0 for all x, with equality ⇔ x =0,sof is
increasing on R. f 00 (x) has the same sign as x,sof is CD on (−∞, 0) and CU on
(0, ∞). f(−x) =−f(x),sof is an odd function; its graph is symmetric about the
origin. f has no local extreme values. Its only IP is at (0, 0).
69.
x 2
a 2 − y2
lim
x→∞
b
a
b =1 ⇒ y = ± b √
x2 − a 2 a
2 .Now
√
x2 − a 2 − b
a x = b a · lim
x→∞
which shows that y = b x is a slant asymptote. Similarly,
a
lim − b √
x2 − a
x→∞ a 2 −
− b
a x = − b a · lim
x→∞
√
x2 − a 2 − x √ x 2 − a 2 + x
√
x2 − a 2 + x = b a · lim
x→∞
−a 2
√
x2 − a 2 + x =0,
−a 2
√
x2 − a 2 + x =0,soy = − b x is a slant asymptote.
a
71. lim
x→±∞
f(x) − x
3 =
lim
x→±∞
x 4 +1
x
− x4
x = lim 1
x→±∞ x =0,sothegraphoff is asymptotic to that of y = x3 .
= −∞ and
A. D = {x | x 6= 0} B. No intercept C. f is symmetric about the origin. D. lim
x 3 + 1
x→0 − x
lim
x 3 + 1
= ∞,sox =0is a vertical asymptote, and as shown above, the graph of f is asymptotic to that of y = x 3 .
x→0 + x
E. f 0 (x) =3x 2 − 1/x 2 > 0 ⇔ x 4 > 1 3
⇔ |x| > 1 4 √ 3 ,sof is increasing on
−∞, − 1 4 √ 3
decreasing on − 4√ 1
, 0 and 0,
3
1
4√
3
. F. Local maximum value
f − 1
1
√ = −4 · 3 −5/4 , local minimum value f
4
3
4√ =4· 3 −5/4
3
G. f 00 (x) =6x +2/x 3 > 0 ⇔ x>0,sof is CU on (0, ∞) and CD
on (−∞, 0). NoIP
H.
1
and √ 4
3 , ∞ and

F.
184 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
4.6 Graphing with Calculus and Calculators
1. f(x) =4x 4 − 32x 3 +89x 2 − 95x +29 ⇒ f 0 (x) =16x 3 − 96x 2 + 178x − 95 ⇒ f 00 (x) =48x 2 − 192x +178.
f(x) =0 ⇔ x ≈ 0.5, 1.60; f 0 (x) =0 ⇔ x ≈ 0.92, 2.5, 2.58 and f 00 (x) =0 ⇔ x ≈ 1.46, 2.54.
From the graphs of f 0 ,weestimatethatf 0 < 0 and that f is decreasing on (−∞, 0.92) and (2.5, 2.58),andthatf 0 > 0 and f
is increasing on (0.92, 2.5) and (2.58, ∞) with local minimum values f(0.92) ≈−5.12 and f(2.58) ≈ 3.998 and local
maximum value f(2.5) = 4. The graphs of f 0 make it clear that f has a maximum and a minimum near x =2.5,shownmore
clearly in the fourth graph.
From the graph of f 00 ,weestimatethatf 00 > 0 and that f is CU on
(−∞, 1.46) and (2.54, ∞),andthatf 00 < 0 and f is CD on (1.46, 2.54).
There are inflection points at about (1.46, −1.40) and (2.54, 3.999).
3. f(x) =x 6 − 10x 5 − 400x 4 +2500x 3 ⇒ f 0 (x) =6x 5 − 50x 4 − 1600x 3 + 7500x 2 ⇒
f 00 (x) =30x 4 − 200x 3 − 4800x 2 +1500x

F.
TX.10SECTION 4.6 GRAPHING WITH CALCULUS AND CALCULATORS ¤ 185
From the graph of f 0 ,weestimatethatf is decreasing on (−∞, −15), increasing on (−15, 4.40), decreasing
on (4.40, 18.93), and increasing on (18.93, ∞), with local minimum values of f(−15) ≈−9,700,000 and
f(18.93) ≈−12,700,000 and local maximum value f(4.40) ≈ 53,800. Fromthegraphoff 00 ,weestimatethatf is CU on
(−∞, −11.34),CDon(−11.34, 0),CUon(0, 2.92),CDon(2.92, 15.08), andCUon(15.08, ∞). There is an inflection
point at (0, 0) and at about (−11.34, −6,250,000), (2.92, 31,800),and(15.08, −8,150,000).
5. f(x) =
x
x 3 − x 2 − 4x +1
⇒ f 0 (x) = −2x3 + x 2 +1
⇒ f 00 (x) = 2(3x5 − 3x 4 +5x 3 − 6x 2 +3x +4)
(x 3 − x 2 − 4x +1) 2 (x 3 − x 2 − 4x +1) 3
We estimate from the graph of f that y =0is a horizontal asymptote, and that there are vertical asymptotes at x = −1.7,
x =0.24,andx =2.46. Fromthegraphoff 0 , we estimate that f is increasing on (−∞, −1.7), (−1.7, 0.24),and(0.24, 1),
and that f is decreasing on (1, 2.46) and (2.46, ∞). There is a local maximum value at f(1) = − 1 3 . From the graph of f 00 ,we
estimate that f is CU on (−∞, −1.7), (−0.506, 0.24),and(2.46, ∞),andthatf is CD on (−1.7, −0.506) and (0.24, 2.46).
There is an inflection point at (−0.506, −0.192).
7. f(x) =x 2 − 4x +7cosx, −4 ≤ x ≤ 4. f 0 (x) =2x − 4 − 7sinx ⇒ f 00 (x) =2− 7cosx.
f(x) =0 ⇔ x ≈ 1.10; f 0 (x) =0 ⇔ x ≈−1.49, −1.07,or2.89; f 00 (x) =0 ⇔ x = ± cos −1 2
7
≈ ±1.28.
From the graphs of f 0 ,weestimatethatf is decreasing (f 0 < 0) on (−4, −1.49), increasing on (−1.49, −1.07), decreasing
on (−1.07, 2.89), and increasing on (2.89, 4), with local minimum values f(−1.49) ≈ 8.75 and f(2.89) ≈−9.99 and local
maximum value f(−1.07) ≈ 8.79 (notice the second graph of f). From the graph of f 00 ,weestimatethatf is CU (f 00 > 0)
on (−4, −1.28), CDon(−1.28, 1.28), and CU on (1.28, 4). Thereareinflection points at about (−1.28, 8.77)
and (1.28, −1.48).

F.
186 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
9. f(x) =1+ 1 x + 8 x 2 + 1 x 3 ⇒ f 0 (x) =− 1 x 2 − 16
x 3 − 3 x 4 = − 1 x 4 (x2 +16x +3) ⇒
f 00 (x) = 2 x 3 + 48
x 4 + 12
x 5 = 2 x 5 (x2 +24x +6).
From the graphs, it appears that f increases on (−15.8, −0.2) and decreases on (−∞, −15.8), (−0.2, 0),and(0, ∞);thatf
has a local minimum value of f(−15.8) ≈ 0.97 and a local maximum value of f(−0.2) ≈ 72;thatf is CD on (−∞, −24)
and (−0.25, 0) and is CU on (−24, −0.25) and (0, ∞);andthatf has IPs at (−24, 0.97) and (−0.25, 60).
To find the exact values, note that f 0 =0 ⇒ x = −16 ± √ 256 − 12
2
= −8 ± √ 61 [≈ −0.19 and −15.81].
f 0 is positive (f is increasing) on −8 − √ 61, −8+ √ 61 and f 0 is negative (f is decreasing) on −∞, −8 − √ 61 ,
−8+
√
61, 0
,and(0, ∞). f 00 =0 ⇒ x = −24 ± √ 576 − 24
2
= −12 ± √ 138 [≈ −0.25 and −23.75]. f 00 is
positive (f is CU) on −12 − √ 138, −12 + √ 138 and (0, ∞) and f 00 is negative (f is CD) on −∞, −12 − √ 138
and −12 + √ 138, 0 .
11. (a) f(x) =x 2 ln x. The domain of f is (0, ∞).
(b)
ln x
lim x2 ln x = lim
H 1/x
= lim
x→0 + x→0 + 1/x 2 x→0 + −2/x = lim
3 x→0 +
There is a hole at (0, 0).
− x2
=0.
2
(c) It appears that there is an IP at about (0.2, −0.06) and a local minimum at (0.6, −0.18). f(x) =x 2 ln x ⇒
f 0 (x) =x 2 (1/x)+(lnx)(2x) =x(2 ln x +1)> 0 ⇔ ln x>− 1 2
⇔ x>e −1/2 ,sof is increasing on
1/ √
e, ∞ , decreasing on 0, 1/ √
e .BytheFDT,f 1/ √
e = −1/(2e) is a local minimum value. This point is
approximately (0.6065, −0.1839), which agrees with our estimate.
f 00 (x) =x(2/x)+(2lnx +1)=2lnx +3> 0 ⇔ ln x>− 3 2
⇔ x>e −3/2 ,sof is CU on (e −3/2 , ∞)
and CD on (0,e −3/2 ). IPis(e −3/2 , −3/(2e 3 )) ≈ (0.2231, −0.0747).

F.
13. f(x) =
lim
x→1
TX.10SECTION 4.6 GRAPHING WITH CALCULUS AND CALCULATORS ¤ 187
(x +4)(x − 3)2
x 4 (x − 1)
f(x) =−∞ and lim f(x) =∞.
− +
f(x) =
x +4
·
x
x 4
x→1
(x − 3)2
x 2
· (x − 1)
x3 as x → ±∞,sof is asymptotic to the x-axis.
has VA at x =0and at x =1since lim
x→0
f(x) =−∞,
dividing numerator
and denominator by x 3
=
(1 + 4/x)(1 − 3/x)2
x(x − 1)
Since f is undefined at x =0,ithasnoy-intercept. f(x) =0 ⇒ (x +4)(x − 3) 2 =0 ⇒ x = −4 or x =3,sof has
x-intercepts −4 and 3. Note, however, that the graph of f is only tangent to the x-axis and does not cross it at x =3,sincef is
positive as x → 3 − and as x → 3 + .
→ 0
From these graphs, it appears that f has three maximum values and one minimum value. The maximum values are
approximately f(−5.6) = 0.0182, f(0.82) = −281.5 and f(5.2) = 0.0145 and we know (since the graph is tangent to the
x-axis at x =3) that the minimum value is f(3) = 0.
15. f(x) =
x 2 (x +1) 3
(x − 2) 2 (x − 4) 4 ⇒ f 0 (x) =− x(x +1)2 (x 3 +18x 2 − 44x − 16)
(x − 2) 3 (x − 4) 5 [from CAS].
From the graphs of f 0 , it seems that the critical points which indicate extrema occur at x ≈−20, −0.3,and2.5, as estimated
in Example 3. (There is another critical point at x = −1,butthesignoff 0 does not change there.) We differentiate again,
obtaining f 00 (x) =2 (x +1)(x6 +36x 5 +6x 4 − 628x 3 + 684x 2 +672x +64)
(x − 2) 4 (x − 4) 6 .
From the graphs of f 00 , it appears that f is CU on (−35.3, −5.0), (−1, −0.5), (−0.1, 2), (2, 4) and (4, ∞) and CD

F.
188 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
on (−∞, −35.3), (−5.0, −1) and (−0.5, −0.1). We check back on the graphs of f to find the y-coordinates of the
inflection points, and find that these points are approximately (−35.3, −0.015), (−5.0, −0.005), (−1, 0), (−0.5, 0.00001),
and (−0.1, 0.0000066).
17. y = f(x) =
√
x
x 2 + x +1 . FromaCAS,y0 = − 3x2 + x − 1
2 √ x (x 2 + x +1) 2 and y00 = 15x4 +10x 3 − 15x 2 − 6x − 1
4x 3/2 (x 2 + x +1) 3 .
f 0 (x) =0 ⇔ x ≈ 0.43,sof is increasing on (0, 0.43) and decreasing on (0.43, ∞). There is a local maximum value of
f(0.43) ≈ 0.41. f 00 (x) =0 ⇔ x ≈ 0.94,sof is CD on (0, 0.94) and CU on (0.94, ∞). There is an inflection point at
(0.94, 0.34).
19. y = f(x) = √ x +5sinx, x ≤ 20.
From a CAS, y 0 =
5cosx +1
2 √ x +5sinx and y00 = − 10 cos x +25sin2 x +10x sin x +26
.
4(x +5sinx) 3/2
We’ll start with a graph of g(x) =x +5sinx. Notethatf(x) = g(x) is only defined if g(x) ≥ 0. g(x) =0 ⇔ x =0
or x ≈−4.91, −4.10, 4.10,and4.91. Thus, the domain of f is [−4.91, −4.10] ∪ [0, 4.10] ∪ [4.91, 20].
From the expression for y 0 , we see that y 0 =0 ⇔ 5cosx +1=0 ⇒ x 1 =cos −1
− 1 5 ≈ 1.77 and
x 2 =2π − x 1 ≈−4.51 (not in the domain of f ). The leftmost zero of f 0 is x 1 − 2π ≈−4.51. Moving to the right, the zeros
of f 0 are x 1 , x 1 +2π, x 2 +2π, x 1 +4π, andx 2 +4π. Thus, f is increasing on (−4.91, −4.51), decreasing on
(−4.51, −4.10),increasingon(0, 1.77), decreasing on (1.77, 4.10),increasingon(4.91, 8.06), decreasing on (8.06, 10.79),
increasing on (10.79, 14.34), decreasing on (14.34, 17.08), and increasing on (17.08, 20). The local maximum values are
f(−4.51) ≈ 0.62, f(1.77) ≈ 2.58, f(8.06) ≈ 3.60,andf(14.34) ≈ 4.39. The local minimum values are f(10.79) ≈ 2.43
and f(17.08) ≈ 3.49.
f is CD on (−4.91, −4.10), (0, 4.10), (4.91, 9.60),CUon(9.60, 12.25),CD
on (12.25, 15.81),CUon(15.81, 18.65), and CD on (18.65, 20). Thereare
inflection points at (9.60, 2.95), (12.25, 3.27), (15.81, 3.91),and(18.65, 4.20).

F.
TX.10SECTION 4.6 GRAPHING WITH CALCULUS AND CALCULATORS ¤ 189
21. y = f(x) = 1 − e1/x
1+e . 2e 1/x
1/x FromaCAS,y0 =
x 2 (1 + e 1/x ) and 2 y00 = −2e1/x (1 − e 1/x +2x +2xe 1/x )
.
x 4 (1 + e 1/x ) 3
f is an odd function defined on (−∞, 0) ∪ (0, ∞). Its graph has no x-ory-intercepts. Since
lim
x→±∞
f(x) =0,thex-axis
is a HA. f 0 (x) > 0 for x 6= 0,sof is increasing on (−∞, 0) and (0, ∞). It has no local extreme values.
f 00 (x) =0for x ≈ ±0.417,sof is CU on (−∞, −0.417),CDon(−0.417, 0),CUon(0, 0.417), and CD on (0.417, ∞).
f has IPs at (−0.417, 0.834) and (0.417, −0.834).
23. (a) f(x) =x 1/x
(b) Recall that a b = e b ln a .
lim x1/x = lim e(1/x)lnx .Asx → 0 + , ln x
x→0 + x→0 + x
→−∞,sox1/x = e (1/x)lnx → 0. This
indicates that there is a hole at (0, 0). Asx →∞, we have the indeterminate form ∞ 0 . lim
x→∞ x1/x = lim
x→∞ e(1/x)lnx ,
ln x H 1/x
but lim = lim =0,so lim
x→∞ x x→∞ 1 x→∞ x1/x = e 0 =1. This indicates that y =1is a HA.
(c) Estimated maximum: (2.72, 1.45). No estimated minimum. We use logarithmic differentiation to find any critical
numbers. y = x 1/x ⇒ ln y = 1 y0
lnx ⇒
x y = 1 x · 1
x +(lnx) − 1
1 − ln x
⇒ y 0 = x 1/x =0 ⇒
x 2 x 2
ln x =1 ⇒ x = e. For0 e, y 0 < 0,sof(e) =e 1/e is a local maximum value. This
point is approximately (2.7183, 1.4447), which agrees with our estimate.
(d) From the graph, we see that f 00 (x) =0at x ≈ 0.58 and x ≈ 4.37. Sincef 00
changes sign at these values, they are x-coordinates of inflection points.

F.
190 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
25.
From the graph of f(x) =sin(x +sin3x) in the viewing rectangle [0,π] by [−1.2, 1.2], it looks like f has two maxima
and two minima. If we calculate and graph f 0 (x) =[cos(x +sin3x)] (1 + 3 cos 3x) on [0, 2π], we see that
the graph of f 0 appears to be almost tangent to the x-axis at about x =0.7. The graph of
f 00 = − [sin(x +sin3x)] (1 + 3 cos 3x) 2 +cos(x +sin3x)(−9sin3x) is even more interesting near this x-value:
it seems to just touch the x-axis.
If we zoom in on this place on the graph of f 00 ,weseethatf 00 actually does cross the axis twice near x =0.65,
indicating a change in concavity for a very short interval. If we look at the graph of f 0 onthesameinterval,weseethatit
changes sign three times near x =0.65, indicating that what we had thought was a broad extremum at about x =0.7 actually
consists of three extrema (two maxima and a minimum). These maximum values are roughly f(0.59) = 1 and f(0.68) = 1,
and the minimum value is roughly f(0.64) = 0.99996. There are also a maximum value of about f(1.96) = 1 and minimum
values of about f(1.46) = 0.49 and f(2.73) = −0.51. The points of inflection on (0,π) are about (0.61, 0.99998),
(0.66, 0.99998), (1.17, 0.72), (1.75, 0.77), and(2.28, 0.34). On(π, 2π), they are about (4.01, −0.34), (4.54, −0.77),
(5.11, −0.72), (5.62, −0.99998),and(5.67, −0.99998). TherearealsoIPat(0, 0) and (π, 0). Note that the function is odd
and periodic with period 2π, and it is also rotationally symmetric about all points of the form ((2n +1)π, 0), n an integer.
27. f(x) =x 4 + cx 2 = x 2 x 2 + c . Note that f is an even function. For c ≥ 0, the only x-intercept is the point (0, 0). We
calculate f 0 (x) =4x 3 +2cx =4x x 2 + 1 c ⇒ f 00 (x) =12x 2 +2c. Ifc ≥ 0, x =0is the only critical point and there
2
is no inflection point. As we can see from the examples, there is no change in the basic shape of the graph for c ≥ 0;itmerely

F.
TX.10SECTION 4.6 GRAPHING WITH CALCULUS AND CALCULATORS ¤ 191
becomes steeper as c increases. For c =0, the graph is the simple curve y = x 4 . For c0, we see that there is a HA at y =1, and that the graph
spreads out as c increases. At first glance there appears to be a minimum at (0, 0),butf(0) is undefined, so there is no
minimum or maximum. For c0,theinflection points spread out as c increases, and if c0, there are IP at
±
2c/3,e −3/2 . Note that the y-coordinate of the IP is constant.
31. Note that c =0is a transitional value at which the graph consists of the x-axis. Also, we can see that if we substitute −c for c,
cx
the function f(x) = will be reflected in the x-axis, so we investigate only positive values of c (except c = −1,asa
1+c 2 x2 demonstration of this reflective property). Also, f is an odd function. lim f(x) =0,soy =0is a horizontal asymptote for
x→±∞
all c. We calculate f 0 (x) = (1 + c2 x 2 )c − cx(2c 2 x)
= − c(c2 x 2 − 1)
(1 + c 2 x 2 ) 2 (1 + c 2 x 2 ) . f 0 (x) =0 ⇔ c 2 x 2 − 1=0 ⇔ x = ±1/c.
2
So there is an absolute maximum value of f(1/c) = 1 and an absolute minimum value of f(−1/c) =− 1 .Theseextrema
2 2
have the same value regardless of c, but the maximum points move closer to the y-axis as c increases.
f 00 (x) = (−2c3 x)(1 + c 2 x 2 ) 2 − (−c 3 x 2 + c)[2(1 + c 2 x 2 )(2c 2 x)]
(1 + c 2 x 2 ) 4
= (−2c3 x)(1 + c 2 x 2 )+(c 3 x 2 − c)(4c 2 x)
= 2c3 x(c 2 x 2 − 3)
(1 + c 2 x 2 ) 3 (1 + c 2 x 2 ) 3
f 00 (x) =0 ⇔ x =0or ± √ 3/c,sothereareinflection points at (0, 0) and
at ± √ 3/c, ± √ 3/4 .Again,they-coordinate of the inflection points does not
depend on c,butasc increases, both inflection points approach the y-axis.

F.
192 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
33. f(x) =cx +sinx ⇒ f 0 (x) =c +cosx ⇒ f 00 (x) =− sin x
f(−x) =−f(x),sof is an odd function and its graph is symmetric with respect to the origin.
f(x) =0 ⇔ sin x = −cx,so0 is always an x-intercept.
f 0 (x) =0 ⇔ cos x = −c, so there is no critical number when |c| > 1. If|c| ≤ 1, then there are infinitely
many critical numbers. If x 1 is the unique solution of cos x = −c in the interval [0,π], then the critical numbers are 2nπ ± x 1 ,
where n ranges over the integers. (Special cases: When c =1, x 1 =0;whenc =0, x = π ;andwhenc = −1, x 2 1 = π.)
f 00 (x) < 0 ⇔ sin x>0, sof is CD on intervals of the form (2nπ, (2n +1)π). f is CU on intervals of the form
((2n − 1)π, 2nπ). Theinflection points of f are the points (2nπ, 2nπc),wheren is an integer.
If c ≥ 1, thenf 0 (x) ≥ 0 for all x,sof is increasing and has no extremum. If c ≤−1,thenf 0 (x) ≤ 0 for all x,sof is
decreasing and has no extremum. If |c| < 1, thenf 0 (x) > 0 ⇔ cos x>−c ⇔ x is in an interval of the form
(2nπ − x 1 , 2nπ + x 1 ) for some integer n. These are the intervals on which f is increasing. Similarly, we
find that f is decreasing on the intervals of the form (2nπ + x 1 , 2(n +1)π − x 1 ).Thus,f has local maxima at the points
2nπ + x 1,wheref has the values c(2nπ + x 1)+sinx 1 = c(2nπ + x √ 1)+ 1 − c 2 ,andf has local minima at the points
2nπ − x 1,wherewehavef(2nπ − x 1)=c(2nπ − x 1) − sin x 1 = c(2nπ − x 1) − √ 1 − c 2 .
The transitional values of c are −1 and 1. Theinflection points move vertically, but not horizontally, when c changes.
When |c| ≥ 1, there is no extremum. For |c| < 1, the maxima are spaced
2π apart horizontally, as are the minima. The horizontal spacing between
maxima and adjacent minima is regular (and equals π)whenc =0,but
the horizontal space between a local maximum and the nearest local
minimum shrinks as |c| approaches 1.
35. If c0,then
lim
x→−∞
If c =0,thenf(x) =x,so
lim
x→−∞ xe−cx =
lim
x→−∞
f(x) =−∞,and lim f(x) H 1
= lim =0.
x→∞ x→∞ cecx lim
x→±∞
f(x) =±∞, respectively.
x H 1
= lim =0,and lim f(x) =∞.
e cx x→−∞ cecx x→∞
So we see that c =0is a transitional value. We now exclude the case c =0, since we know how the function behaves
in that case. To find the maxima and minima of f, we differentiate: f(x) =xe −cx
⇒
f 0 (x) =x(−ce −cx )+e −cx =(1− cx)e −cx .Thisis0 when 1 − cx =0 ⇔ x =1/c. Ifc

F.
TX.10
SECTION 4.7 OPTIMIZATION PROBLEMS ¤ 193
and if c>0, it represents a maximum value. As |c| increases, the maximum or
minimum point gets closer to the origin. To find the inflection points, we
differentiate again: f 0 (x) =e −cx (1 − cx)
⇒
f 00 (x) =e −cx (−c)+(1− cx)(−ce −cx )=(cx − 2)ce −cx . This changes sign
when cx − 2=0 ⇔ x =2/c. Soas|c| increases, the points of inflection get
closer to the origin.
37. (a) f(x) =cx 4 − 2x 2 +1.Forc =0, f(x) =−2x 2 +1, a parabola whose vertex, (0, 1), is the absolute maximum. For
c>0, f(x) =cx 4 − 2x 2 +1opens upward with two minimum points. As c → 0, the minimum points spread apart and
move downward; they are below the x-axis for 0 0, so there is a local minimum at
x = ±1/ √
c.Heref ±1/ √
c = c(1/c 2 ) − 2/c +1=−1/c +1.
But ±1/ √
c, −1/c +1 lies on y =1− x 2 since 1 − ±1/ √ 2
c =1− 1/c.
4.7 Optimization Problems
1. (a)
First Number Second Number Product
1 22 22
2 21 42
3 20 60
4 19 76
5 18 90
6 17 102
7 16 112
8 15 120
9 14 126
10 13 130
11 12 132
We needn’t consider pairs where the first number
is larger than the second, since we can just
interchange the numbers in such cases. The
answer appears to be 11 and 12, but we have
considered only integers in the table.

F.
194 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
(b) Call the two numbers x and y. Thenx + y =23,soy =23− x. Call the product P .Then
P = xy = x(23 − x) =23x − x 2 ,sowewishtomaximizethefunctionP (x) =23x − x 2 .SinceP 0 (x) =23− 2x,
we see that P 0 (x) =0 ⇔ x = 23 2 =11.5. Thus, the maximum value of P is P (11.5) = (11.5)2 = 132.25 and it
occurs when x = y =11.5.
Or: Note that P 00 (x) =−2 < 0 for all x,soP is everywhere concave downward and the local maximum at x =11.5
must be an absolute maximum.
3. The two numbers are x and 100
100
,wherex>0. Minimize f(x) =x +
x x . f 0 (x) =1− 100
x = x2 − 100
. The critical
2 x 2
number is x =10.Sincef 0 (x) < 0 for 0 10, there is an absolute minimum at x =10.
The numbers are 10 and 10.
5. If the rectangle has dimensions x and y, thenitsperimeteris2x +2y = 100 m, so y =50− x. Thus, the area is
A = xy = x(50 − x). We wish to maximize the function A(x) =x(50 − x) =50x − x 2 ,where0

F.
TX.10
SECTION 4.7 OPTIMIZATION PROBLEMS ¤ 195
11. xy =1.5 × 10 6 ,soy =1.5 × 10 6 /x. Minimize the amount of fencing, which is
3x +2y =3x +2(1.5 × 10 6 /x) =3x +3× 10 6 /x = F (x).
F 0 (x) =3− 3 × 10 6 /x 2 =3(x 2 − 10 6 )/x 2 . The critical number is x =10 3 and
F 0 (x) < 0 for 0 10 3 , so the absolute minimum
occurs when x =10 3 and y =1.5 × 10 3 .
The field should be 1000 feet by 1500 feet with the middle fence parallel to the short side of the field.
13. Let b be the length of the base of the box and h the height. The surface area is 1200 = b 2 +4hb ⇒ h = (1200 − b 2 )/(4b).
The volume is V = b 2 h = b 2 (1200 − b 2 )/4b =300b − b 3 /4 ⇒ V 0 (b) =300− 3 4 b2 .
V 0 (b) =0 ⇒ 300 = 3 4 b2 ⇒ b 2 =400 ⇒ b = √ 400 = 20. SinceV 0 (b) > 0 for 0 3 45
. The minimum cost is C 3 45
=32(2.8125) 2/3 +180 √ 2.8125 ≈ $191.28.
16 16
17. The distance from a point (x, y) on the line y =4x +7to the origin is (x − 0) 2 +(y − 0) 2 = x 2 + y 2 .However,itis
easier to work with the square of the distance; that is, D(x) =
x2 + y 2 2
= x 2 + y 2 = x 2 +(4x +7) 2 . Because the
distance is positive, its minimum value will occur at the same point as the minimum value of D.
D 0 (x) =2x +2(4x + 7)(4) = 34x +56,soD 0 (x) =0 ⇔ x = − 28
17 .
D 00 (x) =34> 0,soD is concave upward for all x. Thus,D has an absolute minimum at x = − 28
17
origin is (x, y) = − 28 , 4 − 28
17 17
+7
=
−
28
17 , 7 17
.
. The point closest to the
19. From the figure, we see that there are two points that are farthest away from
S
− 1 3 =
16
,andS(1) = 0, we see that the maximum distance is 3
y = ± 4 − 4
− 1 2
= ±
3
A(1, 0). The distance d from A to an arbitrary point P (x, y) on the ellipse is
d = (x − 1) 2 +(y − 0) 2 and the square of the distance is
S = d 2 = x 2 − 2x +1+y 2 = x 2 − 2x +1+(4− 4x 2 )=−3x 2 − 2x +5.
S 0 = −6x − 2 and S 0 =0 ⇒ x = − 1 3 .NowS00 = −6 < 0, so we know
that S has a maximum at x = − 1 .Since−1 ≤ x ≤ 1, S(−1) = 4,
3
32
= ± √ 4
9 3 2 ≈ ±1.89. The points are −
1
, ± √ 4
3 3 2 .
16
3
. The corresponding y-values are

F.
196 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
21. The area of the rectangle is (2x)(2y) =4xy. Alsor 2 = x 2 + y 2 so
y = r 2 −
and 2y = √ 2 r.
√
1
2
r
2
=
1
2 r2 = 1
y = √ r 2 − x 2 , so the area is A(x) =4x √ r 2 − x 2 .Now
A 0 (x) =4 √r2
− x 2 −
x = 1 √
2
r. Clearly this gives a maximum.
x 2
√ =4 r2 − 2x 2
√ . The critical number is
r2 − x 2 r2 − x2 √
2
r = x, which tells us that the rectangle is a square. The dimensions are 2x = √ 2 r
23. The height h of the equilateral triangle with sides of length L is √ 3
2 L,
since h 2 +(L/2) 2 = L 2 ⇒ h 2 = L 2 − 1 4 L2 = 3 4 L2 ⇒
√
h = √ 3
3
L. Using similar triangles, L − y 2
=
2
x
√
3
2 L
L/2 = √ 3 ⇒
√ √
√ 3
3
3 x =
2 L − y ⇒ y = 2 L − √ √
3
3 x ⇒ y = (L − 2x).
2
The area of the inscribed rectangle is A(x) =(2x)y = √ 3 x(L − 2x) = √ 3 Lx − 2 √ 3 x 2 ,where0 ≤ x ≤ L/2. Now
0=A 0 (x) = √ 3 L − 4 √ 3 x ⇒ x = √ 3 L 4 √ 3 = L/4. SinceA(0) = A(L/2) = 0, the maximum occurs when
x = L/4,andy = √ 3
2 L − √ 3
4 L = √ 3
4 L, so the dimensions are L/2 and √ 3
4 L.
25. Theareaofthetriangleis
A(x) = 1 (2t)(r + x) =t(r + x) =√ r
2 2 − x 2 (r + x). Then
0=A 0 −2x
(x) =r
2 √ r 2 − x + √ −2x
r 2 − x 2 + x
2 2 √ r 2 − x 2
= − x2 + rx
√
r2 − x + √ r 2 − x 2 ⇒
2
x 2 + rx
√
r2 − x 2 = √ r 2 − x 2 ⇒ x 2 + rx = r 2 − x 2 ⇒ 0=2x 2 + rx − r 2 =(2x − r)(x + r) ⇒
x = 1 r or x = −r. NowA(r) =0=A(−r) ⇒ the maximum occurs where x = 1 r, so the triangle has height
2 2
r + 1 r = 3 r and base 2 r
2 2 2 −
1
2 r2 3
=2
4 r2 = √ 3 r.
27. The cylinder has volume V = πy 2 (2x). Alsox 2 + y 2 = r 2 ⇒ y 2 = r 2 − x 2 ,so
V (x) =π(r 2 − x 2 )(2x) =2π(r 2 x − x 3 ),where0 ≤ x ≤ r.
V 0 (x) =2π r 2 − 3x 2 =0 ⇒ x = r/ √ 3.NowV (0) = V (r) =0,sothereisa
maximum when x = r/ √ 3 and V r/ √ 3 = π(r 2 − r 2 /3) 2r/ √ 3 =4πr 3 / 3 √ 3 .

F.
TX.10
SECTION 4.7 OPTIMIZATION PROBLEMS ¤ 197
29. Thecylinderhassurfacearea
2(area of the base) + (lateral surface area) = 2π(radius) 2 +2π(radius)(height)
=2πy 2 +2πy(2x)
Now x 2 + y 2 = r 2 ⇒ y 2 = r 2 − x 2 ⇒ y = √ r 2 − x 2 , so the surface area is
S(x)=2π(r 2 − x 2 )+4πx √ r 2 − x 2 , 0 ≤ x ≤ r
=2πr 2 − 2πx 2 +4π x √ r 2 − x 2
Thus, S 0 (x) =0− 4πx +4π x · 1
2 (r2 − x 2 ) −1/2 (−2x)+(r 2 − x 2 ) 1/2 · 1
x 2
=4π −x − √
r2 − x + √
r 2 − x 2 =4π · −x √ r 2 − x 2 − x 2 + r 2 − x 2
√ 2 r2 − x 2
S 0 (x) =0 ⇒ x √ r 2 − x 2 = r 2 − 2x 2 () ⇒ x √ r 2 − x 2 2
=(r 2 − 2x 2 ) 2 ⇒
x 2 (r 2 − x 2 )=r 4 − 4r 2 x 2 +4x 4 ⇒ r 2 x 2 − x 4 = r 4 − 4r 2 x 2 +4x 4 ⇒ 5x 4 − 5r 2 x 2 + r 4 =0.
This is a quadratic equation in x 2 . By the quadratic formula, x 2 = 5 ± √ 5
r 2 , but we reject the root with the + sign since it
10
5 −
doesn’t satisfy (). [The right side is negative and the left side is positive.] So x =
√ 5
r. SinceS(0) = S(r) =0,the
10
maximum surface area occurs at the critical number and x 2 = 5 − √ 5
10
r 2 ⇒ y 2 = r 2 − 5 − √ 5
10
r 2 = 5+√ 5
10
r 2 ⇒
the surface area is
2π
5+ √ 5
10
r 2 +4π
5 − √
5 5+ √ √ √
5
r 2 = πr 2 2 · 5+√ 5 (5− 5)(5+ 5)
+4
= πr 2 5+ √
5
+ 2√ 20
10 10 10 10
5 5
= πr 2 5+ √ 5+2·2 √
5
= πr 2 5+5 √
5
= πr 2 1+ √ 5 .
5
5
31. xy = 384 ⇒ y =384/x. Total area is
A(x) =(8+x)(12 + 384/x) = 12(40 + x +256/x),so
A 0 (x) = 12(1 − 256/x 2 )=0 ⇒ x =16. There is an absolute minimum
when x =16since A 0 (x) < 0 for 0 16.
When x =16, y =384/16 = 24, so the dimensions are 24 cm and 36 cm.
33. Let x be the length of the wire used for the square. The total area is
x
√
2 1 10 − x 3 10 − x
A(x)= +
4 2 3 2 3
= 1 16 x2 + √ 3
36 (10 − x)2 , 0 ≤ x ≤ 10
A 0 (x) = 1 x − √ √3
3
9
(10 − x) =0 ⇔ x + 4√ 3
x − 40√ 3
=0 ⇔ x = 40√ 3
8 18 72 72 72 9+4 √ .NowA(0) = 3 36
A(10) = 100 =6.25 and A 40 √
3
≈ 2.72,so
16
9+4 √ 3
(a ) The maximum area occurs when x =10m, and all the wire is used for the square.
(b) The minimum area occurs when x = 40√ 3
9+4 √ ≈ 4.35 m.
3
100 ≈ 4.81,

F.
198 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
35. The volume is V = πr 2 h and the surface area is
V
S(r) =πr 2 +2πrh = πr 2 +2πr = πr 2 + 2V πr 2 r .
S 0 (r) =2πr − 2V
r 2 =0 ⇒ 2πr3 =2V ⇒ r = 3
V
π cm.
This gives an absolute minimum since S 0 (r) < 0 for 0 0 for r> 3
V
π .
When r = 3
V
π , h =
V
πr = V V 2 π(V/π) = 3 2/3 π cm.
37. h 2 + r 2 = R 2 ⇒ V = π 3 r2 h = π 3 (R2 − h 2 )h = π 3 (R2 h − h 3 ).
V 0 (h) = π 3 (R2 − 3h 2 )=0when h = 1 √
3
R. This gives an absolute maximum, since
V 0 (h) > 0 for 0 1 √
3
R. The maximum volume is
V
√
1
3
R
= π 3
√
1
3
R 3 − 1
3 √ 3 R3
39. By similar triangles, H R = H − h
r
so we’ll solve (1) for h.
h = H − Hr
R
=
HR − Hr
R
Hr
R = H − h
= 2
9 √ 3 πR3 .
(1). The volume of the inner cone is V = 1 3 πr2 h,
⇒
= H (R − r) (2).
R
Thus, V(r) = π 3 r2 · H πH
(R − r) =
R 3R (Rr2 − r 3 )
V 0 (r) = πH
3R (2Rr − 3r2 )= πH r(2R − 3r).
3R
V 0 (r) =0 ⇒ r =0or 2R =3r ⇒ r = 2 R and from (2), h = H
3 R −
2
R
R = H 1
3
R R = 1 H.
3 3
V 0 (r) changes from positive to negative at r = 2 3
R, so the inner cone has a maximum volume of
V = 1 3 πr2 h = 1 3 π 2
3 R 2 1
3 H = 4 27 · 1
3 πR2 H, which is approximately 15% of the volume of the larger cone.
41. P (R) = E2 R
(R + r) 2 ⇒
P 0 (R)= (R + r)2 · E 2 − E 2 R · 2(R + r)
[(R + r) 2 ] 2
⇒
= (R2 +2Rr + r 2 )E 2 − 2E 2 R 2 − 2E 2 Rr
(R + r) 4
= E2 r 2 − E 2 R 2
= E2 (r 2 − R 2 )
= E2 (r + R)(r − R)
= E2 (r − R)
(R + r) 4 (R + r) 4 (R + r) 4 (R + r) 3
P 0 (R) =0 ⇒ R = r ⇒ P (r) = E2 r
(r + r) = E2 r
2 4r = E2
2 4r .
The expression for P 0 (R) shows that P 0 (R) > 0 for Rr. Thus, the maximum value of the
power is E 2 /(4r), and this occurs when R = r.

F.
TX.10
SECTION 4.7 OPTIMIZATION PROBLEMS ¤ 199
43. S =6sh − 3 2 s2 cot θ +3s 2 √ 3
2 csc θ
(a) dS
dθ = 3 2 s2 csc 2 θ − 3s 2 √ 3
2 csc θ cot θ or 3 2 s2 csc θ csc θ − √ 3cotθ .
(b) dS
dθ =0when csc θ − √ 3cotθ =0 ⇒ 1
sin θ − √ 3 cos θ =0 ⇒ cos θ = √
1
sin θ 3
. The First Derivative Test shows
that the minimum surface area occurs when θ =cos −1 √
1
3
≈ 55 ◦ .
(c) If cos θ = 1 √
3
,thencot θ = 1 √
2
and csc θ = √ 3
√
2
, so the surface area is
S =6sh − 3 2 s2 1
√
2
+3s 2 √ √
3 3
2
=6sh + 6
2 √ 2 s2 =6s h + 1
2 √ s 2
√
2
=6sh − 3
2 √ 2 s2 + 9
2 √ 2 s2
√
x2 +25
45. Here T (x) =
6
+ 5 − x , 0 ≤ x ≤ 5 ⇒ T 0 (x) =
8
x
6 √ x 2 +25 − 1 8 =0 ⇔ 8x =6√ x 2 +25 ⇔
16x 2 =9(x 2 + 25) ⇔ x = 15 √
7
.But 15 √
7
> 5, soT has no critical number. Since T (0) ≈ 1.46 and T (5) ≈ 1.18,he
should row directly to B.
47. There are (6 − x) km over land and √ x 2 +4km under the river.
We need to minimize the cost C (measured in $100,000) of the pipeline.
C(x) =(6− x)(4) + √ x 2 +4 (8)
⇒
C 0 (x) =−4+8· 1
2 (x2 +4) −1/2 8x
(2x) =−4+ √
x2 +4 .
C 0 (x) =0 ⇒ 4=
8x
√
x2 +4
⇒ √ x 2 +4=2x ⇒ x 2 +4=4x 2 ⇒ 4=3x 2 ⇒ x 2 = 4 3
⇒
x =2/ √ 3 [0 ≤ x ≤ 6]. Compare the costs for x =0, 2/ √ 3,and6. C(0) = 24 + 16 = 40,
C 2/ √ 3 =24− 8/ √ 3+32/ √ 3=24+24/ √ 3 ≈ 37.9,andC(6) = 0 + 8 √ 40 ≈ 50.6. So the minimum cost is about
$3.79 million when P is 6 − 2/ √ 3 ≈ 4.85 km east of the refinery.
49. The total illumination is I(x) = 3k
x + k
, 0

F.
200 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
51. Every line segment in the first quadrant passing through (a, b) with endpoints on the x-
and y-axes satisfies an equation of the form y − b = m(x − a),wherem 0 and m− 3 b
.Thus,S has its absolute minimum value when m = − 3 b
a
That value is
S
− 3
b
a
2 2
=
a + b 3 a
+
−a 3 b
− b =
a + 3√ 2
3√ 2
ab
b a 2 + a2 b + b
= a 2 +2a 4/3 b 2/3 + a 2/3 b 4/3 + a 4/3 b 2/3 +2a 2/3 b 4/3 + b 2 = a 2 +3a 4/3 b 2/3 +3a 2/3 b 4/3 + b 2
The last expression is of the form x 3 +3x 2 y +3xy 2 + y 3 [=(x + y) 3 ] withx = a 2/3 and y = b 2/3 ,
so we can write it as (a 2/3 + b 2/3 ) 3 and the shortest such line segment has length √ S =(a 2/3 + b 2/3 ) 3/2 .
53. (a) If c(x) = C(x)
x
, then, by Quotient Rule, we have c0 (x) = xC0 (x) − C(x)
.Nowc 0 (x) =0when xC 0 (x) − C(x) =0
x 2
and this gives C 0 (x) = C(x)
x
= c(x). Therefore, the marginal cost equals the average cost.
(b) (i) C(x) =16,000 + 200x +4x 3/2 , C(1000) = 16,000 + 200,000 + 40,000 √ 10 ≈ 216,000 + 126,491, so
C(1000) ≈ $342,491. c(x) =C(x)/x = 16,000
x
C 0 (1000) = 200 + 60 √ 10 ≈ $389.74/unit.
(ii) We must have C 0 (x) =c(x) ⇔ 200 + 6x 1/2 = 16,000
x
x =(8,000) 2/3 = 400 units. To check that this is a minimum, we calculate
+200+4x 1/2 , c(1000) ≈ $342.49/unit. C 0 (x) = 200 + 6x 1/2 ,
+ 200 + 4x 1/2 ⇔ 2x 3/2 =16,000 ⇔
c 0 (x) = −16,000
x 2 + 2 √
x
= 2 x 2 (x3/2 − 8000).Thisisnegativeforx400,soc is decreasing on (0, 400) andincreasingon(400, ∞). Thus,c has an absolute minimum
at x = 400. [Note: c 00 (x) is not positive for all x>0.]
(iii) The minimum average cost is c(400) = 40 + 200 + 80 = $320/unit.
a .

F.
TX.10
SECTION 4.7 OPTIMIZATION PROBLEMS ¤ 201
55. (a) We are given that the demand function p is linear and p(27,000) = 10, p(33,000) = 8, so the slope is
10 − 8
= − 1 and an equation of the line is y − 10 =
− 1
27,000 − 33,000 3000 3000 (x − 27,000) ⇒
y = p(x) =− 1
3000
x +19=19− (x/3000).
(b) The revenue is R(x) =xp(x) =19x − (x 2 /3000) ⇒ R 0 (x) =19− (x/1500) = 0 when x =28,500. Since
R 00 (x) =−1/1500 < 0, the maximum revenue occurs when x =28,500 ⇒ the price is p(28,500) = $9.50.
57. (a) As in Example 6, we see that the demand function p is linear. We are given that p(1000) = 450 and deduce that
p(1100) = 440,sincea$10 reduction in price increases sales by 100 per week. The slope for p is
so an equation is p − 450 = − 1
1
10
(x − 1000) or p(x) =− x +550.
10
(b) R(x) =xp(x) =− 1 10 x2 +550x. R 0 (x) =− 1 x + 550 = 0 when x = 5(550) = 2750.
5
p(2750) = 275, so the rebate should be 450 − 275 = $175.
440 − 450
1100 − 1000 = − 1 10 ,
(c) C(x) =68,000 + 150x ⇒ P (x) =R(x) − C(x) =− 1
10 x2 + 550x − 68,000 − 150x = − 1
10 x2 +400x − 68,000,
P 0 (x) =− 1 x +400=0 when x =2000. p(2000) = 350. Therefore, the rebate to maximize profits should be
5
450 − 350 = $100.
59. Here s 2 = h 2 + b 2 /4,soh 2 = s 2 − b 2 /4. TheareaisA = 1 2 b s 2 − b 2 /4.
Let the perimeter be p,so2s + b = p or s =(p − b)/2
⇒
A(b) = 1 b (p − b)
2 2 /4 − b 2 /4=b p 2 − 2pb/4. Now
A 0 p2 − 2pb bp/4
(b) =
−
4 p2 − 2pb = −3pb + p2
4 p 2 − 2pb .
Therefore, A 0 (b) =0 ⇒ −3pb + p 2 =0 ⇒ b = p/3. SinceA 0 (b) > 0 for bp/3,there
is an absolute maximum when b = p/3.Butthen2s + p/3 =p,sos = p/3 ⇒ s = b ⇒ the triangle is equilateral.
61. Note that |AD| = |AP | + |PD| ⇒ 5=x + |PD| ⇒ |PD| =5− x.
Using the Pythagorean Theorem for ∆PDB and ∆PDC gives us
L(x)=|AP | + |BP| + |CP| = x + (5 − x) 2 +2 2 + (5 − x) 2 +3 2
= x + √ x 2 − 10x +29+ √ x 2 − 10x +34 ⇒
L 0 (x) =1+
x − 5
√
x2 − 10x +29 + x − 5
√ . From the graphs of L
x2 − 10x +34
and L 0 , it seems that the minimum value of L is about L(3.59) = 9.35 m.

F.
TX.10
202 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
√
3=tan ψ +
π
⇒ ψ = π . 6
6
63. The total time is
T (x)=(time from A to C)+(time from C to B)
√
a2 + x
=
b2 +(d − x)
+
, 0 0 for 0 ≤ t< √ 1
3
and f 0 (t) < 0 for t> √ 1
3
,sof has an absolute maximum when t = √ 1
3
and tan θ =
2 1/ √ 3
1+3 1/ √ 3 2 = √ 1 3
⇒ θ = π 6
. Substituting for t and θ in 3t =tan(ψ + θ) gives us

F.
TX.10
SECTION 4.7 OPTIMIZATION PROBLEMS ¤ 203
69. From the figure, tan α = 5 2
and tan β =
x 3 − x .Since
5
α + β + θ = 180 ◦ = π, θ = π − tan −1 x
2
− 5
1
−
x 2
dθ
dx = − 1
5
1+
x
Now dθ
dx =0 ⇒ 5
x 2 +25 = 2
x 2 − 6x +13
= x2
x 2 +25 ·
1+
5 (3 − x)2
−
x2 (3 − x) 2 +4 ·
2
3 − x
2
− tan −1 3 − x
2
2
(3 − x) 2
2
(3 − x) 2 .
⇒ 2x 2 +50=5x 2 − 30x +65 ⇒
⇒
3x 2 − 30x +15=0 ⇒ x 2 − 10x +5=0 ⇒ x =5± 2 √ 5. We reject the root with the + sign, since it is larger
than 3.
dθ/dx > 0 for x5 − 2 √ 5,soθ is maximized when
|AP | = x =5− 2 √ 5 ≈ 0.53.
71. In the small triangle with sides a and c and hypotenuse W , sin θ = a W and
cos θ = c W . In the triangle with sides b and d and hypotenuse L, sin θ = d L and
cos θ = b .Thus,a = W sin θ, c = W cos θ, d = L sin θ,andb = L cos θ,sothe
L
area of the circumscribed rectangle is
A(θ)=(a + b)(c + d) =(W sin θ + L cos θ)(W cos θ + L sin θ)
= W 2 sin θ cos θ + WLsin 2 θ + LW cos 2 θ + L 2 sin θ cos θ
= LW sin 2 θ + LW cos 2 θ +(L 2 + W 2 )sinθ cos θ
= LW (sin 2 θ +cos 2 θ)+(L 2 + W 2 ) · 1
2 · 2sinθ cos θ = LW + 1 2 (L2 + W 2 )sin2θ, 0 ≤ θ ≤ π 2
This expression shows, without calculus, that the maximum value of A(θ) occurs when sin 2θ =1 ⇔ 2θ = π 2
⇒
θ = π . So the maximum area is A
π
4 4 = LW +
1
2 (L2 + W 2 )= 1 2 (L2 +2LW + W 2 )= 1 (L + W 2 )2 .
73. (a) If k = energy/km over land, then energy/km over water =1.4k.
So the total energy is E =1.4k √ 25 + x 2 + k(13 − x), 0 ≤ x ≤ 13,
and so dE
dx =
1.4kx
− k.
(25 + x 2 1/2
)
Set dE
dx =0: 1.4kx = k(25 + x2 ) 1/2 ⇒ 1.96x 2 = x 2 +25 ⇒ 0.96x 2 =25 ⇒ x = 5 √
0.96
≈ 5.1.
Testing against the value of E at the endpoints: E(0) = 1.4k(5) + 13k =20k, E(5.1) ≈ 17.9k, E(13) ≈ 19.5k.
Thus, to minimize energy, the bird should fly to a point about 5.1 km from B.

F.
204 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
(b) If W/L is large, the bird would flytoapointC that is closer to B than to D to minimize the energy used flying over water.
If W/L is small, the bird would flytoapointC that is closer to D than to B to minimize the distance of the flight.
E = W √ 25 + x 2 + L(13 − x) ⇒ dE
dx =
Wx
√ − L =0when W √
25 + x
25 + x
2 L = 2
. By the same sort of
x
argument as in part (a), this ratio will give the minimal expenditure of energy if the bird heads for the point x km from B.
(c) For flight direct to D, x =13, so from part (b), W/L =
√
25 + 13 2
13
≈ 1.07. There is no value of W/L for which the bird
should fly directly to B. But note that lim =∞, so if the point at which E is a minimum is close to B,then
x→0 +(W/L)
W/L is large.
(d) Assuming that the birds instinctively choose the path that minimizes the energy expenditure, we can use the equation for
dE/dx =0from part (a) with 1.4k = c, x =4,andk =1: c(4) = 1 · (25 + 4 2 ) 1/2 ⇒ c = √ 41/4 ≈ 1.6.
4.8 Newton's Method
1. (a) The tangent line at x =1intersects the x-axis at x ≈ 2.3,so
x 2 ≈ 2.3. The tangent line at x =2.3 intersects the x-axis at
x ≈ 3,sox 3 ≈ 3.0.
(b) x 1 =5would not be a better first approximation than x 1 =1since the tangent line is nearly horizontal. In fact, the second
approximation for x 1 =5appears to be to the left of x =1.
3. Since x 1 =3and y =5x − 4 is tangent to y = f(x) at x =3, we simply need to find where the tangent line intersects the
x-axis. y =0 ⇒ 5x 2 − 4=0 ⇒ x 2 = 4 . 5
5. f(x) =x 3 +2x − 4 ⇒ f 0 (x) =3x 2 +2,sox n+1 = x n − x3 n +2x n − 4
.Nowx
3x 2 1 =1
n +2
⇒
x 2 =1− 1+2− 4 −1
=1−
3 · 1 2 +2 5 =1.2 ⇒ x3 =1.2 − (1.2)3 +2(1.2) − 4
≈ 1.1797.
3(1.2) 2 +2
7. f(x) =x 5 − x − 1 ⇒ f 0 (x) =5x 4 − 1, sox n+1 = x n − x5 n − x n − 1
.Nowx
5x 4 1 =1
n − 1
⇒
x 2 =1− 1 − 1 − 1
5 − 1
=1− − 1 4
=1.25 ⇒ x3 =1.25 − (1.25)5 − 1.25 − 1
5(1.25) 4 − 1
9. f(x) =x 3 + x +3 ⇒ f 0 (x) =3x 2 +1,sox n+1 = x n − x3 n + x n +3
.
3x 2 n +1
Now x 1 = −1
⇒
x 2 = −1 − (−1)3 +(−1) + 3
3(−1) 2 +1
= −1 −
−1 − 1+3
3+1
= −1 − 1 4 = −1.25.
Newton’s method follows the tangent line at (−1, 1) down to its intersection with
the x-axis at (−1.25, 0), giving the second approximation x 2 = −1.25.
≈ 1.1785.

F.
TX.10
SECTION 4.8 NEWTON’S METHOD ¤ 205
11. To approximate x = 5√ 20 (so that x 5 =20),wecantakef(x) =x 5 − 20. Sof 0 (x) =5x 4 , and thus,
x n+1 = x n − x5 n − 20
.Since 5√ 32 = 2 and 32 is reasonably close to 20, we’ll use x
5x 4 1 =2. We need to find approximations
n
until they agree to eight decimal places. x 1 =2 ⇒ x 2 =1.85, x 3 ≈ 1.82148614, x 4 ≈ 1.82056514,
x 5 ≈ 1.82056420 ≈ x 6 .So 5√ 20 ≈ 1.82056420, to eight decimal places.
Here is a quick and easy method for finding the iterations for Newton’s method on a programmable calculator.
(The screens shown are from the TI-84 Plus, but the method is similar on other calculators.) Assign f(x) =x 5 − 20
to Y 1 ,andf 0 (x) =5x 4 to Y 2 .Nowstorex 1 =2in X andthenenterX − Y 1 /Y 2 → X to get x 2 =1.85. By successively
pressing the ENTER key, you get the approximations x 3, x 4, ....
In Derive, load the utility file SOLVE. EnterNEWTON(xˆ5-20,x,2) and then APPROXIMATE to get
[2, 1.85, 1.82148614, 1.82056514, 1.82056420]. You can request a specific iteration by adding a fourth argument. For
example, NEWTON(xˆ5-20,x,2,2) gives [2, 1.85, 1.82148614].
In Maple, make the assignments f := x → xˆ5 − 20;, g := x → x − f(x)/D(f)(x);,andx := 2.;. Repeatedly execute
the command x := g(x); to generate successive approximations.
In Mathematica, make the assignments f[x_ ]:=xˆ5 − 20, g[x_ ]:=x − f[x]/f 0 [x],andx =2. Repeatedly execute the
command x = g[x] to generate successive approximations.
13. f(x) =x 4 − 2x 3 +5x 2 − 6 ⇒ f 0 (x) =4x 3 − 6x 2 +10x ⇒ x n+1 = x n − x4 n − 2x 3 n +5x 2 n − 6
. We need to find
4x 3 n − 6x 2 n +10x n
approximations until they agree to six decimal places. We’ll let x 1 equal the midpoint of the given interval, [1, 2].
x 1 =1.5 ⇒ x 2 =1.2625, x 3 ≈ 1.218808, x 4 ≈ 1.217563, x 5 ≈ 1.217562 ≈ x 6. So the root is 1.217562 to six decimal
places.
15. sin x = x 2 ,sof(x) =sinx − x 2 ⇒ f 0 (x) =cosx − 2x ⇒
x n+1 = x n − sin xn − x2 n
cos x n − 2x n
.Fromthefigure, the positive root of sin x = x 2 is
near 1. x 1 =1 ⇒ x 2 ≈ 0.891396, x 3 ≈ 0.876985, x 4 ≈ 0.876726 ≈ x 5.So
thepositiverootis0.876726, to six decimal places.

F.
206 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
17. From the graph, we see that there appear to be points of intersection near
x = −0.7 and x =1.2. Solving x 4 =1+x is the same as solving
f(x) =x 4 − x − 1=0. f(x) =x 4 − x − 1 ⇒ f 0 (x) =4x 3 − 1,
so x n+1 = x n − x4 n − x n − 1
.
4x 3 n − 1
x 1 = −0.7 x 1 =1.2
x 2 ≈−0.725253 x 2 ≈ 1.221380
x 3 ≈−0.724493 x 3 ≈ 1.220745
x 4 ≈−0.724492 ≈ x 5 x 4 ≈ 1.220744 ≈ x 5
To six decimal places, the roots of the equation are −0.724492 and 1.220744.
19. From the graph, we see that there appear to be points of intersection near
x =1.5 and x =3. Solving (x − 2) 2 =lnx is the same as solving
f(x) =(x − 2) 2 − ln x =0. f(x) =(x − 2) 2 − ln x
⇒
f 0 (x) =2(x − 2) − 1/x,sox n+1 = x n − (x n − 2) 2 − ln x n
2(x n − 2) − 1/x n
.
x 1 =1.5 x 1 =3
x 2 ≈ 1.406721 x 2 ≈ 3.059167
x 3 ≈ 1.412370 x 3 ≈ 3.057106
x 4 ≈ 1.412391 ≈ x 5 x 4 ≈ 3.057104 ≈ x 5
To six decimal places, the roots of the equation are 1.412391 and 3.057104.
21. From the graph, there appears to be a point of intersection near x =0.6.
Solving cos x = √ x is the same as solving f(x) =cosx − √ x =0.
f(x) =cosx − √
x ⇒ f 0 (x) =− sin x − 1/ 2 √
x ,so
cos x n − √ x n
x n+1 = x n −
− sin x n − 1/ 2 √ .Nowx 1 =0.6
x
⇒ x 2 ≈ 0.641928,
x 3 ≈ 0.641714 ≈ x 4 . To six decimal places, the root of the equation is 0.641714.
23. f(x) =x 6 − x 5 − 6x 4 − x 2 + x +10 ⇒
f 0 (x) =6x 5 − 5x 4 − 24x 3 − 2x +1
⇒
x n+1 = x n − x6 n − x 5 n − 6x 4 n − x 2 n + x n +10
6x 5 n − 5x 4 n − 24x 3 n − 2x n +1 .
From the graph of f, there appear to be roots near −1.9, −1.2, 1.1,and3.

F.
TX.10
SECTION 4.8 NEWTON’S METHOD ¤ 207
x 1 = −1.2
x 1 =1.1
x 1 =3
x 1 = −1.9
x 5 ≈−1.93822883 ≈ x 6
x 3 ≈−1.93828380
x 4 ≈−1.93822884
x 3 ≈−1.21997997 ≈ x 4
x 3 ≈ 1.13929741
x 4 ≈ 1.13929375 ≈ x 5
x 3 ≈ 2.98984106
x 4 ≈ 2.98984102 ≈ x 5
x 2 ≈−1.94278290 x 2 ≈−1.22006245 x 2 ≈ 1.14111662 x 2 ≈ 2.99
To eight decimal places, the roots of the equation are −1.93822883, −1.21997997, 1.13929375,and2.98984102.
25. From the graph, y = x 2√ 2 − x − x 2 and y =1intersect twice, at x ≈−2 and
at x ≈−1. f(x) =x 2 √ 2 − x − x 2 − 1
⇒
f 0 (x) =x 2 · 1
2 (2 − x − x2 ) −1/2 (−1 − 2x)+(2− x − x 2 ) 1/2 · 2x
= 1 2 x(2 − x − x2 ) −1/2 [x(−1 − 2x)+4(2− x − x 2 )]
= x(8 − 5x − 6x2 )
2 (2 + x)(1 − x) ,
√
so x n+1 = x n − x2 n 2 − xn − x 2 n − 1
x n(8 − 5x n − 6x 2 n)
try x 1 = −1.95.
2 (2 + x n)(1 − x n)
.Tryingx 1 = −2 won’t work because f 0 (−2) is undefined, so we’ll
x 1 = −1.95
x 1 = −0.8
x 2 ≈−1.98580357
x 2 ≈−0.82674444
x 3 ≈−1.97899778
x 3 ≈−0.82646236
x 4 ≈−1.97807848
x 4 ≈−0.82646233 ≈ x 5
x 5 ≈−1.97806682
x 6 ≈−1.97806681 ≈ x 7
To eight decimal places, the roots of the equation are −1.97806681 and −0.82646233.
27. Solving 4e −x2 sin x = x 2 − x +1is the same as solving
f(x) =4e −x2 sin x − x 2 + x − 1=0.
f 0 (x) =4e −x2 (cos x − 2x sin x) − 2x +1
⇒
x n+1 = x n −
4e −x2 n sin x n − x 2 n + x n − 1
4e −x2 n (cos x n − 2x n sin x n ) − 2x n +1 .
From the figure, we see that the graphs intersect at approximately x =0.2 and x =1.1.
x 1 =0.2
x 1 =1.1
x 2 ≈ 0.21883273
x 2 ≈ 1.08432830
x 3 ≈ 0.21916357
x 3 ≈ 1.08422462 ≈ x 4
x 4 ≈ 0.21916368 ≈ x 5
To eight decimal places, the roots of the equation are 0.21916368 and 1.08422462.

F.
208 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
29. (a) f(x) =x 2 − a ⇒ f 0 (x) =2x, so Newton’s method gives
x n+1 = x n − x2 n − a
2x n
= x n − 1 2 x n + a
2x n
= 1 2 x n +
a
2x n
= 1 2
x n + a
x n
.
(b) Using (a) with a = 1000 and x 1 = √ 900 = 30,wegetx 2 ≈ 31.666667, x 3 ≈ 31.622807,andx 4 ≈ 31.622777 ≈ x 5 .
So √ 1000 ≈ 31.622777.
31. f(x) =x 3 − 3x +6 ⇒ f 0 (x) =3x 2 − 3. Ifx 1 =1,thenf 0 (x 1)=0and the tangent line used for approximating x 2 is
horizontal. Attempting to find x 2 results in trying to divide by zero.
33. For f(x) =x 1/3 , f 0 (x) = 1 3 x−2/3 and
x n+1 = x n − f(x n)
f 0 (x n ) = x n − x1/3 n
1
3 x−2/3 n
= x n − 3x n = −2x n .
Therefore, each successive approximation becomes twice as large as the
previous one in absolute value, so the sequence of approximations fails to
converge to the root, which is 0. Inthefigure, we have x 1 =0.5,
x 2 = −2(0.5) = −1,andx 3 = −2(−1) = 2.
35. (a) f(x) =x 6 − x 4 +3x 3 − 2x ⇒ f 0 (x) =6x 5 − 4x 3 +9x 2 − 2 ⇒
f 00 (x) =30x 4 − 12x 2 +18x. Tofind the critical numbers of f,we’llfind the
zeros of f 0 .Fromthegraphoff 0 , it appears there are zeros at approximately
x = −1.3, −0.4,and0.5. Tryx 1 = −1.3 ⇒
x 2 = x 1 − f 0 (x 1)
≈−1.293344 ⇒
f 00 x3 ≈−1.293227 ≈ x4.
(x 1 )
Now try x 1 = −0.4 ⇒ x 2 ≈−0.443755 ⇒ x 3 ≈−0.441735 ⇒ x 4 ≈−0.441731 ≈ x 5 . Finally try
x 1 =0.5 ⇒ x 2 ≈ 0.507937 ⇒ x 3 ≈ 0.507854 ≈ x 4 . Therefore, x = −1.293227, −0.441731,and0.507854 are
all the critical numbers correct to six decimal places.
(b) There are two critical numbers where f 0 changes from negative to positive, so f changes from decreasing to increasing.
f(−1.293227) ≈−2.0212 and f(0.507854) ≈−0.6721,so−2.0212 is the absolute minimum value of f correct to four
decimal places.
37. From the figure, we see that y = f(x) =e cos x is periodic with period 2π. To
find the x-coordinates of the IP, we only need to approximate the zeros of y 00
on [0,π]. f 0 (x) =−e cos x sin x ⇒ f 00 (x) =e cos x sin 2 x − cos x .Since
e cos x 6=0, we will use Newton’s method with g(x) =sin 2 x − cos x,
g 0 (x) =2sinx cos x +sinx,andx 1 =1. x 2 ≈ 0.904173,
x 3 ≈ 0.904557 ≈ x 4.Thus,(0.904557, 1.855277) is the IP.

F.
TX.10
39. We need to minimize the distance from (0, 0) to an arbitrary point (x, y) on the
SECTION 4.9 ANTIDERIVATIVES ¤ 209
curve y =(x − 1) 2 . d = x 2 + y 2
⇒
d(x) = x 2 +[(x − 1) 2 ] 2 = x 2 +(x − 1) 4 .Whend 0 =0, d will be
minimized and equivalently, s = d 2 will be minimized, so we will use Newton’s
method with f = s 0 and f 0 = s 00 .
f(x) =2x +4(x − 1) 3 ⇒ f 0 (x) = 2 + 12(x − 1) 2 ,sox n+1 = x n −
2xn +4(xn − 1)3
.Tryx1 =0.5
2+12(x n − 1) ⇒
2
x 2 =0.4, x 3 ≈ 0.410127, x 4 ≈ 0.410245 ≈ x 5 .Nowd(0.410245) ≈ 0.537841 is the minimum distance and the point on
theparabolais(0.410245, 0.347810), correct to six decimal places.
41. In this case, A =18,000, R =375,andn =5(12)=60. So the formula A = R i [1 − (1 + i)−n ] becomes
18,000 = 375
x [1 − (1 + x)−60 ] ⇔ 48x =1− (1 + x) −60 [multiplyeachtermby(1 + x) 60 ] ⇔
48x(1 + x) 60 − (1 + x) 60 +1=0. Let the LHS be called f(x),sothat
f 0 (x)=48x(60)(1 + x) 59 +48(1+x) 60 − 60(1 + x) 59
=12(1+x) 59 [4x(60) + 4(1 + x) − 5] = 12(1 + x) 59 (244x − 1)
x n+1 = x n − 48x n(1 + x n ) 60 − (1 + x n ) 60 +1
. An interest rate of 1% permonthseemslikeareasonableestimatefor
12(1 + x n ) 59 (244x n − 1)
x = i. Soletx 1 =1%=0.01,andwegetx 2 ≈ 0.0082202, x 3 ≈ 0.0076802, x 4 ≈ 0.0076291, x 5 ≈ 0.0076286 ≈ x 6 .
Thus, the dealer is charging a monthly interest rate of 0.76286% (or 9.55% per year, compounded monthly).
4.9 Antiderivatives
1. f(x) =x − 3=x 1 − 3 ⇒ F (x) = x1+1
1+1 − 3x + C = 1 2 x2 − 3x + C
Check: F 0 (x) = 1 (2x) − 3+0=x − 3=f(x)
2
3. f(x) = 1 2 + 3 4 x2 − 4 5 x3 ⇒ F (x) = 1 2 x + 3 4
Check: F 0 (x) = 1 2 + 1 4 (3x2 ) − 1 5 (4x3 )+0= 1 2 + 3 4 x2 − 4 5 x3 = f(x)
x 2+1
2+1 − 4 x 3+1
5 3+1 + C = 1 x + 1 2 4 x3 − 1 5 x4 + C
5. f(x) =(x + 1)(2x − 1) = 2x 2 + x − 1 ⇒ F (x) =2 1
3 x3 + 1 2 x2 − x + C = 2 3 x3 + 1 2 x2 − x + C
7. f(x) =5x 1/4 − 7x 3/4 ⇒ F (x) =5 x1/4+1
+1 − 7 x3/4+1
3
+ C =5x5/4
+1 5/4 − 7x7/4
7/4 + C =4x5/4 − 4x 7/4 + C
9. f(x) =6 √ x − 6√ x =6x 1/2 − x 1/6 ⇒
F (x) =6 x1/2+1
+1 − x1/6+1
1
+ C =6x3/2
+1 3/2 − x7/6
7/6 + C =4x3/2 − 6 7 x7/6 + C
1
2
6
1
4
4

F.
210 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION TX.10
⎧
10x
11. f(x) = 10
⎪⎨
−8
+ C 1 = − 5
x 9 =10x−9 has domain (−∞, 0) ∪ (0, ∞),soF (x) =
−8
4x + C 8 1 if x0
See Example 1(b) for a similar problem.
13. f(u) = u4 +3 √ u
u 2
= u4
u 2 + 3u1/2
u 2 = u 2 +3u −3/2 ⇒
F (u) = u3
3
+3
u−3/2+1
−3/2+1 + C = 1 3 u3 +3 u−1/2
−1/2 + C = 1 3 u3 − 6 √ u
+ C
15. g(θ) =cosθ − 5sinθ ⇒ G(θ) =sinθ − 5(− cos θ)+C =sinθ +5cosθ + C
17. f(x) =5e x − 3coshx ⇒ F (x) =5e x − 3sinhx + C
19. f(x) = x5 − x 3 +2x
= x − 1 x 4 x + 2 x = x − 1 3 x +2x−3 ⇒
F (x) = x2
x
−3+1
2 − ln |x| +2 + C = 1 2
−3+1
x2 − ln |x| − 1 x + C 2
21. f(x) =5x 4 − 2x 5 ⇒ F (x) =5· x5
5 − 2 · x6
6 + C = x5 − 1 3 x6 + C.
F (0) = 4 ⇒ 0 5 − 1 3 · 06 + C =4 ⇒ C =4,soF (x) =x 5 − 1 3 x6 +4.
The graph confirms our answer since f(x) =0when F has a local maximum, f is
positive when F is increasing, and f is negative when F is decreasing.
23. f 00 (x) =6x +12x 2 ⇒ f 0 (x) =6· x2
2
f(x) =3· x3
3
25. f 00 (x) = 2 3 x2/3 ⇒ f 0 (x) = 2 3
+12·
x3
3 + C =3x2 +4x 3 + C ⇒
+4·
x4
4 + Cx + D = x3 + x 4 + Cx + D [C and D are just arbitrary constants]
x
5/3
5/3
+ C = 2 5 x5/3 + C ⇒ f(x) = 2 5
x
8/3
8/3
+ Cx + D = 3
20 x8/3 + Cx + D
27. f 000 (t) =e t ⇒ f 00 (t) =e t + C ⇒ f 0 (t) =e t + Ct + D ⇒ f(t) =e t + 1 2 Ct2 + Dt + E
29. f 0 (x) =1− 6x ⇒ f(x) =x − 3x 2 + C. f(0) = C and f(0) = 8 ⇒ C =8,sof(x) =x − 3x 2 +8.
31. f 0 (x) = √ x(6 + 5x) =6x 1/2 +5x 3/2 ⇒ f(x) =4x 3/2 +2x 5/2 + C.
f(1) = 6 + C and f(1) = 10 ⇒ C =4,sof(x) =4x 3/2 +2x 5/2 +4.
33. f 0 (t) =2cost +sec 2 t ⇒ f(t) =2sint +tant + C because −π/2

F.
TX.10
SECTION 4.9 ANTIDERIVATIVES ¤ 211
35. f 0 (x) =x −1/3 has domain (−∞, 0) ∪ (0, ∞) ⇒ f(x) =
3
2 x2/3 + C 1 if x>0
3
2 x2/3 + C 2 if x0
2
Thus, f(x) =
3
2 x2/3 − 5 if x0 ⇒ f 0 (x) =−1/x + C ⇒ f(x) =− ln |x| + Cx + D = − ln x + Cx + D [since x>0].
f(1) = 0 ⇒ C + D =0and f(2) = 0 ⇒ −ln 2 + 2C + D =0 ⇒ −ln 2 + 2C − C =0 [since D = −C] ⇒
− ln 2 + C =0 ⇒ C =ln2and D = − ln 2. Sof(x) =− ln x +(ln2)x − ln 2.
47. Given f 0 (x) =2x +1,wehavef(x) =x 2 + x + C. Sincef passes through (1, 6), f(1) = 6 ⇒ 1 2 +1+C =6 ⇒
C =4. Therefore, f(x) =x 2 + x +4and f(2) = 2 2 +2+4=10.
49. b is the antiderivative of f. Forsmallx, f is negative, so the graph of its antiderivative must be decreasing. But both a and c
are increasing for small x,soonlyb can be f’s antiderivative. Also, f is positive where b is increasing, which supports our
conclusion.
51. The graph of F must start at (0, 1). Where the given graph, y = f(x),hasa
local minimum or maximum, the graph of F will have an inflection point.
Where f is negative (positive), F is decreasing (increasing).
Where f changes from negative to positive, F will have a minimum.
Where f changes from positive to negative, F will have a maximum.
Where f is decreasing (increasing), F is concave downward (upward).

F.
212 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
53.
⎧
2 if 0 ≤ x

F.
TX.10
SECTION 4.9 ANTIDERIVATIVES ¤ 213
(c) The velocity with which the stone strikes the ground is v(t 1)=−9.8 450/4.9 ≈−93.9 m/s.
(d) This is just reworking parts (a) and (b) with v(0) = −5. Usingv(t) =−9.8t + C, v(0) = −5 ⇒ 0+C = −5 ⇒
v(t) =−9.8t − 5. Sos(t) =−4.9t 2 − 5t + D and s(0) = 450 ⇒ D =450 ⇒ s(t) =−4.9t 2 − 5t + 450.
Solving s(t) =0by using the quadratic formula gives us t = 5 ± √ 8845 (−9.8) ⇒ t 1 ≈ 9.09 s.
65. By Exercise 64 with a = −9.8, s(t) =−4.9t 2 + v 0 t + s 0 and v(t) =s 0 (t) =−9.8t + v 0 .So
[v(t)] 2 =(−9.8t + v 0 ) 2 =(9.8) 2 t 2 − 19.6v 0 t + v0 2 = v0 2 +96.04t 2 − 19.6v 0 t = v0 2 − 19.6 −4.9t 2 + v 0 t .
But −4.9t 2 + v 0t is just s(t) without the s 0 term;thatis,s(t) − s 0.Thus,[v(t)] 2 = v0 2 − 19.6[s(t) − s 0].
67. Using Exercise 64 with a = −32, v 0 =0,ands 0 = h (the height of the cliff ), we know that the height at time t is
s(t) =−16t 2 + h. v(t) =s 0 (t) =−32t and v(t) =−120 ⇒ −32t = −120 ⇒ t =3.75, so
0=s(3.75) = −16(3.75) 2 + h ⇒ h = 16(3.75) 2 = 225 ft.
69. Marginal cost =1.92 − 0.002x = C 0 (x) ⇒ C(x) =1.92x − 0.001x 2 + K. ButC(1) = 1.92 − 0.001 + K =562 ⇒
K =560.081. Therefore, C(x) =1.92x − 0.001x 2 +560.081 ⇒ C(100) = 742.081, so the cost of producing
100 items is $742.08.
71. Taking the upward direction to be positive we have that for 0 ≤ t ≤ 10 (using the subscript 1 to refer to 0 ≤ t ≤ 10),
a 1 (t) =− (9 − 0.9t) =v1(t) 0 ⇒ v 1 (t) =−9t +0.45t 2 + v 0 ,butv 1 (0) = v 0 = −10 ⇒
v 1(t) =−9t +0.45t 2 − 10 = s 0 1(t) ⇒ s 1(t) =− 9 2 t2 +0.15t 3 − 10t + s 0.Buts 1(0) = 500 = s 0 ⇒
s 1 (t) =− 9 2 t2 +0.15t 3 − 10t + 500.
s 1 (10) = −450 + 150 − 100 + 500 = 100,soittakes
more than 10 seconds for the raindrop to fall. Now for t>10, a(t) =0=v 0 (t)
⇒
v(t) =constant = v 1 (10) = −9(10) + 0.45(10) 2 − 10 = −55 ⇒ v(t) =−55.
At 55 m/s, it will take 100/55 ≈ 1.8 stofallthelast100 m. Hence, the total time is 10 + 100 = 130 ≈ 11.8 s.
55 11
73. a(t) =k, the initial velocity is 30 mi/h =30· 5280
3600
=44ft/s, and the final velocity (after 5 seconds) is
50 mi/h =50· 5280 = 220 ft/s. So v(t) =kt + C and v(0) = 44 ⇒ C =44.Thus,v(t) =kt +44 ⇒
3600 3
v(5) = 5k +44.Butv(5) = 220
220
,so5k +44= ⇒ 5k = 88 ⇒ k = 88 ≈ 5.87 3 3 3 15 ft/s2 .
75. Let the acceleration be a(t) =k km/h 2 .Wehavev(0) = 100 km/h and we can take the initial position s(0) to be 0.
We want the time t f for which v(t) =0to satisfy s(t) < 0.08 km. In general, v 0 (t) =a(t) =k, sov(t) =kt + C,where
C = v(0) = 100. Nows 0 (t) =v(t) =kt + 100, sos(t) = 1 2 kt2 + 100t + D, whereD = s(0) = 0.
Thus, s(t) = 1 2 kt2 +100t. Sincev(t f )=0,wehavekt f + 100 = 0 or t f = −100/k, so
s(t f )= 1
2 k − 100 2
+100
− 100
k
k
− 5,000
k
1
=10,000
2k − 1
k
= − 5,000
k
. The condition s(t f ) must satisfy is
< 0.08 ⇒ − 5,000
0.08 >k [k is negative] ⇒ k

F.
214 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
77. (a) First note that 90 mi/h =90× 5280
3600 ft/s = 132 ft/s. Then a(t) =4ft/s2 ⇒ v(t) =4t + C,butv(0) = 0 ⇒
C =0.Now4t =132when t = 132
4
=33s, so it takes 33 s to reach 132 ft/s. Therefore, taking s(0) = 0,wehave
s(t) =2t 2 , 0 ≤ t ≤ 33. Sos(33) = 2178 ft. 15 minutes = 15(60) = 900 s, so for 33

F.
TX.10
CHAPTER 4 REVIEW ¤ 215
7. (a) See l’Hospital’s Rule and the three notes that follow it in Section 4.4.
(b) Write fg as
f
1/g or g
1/f .
(c) Convert the difference into a quotient using a common denominator, rationalizing, factoring, or some other method.
(d) Convert the power to a product by taking the natural logarithm of both sides of y = f g or by writing f g as e g ln f .
8. Without calculus you could get misleading graphs that fail to show the most interesting features of a function.
See the discussion at the beginning of Section 4.5 and the first paragraph in Section 4.6.
9. (a)SeeFigure3inSection4.8.
(b) x 2 = x 1 − f(x 1)
f 0 (x 1 )
(c) x n+1 = x n − f(xn)
f 0 (x n)
(d)Newton’smethodislikelytofailortoworkveryslowlywhenf 0 (x 1) is close to 0. It also fails when f 0 (x i) is undefined,
such as with f(x) =1/x − 2 and x 1 =1.
10. (a) See the definition at the beginning of Section 4.9.
(b) If F 1 and F 2 are both antiderivatives of f on an interval I,thentheydifferbyaconstant.
1. False. For example, take f(x) =x 3 ,thenf 0 (x) =3x 2 and f 0 (0) = 0,butf(0) = 0 is not a maximum or minimum;
(0, 0) is an inflection point.
3. False. For example, f(x) =x is continuous on (0, 1) but attains neither a maximum nor a minimum value on (0, 1).
Don’t confuse this with f being continuous on the closed interval [a, b], which would make the statement true.
5. True. This is an example of part (b) of the I/D Test.
7. False. f 0 (x) =g 0 (x) ⇒ f(x) =g(x)+C. Forexample,iff(x) =x +2and g(x) =x +1,thenf 0 (x) =g 0 (x) =1,
but f(x) 6=g(x).
9. True. The graph of one such function is sketched.
11. True. Let x 1

F.
216 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
15. True. Let x 1,x 2 ∈ I and x 1 1
f(x 2 )
[f is positive] ⇒
17. True. If f is periodic, then there is a number p such that f(x + p) =f(p) for all x. Differentiating gives
f 0 (x) =f 0 (x + p) · (x + p) 0 = f 0 (x + p) · 1=f 0 (x + p),sof 0 is periodic.
19. True. By the Mean Value Theorem, there exists a number c in (0, 1) such that f(1) − f(0) = f 0 (c)(1 − 0) = f 0 (c).
Since f 0 (c) is nonzero, f(1) − f(0) 6= 0,sof(1) 6=f(0).
1. f(x) =x 3 − 6x 2 +9x +1, [2, 4]. f 0 (x) =3x 2 − 12x +9=3(x 2 − 4x +3)=3(x − 1)(x − 3). f 0 (x) =0 ⇒
x =1or x =3,but1 is not in the interval. f 0 (x) > 0 for 3

F.
TX.10
CHAPTER 4 REVIEW ¤ 217
13. This limit has the form ∞−∞.
x
lim
x→1 + x − 1 − 1
x ln x − x +1
= lim
ln x x→1 + (x − 1) ln x
H
= lim
x→1 + 1/x
1/x 2 +1/x = 1
1+1 = 1 2
H
= lim
x→1 + x · (1/x)+lnx − 1
(x − 1) · (1/x)+lnx = lim
x→1 +
ln x
1 − 1/x +lnx
15. f(0) = 0, f 0 (−2) = f 0 (1) = f 0 (9) = 0, lim
x→∞
f(x) =0, lim
x→6
f(x) =−∞,
f 0 (x) < 0 on (−∞, −2), (1, 6),and(9, ∞), f 0 (x) > 0 on (−2, 1) and (6, 9),
f 00 (x) > 0 on (−∞, 0) and (12, ∞), f 00 (x) < 0 on (0, 6) and (6, 12)
17. f is odd, f 0 (x) < 0 for 0 2,
f 00 (x) > 0 for 0

F.
218 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
23. y = f(x) =
1
x(x − 3) 2 A. D = {x | x 6= 0, 3} =(−∞, 0) ∪ (0, 3) ∪ (3, ∞) B. No intercepts. C. No symmetry.
1
D. lim
=0,soy =0 is a HA.
x→±∞ x(x − 3) lim
2
1
1
1
= ∞, lim
= −∞, lim
x→0 + x(x − 3)
2
x→0 − x(x − 3)
2 x→3 x(x − 3) = ∞, 2
so x =0and x =3are VA. E. f 0 (x) =− (x − 3)2 +2x(x − 3) 3(1 − x)
= ⇒ f 0 (x) > 0 ⇔ 1 0 ⇔ x>0 ⇒ f is CU on (0, 3) and (3, ∞) and
CD on (−∞, 0). NoIP
25. y = f(x) = x2
64
= x − 8+
x +8 x +8
A. D = {x | x 6=−8} B. Intercepts are 0 C. No symmetry
x 2
64
D. lim = ∞, butf(x) − (x − 8) = → 0 as x →∞,soy = x − 8 is a slant asymptote.
x→∞ x +8 x +8
x 2
lim
x→−8 + x +8 = ∞ and
lim x 2
x→−8 − x +8 = −∞,sox = −8 is a VA. E. f 0 (x) =1−
64 x(x +16)
=
(x +8)
2
(x +8) > 0 ⇔
2
x>0 or x 0 ⇔ x>−8,sof is CU on (−8, ∞) and
CD on (−∞, −8). NoIP
27. y = f(x) =x √ 2+x A. D =[−2, ∞) B. y-intercept: f(0) = 0; x-intercepts: −2 and 0 C. No symmetry
D. No asymptote E. f 0 x
(x) =
2 √ 2+x + √ 1
2+x =
2 √ 3x +4
[x +2(2+x)] =
2+x 2 √ 2+x =0when x = − 4 ,sof is
3
decreasing on
−2, − 4 3 and increasing on −
4
, ∞ . F. Local minimum value f
− 4 3 3 = −
4 2
= − 4√ 6
≈−1.09,
3 3 9
no local maximum
2 √ 1
2+x · 3 − (3x +4) √
G. f 00 2+x
(x) =
4(2 + x)
=
3x +8
4(2 + x) 3/2
=
6(2 + x) − (3x +4)
4(2 + x) 3/2
H.
f 00 (x) > 0 for x>−2,sof is CU on (−2, ∞). NoIP

F.
TX.10
CHAPTER 4 REVIEW ¤ 219
29. y = f(x) =sin 2 x − 2cosx A. D = R B. y-intercept: f(0) = −2 C. f(−x) =f(x),sof is symmetric with respect
to the y-axis. f has period 2π. D. No asymptote E. y 0 =2sinx cos x +2sinx =2sinx (cos x +1). y 0 =0 ⇔
sin x =0or cos x = −1 ⇔ x = nπ or x =(2n +1)π. y 0 > 0 when sin x>0,sincecos x +1≥ 0 for all x.
Therefore, y 0 > 0 [and so f is increasing] on (2nπ, (2n +1)π); y 0 < 0 [and so f is decreasing] on ((2n − 1)π, 2nπ).
F. Local maximum values are f((2n +1)π) =2; local minimum values are f(2nπ) =−2.
G. y 0 =sin2x +2sinx ⇒ y 00 =2cos2x +2cosx =2(2cos 2 x − 1) + 2 cos x =4cos 2 x +2cosx − 2
=2(2cos 2 x +cosx − 1) = 2(2 cos x − 1)(cos x +1)
y 00 =0 ⇔ cos x = 1 or −1 ⇔ x =2nπ ± π or x =(2n +1)π. H.
2 3
y 00 > 0 [and so f is CU] on 2nπ − π , 2nπ + π
3 3 ; y 00 ≤ 0 [and so f is CD]
on 2nπ + π , 2nπ + 5π
3 3 .Thereareinflection points at 2nπ ±
π
, − 1
3 4 .
31. y = f(x) =sin −1 (1/x) A. D = {x | −1 ≤ 1/x ≤ 1} =(−∞, −1] ∪ [1, ∞) . B. No intercept
C. f(−x) =−f(x), symmetric about the origin D. lim
x→±∞ sin−1 (1/x) =sin −1 (0) = 0, soy =0is a HA.
E. f 0 (x) =
1
− 1
−1
= √ < 0,sof is decreasing on (−∞, −1) and (1, ∞) .
1 − (1/x)
2 x 2 x4 − x2 F. No local extreme value, but f(1) = π is the absolute maximum value
2
H.
and f(−1) = − π 2
is the absolute minimum value.
G. f 00 (x) = 4x3 − 2x
2(x 4 − x 2 ) 3/2 = x 2x 2 − 1
(x 4 − x 2 ) 3/2 > 0 for x>1 and f 00 (x) < 0
for x 0 ⇔ −2x +1> 0 ⇔ x< 1 2 and f 0 (x) < 0 ⇔ x> 1 2 ,
so f is increasing on −∞, 1 2
no local minimum value
and decreasing on
1 , ∞ . F. Local maximum value f
1
2 2 =
1
2 e−1 =1/(2e);
G. f 00 (x)=e −2x (−2) + (−2x +1)(−2e −2x )
H.
=2e −2x [−1 − (−2x +1)]=4(x − 1) e −2x .
f 00 (x) > 0 ⇔ x>1 and f 00 (x) < 0 ⇔ x

F.
220 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
35. f(x) = x2 − 1
⇒ f 0 (x) = x3 (2x) − x 2 − 1 3x 2
= 3 − x2
x 3 x 6
x 4
f 00 (x) = x4 (−2x) − 3 − x 2 4x 3
= 2x2 − 12
x 8 x 5
Estimates: From the graphs of f 0 and f 00 ,itappearsthatf is increasing on
(−1.73, 0) and (0, 1.73) and decreasing on (−∞, −1.73) and (1.73, ∞);
f has a local maximum of about f(1.73) = 0.38 and a local minimum of about
f(−1.7) = −0.38; f is CU on (−2.45, 0) and (2.45, ∞), and CD on
(−∞, −2.45) and (0, 2.45);andf has inflection points at about
(−2.45, −0.34) and (2.45, 0.34).
Exact: Now f 0 (x) = 3 − x2
x 4
⇒
is positive for 0

F.
TX.10
CHAPTER 4 REVIEW ¤ 221
f is CU on (−∞, −0.12) and (1.24, ∞) and CD on (−0.12, 1.24);andf has inflection points at about (−0.12, 1.98) and
(1.24, −12.1).
39. From the graph, we estimate the points of inflection to be about (±0.82, 0.22).
41. f(x) =
f(x) =e −1/x2 ⇒ f 0 (x) =2x −3 e −1/x2 ⇒
f 00 (x) =2[x −3 (2x −3 )e −1/x2 + e −1/x2 (−3x −4 )] = 2x −6 e −1/x2 2 − 3x 2 .
This is 0 when 2 − 3x 2 2
=0 ⇔ x = ± ,sotheinflection points
3
are ±
2
3 ,e−3/2 .
cos 2 x
√
x2 + x +1 , −π ≤ x ≤ π ⇒ f 0 (x) =− cos x [(2x +1)cosx +4(x2 + x +1)sinx]
2(x 2 + x +1) 3/2 ⇒
f 00 (x) =− (8x4 +16x 3 +16x 2 +8x +9)cos 2 x − 8(x 2 + x + 1)(2x +1)sinx cos x − 8(x 2 + x +1) 2 sin 2 x
4(x 2 + x +1) 5/2
f(x) =0 ⇔ x = ± π 2 ; f 0 (x) =0 ⇔ x ≈−2.96, −1.57, −0.18, 1.57, 3.01;
f 00 (x) =0 ⇔ x ≈−2.16, −0.75, 0.46,and2.21.
The x-coordinates of the maximum points are the values at which f 0 changes from positive to negative, that is, −2.96, −0.18,
and 3.01. Thex-coordinates of the minimum points are the values at which f 0 changes from negative to positive, that is,
−1.57 and 1.57. Thex-coordinates of the inflection points are the values at which f 00 changes sign, that is, −2.16, −0.75,
0.46,and2.21.
43. The family of functions f(x) =ln(sinx + C) all have the same period and all
have maximum values at x = π +2πn. Since the domain of ln is (0, ∞), f has
2
a graph only if sin x + C>0 somewhere. Since −1 ≤ sin x ≤ 1, this happens
if C>−1,thatis,f has no graph if C ≤−1. Similarly, if C>1,then
sin x + C>0 and f is continuous on (−∞, ∞).AsC increases, the graph of
f is shifted vertically upward and flattens out. If −1

F.
222 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
where sin x + C>0 ⇔ sin x>−C ⇔ sin −1 (−C) 0 implies that f is increasing on R, so there is exactly one zero of f, and hence, exactly one real
root of the equation 3x +2cosx +5=0.
47. Since f is continuous on [32, 33] and differentiable on (32, 33), then by the Mean Value Theorem there exists a number c in
(32, 33) such that f 0 (c) = 1 5 c−4/5 =
5√
33 −
5 √ 32
33 − 32
= 5√ 33 − 2,but 1 5 c−4/5 > 0 ⇒ 5√ 33 − 2 > 0 ⇒ 5√ 33 > 2. Also
f 0 is decreasing, so that f 0 (c) f 0 (c) = 5√ 33 − 2 ⇒ 5√ 33 < 2.0125.
Therefore, 2 < 5√ 33 < 2.0125.
49. (a) g(x) =f(x 2 ) ⇒ g 0 (x) =2xf 0 (x 2 ) by the Chain Rule. Since f 0 (x) > 0 for all x 6= 0,wemusthavef 0 (x 2 ) > 0 for
x 6= 0,sog 0 (x) =0 ⇔ x =0.Nowg 0 (x) changes sign (from negative to positive) at x =0, since one of its factors,
f 0 (x 2 ), is positive for all x, and its other factor, 2x, changes from negative to positive at this point, so by the First
Derivative Test, f has a local and absolute minimum at x =0.
(b) g 0 (x) =2xf 0 (x 2 ) ⇒ g 00 (x) =2[xf 00 (x 2 )(2x)+f 0 (x 2 )] = 4x 2 f 00 (x 2 )+2f 0 (x 2 ) by the Product Rule and the Chain
Rule. But x 2 > 0 for all x 6= 0, f 00 (x 2 ) > 0 [since f is CU for x>0], and f 0 (x 2 ) > 0 for all x 6= 0, so since all of its
factors are positive, g 00 (x) > 0 for x 6= 0.Whetherg 00 (0) is positive or 0 doesn’t matter [since the sign of g 00 does not
change there]; g is concave upward on R.
51. If B =0, the line is vertical and the distance from x = − C A to (x 1,y 1 ) is
x 1 + C A = |Ax 1 + By 1 + C|
√ , so assume
A2 + B 2
B 6= 0. The square of the distance from (x 1,y 1) to the line is f(x) =(x − x 1) 2 +(y − y 1) 2 where Ax + By + C =0,so
we minimize f(x) =(x − x 1) 2 + − A B x − C 2
B − y1 ⇒ f 0 (x) =2(x − x 1)+2
− A B x − C
B − y1 − A
.
B
f 0 (x) =0 ⇒ x = B2 x 1 − ABy 1 − AC
and this gives a minimum since f 00 (x) =2
1+ A2
> 0. Substituting
A 2 + B 2 B 2
this value of x into f(x) and simplifying gives f(x) = (Ax 1 + By 1 + C) 2
, so the minimum distance is
A 2 + B 2
f(x) =
|Ax 1 + By 1 + C|
√
A2 + B 2 .

F.
TX.10
CHAPTER 4 REVIEW ¤ 223
53. By similar triangles, y x = r
√ , so the area of the triangle is
x2 − 2rx
A(x) = 1 2 (2y)x = xy = rx 2
√
x2 − 2rx
⇒
A 0 (x) = 2rx √ x 2 − 2rx − rx 2 (x − r)/ √ x 2 − 2rx
x 2 − 2rx
when x =3r.
= rx2 (x − 3r)
=0
(x 2 3/2
− 2rx)
A 0 (x) < 0 when 2r 3r. Sox =3r gives a minimum and A(3r) = r(9r2 )
√
3 r
=3 √ 3 r 2 .
55. We minimize L(x) =|PA| + |PB| + |PC| =2 √ x 2 +16+(5− x),
57. v = K
L
C + C L ⇒ dv
dL =
0 ≤ x ≤ 5. L 0 (x) =2x √ x 2 +16 − 1=0 ⇔ 2x = √ x 2 +16 ⇔
4x 2 = x 2 +16 ⇔ x = √ 4
3
. L(0) = 13, L
minimum occurs when x = 4 √
3
≈ 2.3.
√
4
3
≈ 11.9, L(5) ≈ 12.8,sothe
K 1
2 (L/C)+(C/L) C − C
=0 ⇔ 1 L 2 C = C ⇔ L 2 = C 2 ⇔ L = C.
L 2
This gives the minimum velocity since v 0 < 0 for 0 C.
59. Let x denote the number of $1 decreases in ticket price. Then the ticket price is $12 − $1(x), and the average attendance is
11,000 + 1000(x). Nowtherevenuepergameis
R(x) =(price per person) × (number of people per game)
=(12− x)(11,000 + 1000x) =−1000x 2 +1000x +132,000
for 0 ≤ x ≤ 4 [since the seating capacity is 15,000] ⇒ R 0 (x) =−2000x +1000=0 ⇔ x =0.5. Thisisa
maximum since R 00 (x) =−2000 < 0 for all x. Now we must check the value of R(x) =(12− x)(11,000 + 1000x) at
x =0.5 and at the endpoints of the domain to see which value of x gives the maximum value of R.
R(0) = (12)(11,000) = 132,000, R(0.5) = (11.5)(11,500) = 132,250,andR(4) = (8)(15,000) = 120,000. Thus,the
maximum revenue of $132,250 per game occurs when the average attendance is 11,500 and the ticket price is $11.50.
61. f(x) =x 5 − x 4 +3x 2 − 3x − 2 ⇒ f 0 (x) =5x 4 − 4x 3 +6x − 3,sox n+1 = x n − x5 n − x 4 n +3x 2 n − 3x n − 2
.
5x 4 n − 4x 3 n +6x n − 3
Now x 1 =1 ⇒ x 2 =1.5 ⇒ x 3 ≈ 1.343860 ⇒ x 4 ≈ 1.300320 ⇒ x 5 ≈ 1.297396 ⇒
x 6 ≈ 1.297383 ≈ x 7 ,sotherootin[1, 2] is 1.297383, to six decimal places.

F.
224 ¤ CHAPTER 4 APPLICATIONS OF DIFFERENTIATION
TX.10
63. f(t) =cost + t − t 2 ⇒ f 0 (t) =− sin t +1− 2t. f 0 (t) exists for all
t,sotofind the maximum of f, we can examine the zeros of f 0 .
From the graph of f 0 ,weseethatagoodchoicefort 1 is t 1 =0.3.
Use g(t) =− sin t +1− 2t and g 0 (t) =− cos t − 2 to obtain
t 2 ≈ 0.33535293, t 3 ≈ 0.33541803 ≈ t 4 .Sincef 00 (t) =− cos t − 2 < 0
for all t, f(0.33541803) ≈ 1.16718557 is the absolute maximum.
65. f 0 (x) =cosx − (1 − x 2 ) −1/2 =cosx −
1
√
1 − x
2
⇒
f(x) =sinx − sin −1 x + C
67. f 0 (x) = √ x 3 + 3√ x 2 = x 3/2 + x 2/3 ⇒ f(x) = x5/2
5/2 + x5/3
5/3 + C = 2 5 x5/2 + 3 5 x5/3 + C
69. f 0 (t) =2t − 3sint ⇒ f(t) =t 2 +3cost + C.
f(0) = 3 + C and f(0) = 5 ⇒ C =2,sof(t) =t 2 +3cost +2.
71. f 00 (x) =1− 6x +48x 2 ⇒ f 0 (x) =x − 3x 2 +16x 3 + C. f 0 (0) = C and f 0 (0) = 2 ⇒ C =2,so
f 0 (x) =x − 3x 2 +16x 3 +2and hence, f(x) = 1 2 x2 − x 3 +4x 4 +2x + D.
f(0) = D and f(0) = 1 ⇒ D =1,sof(x) = 1 2 x2 − x 3 +4x 4 +2x +1.
73. v(t) =s 0 (t) =2t − 1
1+t 2 ⇒ s(t) =t 2 − tan −1 t + C.
s(0) = 0 − 0+C = C and s(0) = 1 ⇒ C =1,sos(t) =t 2 − tan −1 t +1.
75. (a) Since f is 0 just to the left of the y-axis, we must have a minimum of F at the same place since we are increasing through
(0, 0) on F . There must be a local maximum to the left of x = −3,sincef changes from positive to negative there.
(b) f(x) =0.1e x +sinx ⇒
F (x) =0.1e x − cos x + C. F (0) = 0
0.1 − 1+C =0 ⇒ C =0.9,so
F (x) =0.1e x − cos x +0.9.
⇒
(c)
77. Choosing the positive direction to be upward, we have a(t) =−9.8 ⇒ v(t) =−9.8t + v 0 ,butv(0) = 0 = v 0 ⇒
v(t) =−9.8t = s 0 (t) ⇒ s(t) =−4.9t 2 + s 0,buts(0) = s 0 =500 ⇒ s(t) =−4.9t 2 +500.Whens =0,
−4.9t 2 500
+500=0 ⇒ t 1 = ≈ 10.1 ⇒ v(t 500
4.9 1)=−9.8 ≈−98.995 m/s. Since the canister has been
4.9
designed to withstand an impact velocity of 100 m/s, the canister will not burst.

F.
TX.10
CHAPTER 4 REVIEW ¤ 225
79. (a) The cross-sectional area of the rectangular beam is
A =2x · 2y =4xy =4x √ 100 − x 2 , 0 ≤ x ≤ 10,so
dA
dx =4x
1
2 (100 − x 2 ) −1/2 (−2x)+(100− x 2 ) 1/2 · 4
−4x 2
=
(100 − x 2 ) + 4(100 − 1/2 x2 ) 1/2 = 4[−x2 + 100 − x 2 ]
.
(100 − x 2 ) 1/2
dA
dx =0when −x2 + 100 − x 2 =0 ⇒ x 2 =50 ⇒ x = √
50 ≈ 7.07 ⇒ y = 100 − √ 50 2 √
= 50.
Since A(0) = A(10) = 0, the rectangle of maximum area is a square.
(b)
The cross-sectional area of each rectangular plank (shaded in the figure) is
A =2x y − √ 50 =2x √ 100 − x 2 − √ 50 , 0 ≤ x ≤ √ 50,so
dA
dx =2√ 100 − x 2 − √ 50 +2x 1
2
(100 − x 2 ) −1/2 (−2x)
=2(100− x 2 ) 1/2 − 2 √ 50 −
2x 2
(100 − x 2 ) 1/2
Set dA
dx =0: (100 − x2 ) − √ 50 (100 − x 2 ) 1/2 − x 2 =0 ⇒ 100 − 2x 2 = √ 50 (100 − x 2 ) 1/2 ⇒
10,000 − 400x 2 +4x 4 = 50(100 − x 2 ) ⇒ 4x 4 − 350x 2 + 5000 = 0 ⇒ 2x 4 − 175x 2 +2500=0 ⇒
x 2 = 175 ± √ 10,625
4
≈ 69.52 or 17.98 ⇒ x ≈ 8.34 or 4.24. But8.34 > √ 50, sox 1 ≈ 4.24 ⇒
y − √ 50 = 100 − x 2 1 − √ 50 ≈ 1.99. Each plank should have dimensions about 8 1 inches by 2 inches.
2
(c) From the figure in part (a), the width is 2x and the depth is 2y, so the strength is
S = k(2x)(2y) 2 =8kxy 2 =8kx(100 − x 2 )=800kx − 8kx 3 , 0 ≤ x ≤ 10. dS/dx =800k − 24kx 2 =0when
24kx 2 =800k ⇒ x 2 = 100
3
⇒ x = √ 10
200
3
⇒ y =
3
= 10 √ √
2
3
= √ 2 x. SinceS(0) = S(10) = 0, the
maximum strength occurs when x = √ 10
3
. The dimensions should be √ 20
3
≈ 11.55 inches by 20 √ √
2
3
≈ 16.33 inches.
81. We first show that
x
1+x < 2 tan−1 x for x>0. Letf(x) =tan −1 x − x
1+x .Then 2
f 0 (x) = 1
1+x − 1(1 + x2 ) − x(2x)
= (1 + x2 ) − (1 − x 2 ) 2x 2
= > 0 for x>0. Sof(x) is increasing
2 (1 + x 2 ) 2 (1 + x 2 ) 2 (1 + x 2 )
2
on (0, ∞). Hence, 0

F.
TX.10

F.
TX.10
PROBLEMS PLUS
1. Let y = f(x) =e −x2 . The area of the rectangle under the curve from −x to x is A(x) =2xe −x2 where x ≥ 0. We maximize
A(x): A 0 (x) =2e −x2 − 4x 2 e −x2 =2e −x2 1 − 2x 2 =0 ⇒ x = 1 √
2
. This gives a maximum since A 0 (x) > 0
for 0 ≤ x< 1 √
2
and A 0 (x) < 0 for x>
1 √
2
. We next determine the points of inflection of f(x). Notice that
f 0 (x) =−2xe −x2 = −A(x). Sof 00 (x) =−A 0 (x) and hence, f 00 (x) < 0 for − 1 √
2
0 for x 1 √
2
.Sof(x) changes concavity at x = ± 1 √
2
, and the two vertices of the rectangle of largest area are at the inflection
points.
3. First, we recognize some symmetry in the inequality: ex + y
xy
≥ e 2 ⇔ ex x · ey
y
≥ e · e. This suggests that we need to show
that ex x
≥ e for x>0. If we can do this, then the inequality
ey
y ≥ e is true, and the given inequality follows. f(x) =ex x
⇒
f 0 (x) = xex − e x
x 2 = ex (x − 1)
x 2 =0 ⇒ x =1. By the First Derivative Test, we have a minimum of f(1) = e, so
e x /x ≥ e for all x.
ax 2 +sinbx +sincx +sindx
5. Let L =lim
.NowL has the indeterminate form of type 0 , so we can apply l’Hospital’s
x→0 3x 2 +5x 4 +7x 6
0
2ax + b cos bx + c cos cx + d cos dx
Rule. L =lim
. The denominator approaches 0 as x → 0, so the numerator must also
x→0 6x +20x 3 +42x 5
approach 0 (because the limit exists). But the numerator approaches 0+b + c + d,sob + c + d =0. Apply l’Hospital’s Rule
2a − b 2 sin bx − c 2 sin cx − d 2 sin dx
again. L =lim
= 2a − 0
x→0 6+60x 2 +210x 4 6+0 = 2a 6 , which must equal 8. 2a
=8 ⇒ a =24.
6
Thus, a + b + c + d = a +(b + c + d) =24+0=24.
7. Differentiating x 2 + xy + y 2 =12implicitly with respect to x gives 2x + y + x dy dy dy
+2y =0,so
dx dx dx = − 2x + y
x +2y .
At a highest or lowest point, dy =0 ⇔ y = −2x. Substituting −2x for y in the original equation gives
dx
x 2 + x(−2x)+(−2x) 2 =12,so3x 2 =12and x = ±2. Ifx =2,theny = −2x = −4, andifx = −2 then y =4.
Thus, the highest and lowest points are (−2, 4) and (2, −4).
9. y = x 2 ⇒ y 0 =2x, so the slope of the tangent line at P (a, a 2 ) is 2a and the slope of the normal line is − 1 for a 6= 0.
2a
An equation of the normal line is y − a 2 = − 1
2a (x − a). Substitute x2 for y to find the x-coordinates of the two points of
intersection of the parabola and the normal line. x 2 − a 2 = − x 2a + 1 2
⇒ 2ax 2 + x − 2a 3 − a =0 ⇒
227

F.
228 ¤ CHAPTER 4 PROBLEMS PLUS
TX.10
x = −1 ± 1 − 4(2a)(−2a 3 − a)
2(2a)
= −1 ± √ 1+16a 4 +8a 2
4a
= −1 ± (4a 2 +1) 2
4a
= −1 ± (4a2 +1)
4a
= 4a2
4a or −4a2 − 2
, or equivalently, a or −a − 1
4a
2a
So the point Q has coordinates −a − 1
2a , −a − 1 2
. The square S of the distance from P to Q is given by
2a
S = −a − 1 2
2a − a + −a − 1 2 2
− a 2 = −2a − 1 2
+ a 2 +1+ 1 2
− a 2
2a
2a
4a 2
= 4a 2 +2+ 1
+ 1+ 1 2
= 4a 2 +2+ 1
+1+ 2
4a 2 4a 2 4a 2 4a + 1
2 16a 4 =4a2 +3+ 3
4a + 1
2 16a 4
S 0 =8a − 6
4a − 4
3 16a =8a − 3
5 2a − 1
3 4a = 32a6 − 6a 2 − 1
. The only real positive zero of the equation S 0 =0is
5 4a 5
a = 1 √
2
.SinceS 00 =8+ 9
2a 4 + 5
4a 6 > 0, a =
1 √
2
corresponds to the shortest possible length of the line segment PQ.
11. f(x) = a 2 + a − 6 cos 2x +(a − 2)x +cos1 ⇒ f 0 (x) =− a 2 + a − 6 sin 2x (2) + (a − 2). The derivative exists
for all x, so the only possible critical points will occur where f 0 (x) =0 ⇔ 2(a − 2)(a +3)sin2x = a − 2 ⇔
either a =2or 2(a +3)sin2x =1, with the latter implying that sin 2x =
this equation has no solution whenever either
− 7 2 x1.
2
f 0 (x) =0 ⇒ 2x(x 1 − x 2 )=x 2 1 − x 2 2 ⇒ x P = 1 2 (x 1 + x 2 ).
2
f(x P )= 1 2 x
2 1 (x 1 2 2 − x 1 )
+ x 2 1
2 (x 2 2 − x 1 ) + 1 (x 4 1 + x 2 ) 2 (x 1 − x 2 )
= 1 1 (x 2 2 2 − x 1 )
x 2 1 + x 2 2 −
1
(x 4 2 − x 1 )(x 1 + x 2 ) 2 = 1 (x 8 2 − x 1 ) 2
x 2 1 + x2 2 − x
2
1 +2x 1 x 2 + x 2 2
= 1 8 (x2 − x1)
x 2 1 − 2x 1x 2 + x 2 2 =
1
8 (x2 − x1)(x1 − x2)2 = 1 8 (x2 − x1)(x2 − x1)2 = 1 8
(x2 − x1)3

F.
TX.10
CHAPTER 4 PROBLEMS PLUS ¤ 229
To put this in terms of m and b, wesolvethesystemy = x 2 1 and y = mx 1 + b, givingusx 2 1 − mx 1 − b =0
⇒
√
x 1 = 1 2 m − m2 +4b √
. Similarly, x 2 = 1 2 m + m2 +4b . The area is then 1 (x √
8 2 − x 1 ) 3 = 1 m2 +4b 3
8 ,
and is attained at the point P
x P ,x 2 P = P 1 m, 1 m2 .
2 4
Note: Another way to get an expression for f(x) is to use the formula for an area of a triangle in terms of the coordinates of
the vertices: f(x) = 1 2 x2x 2 1 − x 1x2 2 + x1x 2 − xx1 2 + xx
2
2 − x 2x 2 .
15. Suppose that the curve y = a x intersects the line y = x. Thena x 0
= x 0 for some x 0 > 0, and hence a = x 1/x 0
0 .Wefind the
maximum value of g(x) =x 1/x , > 0, because if a is larger than the maximum value of this function, then the curve y = a x
does not intersect the line y = x. g 0 (x) =e (1/x)lnx − 1 x ln x + 1 2 x · 1 1
= x 1/x (1 − ln x). Thisis0 only where
x
x 2
x = e,andfor0 e, f 0 (x) < 0,sog has an absolute maximum of g(e) =e 1/e .Soif
y = a x intersects y = x,wemusthave0

F.
230 ¤ CHAPTER 4 PROBLEMS PLUS
TX.10
(c) Using part (a) with D =1and T 1 =0.26,wehaveT 1 = D √
⇒ c 1 = 1
c ≈ 3.85 km/s. T 4h2 + D
0.26 3 =
2
1
c 1
⇒
4h 2 + D 2 = T 2 3 c 2 1 ⇒ h = 1 2
T
2
3 c 2 1 − D2 = 1 2
from part (b) and T 2 =
sin θ = c 1
c 2
⇒ sec θ =
2hc 2
T 2 =
c 1 c
2
2 − c 2 1
2h sec θ
c 1
+
D − 2h tan θ
c 2
c 2
and tan θ =
c
2
2 − c 2 1
(0.34)2 (1/0.26) 2 − 1 2 ≈ 0.42 km. To find c 2 ,weusesin θ = c1
c 2
from part (a). From the figure,
c 1
,so
c
2
2 − c 2 1
+ D c 2 2 − c2 1
− 2hc1 . Using the values for T 2 [given as 0.32],
c 2 c
2
2 − c 2 1
2hc 2
h, c 1,andD, we can graph Y 1 = T 2 and Y 2 =
c 1 c
2
2 − c 2 1
+ D c 2 2 − c2 1 − 2hc 1
c 2
c
2
2 − c 2 1
and find their intersection points.
Doingsogivesusc 2 ≈ 4.10 and 7.66,butifc 2 =4.10,thenθ =arcsin(c 1 /c 2 ) ≈ 69.6 ◦ , which implies that point S is to
the left of point R in the diagram. So c 2 =7.66 km/s.
21. Let a = |EF| and b = |BF| asshowninthefigure. Since = |BF| + |FD|,
|FD| = − b. Now
|ED| = |EF| + |FD| = a + − b = √ r 2 − x 2 + − (d − x) 2 + a 2
= √
r 2 − x 2 + − (d − x) 2 + √ r 2 − x 2 2
= √ r 2 − x 2 + − √ d 2 − 2dx + x 2 + r 2 − x 2
Let f(x) = √ r 2 − x 2 + − √ d 2 + r 2 − 2dx.
f 0 (x) = 1 2 (r2 − x 2 ) −1/2 (−2x) − 1 2 (d2 + r 2 − 2dx) −1/2 (−2d) =
−x
√
r2 − x + d
√ 2 d2 + r 2 − 2dx .
f 0 (x) =0
⇒
x
√
r2 − x = d
√ 2 d2 + r 2 − 2dx ⇒ x 2
r 2 − x = d 2
2 d 2 + r 2 − 2dx
⇒
d 2 x 2 + r 2 x 2 − 2dx 3 = d 2 r 2 − d 2 x 2 ⇒ 0=2dx 3 − 2d 2 x 2 − r 2 x 2 + d 2 r 2 ⇒
0=2dx 2 (x − d) − r 2 (x 2 − d 2 ) ⇒ 0=2dx 2 (x − d) − r 2 (x + d)(x − d) ⇒ 0=(x − d)[2dx 2 − r 2 (x + d)]
But d>r>x,sox 6=d. Thus, we solve 2dx 2 − r 2 x − dr 2 =0for x:
x = −(−r2 ) ± (−r 2 ) 2 − 4(2d)(−dr 2 )
2(2d)
= r2 ± √ r 4 +8d 2 r 2
. Because √ r
4d
4 +8d 2 r 2 >r 2 , the “negative” can be
discarded. Thus, x = r2 + √ r 2 √ r 2 +8d 2
4d
of |ED| occurs at this value of x.
= r2 + r √ r 2 +8d 2
4d
[r >0] = r √
r + r2 +8d 4d
2 . The maximum value

F.
TX.10
CHAPTER 4 PROBLEMS PLUS ¤ 231
23. V = 4 3 πr3 ⇒ dV
dt =4πr2 dr
dt .
dV
But
dt
is proportional to the surface area, so
dV
dt = k · 4πr2 for some constant k.
Therefore, 4πr 2 dr
dt = k · 4πr2 ⇔ dr = k = constant. An antiderivative of k with respect to t is kt, sor = kt + C.
dt
When t =0, the radius r must equal the original radius r 0,soC = r 0,andr = kt + r 0.Tofind k we use the fact that when
t =3, r =3k + r 0 and V = 1 2 V 0 ⇒ 4 3 π(3k + r 0) 3 = 1 2 · 4
3 πr3 0 ⇒ (3k + r 0 ) 3 = 1 2 r3 0 ⇒ 3k + r 0 = 1 3 √ 2
r 0 ⇒
k = 1 3 r 0
r =0 ⇒ 1 3 r 0
longer.
1
− 1 .Sincer = kt + r
3√ 0 , r = 1 r 1
3 0 − 1 t + r
2
3√ 0 . When the snowball has melted completely we have
2
1
− 1 t + r
3√ 0 =0which gives t = 3 3√ 2
2
3√ . Hence, it takes
2 − 1
3 3√ 2
3√
2 − 1
− 3=
3
3√
2 − 1
≈ 11 h 33 min

F.
TX.10

F.
TX.10
5 INTEGRALS
5.1 Areas and Distances
1. (a) Since f is increasing, we can obtain a lower estimate by using
left endpoints. We are instructed to use five rectangles, so n =5.
L 5 = 5 f(x i−1 ) ∆x
i=1
[∆x = b−a
n = 10 − 0
5
=2]
= f(x 0) · 2+f(x 1) · 2+f(x 2) · 2+f(x 3) · 2+f(x 4) · 2
=2[f(0) + f(2) + f(4) + f(6) + f(8)]
≈ 2(1+3+4.3+5.4+6.3) = 2(20) = 40
Since f is increasing, we can obtain an upper estimate by using
right endpoints.
R 5 = 5 f(x i ) ∆x
i=1
=2[f(x 1 )+f(x 2 )+f(x 3 )+f(x 4 )+f(x 5 )]
=2[f(2) + f(4) + f(6) + f(8) + f(10)]
≈ 2(3 + 4.3+5.4+6.3+7)=2(26)=52
Comparing R 5 to L 5 , we see that we have added the area of the rightmost upper rectangle, f(10) · 2, to the sum and
subtracted the area of the leftmost lower rectangle, f(0) · 2,fromthesum.
(b) L 10 = 10
i=1
f(x i−1 ) ∆x [∆x = 10 − 0
10
=1]
=1[f(x 0)+f(x 1)+···+ f(x 9)]
= f(0) + f(1) + ···+ f (9)
≈ 1+2.1+3+3.7+4.3+4.9+5.4+5.8+6.3+6.7
=43.2
R 10 = 10 f(x i) ∆x = f(1) + f(2) + ···+ f(10)
i=1
= L 10 +1· f(10) − 1 · f(0)
=43.2+7− 1=49.2
add rightmost upper rectangle,
subtract leftmost lower rectangle
233

F.
234 ¤ CHAPTER 5 INTEGRALS
3. (a) R 4 = 4 f(x i ) ∆x ∆x = π/2 − 0 = π
4 8
i=1
=[f(x 1 )+f(x 2 )+f(x 3 )+f(x 4 )] ∆x
TX.10
4
= f(x i ) ∆x
i=1
= cos π 8 +cos 2π 8 +cos3π 8 +cos4π 8
π
8
≈ (0.9239 + 0.7071 + 0.3827 + 0) π 8 ≈ 0.7908
Since f is decreasing on [0,π/2],anunderestimate is obtained by using the right endpoint approximation, R 4 .
(b) L 4 = 4 4
f(x i−1) ∆x = f (x i−1) ∆x
i=1
i=1
=[f(x 0 )+f(x 1 )+f(x 2 )+f(x 3 )] ∆x
= cos 0 + cos π 8 +cos 2π 8 +cos3π 8
π
8
≈ (1 + 0.9239 + 0.7071 + 0.3827) π 8 ≈ 1.1835
L 4 is an overestimate. Alternatively, we could just add the area of the leftmost upper rectangle and subtract the area of the
rightmost lower rectangle; that is, L 4 = R 4 + f(0) · π − f
π
8 2 · π
. 8
5. (a) f(x) =1+x 2 and ∆x = 2 − (−1) =1 ⇒
3
R 3 =1· f(0) + 1 · f(1) + 1 · f(2) = 1 · 1+1· 2+1· 5=8.
∆x = 2 − (−1)
6
=0.5 ⇒
R 6 =0.5[f(−0.5) + f(0) + f(0.5) + f(1) + f(1.5) + f(2)]
=0.5(1.25 + 1 + 1.25 + 2 + 3.25 + 5)
=0.5(13.75) = 6.875
(b) L 3 =1· f(−1) + 1 · f(0) + 1 · f(1) = 1 · 2+1· 1+1· 2=5
L 6 =0.5[f(−1) + f(−0.5) + f(0) + f(0.5) + f(1) + f(1.5)]
=0.5(2 + 1.25 + 1 + 1.25 + 2 + 3.25)
=0.5(10.75) = 5.375
(c) M 3 =1· f(−0.5) + 1 · f(0.5) + 1 · f(1.5)
=1· 1.25 + 1 · 1.25 + 1 · 3.25 = 5.75
M 6 =0.5[f(−0.75) + f(−0.25) + f(0.25)
+ f(0.75) + f(1.25) + f(1.75)]
=0.5(1.5625 + 1.0625 + 1.0625 + 1.5625 + 2.5625 + 4.0625)
=0.5(11.875) = 5.9375
(d) M 6 appears to be the best estimate.

F.
TX.10
SECTION 5.1 AREAS AND DISTANCES ¤ 235
7. Here is one possible algorithm (ordered sequence of operations) for calculating the sums:
1 Let SUM = 0, X_MIN = 0, X_MAX = 1, N = 10 (depending on which sum we are calculating),
DELTA_X = (X_MAX - X_MIN)/N, and RIGHT_ENDPOINT = X_MIN + DELTA_X.
2 Repeat steps 2a, 2b in sequence until RIGHT_ENDPOINT > X_MAX.
2a Add (RIGHT_ENDPOINT)^4 to SUM.
2b Add DELTA_X to RIGHT_ENDPOINT.
At the end of this procedure, (DELTA_X)·(SUM) is equal to the answer we are looking for. We find that
R 10 = 1 10
10
i=1
R 100 = 1
100
i
10
4
≈ 0.2533, R 30 = 1 30
100
i=1
30
i=1
i
30
4
≈ 0.2170, R 50 = 1 50
4 i
≈ 0.2050. It appears that the exact area is 0.2.
100
50
i=1
4 i
≈ 0.2101, and
50
The following display shows the program SUMRIGHT and its output from a TI-83 Plus calculator. To generalize the
program, we have input (rather than assign) values for Xmin, Xmax, and N. Also, the function, x 4 , is assigned to Y 1, enabling
us to evaluate any right sum merely by changing Y 1 and running the program.
9. In Maple, we have to perform a number of steps before getting a numerical answer. After loading the student package
[command: with(student);] weusethecommand
left_sum:=leftsum(1/(xˆ2+1),x=0..1,10 [or 30, or50]); which gives us the expression in summation
notation. To get a numerical approximation to the sum, we use evalf(left_sum);. Mathematica does not have a special
command for these sums, so we must type them in manually. For example, the firstleftsumisgivenby
(1/10)*Sum[1/(((i-1)/10)ˆ2+1)],{i,1,10}], and we use the N command on the resulting output to get a
numerical approximation.
In Derive, we use the LEFT_RIEMANN command to get the left sums, but must define the right sums ourselves.
(We can define a new function using LEFT_RIEMANN with k ranging from 1 to n instead of from 0 to n − 1.)
(a) With f(x) = 1
x 2 +1 , 0 ≤ x ≤ 1, the left sums are of the form Ln = 1 n
L 30 ≈ 0.7937,andL 50 ≈ 0.7904. The right sums are of the form R n = 1 n
R 30 ≈ 0.7770,andR 50 ≈ 0.7804.
n
i=1
i−1
n
n
i=1
1
2
. Specifically, L10 ≈ 0.8100,
+1
1
i
2
.Specifically, R10 ≈ 0.7600,
n
+1

F.
236 ¤ CHAPTER 5 INTEGRALS
TX.10
(b) In Maple, we use the leftbox (with the same arguments as left_sum)andrightbox commands to generate the
graphs.
left endpoints, n =10 left endpoints, n =30 left endpoints, n =50
right endpoints, n =10 right endpoints, n =30 right endpoints, n =50
(c) We know that since y =1/(x 2 +1)is a decreasing function on (0, 1), all of the left sums are larger than the actual area,
and all of the right sums are smaller than the actual area. Since the left sum with n =50is about 0.7904 < 0.791 and the
right sum with n =50is about 0.7804 > 0.780, we conclude that 0.780

F.
TX.10
SECTION 5.2 THEDEFINITEINTEGRAL ¤ 237
21. lim
n
n→∞ i=1
π iπ
tan
4n 4n can be interpreted as the area of the region lying under the graph of y =tanx on the interval
0, π 4 ,
since for y =tanx on
0, π π/4 − 0
4 with ∆x =
n
n
n iπ
A = lim f (x ∗ i ) ∆x = lim tan
n→∞
n→∞ 4n
i=1
i=1
= π iπ
, xi =0+i ∆x =
4n 4n ,andx∗ i = x i, the expression for the area is
π
. Note that this answer is not unique, since the expression for the area is
4n
the same for the function y =tan(x − kπ) on the interval kπ, kπ + π 4
,wherek is any integer.
23. (a) y = f(x) =x 5 . ∆x = 2 − 0
n
(b)
(c)
A = lim Rn = lim
n→∞ n→∞
n
i=1
= 2 n and x i =0+i ∆x = 2i
n .
n
f(x i) ∆x = lim
i=1
i 5 CAS
= n2 (n +1) 2 2n 2 +2n − 1
12
n→∞ i=1
n 2i
n
5
· 2
n = lim
n→∞
n
i=1
32i 5
n · 2
5 n = lim 64
n→∞ n 6
64
lim
n→∞ n · n2 (n +1) 2 2n 2 +2n − 1
= 64
n 2
6 12
12 lim +2n +1 2n 2 +2n − 1
n→∞
n 2 · n 2
= 16 1+
3 lim 2 n→∞ n + 1 2+ 2 n 2 n − 1
n 2
25. y = f(x) =cosx. ∆x = b − 0
n
A = lim R n = lim
n→∞ n→∞
= b n
n
f(x i ) ∆x = lim
i=1
If b = π 2 ,thenA =sinπ 2 =1.
and xi =0+i ∆x =
bi
n .
n→∞ i=1
n bi
cos
n
· b CAS
= lim
n n→∞
⎡
⎢
⎣
= 16 3
1
b sin b
n
i 5 .
i=1
· 1 · 2=
32
3
⎤
2n +1 − b
⎥
b 2n ⎦ CAS
= sinb
2n sin
2n
5.2 The Definite Integral
1. f(x) =3− 1 b − a
x, 2 ≤ x ≤ 14. ∆x =
2
n = 14 − 2 =2.
6
Since we are using left endpoints, x ∗ i = x i−1 .
L 6 = 6 f(x i−1 ) ∆x
i=1
=(∆x)[f(x 0)+f(x 1)+f(x 2)+f(x 3)+f(x 4)+f(x 5)]
=2[f(2) + f(4) + f(6) + f(8) + f(10) + f(12)]
=2[2+1+0+(−1) + (−2) + (−3)] = 2(−3) = −6
The Riemann sum represents the sum of the areas of the two rectangles above the x-axis minus the sum of the areas of the
three rectangles below the x-axis; that is, the net area of the rectangles with respect to the x-axis.

F.
238 ¤ CHAPTER 5 INTEGRALS
TX.10
3. f(x) =e x − 2, 0 ≤ x ≤ 2. ∆x = b − a
n = 2 − 0 = 1 4 2 .
Since we are using midpoints, x ∗ i = x i = 1 2
(xi−1 + xi).
M 4 = 4 f(x i) ∆x =(∆x) [f(x 1)+f(x 2)+f(x 3)+f(x 4)]
i=1
= 1 2 f 1
4 + f 3
4 + f 5
4 + f 7
4
(e 1/4 − 2) + (e 3/4 − 2) + (e 5/4 − 2) + (e 7/4 − 2)
= 1 2
≈ 2.322986
The Riemann sum represents the sum of the areas of the three rectangles above the x-axis minus the area of the rectangle
below the x-axis; that is, the net area of the rectangles with respect to the x-axis.
5. ∆x =(b − a)/n =(8− 0)/4 =8/4 =2.
(a) Using the right endpoints to approximate 8
f(x) dx, wehave
0
4
f(x i) ∆x =2[f(2) + f(4) + f(6) + f(8)] ≈ 2[1+2+(−2) + 1] = 4.
i=1
(b) Using the left endpoints to approximate 8
f(x) dx, wehave
0
4
f(x i−1 ) ∆x =2[f(0) + f(2) + f(4) + f(6)] ≈ 2[2+1+2+(−2)] = 6.
i=1
(c) Using the midpoint of each subinterval to approximate 8
f(x) dx, wehave
0
4
f(x i ) ∆x =2[f(1) + f(3) + f(5) + f(7)] ≈ 2[3+2+1+(−1)] = 10.
i=1
7. Since f is increasing, L 5 ≤ 25
0
f(x) dx ≤ R 5 .
Lower estimate = L 5 = 5 f(x i−1 ) ∆x =5[f(0) + f(5) + f(10) + f(15) + f(20)]
i=1
=5(−42 − 37 − 25 − 6 + 15) = 5(−95) = −475
Upper estimate = R 5 = 5 f(x i) ∆x =5[f(5) + f(10) + f(15) + f(20) + f(25)]
i=1
=5(−37 − 25 − 6 + 15 + 36) = 5(−17) = −85
9. ∆x =(10− 2)/4 =2, so the endpoints are 2, 4, 6, 8,and10, and the midpoints are 3, 5, 7,and9. The Midpoint Rule
gives 10
√
x3 +1dx ≈ 4 f(x
2
i ) ∆x =2 √ 3 3 +1+ √ 5 3 +1+ √ 7 3 +1+ √ 9 3 +1 ≈ 124.1644.
i=1
11. ∆x =(1− 0)/5 =0.2, so the endpoints are 0, 0.2, 0.4, 0.6, 0.8, and 1, and the midpoints are 0.1, 0.3, 0.5, 0.7, and 0.9.
The Midpoint Rule gives
1
0 sin(x2 ) dx ≈ 5 f(x i ) ∆x =0.2 sin(0.1) 2 +sin(0.3) 2 +sin(0.5) 2 +sin(0.7) 2 +sin(0.9) 2 ≈ 0.3084.
i=1

F.
TX.10
SECTION 5.2 THEDEFINITEINTEGRAL ¤ 239
13. In Maple, we use the command with(student); to load the sum and box commands, then
m:=middlesum(sin(xˆ2),x=0..1,5); which gives us the sum in summation notation, then M:=evalf(m); which
gives M 5 ≈ 0.30843908,confirming the result of Exercise 11. The command middlebox(sin(xˆ2),x=0..1,5)
generates the graph. Repeating for n =10and n =20gives M 10 ≈ 0.30981629 and M 20 ≈ 0.31015563.
15. We’ll create the table of values to approximate π
sin xdxby using the
0
program in the solution to Exercise 5.1.7 with Y 1 =sinx, Xmin = 0,
Xmax = π,andn =5, 10, 50,and100.
The values of R n appear to be approaching 2.
17. On [2, 6], lim
n
n→∞ i=1
x i ln(1 + x 2 i ) ∆x = 6
2 x ln(1 + x2 ) dx.
n
R n
5 1.933766
10 1.983524
50 1.999342
100 1.999836
19. On [1, 8], lim
n
n→∞ i=1
21. Note that ∆x = 5 − (−1)
n
5
−1
23. Note that ∆x = 2 − 0
n
2
0
2x
∗
i +(x ∗ i )2 ∆x = 8
1
√
2x + x2 dx.
(1 + 3x) dx = lim
2 − x
2 dx = lim
= 6 n and x i = −1+i ∆x = −1+ 6i
n .
n→∞ i=1
6 n
= lim
n→∞ n
n
f(x i) ∆x = lim
i=1
(−2) + n
6
= lim −2n + 18
n→∞ n n
= lim
n→∞
n
n→∞ i=1
i=1
−12 + 54 n +1
n
= 2
n and x i =0+i ∆x = 2i
n .
n→∞ i=1
= lim
n→∞
n
f(x i ) ∆x = lim
2
n
2n − 4 n 2
n
i=1
= lim 4 − 4
n→∞ 3 · n +1
n
18i
n
1+3
= lim
n→∞
n(n +1) · = lim
2
n→∞
n→∞ i=1
−1+ 6i
n
6
−2n + 18 n n
6
n = lim 6
n→∞ n
n
i
i=1
−12 + 108 n(n +1) ·
n2 2
n
−2+ 18i
n
i=1
= lim −12 + 54 1+ 1
= −12 + 54 · 1=42
n→∞
n
n
2 − 4i2 2
n 2 n
i 2
= lim
n→∞
· 2n +1
n
= lim
n→∞
4 − 8 n(n + 1)(2n +1)
·
n3 6
2 n
2 − 4
n
i 2
n i=1 n 2 i=1
= lim 4 − 4
1+ 1
2+ 1
=4− 4
n→∞
3
3 n n
· 1 · 2= 4 3

F.
240 ¤ CHAPTER 5 INTEGRALS
TX.10
25. Note that ∆x = 2 − 1
n
= 1 and xi =1+i ∆x =1+i(1/n) =1+i/n.
n
2
n
n
x 3 dx = lim f(x
n→∞ i) ∆x = lim 1+ i 3
1 1 n n + i
= lim
n→∞ n n n→∞ n n
27.
b
a
1
i=1
1
= lim
n→∞ n 4
= lim
n→∞
= lim
n→∞
i=1
n
n 3 +3n 2 i +3ni 2 + i 3 = lim
i=1
1
n · n 3 +3n 2 n
n 4 i=1
1+ 3 n(n +1) ·
n2 2
= lim 1+ 3
n→∞ 2 · n +1
n + 1 2 · n +1
n
= lim
n→∞
b − a
xdx = lim
n→∞ n
1+ 3 2
n→∞
i +3n n i 2
+ n
i=1
i=1
+ 3 n(n + 1)(2n +1)
·
n3 6
i=1
3
1 n
n 3 + n
3n 2 i + n
3ni 2 + n i 3
n 4 i=1 i=1
i=1
i=1
i 3
+ 1
n · n2 (n +1) 2
4 4
(n +1)2 ·
n 2
2n +1 · + 1 n 4
1+ 1
+ 1
1+ 1
2+ 1
+ 1
1+ 1 2
=1+ 3 n 2 n n 4 n
2 + 1 · 2+ 1 =3.75
2 4
n
a + b − a
n
i
i=1
a(b − a)
= lim n +
n→∞ n
(b − a)2
n 2 ·
a(b − a)
= lim
n→∞ n
n
1+
i=1
(b − a)2
n 2
n
i=1
i
n(n +1)
(b − a) 2
= a (b − a)+ lim
1+ 1
2
n→∞ 2 n
= a(b − a)+ 1 2 (b − a)2 =(b − a) a + 1 2 b − 1 2 a =(b − a) 1 2 (b + a) = 1 2
b 2 − a 2
29. f(x) = x
6 − 2
, a =2, b =6,and∆x = = 4 1+x5 n n . Using Theorem 4, we get x∗ i = x i =2+i ∆x =2+ 4i
n ,
6
x
so
dx = lim
2 1+x R 5
n = lim
n→∞ n→∞
n 2+ 4i
n
i=1
1+ 2+ 4i
n
5 · 4
n .
31. ∆x =(π − 0)/n = π/n and x ∗ i = x i = πi/n.
π
n
π n
sin 5xdx= lim (sin 5x i ) = lim
n
0
n→∞ i=1
n→∞ i=1
sin 5πi
n
π
n
CAS 1 5π
= π lim
n→∞ n cot CAS 2
= π = 2 2n 5π 5
33. (a) Think of 2
f(x) dx as the area of a trapezoid with bases 1 and 3 and height 2. The area of a trapezoid is A = 1
0 2
(b + B)h,
so 2
f(x) dx = 1 (1 + 3)2 = 4.
0 2
(b) 5
f(x) dx = 2
f(x) dx
0 0
trapezoid
+ 3
f(x) dx
2
rectangle
+ 5
f(x) dx
3
triangle
= 1 2 (1 + 3)2 + 3 · 1 + 1 2 · 2 · 3 =4+3+3=10
(c) 7
5 f(x) dx is the negative of the area of the triangle with base 2 and height 3. 7
5 f(x) dx = − 1 2 · 2 · 3=−3.
(d) 9
f(x) dx is the negative of the area of a trapezoid with bases 3 and 2 and height 2, so it equals
7
− 1 (B + b)h = − 1 (3 + 2)2 = −5. Thus,
2 2
9 f(x) dx = 5
f(x) dx + 7
f(x) dx + 9
0 0 5 7
f(x) dx =10+(−3) + (−5) = 2.

F.
TX.10
SECTION 5.2 THEDEFINITEINTEGRAL ¤ 241
35.
3
0
1
2 x − 1 dx can be interpreted as the area of the triangle above the x-axis
minus the area of the triangle below the x-axis; that is,
1
(1)
1
2 2 −
1
(2)(1) = 1 − 1=− 3 . 0
2 4 4
37.
0
√
−3 1+ 9 − x
2
dx can be interpreted as the area under the graph of
f(x) =1+ √ 9 − x 2 between x = −3 and x =0. This is equal to one-quarter
the area of the circle with radius 3, plus the area of the rectangle, so
0
√
−3 1+ 9 − x
2
dx = 1 π · 4 32 +1· 3=3+ 9 π. 4
39.
2
|x| dx can be interpreted as the sum of the areas of the two shaded
−1
triangles; that is, 1 2 (1)(1) + 1 2 (2)(2) = 1 2 + 4 2 = 5 2 . 0
π
41.
π sin2 x cos 4 xdx=0since the limits of intergration are equal.
43.
45.
47.
1 (5 − 0 6x2 ) dx = 1
5 dx − 6 1
0 0 x2 dx =5(1− 0) − 6
1
3 =5− 2=3
3
1 ex +2 dx = 3
1 ex · e 2 dx = e 2 3
1 ex dx = e 2 (e 3 − e) =e 5 − e 3
2 f(x) dx + 5
f(x) dx − −1
f(x) dx = 5
f(x) dx + −2
f(x) dx [byProperty5andreversinglimits]
−2 2 −2 −2 −1
= 5
f(x) dx [Property 5]
−1
9
49. [2f(x)+3g(x)] dx =2 9
f(x) dx +3 9
g(x) dx = 2(37) + 3(16) = 122
0 0 0
51. Using Integral Comparison Property 8, m ≤ f(x) ≤ M ⇒ m(2 − 0) ≤ 2
f(x) dx ≤ M(2 − 0) ⇒
0
2m ≤ 2
f(x) dx ≤ 2M.
0
53. If −1 ≤ x ≤ 1, then0 ≤ x 2 ≤ 1 and 1 ≤ 1+x 2 ≤ 2, so1 ≤ √ 1+x 2 ≤ √ 2 and
1[1 − (−1)] ≤ 1
√
1+x2 dx ≤ √ 2[1− (−1)] [Property 8]; that is, 2 ≤ 1
√
1+x2 dx ≤ 2 √ 2.
−1
−1
55. If 1 ≤ x ≤ 4,then1 ≤ √ x ≤ 2, so 1(4 − 1) ≤ 4 √ 4 √
1 xdx≤ 2(4 − 1); that is, 3 ≤
1 xdx≤ 6.
57. If π ≤ x ≤ π ,then1 ≤ tan x ≤ √ 3,so1 π
− π π/3
4 3 3 4 ≤ tan xdx≤ √ 3 π
− π
π/4 3 4 or
π
≤ π/3
tan xdx≤ √ π
12 π/4 12 3.
59. The only critical number of f(x) =xe −x on [0, 2] is x =1.Sincef(0) = 0, f(1) = e −1 ≈ 0.368, and
61.
f(2) = 2e −2 ≈ 0.271, we know that the absolute minimum value of f on [0, 2] is 0, and the absolute maximum is e −1 .By
Property 8, 0 ≤ xe −x ≤ e −1 for 0 ≤ x ≤ 2 ⇒ 0(2 − 0) ≤ 2
0 xe−x dx ≤ e −1 (2 − 0) ⇒ 0 ≤ 2
0 xe−x dx ≤ 2/e.
√
x4 +1≥ √ x 4 = x 2 ,so 3
√
x4 +1dx ≥ 3
1
1 x2 dx = 1 3 3 3 − 1 3 = 26 . 3

F.
242 ¤ CHAPTER 5 INTEGRALS
TX.10
63. Using right endpoints as in the proof of Property 2, we calculate
b
n
cf(x) dx = lim cf(x
a i ) ∆x = lim c n f(x i ) ∆x = c lim
n→∞
n→∞
i=1
65. Since − |f(x)| ≤ f(x) ≤ |f(x)|, it follows from Property 7 that
i=1
n
n→∞ i=1
f(x i ) ∆x = c b
f(x) dx.
a
− b
|f(x)| dx ≤ b
f(x) dx ≤
b
|f(x)| dx ⇒
b f(x) dx b
a a a a ≤ |f(x)| dx
a
Note that the definite integral is a real number, and so the following property applies: −a ≤ b ≤ a ⇒ |b| ≤ a for all real
numbers b and nonnegative numbers a.
67. To show that f is integrable on [0, 1] , we must show that lim
the interval [0, 1] into n equal subintervals 0, 1
,
n
subinterval, then we obtain the Riemann sum n
i=1
choose x ∗ i to be an irrational number. Then we get
lim
n
n→∞ i=1
n→∞ i=1
1
n , 2 n
, ... ,
n
f(x ∗ i ) ∆x exists. Let n denote a positive integer and divide
n − 1
n
f(x ∗ i ) · 1 =0,so lim
n n→∞
n
i=1
f(x ∗ i ) · 1
n = lim 1=1. Since the value of lim
n→∞ n→∞
limit does not exist, and f is not integrable on [0, 1].
69. lim
n
n→∞ i=1
lim
n→∞ i=1
i 4
n = lim
5 n→∞
n
i=1
i 4
n · 1
4 n = lim
n→∞
n
i=1
i
n
f(x ∗ i ) · 1
n = n
i=1
, 1
. If we choose x ∗ i to be a rational number in the ith
n
i=1
f(x ∗ i ) · 1
n = lim 0=0. Now suppose we
n→∞
1 · 1
n = n · 1 =1for each n, so
n
n
f(x ∗ i ) ∆x depends on the choice of the sample points x ∗ i ,the
i=1
4
1
. At this point, we need to recognize the limit as being of the form
n
n
f(x i ) ∆x,where∆x =(1− 0)/n =1/n, x i =0+i ∆x = i/n,andf(x) =x 4 . Thus, the definite integral
is 1
0 x4 dx.
71. Choose x i =1+ i n and x∗ i = √ x i−1 x i =
2 1
1 x−2 dx = lim
n→∞ n
n
i=1
= lim n n
n→∞ i=1
1+
i − 1
n
1+ i − 1
1+ i
.Then
n n
1
1+
i
n
1
n + i − 1 − 1
n + i
= lim
n→∞ n 1
n + 1
n +1 + ···+ 1
= lim
n→∞ n 1
n − 1
2n
= lim n n
n→∞ i=1
2n − 1
[by the hint]
= lim 1 −
1
n→∞ 2 =
1
2
1
(n + i − 1)(n + i)
n−1
= lim n 1
n→∞ i=0 n + i − n
1
−
n +1 + ···+ 1
2n − 1 + 1
2n
1
i=1 n + i
5.3 The Fundamental Theorem of Calculus
1. Oneprocessundoeswhattheotheronedoes.Thepreciseversion of this statement is given by the Fundamental Theorem of
Calculus. See the statement of this theorem and the paragraph that follows it on page 387.

F.
TX.10
SECTION 5.3 THE FUNDAMENTAL THEOREM OF CALCULUS ¤ 243
3. (a) g(x) = x
f(t) dt.
0
g(0) = 0
f(t) dt =0
0
g(1) = 1
f(t) dt =1· 2=2 [rectangle],
0
g(2) = 2
f(t) dt = 1
f(t) dt + 2
f(t) dt = g(1) + 2
f(t) dt
0 0 1 1
=2+1· 2+ 1 · 1 · 2=5 [rectangle plus triangle],
2
(d)
g(3) = 3
0 f(t) dt = g(2) + 3
2 f(t) dt =5+1 2 · 1 · 4=7,
g(6) = g(3) + 6
f(t) dt [the integral is negative since f lies under the x-axis]
3
=7+ − 1
2 · 2 · 2+1· 2 =7− 4=3
(b) g is increasing on (0, 3) because as x increases from 0 to 3,wekeepaddingmorearea.
(c) g has a maximum value when we start subtracting area; that is, at x =3.
5. (a)ByFTC1withf(t) =t 2 and a =1, g(x) = x
1 t2 dt ⇒
g 0 (x) =f(x) =x 2 .
(b) Using FTC2, g(x) = x
1 t2 dt = 1
3 t3 x
1 = 1 3 x3 − 1 3
⇒ g 0 (x) =x 2 .
7. f(t) = 1
x
t 3 +1 and g(x) = 1
t 3 +1 dt,sobyFTC1, g0 (x) =f(x) = 1 . Note that the lower limit, 1, could be any
x 3 +1
real number greater than −1 and not affect this answer.
1
9. f(t) =t 2 sin t and g(y) = y
2 t2 sin tdt,sobyFTC1,g 0 (y) =f(y) =y 2 sin y.
11. F (x) =
π
x
√
1+sectdt= −
x
π
√
1+sectdt ⇒ F 0 (x) =− d
dx
13. Let u = 1 x .Thendu dx = − 1 x .Also,dh
2 dx = dh du
du dx ,so
h 0 (x) = d
dx
0
1/x
2
arctan tdt= d
du
u
2
x
π
√
1+sectdt= −
√
1+secx
arctan tdt· du
du
=arctanu
dx dx = − arctan(1/x) .
x 2
15. Let u =tanx. Then du
dx =sec2 x.Also, dy
dx = dy du
du dx ,so
y 0 = d tan x
t + √ tdt= d u
t + √ tdt· du
dx
du
dx = u + √ u du
dx = tan x + √ tan x sec 2 x.
17. Let w =1− 3x. Then dw = −3. Also,dy
dx dx = dy
dw
y 0 = d 1
u 3
dx 1−3x 1+u du = d 1
2 dw w
19.
2
−1
0
dw
dx ,so
u 3 dw
du ·
1+u2 dx = − d
dw
w
1
u 3 dw
du ·
1+u2 dx = −
w3
3(1 − 3x)3
(−3) =
1+w2 1+(1− 3x) 2
x 3 − 2x x
4
2 2
4
(−1)
4
dx =
4 − x2 =
−1
4 − 22 − − (−1) 2 =(4− 4) − 1
4
− 1 =0−
− 3 4 4 =
3
4

F.
244 ¤ CHAPTER 5 INTEGRALS
TX.10
21.
23.
25.
27.
4 (5 − 2t 1 +3t2 ) dt = 5t − t 2 + t 3 4
=(20− 16 + 64) − (5 − 1+1)=68− 5=63
1
1 1
0 x4/5 5
dx =
9 x9/5 = 5 − 0= 5 9 9
0
2
1
3
2
t dt =3 4
1
t
t −4 −3 2
dt =3 = 3 2
1 1
= −1
−3
1
−3 t 3 1
8 − 1 = 7 8
2 x(2 + 0 x5 ) dx = 2
(2x + 0 x6 ) dx = x 2 + 1 x7 2
=
4+ 128
7 0 7 − (0 + 0) =
156
7
29.
9
1
x − 1
√
x
dx =
9
1
x√x
− 1 √
x
dx =
9
1
(x 1/2 − x −1/2 ) dx =
= 2 · 27 − 2 · 3 − 2
− 2 =12−
− 4 3 3 3 =
40
3
2
3 x3/2 − 2x 1/2 9
1
31.
33.
35.
37.
39.
π/4
sec 2 tdt= tan t π/4
=tan π 0 0 4
− tan 0 = 1 − 0=1
2
1 (1 + 2y)2 dy = 2
1 (1 + 4y +4y2 ) dy = y +2y 2 + 4 3 y3 2
1 = 2+8+ 32 3
9
1
√ 3/2
1/2
1
2x dx = 1 9
1
9
2 1 x dx = 1 ln |x| = 1 (ln 9 − ln 1) = 1 ln 9 − 2
2 2 0=ln91/2 =ln3
1
6
√ 3/2
√ dt =6 1 − t
2
1/2
1
√ dt =6 sin −1 t √
3/2
=6
1 − t
2 1/2
sin −1 √
3
2
1
−1 eu+1 du = e u+1 1
−1 = e2 − e 0 = e 2 − 1 [or start with e u+1 = e u e 1 ]
− 1+2+
4
3 =
62
− 13 = 49
3 3 3
− sin −1 1
=6 π
− π
2
3 6 =6 π
6 = π
sin x
41. If f(x) =
cos x
if 0 ≤ x

F.
TX.10
SECTION 5.3 THE FUNDAMENTAL THEOREM OF CALCULUS ¤ 245
49. It appears that the area under the graph is about 2 3 oftheareaoftheviewing
rectangle, or about 2 π ≈ 2.1. The actual area is
3
π sin xdx=[− cos 0 x]π 0
=(− cos π) − (− cos 0) = − (−1) + 1 = 2.
51.
2
−1 x3 dx = 1
4 x4 2
−1 =4− 1 4 = 15 4 =3.75
53. g(x) =
3x
2x
u 2
− 1
0
u 2 +1 du =
g 0 (x) =− (2x)2 − 1
(2x) 2 +1 ·
2x
u 2
− 1
3x
u 2 +1 du +
d
− 1
dx (2x)+(3x)2 (3x) 2 +1 ·
0
u 2
− 1
2x
u 2 +1 du = −
0
u 2
− 1
3x
u 2 +1 du +
d
dx (3x) =−2 · 4x2 − 1
4x 2 +1 +3· 9x2 − 1
9x 2 +1
0
u 2 − 1
u 2 +1 du
⇒
55. y = x 3
√ x
√
t sin tdt=
1√x
√
t sin tdt+
x
3
y 0 = − 4√ x (sin √ x ) ·
57. F (x) =
=3x 7/2 sin(x 3 ) − sin √ x
2 4√ x
x
1
F 00 (x) =f 0 (x) =
1
√ √ x
√ x
3 √
t sin tdt= −
1 t sin tdt+
1 t sin tdt
d
dx (√ x )+x 3/2 sin(x 3 d
) ·
x
3
= − 4√ x sin √ x
dx
2 √ + x 3/2 sin(x 3 )(3x 2 )
x
f(t) dt ⇒ F 0 (x) =f(x) =
1+(x2 ) 4
x 2 ·
x
2
d √
x
2 1+x
8
=
dx
1
√
1+u
4
du since f(t) =
u
· 2x = 2 √ 1+x 8
x 2 x
59. By FTC2, 4
1 f 0 (x) dx = f(4) − f(1),so17 = f(4) − 12 ⇒ f(4) = 17 + 12 = 29.
t
2
1
⇒
√
1+u
4
du
u
⇒
.SoF 00 (2) = √ 1+2 8 = √ 257.
61. (a) The Fresnel function S(x) = x
0 sin π
2 t2 dt has local maximum values where 0=S 0 (x) =sin π
2 t2 and
S 0 changes from positive to negative. For x>0, this happens when π 2 x2 =(2n − 1)π [odd multiples of π] ⇔
x 2 =2(2n − 1) ⇔ x = √ 4n − 2, n any positive integer. For x 0. Differentiating our expression for S 0 (x), weget
S 00 (x) =cos π
x2 2 π x = πx cos π
x2 .Forx>0, S 00 (x) > 0 where cos( π 2 2 2 2 x2 ) > 0 ⇔ 0 < π 2 x2 < π or 2
2n −
1
2 π<
π
2 x2 < √ √
2n + 1 2 π, n any integer ⇔ 0

F.
246 ¤ CHAPTER 5 INTEGRALS
TX.10
− √ 4n − 3 >x>− √ 4n − 1, so the intervals of upward concavity for x9.
(b) We can see from the graph that
1 fdt
3
0 < fdt
5
1 < fdt
7
3 < fdt
9 .
5 < fdt
1 ,
7 Sog(1) = fdt 0
g(5) = 5
0 fdt= g(1) − 3
1 fdt +
5
3 fdt , andg(9) =
9
0 fdt= g(5) − 7
5 fdt +
9
7 fdt . Thus,
g(1)

F.
SECTION TX.105.4 INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM ¤ 247
69. (a) Let f(x) = √ x ⇒ f 0 (x) =1/(2 √ x ) > 0 for x>0 ⇒ f is increasing on (0, ∞).Ifx ≥ 0, thenx 3 ≥ 0, so
1+x 3 ≥ 1 and since f is increasing, this means that f 1+x 3 ≥ f(1) ⇒ √ 1+x 3 ≥ 1 for x ≥ 0. Nextlet
g(t) =t 2 − t ⇒ g 0 (t) =2t − 1 ⇒ g 0 (t) > 0 when t ≥ 1. Thus, g is increasing on (1, ∞). Andsinceg(1) = 0,
g(t) ≥ 0 when t ≥ 1.Nowlett = √ 1+x 3 ,wherex ≥ 0.
√
1+x3 ≥ 1 (from above) ⇒ t ≥ 1 ⇒ g(t) ≥ 0 ⇒
1+x
3 − √ 1+x 3 ≥ 0 for x ≥ 0. Therefore, 1 ≤ √ 1+x 3 ≤ 1+x 3 for x ≥ 0.
(b) From part (a) and Property 7: 1
1 dx ≤ 1
√
1+x3 dx ≤ 1
(1 + 0 0
0 x3 ) dx ⇔
1
x ≤ 1
√
1+x3 dx ≤ x + 1 x4 1
0 0
4
⇔ 1 ≤ 1
√
1+x3 dx ≤ 1+ 1 =1.25.
0 0 4
71. 0 <
10
0 ≤
5
x 2
x 4 + x 2 +1 < x2
x = 1 on [5, 10], so
4 x2 x 2 10
x 4 + x 2 +1 dx <
5
1
−
x dx = 1 10
= − 1
2 x
5
10 − − 1
= 1 5 10 =0.1.
x
f(t)
73. Using FTC1, we differentiate both sides of 6+ dt =2 √ x to get f(x)
a t 2
x 2 =2
1
2 √ x ⇒ f(x) =x3/2 .
a
f(t)
To find a, we substitute x = a in the original equation to obtain 6+ dt =2 √ a ⇒ 6+0=2 √ a ⇒
a t 2
3= √ a ⇒ a =9.
75. (a) Let F (t) = t
0 f(s) ds. Then, by FTC1, F 0 (t) =f(t) =rate of depreciation, so F (t) represents the loss in value over the
interval [0,t].
(b) C(t) = 1 t
A +
t
0
f(s) ds = A + F (t) represents the average expenditure per unit of t during the interval [0,t],
t
assuming that there has been only one overhaul during that time period. The company wants to minimize average
expenditure.
(c) C(t) = 1 t
A +
t
C 0 (t) =0 ⇒ tf(t) =A +
0
f(s) ds
. UsingFTC1,wehaveC 0 (t) =− 1
A +
t 2
t
0
f(s) ds ⇒ f(t) = 1 t
A +
t
0
t
0
f(s) ds + 1 t f(t).
f(s) ds = C(t).
5.4 Indefinite Integrals and the Net Change Theorem
1.
3.
d √
x2 +1+C = d x 2 +1 1/2
+ C
= 1 2 x 2 +1 −1/2 x
· 2x +0= √
dx
dx
x2 +1
5.
d
sin x −
1
dx
3 sin3 x + C = d
sin x −
1
dx
(sin 3 x)3 + C =cosx − 1 · 3(sin 3 x)2 (cos x)+0
=cosx(1 − sin 2 x)=cosx(cos 2 x)=cos 3 x
(x 2 + x −2 ) dx = x3
3 + x−1
−1 + C = 1 3 x3 − 1 x + C

F.
248 ¤ CHAPTER 5 INTEGRALS
TX.10
7.
x 4 − 1 2 x3 + 1 x − 2 dx = x5
4
5 − 1 x 4
2 4 + 1 x 2
4 2 − 2x + C = 1 5 x5 − 1 8 x4 + 1 8 x2 − 2x + C
9.
(1 − t)(2 + t 2 ) dt = (2 − 2t + t 2 − t 3 ) dt =2t − 2 t2 2 + t3 3 − t4 4 + C =2t − t2 + 1 3 t3 − 1 4 t4 + C
x 3 − 2 √
x x
3
11.
dx =
x
x − 2x1/2
x
13. (sin x +sinhx) dx = − cos x +coshx + C
15.
(θ − csc θ cot θ) dθ =
1
2 θ2 +cscθ + C
17. (1 + tan 2 α) dα = sec 2 αdα=tanα + C
dx = (x 2 − 2x −1/2 ) dx = x3
3 − 2 x1/2
1/2 + C = 1 3 x3 − 4 √ x + C
19.
cos x +
1
2 x dx =sinx + 1 4 x2 + C. The members of the family
in the figure correspond to C = −5, 0, 5,and10.
21.
23.
25.
27.
29.
2
0 (6x2 − 4x +5)dx = 6 · 1
3 x3 − 4 · 1
2 x2 +5x 2
= 2x 3 − 2x 2 +5x 2
=(16− 8+10)− 0=18
0 0
0
−1 (2x − ex ) dx = x 2 − e x 0
−1 =(0− 1) − 1 − e −1 = −2+1/e
2 (3u −2 +1)2 du = 2
−2 9u 2 +6u +1 du = 9 · 1
3 u3 +6· 1
2 u2 + u 2
= 3u 3 +3u 2 + u 2
−2 −2
4
1
−1
−2
= (24 + 12 + 2) − (−24 + 12 − 2) = 38 − (−14) = 52
√
4 4 t (1 + t) dt =
1 (t1/2 + t 3/2 2
) dt =
3 t3/2 + 2 5 t5/2 = 16
+ 64
3 5 − 2 + 2
3 5 =
14
+ 62 = 256
3 5 15
1
4y 3 + 2
−1
dy = 4 · 1 1
y 3 4 y4 +2·
−2 y−2 =
y 4 − 1 −1
=(1− 1) −
16 − 1
y 2 4 = −
63
4
1
31. x √ √
3 4
0 x + x
33.
4
1
dx = 1
0 (x4/3 + x 5/4 ) dx =
−2
−2
1
3
7 x7/3 + 4 9 x9/4 = 3
+ 4
7 9 − 0=
55
63
0
√ 4 5/x dx = 5
1 x−1/2 dx = √
5 2 √ 4
x = √ 5(2· 2 − 2 · 1) = 2 √ 5
1
35.
37.
π (4 sin θ − 3cosθ) dθ = − 4cosθ − 3sinθ π
=(4− 0) − (−4 − 0) = 8
0 0
π/4
0
1+cos 2 θ
cos 2 θ
dθ =
π/4
0
1
cos 2 θ + cos2 θ
dθ =
cos 2 θ
= tan θ + θ π/4
0
= tan π 4 + π 4
π/4
0
(sec 2 θ +1)dθ
− (0 + 0) = 1 +
π
4

F.
SECTION TX.105.4 INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM ¤ 249
64
1+ 3√
x
64
1
64
x1/3
39. √ dx =
+ dx = x −1/2 + x (1/3) − (1/2) 64
dx = (x −1/2 + x −1/6 ) dx
1 x 1 x1/2 x 1/2 1
1
64
=
2x 1/2 + 6 5 x5/6 =
16 + 192
5 − 2+
6
5 =14+
186
= 256
5 5
1
√
1/ 3
t 2
√
− 1
1/ 3
41.
0 t 4 − 1 dt = t 2
√
− 1
1/ 3
0 (t 2 +1)(t 2 − 1) dt = 1
0 t 2 +1 dt = arctan t 1/ √
3
=arctan 1/ √
3 − arctan 0
0
= π − 0= π 6 6
2
43. (x − 2 |x|) dx = 0
[x − 2(−x)] dx + 2
[x − 2(x)] dx = 0
3xdx+ 2
(−x) dx =3 1
x2 0
− 1
x2 2
−1 −1 0 −1 0 2 −1 2 0
=3
0 − 1 2 − (2 − 0) = −
7
= −3.5
2
45. The graph shows that y = x + x 2 − x 4 has x-intercepts at x =0and at
x = a ≈ 1.32. So the area of the region that lies under the curve and above the
x-axis is
a (x + 0 x2 − x 4 ) dx = 1
2 x2 + 1 3 x3 − 1 x5 a
5 0
= 1
2 a2 + 1 3 a3 − 1 a5 − 0 ≈ 0.84
5
47. A = 2
0 2y − y
2
dy = y 2 − 1 y3 2
=
3
4 − 8 0 3 − 0=
4
3
49. If w 0 (t) is the rate of change of weight in pounds per year, then w(t) represents the weight in pounds of the child at age t. We
know from the Net Change Theorem that 10
w 0 (t) dt = w(10) − w(5), so the integral represents the increase in the child’s
5
weight (in pounds) between the ages of 5 and 10.
51. Since r(t) istherateatwhichoilleaks,wecanwriter(t) =−V 0 (t),whereV (t) is the volume of oil at time t. [Note that the
minus sign is needed because V is decreasing, so V 0 (t) is negative, but r(t) is positive.] Thus, by the Net Change Theorem,
120
r(t) dt = − 120
V 0 (t) dt = − [V (120) − V (0)] = V (0) − V (120), which is the number of gallons of oil that leaked
0 0
from the tank in the first two hours (120 minutes).
53. By the Net Change Theorem, 5000
1000 R0 (x) dx = R(5000) − R(1000), so it represents the increase in revenue when
production is increased from 1000 units to 5000 units.
55. In general, the unit of measurement for b
f(x) dx is the product of the unit for f(x) and the unit for x. Sincef(x) is
a
measured in newtons and x is measured in meters, the units for 100
f(x) dx are newton-meters. (A newton-meter is
0
abbreviated N·m and is called a joule.)
57. (a) Displacement = 3
(3t − 5) dt = 3
0 2 t2 − 5t 3
= 27 − 15 = − 3 m
0 2 2
(b) Distance traveled = 3
|3t − 5| dt = 5/3
(5 − 3t) dt + 3
(3t − 5) dt
0 0 5/3
= 5t − 3 t2 5/3
+ 3
2 0 2 t2 − 5t 3
= 25 − 3 · 25 + 27 − 15 − 3 · 25 − 25
5/3 3 2 9 2 2 9 3 =
41
m 6
59. (a) v 0 (t) =a(t) =t +4 ⇒ v(t) = 1 2 t2 +4t + C ⇒ v(0) = C =5 ⇒ v(t) = 1 2 t2 +4t +5m/s
(b) Distance traveled = 10
|v(t)| dt = 10
1
0 0 2 t2 +4t +5 dt = 10 1
0 2 t2 +4t +5 dt = 1
6 t3 +2t 2 +5t 10
0
= 500
3 +200+50=4162 m 3

F.
250 ¤ CHAPTER 5 INTEGRALS
TX.10
61. Since m 0 (x) =ρ(x), m = 4
ρ(x) dx =
4
9+2 √ 4
x dx = 9x + 4 0 0
3 x3/2 =36+ 32 3
0
− 0=
140
3
=46 2 3 kg.
63. Let s be the position of the car. We know from Equation 2 that s(100) − s(0) = 100
0
v(t) dt. We use the Midpoint Rule for
0 ≤ t ≤ 100 with n =5. Note that the length of each of the five time intervals is 20 seconds = 20 hour = 1 hour.
3600 180
So the distance traveled is
100
v(t) dt ≈ 1
1
247
[v(10) + v(30) + v(50) + v(70) + v(90)] = (38 + 58 + 51 + 53 + 47) = ≈ 1.4 miles.
0 180 180 180
65. From the Net Change Theorem, the increase in cost if the production level is raised
from 2000 yards to 4000 yards is C(4000) − C(2000) = 4000
2000 C0 (x) dx.
4000
2000 C0 (x) dx = 4000
2000 3 − 0.01x +0.000006x
2
dx = 3x − 0.005x 2 +0.000002x 3 4000
=60,000 −2,000 = $58,000
2000
67. (a) We can find the area between the Lorenz curve and the line y = x by subtracting the area under y = L(x) from the area
under y = x. Thus,
area between Lorenz curve and line y = x
coefficient of inequality = =
area under line y = x
=
1
0
[x − L(x)] dx
[x 2 /2] 1 =
0
1
0
[x − L(x)] dx
1/2
1
0
[x − L(x)] dx
1
0 xdx
=2 1
[x − L(x)] dx
0
(b) L(x) = 5
12 x2 + 7 x ⇒ L(50%) = L
1
12 2 =
5
+ 7 = 19 =0.39583, so the bottom 50% of the households receive
48 24 48
at most about 40% of the income. Using the result in part (a),
coefficient of inequality =2 1
[x − L(x)] dx =2 1 0 0 x −
5
12 x2 − 7 x 12
dx =2 1
5
x − 5 x2 dx
0 12 12
=2 1 5
(x −
0 12 x2 ) dx = 5 1
6 2 x2 − 1 x3 1
= 5 1 − 1
3 0 6 2 3 =
5 1
6 6 =
5
36
5.5 The Substitution Rule
1. Let u = −x. Thendu = − dx, so dx = − du. Thus, e −x dx = e u (−du) =−e u + C = −e −x + C. Don’t forget that it
is often very easy to check an indefinite integration by differentiating your answer. In this case,
d
dx (−e−x + C) =−[e −x (−1)] = e −x , the desired result.
3. Let u = x 3 +1.Thendu =3x 2 dx and x 2 dx = 1 du,so 3
x 2 √u
x 3 +1dx =
1 du = 1 u 3/2
3
3 3/2 + C = 1 3 · 2
3 u3/2 + C = 2 9 (x3 +1) 3/2 + C.
5. Let u =cosθ. Thendu = − sin θdθand sin θdθ = −du, so
cos 3 θ sin θdθ= u 3 (−du) =− u4
4 + C = − 1 4 cos4 θ + C.
7. Let u = x 2 .Thendu =2xdxand xdx= 1 2 du,so x sin(x 2 ) dx = sin u 1
2 du = − 1 2 cos u + C = − 1 2 cos(x2 )+C.

F.
TX.10
SECTION 5.5 THE SUBSTITUTION RULE ¤ 251
9. Let u =3x − 2. Thendu =3dx and dx = 1 3 du,so (3x − 2) 20 dx = u 20 1
3 du = 1 3 · 1
21 u21 + C = 1
63 (3x − 2)21 + C.
11. Let u =2x + x 2 .Thendu =(2+2x) dx =2(1+x) dx and (x +1)dx = 1 du, so
2
(x +1) √u
2x + x 2 dx =
1 du = 1 u 3/2
2
2 3/2 + C = 1
3 2x + x
2 3/2
+ C.
Or: Letu = √ 2x + x 2 .Thenu 2 =2x + x 2 ⇒ 2udu =(2+2x) dx ⇒ udu =(1+x) dx, so
(x +1)
√
2x + x2 dx = u · udu= u 2 du = 1 3 u3 + C = 1 3 (2x + x2 ) 3/2 + C.
13. Let u =5− 3x. Thendu = −3 dx and dx = − 1 3
du, so
dx 1
5 − 3x =
−
1
u
du = − 1 ln |u| + C = − 1 ln |5 − 3x| + C.
3 3 3
15. Let u = πt. Thendu = πdtand dt = 1 du,so sin πt dt = sin u 1
du = 1 (− cos u)+C = − 1 cos πt + C.
π π π π
17. Let u =3ax + bx 3 .Thendu =(3a +3bx 2 ) dx =3(a + bx 2 ) dx, so
a + bx 2
√ dx = 3ax + bx
3
1 du 3
u = 1
1/2 3
u −1/2 du = 1 3 · 2u2 + C = 2 3
3ax + bx3 + C.
19. Let u =lnx. Thendu = dx (ln x)
2
x ,so dx = u 2 du = 1 3
x
u3 + C = 1 (ln 3 x)3 + C.
21. Let u = √ t.Thendu = dt
2 √ t and √ 1 √
cos t
dt =2du,so √ dt = cos u (2 du) =2sinu + C =2sin √ t + Ċ.
t t
23. Let u =sinθ. Thendu =cosθdθ,so cos θ sin 6 θdθ= u 6 du = 1 7 u7 + C = 1 7 sin7 θ + C.
25. Let u =1+e x .Thendu = e x dx,so e x√ 1+e x dx = √ udu= 2 3 u3/2 + C = 2 3 (1 + ex ) 3/2 + C.
Or: Let u = √ 1+e x .Thenu 2 =1+e x and 2udu = e x dx, so
e
x √ 1+e x dx = u · 2udu= 2 3 u3 + C = 2 3 (1 + ex ) 3/2 + C.
27. Let u =1+z 3 .Thendu =3z 2 dz and z 2 dz = 1 3 du,so
z 2
3√ dz = 1+z
3
u −1/3 1
3 du = 1 3 · 3
2 u2/3 + C = 1 2 (1 + z3 ) 2/3 + C.
29. Let u =tanx. Thendu =sec 2 xdx,so e tan x sec 2 xdx= e u du = e u + C = e tan x + C.
31. Let u =sinx. Thendu =cosxdx,so
[or −csc x + C ].
cos x
sin 2 x dx =
1
u 2 du =
u −2 du = u−1
−1 + C = − 1 u + C = − 1
sin x + C
33. Let u =cotx. Thendu = − csc 2 xdxand csc 2 xdx= −du,so
√cot
x csc 2 xdx=
√u
(−du) =−
u 3/2
3/2 + C = − 2 3 (cot x)3/2 + C.

F.
252 ¤ CHAPTER 5 INTEGRALS
TX.10
sin 2x
sin x cos x
35.
1+cos 2 x dx =2 dx =2I. Letu =cosx. Thendu = − sin xdx,so
1+cos 2 x
udu
2I = −2
1+u = −2 · 1 ln(1 + 2 2 u2 )+C = − ln(1 + u 2 )+C = − ln(1 + cos 2 x)+C.
37.
Or: Let u =1+cos 2 x.
cos x
cot xdx=
sin x dx. Letu =sinx. Thendu =cosxdx,so
39. Let u =secx. Thendu =secx tan xdx,so
sec 3 x tan xdx= sec 2 x (sec x tan x) dx = u 2 du = 1 3 u3 + C = 1 3 sec3 x + C.
41. Let u =sin −1 x.Thendu =
1
√ dx,so 1 − x
2
1
cot xdx= du =ln|u| + C =ln|sin x| + C.
u
dx
√
1 − x2 sin −1 x = 1
u du =ln|u| + C =ln sin −1 x + C.
43. Let u =1+x 2 .Thendu =2xdx,so
1+x
1+x 2 dx =
1
1+x dx + 2
x
1+x dx 2 =tan−1 x +
1
2 du
u
=tan−1 x + 1 2 ln|u| + C
=tan −1 x + 1 2 ln 1+x 2 + C =tan −1 x + 1 2 ln 1+x 2 + C [since 1+x 2 > 0].
45. Let u = x +2.Thendu = dx,so
x
u − 2
4√ dx = x +2
4√ du = (u 3/4 − 2u −1/4 ) du = 4 7 u7/4 − 2 · 4
3 u3/4 + C
u
= 4 7 (x +2)7/4 − 8 3 (x +2)3/4 + C
In Exercises 47–50, let f(x) denote the integrand and F (x) its antiderivative (with C =0).
47. f(x) =x(x 2 − 1) 3 . u = x 2 − 1 ⇒ du =2xdx,so
x(x 2 − 1) 3 dx = u 3 1
2 du = 1 8 u4 + C = 1 8 (x2 − 1) 4 + C
Where f is positive (negative), F is increasing (decreasing). Where f
changes from negative to positive (positive to negative), F has a local
minimum (maximum).
49. f(x) =sin 3 x cos x. u =sinx ⇒ du =cosxdx,so
sin 3 x cos xdx= u 3 du = 1 4 u4 + C = 1 4 sin4 x + C
Note that at x = π 2
, f changes from positive to negative and F has a local
maximum. Also, both f and F are periodic with period π,soatx =0and
at x = π, f changes from negative to positive and F has local minima.
51. Let u = x − 1, sodu = dx. Whenx =0, u = −1; whenx =2, u =1. Thus, 2
0 (x − 1)25 dx = 1
−1 u25 du =0by
Theorem 7(b), since f(u) =u 25 is an odd function.

F.
TX.10
SECTION 5.5 THE SUBSTITUTION RULE ¤ 253
53. Let u =1+2x 3 ,sodu =6x 2 dx. Whenx =0, u =1;whenx =1, u =3.Thus,
1 x2 1+2x 35 dx = 3
u5 1
du
= 1 1 u6 3
= 1
0 1 6 6 6 1 36 (36 − 1 6 )= 1 728
(729 − 1) = = 182 .
36 36 9
55. Let u = t/4, sodu = 1 dt. Whent =0, u =0;whent = π, u = π/4. Thus,
4
π
0 sec2 (t/4) dt = π/4
sec 2 u (4 du) =4 tan u π/4
=4 tan π − tan 0 =4(1− 0) = 4.
0 0
4
π/6
57.
−π/6 tan3 θdθ=0by Theorem 7(b), since f(θ) =tan 3 θ is an odd function.
59. Let u =1/x, sodu = −1/x 2 dx. Whenx =1, u =1;whenx =2, u = 1 2 . Thus,
2
e 1/x
1
x 2 dx = 1/2
1
e u (−du) =− e u 1/2
1
= −(e 1/2 − e) =e − √ e.
61. Let u =1+2x,sodu =2dx. Whenx =0, u =1;whenx =13, u =27.Thus,
13
dx
27
= u (1 −2/3
1
du 1
= · 3u1/3 27
= 3 (3 − 1) = 3.
+ 2x) 2 2 2 2
1
0
3
1
63. Let u = x 2 + a 2 ,sodu =2xdxand xdx= 1 2 du. Whenx =0, u = a2 ;whenx = a, u =2a 2 . Thus,
a
0
1
x 2a
2
x 2 + a 2 dx = u 1/2 1
du
a 2 2
= 1 2
0
0
2
3 u3/2 2a 2
a 2 =
1
u3/2 2a 2
3
= 1 (2a 2
a 2 3
) 3/2 − (a 2 ) 3/2
= 1 2 √
3
2 − 1 a 3
65. Let u = x − 1,sou +1=x and du = dx. Whenx =1, u =0;whenx =2, u =1. Thus,
2
x √ 1
x − 1 dx = (u +1) √ 1
udu= (u 3/2 + u 1/2 2
) du =
5 u5/2 + 2 u3/2 1
= 2 + 2 = 16 .
3 5 3 15
67. Let u =lnx, sodu = dx x .Whenx = e, u =1;whenx = e4 ; u =4.Thus,
0
e
4
e
0
dx
4
x √ ln x = u −1/2 du =2
u 1/2 4
=2(2− 1) = 2.
1
1
69. Let u = e z + z,sodu =(e z +1)dz. Whenz =0, u =1;whenz =1, u = e +1. Thus,
1
e z
+1
e+1
e z + z dz = 1
e+1
u du = ln |u| =ln|e +1| − ln |1| =ln(e +1).
1
1
71. From the graph, it appears that the area under the curve is about
1+ a little more than 1 2 · 1 · 0.7 , or about 1.4. The exact area is given by
A = 1 √
0 2x +1dx. Letu =2x +1,sodu =2dx. The limits change to
2 · 0+1=1and 2 · 1+1=3,and
A = √
3
3
1 u 1 du √ √
2
= 1 2
2 3 u3/2 = 1 3 3 3 − 1 = 3 −
1
≈ 1.399.
3
1
73. First write the integral as a sum of two integrals:
I = 2
−2 (x +3)√ 4 − x 2 dx = I 1 + I 2 = 2
−2 x √ 4 − x 2 dx + 2
−2 3 √ 4 − x 2 dx. I 1 =0by Theorem 7(b), since
f(x) =x √ 4 − x 2 is an odd function and we are integrating from x = −2 to x =2. We interpret I 2 as three times the area of
a semicircle with radius 2,soI =0+3· 1
2
π · 2
2 =6π.

F.
254 ¤ CHAPTER 5 INTEGRALS
TX.10
75. First Figure Let u = √ x,sox = u 2 and dx =2udu.Whenx =0, u =0;whenx =1, u =1. Thus,
A 1 = 1
0 e√x dx = 1
0 eu (2udu)=2 1
0 ueu du.
Second Figure A 2 = 1
0 2xex dx =2 1
0 ueu du.
Third Figure
Let u =sinx,sodu =cosxdx.Whenx =0, u =0;whenx = π 2 , u =1.Thus,
A 3 = π/2
0
e sin x sin 2xdx= π/2
0
e sin x (2 sin x cos x) dx = 1
0 eu (2udu)=2 1
0 ueu du.
Since A 1 = A 2 = A 3 , all three areas are equal.
77. The rate is measured in liters per minute. Integrating from t =0minutes to t =60minutes will give us the total amount of oil
that leaks out (in liters) during the first hour.
60
r(t) dt = 60
0 0 100e−0.01t dt [u = −0.01t, du = −0.01dt]
= 100 −0.6
e u (−100 du) =−10,000 e u −0.6
= −10,000(e −0.6 − 1) ≈ 4511.9 ≈ 4512 liters
0 0
79. The volume of inhaled air in the lungs at time t is
V (t)= t
f(u) du = t 1
sin 2π
u du = 2πt/5 1
sin v 5
0 0 2 5 0 2
= 5
4π
− cos v
2πt/5
0
= 5
4π
− cos
2π
5 t +1 = 5
4π
1 − cos
2π
5 t liters
2π dv substitute v = 2π 5 u, dv = 2π 5 du
81. Let u =2x. Thendu =2dx,so 2
f(2x) dx = 4
f(u) 1
du
= 1 4 f(u) du = 1 (10) = 5.
0 0 2 2 0 2
83. Let u = −x. Thendu = −dx,so
b f(−x) dx = −b
a
f(u)(−du) = −a
−a −b
f(u) du = −a
f(x) dx
−b
From the diagram, we see that the equality follows from the fact that we are
reflecting the graph of f, and the limits of integration, about the y-axis.
85. Let u =1− x. Thenx =1− u and dx = −du,so
1
0 xa (1 − x) b dx = 0
(1 − 1 u)a u b (−du) = 1
0 ub (1 − u) a du = 1
0 xb (1 − x) a dx.
87.
x sin x
1+cos 2 x = x · sin x
t
2 − sin 2 = xf(sin x),wheref(t) = . By Exercise 86,
x 2 − t2 π
0
x sin x
π
1+cos 2 x dx = xf(sin x) dx = π 2
0
π
0
f(sin x) dx = π 2
Let u =cosx. Thendu = − sin xdx.Whenx = π, u = −1 and when x =0, u =1.So
π π
2 0
sin x
1+cos 2 x dx = − π −1
2 1
du
1+u = π 1
2 2 −1
= π 2 [tan−1 1 − tan −1 (−1)] = π 2
π
0
sin x
1+cos 2 x dx
du
1+u = π
tan −1 u 1
2 2
−1
π
4 − − π
= π2
4 4

F.
TX.10
CHAPTER 5 REVIEW ¤ 255
5 Review
1. (a) n
i=1 f(x∗ i ) ∆x is an expression for a Riemann sum of a function f.
x ∗ i is a point in the ith subinterval [x i−1 ,x i ] and ∆x is the length of the subintervals.
(b)SeeFigure1inSection5.2.
(c) In Section 5.2, see Figure 3 and the paragraph beside it.
2. (a) See Definition 5.2.2.
(b)SeeFigure2inSection5.2.
(c) In Section 5.2, see Figure 4 and the paragraph by it (contains “net area”).
3. See the Fundamental Theorem of Calculus after Example 9 in Section 5.3.
4. (a) See the Net Change Theorem after Example 5 in Section 5.4.
(b) t 2
t 1
r(t) dt represents the change in the amount of water in the reservoir between time t 1 and time t 2.
5. (a) 120
60
v(t) dt represents the change in position of the particle from t =60to t =120seconds.
(b) 120
60
|v(t)| dt represents the total distance traveled by the particle from t =60to 120 seconds.
(c) 120
60
a(t) dt represents the change in the velocity of the particle from t =60to t =120seconds.
6. (a) f(x) dx is the family of functions {F | F 0 = f}. Any two such functions differ by a constant.
(b) The connection is given by the Net Change Theorem: b
f(x) dx = f(x) dx b
if f is continuous.
a a
7. The precise version of this statement is given by the Fundamental Theorem of Calculus. See the statement of this theorem and
the paragraph that follows it at the end of Section 5.3.
8. See the Substitution Rule (5.5.4). This says that it is permissible to operate with the dx afteranintegralsignasifitwerea
differential.
1. True by Property 2 of the Integral in Section 5.2.
3. True by Property 3 of the Integral in Section 5.2.
5. False. For example, let f(x) =x 2 .Then 1
√
x2 dx =
1
xdx= 1 ,but 1
0
0 2 0 x2 1
dx = = √ 1
3 3
.
7. True by Comparison Property 7 of the Integral in Section 5.2.
9. True. The integrand is an odd function that is continuous on [−1, 1], so the result follows from Theorem 5.5.7(b).

F.
256 ¤ CHAPTER 5 INTEGRALS
TX.10
11. False. The function f(x) =1/x 4 is not bounded on the interval [−2, 1]. It has an infinite discontinuity at x =0,soitis
not integrable on the interval. (If the integral were to exist, a positive value would be expected, by Comparison
Property 6 of Integrals.)
13. False. For example, the function y = |x| is continuous on R, but has no derivative at x =0.
15. False.
b d
b
f(x) dx is a constant, so
a
dx
f(x) dx =0, not f(x) [unless f(x) =0]. Compare the given statement
a
carefully with FTC1, in which the upper limit in the integral is x.
1. (a) L 6 = 6
i=1
f(x i−1 ) ∆x [∆x = 6 − 0
6
=1]
= f(x 0 ) · 1+f(x 1 ) · 1+f(x 2 ) · 1+f(x 3 ) · 1+f(x 4 ) · 1+f(x 5 ) · 1
≈ 2+3.5+4+2+(−1) + (−2.5) = 8
(b)
The Riemann sum represents the sum of the areas of the four rectangles
above the x-axis minus the sum of the areas of the two rectangles below the
x-axis.
M 6 = 6
i=1
f(x i ) ∆x [∆x = 6 − 0
6
=1]
= f(x 1) · 1+f(x 2) · 1+f(x 3) · 1+f(x 4) · 1+f(x 5) · 1+f(x 6) · 1
= f(0.5) + f(1.5) + f(2.5) + f(3.5) + f(4.5) + f(5.5)
≈ 3+3.9+3.4+0.3+(−2) + (−2.9) = 5.7
3.
1
√
0 x + 1 − x
2
dx = 1
xdx+ 1
√
0 0 1 − x2 dx = I 1 + I 2 .
I 1 can be interpreted as the area of the triangle shown in the figure
and I 2 can be interpreted as the area of the quarter-circle.
Area = 1 2 (1)(1) + 1 4 (π)(1)2 = 1 2 + π 4 .
5.
6 f(x) dx = 4
f(x) dx + 6
f(x) dx ⇒ 10 = 7 + 6
f(x) dx ⇒ 6
0 0 4 4 4
f(x) dx =10− 7=3
7. First note that either a or b must be the graph of x
f(t) dt,since 0
f(t) dt =0,andc(0) 6= 0. Now notice that b>0 when c
0 0
is increasing, and that c>0 when a is increasing. It follows that c is the graph of f(x), b is the graph of f 0 (x),anda is the
graph of x
f(t) dt.
0
2
9.
1 8x 3 +3x 2 dx = 8 · 1
4 x4 +3· 1 x3 2
=
3
2x 4 + x 3 2
= 2 · 2 4 +2 3 − (2 + 1) = 40 − 3=37
1 1
1
11.
0 1 − x
9
dx = x − 1 x10 1
=
1 − 1 10 0 10 − 0=
9
10

F.
TX.10
CHAPTER 5 REVIEW ¤ 257
13.
9
1
√ u − 2u
2
du =
u
9
1
(u −1/2 − 2u) du =
2u 1/2 − u 2 9
=(6− 81) − (2 − 1) = −76
15. Let u = y 2 +1,sodu =2ydyand ydy = 1 du. Wheny =0, u =1;wheny =1, u =2.Thus,
2
1
0 y(y2 +1) 5 dy = 2
u5 1
du
= 1 1 u6 2
= 1 63
(64 − 1) = = 21 .
1 2 2 6 1 12 12 4
17.
5
1
dt
(t − 4) does not exist because the function f(t) = 1
has an infinite discontinuity at t =4;
2 (t − 4)
2
that is, f is discontinuous on the interval [1, 5].
19. Let u = v 3 ,sodu =3v 2 dv. Whenv =0, u =0;whenv =1, u =1.Thus,
1
0 v2 cos(v 3 ) dv = 1
cos u 1
du
0 3
= 1 1
3 sin u = 1 (sin 1 − 0) = 1 0 3 3
sin 1.
21.
23.
π/4
−π/4
t 4 tan t
2+cost dt =0by Theorem 5.5.7(b), since f(t) = t4 tan t
is an odd function.
2+cost
2 2 1 − x
1 1
dx =
x
x − 1 dx =
x − 2
2 x +1 dx = − 1 x − 2ln|x| + x + C
25. Let u = x 2 +4x. Thendu =(2x +4)dx =2(x +2)dx, so
x +2
√
x2 +4x dx = u −1/2 1
du = 1 · 2 2 2u1/2 + C = √ u + C = x 2 +4x + C.
1
27. Let u =sinπt. Thendu = π cos πt dt,so sin πt cos πt dt = u 1
π du = 1 π · 1
2 u2 + C = 1
2π (sin πt)2 + C.
29. Let u = √ x.Thendu = dx √
e
x
2 √ x ,so √ dx =2 x
e u du =2e u + C =2e √x + C.
31. Let u =ln(cosx). Thendu = − sin x dx = − tan xdx,so
cos x
tan x ln(cos x) dx = − udu= −
1
2 u2 + C = − 1 [ln(cos 2 x)]2 + C.
33. Let u =1+x 4 .Thendu =4x 3 dx,so
x 3
1+x dx = 1 1 4 4 u du = 1 ln|u| + C = 1 ln 1+x 4 + C.
4 4
35. Let u =1+secθ. Thendu =secθ tan θdθ,so
sec θ tan θ
1+secθ dθ = 1
1
1+secθ (sec θ tan θdθ)= du =ln|u| + C =ln|1+secθ| + C.
u
37. Since x 2 − 4 < 0 for 0 ≤ x 0 for 2

F.
258 ¤ CHAPTER 5 INTEGRALS
TX.10
In Exercises 39 and 40, let f(x) denote the integrand and F (x) its antiderivative (with C =0).
39. Let u =1+sinx. Thendu =cosxdx,so
cos xdx
√ = u −1/2 du =2u 1/2 + C =2 √ 1+sinx + C.
1+sinx
41. From the graph, it appears that the area under the curve y = x √ x between x =0
and x =4is somewhat less than half the area of an 8 × 4 rectangle, so perhaps
about 13 or 14. Tofind the exact value, we evaluate
4
0 x √ xdx= 4
0 x3/2 dx =
4
2
5 x5/2 = 2 5 (4)5/2 = 64 =12.8.
5
0
x
43. F (x) =
0
t 2
1+t dt ⇒ F 0 (x) = d x
3 dx 0
t 2 x2
dt =
1+t3 1+x 3
45. Let u = x 4 .Then du
dx =4x3 .Also, dg
dx = dg du
du dx ,so
g 0 (x) = d x
4
cos(t 2 ) dt = d u
cos(t 2 ) dt · du
dx 0
du 0
dx =cos(u2 ) du
dx =4x3 cos(x 8 ).
47. y =
x
e t 1
√ x t dt = e t x
√ x t dt +
1
e t
t dt = − √ x
1
e t
t dt + x
dy
dx = − d
√
x
e t
dx 1 t dt + d x
e t
dx 1 t dt .Letu = √ x.Then
1
e t
t dt
⇒
so dy
dx = −e√ x
2x + ex x .
d
√ x
e t
dx 1 t dt = d u
e t
dx 1 t dt = d u
e t du
du 1 t dt dx = eu u ·
1
2 √ x = e√ x
√ ·
x
1
2 √ x = e√ x
2x ,
49. If 1 ≤ x ≤ 3, then √ 1 2 +3≤ √ x 2 +3≤ √ 3 2 +3 ⇒ 2 ≤ √ x 2 +3≤ 2 √ 3,so
2(3 − 1) ≤ 3
√
x2 +3dx ≤ 2 √ 3(3 − 1); thatis,4 ≤ 3
√
x2 +3dx ≤ 4 √ 3.
1
1
51. 0 ≤ x ≤ 1 ⇒ 0 ≤ cos x ≤ 1 ⇒ x 2 cos x ≤ x 2 ⇒ 1
0 x2 cos xdx≤ 1
0 x2 dx = 1 3 x
3 1
= 1 [Property 7].
0 3
53. cos x ≤ 1 ⇒ e x cos x ≤ e x ⇒ 1
0 ex cos xdx≤ 1
0 ex dx =[e x ] 1 0 = e − 1
55. ∆x =(3− 0)/6 = 1 2 , so the endpoints are 0, 1 2 , 1, 3 2 , 2, 5 2 ,and3, and the midpoints are 1 4 , 3 4 , 5 4 , 7 4 , 9 4 ,and 11 4 .
The Midpoint Rule gives
3
0 sin(x3 ) dx ≈
6
i=1
f(x i) ∆x = 1 2
sin 1
4
3
+sin 3
4
3
+sin 5
4
3
+sin 7
4
3
+sin 9
4
3
+sin
11 3
≈ 0.280981.
4

F.
CHAPTER 5 REVIEW ¤ 259
57. Note that r(t) =b 0 (t),whereb(t) =the number of barrels of oil consumed up to time t. So, by the Net Change Theorem,
8
0
r(t) dt = b(8) − b(0) represents the number of barrels of oil consumed from Jan. 1, 2000, through Jan. 1, 2008.
59. We use the Midpoint Rule with n =6and ∆t = 24 − 0
6
=4. The increase in the bee population was
TX.10
24
0 f −1 (y) dy = 2 . 0 3
r(t) dt ≈ M6 =4[r(2) + r(6) + r(10) + r(14) + r(18) + r(22)]
≈ 4[50 + 1000 + 7000 + 8550 + 1350 + 150] = 4(18,100) = 72,400
61. Let u =2sinθ. Thendu =2cosθdθand when θ =0, u =0;whenθ = π 2 , u =2.Thus,
π/2
f(2 sin θ)cosθdθ= 2
f(u) 1
du
= 1 2 f(u) du = 1 2 f(x) dx = 1 (6) = 3.
0 0 2 2 0 2 0 2
63. Area under the curve y =sinhcx between x =0and x =1is equal to 1 ⇒
1 sinh cx dx =1 ⇒ 1 1
0 c cosh cx =1 ⇒ 1 (cosh c − 1) = 1 ⇒
0 c
cosh c − 1=c ⇒ cosh c = c +1. From the graph, we get c =0and
c ≈ 1.6161,butc =0isn’t a solution for this problem since the curve
y =sinhcx becomes y =0and the area under it is 0. Thus,c ≈ 1.6161.
65. Using FTC1, we differentiate both sides of the given equation, x
0 f(t) dt = xe2x + x
0 e−t f(t) dt, and get
f(x) =e 2x +2xe 2x + e −x f(x) ⇒ f(x) 1 − e −x = e 2x +2xe 2x ⇒ f(x) = e2x (1 + 2x)
1 − e −x .
67. Let u = f(x) and du = f 0 (x) dx. So2 b
a f(x)f 0 (x) dx =2 f(b)
f(a) udu= u 2 f(b)
f(a) =[f(b)]2 − [f(a)] 2 .
69. Let u =1− x. Thendu = −dx,so 1
0 f(1 − x) dx = 0
1 f(u)(−du) = 1
71. The shaded region has area 1
0 f(x) dx = 1 3 . The integral 1
0 f −1 (y) dy
gives the area of the unshaded region, which we know to be 1 − 1 3 = 2 3 .
So 1
f(u) du = 1
f(x) dx.
0 0

F.
TX.10

F.
TX.10
PROBLEMS PLUS
1. Differentiating both sides of the equation x sin πx = x 2
f(t) dt (using FTC1 and the Chain Rule for the right side) gives
0
sin πx + πx cos πx =2xf(x 2 ). Letting x =2so that f(x 2 )=f(4), we obtain sin 2π +2π cos 2π =4f(4), so
f(4) = 1 (0 + 2π · 1) = π .
4 2
3. Differentiating the given equation, x
0 f(t) dt =[f(x)]2 , using FTC1 gives f(x) =2f(x) f 0 (x) ⇒
f(x)[2f 0 (x) − 1] = 0,sof(x) =0or f 0 (x) = 1 2 .Sincef(x) is never 0,wemusthavef 0 (x) = 1 2 and f 0 (x) = 1 2
⇒
f(x) = 1 x + C. Tofind C, we substitute into the given equation to get x
1 t + C 2 0 2
dt = 1
x + C 2
2
⇔
1
4 x2 + Cx = 1 4 x2 + Cx + C 2 . It follows that C 2 =0,soC =0,andf(x) = 1 x. 2
5. f(x) =
f 0 (x) =
g(x)
0
1
cos x
√ dt, whereg(x) = [1 + sin(t 2 )] dt. Using FTC1 and the Chain Rule (twice) we have
1+t
3
1
1+[g(x)]
3 g0 (x) =
f 0
π 1
2 = √ (1 + sin 0)(−1) = 1 · 1 · (−1) = −1.
1+0
0
1
[1 + 1+[g(x)]
3 sin(cos2 x)](− sin x). Nowg
0
π = [1 + sin(t 2 )] dt =0,so
2
0
7. By l’Hospital’s Rule and the Fundamental Theorem, using the notation exp(y) =e y ,
lim
x→0
x
0 (1 − tan 2t)1/t dt
x
H (1 − tan 2x) 1/x
=lim
x→0 1
H
=exp
lim
x→0
−2sec 2 2x
1 − tan 2x
=exp lim
x→0
ln(1 − tan 2x)
x
−2 · 1
2
=exp
= e −2
1 − 0
9. f(x) =2+x − x 2 =(−x +2)(x +1)=0 ⇔ x =2or x = −1. f(x) ≥ 0 for x ∈ [−1, 2] and f(x) < 0 everywhere
else. The integral b
a (2 + x − x2 ) dx has a maximum on the interval where the integrand is positive, which is [−1, 2]. So
a = −1, b =2. (Any larger interval gives a smaller integral since f(x) < 0 outside [−1, 2]. Any smaller interval also gives a
smaller integral since f(x) ≥ 0 in [−1, 2].)
11. (a) We can split the integral n
[x] dx into the sum n
0
i=1
i
[x] dx . But on each of the intervals [i − 1,i) of integration,
i−1
[x] is a constant function, namely i − 1. Sotheith integral in the sum is equal to (i − 1)[i − (i − 1)] = (i − 1).Sothe
original integral is equal to n
i=1
(i − 1) = n−1
i=1
i =
(n − 1)n
.
2
261

F.
262 ¤ PROBLEMS PLUS
TX.10
(b) We can write b
[x] dx = b
[x] dx − a
[x] dx.
a 0 0
Now b
[x] dx = [b ]
[x] dx + b
[x] dx. Thefirstoftheseintegralsisequalto 1 ([b] − 1) [b],
0 0 [b ] 2
by part (a), and since [x]] = [b] on [[b] ,b], the second integral is just [b] (b − [b]).So
b [x] dx = 1 ([b]] − 1) [b] + [[b] (b − [b]) = 1 [b] (2b − [b] − 1) and similarly a
[[x] dx = 1
0 2 2 0 2
[a] (2a − [a] − 1).
Therefore, b
[x] dx = 1 [b] (2b − [b] − 1) − 1
a 2 2
[a] (2a − [a] − 1).
13. Let Q(x) =
x
0
P (t) dt = at + b 2 t2 + c 3 t3 + d x
4 t4 = ax + b
0
2 x2 + c 3 x3 + d 4 x4 .ThenQ(0) = 0,andQ(1) = 0 by the
given condition, a + b 2 + c 3 + d 4 =0.Also,Q0 (x) =P (x) =a + bx + cx 2 + dx 3 by FTC1. By Rolle’s Theorem, applied to
Q on [0, 1], there is a number r in (0, 1) such that Q 0 (r) =0, that is, such that P (r) =0. Thus, the equation P (x) =0has a
root between 0 and 1.
More generally, if P (x) =a 0 + a 1 x + a 2 x 2 + ···+ a n x n and if a 0 + a1
2 + a2 an
+ ···+ =0, then the equation
3 n +1
P(x) =0has a root between 0 and 1. Theproofisthesameasbefore:
Let Q(x) =
x
0
P (t) dt = a 0x + a 1
2 x2 + a 2
3 x3 + ···+ a n
n +1 xn .ThenQ(0) = Q(1) = 0 and Q 0 (x) =P (x). By
Rolle’s Theorem applied to Q on [0, 1], there is a number r in (0, 1) such that Q 0 (r) =0, that is, such that P (r) =0.
15. Note that
d
dx
x
u
0
0
f(t) dt du =
d
dx
x
x
f(u)(x − u) du
0
0
f(t) dt by FTC1, while
= d
x
dx
x
0
f(u) du − d
dx
x
0
f(u)udu
= x
f(u) du + xf(x) − f(x)x = x
f(u) du
0 0
Hence, x
f(u)(x − u) du = x
u f(t) dt du + C. Setting x =0gives C =0.
0 0 0
17. lim
n→∞
1
√ n
√ n +1
+
1
√ n
√ n +2
+ ···+
= lim
n→∞
= lim
n→∞
1
= lim
n→∞ n
=
1
0
1
√ √ n n + n
1 n n
n
n n +1 + n +2 + ···+ n + n
1 1
1
1
+ + ···+ √
n 1+1/n 1+2/n 1+1
n
i=1
i 1
f
where f(x) = √
n
1+x
1
√ 1+x
dx = 2 √ 1+x 1
0 =2 √
2 − 1

F.
TX.10
6 APPLICATIONS OF INTEGRATION
6.1 Areas Between Curves
1. A =
3. A =
x=4
x=0
y=1
y=−1
(y T − y B) dx =
4
1
(x R − x L ) dy =
0
−1
(5x − x 2 ) − x 4
dx = (4x − x 2 ) dx = 2x 2 − 1 x3 4
=
3
32 − 64 0 3
e y − (y 2 − 2) 1
dy = e y − y 2 +2 dy
= e y − 1 3 y3 +2y 1
−1 = e 1 − 1 3 +2 − e −1 + 1 3 − 2 = e − 1 e + 10 3
0
−1
− (0) =
32
3
2
5. A = (9 − x 2 ) − (x +1) dx
−1
=
2
−1
(8 − x − x 2 ) dx
2
=
8x − x2
2 − x3
3
−1
=
16 − 2 − 8 3 − −8 −
1
+
1
2 3
=22− 3+ 1 2 = 39 2
7. The curves intersect when x = x 2 ⇔ x 2 − x =0 ⇔
x(x − 1) = 0 ⇔ x =0or 1.
A =
1
0
(x − x 2 ) dx = 1
2 x2 − 1 3 x3 1
0 = 1 2 − 1 3 = 1 6
9. A =
2
1
1
x − 1
dx = ln x + 1 2
x 2 x
1
= ln 2 + 1 2
− (ln 1 + 1)
=ln2− 1 2 ≈ 0.19
11. A =
=
1
0
√ x − x
2 dx
2
3 x3/2 − 1 3 x3 1
0
= 2 3 − 1 3 = 1 3
263

F.
264 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
13. 12 − x 2 = x 2 − 6 ⇔ 2x 2 =18 ⇔
x 2 =9 ⇔ x = ±3,so
A =
3
−3
(12 − x 2 ) − (x 2 − 6) dx
=2
3
0
18 − 2x
2 dx
[by symmetry]
=2 18x − 2 x3 3
3
=2[(54− 18) − 0]
0
=2(36)=72
15. The curves intersect when tan x =2sinx (on [−π/3,π/3]) ⇔ sin x =2sinx cos x ⇔
2sinx cos x − sin x =0 ⇔ sin x (2 cos x − 1) = 0 ⇔ sin x =0or cos x = 1 2
⇔ x =0or x = ± π 3 .
A =
=2
=2
π/3
−π/3
π/3
0
(2 sin x − tan x) dx
(2 sin x − tan x) dx [by symmetry]
−2cosx − ln |sec x|
π/3
=2[(−1 − ln 2) − (−2 − 0)]
=2(1− ln 2) = 2 − 2ln2
0
17.
1
x = √ x ⇒ 1 2 4 x2 = x ⇒ x 2 − 4x =0 ⇒ x(x − 4) = 0 ⇒ x =0or 4,so
A =
4
0
√ x −
1
2 x dx +
9
4
1
2 x − √ x dx =
2
3 x3/2 − 1 4 x2 4
= 16
− 4 3
− 0 + 81
− 18 4
−
4 − 16
3 =
81
+ 32
59
4 3
− 26 =
12
0
+
1
4 x2 − 2 3 x3/2 9
4

F.
TX.10
SECTION 6.1 AREAS BETWEEN CURVES ¤ 265
19. 2y 2 =4+y 2 ⇔ y 2 =4 ⇔ y = ±2,so
2
A = (4 + y 2 ) − 2y 2 dy
−2
=2
2
0
(4 − y 2 ) dy [by symmetry]
=2 4y − 1 y3 2
=2
3
8 − 8 0 3 =
32
3
21. The curves intersect when 1 − y 2 = y 2 − 1 ⇔ 2=2y 2 ⇔ y 2 =1 ⇔ y = ±1.
1
A = (1 − y 2 ) − (y 2 − 1) dy
−1
1
= 2(1 − y 2 ) dy
−1
=2· 2
1
0
(1 − y 2 ) dy
=4 y − 1 y3 1
=4
1 − 1 3 0 3 =
8
3
23. Notice that cos x =sin2x =2sinx cos x ⇔
2sinx cos x − cos x =0 ⇔ cos x (2 sin x − 1) = 0 ⇔
2sinx =1or cos x =0 ⇔ x = π 6 or π 2 .
A =
π/6
0
(cos x − sin 2x) dx +
= sin x + 1 2 cos 2x π/6
0
π/2
π/6
(sin 2x − cos x) dx
+ − 1 2 cos 2x − sin x π/2
π/6
= 1 2 + 1 2 · 1
2 − 0+ 1 2 · 1 + 1
2 − 1 − − 1 2 · 1
2 − 1 2
=
1
2
25. The curves intersect when x 2 = 2
x 2 +1
x 4 + x 2 =2 ⇔ x 4 + x 2 − 2=0 ⇔
(x 2 +2)(x 2 − 1) = 0 ⇔ x 2 =1 ⇔ x = ±1.
1
2
A =
−1 x 2 +1 − x2 dx =2
1
=2 2tan −1 x − 1 3 x3 0
1
0
⇔
2
x 2 +1 − x2 dx
=2 2 · π − 1
4 3 = π −
2
≈ 2.47
3

F.
266 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
27. 1/x = x ⇔ 1=x 2 ⇔ x = ±1 and 1/x = 1 4 x ⇔
4=x 2 ⇔ x = ±2,soforx>0,
1
A =
x − 1 2
1
4 x dx +
x − 1
4 x dx
=
0
1
0
3
4 x
dx +
2
1
1
1
x − 1 4 x
dx
= 3
x2 1
+ ln |x| − 1 x2 2
8 0 8 1
= 3 +
ln 2 − 1 8 2 − 0 −
1
8 =ln2
29. An equation of the line through (0, 0) and (2, 1) is y = 1 x; through (0, 0)
2
and (−1, 6) is y = −6x; through (2, 1) and (−1, 6) is y = − 5 3 x + 13 3 .
0
A = −
5
x + 13 3 3
−1
0
=
−1
13
3 x + 13
3
− (−6x)
dx +
2
dx +
2
0
0
−
13
x + 13
6 3 dx
−
5
x + 13
3 3 −
1
x 2
dx
= 13 3
0
−1
(x +1)dx + 13 3
2
0
−
1
2 x +1 dx
= 13 3
= 13 3
1
2 x2 + x 0
−1 + 13 3
−
1
4 x2 + x 2
0
0 −
1
2 − 1 + 13
3 [(−1+2)− 0] = 13 3 · 1
2 + 13 3 · 1= 13 2
31. The curves intersect when sin x =cos2x (on [0,π/2]) ⇔ sin x =1− 2sin 2 x ⇔ 2sin 2 x +sinx − 1=0 ⇔
(2 sin x − 1)(sin x +1)=0 ⇒ sin x = 1 2
⇒ x = π 6 .
A =
=
π/2
0
π/6
0
|sin x − cos 2x| dx
(cos 2x − sin x) dx +
π/2
π/6
(sin x − cos 2x) dx
= 1
sin 2x +cosx π/6
2
+ − cos x − 1 sin 2x π/2
0 2 π/6
= √
1
4 3+
1
2
= 3 2
√
3 − 1
πx
33. Let f(x) =cos 2 4
The shaded area is given by
√
3
− (0 + 1) + (0 − 0) −
−
1
2
√
3 −
1
4
πx
− sin 2 and ∆x = 1 − 0
4
4 .
A = 1
f(x) dx ≈ M 0 4
= 1 4 f 1
8 + f 3
8 + f 5
8 + f 7
8
≈ 0.6407
√
3

F.
TX.10
SECTION 6.1 AREAS BETWEEN CURVES ¤ 267
35. From the graph, we see that the curves intersect at x =0and x = a ≈ 0.896,with
x sin(x 2 ) >x 4 on (0,a). So the area A of the region bounded by the curves is
A =
a
x sin(x 2 ) − x 4 dx = − 1 2 cos(x2 ) − 1 5 x5 a
0
0
= − 1 2 cos(a2 ) − 1 5 a5 + 1 2 ≈ 0.037
37. From the graph, we see that the curves intersect at
x = a ≈ −1.11,x= b ≈ 1.25, and x = c ≈ 2.86, with
x 3 − 3x +4> 3x 2 − 2x on (a, b) and 3x 2 − 2x >x 3 − 3x +4
on (b, c). So the area of the region bounded by the curves is
A =
=
b
a
b
a
(x 3 − 3x +4)− (3x 2 − 2x) c
dx + (3x 2 − 2x) − (x 3 − 3x +4) dx
(x 3 − 3x 2 − x +4)dx +
c
b
b
(−x 3 +3x 2 + x − 4) dx
= 1
4 x4 − x 3 − 1 2 x2 +4x b
a + − 1 4 x4 + x 3 + 1 2 x2 − 4x c
b ≈ 8.38
39. As the figure illustrates, the curves y = x and y = x 5 − 6x 3 +4x
enclose a four-part region symmetric about the origin (since
x 5 − 6x 3 +4x and x are odd functions of x). The curves intersect
at values of x where x 5 − 6x 3 +4x = x;thatis,where
x(x 4 − 6x 2 +3)=0. That happens at x =0and where
x 2 = 6 ± √ 36 − 12
=3± √ 6;thatis,atx = − 3+ √ 6, − 3 − √ 6, 0, 3 − √ 6,and 3+ √ 6.
2
The exact area is
√ 3+ √ 6
2 (x 5 − 6x 3 +4x) − x √ 3+ √ 6
dx =2 x 5 − 6x 3 +3x dx
0
0
√ 3− √ 6
√
=2 (x 5 − 6x 3 3+ √ 6
+3x) dx +2 (−x 5 +6x 3 − 3x) dx
0
CAS
= 12 √ 6 − 9
√
3− √ 6

F.
268 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
41. 1 second = 1
1
3600
hour, so 10 s =
360
h. With the given data, we can take n =5to use the Midpoint Rule.
∆t = 1/360−0
5
= 1
1800 ,so
distance Kelly − distance Chris = 1/360
0
v K dt − 1/360
0
v C dt = 1/360
0
(v K − v C) dt
43. Let h(x) denote the height of the wing at x cm from the left end.
A ≈ M 5 = 200 − 0
5
≈ M 5 = 1
1800 [(v K − v C )(1) + (v K − v C )(3) + (v K − v C )(5)
+(v K − v C )(7) + (v K − v C )(9)]
= 1 [(22 − 20) + (52 − 46) + (71 − 62) + (86 − 75) + (98 − 86)]
1800
= 1
1800 (2+6+9+11+12)= 1
1800 (40) = 1 45 mile, or 117 1 3 feet
[h(20) + h(60) + h(100) + h(140) + h(180)]
=40(20.3+29.0+27.3+20.5+8.7) = 40(105.8) = 4232 cm 2
45. We know that the area under curve A between t =0and t = x is x
vA(t) dt = sA(x),wherevA(t) is the velocity of car A
0
and s A is its displacement. Similarly, the area under curve B between t =0and t = x is x
0 v B(t) dt = s B (x).
(a)Afteroneminute,theareaundercurveA is greater than the area under curve B. So car A is ahead after one minute.
(b) The area of the shaded region has numerical value s A(1) − s B(1), which is the distance by which A is ahead of B after
1 minute.
(c) After two minutes, car B is traveling faster than car A and has gained some ground, but the area under curve A from t =0
to t =2is still greater than the corresponding area for curve B, so car A is still ahead.
(d) From the graph, it appears that the area between curves A and B for 0 ≤ t ≤ 1 (when car A is going faster), which
corresponds to the distance by which car A is ahead, seems to be about 3 squares. Therefore, the cars will be side by side
at the time x where the area between the curves for 1 ≤ t ≤ x (when car B is going faster) is the same as the area for
0 ≤ t ≤ 1. From the graph, it appears that this time is x ≈ 2.2. So the cars are side by side when t ≈ 2.2 minutes.
47. To graph this function, we must first express it as a combination of explicit
functions of y;namely,y = ±x √ x +3. We can see from the graph that the loop
extends from x = −3 to x =0, and that by symmetry, the area we seek is just
twice the area under the top half of the curve on this interval, the equation of the
top half being y = −x √ x +3. So the area is A =2 0
√
−3 −x x +3 dx. We
substitute u = x +3,sodu = dx and the limits change to 0 and 3,andweget
A = −2 3
[(u − 3)√ u ] du = −2 3
0 0 (u3/2 − 3u 1/2 ) du
= −2
2
5 u5/2 − 2u 3/2 3
0
= −2 √ 2
5 3
2
3 − 2 3 √ 3 √
= 24
5 3

F.
TX.10
SECTION 6.1 AREAS BETWEEN CURVES ¤ 269
49. By the symmetry of the problem, we consider only the first quadrant, where
y = x 2 ⇒ x = y. We are looking for a number b such that
b
0
ydy=
4
b
b 4
ydy ⇒
2
y 3/2 = 2 y 3/2 3
3
0 b
b 3/2 =4 3/2 − b 3/2 ⇒ 2b 3/2 =8 ⇒ b 3/2 =4 ⇒ b =4 2/3 ≈ 2.52.
⇒
51. We first assume that c>0, sincec can be replaced by −c in both equations without changing the graphs, and if c =0the
curves do not enclose a region. We see from the graph that the enclosed area A lies between x = −c and x = c, and by
symmetry, it is equal to four times the area in the first quadrant. The enclosed area is
A =4 c
0 (c2 − x 2 ) dx =4 c 2 x − 1 3 x3 c
0 =4 c 3 − 1 3 c3 =4 2
3 c3 = 8 3 c3
So A = 576 ⇔ 8 3 c3 =576 ⇔ c 3 =216 ⇔ c = 3√ 216 = 6.
Note that c = −6 is another solution, since the graphs are the same.
53. The curve and the line will determine a region when they intersect at two or
more points. So we solve the equation x/(x 2 +1)=mx
⇒
x = x(mx 2 + m) ⇒ x(mx 2 + m) − x =0 ⇒
x(mx 2 + m − 1) = 0 ⇒ x =0 or mx 2 + m − 1=0 ⇒
x =0or x 2 = 1 − m
m
1
⇒ x =0or x = ±
− 1. Notethatifm =1, this has only the solution x =0, and no region
m
is determined. But if 1/m − 1 > 0 ⇔ 1/m > 1 ⇔ 0

F.
270 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
6.2 Volumes
1. A cross-section is a disk with radius 2 − 1 2 x, so its area is A(x) =π 2 − 1 2 x2 .
V =
= π
2
1
2
1
A(x) dx =
2
4 − 2x +
1
4 x2 dx
= π 4x − x 2 + 1
12 x3 2
1
= π 8 − 4+ 8
12
= π 1+ 7 12
1
=
19
12 π
π 2 − 1 2 x2 dx
−
4 − 1+
1
12
3. A cross-section is a disk with radius 1/x, so its area is
A(x) =π(1/x) 2 .
V =
2
1
A(x) dx =
2
1
2 1
π dx
x
2
1
= π
−
1 x dx = π 1 2 2 x
1
= π − 1 2 − (−1) = π 2
5. A cross-section is a disk with radius 2 y, so its area is
A(y) =π 2 2.
y
V =
9
0
A(y) dy =
9
=4π 1
y2 9
=2π(81) = 162π
2 0
0
π 2 2
9
y dy =4π
7. A cross-section is a washer (annulus) with inner
radius x 3 and outer radius x, so its area is
A(x) =π(x) 2 − π(x 3 ) 2 = π(x 2 − x 6 ).
V =
1
0
A(x) dx =
1
0
π(x 2 − x 6 ) dx
= π 1
3 x3 − 1 7 x7 1
0 = π 1
3 − 1 7
=
4
TX.10
ydy
0
π 21

F.
9. A cross-section is a washer with inner radius y 2
and outer radius 2y, so its area is
A(y) =π(2y) 2 − π(y 2 ) 2 = π(4y 2 − y 4 ).
V =
2
0
TX.10
2
(4y 2 − y 4 ) dy
0
=
64
π 15
1 − √ 2
x
1 − 2 √
x + x
.
1
−3x + x 2 +2 √
x dx
0
=
π
6
π/3
=2π 4π
− √ 3 − 0
3
0
1
π(1 − y 2 ) 2 dy
0
π 15
A(y) dy = π
= π 4
3 y3 − 1 5 y5 2
0 = π 32
3 − 32 5
11. A cross-section is a washer with inner radius 1 − √ x and outer radius 1 − x, so its area is
A(x) =π(1 − x) 2 − π
= π (1 − 2x + x 2 ) −
= π −3x + x 2 +2 √ x
V =
1
0
A(x) dx = π
1
= π − 3 2 x2 + 1 3 x3 + 4 3 x3/2 = π − 3 + 5 2 3
0
SECTION 6.2 VOLUMES ¤ 271
13. A cross-section is a washer with inner radius (1 + sec x) − 1=secx and outer radius 3 − 1=2, so its area is
A(x) =π 2 2 − (sec x) 2 = π(4 − sec 2 x).
π/3
π/3
V = A(x) dx = π(4 − sec 2 x) dx
−π/3
−π/3
=2π
π/3
0
=2π 4x − tan x
=2π 4π
3 − √ 3
1
15. V = π(1 − y 2 ) 2 dy =2
−1
=2π
1
0
(4 − sec 2 x) dx [by symmetry]
(1 − 2y 2 + y 4 ) dy
=2π y − 2 3 y3 + 1 5 y5 1
0
=2π · 8
15 = 16

F.
272 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
17. y = x 2 ⇒ x = √ y for x ≥ 0. The outer radius is the distance from x = −1 to x = y and the inner radius is the
distance from x = −1 to x = y 2 .
V =
= π
1
= π
0
1
0
2
π y − (−1) − y 2 − (−1) 2
dy = π
y +2
y +1− y 4 − 2y 2 − 1 dy = π
1
2 y2 + 4 3 y3/2 − 1 5 y5 − 2 3 y3 1
0
1
0
1
= π 1
+ 4 − 1 − 2
2 3 5 3 =
29
π 30
0
y +1
2
− (y 2 +1) 2
dy
y +2 y − y 4 − 2y 2 dy
19. R 1 about OA (the line y =0): V =
21. R 1 about AB (the line x =1):
V =
1
0
A(y) dy =
= π 1 − 3 2 + 3 5
1
0
=
π
10
1
0
A(x) dx =
0
1
0
π(x 3 ) 2 dx = π
1
0
1 1
x 6 dx = π
7 x7 = π
0
7
π 1 − 3 2
1
y dy = π (1 − 2y 1/3 + y 2/3 ) dy = π
y − 3 2 y4/3 + 3 y5/3 1
5
23. R 2 about OA (the line y =0):
1
1
√
2
V = A(x) dx = π(1) 2 − π x dx = π
0
25. R 2 about AB (the line x =1):
1
V =
0
= π
A(y) dy =
2
3 y3 − 1 5 y5 1
0
0
1
0
1
0
(1 − x) dx = π x − 1 x2 1
= π
2
1 − 1 0 2 =
π
2
π(1) 2 − π(1 − y 2 ) 2 1
dy = π 1 − (1 − 2y 2 + y 4 ) 1
dy = π (2y 2 − y 4 ) dy
= π 2
3 − 1 5
=
7
15 π
27. R 3 about OA (the line y =0):
1
1
√
2
V = A(x) dx = π x − π(x 3 ) 2 dx = π
0
0
0
1
0
(x − x 6 ) dx = π 1
2 x2 − 1 x7 1
= π 1
− 1
7 0 2 7 =
5
π. 14
Note: Let R = R 1 + R 2 + R 3 .IfwerotateR about any of the segments OA, OC, AB,orBC, we obtain a right circular
cylinder of height 1 and radius 1. Its volume is πr 2 h = π(1) 2 · 1=π. As a check for Exercises 19, 23, and 27, we can add the
answers, and that sum must equal π. Thus, π + π + 5π =
2+7+5
7 2 14 14 π = π.
0
0

F.
TX.10
SECTION 6.2 VOLUMES ¤ 273
29. R 3 about AB (the line x =1):
1
1
V = A(y) dy = π(1 − y 2 ) 2 − π 1 − 3 2
y dy = π
0
1
0
1
0
(1 − 2y 2 + y 4 ) − (1 − 2y 1/3 + y 2/3 ) dy
= π (−2y 2 + y 4 +2y 1/3 − y 2/3 ) dy = π
− 2 3 y3 + 1 5 y5 + 3 2 y4/3 − 3 y5/3 1
5
= π − 2 + 1 + 3 − 3 3 5 2 5
0
0
Note: See the note in Exercise 27. For Exercises 21, 25, and 29, we have π + 7π + 13π =
3+14+13
10 15 30 30 π = π.
31. V = π
π/4
0
(1 − tan 3 x) 2 dx
=
13
30 π
33. V = π
= π
π
0
π
0
(1 − 0) 2 − (1 − sin x) 2 dx
1 2 − (1 − sin x) 2 dx
√ 8
2
35. V = π [3 − (−2)] 2 − y2 +1− (−2) dy
− √ 8
√
2 2
2
= π 5 2 − 1+y2 +2 dy
−2 √ 2
37. y =2+x 2 cos x and y = x 4 + x +1intersect at
x = a ≈ −1.288 and x = b ≈ 0.884.
V = π
b
a
[(2 + x 2 cos x) 2 − (x 4 + x +1) 2 ]dx ≈ 23.780

F.
274 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
39. V = π
π
0
CAS
= 11 8 π2
sin 2 x − (−1) 2
− [0 − (−1)]
2 dx
41. π π/2
cos 2 xdxdescribes the volume of the solid obtained by rotating the region R = (x, y) | 0 ≤ x ≤ π 0 2
, 0 ≤ y ≤ cos x
43. π
of the xy-plane about the x-axis.
1
0
(y 4 − y 8 ) dy = π
1
0
(y 2 ) 2 − (y 4 ) 2 dy describes the volume of the solid obtained by rotating the region
R = (x, y) | 0 ≤ y ≤ 1,y 4 ≤ x ≤ y 2 of the xy-plane about the y-axis.
45. There are 10 subintervals over the 15-cm length, so we’ll use n =10/2 =5for the Midpoint Rule.
V = 15
0
A(x) dx ≈ M 5 = 15−0
5
[A(1.5) + A(4.5) + A(7.5) + A(10.5) + A(13.5)]
= 3(18 + 79 + 106 + 128 + 39) = 3 · 370 = 1110 cm 3
47. (a) V = 10
π [f(x)] 2 dx ≈ π 10 − 2
2 4 [f(3)] 2 +[f(5)] 2 +[f(7)] 2 +[f(9)] 2
≈ 2π (1.5) 2 +(2.2) 2 +(3.8) 2 +(3.1) 2 ≈ 196 units 3
(b) V = 4
π (outer radius) 2 − (inner radius) 2 dy
0
≈ π 4 − 0
4 (9.9) 2 − (2.2) 2 + (9.7) 2 − (3.0) 2 + (9.3) 2 − (5.6) 2 + (8.7) 2 − (6.5) 2
≈ 838 units 3
49. We’ll form a right circular cone with height h and base radius r by
revolving the line y = r x about the x-axis.
h
V = π
h
0
r
h x 2dx
= π
h
= π r2
h 2 1
3 h3
= 1 3 πr2 h
0
r 2
h 2 x2 dx = π r2
h 2 1
3 x3 h
0
Another solution: Revolve x = − r y + r about the y-axis.
h
h
V = π − r 2dy
h
h y + r = ∗ r
2
π
h 2 y2 − 2r2
h y + r2 dy
0
0
r
2
h
= π
3h 2 y3 − r2
h y2 + r y 2 = π 1
3 r2 h − r 2 h + r 2 h = 1 3 πr2 h
0
∗ Or use substitution with u = r − r h y and du = − r dy to get
h
0
π u 2 − h
r r du = −π h 0 1
r 3 u3 = −π h − 1
r
r 3 r3 = 1 3 πr2 h.

F.
TX.10
SECTION 6.2 VOLUMES ¤ 275
51. x 2 + y 2 = r 2 ⇔ x 2 = r 2 − y 2
V = π
= π
r
r−h
r 2 − y 2 dy = π
r 3 − r3
3
r 2 y − y3
3
−
r 2 (r − h) −
= π 2
3 r3 − 1 3 (r − h) 3r 2 − (r − h) 2
r
r−h
(r − h)3
3
= 1 3 π 2r 3 − (r − h) 3r 2 − r 2 − 2rh + h 2
= 1 3 π 2r 3 − (r − h) 2r 2 +2rh − h 2
= 1 π 2r 3 − 2r 3 − 2r 2 h + rh 2 +2r 2 h +2rh 2 − h 3
3
= 1 π 3rh 2 − h 3
= 1 3 3 πh2 (3r − h), or, equivalently, πh 2 r − h
3
53. For a cross-section at height y, we see from similar triangles that α/2
b/2 = h − y
1
h ,soα = b − y h
Similarly, for cross-sections having 2b as their base and β replacing α, β =2b 1 − y
.So
h
V =
=
h
0
h
0
A(y) dy =
2b 2 1 − y h
h
b 1 − y
2b 1 − y
dy
0 h
h
2dy
h
=2b
2
1 − 2y
h + y2
dy
h 2
h
=2b 2 y − y2
h + y3
=2b 2 h − h + 1
3h h
2 3
0
0
= 2 3 b2 h [ = 1 Bh where B is the area of the base, as with any pyramid.]
3
55. A cross-section at height z is a triangle similar to the base, so we’ll multiply the legs of the base triangle, 3 and 4, by a
proportionality factor of (5 − z)/5.Thus,thetriangleatheightz has area
A(z) = 1
5 − z 5 − z
2 · 3 · 4 =6 1 − z 2,so
5 5
5
V =
5
0
A(z) dz =6
5
0
1 − z 2
0
dz =6
5
= −30 1
u3 0
= −30
− 1 3 1 3 =10cm
3
1
u 2 (−5 du)
u =1− z/5,
du = − 1 dz 5
.
57. If l is a leg of the isosceles right triangle and 2y is the hypotenuse,
then l 2 + l 2 =(2y) 2 ⇒ 2l 2 =4y 2 ⇒ l 2 =2y 2 .
V = 2
A(x) dx =2 2
A(x) dx =2 2 1
(l)(l) dx =2 2
−2 0 0 2
2 (4 − 0 x2 ) dx
=2 2
0
= 9 2
1
4 (36 − 9x2 ) dx = 9 2
4x −
1
3 x3 2
0 = 9 2
8 −
8
3
=24
0 y2 dx

F.
276 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
59. The cross-section of the base corresponding to the coordinate x has length
y =1− x. The corresponding square with side s has area
A(x) =s 2 =(1− x) 2 =1− 2x + x 2 . Therefore,
V =
Or:
1
0
A(x) dx =
1
0
(1 − 2x + x 2 ) dx
= x − x 2 + 1 x3 1
=
1 − 1+ 1 3 0 3 − 0=
1
3
1
0
(1 − x) 2 dx =
0
1
u 2 (−du) [u =1− x] = 1
3 u3 1
0 = 1 3
61. The cross-section of the base b corresponding to the coordinate x has length 1 − x 2 . The height h also has length 1 − x 2 ,
so the corresponding isosceles triangle has area A(x) = 1 2 bh = 1 2 (1 − x2 ) 2 . Therefore,
V =
1
1
(1 − 2 x2 ) 2 dx
−1
1
=2· 1
2
0
(1 − 2x 2 + x 4 ) dx [by symmetry]
= x − 2 3 x3 + 1 x5 1
= 1 − 2 + 1
5 0 3 5 − 0=
8
15
63. (a) The torus is obtainedbyrotatingthecircle(x − R) 2 + y 2 = r 2 about
the y-axis. Solving for x, we see that the right half of the circle is given by
x = R + r 2 − y 2 = f(y) and the left half by x = R − r 2 − y 2 = g(y). So
V = π r
−r [f(y)] 2 − [g(y)] 2 dy
=2π r
0
R 2 +2R r 2 − y 2 + r 2 − y 2
−
=2π r
0 4R r 2 − y 2 dy =8πR r
0
r2 − y 2 dy
R 2 − 2R r 2 − y 2 + r 2 − y 2
dy
(b) Observe that the integral represents a quarter of the area of a circle with radius r, so
8πR r
r2 − y
0
2 dy =8πR · 1
4 πr2 =2π 2 r 2 R.
65. (a) Volume(S 1 )= h
0 A(z) dz = Volume(S 2) since the cross-sectional area A(z) at height z is the same for both solids.
(b) By Cavalieri’s Principle, the volume of the cylinder in the figure is the same as that of a right circular cylinder with radius r
and height h,thatis,πr 2 h.

F.
TX.10
SECTION 6.2 VOLUMES ¤ 277
67. The volume is obtained by rotating the area common to two circles of radius r,as
shown. The volume of the right half is
V right = π r/2
y 2 dx = π r/2
0 0
r 2 − 1
2 r + x 2 dx
= π r 2 x − 1 1 r + x 3
r/2
= π 1
3 2 2 r3 − 1 r3 − 0 − 1 r3 = 5 3 24 24 πr3
0
So by symmetry, the total volume is twice this, or 5 12 πr3 .
Another solution: We observe that the volume is the twice the volume of a cap of a sphere, so we can use the formula from
Exercise 51 with h = 1 1
r: V =2·
2 3 πh2 (3r − h) = 2 π 1
r 2
3 2 3r −
1
r = 5
2 12 πr3 .
69. Take the x-axis to be the axis of the cylindrical hole of radius r.
A quarter of the cross-section through y, perpendicular to the
y-axis, is the rectangle shown. Using the Pythagorean Theorem
twice, we see that the dimensions of this rectangle are
x = R 2 − y 2 and z = r 2 − y 2 ,so
1
A(y) =xz = r
4 2 − y 2 R 2 − y 2 ,and
V = r
−r A(y) dy = r
−r 4 r 2 − y 2 R 2 − y 2 dy =8 r
0
r2 − y 2 R 2 − y 2 dy
71. (a) The radius of the barrel is the same at each end by symmetry, since the
function y = R − cx 2 is even. Since the barrel is obtained by rotating
the graph of the function y about the x-axis, this radius is equal to the
value of y at x = 1 2 h,whichisR − c 1
2 h2 = R − d = r.
(b) The barrel is symmetric about the y-axis, so its volume is twice the volume of that part of the barrel for x>0. Also,the
barrel is a volume of rotation, so
V =2
h/2
0
πy 2 dx =2π
h/2
=2π 1
2 R2 h − 1 12 Rch3 + 1
160 c2 h 5
0
R − cx
2 2
dx =2π
R 2 x − 2 3 Rcx3 + 1 5 c2 x 5 h/2
Trying to make this look more like the expression we want, we rewrite it as V = 1 3 πh 2R 2 + R 2 − 1 2 Rch2 + 3
80 c2 h 4 .
But R 2 − 1 2 Rch2 + 3 80 c2 h 4 = R − 1 4 ch22 − 1 40 c2 h 4 =(R − d) 2 − 2 5
1
4 ch22 = r 2 − 2 5 d2 .
Substituting this back into V ,weseethatV = 1 3 πh 2R 2 + r 2 − 2 5 d 2 , as required.
0

F.
278 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
6.3 Volumes by Cylindrical Shells
1. If we were to use the “washer” method, we would first have to locate the
local maximum point (a, b) of y = x(x − 1) 2 using the methods of
Chapter 4. Then we would have to solve the equation y = x(x − 1) 2
for x in terms of y to obtain the functions x = g 1(y) and x = g 2(y)
showninthefirst figure. This step would be difficult because it involves
the cubic formula. Finally we would find the volume using
V = π b
[g1 (y)] 2 − [g
0
2 (y)] 2 dy.
Using shells, we find that a typical approximating shell has radius x, so its circumference is 2πx. Its height is y,thatis,
x(x − 1) 2 . So the total volume is
3. V =
2
1
V =
1
0
2πx · 1 2
x dx =2π 1 dx
=2π x 2
=2π(2 − 1) = 2π
1
2πx x(x − 1) 2 dx =2π
1
1
0
x 4 − 2x 3 + x 2 dx =2π x
5
5 − 2 x4
4 + x3
3
1
0
= π 15
5. V = 1
0 2πxe−x2 dx. Letu = x 2 .
Thus, du =2xdx,so
V = π 1
0 e−u du = π −e −u 1
= π(1 − 1/e).
0
7. The curves intersect when 4(x − 2) 2 = x 2 − 4x +7 ⇔ 4x 2 − 16x +16=x 2 − 4x +7 ⇔
3x 2 − 12x +9=0 ⇔ 3(x 2 − 4x +3)=0 ⇔ 3(x − 1)(x − 3) = 0,sox =1 or 3.
V =2π 3
1
=2π 3
1
x x 2 − 4x +7 − 4(x − 2) 2 dx =2π 3
1 x(x 2 − 4x +7− 4x 2 +16x − 16) dx
x(−3x 2 +12x − 9) dx =2π(−3) 3
1 (x3 − 4x 2 +3x) dx = −6π 1
4 x4 − 4 3 x3 + 3 2 x2 3
1
= −6π 81
4 − 36 + 27 2
−
1
4 − 4 3 + 3 2
= −6π
20 − 36 + 12 +
4
3
= −6π
−
8
3
=16π

F.
TX.10
SECTION 6.3 VOLUMESBYCYLINDRICALSHELLS ¤ 279
9. V = 2
2πy(1 + 1 y2 ) dy =2π 2
(y + 1 y3 ) dy =2π 1
2 y2 + 1 y4 2
4 1
=2π (2 + 4) − 1
+ 1
2 4 =2π 21
4 =
21
π 2
8
11. V =2π y( 3
y − 0) dy
0
8
=2π y 4/3 3
dy =2π y7/3 8
7
0
0
= 6π 7 (87/3 )= 6π 7 (27 )= 768
7 π
13. Theheightoftheshellis2 − 1+(y − 2) 2 =1− (y − 2) 2 =1− y 2 − 4y +4 = −y 2 +4y − 3.
V =2π 3
1 y(−y2 +4y − 3) dy
=2π 3
1 (−y3 +4y 2 − 3y) dy
=2π − 1 4 y4 + 4 3 y3 − 3 y2 3
2 1
=2π − 81 +36− 27
4 2 − −
1
+ 4 −
3
4 3 2
=2π
8
3 =
16
π 3
15. The shell has radius 2 − x, circumference 2π(2 − x), and height x 4 .
V = 1
2π(2 − 0 x)x4 dx
=2π 1
0 (2x4 − x 5 ) dx
=2π 2
5 x5 − 1 x6 1
6 0
=2π 2
− 1
5 6 − 0 =2π 7
30 =
7
π 15

F.
280 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
17. The shell has radius x − 1, circumference 2π(x − 1), and height (4x − x 2 ) − 3=−x 2 +4x − 3.
V = 3
1 2π(x − 1)(−x2 +4x − 3) dx
=2π 3
1 (−x3 +5x 2 − 7x +3)dx
=2π − 1 4 x4 + 5 3 x3 − 7 2 x2 +3x 3
1
=2π − 81 +45− 63 +9 4 2
− − 1 + 5 − 7 +3
4 3 2
=2π
4
3 =
8
π 3
19. The shell has radius 1 − y, circumference 2π(1 − y), and height 1 − 3 y
y = x 3 ⇔ x = 3
y .
V = 1
2π(1 − y)(1 − 0 y1/3 ) dy
=2π 1
(1 − y − 0 y1/3 + y 4/3 ) dy
1
=2π y − 1 2 y2 − 3 4 y4/3 + 3 7 y7/3 0
=2π 1 − 1 − 3 +
3
2 4 7 − 0
=2π
5
28 =
5
π 14
21. V = 2
1 2πx ln xdx 23. V = 1
0 2π[x − (−1)] sin π 2 x − x4 dx
25. V = π
0 2π(4 − y) √ sin ydy

F.
TX.10
SECTION 6.3 VOLUMESBYCYLINDRICALSHELLS ¤ 281
27. V = 1
0 2πx √ 1+x 3 dx. Letf(x) =x √ 1+x 3 .
Then the Midpoint Rule with n =5gives
1
0
f(x) dx ≈
1−0
5
[f(0.1) + f(0.3) + f(0.5) + f(0.7) + f(0.9)]
≈ 0.2(2.9290)
Multiplying by 2π gives V ≈ 3.68.
29.
3
0 2πx5 dx =2π 3
0 x(x4 ) dx. The solid is obtained by rotating the region 0 ≤ y ≤ x 4 , 0 ≤ x ≤ 3 about the y-axis using
cylindrical shells.
1
31. 2π(3 − y)(1 − 0 y2 ) dy. The solid is obtained by rotating the region bounded by (i) x =1− y 2 , x =0,andy =0or
(ii) x = y 2 , x =1,andy =0about the line y =3using cylindrical shells.
33. From the graph, the curves intersect at x =0and x = a ≈ 0.56,
with √ x +1>e x on the interval (0,a). So the volume of the solid
obtained by rotating the region about the y-axis is
V =2π √
a
x 0 x +1 − e x dx ≈ 0.13.
35. V =2π
π/2
0
CAS
= 1 32 π3
π
2 − x sin 2 x − sin 4 x dx
37. Use shells:
V = 4
2 2πx(−x2 +6x − 8) dx =2π 4
2 (−x3 +6x 2 − 8x) dx
=2π − 1 4 x4 +2x 3 − 4x 2 4
2
=2π[(−64 + 128 − 64) − (−4+16− 16)]
=2π(4) = 8π
39. Use shells:
V = 4
2π[x − (−1)][5 − (x +4/x)] dx
1
=2π 4
(x + 1)(5 − x − 4/x) dx
1
=2π 4
1 (5x − x2 − 4+5− x − 4/x) dx
=2π 4
1 (−x2 +4x +1− 4/x) dx =2π − 1 3 x3 +2x 2 + x − 4lnx 4
1
=2π − 64 +32+4− 4ln4 − − 1 +2+1− 0
3 3
=2π(12 − 4ln4)=8π(3 − ln 4)

F.
282 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
41. Use disks: x 2 +(y − 1) 2 =1 ⇔ x = ± 1 − (y − 1) 2
V = π
2
0
2
2
1 − (y − 1)
2 dy = π (2y − y 2 ) dy
= π y 2 − 1 3 y3 2
0 = π 4 − 8 3
=
4
3 π
43. Use shells:
V =2 r
2πx √ r
0 2 − x 2 dx
= −2π r
0 (r2 − x 2 ) 1/2 (−2x) dx
r
= −2π · 2
3 (r2 − x 2 ) 3/2 0
45. V =2π
6.4 Work
= − 4 3 π(0 − r3 )= 4 3 πr3
r
0
=2πh
− x3
3r + x2
2
x
− h
r x + h dx =2πh
r
=2πh r2
0
6 = πr2 h
3
1. W = Fd = mgd =(40)(9.8)(1.5) = 588 J
3. W =
b
a
9
f(x) dx =
0
10
10
(1 + x) dx =10 2
TX.10
0
r
− x2
0 r + x dx
1
1
− 1
100 400 =
25
24 m =10.8 cm 2500
1
−
u du [u =1+x, du = dx] =10 1 10
=10 − 1
2 u
+1 =9ft-lb
10
1
5. The force function is given by F (x) (in newtons) and the work (in joules) is the area under the curve, given by
8 F (x) dx = 4
F (x) dx + 8
F (x) dx = 1 (4)(30) + (4)(30) = 180 J.
0 0 4 2
7. 10 = f(x) =kx = 1 k [4 inches = 1 foot], so k =30lb/ft and f(x) =30x. Now6 inches = 1 3 3 2
foot, so
W = 1/2
30xdx= 15x 2 1/2
= 15 ft-lb.
0 0 4
9. (a) If 0.12
kx dx =2J, then 2= 1
kx2 0.12
0 2
= 1 2
0 2
k(0.0144) = 0.0072k and k = = 2500
0.0072 9
≈ 277.78 N/m.
Thus, the work needed to stretch the spring from 35 cm to 40 cm is
0.10
0.05
2500
xdx= 1250
x2 1/10
= 1250
9 9 1/20 9
(b) f(x) =kx,so30 = 2500
270
9
x and x =
≈ 1.04 J.

F.
TX.10
SECTION 6.4 WORK ¤ 283
11. Thedistancefrom20 cm to 30 cm is 0.1 m, so with f(x) =kx,wegetW 1 = 0.1
kx dx = k 1
x2 0.1
0 2
= 1
0
Now W 2 = 0.2
kx dx = k 1
x2 0.2
= k 4
− 1
0.1 2 0.1 200 200 =
3
k.Thus,W 200 2 =3W 1 .
In Exercises 13 – 20, n is the number of subintervals of length ∆x, andx ∗ i isasamplepointintheith subinterval [x i−1,x i].
13. (a) The portion of the rope from x ft to (x + ∆x) ft below the top of the building weighs 1 2 ∆x lb and must be lifted x∗ i ft,
so its contribution to the total work is 1 2 x∗ i ∆x ft-lb. The total work is
W = lim
n
n→∞ i=1
1
2 x∗ i ∆x = 50 1
xdx= 1
x2 50
= 2500
0 2 4 0 4
=625ft-lb
Notice that the exact height of the building does not matter (as long as it is more than 50 ft).
(b) When half the rope is pulled to the top of the building, the work to lift the top half of the rope is
W 1 = 25
0
that is W 2 = 50
25
1
xdx= 1
x2 25
= 625 ft-lb. The bottom half of the rope is lifted 25 ft and the work needed to accomplish
2 4 0 4
1
2
200 k.
25 50
· 25 dx =
2 x = 625 ft-lb. The total work done in pulling half the rope to the top of the building
25 2
is W = W 1 + W 2 = 625 + 625 = 3 1875
· 625 = ft-lb.
2 4 4 4
15. Theworkneededtoliftthecableis lim n
n→∞
i=1 2x∗ i ∆x = 500
2xdx= x 2 500
= 250,000 ft-lb.Theworkneededtolift
0 0
the coal is 800 lb · 500 ft = 400,000 ft-lb. Thus, the total work required is 250,000 + 400,000 = 650,000 ft-lb.
17. At a height of x meters (0 ≤ x ≤ 12), the mass of the rope is (0.8 kg/m)(12 − x m) =(9.6 − 0.8x) kg and the mass of the
water is 36
12 kg/m (12 − x m) = (36 − 3x) kg. The mass of the bucket is 10 kg, so the total mass is
(9.6 − 0.8x)+(36− 3x)+10=(55.6 − 3.8x) kg, and hence, the total force is 9.8(55.6 − 3.8x) N.Theworkneededtolift
the bucket ∆x m through the ith subinterval of [0, 12] is 9.8(55.6 − 3.8x ∗ i )∆x, so the total work is
W = lim
n
n→∞ i=1
9.8(55.6 − 3.8x ∗ i ) ∆x = 12
0 (9.8)(55.6 − 3.8x) dx =9.8
55.6x − 1.9x 2 12
0
=9.8(393.6) ≈ 3857 J
19. A“slice”ofwater∆x m thick and lying at a depth of x ∗ i m(where0 ≤ x ∗ i ≤ 1 2 )hasvolume(2 × 1 × ∆x) m3 ,amassof
2000 ∆x kg, weighs about (9.8)(2000 ∆x) =19,600 ∆x N, and thus requires about 19,600x ∗ i ∆x J of work for its removal.
So W = lim
n
n→∞ i=1
19,600x ∗ i ∆x = 1/2
0
19,600xdx= 9800x 2 1/2
0
=2450J.
21. A rectangular “slice” of water ∆x m thick and lying x m above the bottom has width x mandvolume8x ∆x m 3 .Itweighs
about (9.8 × 1000)(8x ∆x) N, and must be lifted (5 − x) m by the pump, so the work needed is about
(9.8 × 10 3 )(5 − x)(8x ∆x) J. The total work required is
W ≈ 3
0 (9.8 × 103 )(5 − x)8xdx=(9.8 × 10 3 ) 3
0 (40x − 8x2 ) dx =(9.8 × 10 3 ) 20x 2 − 8 3 x3 3
0
=(9.8 × 10 3 )(180 − 72) = (9.8 × 10 3 )(108) = 1058.4 × 10 3 ≈ 1.06 × 10 6 J

F.
284 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
23. Let x measure depth (in feet) below the spout at the top of the tank. A horizontal
disk-shaped “slice” of water ∆x ft thick and lying at coordinate x has radius
3
(16 − x) ft () 8 andvolumeπr2 ∆x = π · 9 (16 − 64 x)2 ∆x ft 3 .Itweighs
about (62.5) 9π
64 (16 − x)2 ∆x lb and must be lifted x ft by the pump, so the
work needed to pump it out is about (62.5)x 9π
64 (16 − x)2 ∆x ft-lb. The total
work required is
W ≈ 8
9π
(62.5)x (16 − 0 64 x)2 dx =(62.5) 9π
64
=(62.5) 9π
64
=(62.5) 9π
64
8
0 x(256 − 32x + x2 ) dx
8 (256x −
0 32x2 + x 3 ) dx =(62.5) 9π
64 128x 2 − 32
3 x3 + 1 x4 8
4 0
11, 264
3
=33,000π ≈ 1.04 × 10 5 ft-lb
d
() From similar triangles,
8 − x = 3 . 8
So r =3+d =3+ 3 8 (8 − x)
= 3(8)
8
= 3 8 (16 − x)
+ 3 (8 − x)
8
25. If only 4.7 × 10 5 J of work is done, then only the water above a certain level (call
it h) will be pumped out. So we use the same formula as in Exercise 21, except that
the work is fixed, and we are trying to find the lower limit of integration:
4.7 × 10 5 ≈ 3
h (9.8 × 103 )(5 − x)8xdx= 9.8 × 10 3 20x 2 − 8 3 x3 3
h
⇔
4.7
× 9.8 102 ≈ 48 = 20 · 3 2 − 8 · 33 − 20h 2 − 8 h3 ⇔
3 3
2h 3 − 15h 2 +45=0.Tofind the solution of this equation, we plot 2h 3 − 15h 2 +45between h =0and h =3.
We see that the equation is satisfied for h ≈ 2.0.Sothedepthofwaterremaininginthetankisabout2.0 m.
27. V = πr 2 x,soV is a function of x and P canalsoberegardedasafunctionofx. IfV 1 = πr 2 x 1 and V 2 = πr 2 x 2,then
W =
=
29. W =
x2
x 1
F (x) dx =
V2
x2
x 1
πr 2 P (V (x)) dx =
V 1
P (V ) dV by the Substitution Rule.
b
a
F (r) dr =
b
a
G m 1m 2
r 2
x2
x 1
P (V (x)) dV (x) [Let V (x) =πr 2 x,sodV (x) =πr 2 dx.]
b −1
1
dr = Gm 1 m 2 = Gm 1 m 2
r
a
a − 1
b
6.5 Average Value of a Function
1. f ave = 1 b
1 4
f(x) dx = (4x −
b − a a 4 − 0 0 x2 ) dx = 1 4 2x 2 − 1 x3 4
= 1
3 0 4
32 −
64
3
− 0
=
1
4
32
3 =
8
3
3. g ave = 1 b
1 8
g(x) dx =
b − a a 8 − 1 1
3√ 8
xdx=
1 3
7 4 x4/3 = 3 45
(16 − 1) =
28 28
1
5. f ave = 1 5
5 − 0 0 te−t2 dt = 1 −25
e u − 1 du u = −t 2 , du = −2tdt, tdt= − 1 du
5 0 2 2
= − 1 10 e
u −25
= − 1 10 (e−25 − 1) = 1 (1 − 10 e−25 )
0

F.
TX.10
SECTION 6.5 AVERAGE VALUE OF A FUNCTION ¤ 285
7. h ave = 1 π
π − 0 0
= 1 π
cos4 x sin xdx= 1 π
−1
1
u 4 (−du) [u =cosx, du = − sin xdx]
1 u5 1
5 0 5π
1
−1 u4 du = 1 π · 2 1
0 u4 du [by Theorem 5.5.7] = 2 π
9. (a) f ave = 1
5 − 2
5
2
(x − 3) 2 dx = 1 3
1
3 (x − 3)3 5
2
= 1 9
(c)
2 3 − (−1) 3 = 1 (8 + 1) = 1
9
(b) f(c) =f ave ⇔ (c − 3) 2 =1 ⇔
c − 3=±1 ⇔ c =2 or 4
11. (a) f ave = 1
π − 0
= 1 π
= 1 π
π
0
(2 sin x − sin 2x) dx
−2cosx +
1
2 cos 2x π
0
2+
1
2
−
−2+
1
2
=
4
π
(c)
(b) f(c) =f ave ⇔ 2sinc − sin 2c = 4 π
⇔
c 1 ≈ 1.238 or c 2 ≈ 2.808
13. f is continuous on [1, 3], so by the Mean Value Theorem for Integrals there exists a number c in [1, 3] such that
3 f(x) dx = f(c)(3 − 1) ⇒ 8=2f(c); that is, there is a number c such that f(c) = 8 =4.
1 2
15. f ave =
1
50 − 20
50
20
f(x) dx ≈ 1 30 M 3 = 1 30
17. Let t =0and t =12correspond to 9 AM and 9 PM, respectively.
19. ρ ave = 1 8
8
0
12
√ x +1
dx = 3 2
21. V ave = 1 5 V (t) dt = 1
5 0 5
T ave = 1
12 − 0
5
0
= 1 12
8
0
50 − 20 · [f(25) + f(35) + f(45)] = 1 115
(38 + 29 + 48) = =38 1 3 3 3
3
12
0 50 + 14 sin
1
πt
dt = 1
12 12 50t − 14 · 12
cos 1 πt 12
π 12 0
◦
F ≈ 59 ◦ F
50 · 12 + 14 · 12
π
+14· 12
π
=
50 +
28
π
(x +1) −1/2 dx = 3 √ x +1 8
=9− 3=6kg/m
5
4π 1 − cos 2 πt
5
dt = 1 5
4π 0 1 − cos 2 πt 5
dt
= 1
4π t −
5
sin 2
πt 5
= 1
5
[(5 − 0) − 0] = ≈ 0.4 L
2π 5 0 4π 4π
23. Let F (x) = x
f(t) dt for x in [a, b]. ThenF is continuous on [a, b] and differentiable on (a, b), sobytheMeanValue
a
Theorem there is a number c in (a, b) such that F (b) − F (a) =F 0 (c)(b − a). ButF 0 (x) =f(x) by the Fundamental
Theorem of Calculus. Therefore, b
f(t) dt − 0=f(c)(b − a).
a
0

F.
286 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
6 Review
1. (a) See Section 6.1, Figure 2 and Equations 6.1.1 and 6.1.2.
(b) Instead of using “top minus bottom” and integrating from left to right, we use “right minus left” and integrate from bottom
to top. See Figures 11 and 12 in Section 6.1.
2. The numerical value of the area represents the number of meters by which Sue is ahead of Kathy after 1 minute.
3. (a) See the discussion in Section 6.2, near Figures 2 and 3, ending in the Definition of Volume.
(b) See the discussion between Examples 5 and 6 in Section 6.2. If the cross-section is a disk, find the radius in terms of x or y
and use A = π(radius) 2 . If the cross-section is a washer, find the inner radius r in and outer radius r out and use
A = π r 2 out
− π
r
2
in
.
4. (a) V =2πrh ∆r =(circumference)(height)(thickness)
5.
(b) For a typical shell, find the circumference and height in terms of x or y and calculate
V = b
(circumference)(height)(dx or dy),wherea and b are the limits on x or y.
a
(c) Sometimes slicing produces washers or disks whose radii are difficult (or impossible) to find explicitly. On other
6
0
occasions, the cylindrical shell method leads to an easier integral than slicing does.
f(x) dx represents the amount of work done. Its units are newton-meters, or joules.
6. (a) The average value of a function f on an interval [a, b] is f ave = 1
b − a
b
a
f(x) dx.
(b) The Mean Value Theorem for Integrals says that there is a number c at which the value of f is exactly equal to the average
value of the function, that is, f(c) =f ave . For a geometric interpretation of the Mean Value Theorem for Integrals, see
Figure 2 in Section 6.5 and the discussion that accompanies it.
1. The curves intersect when x 2 =4x − x 2 ⇔ 2x 2 − 4x =0 ⇔
2x(x − 2) = 0 ⇔ x =0 or 2.
A = 2
0
(4x − x 2 ) − x 2 dx = 2
0 (4x − 2x2 ) dx
= 2x 2 − 2 x3 2
=
8 − 16
3 0 3 − 0 =
8
3
3. If x ≥ 0,then| x | = x, and the graphs intersect when x =1− 2x 2 ⇔ 2x 2 + x − 1=0 ⇔ (2x − 1)(x +1)=0 ⇔
x = 1 2 or −1,but−1 < 0. By symmetry, we can double the area from x =0to x = 1 2 .
A =2 1/2
0 (1 − 2x 2 ) − x dx =2 1/2
(−2x 2 − x +1)dx
0
=2 − 2 3 x3 − 1 2 x2 + x 1/2
0
=2 7
24
=
7
12
=2 − 1 − 1 +
1
12 8 2 − 0

F.
5. A =
2
=
0
sin
πx
2
− 2 π cos πx
2
− (x 2 − 2x) dx
− 1 3 x3 + x 2 2
0
= 2
π − 8 3 +4 − − 2 π − 0+0 = 4 3 + 4 π
TX.10
CHAPTER 6 REVIEW ¤ 287
7. Using washers with inner radius x 2 and outer radius 2x,wehave
V = π 2
0 (2x) 2 − (x 2 ) 2 dx = π 2
0 (4x2 − x 4 ) dx
= π 4
3 x3 − 1 x5 2
= π 32
−
32
5 0 3 5
=32π · 2
15 = 64
15 π
9. V = π
3 (9
−3 − y 2 ) − (−1) 2 − [0 − (−1)] 2 dy
=2π 3
0 (10 − y 2 ) 2 − 1 dy =2π 3
(100 − 0 20y2 + y 4 − 1) dy
=2π 3
0 (99 − 20y2 + y 4 ) dy =2π 99y − 20
3 y3 + 1 5 y5 3
0
=2π 297 − 180 + 243
5
=
1656
5 π
11. The graph of x 2 − y 2 = a 2 is a hyperbola with right and left branches.
Solving for y gives us y 2 = x 2 − a 2 ⇒ y = ± √ x 2 − a 2 .
We’ll use shells and the height of each shell is
√
x2 − a 2 − − √ x 2 − a 2 =2 √ x 2 − a 2 .
The volume is V = a+h
a
2πx · 2 √ x 2 − a 2 dx. Toevaluate,letu = x 2 − a 2 ,
so du =2xdxand xdx= 1 2
du. Whenx = a, u =0,andwhenx = a + h,
u =(a + h) 2 − a 2 = a 2 +2ah + h 2 − a 2 =2ah + h 2 .
2ah+h
2
√
2ah+h
2
1 2
Thus, V =4π
u
0
2 du =2π
3 u3/2 = 4
0
3 π 2ah + h 2 3/2
.
13. A shell has radius π 2 − x, circumference 2π π
2 − x , and height cos 2 x − 1 4 .
y =cos 2 x intersects y = 1 4 when cos2 x = 1 4
cos x = ± 1 2
[ |x| ≤ π/2] ⇔ x = ± π 3 .
V =
π/3
−π/3
π
2π
2 − x cos 2 x − 1
dx
4
⇔

F.
288 ¤ CHAPTER 6 APPLICATIONS OF INTEGRATION
TX.10
15. (a) A cross-section is a washer with inner radius x 2 and outer radius x.
V = 1
π (x) 2 − (x 2 ) 2 dx = 1
0 0 π(x2 − x 4 ) dx = π 1
3 x3 − 1 x5 1
= π 1
− 1
5 0 3 5 =
2
π 15
(b) A cross-section is a washer with inner radius y and outer radius y.
V =
1
π 2
0 y − y
2
dy = 1
π(y − 0 y2 ) dy = π 1
2 y2 − 1 y3 1
= π 1
− 1
3 0 2 3 =
π
6
(c) A cross-section is a washer with inner radius 2 − x and outer radius 2 − x 2 .
V = 1
0 π (2 − x 2 ) 2 − (2 − x) 2 dx = 1
0 π(x4 − 5x 2 +4x) dx = π 1
5 x5 − 5 3 x3 +2x 2 1
0 = π 1
5 − 5 3 +2 = 8
15 π
17. (a) Using the Midpoint Rule on [0, 1] with f(x) =tan(x 2 ) and n =4,weestimate
A =
1
0 tan(x2 ) dx ≈ tan 1 1
2
+tan
3
2
+tan
5
2
+tan
7
2
≈ 1 (1.53) ≈ 0.38
4 8
8
8
8
4
(b) Using the Midpoint Rule on [0, 1] with f(x) =π tan 2 (x 2 ) (fordisks)andn =4,weestimate
V =
1
tan f(x) dx ≈ 1 π 2 1
2
+tan 2 3
2
+tan 2 5
2
+tan 2 7
2
≈ π (1.114) ≈ 0.87
0 4 8
8
8
8
4
19.
π/2
0
2πx cos xdx= π/2
0
(2πx)cosxdx
The solid is obtained by rotating the region R = (x, y) | 0 ≤ x ≤ π 2 , 0 ≤ y ≤ cos x about the y-axis.
21.
π
0
π(2 − sin x)2 dx
The solid is obtained by rotating the region R = {(x, y) | 0 ≤ x ≤ π, 0 ≤ y ≤ 2 − sin x} about the x-axis.
23. Takethebasetobethediskx 2 + y 2 ≤ 9. ThenV = 3
−3 A(x) dx,whereA(x 0) is the area of the isosceles right triangle
whose hypotenuse lies along the line x = x 0 in the xy-plane. The length of the hypotenuse is 2 √ 9 − x 2 and the length of
each leg is √ 2 √ 9 − x 2 . A(x) = 1 2
√
2
√
9 − x
2 2
=9− x 2 ,so
V =2 3
A(x) dx =2 3
(9 − 0 0 x2 ) dx =2 9x − 1 x3 3
=2(27− 9) = 36
3 0
25. Equilateral triangles with sides measuring 1 x meters have height 1 x sin 4 4 60◦ = √ 3
8
x. Therefore,
A(x) = 1 · 1 x · √3
x = √ 3
2 4 8 64 x2 . V = 20
A(x) dx = √
3 20
x 2 dx = √
3 1 x3 20
= 8000 √ 3
= 125 √ 3
m 3 .
0 64 0 64 3 0 64 · 3 3
27. f(x) =kx ⇒ 30 N = k(15 − 12) cm ⇒ k =10N/cm = 1000 N/m. 20 cm − 12 cm =0.08 m ⇒
W = 0.08
kx dx =1000 0.08
xdx=500 x 2 0.08
= 500(0.08) 2 =3.2 N·m =3.2 J.
0 0 0

F.
TX.10
CHAPTER 6 REVIEW ¤ 289
29. (a) The parabola has equation y = ax 2 with vertex at the origin and passing through
(4, 4). 4=a · 4 2 ⇒ a = 1 4
⇒ y = 1 4 x2 ⇒ x 2 =4y ⇒
x =2 y. Each circular disk has radius 2 y and is moved 4 − y ft.
W = 4
2 π 2 4
y
0 62.5(4 − y) dy = 250π y(4 − y) dy
0
= 250π 2y 2 − 1 y3 4
=250π
3
32 − 64
0 3 =
8000π
3
≈ 8378 ft-lb
(b) In part (a) we knew the final water level (0) but not the amount of work done. Here
we use the same equation, except with the work fixed, and the lower limit of
integration (that is, the final water level — call it h) unknown: W = 4000
250π 2y 2 − 1 y3 4
16
3
=4000 ⇔ =
h π
32 − 64 3 − 2h 2 − 1 h3 3
⇔
⇔
h 3 − 6h 2 +32− 48 =0. We graph the function f(h) π =h3 − 6h 2 +32− 48
π
on the interval [0, 4] to see where it is 0. From the graph, f(h) =0for h ≈ 2.1.
So the depth of water remaining is about 2.1 ft.
31. lim
h→0
f ave =lim
h→0
1
(x + h) − x
x+h
x
F (x + h) − F (x)
f(t) dt =lim
,whereF (x) = x
f(t) dt. But we recognize this
h→0 h
a
limit as being F 0 (x) by the definition of a derivative. Therefore, lim
h→0
f ave = F 0 (x) =f(x) by FTC1.

F.
TX.10

F.
TX.10
PROBLEMS PLUS
1. (a) The area under the graph of f from 0 to t is equal to t
0 f(x) dx, so the requirement is that t
0 f(x) dx = t3 for all t. We
differentiate both sides of this equation with respect to t (with the help of FTC1) to get f(t) =3t 2 . This function is
positive and continuous, as required.
(b) The volume generated from x =0to x = b is b
0 π[f(x)]2 dx. Hence, we are given that b 2 = b
0 π[f(x)]2 dx for all
b>0. Differentiating both sides of this equation with respect to b using the Fundamental Theorem of Calculus gives
2b = π[f(b)] 2 ⇒ f(b) = 2b/π,sincef is positive. Therefore, f(x) = 2x/π.
3. Let a and b be the x-coordinates of the points where the line intersects the
curve. From the figure, R 1 = R 2
⇒
a
0
c − 8x − 27x
3
dx = b
a 8x − 27x
3
− c dx
cx − 4x 2 + 27
4 x4 a
0 = 4x 2 − 27
4 x4 − cx b
a
ac − 4a 2 + 27
4 a4 = 4b 2 − 27
4 b4 − bc − 4a 2 − 27 4 a4 − ac
0=4b 2 − 27 4 b4 − bc =4b 2 − 27 4 b4 − b 8b − 27b 3
=4b 2 − 27 4 b4 − 8b 2 +27b 4 = 81
4 b4 − 4b 2
= b 2 81
4 b2 − 4
So for b>0, b 2 = 16 ⇒ b = 4 . Thus, c =8b − 81 9 27b3 =8
4
9 − 27 64
729 =
32
− 64 = 32 .
9 27 27
5. (a) V = πh 2 (r − h/3) = 1 3 πh2 (3r − h). See the solution to Exercise 6.2.51.
(b) The smaller segment has height h =1− x andsobypart(a)itsvolumeis
V = 1 π(1 − 3 x)2 [3(1) − (1 − x)] = 1 π(x − 3 1)2 (x +2). This volume must be 1 of the total volume of the sphere,
3
which is 4 3 π(1)3 .So 1 3 π(x − 1)2 (x +2)= 1 3
4
3 π ⇒ (x 2 − 2x +1)(x +2)= 4 3
⇒ x 3 − 3x +2= 4 3
⇒
3x 3 − 9x +2=0. Using Newton’s method with f(x) =3x 3 − 9x +2, f 0 (x) =9x 2 − 9, weget
x n+1 = x n − 3x3 n − 9x n +2
.Takingx
9x 2 1 =0,wegetx 2 ≈ 0.2222,andx 3 ≈ 0.2261 ≈ x 4 , so, correct to four decimal
n − 9
places, x ≈ 0.2261.
(c) With r =0.5 and s =0.75, the equation x 3 − 3rx 2 +4r 3 s =0becomes x 3 − 3(0.5)x 2 +4(0.5) 3 (0.75) = 0
⇒
x 3 − 3 2 x2 +4 1
8
3
4 =0 ⇒ 8x3 − 12x 2 +3=0. We use Newton’s method with f(x) =8x 3 − 12x 2 +3,
f 0 (x) =24x 2 − 24x,sox n+1 = x n − 8x3 n − 12x 2 n +3
24x 2 n − 24x n
.Takex 1 =0.5. Thenx 2 ≈ 0.6667,andx 3 ≈ 0.6736 ≈ x 4.
So to four decimal places the depth is 0.6736 m.
291

F.
292 ¤ CHAPTER 6 PROBLEMS PLUS
(d) (i) From part (a) with r =5in., the volume of water in the bowl is
V = 1 3 πh2 (3r − h) = 1 3 πh2 (15 − h) =5πh 2 − 1 3 πh3 . We are given that dV
dt =0.2 in3 /s and we want to find dh
dt
when h =3.Now dV
dt
dh
dt = 0.2
π(10 · 3 − 3 2 ) = 1 ≈ 0.003 in/s.
105π
dh
=10πh
dt − dh
πh2 dt ,sodh dt = 0.2
.Whenh =3,wehave
π(10h − h 2 )
(ii) From part (a), the volume of water required to fill the bowl from the instant that the water is 4 in. deep is
V = 1 2 · 4
3 π(5)3 − 1 3 π(4)2 (15 − 4) = 2 3 · 125π − 16 3
this volume by the rate: Time = 74π/3
0.2
= 370π
3
≈ 387 s ≈ 6.5 min.
7. We are given that the rate of change of the volume of water is dV
dt
74 · 11π =
3
π.Tofind the time required to fill the bowl we divide
= −kA(x),wherek is some positive constant and A(x) is
the area of the surface when the water has depth x. Now we are concerned with the rate of change of the depth of the water
with respect to time, that is, dx
dV
. But by the Chain Rule,
dt dt = dV
dx
dx
,sothefirst equation can be written
dt
dV dx
dx dt = −kA(x) (). Also, we know that the total volume of water up to a depth x is V (x) = x
A(s) ds,whereA(s) is
0
the area of a cross-section of the water at a depth s. Differentiating this equation with respect to x,wegetdV/dx = A(x).
Substituting this into equation ,wegetA(x)(dx/dt) =−kA(x) ⇒ dx/dt = −k, a constant.
9. We must find expressions for the areas A and B, and then set them equal and see what this says about the curve C. If
P = a, 2a 2 ,thenareaA is just a
0 (2x2 − x 2 ) dx = a
0 x2 dx = 1 3 a3 .Tofind area B, weusey as the variable of
integration. So we find the equation of the middle curve as a function of y: y =2x 2 ⇔ x = y/2, sinceweare
concerned with the first quadrant only. We can express area B as
2a
2
0
2a
2 4 2a
2
y/2 − C(y) dy =
3 (y/2)3/2 − C(y) dy = 4 2a
2
0 0
3 a3 − C(y) dy
0
where C(y) is the function with graph C. SettingA = B,weget 1 3 a3 = 4 3 a3 − 2a 2
C(y) dy ⇔ 2a 2
C(y) dy = a 3 .
0 0
Now we differentiate this equation with respect to a using the Chain Rule and the Fundamental Theorem:
C(2a 2 )(4a) =3a 2 ⇒ C(y) = 3 4
y/2,wherey =2a 2 .Nowwecansolvefory: x = 3 4
y/2 ⇒
x 2 = 9 32
(y/2) ⇒ y =
16 9 x2 .
11. (a) Stacking disks along the y-axis gives us V = h
0 π [f(y)]2 dy.
(b) Using the Chain Rule, dV
dt = dV
dh · dh
dt = π [f(h)]2 dh
dt . TX.10

F.
TX.10
CHAPTER 6 PROBLEMS PLUS ¤ 293
(c) kA √ h = π[f(h)] 2 dh
dt .Setdh dt = C: π[f(h)]2 C = kA √ h ⇒ [f(h)] 2 = kA
πC
is, f(y) =
√
h ⇒ f(h) =
kA
πC h1/4 ;that
kA
πC y1/4 . The advantage of having dh = C is that the markings on the container are equally spaced.
dt
13. The cubic polynomial passes through the origin, so let its equation be
y = px 3 + qx 2 + rx. The curves intersect when px 3 + qx 2 + rx = x 2
⇔
px 3 +(q − 1)x 2 + rx =0.Calltheleftsidef(x). Sincef(a) =f(b) =0,
another form of f is
f(x)=px(x − a)(x − b) =px[x 2 − (a + b)x + ab]
= p[x 3 − (a + b)x 2 + abx]
Since the two areas are equal, we must have a
f(x) dx = − b
f(x) dx ⇒
0 a
[F (x)] a 0 =[F (x)]a b
⇒ F (a) − F (0) = F (a) − F (b) ⇒ F (0) = F (b),whereF is an antiderivative of f.
Now F (x) = f(x) dx = p[x 3 − (a + b)x 2 + abx] dx = p 1
4 x4 − 1 3 (a + b)x3 + 1 2 abx2 + C,so
F (0) = F (b) ⇒ C = p 1
4 b4 − 1 3 (a + b)b3 + 1 2 ab3 + C ⇒ 0=p 1
4 b4 − 1 3 (a + b)b3 + 1 2 ab3 ⇒
0=3b − 4(a + b)+6a [multiply by 12/(pb 3 ), b 6= 0] ⇒ 0=3b − 4a − 4b +6a ⇒ b =2a.
Hence, b is twice the value of a.
15. We assume that P lies in the region of positive x. Sincey = x 3 is an odd
function, this assumption will not affect the result of the calculation. Let
P = a, a 3 . The slope of the tangent to the curve y = x 3 at P is 3a 2 ,andso
the equation of the tangent is y − a 3 =3a 2 (x − a) ⇔ y =3a 2 x − 2a 3 .
We solve this simultaneously with y = x 3 to find the other point of intersection:
x 3 =3a 2 x − 2a 3 ⇔ (x − a) 2 (x +2a) =0.SoQ = −2a, −8a 3 is
the other point of intersection. The equation of the tangent at Q is
y − (−8a 3 )=12a 2 [x − (−2a)] ⇔ y =12a 2 x +16a 3 . By symmetry,
this tangent will intersect the curve again at x = −2(−2a) =4a.Thecurveliesabovethefirst tangent, and
below the second, so we are looking for a relationship between A = a
−2a x 3 − (3a 2 x − 2a 3 ) dx and
B = 4a
−2a (12a 2 x +16a 3 ) − x 3 dx. We calculate A = 1
4 x4 − 3 2 a2 x 2 +2a 3 x a
= 3 −2a 4 a4 − (−6a 4 )= 27
4 a4 ,and
B = 6a 2 x 2 +16a 3 x − 1 4 x4 4a
−2a =96a4 − (−12a 4 ) = 108a 4 . We see that B =16A =2 4 A.Thisisbecauseour
calculation of area B was essentially the same as that of area A,witha replaced by −2a, so if we replace a with −2a in our
expression for A,weget 27 4 (−2a)4 =108a 4 = B.

F.
TX.10

F.
TX.10
7 TECHNIQUES OF INTEGRATION
7.1 Integration by Parts
1. Let u =lnx, dv = x 2 dx ⇒ du = 1 dx, v = 1 x 3 x3 .ThenbyEquation2,
x 2 ln xdx=(lnx) 1
x3 − 1
x3
1
3 3 x dx =
1
3 x3 ln x − 1 3 x 2 dx = 1 3 x3 ln x − 1 1 x3 + C
3 3
= 1 3 x3 ln x − 1 9 x3 + C or 1 x3
ln x − 1 3 3 + C
Note: A mnemonic device which is helpful for selecting u when using integration by parts is the LIATE principle of precedence for u:
Logarithmic
Inverse trigonometric
Algebraic
Trigonometric
Exponential
If the integrand has several factors, then we try to choose among them a u which appears as high as possible on the list. For example, in xe 2x dx
the integrand is xe 2x , which is the product of an algebraic function (x) and an exponential function (e 2x ). Since Algebraic appears before Exponential,
we choose u = x. Sometimes the integration turns out to be similar regardless of the selection of u and dv, but it is advisable to refer to LIATE when in
doubt.
3. Let u = x, dv =cos5xdx ⇒ du = dx, v = 1 5
sin 5x. ThenbyEquation2,
x cos 5xdx=
1
x sin 5x − 1
sin 5xdx= 1 1
5 5 5
x sin 5x +
25
cos 5x + C.
5. Let u = r, dv = e r/2 dr ⇒ du = dr, v =2e r/2 .Then re r/2 dr =2re r/2 − 2e r/2 dr =2re r/2 − 4e r/2 + C.
7. Let u = x 2 , dv =sinπxdx ⇒ du =2xdxand v = − 1 π
cos πx. Then
I =
x 2 sin πxdx = − 1 π x2 cos πx + 2 π x cos πxdx (). Next let U = x, dV =cosπx dx ⇒ dU = dx,
V = 1 sin πx, so π
x cos πxdx = 1 x sin πx − 1
π π sin πx dx =
1
1
π
x sin πx + cos πx + C
π 1.
2
Substituting for x cos πx dx in (), we get
I = − 1 π x2 cos πx + 2 1
1
x sin πx + cos πx + C
π π π 2 1 = −
1
π x2 cos πx + 2 x sin πx + 2 cos πx + C,whereC = 2 C π 2 π 3 π 1.
9. Let u =ln(2x +1), dv = dx ⇒ du = 2 dx, v = x. Then
2x +1
2x
(2x +1)− 1
ln(2x +1)dx = x ln(2x +1)−
2x +1 dx = x ln(2x +1)− dx
2x +1
= x ln(2x +1)− 1 − 1
dx = x ln(2x +1)− x + 1 ln(2x +1)+C
2
2x +1
11. Let u =arctan4t, dv = dt ⇒ du =
arctan 4tdt= t arctan 4t −
= 1 2
(2x +1)ln(2x +1)− x + C
4
1+(4t) dt = 4
dt, v = t. Then
2 1+16t2 4t
1+16t 2 dt = t arctan 4t − 1 8
32t
1+16t 2 dt = t arctan 4t − 1 8 ln(1 + 16t2 )+C.
295

F.
296 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
13. Let u = t, dv =sec 2 2tdt ⇒ du = dt, v = 1 2
TX.10
tan 2t. Then
t sec 2 2tdt= 1 2 t tan 2t − 1 2
tan 2tdt=
1
t tan 2t − 1 ln |sec 2t| + C.
2 4
15. First let u =(lnx) 2 , dv = dx ⇒ du =2lnx · 1 dx, v = x.ThenbyEquation2,
x
I = (ln x) 2 dx = x(ln x) 2 − 2 x ln x · 1
x dx = x(ln x)2 − 2 ln xdx.NextletU =lnx, dV = dx
⇒
dU =1/x dx, V = x to get ln xdx = x ln x − x · (1/x) dx = x ln x − dx = x ln x − x + C 1.Thus,
I = x(ln x) 2 − 2(x ln x − x + C 1 )=x(ln x) 2 − 2x ln x +2x + C,whereC = −2C 1 .
17. First let u =sin3θ, dv = e 2θ dθ ⇒ du =3cos3θdθ, v = 1 2 e2θ .Then
I =
e 2θ sin 3θdθ= 1 2 e2θ sin 3θ − 3 2 e 2θ cos 3θdθ.NextletU =cos3θ, dV = e 2θ dθ ⇒ dU = −3sin3θdθ,
V = 1 2 e2θ to get
e 2θ cos 3θdθ= 1 2 e2θ cos 3θ + 3 2 e 2θ sin 3θdθ. Substituting in the previous formula gives
I = 1 2 e2θ sin 3θ − 3 4 e2θ cos 3θ − 9 4
e 2θ sin 3θdθ = 1 2 e2θ sin 3θ − 3 4 e2θ cos 3θ − 9 4 I
⇒
13
I = 1
4 2 e2θ sin 3θ − 3 4 e2θ cos 3θ + C 1 . Hence, I= 1 13 e2θ (2 sin 3θ − 3cos3θ)+C,whereC = 4 C 13 1.
19. Let u = t, dv =sin3tdt ⇒ du = dt, v = − 1 cos 3t. Then
3
π
0
t sin 3tdt= − 1 t cos 3t π
+ 1 π cos 3tdt= 1
π − 0
+ 1 π
3 0 3 0 3 9 sin 3t = π . 0 3
21. Let u = t, dv =coshtdt ⇒ du = dt, v =sinht. Then
1 t cosh tdt= t sinh t 1
− 1
sinh tdt=(sinh1− sinh 0) − cosh t 1
=sinh1− (cosh 1 − cosh 0)
0 0 0 0
=sinh1− cosh 1 + 1.
We can use the definitions of sinh and cosh to write the answer in terms of e:
sinh 1 − cosh 1 + 1 = 1 2 (e1 − e −1 ) − 1 2 (e1 + e −1 )+1=−e −1 +1=1− 1/e.
23. Let u =lnx, dv = x −2 dx ⇒ du = 1 x dx, v = −x−1 .By(6),
2
ln x
1 x dx = − ln x 2 2
+ x −2 dx = − 1 2
x
ln 2 + ln 1 + − 1 2
= − 1 2
1 1
x
ln 2 + 0 − 1 +1= 1 − 1 ln 2.
2 2 2 2
1
25. Let u = y, dv = dy
e = 2y e−2y dy ⇒ du = dy, v = − 1 2 e−2y .Then
1
0
y
e dy = − 1 ye−2y 1
2y 2
0
+ 1 2
1
0
e −2y dy = − 1 2 e−2 +0 − 1 4
27. Let u =cos −1 x, dv = dx ⇒ du = −√ dx , v = x. Then
1 − x
2
e −2y 1
0
= − 1 2 e−2 − 1 4 e−2 + 1 4 = 1 4 − 3 4 e−2 .
I =
1/2
0
cos −1 xdx= x cos −1 x 1/2
1/2
+
0
0
dt = −2xdx.Thus,I = π 6 + 1 2
xdx
√
1 − x
2 = 1 2 · π
3/4
+ t −1/2 − 1 dt ,wheret =1− x 2
3 2
1
3/4 t−1/2 dt = π 6 + √ t 1
3/4 = π 6 +1− √ 3
2 = 1 6
π +6− 3
√
3
.
1
⇒

F.
TX.10
SECTION 7.1 INTEGRATION BY PARTS ¤ 297
29. Let u =ln(sinx), dv =cosxdx ⇒ du = cos x dx, v =sinx. Then
sin x
I = cos x ln(sin x) dx =sinx ln(sin x) − cos xdx=sinx ln(sin x) − sin x + C.
Another method: Substitute t =sinx,sodt =cosxdx.ThenI = ln tdt = t ln t − t + C (see Example 2) and so
I =sinx (ln sin x − 1) + C.
31. Let u =(lnx) 2 , dv = x 4 dx ⇒ du =2 ln x x5
dx, v =
x 5 .By(6),
2
x
x 4 (ln x) 2 5
2 2
dx =
1
5 (ln x 4
2
x)2 32
− 2 ln xdx=
1 1 5 (ln x 4
5 2)2 − 0 − 2 ln xdx.
1 5
dx, V =
x5
25 .
Let U =lnx, dV = x4
5 dx ⇒ dU = 1 x
2
x 4
x
5 2 2
Then
1 5 ln xdx= 25 ln x x 4
32
x
5
− dx =
1 1 25 ln 2 − 0 − 25
So 2
1 x4 (ln x) 2 dx = 32 (ln 5 2)2 − 2 32
31
25
ln 2 −
125
125
2
1
=
32
(ln 5 2)2 − 64
62
25
ln 2 + . 125
= 32
25 ln 2 − 32
125 − 1
125
.
33. Let y = √ x,sothatdy = 1 2 x−1/2 dx = 1
2 √ x dx = 1
2y dx. Thus, cos √ xdx= cos y (2ydy)=2 y cos ydy.Now
use parts with u = y, dv =cosydy, du = dy, v =siny to get y cos ydy= y sin y − sin ydy= y sin y +cosy + C 1 ,
so cos √ xdx=2y sin y +2cosy + C =2 √ x sin √ x +2cos √ x + C.
√
35. Let x = θ 2 π
,sothatdx =2θdθ.Thus, θ 3 cos θ 2 √ π
dθ = θ 2 cos θ 2 π
· 1 (2θdθ)= 1 x cos xdx.Nowuse
√ √ 2 2
π/2 π/2 π/2
parts with u = x, dv =cosxdx, du = dx, v =sinx to get
π
1
x π
x cos xdx= 1 π
2
2 sin x − sin xdx
π/2
π/2
π/2
= 1 2
= 1 2 (π sin π +cosπ) − 1 2
π
2 sin π 2 +cos π 2
x sin x +cosx
π
π/2
=
1
2 (π · 0 − 1) − 1 2
π
2 · 1+0 = − 1 2 − π 4
37. Let y =1+x, so that dy = dx. Thus, x ln(1 + x) dx = (y − 1) ln ydy.Nowusepartswithu =lny, dv =(y − 1) dy,
du = 1 dy, v = 1 y 2 y2 − y to get
(y − 1) ln ydy= 1
2 y2 − y ln y − 1
y − 1 dy = 1 y(y − 2) ln y − 1 2 2 4 y2 + y + C
= 1 2 (1 + x)(x − 1) ln(1 + x) − 1 4 (1 + x)2 +1+x + C,
which can be written as 1 2 (x2 − 1) ln(1 + x) − 1 4 x2 + 1 2 x + 3 4 + C.
In Exercises 39 – 42, let f(x) denote the integrand and F (x) its antiderivative (with C =0).
39. Let u =2x +3, dv = e x dx ⇒ du =2dx, v = e x .Then
(2x +3)e x dx =(2x +3)e x − 2 e x dx =(2x +3)e x − 2e x + C
=(2x +1)e x + C
We see from the graph that this is reasonable, since F has a minimum where
f changes from negative to positive.

F.
298 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
41. Let u = 1 2 x2 , dv =2x √ 1+x 2 dx ⇒ du = xdx, v = 2 3 (1 + x2 ) 3/2 .
Then
√
x
3
1+x 2 dx = 1 2 x2 2
(1 +
3 x2 ) 3/2 − 2 3 x(1 + x 2 ) 3/2 dx
= 1 3 x2 (1 + x 2 ) 3/2 − 2 3 · 2
5 · 1
2 (1 + x2 ) 5/2 + C
= 1 3 x2 (1 + x 2 ) 3/2 − 2
15 (1 + x2 ) 5/2 + C
Another method: Use substitution with u =1+x 2 to get 1 5 (1 + x2 ) 5/2 − 1 3 (1 + x2 ) 3/2 + C.
43. (a) Take n =2in Example 6 to get
0
sin 2 xdx= − 1 2 cos x sin x + 1 2
1 dx = x 2
−
sin 2x
4
(b) sin 4 xdx= − 1 cos x
4 sin3 x + 3 4 sin 2 xdx= − 1 cos x 4 sin3 x + 3 x − 3 sin 2x + C.
8 16
45. (a) From Example 6, sin n xdx= − 1 n cos x sinn−1 x + n − 1
sin n−2 xdx. Using (6),
n
π/2
sin n xdx= − cos x π/2
sinn−1 x
+ n − 1 π/2
sin n−2 xdx
n
n
=(0− 0) + n − 1
n
0
π/2
0
sin n−2 xdx= n − 1
n
(b) Using n =3in part (a), we have π/2
sin 3 xdx= 2 π/2
sin xdx= − 2 cos x π/2
0 3 0 3
Using n =5in part (a), we have π/2
0
sin 5 xdx= 4 5
π/2
0
sin 3 xdx= 4 5 · 2
3 = 8 15 .
0
π/2
0
0
= 2 3 .
+ C.
sin n−2 xdx
(c) The formula holds for n =1(that is, 2n +1=3) by (b). Assume it holds for some k ≥ 1. Then
π/2
0
sin 2k+1 xdx=
π/2
0
2 · 4 · 6 ·····(2k)
3 · 5 · 7 ·····(2k +1) .ByExample6,
sin 2k+3 xdx=
=
2k +2
2k +3
π/2
0
sin 2k+1 xdx=
2 · 4 · 6 ·····(2k)[2 (k +1)]
3 · 5 · 7 ·····(2k +1)[2(k +1)+1] ,
so the formula holds for n = k +1. By induction, the formula holds for all n ≥ 1.
47. Let u =(lnx) n , dv = dx ⇒ du = n(ln x) n−1 (dx/x), v = x. By Equation 2,
(ln x) n dx = x(ln x) n − nx(ln x) n−1 (dx/x) =x(ln x) n − n (ln x) n−1 dx.
2k +2
2k +3 · 2 · 4 · 6 ·····(2k)
3 · 5 · 7 ·····(2k +1)
49.
tan n xdx= tan n−2 x tan 2 xdx= tan n−2 x (sec 2 x − 1) dx = tan n−2 x sec 2 xdx− tan n−2 xdx
= I − tan n−2 xdx.
Let u =tan n−2 x, dv =sec 2 xdx ⇒ du =(n − 2) tan n−3 x sec 2 xdx, v =tanx. Then, by Equation 2,
I =tan n−1 x − (n − 2) tan n−2 x sec 2 xdx
1I =tan n−1 x − (n − 2)I
(n − 1)I =tan n−1 x
I = tann−1 x
n − 1
Returning to the original integral, tan n xdx= tann−1 x
n − 1
− tan n−2 xdx.

F.
TX.10
SECTION 7.1 INTEGRATION BY PARTS ¤ 299
51. By repeated applications of the reduction formula in Exercise 47,
(ln x) 3 dx = x (ln x) 3 − 3 (ln x) 2 dx = x(ln x) 3 − 3 x(ln x) 2 − 2 (ln x) 1 dx
= x (ln x) 3 − 3x(ln x) 2 +6 x(ln x) 1 − 1 (ln x) 0 dx
= x (ln x) 3 − 3x(ln x) 2 +6x ln x − 6 1 dx = x (ln x) 3 − 3x(ln x) 2 +6x ln x − 6x + C
53. Area = 5
0 xe−0.4x dx. Letu = x, dv = e −0.4x dx ⇒
du = dx, v = −2.5e −0.4x .Then
area = −2.5xe −0.4x 5
0 +2.5 5
0 e−0.4x dx
= −12.5e −2 +0+2.5 −2.5e −0.4x 5
0
= −12.5e −2 − 6.25(e −2 − 1) = 6.25 − 18.75e −2 or 25 − 75 4 4 e−2
55. The curves y = x sin x and y =(x − 2) 2 intersect at a ≈ 1.04748 and
b ≈ 2.87307,so
area = b
a [x sin x − (x − 2)2 ] dx
= −x cos x +sinx − 1 (x − 2)3 b
3 a
≈ 2.81358 − 0.63075 = 2.18283
[by Example 1]
57. V = 1
2πx cos(πx/2) dx. Letu = x, dv =cos(πx/2) dx ⇒ du = dx, v = 2 sin(πx/2).
0 π
2
πx
1
V =2π
π x sin − 2π · 2 1 πx
2
sin dx =2π
2
π 2
π − 0 − 4 − 2 πx
1
π cos =4+ 8 2
π (0 − 1) = 4 − 8 π .
0
0
59. Volume = 0
−1 2π(1 − x)e−x dx.Letu =1− x, dv = e −x dx ⇒ du = − dx, v = −e −x .
V =2π (1 − x)(−e −x ) 0
− 2π 0
−1 −1 e−x dx =2π (x − 1)(e −x )+e −x 0
=2π xe −x 0
=2π(0 + e) =2πe
−1 −1
61. Theaveragevalueoff(x) =x 2 ln x on the interval [1, 3] is f ave = 1
3 − 1
Let u =lnx, dv = x 2 dx ⇒ du =(1/x) dx, v = 1 3 x3 .
So I = 1
3 x3 ln x 3
− 3
1 1
Thus, f ave = 1 2 I = 1 2
9ln3−
26
9
3
1
3 x2 dx =(9ln3− 0) − 1
x3 3
=9ln3−
9
3 − 1 1 9
=
9
ln 3 − 13 .
2 9
1
0
x 2 ln xdx= 1 2 I.
=9ln3−
26
9 .
63. Since v(t) > 0 for all t, the desired distance is s(t) = t
t
0 0 w2 e −w dw.
First let u = w 2 , dv = e −w dw ⇒ du =2wdw, v = −e −w .Thens(t) = −w 2 e −w t
t
0 0 we−w dw.
Next let U = w, dV = e −w dw ⇒ dU = dw, V = −e −w .Then
−we
s(t) =−t 2 e −t −w
+2
t
t
0 0 e−w dw = −t 2 e −t +2 −te −t +0+ −e −w
t
0
= −t 2 e −t +2(−te −t − e −t +1)=−t 2 e −t − 2te −t − 2e −t +2=2− e −t (t 2 +2t +2)meters
65. For I = 4
xf 00 (x) dx, letu = x, dv = f 00 (x) dx ⇒ du = dx, v = f 0 (x). Then
1
I = xf 0 (x) 4
− 4
f 0 (x) dx =4f 0 (4) − 1 · f 0 (1) − [f(4) − f(1)] = 4 · 3 − 1 · 5 − (7 − 2) = 12 − 5 − 5=2.
1 1
We used the fact that f 00 is continuous to guarantee that I exists.

F.
300 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
67. Using the formula for volumes of rotation and the figure, we see that
Volume = d
0 πb2 dy − c
0 πa2 dy − d
c π[g(y)]2 dy = πb 2 d − πa 2 c − d
c π[g(y)]2 dy. Lety = f(x),
which gives dy = f 0 (x) dx and g(y) =x, sothatV = πb 2 d − πa 2 c − π b
a x2 f 0 (x) dx.
Now integrate by parts with u = x 2 ,anddv = f 0 (x) dx ⇒ du =2xdx, v = f(x), and
b
a x2 f 0 (x) dx = x 2 f(x) b
a − b
V = πb 2 d − πa 2 c − π b 2 d − a 2 c −
b
2xf(x) dx a
2xf(x) dx = a b2 f(b) − a 2 f(a) − b
2xf(x) dx,butf(a) =c and f(b) =d ⇒
a
= b
2πx f(x) dx.
a
7.2 Trigonometric Integrals
The symbols
s = and c = indicate the use of the substitutions {u =sinx, du =cosxdx} and {u =cosx, du = − sin xdx}, respectively.
1. sin 3 x cos 2 xdx = sin 2 x cos 2 x sin xdx= (1 − cos 2 x)cos 2 x sin xdx= c (1 − u 2 )u 2 (−du)
= (u 2 − 1)u 2 du = (u 4 − u 2 ) du = 1 5 u5 − 1 3 u3 + C = 1 5 cos5 x − 1 3 cos3 x + C
3.
3π/4
sin 5 x cos 3 xdx = 3π/4
sin 5 x cos 2 x cos xdx= 3π/4
sin 5 x (1 − sin 2 x)cosxdx= s √ 2/2
u 5 (1 − u 2 ) du
π/2 π/2 π/2 1
= √ 2/2
(u 5 − u 7 ) du = 1
1 6 u6 − 1 8 u8√ 2/2
=
1
1/8
− 1/16
6 8
5. Let y = πx,sody = πdxand
sin 2 (πx) cos 5 (πx) dx = 1 π sin 2 y cos 5 ydy= 1 π sin 2 y cos 4 y cos ydy
− 1
− 1
6 8 = −
11
384
= 1 π
= 1 π
sin 2 y (1 − sin 2 y) 2 cos ydy = s 1
π u 2 (1 − u 2 ) 2 du = 1 π (u 2 − 2u 4 + u 6 ) du
1
3 u3 − 2 5 u5 + 1 7 u7 + C = 1
3π sin3 y − 2
5π sin5 y + 1
7π sin7 y + C
= 1
3π sin3 (πx) − 2
5π sin5 (πx)+ 1
7π sin7 (πx)+C
7.
π/2
cos 2 θdθ = π/2
0 0
= 1 2
1
(1 + cos 2θ) dθ [half-angle identity]
2
θ +
1
sin 2θ π/2
= 1 π +0 − (0 + 0) = π 2 0 2 2 4
9.
π
0
sin4 (3t) dt = π
0
= 1 π
4 0
= 1 4
sin 2 (3t) 2 dt = π
1 (1 − cos
0 2 6t)2 dt = 1 π (1 − 2cos6t 4 0 +cos2 6t) dt
3 − 2cos6t + 1 cos 12t 2 2
dt
1 − 2cos6t +
1
2 (1 + cos 12t) dt = 1 4
3
2 t − 1 3
sin 6t +
1
24 sin 12t π
0 = 1 4
3π
2 − 0+0 − (0 − 0+0) = 3π 8
π
0
11. (1 + cos θ) 2 dθ =
(1 + 2 cos θ +cos 2 θ) dθ = θ +2sinθ + 1 2 (1 + cos 2θ) dθ
= θ +2sinθ + 1 θ + 1 sin 2θ + C = 3 θ +2sinθ + 1 sin 2θ + C
2 4 2 4
13.
π/2
sin 2 x cos 2 xdx = π/2 1
(4 0 0 4 sin2 x cos 2 x) dx = π/2 1
(2 sin x cos
0 4 x)2 dx = 1 π/2
4
= 1 4
π/2
0
1
2 (1 − cos 4x) dx = 1 8
π/2
0
(1 − cos 4x) dx = 1 8
sin 2 2xdx
0
x −
1
sin 4x π/2
= 1 π
4 0 8 2
=
π
16

F.
15.
cos 5
α
√ dα =
sin α
TX.10
cos 4
α
1 − sin 2 α 2
√ cos αdα= √ cos αdα s (1 − u 2 ) 2
= √ du
sin α sin α u
SECTION 7.2 TRIGONOMETRIC INTEGRALS ¤ 301
1 − 2u 2 + u 4
=
du = (u −1/2 − 2u 3/2 + u 7/2 ) du =2u 1/2 − 4
u 1/2 5 u5/2 + 2 9 u9/2 + C
17.
= 2
45 u1/2 (45 − 18u 2 +5u 4 )+C = 2 45
√
sin α (45 − 18 sin 2 α +5sin 4 α)+C
sin
cos 2 x tan 3 3
x
xdx =
cos x dx =
c (1 − u 2
)(−du) −1
=
u
u + u du
= − ln |u| + 1 2 u2 + C = 1 2 cos2 x − ln |cos x| + C
19.
cos x +sin2x
sin x
cos x +2sinx cos x cos x
dx =
dx =
sin x
sin x dx +
=ln|u| +2sinx + C =ln|sin x| +2sinx + C
2cosxdx= s 1
du +2sinx
u
Or: Use the formula cot xdx=ln|sin x| + C.
21. Let u =tanx, du =sec 2 xdx.Then sec 2 x tan xdx= udu= 1 2 u2 + C = 1 2 tan2 x + C.
Or: Let v =secx, dv =secx tan xdx.Then sec 2 x tan xdx= vdv = 1 2 v2 + C = 1 2 sec2 x + C.
23. tan 2 xdx= (sec 2 x − 1) dx =tanx − x + C
25. sec 6 tdt = sec 4 t · sec 2 tdt= (tan 2 t +1) 2 sec 2 tdt= (u 2 +1) 2 du
[u =tant, du =sec 2 tdt]
= (u 4 +2u 2 +1)du = 1 5 u5 + 2 3 u3 + u + C = 1 5 tan5 t + 2 3 tan3 t +tant + C
27.
π/3
0
tan 5 x sec 4 xdx = π/3
0
tan 5 x (tan 2 x +1)sec 2 xdx= √ 3
0
u 5 (u 2 +1)du [u =tanx, du =sec 2 xdx]
= √ 3
(u 7 + u 5 ) du = 1
0 8 u8 + 1 6 u6√ 3
= 81 + 27 = 81 + 9 = 81 + 36 = 117
0 8 6 8 2 8 8 8
Alternate solution:
π/3
0
tan 5 x sec 4 xdx = π/3
0
tan 4 x sec 3 x sec x tan xdx= π/3
0
(sec 2 x − 1) 2 sec 3 x sec x tan xdx
= 2
1 (u2 − 1) 2 u 3 du [u =secx, du =secx tan xdx] = 2
1 (u4 − 2u 2 +1)u 3 du
= 2
1 (u7 − 2u 5 + u 3 ) du = 1
8 u8 − 1 3 u6 + 1 4 u4 2
1 = 32 − 64 3 +4 − 1
8 − 1 3 + 1 4
29. tan 3 x sec xdx = tan 2 x sec x tan xdx= (sec 2 x − 1) sec x tan xdx
=
117
8
= (u 2 − 1) du [u =secx, du =secx tan xdx] = 1 3 u3 − u + C = 1 3 sec3 x − sec x + C
31. tan 5 xdx = (sec 2 x − 1) 2 tan xdx= sec 4 x tan xdx− 2 sec 2 x tan xdx+ tan xdx
= sec 3 x sec x tan xdx− 2 tan x sec 2 xdx+ tan xdx
= 1 4 sec4 x − tan 2 x +ln|sec x| + C [or 1 4 sec4 x − sec 2 x +ln|sec x| + C ]
33.
tan 3 θ
cos 4 θ dθ =
tan 3 θ sec 4 θdθ=
tan 3 θ · (tan 2 θ +1)· sec 2 θdθ
= u 3 (u 2 +1)du [u =tanθ, du =sec 2 θdθ]
= (u 5 + u 3 ) du = 1 6 u6 + 1 4 u4 + C = 1 6 tan6 θ + 1 4 tan4 θ + C

F.
302 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
35. Let u = x, dv =secx tan xdx ⇒ du = dx, v =secx. Then
x sec x tan xdx= x sec x −
sec xdx= x sec x − ln |sec x +tanx| + C.
37.
π/2
π/6 cot2 xdx= π/2
π/6 (csc2 x − 1) dx = − cot x − x π/2
= √ √
0 − π π/6 2 − − 3 −
π
6 = 3 −
π
3
39. cot 3 α csc 3 αdα = cot 2 α csc 2 α · csc α cot αdα= (csc 2 α − 1) csc 2 α · csc α cot αdα
= (u 2 − 1)u 2 · (−du) [u =cscα, du = − csc α cot αdα]
= (u 2 − u 4 ) du = 1 3 u3 − 1 5 u5 + C = 1 3 csc3 α − 1 5 csc5 α + C
41. I = csc xdx =
csc x (csc x − cot x)
csc x − cot x
dx =
− csc x cot x +csc 2 x
csc x − cot x
du =(− csc x cot x +csc 2 x) dx. ThenI = du/u =ln|u| =ln|csc x − cot x| + C.
dx. Letu =cscx − cot x
⇒
43.
2a
sin 8x cos 5xdx = 1
[sin(8x − 5x)+sin(8x +5x)] dx = 1
2 2 sin 3xdx+
1
2 sin 13xdx
= − 1 6
1
cos 3x − cos 13x + C
26
45.
47.
2b
sin 5θ sin θdθ = 1
[cos(5θ − θ) − cos(5θ + θ)] dθ = 1
2 2 cos 4θdθ−
1
2 cos 6θdθ=
1
sin 4θ − 1 sin 6θ + C
8 12
1 − tan 2 x
sec 2 x
cos
dx = 2 x − sin 2 x
dx =
cos 2xdx= 1 sin 2x + C
2
49. Let u =tan(t 2 ) ⇒ du =2t sec 2 (t 2 ) dt. Then t sec 2 (t 2 )tan 4 (t 2 ) dt = u 4 1
2 du = 1 10 u5 + C = 1 10 tan5 (t 2 )+C.
In Exercises 51–54, let f(x) denote the integrand and F (x) its antiderivative (with C =0).
51. Let u = x 2 ,sothatdu =2xdx.Then
x sin 2 (x 2 ) dx = sin 2 u 1
du
2
= 1 1
2 2
(1 − cos 2u) du
= 1 4 u −
1
sin 2u + C = 1 u − 1 1 · 2sinu cos u + C
2 4 4 2
= 1 4 x2 − 1 4 sin(x2 )cos(x 2 )+C
We see from the graph that this is reasonable, since F increases where f is positive and F decreases where f is negative.
Note also that f is an odd function and F is an even function.
53. sin 3x sin 6xdx = 1
[cos(3x − 6x) − cos(3x +6x)] dx
2
(cos 3x − cos 9x) dx
= 1 2
= 1 6
1
sin 3x − sin 9x + C
18
Notice that f(x) =0whenever F has a horizontal tangent.
55. f ave = 1 π
2π −π sin2 x cos 3 xdx= 1 π
2π −π sin2 x (1 − sin 2 x)cosxdx
= 1
2π
=0
0
0 u2 (1 − u 2 ) du
[where u =sinx]

F.
TX.10
SECTION 7.2 TRIGONOMETRIC INTEGRALS ¤ 303
57. A =
=2
π/4
−π/4
π/4
0
=1− 0=1
π/4
(cos 2 x − sin 2 x) dx =
−π/4
cos 2xdx
π/4
1
cos 2xdx=2
2 sin 2x = sin 2x π/4
0
0
59. It seems from the graph that 2π
0
cos 3 xdx=0, since the area below the
x-axis and above the graph looks about equal to the area above the axis and
below the graph. By Example 1, the integral is sin x − 1 3 sin3 x 2π
0
=0.
Note that due to symmetry, the integral of any odd power of sin x or cos x
between limits which differ by 2nπ (n any integer) is 0.
61. Using disks, V = π
π π/2 sin2 xdx= π π 1
(1 − cos 2x) dx = π 1
x − 1 sin 2x π
= π π
− 0 − π +0 = π2
π/2 2 2 4 π/2 2 4 4
63. Using washers,
V = π/4
π (1 − sin x) 2 − (1 − cos x) 2 dx
0
= π π/4
0 (1 − 2sinx +sin 2 x) − (1 − 2cosx +cos 2 x) dx
= π π/4
(2 cos x − 2sinx +sin 2 x − cos 2 x) dx
0
= π π/4
(2 cos x − 2sinx − cos 2x) dx = π 2sinx +2cosx − 1 sin 2x π/4
0 2 0
= π √ 2+ √ √
2 − 1 2 − (0 + 2 − 0) = π 2 2 −
5
2
65. s = f(t) = t
sin ωu 0 cos2 ωu du. Lety =cosωu ⇒ dy = −ω sin ωu du. Then
s = − 1 cos ωt
y 2 dy = − 1 1 y3 cos ωt
= 1 (1 − ω 1 ω 3 3ω cos3 ωt).
67. Just note that the integrand is odd [f(−x) =−f(x)].
69.
Or: If m 6= n, calculate
π
−π
1
π
1
sin mx cos nx dx = [sin(m − n)x +sin(m + n)x] dx = 1
cos(m − n)x
− −
2
−π
2 m − n
If m = n,thenthefirst term in each set of brackets is zero.
π cos mx cos nx dx = π 1
[cos(m − n)x +cos(m + n)x] dx.
−π −π 2
If m 6= n, this is equal to 1 2
sin(m − n)x
m − n
+
π
sin(m + n)x
=0.
m + n
−π
If m = n,weget π 1
[1 + cos(m + n)x] dx = π
1
x π sin(m + n)x
+ −π 2 2
= π +0=π.
−π
2(m + n)
−π
π
cos(m + n)x
=0
m + n
−π

F.
304 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
7.3 Trigonometric Substitution
1. Let x =3secθ,where0 ≤ θ< π 2 or π ≤ θ< 3π 2 .Then
dx =3secθ tan θdθand
√
x2 − 9= √ 9sec 2 θ − 9= 9(sec 2 θ − 1) = √ 9tan 2 θ
=3|tan θ| =3tanθ for the relevant values of θ.
1
x √ 2 x 2 − 9 dx =
1
9sec 2 θ · 3tanθ 3secθ tan θdθ= 1 9
cos θdθ= 1 sin θ + C = 1 √
x2 − 9
+ C
9
9 x
Note that − sec(θ + π) =secθ,sothefigure is sufficient for the case π ≤ θ< 3π 2 .
3. Let x =3tanθ,where− π 2

F.
TX.10
SECTION 7.3 TRIGONOMETRIC SUBSTITUTION ¤ 305
9. Let x =4tanθ,where− π 2 0, we don’t need the absolute value.)
11. Let 2x =sinθ,where− π ≤ θ ≤ π . Thenx = 1 sin θ, dx = 1 cos θdθ,
2 2 2 2
and √ 1 − 4x 2 = 1 − (2x) 2 =cosθ.
√
1 − 4x2 dx = cos θ 1
cos θ
dθ = 1 2 4 (1 + cos 2θ) dθ
= 1 4 θ +
1
sin 2θ + C = 1 (θ +sinθ cos θ)+C
2 4
= 1 4
sin −1 (2x)+2x √ 1 − 4x 2 + C
13. Let x =3secθ,where0 ≤ θ< π 2 or π ≤ θ< 3π 2 .Then
dx =3secθ tan θdθand √ x 2 − 9=3tanθ,so
√
x2 − 9
dx =
x 3
= 1 3
3tanθ
27 sec 3 θ 3secθ tan θdθ= 1 3
sin 2 θdθ= 1 3
= 1 6 sec−1 x
3
tan 2 θ
sec 2 θ dθ
1 (1 − cos 2θ) dθ = 1 θ − 1 sin 2θ + C = 1 θ − 1 2 6 12 6 6
sin θ cos θ + C
− 1 √
x2 − 9 3
6 x x + C = 1 x
√
x2 − 9
6 sec−1 − + C
3 2x 2
15. Let x = a sin θ, dx = a cos θdθ, x =0 ⇒ θ =0and x = a ⇒ θ = π 2 .Then
√ a
0 x2 a 2 − x 2 dx = π/2
a 2 sin 2 θ (a cos θ) a cos θdθ= a 4 π/2
sin 2 θ cos 2 θdθ= a 4 π/2
1 (2 sin θ cos 0 0 0 2 θ)2 dθ
θ − 1 π/2
4 sin 4θ
= a4
4
π/2
0
sin 2 2θdθ= a4
4
= a4 π
8 2 − 0 − 0 = π 16 a4
17. Let u = x 2 − 7,sodu =2xdx.Then
π/2
0
1
2
x
√
x2 − 7 dx = 1
2
(1 − cos 4θ) dθ =
a4
8
1
√ u
du = 1 2 · 2 √ u + C = x 2 − 7+C.
0

F.
306 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
19. Let x =tanθ,where− π 2

F.
√ 3
x
√
x2 + x +1 dx = tan θ − √ 1
2 2 3
√
3
sec θ 2 sec2 θdθ
2
= √
3
2 tan θ − 1 2
TX.10
sec θdθ= √
3
tan θ sec θdθ− 1
2 2
sec θdθ
SECTION 7.3 TRIGONOMETRIC SUBSTITUTION ¤ 307
= √ 3
2 sec θ − 1 2 ln |sec θ +tanθ| + C 1
= √ x 2 + x +1− 1 2 ln 2 √3
√
x2 + x +1+ 2 √
3
x +
1
2
+ C1
= √ x 2 + x +1− 1 2 ln 2 √3
√
x2 + x +1+ x + 1 2
+ C1
= √ x 2 + x +1− 1 2 ln 2 √
3
− 1 2 ln √ x 2 + x +1+x + 1 2
+ C1
= √ x 2 + x +1− 1 2 ln√ x 2 + x +1+x + 1 2
+ C, whereC = C1 − 1 2 ln 2 √
3
27. x 2 +2x =(x 2 +2x +1)− 1=(x +1) 2 − 1. Letx +1=1secθ,
so dx =secθ tan θdθand √ x 2 +2x =tanθ. Then
√
x2 +2xdx= tan θ (sec θ tan θdθ)= tan 2 θ sec θdθ
= (sec 2 θ − 1) sec θdθ= sec 3 θdθ− sec θdθ
= 1 sec θ tan θ + 1 ln |sec θ +tanθ| − ln |sec θ +tanθ| + C
2 2
= 1 sec θ tan θ − 1 ln |sec θ +tanθ| + C = 1 (x +1)√ 2 2 2
x 2 +2x − 1 ln √ 2 x +1+ x2 +2x + C
29. Let u = x 2 , du =2xdx.Then
x
√
1 − x4 dx = √ 1 − u 2 1
2 du = 1 2
= 1 2
where u =sinθ, du =cosθdθ,
cos θ · cos θdθ
and √ 1 − u 2 =cosθ
1 (1 + cos 2θ) dθ = 1 θ + 1 sin 2θ + C = 1 θ + 1 sin θ cos θ + C
2 4 8 4 4
= 1 4 sin−1 u + 1 4 u √ 1 − u 2 + C = 1 4 sin−1 (x 2 )+ 1 4 x2 √ 1 − x 4 + C
31. (a) Let x = a tan θ,where− π 2

F.
308 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
35. Area of 4POQ = 1 2 (r cos θ)(r sin θ) = 1 2 r2 sin θ cos θ. AreaofregionPQR = r
r cos θ
√
r2 − x 2 dx.
Let x = r cos u ⇒ dx = −r sin udufor θ ≤ u ≤ π . Then we obtain
2
√
r2 − x 2 dx = r sin u (−r sin u) du = −r 2 sin 2 udu= − 1 2 r2 (u − sin u cos u)+C
so area of region PQR= 1 2
= − 1 2 r2 cos −1 (x/r)+ 1 2 x √ r 2 − x 2 + C
−r 2 cos −1 (x/r)+x √ r 2 − x 2 r
r cos θ
= 1 2
0 − (−r 2 θ + r cos θrsin θ) = 1 2 r2 θ − 1 2 r2 sin θ cos θ
and thus, (area of sector POR)=(area of 4POQ)+(area of region PQR)= 1 2 r2 θ.
37. From the graph, it appears that the curve y = x 2 √ 4 − x 2 and the
line y =2− x intersectataboutx = a ≈ 0.81 and x =2, with
x √ 2 4 − x 2 > 2 − x on (a, 2). So the area bounded by the curve and the line is
A ≈ 2
√
a x
2
4 − x 2 − (2 − x) dx = 2
√
a x2 4 − x 2 dx − 2x − 1 x2 2
. 2 a
To evaluate the integral, we put x =2sinθ,where− π 2 ≤ θ ≤ π 2 .Then
dx =2cosθdθ, x =2 ⇒ θ =sin −1 1= π ,andx = a ⇒ θ = α 2 =sin−1 (a/2) ≈ 0.416.So
2 x2√ 4 − x
a 2 dx ≈ π/2
4sin 2 θ (2 cos θ)(2 cos θdθ)=4 π/2
sin 2 2θdθ=4 π/2 1
(1 − cos 4θ) dθ
α α α 2
=2 θ − 1 4 sin 4θ π/2
α
Thus, A ≈ 2.81 − 2 · 2 − 1 2 · 22 − 2a − 1 2 a2 ≈ 2.10.
=2 π
2 − 0 − α − 1 4 (0.996) ≈ 2.81
39. (a) Let t = a sin θ, dt = a cos θdθ, t =0 ⇒ θ =0and t = x ⇒ θ =sin −1 (x/a). Then
x
0
sin
−1 (x/a)
a2 − t 2 dt = a cos θ (a cos θdθ)
0
= a 2 sin −1 (x/a)
0
cos 2 θdθ= a2
2
=
θ a2
sin
−1 (x/a)
+ 1 2
2
sin 2θ = a2
0
2
√
a2 − x 2
= a2
2
sin −1 x
a
+ x a ·
= 1 2 a2 sin −1 (x/a)+ 1 2 x √ a 2 − x 2
a
sin −1 (x/a)
0
(1 + cos 2θ) dθ
sin
−1 (x/a)
θ +sinθ cos θ
0
− 0
(b) The integral x
√
a2 − t
0
2 dt represents the area under the curve y = √ a 2 − t 2 between the vertical lines t =0and t = x.
The figure shows that this area consists of a triangular region and a sector of the circle t 2 + y 2 = a 2 . The triangular region
has base x and height √ a 2 − x 2 , so its area is 1 2 x √ a 2 − x 2 . The sector has area 1 2 a2 θ = 1 2 a2 sin −1 (x/a).

F.
TX.10
SECTION 7.3 TRIGONOMETRIC SUBSTITUTION ¤ 309
41. Let the equation of the large circle be x 2 + y 2 = R 2 . Then the equation of
the small circle is x 2 +(y − b) 2 = r 2 ,whereb = √ R 2 − r 2 is the distance
between the centers of the circles. The desired area is
A = r
√
−r b + r2 − x 2 − √ R 2 − x 2 dx
=2 r √
0 b + r2 − x 2 − √ R 2 − x 2 dx
=2 r
bdx+2 r
0 0
√
r2 − x 2 dx − 2 r
0
√
R2 − x 2 dx
The firstintegralisjust2br =2r √ R 2 − r 2 . The second integral represents the area of a quarter-circle of radius r,soitsvalue
is 1 4 πr2 . To evaluate the other integral, note that
Thus, the desired area is
√
a2 − x 2 dx = a 2 cos 2 θdθ [x = a sin θ, dx = a cos θdθ] = 1
2 a2 (1 + cos 2θ) dθ
= 1 2 a2 θ + 1 2 sin 2θ + C = 1 2 a2 (θ +sinθ cos θ)+C
= a2 x
2 arcsin + a2 x
√ a 2 − x 2
+ C = a2 x
a 2 a a
2 arcsin + x √
a2 − x
a 2 2 + C
A =2r √ R 2 − r 2 +2 1
4 πr2 − R 2 arcsin(x/R)+x √ R 2 − x 2 r
0
=2r √ R 2 − r 2 + 1 2 πr2 − R 2 arcsin(r/R)+r √ R 2 − r 2 = r √ R 2 − r 2 + π 2 r2 − R 2 arcsin(r/R)
43. We use cylindrical shells and assume that R>r. x 2 = r 2 − (y − R) 2 ⇒ x = ± r 2 − (y − R) 2 ,
so g(y) =2 r 2 − (y − R) 2 and
V = R+r
2πy · 2 r
R−r 2 − (y − R) 2 dy = r
4π(u + R)√ r
−r 2 − u 2 du [where u = y − R]
=4π r
u √ r
−r 2 − u 2 du +4πR
r
√
r2 − u
−r 2 where u = r sin θ , du = r cos θdθ
du
in the second integral
r
=4π − 1 3 (r2 − u 2 ) 3/2 +4πR π/2
−π/2 r2 cos 2 θdθ= − 4π (0 − 0) + 3 4πRr2 π/2
−π/2 cos2 θdθ
−r
=2πRr 2 π/2
−π/2 (1 + cos 2θ) dθ =2πRr2 θ + 1 2 sin 2θ π/2
−π/2 =2π2 Rr 2
Another method: Use washers instead of shells, so V =8πR r
r2 − y
0 2 dy as in Exercise 6.2.63(a), but evaluate the
integral using y = r sin θ.

F.
310 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
7.4 Integration of Rational Functions by Partial Fractions
1. (a)
(b)
3. (a)
(b)
5. (a)
(b)
7.
9.
11.
2x
(x + 3)(3x +1) = A
x +3 + B
3x +1
1
x 3 +2x 2 + x = 1
x(x 2 +2x +1) = 1
x(x +1) 2 = A x +
x 4 +1
x 5 +4x = x4 +1
3 x 3 (x 2 +4) = A x + B x + C 2 x + Dx + E
3 x 2 +4
B
x +1 +
1
(x 2 − 9) 2 = 1
[(x +3)(x − 3)] 2 = 1
(x +3) 2 (x − 3) 2 = A
x +3 +
x 4
x 4 − 1 = (x4 − 1) + 1
=1+ 1
x 4 − 1 x 4 − 1
=1+
C
(x +1) 2
[or use long division] =1+
1
(x − 1)(x +1)(x 2 +1) =1+ A
x − 1 + B
x +1 + Cx + D
x 2 +1
t 4 + t 2 +1
(t 2 +1)(t 2 +4) 2 = At + B
t 2 +1 + Ct + D
t 2 +4 + Et + F
(t 2 +4) 2
B
(x +3) 2 + C
x − 3 +
x (x − 6) + 6
x − 6 dx = dx = 1+ 6
dx = x +6ln|x − 6| + C
x − 6
x − 6
1
(x 2 − 1)(x 2 +1)
D
(x − 3) 2
x − 9
(x +5)(x − 2) = A
x +5 + B . Multiply both sides by (x +5)(x − 2) to get x − 9=A(x − 2) + B(x +5)(∗),or
x − 2
equivalently, x − 9=(A + B)x − 2A +5B. Equating coefficients of x on each side of the equation gives us 1=A + B (1)
and equating constants gives us −9 =−2A +5B (2). Adding two times (1)to(2)givesus−7 =7B ⇔ B = −1 and
hence, A =2. [Alternatively, to find the coefficients A and B, we may use substitution as follows: substitute 2 for x in (∗) to
get −7 =7B ⇔ B = −1, then substitute −5 for x in (∗) to get −14 = −7A ⇔ A =2.] Thus,
x − 9
2
(x +5)(x − 2) dx = x +5 + −1
dx =2ln|x +5| − ln |x − 2| + C.
x − 2
1
x 2 − 1 = 1
(x +1)(x − 1) = A
x +1 + B . Multiply both sides by (x +1)(x − 1) to get 1=A(x − 1) + B(x +1).
x − 1
Substituting 1 for x gives 1=2B ⇔ B = 1 2 . Substituting −1 for x gives 1=−2A ⇔ A = − 1 2 . Thus,
3
2
13.
1
3
x 2 − 1 dx =
ax
x 2 − bx dx =
2
−1/2
x +1 + 1/2
dx = − 1 2
x − 1
ln |x +1| + 1 ln |x − 1| 3
2 2
= − 1 2 ln 4 + 1 2 ln 2 − − 1 2 ln 3 + 1 2 ln 1 = 1 2 (ln2+ln3− ln 4) or
1
2 ln 3 2
ax
x(x − b) dx =
a
dx = a ln |x − b| + C
x − b

F.
SECTIONTX.10
7.4 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS ¤ 311
15.
x 3 − 2x 2 − 4
x 3 − 2x 2 =1+
−4
x 2 (x − 2) .Write
−4
x 2 (x − 2) = A x + B x 2 +
C
x − 2 . Multiplying both sides by x2 (x − 2) gives
−4 =Ax(x − 2) + B(x − 2) + Cx 2 . Substituting 0 for x gives −4 =−2B ⇔ B =2. Substituting 2 for x gives
−4 =4C ⇔ C = −1. Equating coefficients of x 2 ,weget0=A + C,soA =1.Thus,
4
3
x 3 − 2x 2 − 4
x 3 − 2x 2 dx =
4
3
1+ 1 x + 2 x − 1
dx = x +ln|x| − 2 4 2 x − 2
x − ln |x − 2| 3
= 4+ln4− 1 2 − ln 2 − 3+ln3− 2 3 − 0 = 7 6 +ln2 3
17.
4y 2 − 7y − 12
y(y +2)(y − 3) = A y +
B
y +2 +
C
y − 3
⇒
4y 2 − 7y − 12 = A(y +2)(y − 3) + By(y − 3) + Cy(y +2).Setting
y =0gives −12 = −6A,soA =2.Settingy = −2 gives 18 = 10B,soB = 9 5 .Settingy =3gives 3=15C,soC = 1 5 .
Now
2
1
4y 2
− 7y − 12
2
y(y +2)(y − 3) dy =
1
2
y + 9/5
y +2 + 1/5
dy = 2ln|y| + 9
y − 3
ln |y +2| + 1 ln |y − 3| 2
5 5 1
=2ln2+ 9 5 ln 4 + 1 5 ln 1 − 2ln1− 9 5 ln 3 − 1 5 ln 2
=2ln2+ 18 ln 2 − 1 ln 2 − 9 27
ln 3 = ln 2 − 9 ln 3 = 9 (3 ln 2 − ln 3) = 9 ln 8 5 5 5 5 5 5 5 3
19.
1
(x +5) 2 (x − 1) = A
x +5 + B
(x +5) 2 + C
x − 1
⇒ 1=A(x +5)(x − 1) + B(x − 1) + C(x +5) 2 .
Setting x = −5 gives 1=−6B, soB = − 1 .Settingx =1gives 1=36C, soC = 1 6 36
.Settingx = −2 gives
1=A(3)(−3) + B(−3) + C 3 2 = −9A − 3B +9C = −9A + 1 + 1 = −9A + 3 ,so9A = − 1 and A = − 1 .Now
2 4 4 4 36
1
−1/36
(x +5) 2 (x − 1) dx = x +5 − 1/6
(x +5) 2 + 1/36
dx = − 1
x − 1 36 ln |x +5| + 1
6(x +5) + 1 ln |x − 1| + C.
36
21. x
x 2 +4 x 3 + 0x 2 + 0x + 4
x 3 + 4x
−4x +4
x 3 +4
x 2 +4 dx =
x +
By long division, x3 +4 −4x +4
= x +
x 2 +4 x 2 +4 .Thus,
−4x +4
dx = x −
x 2 +4
= 1 2 x2 − 4 · 1
2 ln x 2 +4 +4· 1
2 tan−1 x
2
4x
x 2 +4 + 4
dx
x 2 +2 2
+ C = 1 2 x2 − 2ln(x 2 +4)+2tan −1 x
2
+ C
23.
5x 2 +3x − 2
= 5x2 +3x − 2
= A x 3 +2x 2 x 2 (x +2) x + B x + C
2 x +2 . Multiply by x2 (x +2)to
get 5x 2 +3x − 2=Ax(x +2)+B(x +2)+Cx 2 .Setx = −2 to get C =3,andtake
x =0to get B = −1. Equating the coefficients of x 2 gives 5=A + C ⇒ A =2.So
5x 2
+3x − 2 2
dx =
x 3 +2x 2 x − 1 x + 3
dx =2ln|x| + 1 +3ln|x +2| + C.
2 x +2
x

F.
312 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
25.
10
(x − 1)(x 2 +9) = A
x − 1 + Bx + C
x 2 +9 . Multiply both sides by (x − 1) x 2 +9 to get
10 = A x 2 +9 +(Bx + C)(x − 1) (). Substituting 1 for x gives 10 = 10A ⇔ A =1.Substituting0 for x gives
10 = 9A − C ⇒ C =9(1)− 10 = −1. The coefficients of the x 2 -terms in () must be equal, so 0=A + B ⇒
B = −1. Thus,
10
1
(x − 1)(x 2 +9) dx = x − 1 + −x − 1 1
dx =
x 2 +9
x − 1 − x
x 2 +9 − 1
dx
x 2 +9
27.
=ln|x − 1| − 1 2 ln(x2 +9)− 1 3 tan−1 x
3
+ C
In the second term we used the substitution u = x 2 +9and in the last term we used Formula 10.
x 3 + x 2 +2x +1
(x 2 +1)(x 2 +2) = Ax + B
x 2 +1 + Cx + D
x 2 +2 . Multiply both sides by x 2 +1 x 2 +2 to get
x 3 + x 2 +2x +1=(Ax + B) x 2 +2 +(Cx + D) x 2 +1
x 3 + x 2 +2x +1= Ax 3 + Bx 2 +2Ax +2B + Cx 3 + Dx 2 + Cx + D
x 3 + x 2 +2x +1=(A + C)x 3 +(B + D)x 2 +(2A + C)x +(2B + D). Comparing coefficients gives us the following
system of equations:
⇔
A + C =1 (1) B + D =1 (2)
2A + C =2 (3) 2B + D =1 (4)
Subtracting equation (1) from equation (3) gives us A =1,soC =0. Subtracting equation (2) from equation (4) gives us
x 3 + x 2
+2x +1
x
B =0,soD =1.Thus,I =
(x 2 +1)(x 2 +2) dx = x 2 +1 + 1
x
so du =2xdxand then
x 2 +1 dx = 1 2
Formula 10 with a = √
2.So
1
x 2 +2 dx =
Thus, I = 1 2 ln x 2 +1 + 1 √
2
tan −1 x √2 + C.
x
dx. For
x 2 +2
x 2 +1 dx,letu = x2 +1
1
u du = 1 2 ln |u| + C = 1 2 ln x 2 +1 + C. For
⇔
1
x 2 + √ 2 2 dx = 1 √
2
tan −1 x √2 + C.
1
x 2 +2 dx,use
29.
x +4
x 2 +2x +5 dx = x +1
x 2 +2x +5 dx + 3
x 2 +2x +5 dx = 1
2
= 1 2 ln x 2 +2x +5
2 du
+3
4(u 2 +1)
(2x +2)dx
x 2 +2x +5 +
where x +1=2u,
and dx =2du
3 dx
(x +1) 2 +4
= 1 2 ln(x2 +2x +5)+ 3 2 tan−1 u + C = 1 2 ln(x2 +2x +5)+ 3 2 tan−1 x +1
2
+ C
31.
1
x 3 − 1 = 1
(x − 1)(x 2 + x +1) = A
x − 1 + Bx + C
x 2 + x +1
⇒ 1=A x 2 + x +1 +(Bx + C)(x − 1).
Take x =1to get A = 1 3 .Equatingcoefficients of x2 and then comparing the constant terms, we get 0= 1 3 + B, 1= 1 3 − C,

F.
SECTIONTX.10
7.4 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS ¤ 313
so B = − 1 3 , C = − 2 3
⇒
1
1
x 3 − 1 dx = 3
−
1
x − 1 dx + x − 2 3 3
x 2 + x +1 dx = 1 ln |x − 1| − 1
3
3
= 1 ln |x − 1| − 1
x +1/2
3
3 x 2 + x +1 dx − 1
3
(3/2) dx
(x +1/2) 2 +3/4
x +2
x 2 + x +1 dx
= 1 3 ln |x − 1| − 1 6 ln x 2 + x +1 − 1 2
2√3
tan −1
x +
1
2
√
3
2
+ K
= 1 ln |x − 1| − 1 3 6 ln(x2 + x +1)− √ 1
3
tan −1 √
1
3
(2x +1) + K
33. Let u = x 4 +4x 2 +3, so that du =(4x 3 +8x) dx =4(x 3 +2x) dx, x =0 ⇒ u =3,andx =1 ⇒ u =8.
35.
37.
1
Then
0
x 3
+2x
8
x 4 +4x 2 +3 dx =
3
1 1
u 4 du = 1 8
ln |u|
4 = 1 3
4 (ln 8 − ln 3) = 1 4 ln 8 3 .
1
x(x 2 +4) 2 = A x + Bx + C
x 2 +4 + Dx + E
(x 2 +4) 2 ⇒ 1=A(x 2 +4) 2 +(Bx + C)x(x 2 +4)+(Dx + E)x. Setting x =0
gives 1=16A,soA = 1 . Now compare coefficients.
16
1= 1
16 (x4 +8x 2 +16)+(Bx 2 + Cx)(x 2 +4)+Dx 2 + Ex
1= 1
16 x4 + 1 2 x2 +1+Bx 4 + Cx 3 +4Bx 2 +4Cx + Dx 2 + Ex
1= 1
16 + B x 4 + Cx 3 + 1
2 +4B + D x 2 +(4C + E)x +1
So B + 1 =0 ⇒ B = − 1 , C =0, 1 +4B + D =0 ⇒ D = − 1 16 16 2 4
,and4C + E =0 ⇒ E =0.Thus,
dx
1
x(x 2 +4) = 16
2 x + − 1 x 16
x 2 +4 + − 1 x
4
dx = 1 (x 2 +4) 2 16 ln |x| − 1 16 · 1
2 ln x 2 +4
1 − − 1 1
4 2 x 2 +4 + C
= 1 16
1
1
ln |x| −
32 ln(x2 +4)+
8(x 2 +4) + C
x 2 − 3x +7
(x 2 − 4x +6) = Ax + B
2 x 2 − 4x +6 + Cx + D
⇒ x 2 − 3x +7=(Ax + B)(x 2 − 4x +6)+Cx + D ⇒
(x 2 − 4x +6) 2
x 2 − 3x +7=Ax 3 +(−4A + B)x 2 +(6A − 4B + C)x +(6B + D). SoA =0, −4A + B =1 ⇒ B =1,
6A − 4B + C = −3 ⇒ C =1, 6B + D =7 ⇒ D =1.Thus,
I =
=
x 2
− 3x +7
(x 2 − 4x +6) dx = 1
2 x 2 − 4x +6 + x +1
dx
(x 2 − 4x +6) 2
1
(x − 2) 2 +2 dx + x − 2
(x 2 − 4x +6) dx + 3
2 (x 2 − 4x +6) dx 2
= I 1 + I 2 + I 3.
[continued]

F.
314 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
I 1 =
1
(x − 2) 2 + √ 2 2 dx = √ 1
tan −1 x − 2
√ + C 1
2 2
I 2 = 1
2x − 4
2 (x 2 − 4x +6) dx = 1 1 2 2 u du = 1 −
2
1
1
+ C 2 2 = −
u
2(x 2 − 4x +6) + C 2
I 3 =3
= 3 √ 2
4
1
(x − 2) 2 + √ 2 2 2 dx =3
sec 2 θ
sec 4 θ dθ = 3 √
2
4
1
[2(tan 2 θ +1)] 2 √
2sec 2 θdθ
cos 2 θdθ= 3 √ 2
4
1
(1 + cos 2θ) dθ
2
= 3 √ 2 θ +
1
2
8
sin 2θ + C 3 = 3 √
2 x − 2
tan −1 √ + 3 √ 2
8
2 8
= 3 √ 2
8
= 3 √ 2
8
x − 2
tan −1 √ + 3 √ 2
·
2 8
x − 2
√
x2 − 4x +6 ·
x − 2
tan −1 3(x − 2)
√ +
2 4(x 2 − 4x +6) + C3
So I = I 1 + I 2 + I 3 [C = C 1 + C 2 + C 3]
= 1 √
2
tan −1 x − 2
√
2
+
√
4 2
=
8
+ 3 √
2 x − 2
tan −1 √ +
8
2
TX.10
√
2
√
x2 − 4x +6 + C 3
−1
2(x 2 − 4x +6) + 3 √
2 x − 2
tan −1 √ +
8
2
x − 2= √ 2tanθ,
dx = √ 2sec 2 θdθ
1
2 · 2sinθ cos θ + C 3
3(x − 2)
4(x 2 − 4x +6) + C
3(x − 2) − 2
4(x 2 − 4x +6) + C = 7 √
2 x − 2
tan −1 √ +
8
2
3x − 8
4(x 2 − 4x +6) + C
39. Let u = √ x +1.Thenx = u 2 − 1, dx =2udu ⇒
√
dx
x √ x +1 = 2udu
(u 2 − 1) u =2 du
u 2 − 1 =ln u − 1
x +1− 1
u +1 + C =ln √ + C.
x +1+1
41. Let u = √ x,sou 2 = x and dx =2udu.Thus,
Multiply
16
9
√ x
x − 4 dx = 4
3
=2+8
u
4
u 2 − 4 2udu=2
4
3
du
(u +2)(u − 2) ()
3
u 2 4
u 2 − 4 du =2
3
1+ 4
du
u 2 − 4
[by long division]
1
(u +2)(u − 2) = A
u +2 + B by (u +2)(u − 2) to get 1=A(u − 2) + B(u +2). Equating coefficients we
u − 2
get A + B =0and −2A +2B =1. Solving gives us B = 1 and A = − 1 ,so 1
4 4
(u +2)(u − 2) = −1/4
u +2 + 1/4 and ()is
u − 2
2+8
4
3
−1/4
u +2 + 1/4
4
4
du =2+8 − 1
u − 2
ln |u +2| + 1 ln |u − 2| =2+ 2ln|u − 2| − 2ln|u +2|
4 4
3 3
=2+2
ln
u − 2
u +2
4
3
=2+2 ln 2 6 − ln 1 5
=2+2ln 5 or 2+ln
5 2
=2+ln 25
3 3
9
=2+2ln
2/6
1/5

F.
43. Let u = 3√ x 2 +1.Thenx 2 = u 3 − 1, 2xdx=3u 2 du ⇒
x 3
dx (u 3
3√
x2 +1 = − 1) 3 2 u2 du
= 3
(u 4 − u) du
u 2
SECTIONTX.10
7.4 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS ¤ 315
= 3 10 u5 − 3 4 u2 + C = 3 10 (x2 +1) 5/3 − 3 4 (x2 +1) 2/3 + C
45. If we were to substitute u = √ x, then the square root would disappear but a cube root would remain. On the other hand, the
substitution u = 3√ x would eliminate the cube root but leave a square root. We can eliminate both roots by means of the
substitution u = 6√ x. (Note that 6 is the least common multiple of 2 and 3.)
Let u = 6√ x.Thenx = u 6 ,sodx =6u 5 du and √ 3√
x = u 3 , x = u 2 .Thus,
dx 6u 5
√ √ 3
x − x = du
u 3 − u =6 u 5
2 u 2 (u − 1) du =6 u 3
u − 1 du
=6 u 2 + u +1+ 1
du [bylongdivision]
u − 1
=6 1
3 u3 + 1 2 u2 + u +ln|u − 1| + C =2 √ x +3 3√ x +6 6√ x +6ln 6√ x − 1 + C
47. Let u = e x .Thenx =lnu, dx = du u
⇒
e 2x
dx
e 2x +3e x +2 =
u 2
(du/u)
u 2 +3u +2 =
udu
−1
(u +1)(u +2) = u +1 + 2
du
u +2
49. Let u =tant, sothatdu =sec 2 tdt.Then
Now
1
(u +1)(u +2) = A
u +1 + B
u +2
=2ln|u +2| − ln |u +1| + C =ln (ex +2) 2
e x +1
+ C
sec 2
t
tan 2 t +3tant +2 dt =
⇒ 1=A(u +2)+B(u +1).
1
u 2 +3u +2 du =
1
(u +1)(u +2) du.
Setting u = −2 gives 1=−B, soB = −1. Setting u = −1 gives 1=A.
1
1
Thus,
(u +1)(u +2) du = u +1 − 1
du =ln|u +1| −ln |u +2|+C =ln|tan t +1|−ln |tan t +2|+C.
u +2
51. Let u =ln(x 2 − x +2), dv = dx. Thendu = 2x − 1 dx, v = x, and (by integration by parts)
x 2 − x +2
ln(x 2 − x +2)dx = x ln(x 2 2x 2
− x
− x +2)−
x 2 − x +2 dx = x ln(x2 − x +2)− 2+ x − 4
dx
x 2 − x +2
1
= x ln(x 2 (2x − 1)
2
− x +2)− 2x −
x 2 − x +2 dx + 7
dx
2 (x − 1 2 )2 + 7 4
= x ln(x 2 − x +2)− 2x − 1 2 ln(x2 − x +2)+ 7
√
⎡
7
du 2 ⎣
7
2
4 (u2 +1)
=(x − 1 2 )ln(x2 − x +2)− 2x + √ 7tan −1 u + C
=(x − 1 2 )ln(x2 − x +2)− 2x + √ 7tan −1 2x − 1
√ + C
7
where x − 1 2 = √ 7
2 u,
dx = √ 7
2 du,
(x − 1 2 )2 + 7 4 = 7 4 (u2 +1)
⎤
⎦

F.
316 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
53. From the graph, we see that the integral will be negative, and we guess
that the area is about the same as that of a rectangle with width 2 and
height 0.3,soweestimatetheintegraltobe−(2 · 0.3) = −0.6. Now
1
x 2 − 2x − 3 = 1
(x − 3)(x +1) = A
x − 3 + B ⇔
x +1
TX.10
1=(A + B)x + A − 3B,soA = −B and A − 3B =1 ⇔ A = 1 4
and B = − 1 , so the integral becomes
4
2
dx
0 x 2 − 2x − 3 = 1 2
dx
4 0 x − 3 − 1 2
dx
4 0 x +1 = 1
2
ln |x − 3| − ln |x +1| = 1
ln
4
0 4 x − 3
2 x +1
0
= 1 4 ln
1
− ln 3 = − 1 ln 3 ≈−0.55
3 2
dx
55.
x 2 − 2x = dx
(x − 1) 2 − 1 = du
[put u = x − 1]
u 2 − 1
= 1
2 ln u − 1
u +1 + C [by Equation 6] = 1
2 ln x − 2
x + C
x
57. (a) If t =tan ,then x 2 2 =tan−1 t.Thefigure gives
x
1
x
cos = √
2 and sin t
= √
1+t
2 2
. 1+t
2
(b) cos x =cos 2 · x x
=2cos 2 − 1
2
2
2
1
=2 √ − 1= 2 1 − t2
− 1= 1+t
2 1+t2 1+t 2
(c) x 2
=arctant ⇒ x = 2 arctan t ⇒ dx =
2 1+t dt 2
59. Let t =tan(x/2). Then, using the expressions in Exercise 57, we have
1
3sinx − 4cosx dx = 1
2t 1 − t
2
2 dt
1+t =2 dt
3 − 4
2 3(2t) − 4(1 − t 2 ) = dt
2t 2 +3t − 2
1+t 2 1+t 2
dt
2
=
(2t − 1)(t +2) = 1
5 2t − 1 − 1
1
dt [using partial fractions]
5 t +2
= 1 ln |2t − 1| − ln |t +2| + C = 1
5
5 ln 2t − 1
t +2 + C = 1
5 ln 2tan(x/2) − 1
tan (x/2) + 2 + C
61. Let t =tan(x/2). Then, by Exercise 57,
π/2
sin 2x
π/2
2t
0 2+cosx dx = 2sinx cos x
1 2 ·
0 2+cosx dx = 1+t · 1 − t2
8t(1 − t 2 )
2 1+t 2 2
1
0
2+ 1 − t2 1+t dt = (1 + t 2 ) 2
2 0 2(1 + t 2 )+(1− t 2 ) dt
1+t 2
1
1 − t 2
= 8t ·
(t 2 +3)(t 2 +1) dt = I 2
0

F.
If we now let u = t 2 ,then
SECTIONTX.10
7.4 INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS ¤ 317
1 − t 2
(t 2 +3)(t 2 +1) 2 = 1 − u
(u +3)(u +1) 2 = A
u +3 +
B
u +1 +
C
(u +1) 2 ⇒
1 − u = A(u +1) 2 + B(u +3)(u +1)+C(u +3).Setu = −1 to get 2=2C,soC =1.Setu = −3 to get 4=4A,so
A =1.Setu =0to get 1=1+3B +3,soB = −1. So
I =
1
0
8t
t 2 +3 −
8t
t 2 +1 +
8t
(t 2 +1) 2 dt = 4ln(t 2 +3)− 4ln(t 2 +1)− 4 1
t +1 2 0
=(4ln4− 4ln2− 2) − (4 ln 3 − 0 − 4) = 8 ln 2 − 4ln2− 4ln3+2=4ln 2 3 +2
63. By long division,
x 2 +1 3x +1
= −1+
3x − x2 3x − x .Now 2
3x +1 3x +1
=
3x − x2 x(3 − x) = A x + B
3 − x ⇒ 3x +1=A(3 − x)+Bx. Setx =3to get 10 = 3B,soB = 10 .Setx =0to
3
get 1=3A,soA = 1 3
. Thus, the area is
2
1
x 2
+1
2
1 10
3x − x dx = 3
−1+
2 x + 3
1
3 − x
dx = −x + 1 10
ln |x| − ln |3 − x| 2
3 3 1
= −2+ 1 3 ln 2 − 0 − −1+0− 10 3 ln 2 = −1+ 11 3 ln 2
65.
P + S
P [(r − 1)P − S] = A P + B
(r − 1)P − S
⇒ P + S = A [(r − 1)P − S]+BP =[(r − 1)A + B] P − AS ⇒
(r − 1)A + B =1, −A =1 ⇒ A = −1, B = r. Now
t =
so t = − ln P +
P + S
−1
P [(r − 1)P − S] dP = P + r
dP
dP = −
(r − 1)P − S
P + r
r − 1
r ln|(r − 1)P − S| + C. Herer =0.10 and S =900,so
r − 1
r − 1
(r − 1)P − S dP
t = − ln P + 0.1 ln|−0.9P − 900| + C = − ln P − 1 ln(|−1||0.9P + 900|) =− ln P − 1 ln(0.9P + 900) + C.
−0.9 9 9
When t =0, P =10,000,so0=− ln 10,000 − 1 ln(9900) + C. Thus,C =ln10,000 + 1 9 9
ln 9900 [≈ 10.2326], so our
equation becomes
10,000
ln(0.9P + 900) = ln
P
+ 1 9 ln 1100
0.1P + 100 =ln10,000 + 1 11,000
ln
P 9 P +1000
t =ln10,000 − ln P + 1 9 ln 9900 − 1 9
=ln 10,000
P
67. (a) In Maple, we define f(x),andthenuseconvert(f,parfrac,x); to obtain
f(x) = 24,110/4879
5x +2
+ 1 9 ln 9900
0.9P +900
− 668/323
2x +1 − 9438/80,155 (22,098x +48,935)/260,015
+
3x − 7
x 2 + x +5
In Mathematica, we use the command Apart,andinDerive,weuseExpand.

F.
318 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
(b) f(x) dx = 24,110 · 1
668
ln|5x +2| − · 1
9438
ln|2x +1| − · 1 ln |3x − 7|
4879 5 323 2 80,155 3
1 22,098 x +
1
2 +37,886
+
260,015 x +
1 2
dx + C
+ 19
2 4
= 24,110 · 1
4879 5
Using a CAS, we get
+ 1
260,015
668
ln|5x +2| − · 1
9438
ln|2x +1| − · 1
323 2 80,155 3
22,098 · 1 ln x 2 + x +5 +37,886 ·
2
= 4822
334
3146
ln|5x +2| − ln|2x +1| −
4879 323
+ 75,772
260,015 √ 19 tan−1 √
1
19
(2x +1)
4822 ln(5x +2)
4879
−
80,155
334 ln(2x +1)
323
+ C
−
+ 11,049 ln(x2 + x +5)
260,015
ln|3x − 7|
4
19 tan−1 √
1 x +
1
+ C
19/4 2
ln|3x − 7| +
11,049
260,015 ln x 2 + x +5
3146 ln(3x − 7)
80,155
+ 3988 √ 19
260,015 tan−1 √
19
19 (2x +1)
The main difference in this answer is that the absolute value signs and the constant of integration have been omitted. Also,
the fractions have been reduced and the denominators rationalized.
69. There are only finitely many values of x where Q(x) =0(assuming that Q is not the zero polynomial). At all other values of
x, F (x)/Q(x) =G(x)/Q(x),soF (x) =G(x). In other words, the values of F and G agree at all except perhaps finitely
many values of x. By continuity of F and G, the polynomials F and G must agree at those values of x too.
More explicitly: if a is a value of x such that Q(a) =0,thenQ(x) 6= 0for all x sufficiently close to a. Thus,
F (a)= lim
x→a
F (x) [by continuity of F ]
=lim
x→a
G(x) [whenever Q(x) 6= 0]
= G(a) [by continuity of G]
7.5 Strategy for Integration
1. Let u =sinx,sothatdu =cosxdx.Then cos x(1 + sin 2 x) dx = (1 + u 2 ) du = u + 1 3 u3 + C =sinx + 1 3 sin3 x + C.
3.
5.
sin x +secx
tan x
2
0
2t
−1
(t − 3) dt = 2
sin x
dx =
tan x + sec x
dx = (cos x +cscx) dx =sinx +ln|csc x − cot x| + C
tan x
−3
2(u +3)
u 2
du
u = t − 3,
du = dt
=
−1
=(2ln1+6)− (2 ln 3 + 2) = 4 − 2ln3 or 4 − ln 9
−3
2
u + 6
du = 2ln|u| − 6 −1
u 2 u
−3
7. Let u =arctany. Thendu = dy
1+y 2 ⇒
1
−1
e arctan y
1+y 2 dy = π/4
−π/4
e u du = e u π/4
−π/4 = eπ/4 − e −π/4 .

F.
3
9.
1 r4 ln rdr
11.
u =lnr,
du = dr
r
dv = r 4 dr,
v = 1 5 r5
TX.10
= 1
5 r5 ln r 3
− 3 1
1 1 5 r4 dr = 243 ln 3 − 0 − 1
r5 3
5 25 1
= 243 ln 3 − 243
− 1
5 25 25 =
243 242
5
ln 3 −
25
x − 1
(x − 2) + 1
u
x 2 − 4x +5 dx = (x − 2) 2 +1 dx = u 2 +1 + 1
du
u 2 +1
SECTION 7.5 STRATEGY FOR INTEGRATION ¤ 319
[u = x − 2, du = dx]
= 1 2 ln(u2 +1)+tan −1 u + C = 1 2 ln(x2 − 4x +5)+tan −1 (x − 2) + C
13. sin 3 θ cos 5 θdθ = cos 5 θ sin 2 θ sin θdθ= − cos 5 θ (1 − cos 2 θ)(− sin θ) dθ
= −
u 5 (1 − u 2 u =cosθ,
) du
du = − sin θdθ
= (u 7 − u 5 ) du = 1 8 u8 − 1 6 u6 + C = 1 8 cos8 θ − 1 6 cos6 θ + C
Another solution:
sin 3 θ cos 5 θdθ = sin 3 θ (cos 2 θ) 2 cos θdθ= sin 3 θ (1 − sin 2 θ) 2 cos θdθ
= u 3 (1 − u 2 ) 2 du
u =sinθ,
du =cosθdθ
= u 3 (1 − 2u 2 + u 4 ) du
= (u 3 − 2u 5 + u 7 ) du = 1 4 u4 − 1 3 u6 + 1 8 u8 + C = 1 4 sin4 θ − 1 3 sin6 θ + 1 8 sin8 θ + C
15. Let x =sinθ,where− π 2 ≤ θ ≤ π 2 .Thendx =cosθdθand (1 − x2 ) 1/2 =cosθ,
so
dx cos θdθ
(1 − x 2 ) = 3/2 (cos θ) = 3
sec 2 θdθ=tanθ + C =
x
√
1 − x
2 + C.
17. x sin 2 xdx
u = x,
du = dx
dv =sin 2 xdx,
v = sin 2 xdx = 1
2 (1 − cos 2x) dx = 1 2 x − 1 2 sin x cos x
= 1 2 x2 − 1 2 x sin x cos x − 1
2 x − 1 2 sin x cos x dx
= 1 2 x2 − 1 2 x sin x cos x − 1 4 x2 + 1 4 sin2 x + C = 1 4 x2 − 1 2 x sin x cos x + 1 4 sin2 x + C
Note: sin x cos xdx= sds= 1 2 s2 + C
[where s =sinx, ds =cosxdx].
A slightly different method is to write x sin 2 xdx= x · 1
2 (1 − cos 2x) dx = 1 2
the second integral by parts, we arrive at the equivalent answer 1 4 x2 − 1 x sin 2x − 1 4 8
cos 2x + C.
19. Let u = e x . Then e x+ex dx = e ex e x dx = e u du = e u + C = e ex + C.
xdx−
1
2 x cos 2xdx.Ifweevaluate
21. Let t = √ x,sothatt 2 = x and 2tdt = dx. Then arctan √ xdx = arctan t (2tdt)=I. Nowusepartswith
u =arctant, dv =2tdt ⇒ du = 1
1+t 2 dt, v = t2 . Thus,
I = t 2 arctan t −
t 2
1+t dt = 2 t2 arctan t − 1 − 1
dt = t 2 arctan t − t +arctant + C
1+t 2
= x arctan √ x − √ x +arctan √ x + C
or (x +1)arctan √ x − √
x + C

F.
320 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
23. Let u =1+ √ x.Thenx =(u − 1) 2 , dx =2(u − 1) du ⇒
1
1+ √ 8 2
x
0
dx =
1 u8 · 2(u − 1) du =2 2
1 (u9 − u 8 ) du = 1
5 u10 − 2 · 1 u9 2
= 1024 − 1024 − 1 + 2 = 4097 .
9 1 5 9 5 9 45
25.
3x 2 − 2
x 2 − 2x − 8 =3+ 6x +22
(x − 4)(x +2) =3+ A
x − 4 + B
x +2
⇒
6x +22=A(x +2)+B(x − 4). Setting
x =4gives 46 = 6A, soA = 23 .Settingx = −2 gives 10 = −6B, soB = − 5 .Now
3 3
3x 2
− 2
x 2 − 2x − 8 dx = 3+ 23/3
x − 4 − 5/3
dx =3x + 23 3
x +2
ln |x − 4| − 5 ln |x +2| + C.
3
27. Let u =1+e x ,sothatdu = e x dx =(u − 1) dx. Then
29.
1
u(u − 1) = A u +
Thus, I =
B
u − 1
−1
u + 1
u − 1
1
1
1+e dx = x u · du
u − 1 =
1
du = I. Now
u(u − 1)
⇒ 1=A(u − 1) + Bu. Setu =1to get 1=B. Setu =0to get 1=−A, soA = −1.
du = − ln |u| +ln|u − 1| + C = − ln(1 + e x )+lne x + C = x − ln(1 + e x )+C.
Another method: Multiply numerator and denominator by e −x and let u = e −x +1. This gives the answer in the
form − ln(e −x +1)+C.
5
3w − 1
5
w +2 dw = 3 − 7
5
dw = 3w − 7ln|w +2| =15− 7ln7+7ln2
w +2
0
0
0
=15+7(ln2− ln 7) = 15 + 7 ln 2 7
31. As in Example 5,
√ √
1+x 1+x 1+x
1 − x dx = √ · √ dx =
1 − x 1+x
Another method: Substitute u = (1 + x)/(1 − x).
1+x
√ dx = 1 − x
2
33. 3 − 2x − x 2 = −(x 2 +2x +1)+4=4− (x +1) 2 .Letx +1=2sinθ,
where − π ≤ θ ≤ π .Thendx =2cosθdθand
2 2
√
3 − 2x − x2 dx = 4 − (x +1) 2 dx =
4 − 4sin 2 θ 2cosθdθ
dx
√ + 1 − x
2
xdx
√
1 − x
2 =sin−1 x − 1 − x 2 + C.
=4 cos 2 θdθ=2 (1 + cos 2θ) dθ
=2θ +sin2θ + C =2θ +2sinθ cos θ + C
x +1
=2sin −1 +2· x +1
√
3 − 2x − x
2
·
+ C
2
2 2
x +1
=2sin −1 + x +1 √
3 − 2x − x2 + C
2 2
35. Because f(x) =x 8 sin x is the product of an even function and an odd function, it is odd.
Therefore, 1
−1 x8 sin xdx=0
[by (5.5.7)(b)].
37.
π/4
cos 2 θ tan 2 θdθ= π/4
sin 2 θdθ= π/4 1
(1 − cos 2θ) dθ = 1
θ − 1 sin 2θ π/4
= π
− 1
0 0 0 2 2 4 0 8 4 − (0 − 0) =
π
− 1 8 4

F.
39. Let u =secθ, sothatdu =secθ tan θdθ.Then
1
u(u − 1) = A u +
Thus, I =
B
u − 1
−1
u + 1
u − 1
TX.10
SECTION 7.5 STRATEGY FOR INTEGRATION ¤ 321
sec θ tan θ
sec 2 θ − sec θ dθ = 1
u 2 − u du = 1
du = I. Now
u(u − 1)
⇒ 1=A(u − 1) + Bu. Setu =1to get 1=B. Setu =0to get 1=−A, soA = −1.
du = − ln |u| +ln|u − 1| + C =ln|sec θ − 1| − ln |sec θ| + C [or ln |1 − cos θ| + C].
41. Let u = θ, dv =tan 2 θdθ= sec 2 θ − 1 dθ ⇒ du = dθ and v =tanθ − θ. So
θ tan 2 θdθ= θ(tan θ − θ) − (tan θ − θ) dθ = θ tan θ − θ 2 − ln |sec θ| + 1 2 θ2 + C
= θ tan θ − 1 2 θ2 − ln |sec θ| + C
43. Let u =1+e x ,sothatdu = e x dx. Then e x √ 1+e x dx = u 1/2 du = 2 3 u3/2 + C = 2 3 (1 + ex ) 3/2 + C.
Or:
Let u = √ 1+e x ,sothatu 2 =1+e x and 2udu = e x dx. Then
e
x √ 1+e x dx = u · 2udu= 2u 2 du = 2 3 u3 + C = 2 3 (1 + ex ) 3/2 + C.
45. Let t = x 3 .Thendt =3x 2 dx ⇒ I =
x 5 e −x3 dx = 1 3 te −t dt. Now integrate by parts with u = t, dv = e −t dt:
I = − 1 3 te−t + 1 3 e −t dt = − 1 3 te−t − 1 3 e−t + C = − 1 3 e−x3 (x 3 +1)+C.
47. Let u = x − 1,sothatdu = dx. Then
x 3 (x − 1) −4 dx = (u +1) 3 u −4 du = (u 3 +3u 2 +3u +1)u −4 du = (u −1 +3u −2 +3u −3 + u −4 ) du
=ln|u| − 3u −1 − 3 2 u−2 − 1 3 u−3 + C =ln|x − 1| − 3(x − 1) −1 − 3 2 (x − 1)−2 − 1 3 (x − 1)−3 + C
49. Let u = √ 4x +1 ⇒ u 2 =4x +1 ⇒ 2udu=4dx ⇒ dx = 1 2 udu.So
1
1
x √ 4x +1 dx = udu
2
1
4 (u2 − 1) u =2 √ =ln
4x +1− 1
√ + C
4x +1+1
du
u 2 − 1 =2 1
2
u − 1
ln u +1 + C [by Formula 19]
51. Let 2x =tanθ ⇒ x = 1 2 tan θ, dx = 1 2 sec2 θdθ, √ 4x 2 +1=secθ,so
53.
1
dx
x √ 4x 2 +1 = 2 sec2 θdθ sec θ
1
tan θ sec θ = tan θ dθ = csc θdθ
2
= − ln |csc θ +cotθ| + C [or ln |csc θ − cot θ| + C]
√ = − ln
4x2 +1
+ 1
√ 2x 2x
or + C ln
4x2 +1
− 1
2x 2x + C
x 2 sinh(mx)dx = 1 m x2 cosh(mx) − 2 m
x cosh(mx) dx
= 1 m x2 cosh(mx) − 2
1
m
m
x sinh(mx) − 1
sinh(mx) dx
m
= 1 m x2 cosh(mx) − 2 m 2 x sinh(mx)+ 2 m 3 cosh(mx)+C
u = x 2 ,
du =2xdx
U = x,
dU = dx
dv =sinh(mx) dx,
v = 1 m cosh(mx)
dV =cosh(mx) dx,
V = m 1 sinh(mx)

F.
322 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
55. Let u = √
x,sothatx = u 2 and dx =2udu.Then
TX.10
dx
x + x √ x =
2udu
u 2 + u 2 · u =
2
du = I.
u(1 + u)
2
Now
u(1 + u) = A u + B ⇒ 2=A(1 + u)+Bu. Setu = −1 to get 2=−B,soB = −2. Setu =0to get 2=A.
1+u
2
Thus, I =
u − 2
du =2ln|u| − 2ln|1+u| + C =2ln √
x − 2ln 1+ √
x + C.
1+u
57. Let u = 3√ x + c. Thenx = u 3 − c ⇒
x
3 √ x + cdx= (u 3 − c)u · 3u 2 du =3 (u 6 − cu 3 ) du = 3 7 u7 − 3 4 cu4 + C = 3 7 (x + c)7/3 − 3 4 c(x + c)4/3 + C
59. Let u =sinx,sothatdu =cosxdx.Then
cos x cos 3 (sin x) dx = cos 3 udu= cos 2 u cos udu= (1 − sin 2 u)cosudu
= (cos u − sin 2 u cos u) du =sinu − 1 3 sin3 u + C = sin(sin x) − 1 3 sin3 (sin x)+C
61. Let y = √ x so that dy = 1
2 √ x dx ⇒ dx =2√ xdy =2ydy.Then
√xe √ x
dx =
ye y (2ydy)=
=2y 2 e y −
4ye y dy
2y 2 e y dy
U =4y,
dU =4dy
u =2y 2 ,
du =4ydy
dV = e y dy,
V = e y
dv = e y dy,
v = e y
=2y 2 e y − 4ye y − 4e y dy =2y 2 e y − 4ye y +4e y + C
=2(y 2 − 2y +2)e y + C =2 x − 2 √
x +2 e √x + C
63. Let u =cos 2 x,sothatdu =2cosx (− sin x) dx. Then
sin 2x
1+cos 4 x dx =
2sinx cos x
1+(cos 2 x) dx = 2
1
1+u 2 (−du) =− tan−1 u + C = − tan −1 (cos 2 x)+C.
65.
√ √
dx
1 x +1− x x
√
√ √ = √ √ · √ √ dx = x +1− x dx
x +1+ x x +1+ x x +1− x
= 2 3
(x +1) 3/2 − x 3/2
+ C
67. Let x =tanθ,sothatdx =sec 2 θdθ, x = √ 3 ⇒ θ = π 3 ,andx =1 ⇒ θ = π 4 .Then
√ 3
1
√ 1+x
2
π/3
sec θ
π/3
sec θ (tan 2 θ +1)
dx =
x 2 π/4 tan 2 θ sec2 θdθ=
dθ =
π/4 tan 2 θ
=
=
π/3
π/3
π/4
sec θ tan 2 θ
tan 2 θ
+ sec θ
dθ
tan 2 θ
π/3
(sec θ +cscθ cot θ) dθ = ln |sec θ +tanθ| − csc θ
π/4
π/4
ln √ 2+ 3 − √3
2
− ln √ 2+1 √ √ − 2 = 2 − √3
2
+ln 2+ √ 3 − ln 1+ √ 2

F.
TX.10
SECTION 7.5 STRATEGY FOR INTEGRATION ¤ 323
69. Let u = e x .Thenx =lnu, dx = du/u ⇒
e 2x
1+e dx = x
u 2
du
1+u u =
71. Let θ =arcsinx,sothatdθ =
x +arcsinx
√
1 − x
2
u
1+u du = 1 − 1
du = u − ln|1+u| + C = e x − ln(1 + e x )+C.
1+u
1
√ dx and x =sinθ. Then
1 − x
2
dx = (sin θ + θ) dθ = − cos θ + 1 2 θ2 + C
= − √ 1 − x 2 + 1 2 (arcsin x)2 + C
73.
1
(x − 2)(x 2 +4) = A
x − 2 + Bx + C
x 2 +4
⇒
1=A(x 2 +4)+(Bx+ C)(x − 2) = (A + B)x 2 +(C − 2B)x +(4A − 2C).
So 0=A + B = C − 2B, 1=4A − 2C. Settingx =2gives A = 1 8
⇒ B = − 1 8 and C = − 1 4 .So
1
1
(x − 2)(x 2 +4) dx = 8
x − 2 + − 1 x − 1
8 4
dx = 1
x 2 +4 8
= 1 8
dx
x − 2 − 1
16
ln|x − 2| −
1
16 ln(x2 +4)− 1 8 tan−1 (x/2) + C
2xdx
x 2 +4 − 1
4
dx
x 2 +4
75. Let y = √ 1+e x ,sothaty 2 =1+e x , 2ydy= e x dx, e x = y 2 − 1,andx =ln(y 2 − 1). Then
xe x ln(y 2
√ dx = − 1)
(2ydy)=2 [ln(y +1)+ln(y − 1)] dy
1+e
x y
=2[(y +1)ln(y +1)− (y +1)+(y − 1) ln(y − 1) − (y − 1)] + C [by Example 7.1.2]
=2[y ln(y +1)+ln(y +1)− y − 1+y ln(y − 1) − ln(y − 1) − y +1]+C
=2[y(ln(y +1)+ln(y − 1)) + ln(y +1)− ln(y − 1) − 2y]+C
=2 y ln(y 2 − 1) + ln y +1
√
y − 1 − 2y √1+ex
+ C =2
ln(e x 1+ex +1
)+ln√ 1+ex − 1 − 2 √
1+e x + C
=2x √ √ 1+ex +1
1+e x +2ln√ 1+ex − 1 − 4 √ 1+e x + C =2(x − 2) √ √ 1+ex +1
1+e x +2ln√ 1+ex − 1 + C
77. Let u = x 3/2 so that u 2 = x 3 and du = 3 2 x1/2 dx ⇒ √ xdx= 2 du. Then
3
√ x
2
1+x dx = 3
3 1+u du = 2 2 3 tan−1 u + C = 2 3 tan−1 (x 3/2 )+C.
79. Let u = x, dv =sin 2 x cos xdx ⇒ du = dx, v = 1 3 sin3 x.Then
x sin 2 x cos xdx = 1 x 3 sin3 x − 1
3 sin3 xdx= 1 x
3 sin3 x − 1 3 (1 − cos 2 x)sinxdx
= 1 3 x sin3 x + 1
(1 − y 2 u =cosx,
) dy
3
du = − sin xdx
= 1 3 x sin3 x + 1 3 y − 1 9 y3 + C = 1 3 x sin3 x + 1 3 cos x − 1 9 cos3 x + C

F.
324 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
81. The function y =2xe x2 does have an elementary antiderivative, so we’ll use this fact to help evaluate the integral.
(2x 2 +1)e x2 dx = 2x 2 e x2 dx + e x2 dx = x
2xe x2 dx + e x2 dx
= xe x2 − e x2 dx + e x2 dx
u = x,
du = dx
dv= 2xe x2 dx,
v= e x2
= xe x2 + C
7.6 Integration Using Tables and Computer Algebra Systems
Keep in mind that there are several ways to approach many of these exercises, and different methods can lead to different forms of the answer.
1. We could make the substitution u = √ 2 x to obtain the radical √ 7 − u 2 andthenuseFormula33witha = √ 7.
Alternatively, we will factor √
7
2 out of the radical and use a = . 2
√
7 − 2x
2
dx = √ ⎡
⎤
7
2
2
− x2
dx = 33 √ 2⎣− 1
7
x 2 x 2 2
x
− x2 − sin −1 x
⎦ + C
= − 1 x
7
2
√
7 − 2x2 − √ 2sin −1
2
7 x
+ C
3. Let u = πx ⇒ du = πdx,so
sec 3 (πx) dx = 1 π sec 3 udu= 71 1 1 sec u tan u + 1 ln |sec u +tanu| + C
π 2 2
5.
1
0
= 1
1
sec πx tan πx + ln |sec πx +tanπx| + C
2π 2π
2x cos −1 xdx=2
91 2x 2 − 1
cos −1 x − x √ 1
1 − x 2
=2 1
4
4
· 0 − 0 4
− − 1 · π − 0 4 2
=2
π
8 =
π
4
0
7. Let u = πx,sothatdu = πdx.Then
tan 3 (πx) dx = tan 3 u 1
du
π
= 1 π tan 3 udu 69
= 1 1
π 2 tan2 u +ln|cos u| + C
= 1
2π tan2 (πx)+ 1 π
ln |cos (πx)| + C
9. Let u =2x and a =3.Thendu =2dx and
1
dx
x √ 2 4x 2 +9 = du
2
u 2 =2
√
u2 + a 2
11.
0
−1
4
√
4x2 +9
= −2
9 · 2x
√
du
u √ 28 a2 + u
= −2
2
+ C
2 a 2 + u 2 a 2 u
√
4x2 +9
+ C = − + C
9x
0
t 2 e −t dt =
97 1
−1 t2 e −t − 2 0
0
0
te −t dt = e +2 te −t dt 96 1
= e +2
−1
−1 −1
−1
(−1) (−t − 1) 2 e−t −1
= e +2 −e 0 +0 = e − 2
tan 3 (1/z)
13.
dz
z 2
u =1/z,
du = −dz/z 2
= −
tan 3 udu 69 = − 1 2 tan2 u − ln |cos u| + C
= − 1 tan2
1
2 z − ln cos
1 + C
z

F.
15. Let u = e x ,sothatdu = e x dx and e 2x = u 2 .Then
du
e 2x arctan(e x ) dx = u 2 arctan u
u
SECTION 7.6 TX.10 INTEGRATION USING TABLES AND COMPUTER ALGEBRA SYSTEMS ¤ 325
92
= u2 +1
2
=
u arctan udu
arctan u − u 2 + C = 1 2 (e2x +1)arctan(e x ) − 1 2 ex + C
17. Let z =6+4y − 4y 2 =6− (4y 2 − 4y +1)+1=7− (2y − 1) 2 , u =2y − 1,anda = √ 7.Thenz = a 2 − u 2 , du =2dy,
and
y 6+4y − 4y2 dy = y √ zdy= 1
(u +1)√ a
2 2 − u 2 1
du = √ √ 1
2 4 u a2 − u 2 du + 1 a2 − u
4
2 du
√ √
= 1 a2 − u
4
2 du − 1 8 (−2u) a2 − u 2 du
This can be rewritten as
6+4y − 4y
2
30
= u √
a2 − u
8
2 + a2 u
8 sin−1 − 1 a 8
√wdw
w = a 2 − u 2 ,
dw = −2udu
= 2y − 1
6+4y − 4y2 + 7 2y − 1
8
8 sin−1 √ − 1
7 8 · 2
3 w3/2 + C
= 2y − 1
6+4y − 4y2 + 7 2y − 1
8
8 sin−1 √ − 1
7 12 (6 + 4y − 4y2 ) 3/2 + C
1
8 (2y − 1) − 1 )
12 (6 + 4y − 4y2 + 7 2y − 1
8 sin−1 √ + C
7
1
=
3 y2 − 1 12 y − 5 6+4y
− 4y2 + 7 2y − 1
8
8 sin−1 √ + C
7
= 1 24 (8y2 − 2y − 15) 6+4y − 4y 2 + 7 8 sin−1 2y − 1
√
7
+ C
19. Let u =sinx. Thendu =cosxdx,so
sin 2 x cos x ln(sin x) dx =
u 2 ln udu 101
= u2+1
(2 + 1) 2 [(2 + 1) ln u − 1] + C = 1 9 u3 (3 ln u − 1) + C
= 1 9 sin3 x [3 ln(sin x) − 1] + C
21. Let u = e x and a = √ 3.Thendu = e x dx and
e x
3 − e dx = du 19
= 1 2x a 2 − u 2 2a ln u + a
u − a + C = 1
2 √ 3 ln e x + √ 3
e x − √ 3
+ C.
23. sec 5 xdx = 77 1 tan x
4 sec3 x + 3 4 sec 3 xdx= 77 1 tan x
4 sec3 x + 3 1 tan x sec x +
1
4 2 2 sec xdx
14
= 1 tan x 4 sec3 x + 3 tan x sec x + 3 ln|sec x +tanx| + C
8 8
25. Let u =lnx and a =2.Thendu = dx/x and
4+(lnx)
2
x
a2
dx = + u 2 du = 21 u
a2 + u
2 2 + a2
2 ln u +
a 2 + u 2 + C
= 1 (ln x) 4+(lnx)
2 2 +2ln
ln x +
4+(lnx) 2 + C

F.
326 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
27. Let u = e x .Thenx =lnu, dx = du/u, so
29.
e √
u2 − 1
2x
− 1 dx =
du = 41 u
u
2 − 1 − cos −1 (1/u)+C = e 2x − 1 − cos −1 (e −x )+C.
x 4
dx
√
x
10
− 2 =
x 4 dx
(x5 ) 2 − 2 = 1
5
31. Using cylindrical shells, we get
33. (a)
du
√
u2 − 2
u= x 5 ,
du= 5x 4 dx
43
= 1 5 ln u +
√
u2 − 2 + C =
1
5 ln x 5 + √ x 10 − 2 + C
V =2π
2
0
x · x 4 − x 2 dx =2π
2
=2π[(0 + 2 sin −1 1) − (0 + 2 sin −1 0] = 2π
0
x 2 4 − x 2 dx =2π
31 x
8 (2x2 − 4) 4 − x 2 + 16 x
8 sin−1
2 · π
2
=2π 2
d 1
a + bu − a2
du b 3 a + bu − 2a ln |a + bu| + C = 1
ba 2
b +
b 3 (a + bu) − 2ab
2 (a + bu)
= 1
b(a + bu) 2 + ba 2 − (a + bu)2ab
b 3 (a + bu) 2
= 1
b 3 u 2
=
b 3 (a + bu) 2
(b) Let t = a + bu ⇒ dt = bdu.Notethatu = t − a and du = 1 b
b dt.
u 2 du
(a + bu) = 1 (t − a)
2
dt = 1 t 2 − 2at + a 2
dt = 1
1 − 2a
2 b 3 t 2 b 3 t 2
b 3 t + a2
dt
t 2
= 1
t − 2a ln |t| − a2
+ C = 1
a + bu − a2
b 3 t b 3 a + bu − 2a ln |a + bu| + C
35. Maple and Mathematica both give sec 4 xdx= 2 3 tan x + 1 3 tan x sec2 x, while Derive gives the second
sin x
term as
3cos 3 x = 1 sin x 1
3 cos x cos 2 x = 1 3 tan x sec2 x. Using Formula 77, we get
sec 4 xdx= 1 tan x
3 sec2 x + 2 3 sec 2 xdx= 1 tan x 3 sec2 x + 2 3
tan x + C.
2
2
0
u 2
(a + bu) 2
37. Derive gives x 2 √ x 2 +4dx = 1 4 x(x2 +2) √ x 2 +4− 2ln √ x 2 +4+x .Maplegives
1
4 x(x2 +4) 3/2 − 1 x √ 2
x 2 +4− 2arcsinh 1
x 2
. Applying the command convert(%,ln); yields
1
4 x(x2 +4) 3/2 − 1 x √ x
2 2 +4− 2ln 1
x + 1 2 2
√
x2 +4 = 1 4 x(x2 +4) 1/2 (x 2 +4)− 2 − 2ln x + √ x 2 +4 /2
= 1 4 x(x2 +2) √ x 2 +4− 2ln √ x 2 +4+x +2ln2
Mathematica gives 1 4 x(2 + x2 ) √ 3+x 2 − 2arcsinh(x/2). Applying the TrigToExp and Simplify commands gives
1
4 x(2 + x 2 ) √ 4+x 2 − 8log √ 1
2 x + 4+x
2
= 1 4 x(x2 +2) √ x 2 +4− 2ln x + √ 4+x 2 +2ln2,soallare
equivalent (without constant).

F.
SECTION 7.6 TX.10 INTEGRATION USING TABLES AND COMPUTER ALGEBRA SYSTEMS ¤ 327
NowuseFormula22toget
x 2 2 2 + x 2 dx = x 8 (22 +2x 2 ) √ 2 2 + x 2 − 24
8 ln x + √ 2 2 + x 2 + C
= x 8 (2)(2 + x2 ) √ 4+x 2 − 2ln x + √ 4+x 2 + C
= 1 4 x(x2 +2) √ x 2 +4− 2ln √ x 2 +4+x + C
39. Maple gives x √ 1+2xdx= 1 10 (1 + 2x)5/2 − 1 6 (1 + 2x)3/2 , Mathematica gives √ 1+2x 2
5 x2 + 1
15 x − 1 15
,andDerive
gives 1 15 (1 + 2x)3/2 (3x − 1).Thefirst two expressions can be simplified to Derive’s result. If we use Formula 54, we get
x √ 1+2xdx= 2
15(2) 2 (3 · 2x − 2 · 1)(1 + 2x)3/2 + C = 1 30 (6x − 2)(1 + 2x)3/2 + C = 1
15 (3x − 1)(1 + 2x)3/2 .
41. Maple gives tan 5 xdx = 1 4 tan4 x − 1 2 tan2 x + 1 2 ln(1 + tan2 x), Mathematicagives
tan 5 xdx= 1 4 [−1 − 2cos(2x)] sec4 x − ln(cos x), and Derive gives tan 5 xdx= 1 4 tan4 x − 1 2 tan2 x − ln(cos x).
These expressions are equivalent, and none includes absolute value bars or a constant of integration. Note that Mathematica’s
and Derive’s expressions suggest that the integral is undefined where cos x 0 = {x | x 6= 0, |x| < 1} =(−1, 0) ∪ (0, 1). F has the same domain.
(b) Derive gives F (x) =ln √ 1 − x 2 − 1 − ln x and Mathematica gives F (x) =lnx − ln 1+ √ 1 − x 2 .
Both are correct if you take absolute values of the logarithm arguments, and both would then have the
same domain. Maple gives F (x) =− arctanh 1/ √ 1 − x 2 . This function has domain
x |x| < 1, −1 < 1/ √ 1 − x 2 < 1 = x |x| < 1, 1/ √ 1 − x 2 < 1 = x |x| < 1, √ 1 − x 2 > 1 = ∅,
the empty set! If we apply the command convert(%,ln); to Maple’s answer, we get
− 1
2 ln 1
√ +1 + 1
1 1 − x
2 2 ln 1
− √ , which has the same domain, ∅.
1 − x
2
45. Maple gives the antiderivative
x 2 − 1
F (x) =
x 4 + x 2 +1 dx = − 1 2 ln(x2 + x +1)+ 1 2 ln(x2 − x +1).
We can see that at 0, this antiderivative is 0. From the graphs, it appears that F has
amaximumatx = −1 and a minimum at x =1[since F 0 (x) =f(x) changes
sign at these x-values], and that F has inflection points at x ≈−1.7, x =0,and
x ≈ 1.7 [since f(x) has extrema at these x-values].

F.
328 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
47. Since f(x) =sin 4 x cos 6 x is everywhere positive, we know that its antiderivative F is increasing. Maple gives
f(x) dx = −
1
10 sin3 x cos 7 x − 3 sin x 80 cos7 x + 1
160 cos5 x sin x + 1
128 cos3 x sin x + 3
3
cos x sin x + x
256 256
and this expression is 0 at x =0.
F has a minimum at x =0and a maximum at x = π.
F has inflection points where f 0 changes sign, that is, at x ≈ 0.7, x = π/2,
and x ≈ 2.5.
7.7 Approximate Integration
1. (a) ∆x =(b − a)/n =(4− 0)/2 =2
L 2 = 2 f(x i−1) ∆x = f(x 0) · 2+f(x 1) · 2=2[f(0) + f(2)] = 2(0.5+2.5) = 6
i=1
R 2 = 2 f(x i ) ∆x = f(x 1 ) · 2+f(x 2 ) · 2=2[f(2) + f(4)] = 2(2.5+3.5) = 12
i=1
M 2 = 2 f(x i)∆x = f(x 1) · 2+f(x 2) · 2=2[f(1) + f(3)] ≈ 2(1.6+3.2) = 9.6
i=1
(b)
L 2 is an underestimate, since the area under the small rectangles is less than
the area under the curve, and R 2 is an overestimate, since the area under the
large rectangles is greater than the area under the curve. It appears that M 2
is an overestimate, though it is fairly close to I. See the solution to
Exercise 45 for a proof of the fact that if f is concave down on [a, b],then
the Midpoint Rule is an overestimate of b
f(x) dx.
a
(c) T 2 = 1
∆x [f(x
2 0 )+2f(x 1 )+f(x 2 )] = 2 [f(0) + 2f(2) + f(4)] = 0.5+2(2.5) + 3.5 =9.
2
This approximation is an underestimate, since the graph is concave down. Thus, T 2 =9

F.
TX.10
SECTION 7.7 APPROXIMATE INTEGRATION ¤ 329
5. f(x) =x 2 sin x, ∆x = b − a
n = π − 0 = π 8 8
(a) M 8 = π 8 f π
16 + f 3π
16 + f 5π
16 + ···+ f 15π
16 ≈ 5.932957
(b) S 8 =
π
8 · 3 f(0) + 4f π
8 +2f 2π
8 +4f 3π
8 +2f 4π
8 +4f 5π
8 +2f 6π
8 +4f 7π
8 + f(π)
≈ 5.869247
Actual: π
0 x2 sin xdx = 84 −x 2 cos x π
+2 π
x cos xdx = 83 −π 2 (−1) − 0 +2 cos x + x sin x π
0 0 0
= π 2 +2[(−1+0)− (1 + 0)] = π 2 − 4 ≈ 5.869604
Errors: E M = actual − M 8 = π
0 x2 sin xdx− M 8 ≈−0.063353
E S = actual − S 8 = π
0 x2 sin xdx− S 8 ≈ 0.000357
7. f(x) = 4√ 1+x 2 , ∆x = 2 − 0 = 1 8 4
(a) T 8 = 1
4 · 2 f(0) + 2f 1
4 +2f 1
2 + ···+2f 3
2 +2f 7
4 + f(2) ≈ 2.413790
(b) M 8 = 1 4 f 1
8 + f 3
8 + ···+ f 13
8 + f 15
8 ≈ 2.411453
(c) S 8 = 1
4 · 3
f(0) + 4f
1
4
+2f
1
2
+4f
3
4
+2f(1) + 4f
5
4
+2f
3
2
+4f
7
4
+ f(2)
≈ 2.412232
9. f(x) = ln x
1+x , ∆x = 2 − 1 = 1 10 10
(a) T 10 = 1 [f(1) + 2f(1.1) + 2f(1.2) + ···+2f(1.8) + 2f(1.9) + f(2)] ≈ 0.146879
10 · 2
(b) M 10 = 1 [f(1.05) + f(1.15) + ···+ f(1.85) + f(1.95)] ≈ 0.147391
10
(c) S 10 = 1 [f(1) + 4f(1.1) + 2f(1.2) + 4f(1.3) + 2f(1.4) + 4f(1.5) + 2f(1.6) + 4f(1.7)
10 · 3
+2f(1.8) + 4f(1.9) + f(2)]
≈ 0.147219
1
11. f(t) =sin(e t/2 2
), ∆t =
− 0 = 1
8 16
(a) T 8 = 1
16 · 2 f(0) + 2f 1
16 +2f 2
16 + ···+2f 7
16 + f 1
2 ≈ 0.451948
(b) M 8 = 1
16 f 1
32 + f 3
32 + f 5
32 + ···+ f 13
32 + f 15
32 ≈ 0.451991
(c) S 8 = 1
16 · 3 f(0) + 4f 1
16 +2f 2
16 + ···+4f 7
16 + f 1
2 ≈ 0.451976
13. f(t) =e √t sin t, ∆t = 4 − 0 = 1 8 2
(a) T 8 = 1
2 · 2 f(0) + 2f 1
2 +2f(1) + 2f 3
2 +2f(2) + 2f 5
2 +2f(3) + 2f 7
2 + f(4) ≈ 4.513618
(b) M 8 = 1 2 f 1
4 + f 3
4 + f 5
4 + f 7
4 + f 9
4 + f 11
4 + f 13
4 + f 15
4 ≈ 4.748256
(c) S 8 = 1
2 · 3
f(0) + 4f
1
2
+2f(1) + 4f
3
2
+2f(2) + 4f
5
2
+2f(3) + 4f
7
2
+ f(4)
≈ 4.675111
15. f(x) = cos x
x , ∆x = 5 − 1 = 1 8 2
(a) T 8 = 1
2 · 2 f(1) + 2f 3
2 +2f(2) + ···+2f(4) + 2f 9
2 + f(5) ≈−0.495333
(b) M 8 = 1 2 f 5
4 + f 7
4 + f 9
4 + f 11
4 + f 13
4 + f 15
4 + f 17
4 + f 19
4 ≈−0.543321
(c) S 8 = 1
2 · 3
f(1) + 4f
3
2
+2f(2) + 4f
5
2
+2f(3) + 4f
7
2
+2f(4) + 4f
9
2
+ f(5)
≈−0.526123

F.
330 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
17. f(y) = 1
1+y , ∆y = 3 − 0 = 1 5 6 2
(a) T 6 = 1
2 · 2 f(0) + 2f 1
2 +2f 2
2 +2f 3
2 +2f 4
2 +2f 5
2 + f(3) ≈ 1.064275
(b) M 6 = 1 2 f 1
4 + f 3
4 + f 5
4 + f 7
4 + f 9
4 + f 11
4 ≈ 1.067416
(c) S 6 = 1
2 · 3
f(0) + 4f
1
2
+2f
2
2
+4f
3
2
+2f
4
2
+4f
5
2
+ f(3)
≈ 1.074915
19. f(x) =cos(x 2 ), ∆x = 1 − 0
8
= 1 8
(a) T 8 = 1
8 · 2 f(0) + 2 f 1
8 + f 2
8 + ···+ f 7
8 + f(1) ≈ 0.902333
M 8 = 1 8 f 1
16 + f 3
16 + f 5
16 + ···+ f 15
16 =0.905620
(b) f(x) =cos(x 2 ), f 0 (x) =−2x sin(x 2 ), f 00 (x) =−2sin(x 2 ) − 4x 2 cos(x 2 ).For0 ≤ x ≤ 1, sin and cos are positive,
so |f 00 (x)| =2sin(x 2 )+4x 2 cos(x 2 ) ≤ 2 · 1+4· 1 · 1=6since sin(x 2 ) ≤ 1 and cos x 2 ≤ 1 for all x,
and x 2 ≤ 1 for 0 ≤ x ≤ 1. Soforn =8,wetakeK =6, a =0,andb =1in Theorem 3, to get
|E T | ≤ 6 · 1 3 /(12 · 8 2 )= 1 =0.0078125 and |E 128 M| ≤ 1 =0.00390625. [A better estimate is obtained by noting
256
from a graph of f 00 that |f 00 (x)| ≤ 4 for 0 ≤ x ≤ 1.]
(c) Take K =6[as in part (b)] in Theorem 3. |E T | ≤
K(b − a)3
12n 2 ≤ 0.0001 ⇔
6(1 − 0)3
12n 2 ≤ 10 −4 ⇔
1
2n ≤ 1 ⇔ 2n 2 ≥ 10 4 ⇔ n 2 ≥ 5000 ⇔ n ≥ 71. Taken =71for T 2 10 4 n .ForE M , again take K =6in
Theorem 3 to get |E M| ≤ 10 −4 ⇔ 4n 2 ≥ 10 4 ⇔ n 2 ≥ 2500 ⇔ n ≥ 50. Taken =50for M n.
21. f(x) =sinx, ∆x = π − 0
10
= π 10
(a) T 10 =
π
10 · 2
f(0) + 2f
π
10
M 10 = π 10
f
π
20
S 10 =
+ f
3π
20
π
10 · 3
f(0) + 4f
π
10
+2f 2π
10
+ f
5π
20
+2f
2π
10
+ ···+2f 9π
10 + f(π) ≈ 1.983524
+ ···+ f
19π
20 ≈ 2.008248
+4f
3π
10
+ ···+4f
9π
10
+ f(π)
≈ 2.000110
Since I = π
sin xdx = − cos x π
=1− (−1) = 2, E 0 0
T = I − T 10 ≈ 0.016476, E M = I − M 10 ≈−0.008248,
and E S = I − S 10 ≈−0.000110.
(b) f(x) =sinx ⇒ f (n) (x) ≤ 1,sotakeK =1forallerrorestimates.
|E T | ≤
|E S | ≤
K(b − a)3 1(π − 0)3
= = π3
12n 2 12(10) 2 1200 ≈ 0.025839. |E M| ≤ |E T |
= π3
2 2400 ≈ 0.012919.
K(b − a)5 1(π − 0)5
=
180n 4 180(10) = π 5
4 1,800,000 ≈ 0.000170.
The actual error is about 64% of the error estimate in all three cases.
(c) |E T | ≤ 0.00001 ⇔ π3
12n ≤ 1 ⇔ n 2 ≥ 105 π 3
2 10 5 12
|E M | ≤ 0.00001 ⇔ π3
24n ≤ 1 ⇔ n 2 ≥ 105 π 3
2 10 5 24
|E S | ≤ 0.00001 ⇔ π5
180n ≤ 1 ⇔ n 4 ≥ 105 π 5
4 10 5 180
Take n =22for S n (since n must be even).
⇒ n ≥ 508.3. Taken = 509 for T n.
⇒ n ≥ 359.4. Taken = 360 for M n.
⇒ n ≥ 20.3.

F.
TX.10
SECTION 7.7 APPROXIMATE INTEGRATION ¤ 331
23. (a) Using a CAS, we differentiate f(x) =e cos x twice, and find that
f 00 (x) =e cos x (sin 2 x − cos x). From the graph, we see that the maximum
value of |f 00 (x)| occurs at the endpoints of the interval [0, 2π].
Since f 00 (0) = −e,wecanuseK = e or K =2.8.
(b) A CAS gives M 10 ≈ 7.954926518. (InMaple,usestudent[middlesum].)
(c) Using Theorem 3 for the Midpoint Rule, with K = e,weget|E M | ≤
With K =2.8,weget|E M| ≤
(d) A CAS gives I ≈ 7.954926521.
2.8(2π − 0)3
24 · 10 2 =0. 289391916.
(e) The actual error is only about 3 × 10 −9 ,muchlessthantheestimateinpart(c).
(f) We use the CAS to differentiate twice more, and then graph
f (4) (x) =e cos x (sin 4 x − 6sin 2 x cos x +3− 7sin 2 x +cosx).
From the graph, we see that the maximum value of f (4) (x) occurs at the
endpoints of the interval [0, 2π]. Sincef (4) (0) = 4e, we can use K =4e
or K =10.9.
(g) A CAS gives S 10 ≈ 7.953789422. (In Maple, use student[simpson].)
(h) Using Theorem 4 with K =4e,weget|E S | ≤
With K =10.9,weget|E S| ≤
4e(2π − 0)5
180 · 10 4 ≈ 0.059153618.
10.9(2π − 0)5
180 · 10 4 ≈ 0.059299814.
e(2π − 0)3
24 · 10 2 ≈ 0.280945995.
(i) The actual error is about 7.954926521 − 7.953789422 ≈ 0.00114. This is quite a bit smaller than the estimate in part (h),
though the difference is not nearly as great as it was in the case of the Midpoint Rule.
( j) To ensure that |E S | ≤ 0.0001, weuseTheorem4:|E S | ≤ 4e(2π)5
4e(2π)5
≤ 0.0001 ⇒
180 · n4 180 · 0.0001 ≤ n4 ⇒
n 4 ≥ 5,915,362 ⇔ n ≥ 49.3. Sowemusttaken ≥ 50 to ensure that |I − S n | ≤ 0.0001.
(K =10.9 leads to the same value of n.)
25. I = 1
0 xex dx =[(x − 1)e x ] 1 0
[parts or Formula 96] =0− (−1) = 1, f(x) =xe x , ∆x =1/n
n =5: L 5 = 1 [f(0) + f(0.2) + f(0.4) + f(0.6) + f(0.8)] ≈ 0.742943
5
R 5 = 1 [f(0.2) + f(0.4) + f(0.6) + f(0.8) + f(1)] ≈ 1.286599
5
T 5 = 1
5 · 2
[f(0) + 2f(0.2) + 2f(0.4) + 2f(0.6) + 2f(0.8) + f(1)] ≈ 1.014771
M 5 = 1 5
[f(0.1) + f(0.3) + f(0.5) + f(0.7) + f(0.9)] ≈ 0.992621
E L = I − L 5 ≈ 1 − 0.742943 = 0.257057
E R ≈ 1 − 1.286599 = −0.286599
E T ≈ 1 − 1.014771 = −0.014771
E M ≈ 1 − 0.992621 = 0.007379

F.
332 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
n =10: L 10 = 1
10
[f(0) + f(0.1) + f(0.2) + ···+ f(0.9)] ≈ 0.867782
R 10 = 1
10
[f(0.1) + f(0.2) + ···+ f(0.9) + f(1)] ≈ 1.139610
T 10 = 1 {f(0) + 2[f(0.1) + f(0.2) + ···+ f(0.9)] + f(1)} ≈ 1.003696
10 · 2
M 10 = 1 [f(0.05) + f(0.15) + ···+ f(0.85) + f(0.95)] ≈ 0.998152
10
E L = I − L 10 ≈ 1 − 0.867782 = 0.132218
E R ≈ 1 − 1.139610 = −0.139610
E T ≈ 1 − 1.003696 = −0.003696
E M ≈ 1 − 0.998152 = 0.001848
n =20: L 20 = 1 [f(0) + f(0.05) + f(0.10) + ···+ f(0.95)] ≈ 0.932967
20
R 20 = 1
20
[f(0.05) + f(0.10) + ···+ f(0.95) + f(1)] ≈ 1.068881
T 20 = 1 {f(0) + 2[f(0.05) + f(0.10) + ···+ f(0.95)] + f(1)} ≈ 1.000924
20 · 2
M 20 = 1 [f(0.025) + f(0.075) + f(0.125) + ···+ f(0.975)] ≈ 0.999538
20
E L = I − L 20 ≈ 1 − 0.932967 = 0.067033
E R ≈ 1 − 1.068881 = −0.068881
E T ≈ 1 − 1.000924 = −0.000924
E M ≈ 1 − 0.999538 = 0.000462
n L n R n T n M n
5 0.742943 1.286599 1.014771 0.992621
10 0.867782 1.139610 1.003696 0.998152
20 0.932967 1.068881 1.000924 0.999538
n E L E R E T E M
5 0.257057 −0.286599 −0.014771 0.007379
10 0.132218 −0.139610 −0.003696 0.001848
20 0.067033 −0.068881 −0.000924 0.000462
Observations:
1. E L and E R are always opposite in sign, as are E T and E M.
2. As n is doubled, E L and E R are decreased by about a factor of 2,andE T and E M are decreased by a factor of about 4.
3. The Midpoint approximation is about twice as accurate as the Trapezoidal approximation.
4. All the approximations become more accurate as the value of n increases.
5. The Midpoint and Trapezoidal approximations are much more accurate than the endpoint approximations.
27. I = 2
0 x4 dx = 1
x5 2
= 32 − 0=6.4, f(x) 5 0 5 =x4 , ∆x = 2 − 0
n
= 2 n
n =6: T 6 = 2
6 · 2 f(0) + 2 f 1
3 + f 2
3 + f 3
3 + f 4
3 + f 5
3 + f(2) ≈ 6.695473
M 6 = 2 6 f 1
6 + f 3
6 + f 5
6 + f 7
6 + f 9
6 + f 11
6 ≈ 6.252572
S 6 = 2
6 · 3
f(0) + 4f
1
3
+2f
2
3
+4f
3
3
+2f
4
3
+4f
5
3
+ f(2)
≈ 6.403292
E T = I − T 6 ≈ 6.4 − 6.695473 = −0.295473
E M ≈ 6.4 − 6.252572 = 0.147428
E S ≈ 6.4 − 6.403292 = −0.003292

F.
SECTION 7.7 APPROXIMATE INTEGRATION ¤ 333
n =12: T 12 = 2
12 · 2 f(0) + 2 f 1
6 + f 2
6 + f 3
6 + ···+ f 11
6 + f(2) ≈ 6.474023
M 6 = 2 12 f 1
12 + f 3
12 + f 5
12 + ···+ f 23
12 ≈ 6.363008
S 6 = 2
12 · 3 f(0) + 4f 1
6 +2f 2
6 +4f 3
6 +2f 4
6 + ···+4f 11
6 + f(2) ≈ 6.400206
Observations:
E T = I − T 12 ≈ 6.4 − 6.474023 = −0.074023
E M ≈ 6.4 − 6.363008 = 0.036992
E S ≈ 6.4 − 6.400206 = −0.000206
n T n M n S n
6 6.695473 6.252572 6.403292
12 6.474023 6.363008 6.400206
1. E T and E M are opposite in sign and decrease by a factor of about 4 as n is doubled.
n E T E M E S
6 −0.295473 0.147428 −0.003292
12 −0.074023 0.036992 −0.000206
2. The Simpson’s approximation is much more accurate than the Midpoint and Trapezoidal approximations, and E S seems to
decrease by a factor of about 16 as n is doubled.
29. ∆x =(b − a)/n =(6− 0)/6 =1
(a) T 6 = ∆x
2
[f(0) + 2f(1) + 2f(2) + 2f(3) + 2f(4) + 2f(5) + f(6)]
≈ 1 [3+2(5)+2(4)+2(2)+2(2.8) + 2(4) + 1]
2
= 1 (39.6) = 19.8
2
(b) M 6 = ∆x[f(0.5) + f(1.5) + f(2.5) + f(3.5) + f(4.5) + f(5.5)]
≈ 1[4.5+4.7+2.6+2.2+3.4+3.2]
=20.6
(c) S 6 = ∆x [f(0) + 4f(1) + 2f(2) + 4f(3) + 2f(4) + 4f(5) + f(6)]
3
≈ 1 [3+4(5)+2(4)+4(2)+2(2.8) + 4(4) + 1]
3
= 1 (61.6) = 20.53
3
31. (a) We are given the function values at the endpoints of 8 intervals of length 0.4, so we’ll use the Midpoint Rule with
n =8/2 =4and ∆x =(3.2 − 0)/4 =0.8.
3.2
0
f(x) dx ≈ M 4 =0.8[f(0.4) + f(1.2) + f(2.0) + f(2.8)] = 0.8[6.5+6.4+7.6+8.8]
=0.8(29.3) = 23.44
(b) −4 ≤ f 00 (x) ≤ 1 ⇒ |f 00 (x)| ≤ 4, souseK =4, a =0, b =3.2, andn =4in Theorem 3.
So |E M| ≤
4(3.2 − 0)3
24(4) 2 = 128
375 =0.3413. TX.10

F.
334 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
33. By the Net Change Theorem, the increase in velocity is equal to 6
a(t) dt. We use Simpson’s Rule with n =6and
0
∆t =(6− 0)/6 =1to estimate this integral:
6 a(t) dt ≈ S 0 6 = 1 [a(0) + 4a(1) + 2a(2) + 4a(3) + 2a(4) + 4a(5) + a(6)]
3
≈ 1 [0 + 4(0.5) + 2(4.1) + 4(9.8) + 2(12.9) + 4(9.5) + 0] = 1 (113.2) = 37.73 ft/s
3 3
35. By the Net Change Theorem, the energy used is equal to 6
P (t) dt. We use Simpson’s Rule with n =12and
0
∆t = 6 − 0 = 1 to estimate this integral:
12 2
6
0
1/2
P (t) dt ≈ S12 = [P (0) + 4P (0.5) + 2P (1) + 4P (1.5) + 2P (2) + 4P (2.5) + 2P (3)
3
+4P (3.5) + 2P (4) + 4P (4.5) + 2P (5) + 4P (5.5) + P (6)]
= 1 [1814 + 4(1735) + 2(1686) + 4(1646) + 2(1637) + 4(1609) + 2(1604)
6
+ 4(1611) + 2(1621) + 4(1666) + 2(1745) + 4(1886) + 2052]
= 1 (61,064) = 10,177.3 megawatt-hours
6
37. Let y = f(x) denote the curve. Using cylindrical shells, V = 10
2
2πxf(x) dx =2π 10
2
xf(x) dx =2πI 1 .
Now use Simpson’s Rule to approximate I 1 :
I 1 ≈ S 8 = 10 − 2 [2f(2) + 4 · 3f(3) + 2 · 4f(4) + 4 · 5f(5) + 2 · 6f(6) + 4 · 7f(7) + 2 · 8f(8) + 4 · 9f(9) + 10f(10)]
3(8)
≈ 1 3
[2(0) + 12(1.5) + 8(1.9) + 20(2.2) + 12(3.0) + 28(3.8) + 16(4.0) + 36(3.1) + 10(0)]
= 1 3 (395.2)
Thus, V ≈ 2π · 1 (395.2) ≈ 827.7 or 828 cubic units.
3
39. Using disks, V = 5
1 π(e−1/x ) 2 dx = π 5
1 e−2/x dx = πI 1 . Now use Simpson’s Rule with f(x) =e −2/x to approximate
I 1 . I 1 ≈ S 8 = 5 − 1
3(8) [f(1) + 4f(1.5) + 2f(2) + 4f(2.5) + 2f(3) + 4f(3.5) + 2f(4) + 4f(4.5) + f(5)] ≈ 1 6 (11.4566)
Thus, V ≈ π · 1 (11.4566) ≈ 6.0 cubic units.
6
41. I(θ) = N 2 sin 2 k
,wherek =
k 2
πNd sin θ
, N =10,000, d =10 −4 ,andλ = 632.8 × 10 −9 .SoI(θ) = (104 ) 2 sin 2 k
,
λ
k 2
where k = π(104 )(10 −4 )sinθ
.Nown =10and ∆θ = 10−6 − (−10 −6 )
=2× 10 −7 ,so
632.8 × 10 −9 10
M 10 =2× 10 −7 [I(−0.0000009) + I(−0.0000007) + ···+ I(0.0000009)] ≈ 59.4.
43. Consider the function f whose graph is shown. The area 2
f(x) dx
0
is close to 2. The Trapezoidal Rule gives
T 2 = 2 − 0 [f(0) + 2f(1) + f(2)] = 1 [1 + 2 · 1+1]=2.
2 · 2 2
The Midpoint Rule gives M 2 = 2 − 0
2
[f(0.5) + f(1.5)] = 1[0 + 0] = 0,
so the Trapezoidal Rule is more accurate.

F.
TX.10
SECTION 7.8 IMPROPER INTEGRALS ¤ 335
45. Since the Trapezoidal and Midpoint approximations on the interval [a, b] are the sums of the Trapezoidal and Midpoint
approximations on the subintervals [x i−1 ,x i ], i =1, 2,...,n, we can focus our attention on one such interval. The condition
f 00 (x) < 0 for a ≤ x ≤ b means that the graph of f isconcavedownasinFigure5.Inthatfigure, T n is the area of the
trapezoid AQRD, b
a f(x) dx istheareaoftheregionAQP RD, andM n is the area of the trapezoid ABCD, so
T n < b
a f(x) dx < Mn. In general, the condition f 00 < 0 implies that the graph of f on [a, b] lies above the chord joining the
points (a, f(a)) and (b, f(b)). Thus, b
a f(x) dx > Tn. SinceMn is the area under a tangent to the graph, and since f 00 < 0
implies that the tangent lies above the graph, we also have M n > b
f(x) dx. Thus, a Tn < b
f(x) dx < Mn.
a
47. T n = 1 2 ∆x [f(x 0)+2f(x 1 )+···+2f(x n−1 )+f(x n )] and
M n = ∆x [f(x 1)+f(x 2)+···+ f(x n−1)+f(x n)],wherex i = 1 (xi−1 + xi). Now
2
T 2n = 1 2
1
2 ∆x [f(x 0 )+2f(x 1 )+2f(x 1 )+2f(x 2 )+2f(x 2 )+···+2f(x n−1 )+2f(x n−1 )+2f(x n )+f(x n )]
so
1
2 (T n + M n )= 1 2 T n + 1 2 M n
= 1 4 ∆x[f(x0)+2f(x1)+···+2f(xn−1)+f(xn)] + 1 4 ∆x[2f(x1)+2f(x2)+···+2f(xn−1)+2f(xn)]
= T 2n
7.8 Improper Integrals
1. (a) Since ∞
1
x 4 e −x4 dx has an infinite interval of integration, it is an improper integral of Type I.
(b) Since y =secx has an infinite discontinuity at x = π 2 , π/2
0
sec xdxis a Type II improper integral.
(c) Since y =
x
2
(x − 2)(x − 3) has an infinite discontinuity at x =2, x
dx is a Type II improper integral.
x 2 − 5x +6
0
1
(d) Since
dx has an infinite interval of integration, it is an improper integral of Type I.
−∞ x 2 +5
0
3. Theareaunderthegraphofy =1/x 3 = x −3 between x =1and x = t is
A(t) = t
1 x−3 dx = − 1 x−2 t
= − 1
2 1 2 t−2 −
− 1 2 =
1
− 1 2t 2 . So the area for 1 ≤ x ≤ 10 is
2
A(10) = 0.5 − 0.005 = 0.495, theareafor1 ≤ x ≤ 100 is A(100) = 0.5 − 0.00005 = 0.49995, and the area for
1 ≤ x ≤ 1000 is A(1000) = 0.5 − 0.0000005 = 0.4999995. The total area under the curve for x ≥ 1 is
lim A(t) = lim 1 −
t→∞ t→∞ 2 1/(2t2 ) = 1 . 2

F.
TX.10
336 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
= 1 25 Convergent
∞
1
t
1
5. I =
dx = lim
dx. Now
1 (3x +1)
2 t→∞
1 (3x +1)
2
1
(3x +1) dx = 1 2 3
1
du
u2 [u =3x +1, du =3dx]
1
= −
3u + C = − 1
3(3x +1) + C,
t
1
1
so I = lim −
= lim −
t→∞ 3(3x +1)
t→∞
1
3(3t +1) + 1
=0+ 1 12 12 = 1
12 . Convergent
−1
1
−1
1
√ −1
7. √ dw = lim √ dw = lim −2 2 − w
−∞ 2 − w t→−∞
t 2 − w t→−∞
t
[u =2− w, du = −dw]
√ √
= lim −2 3+2 2 − t = ∞.
t→−∞
Divergent
∞
t
9. e −y/2 t
dy = lim
4
t→∞ 4 e−y/2 dy = lim
−2e −y/2 = lim (−2e −t/2 +2e −2 )=0+2e −2 =2e −2 .
t→∞ 4 t→∞
Convergent
∞
xdx 0
11.
−∞ 1+x = xdx ∞
2 −∞ 1+x + xdx
2 0 1+x and 2
0
xdx
−∞ 1+x = lim 1 ln 1+x 2 0
= lim
2 t→−∞ 2 t 0 −
1
1+t 2 = −∞.
t→−∞ 2
Divergent
∞
13.
−∞ xe−x2 dx = 0
−∞ xe−x2 dx + ∞
xe −x2 dx.
0
0
−∞ xe−x2 dx = lim −
1
e −x2 0 = lim
t→−∞ 2
−
1
1 − e −t2 = − 1 · 1=− 1 ,and
t t→−∞ 2 2 2
∞
xe −x2 dx = lim
0
−
1
e −x2 t
= lim
t→∞ 2
−
1
e −t2 − 1 = − 1 · (−1) = 1 .
0 t→∞ 2 2 2
Therefore, ∞
−∞ xe−x2 dx = − 1 + 1 =0.
2 2 Convergent
∞
t
t
15. sin θdθ= lim sin θdθ = lim
2π t→∞ 2π − cos θ = lim (− cos t +1). This limit does not exist, so the integral is
t→∞
2π t→∞
divergent. Divergent
∞
x +1
t 1
(2x +2)
t
2
17.
dx = lim
1 x 2 +2x t→∞
1 x 2 +2x dx = 1 lim ln(x 2 +2x) = 1 lim ln(t 2 +2t) − ln 3 = ∞.
2 t→∞
2
1 t→∞
Divergent
∞
t
19. se −5s ds = lim se −5s
ds = lim −
1
0
t→∞
0
t→∞ 5 se−5s − 1 e−5s by integration by
25 parts with u = s
= lim −
1
t→∞ 5 te−5t − 1 25 e−5t + 1 25 =0− 0+
1
25
[by l’Hospital’s Rule]

F.
21.
∞
1
ln x
x
dx = lim
t→∞
(ln x)
2
2
t
1
TX.10
by substitution with
u =lnx, du = dx/x
(ln t) 2
= lim = ∞. Divergent
t→∞ 2
SECTION 7.8 IMPROPER INTEGRALS ¤ 337
23.
∞
−∞
Now
x 2 0
9+x dx = 6
∞
so 2
0
Convergent
x 2 dx
9+x 6
−∞
x 2 ∞
9+x dx + 6
u = x 3
du =3x 2 dx
x 2
t
dx = 2 lim
9+x6 t→∞
0
0
x 2 ∞
9+x dx =2 6
0
x 2
dx [since the integrand is even].
9+x6 1
3
=
du 1
u =3v
(3 dv)
3
=
9+u 2 du =3dv 9+9v = 1
dv
2 9 1+v 2
= 1 9 tan−1 v + C = 1 u
9 tan−1 + C = 1 x
3
3 9 tan−1 + C,
3
x 2
dx = 2 lim
9+x6 t→∞
1
9 tan−1 x
3
3
t
0
1 t
3
=2 lim
t→∞ 9 tan−1 = 2 3 9 · π
2 = π 9 .
25.
∞
e
1
t
dx = lim
x(ln x)
3 t→∞
e
1
dx = lim
x(ln x)
3 t→∞
= lim − 1
t→∞ 2(lnt) 2 + 1 2
ln t
1
u −3 du
u =lnx,
du = dx/x
=0+ 1 2 = 1 2 . Convergent
= lim − 1 ln t
t→∞ 2u 2 1
27.
1
0
3
dx = lim
x5 t→0 + 1
t
3x −5 dx = lim − 3 1
= − 3
t→0 + 4x 4 t
4 lim 1 − 1
= ∞. Divergent
t→0 + t 4
29.
14
−2
dx
4√ x +2
=
14
14
lim (x +2) −1/4 4
dx = lim
t→−2 + t→−2 + 3 (x +2)3/4
t
= 4 32
(8 − 0) = . Convergent
3 3
t
= 4
3 lim 16 3/4 − (t +2) 3/4
t→−2 +
31.
3
−2
dx 0
x = 4
−2
dx 3
x + 4
0
dx 0
x ,but 4
−2
dx
x 4
= lim
t→0 −
t
− x−3
3
−2
= lim − 1
t→0 − 3t − 1
= ∞. Divergent
3 24
33. There is an infinite discontinuity at x =1.
1
0 (x − 1)−1/5 dx = lim
t→1 − t
0 (x − 1)−1/5 dx = lim
33
1 (x − 1)−1/5 dx = lim
33
t→1 + t
33
0 (x − 1)−1/5 dx = 1
0 (x − 1)−1/5 dx + 33
1 (x − 1)−1/5 dx. Here
t→1 −
5
(x − 1) −1/5 dx = lim
5
t→1 +
Thus, 33
(x − 0 1)−1/5 dx = − 5 75
4
+20= . Convergent
4
5
t→1 −
4 (x − 1)4/5 t
0
= lim
33
(x − 4 1)4/5 = lim
t
(t − 4 1)4/5 − 5 = − 5 and
4 4
5
t→1 +
3
dx
3
35. I =
0 x 2 − 6x +5 = dx
1
0 (x − 1)(x − 5) = I 1 + I 2 =
0
1
Now
(x − 1)(x − 5) = A
x − 1 + B ⇒ 1=A(x − 5) + B(x − 1).
x − 5
Set x =5to get 1=4B,soB = 1 .Setx =1to get 1=−4A,soA = − 1 .Thus
4 4
dx
3
(x − 1)(x − 5) +
4 · 16 − 5 4 (t − 1)4/5
=20.
1
dx
(x − 1)(x − 5) .

F.
338 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
37.
Since I 1 is divergent, I is divergent.
0
−1
e 1/x
x 3
t
−
1 1
4
I 1 = lim
t→1 − 0 x − 1 + 4
t
1
dx = lim
t→0 − −1 x e1/x ·
x − 5
dx = lim
t→1 − − 1 4 ln |x − 1| + 1 4 ln |x − 5| t
0
= lim −
1
ln |t − 1| + 1 ln |t − 5|
t→1 − 4 4
− − 1 ln |−1| +
1
4 4
ln |−5|
= ∞, since lim
t→1 −
−
1
4 ln |t − 1| = ∞.
= lim
t→0 −
(u − 1)e
u −1
1/t
39. I = 2
0 z2 ln zdz= lim
1
dx = lim
x2
t→0 − 1/t
use parts
or Formula 96
−1
ue u (−du)
u =1/x,
du = −dx/x 2
1
= lim −2e −1 −
e
t→0 − t − 1 1/t
= − 2 e − lim (s − s→−∞ 1)es [s =1/t] = − 2 e − lim s − 1
= H − 2
s→−∞ e −s e − lim 1
s→−∞ −e −s
= − 2 e − 0=− 2 e . Convergent
2
t→0 + t
z
3
2 z2 ln zdz= lim
t→0 + 3 (3 ln z − 1) 2 t
integrate by parts
or use Formula 101
= lim 8 (3 ln 2 − 1) − 1
t→0 + 9 9 t3 (3 ln t − 1) = 8 ln 2 − 8 − 1 lim
3 9 9 t 3 (3 ln t − 1) = 8 ln 2 − 8 − 1 L.
t→0 + 3 9 9
Now L = lim
t→0 +
t 3 (3 ln t − 1) = lim
t→0 + 3lnt − 1
t −3
TX.10
H
= lim
t→0 + 3/t
−3/t 4 = lim
t→0 +
−t
3 =0.
Thus, L =0and I = 8 3 ln 2 − 8 9 .
Convergent
41. Area = 1
−∞ ex dx =
lim
t→−∞
e
x 1
= e − lim
t t→−∞ et = e
43. Area =
∞
−∞
2
∞
x 2 +9 dx =2· 2
t 1 x
=4 lim
t→∞ 3 tan−1 = 4 3
0
3 lim
t→∞
0
1
x 2 +9
t
dx = 4 lim
t→∞
0
1
x 2 +9 dx
tan −1 t
3 − 0
= 4 3 · π
2 = 2π 3
45. Area = π/2
sec 2 t
xdx= lim
0
t→(π/2) − 0 sec2 xdx=
= lim t − 0) = ∞
t→(π/2) −(tan
Infinite area
lim [tan x]t
t→(π/2) − 0

F.
SECTION 7.8 IMPROPER INTEGRALS ¤ 339
47. (a)
t
t
1
g(x) dx
2 0.447453
5 0.577101
10 0.621306
100 0.668479
1000 0.672957
10,000 0.673407
g(x) = sin2 x
x 2 .
It appears that the integral is convergent.
(b) −1 ≤ sin x ≤ 1 ⇒ 0 ≤ sin 2 x ≤ 1 ⇒ 0 ≤ sin2 x
≤ 1 ∞
x 2 x .Since 1
dx is convergent
2 x2 [Equation 2 with p =2> 1],
∞
1
sin 2 x
x 2
dx is convergent by the Comparison Theorem.
1
(c)
Since ∞
1
f(x) dx is finite and the area under g(x) is less than the area under f(x)
on any interval [1,t], ∞
1
g(x) dx must be finite; that is, the integral is convergent.
49. For x>0,
x
x 3 +1 < x x 3 = 1 x 2 .
∞
1
1
∞
x dx is convergent by Equation 2 with p =2> 1,so x
dx is convergent
2 x 3 +1
1
by the Comparison Theorem.
convergent.
1
0
x
∞
x 3 +1 dx is a constant, so
0
x
1
x 3 +1 dx =
0
x
∞
x 3 +1 dx +
1
x
dx is also
x 3 +1
51. For x>1, f(x) = √ x +1
x4 − x > x √ +1 > x x
4 x = 1 ∞
∞
2 x ,so 1
f(x) dx diverges by comparison with dx, which diverges
2
2 x
by Equation 2 with p =1≤ 1. Thus, ∞
1
f(x) dx = 2
1 f(x) dx + ∞
2
f(x) dx also diverges.
53. For 0 1 .Now
x3/2 I =
1
0
x −3/2 dx = lim
t→0 + 1
TX.10
1
sec 2 x
comparison,
0 x √ is divergent.
x
t
x −3/2 dx = lim − 2x −1/2
1
= lim −2+ 2
√ = ∞, soI is divergent, and by
t→0 + t t→0 + t

F.
340 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
55.
∞
0
∞
0
dx
1
√ =
x (1 + x) 0
dx
√ =
x (1 + x)
dx
∞
√ +
x (1 + x) 1
2udu
u(1 + u 2 )
u = √ x, x = u 2 ,
dx =2udu
dx
1
√ = lim
x (1 + x) t→0 + t
=2
dx
t
√ +lim
x (1 + x) t→∞
1
dx
√ x (1 + x)
.Now
du
1+u 2 =2tan−1 u + C =2tan −1√ x + C,so
dx
√ = lim 2tan
−1 √ x 1
+lim
x (1 + x) t→0 + t 2tan
−1 √ x t
t→∞
1
= lim 2 π
√
t→0 + 4 − 2tan
−1
t
+ lim
√ 2tan
−1
t − 2
π
t→∞ 4 =
π
− 0+2
π
2 2 −
π
= π. 2
1
dx
1
57. If p =1,then
0 x = lim dx
p t→0 + t x = lim [ln x]1 t→0 + t = ∞.
Divergent.
1
dx
1
If p 6= 1,then
0 x = lim dx
p t→0 + t x p
x
−p+1 1
1
= lim
= lim
t→0 + −p +1
t
t→0 + 1 − p
[note that the integral is not improper if p1,thenp − 1 > 0,so
t →∞as t → p−1 0+ , and the integral diverges.
1
1
If p−1 and diverges otherwise.
(p +1)
2
1/t
lim
t→0 + −(p +1)t −(p+2)

F.
TX.10
SECTION 7.8 IMPROPER INTEGRALS ¤ 341
61. (a) I = ∞
xdx= 0
xdx+ ∞
xdx,and ∞
t
xdx= lim xdx= lim 1 x2 t
= lim
1
−∞ −∞ 0 0 t→∞ 0 t→∞ 2 0 t→∞ 2 t2 − 0 = ∞,
so I is divergent.
(b) t
xdx= 1
x2 t
= 1
−t 2 −t 2 t2 − 1 t
2 t2 =0,so lim xdx=0. Therefore, ∞
t
xdx6= lim xdx.
t→∞ −t −∞ t→∞ −t
63. Volume =
65. Work =
∞
1
∞
R
2 1 t
dx
π dx = π lim
x
t→∞
1 x = π lim − 1 t
= π lim
1 − 1
= π0}.
s
(b) F (s)=
∞
0
= lim
n→∞
f(t)e −st dt =
∞
0
e
(1−s)n
1 − s − 1
1 − s
e t e −st dt = lim
n→∞
n − e−st
s
0
n
0
e
−sn
= lim
n→∞ −s + 1
. This converges to 1 only if s>0.
s
s
n 1
e t(1−s) dt = lim
n→∞ 1 − s et(1−s) 0
This converges only if 1 − s1,inwhichcaseF (s) = 1 with domain {s | s>1}.
s − 1

F.
342 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
(c) F (s) = ∞
f(t)e −st n
dt = lim
0 n→∞ 0 te−st dt. Use integration by parts: let u = t, dv = e −st dt ⇒ du = dt,
v = − e−st
s
.ThenF (s) = lim
n→∞
Therefore, F (s) = 1 and the domain of F is {s | s>0}.
s2
− t s e−st − 1 n −n
s 2 e−st = lim
n→∞
0
se − 1
sn s 2 e +0+ 1
= 1 only if s>0.
sn s 2 s2 73. G(s) = ∞
0
f 0 (t)e −st dt. Integrate by parts with u = e −st , dv = f 0 (t) dt ⇒ du = −se −st , v = f(t):
G(s) = lim
f(t)e
−st n
+ s ∞
f(t)e −st dt = lim
n→∞
0 0 n→∞ f(n)e−sn − f(0) + sF (s)
But 0 ≤ f(t) ≤ Me at ⇒ 0 ≤ f(t)e −st ≤ Me at e −st and lim
t→∞
Me t(a−s) =0for s>a. So by the Squeeze Theorem,
lim
t→∞ f(t)e−st =0for s>a ⇒ G(s) =0− f(0) + sF (s) =sF (s) − f(0) for s>a.
75. We use integration by parts: let u = x, dv = xe −x2 dx ⇒ du = dx, v = − 1 2 e−x2 .So
∞
0
x 2 e −x2 dx = lim
t→∞
− 1 2 xe−x2 t
0
+ 1 2
(The limit is 0 by l’Hospital’s Rule.)
∞
0
e −x2 dx = lim −
t
+ 1
t→∞ 2e t2 2
77. For the firstpartoftheintegral,letx =2tanθ ⇒ dx =2sec 2 θdθ.
1
2sec 2
√
x2 +4 dx = θ
2secθ dθ = sec θdθ=ln|sec θ +tanθ|.
∞
0
e −x2 dx = 1 2
From the figure, tan θ = x √
2 ,andsec θ = x2 +4
.So
2
∞
1
I = √
0 x2 +4 − C
√ dx = lim ln
x2 +4
x +2
t→∞ + x t 2 2 − C ln|x +2| 0
√
t2 +4+t
= lim ln
− C ln(t +2)− (ln 1 − C ln 2)
t→∞ 2
t + √ t
Now L = lim
2 +4 H 1+t/ √ t
= lim
2 +4
t→∞ (t +2) C t→∞ C (t +2) = 2
C−1 C lim (t +2) . C−1
t→∞
If C1, L =0and I diverges to −∞.
∞
0
e −x2 dx
√
t2 +4+t
= lim ln
+ln2 C t + √
t
=ln lim
2 +4
+ln2 C−1
t→∞ 2(t +2) C t→∞ (t +2) C
79. No, I = ∞
0
f(x) dx must be divergent. Since lim
x→∞ f(x) =1,theremustexistanN such that if x ≥ N,thenf(x) ≥ 1 2 .
Thus, I = I 1 + I 2 = N
f(x) dx + ∞
f(x) dx,whereI1 is an ordinary definite integral that has a finite value, and I2 is
0 N
improper and diverges by comparison with the divergent integral ∞
N
1
2 dx.

F.
TX.10
CHAPTER 7 REVIEW ¤ 343
7 Review
1. See Formula 7.1.1 or 7.1.2. We try to choose u = f(x) to be a function that becomes simpler when differentiated (or at least
not more complicated) as long as dv = g 0 (x) dx can be readily integrated to give v.
2. See the Strategy for Evaluating sin m x cos n xdxon page 462.
3. If √ a 2 − x 2 occurs, try x = a sin θ; if √ a 2 + x 2 occurs, try x = a tan θ,andif √ x 2 − a 2 occurs, try x = a sec θ. Seethe
Table of Trigonometric Substitutions on page 467.
4. SeeEquation2andExpressions7,9,and11inSection7.4.
5. See the Midpoint Rule, the Trapezoidal Rule, and Simpson’s Rule, as well as their associated error bounds, all in Section 7.7.
We would expect the best estimate to be given by Simpson’s Rule.
6. See Definitions 1(a), (b), and (c) in Section 7.8.
7. See Definitions 3(b), (a), and (c) in Section 7.8.
8. See the Comparison Theorem after Example 8 in Section 7.8.
1. False. Since the numerator has a higher degree than the denominator, x x 2 +4
3. False. It can be put in the form A x + B x 2 + C
x − 4 .
x 2 − 4
5. False. This is an improper integral, since the denominator vanishes at x =1.
4
0
1
0
x
1
x 2 − 1 dx =
x
t
dx = lim
x 2 − 1 t→1 − 0
So the integral diverges.
0
x
4
x 2 − 1 dx +
1
x
dx = lim
x 2 − 1
x
dx and
x 2 − 1
1
t→1 −
2 ln x 2 − 1 t
0
= x + 8x
x 2 − 4 = x +
1
= lim ln
t→1 − 2 t 2 − 1 = ∞
A
x +2 +
B
x − 2 .
7. False. See Exercise 61 in Section 7.8.
9. (a) True. See the end of Section 7.5.
(b) False.
Examples include the functions f(x) =e x2 , g(x) =sin(x 2 ),andh(x) = sin x
x .
11. False. If f(x) =1/x,thenf is continuous and decreasing on [1, ∞) with lim
x→∞ f(x) =0,but ∞
1
f(x) dx is divergent.
13. False. Take f(x) =1for all x and g(x) =−1 for all x. Then ∞
f(x) dx = ∞ [divergent]
a
and ∞
g(x) dx = −∞ [divergent], but ∞
[f(x)+g(x)] dx =0 [convergent].
a a

F.
344 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
1.
3.
5
0
π/2
0
x
5
x +10 dx =
0
1 − 10
dx = x − 10 ln(x +10) 5
=5− 10 ln 15 + 10 ln 10
x +10
0
=5+10ln 10
15 =5+10ln2 3
cos θ
π/2
1+sinθ dθ = ln(1 + sin θ) =ln2− ln 1 = ln 2
0
π/2
5. sin 3 θ cos 2 θdθ= π/2
(1 − cos 2 θ)cos 2 θ sin θdθ= 0
(1 − 0 0 1 u2 )u 2 (−du)
= 1
0 (u2 − u 4 ) du = 1
3 u3 − 1 u5 1
= 1
− 1
5 0 3 5 − 0=
2
15
sin(ln t)
7. Let u =lnt, du = dt/t. Then
dt =
t
9.
4
1
x 3/2 ln xdx
11. Let x =secθ.Then
2
√
x2 − 1
dx =
x
1
u =lnx,
du = dx/x
π/3
0
dv = x 3/2 dx,
v = 2 5 x5/2
= 2 5
u =cosθ,
du = − sin θdθ
sin udu= − cos u + C = − cos(ln t)+C.
4
x 5/2 ln x − 2
1 5
4
1
x 3/2 dx = 2 5 (32 ln 4 − ln 1) − 2 5
= 2 4
128 124
(64 ln 2) − (32 − 1) = ln 2 −
5 25 5 25
tan θ
π/3
sec θ sec θ tan θdθ= tan 2 θdθ=
0
π/3
0
or
64
5
2
5 x5/2 4
1
124
ln 4 −
25
(sec 2 θ − 1) dθ = tan θ − θ π/3
0
= √ 3 − π 3 .
13. Let t = 3√ x.Thent 3 = x and 3t 2 dt = dx,so e 3√x dx = e t · 3t 2 dt =3I. ToevaluateI,letu = t 2 ,
dv = e t dt ⇒ du =2tdt, v = e t ,soI = t 2 e t dt = t 2 e t − 2te t dt. NowletU = t, dV = e t dt ⇒
dU = dt, V = e t .Thus,I = t 2 e t − 2 te t − e t dt = t 2 e t − 2te t +2e t + C 1 , and hence
3I =3e t (t 2 − 2t +2)+C =3e 3√x (x 2/3 − 2x 1/3 +2)+C.
15.
x − 1
x 2 +2x = x − 1
x(x +2) = A x + B
x +2
to get −1 =2A,soA = − 1 .Thus, 2
⇒ x − 1=A(x +2)+Bx. Setx = −2 to get −3 =−2B,soB = 3 .Setx =0
2
x − 1
−
1 3
x 2 +2x dx = 2
x + 2
dx = − 1 x +2 2 ln |x| + 3 ln |x +2| + C.
2
17. Integrate by parts with u = x, dv =secx tan xdx ⇒ du = dx, v =secx:
14
x sec x tan xdx= x sec x − sec xdx = x sec x − ln|sec x +tanx| + C.
19.
x +1
9x 2 +6x +5 dx =
x +1
(9x 2 +6x +1)+4 dx =
1
3
=
(u − 1)
+1 1
u 2 +4 3 du
= 1
u
9 u 2 +4 du + 1
9
x +1
(3x +1) 2 +4 dx
= 1 3 · 1 (u − 1) + 3
du
3 u 2 +4
u= 3x +1,
du= 3dx
2
u 2 +2 du = 1 2 9 · 1
2 ln(u2 +4)+ 2 9 · 1 1
2 tan−1 2 u + C
= 1
18 ln(9x2 +6x +5)+ 1 9 tan−1 1
2 (3x +1) + C

F.
21.
dx
√
x2 − 4x =
dx
(x2 − 4x +4)− 4 =
2secθ tan θdθ
=
2tanθ
TX.10
dx
(x − 2) 2 − 2 2
x − 2=2secθ,
dx =2secθ tan θdθ
CHAPTER 7 REVIEW ¤ 345
= sec θdθ=ln|sec θ +tanθ| + C 1
=ln
x − 2
√ x2 − 4x
+
2 2 + C1
=ln x − 2+ √ x 2 − 4x + C,whereC = C 1 − ln 2
23. Let x =tanθ,sothatdx =sec 2 θdθ.Then
25.
27.
dx
x √ x 2 +1 =
sec 2
θdθ
tan θ sec θ =
sec θ
tan θ dθ
= csc θdθ=ln|csc θ − cot θ| + C
√ =ln
x2 +1
− 1 √ x x + C =ln x2 +1− 1
x + C
3x 3 − x 2 +6x − 4
(x 2 +1)(x 2 +2) = Ax + B
x 2 +1 + Cx + D
x 2 +2
⇒ 3x 3 − x 2 +6x − 4=(Ax + B) x 2 +2 +(Cx + D) x 2 +1 .
Equating the coefficients gives A + C =3, B + D = −1, 2A + C =6,and2B + D = −4
A =3, C =0, B = −3,andD =2.Now
3x 3 − x 2 +6x − 4
(x 2 +1)(x 2 +2) dx =3 x − 1
x 2 +1 dx +2
dx
x 2 +2 = 3 2 ln x 2 +1 − 3tan −1 x + √ x√2
2tan −1 + C.
π/2
cos 3 x sin 2xdx= π/2
cos 3 x (2 sin x cos x) dx = π/2
2cos 4 x sin xdx= − 2 0 0 0 5 cos5 x π/2
= 2 0 5
29. The product of an odd function and an even function is an odd function, so f(x) =x 5 sec x is an odd function.
By Theorem 5.5.7(b), 1
−1 x5 sec xdx=0.
31. Let u = √ e x − 1. Thenu 2 = e x − 1 and 2udu= e x dx. Also,e x +8=u 2 +9.Thus,
ln 10
0
e x√ e x − 1
e x +8
3
u · 2udu 3
dx =
0 u 2 +9 =2 0
u 2 3
u 2 +9 du =2
0
⇒
1 − 9
du
u 2 +9
=2 u − 9 u
3
3 tan−1 =2 (3 − 3tan −1 1) − 0
=2
3
0
3 − 3 · π
4
=6− 3π 2
33. Let x =2sinθ ⇒ 4 − x 2 3/2 =(2cosθ) 3 , dx =2cosθdθ,so
x 2
(4 − x 2 ) 3/2 dx = 4sin 2 θ
8cos 3 θ 2cosθdθ=
=tanθ − θ + C =
tan 2 θdθ=
sec 2 θ − 1 dθ
x
x
√ − 4 − x
2 sin−1 + C
2

F.
346 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
35.
1
√ dx = x + x
3/2
dx
√ =
x (1 + x )
=4 √ u + C =4 1+ √ x + C
dx
√ x
1+
√ x
TX.10
⎡
⎣
u =1+ √ x,
du =
dx
2 √ x
⎤
⎦ =
2 du
√ = u
2u −1/2 du
37.
(cos x +sinx) 2 cos 2xdx= cos 2 x +2sinx cos x +sin 2 x cos 2xdx= (1 + sin 2x)cos2xdx
= cos 2xdx+ 1 2
sin 4xdx=
1
sin 2x − 1 cos 4x + C
2 8
Or: (cos x +sinx) 2 cos 2xdx= (cos x +sinx) 2 (cos 2 x − sin 2 x) dx
39. We’ll integrate I =
41.
and v = − 1 2 ·
1/2
Thus,
0
∞
1
43.
45.
∞
2
4
0
I = − 1 2 ·
1
1+2x ,so
= (cos x +sinx) 3 (cos x − sin x) dx = 1 4 (cos x +sinx)4 + C 1
xe 2x
(1 + 2x) dx by parts with u = dx
2 xe2x and dv =
(1 + 2x) .Thendu =(x · 2 2e2x + e 2x · 1) dx
xe 2x
1+2x −
xe 2x
1
(1 + 2x) dx = e 2x 2 4 −
1
t
3
dx = lim
(2x +1) t→∞
1
dx
x ln x
u =lnx,
du = dx/x
dx
x ln x = lim
t→∞
= − 1 4 lim
t→∞
t
2
=
du
u
− 1 2 · e2x (2x +1)
1+2x
dx = − xe2x
4x +2 + 1 2 · 1
2 e2x + C = e 2x 1
4 − x
+ C
4x +2
x 1/2 1
= e
4x +2
0
4 − 1 1
− 1
8
4 − 0 = 1 8 e − 1 4 .
t
1
dx = lim
(2x +1)
3 t→∞
1
1
(2t +1) 2 − 1
= − 1 9 4
1
(2x 2 +1)−3 2 dx = lim
t→∞
0 − 1
= 1
9 36
=ln|u| + C =ln|ln x| + C, so
t
1
−
4(2x +1) 2 1
dx
t
x ln x = lim ln |ln x| = lim [ln(ln t) − ln(ln 2)] = ∞, so the integral is divergent.
t→∞
2 t→∞
ln x
4
ln x
√ dx = lim √ dx = ∗ lim 2 √ x ln x − 4 √ 4
x
x t→0 + t x t→0 + t
= lim
t→0 +
(2 · 2ln4− 4 · 2) −
2
√
t ln t − 4
√
t
∗∗
=(4ln4− 8) − (0 − 0) = 4 ln 4 − 8
(∗) Letu =lnx, dv = 1 √
x
dx ⇒ du = 1 x dx, v =2√ x.Then
ln x
√ dx =2 √
x ln x − 2
x
dx
√
x
=2 √ x ln x − 4 √ x + C
(∗∗)
√ 2lnt
lim 2 t ln t = lim
t→0 + t→0 + t −1/2
= H 2/t √
lim = lim −4 t =0
t→0 + t−3/2 t→0 +
− 1 2
47.
1
0
x − 1
1
x√x
√ dx = lim − √ 1 dx = lim
x t→0 + t
x
= lim
t→0 + 2
3 − 2 −
1
t→0 + t
(x 1/2 − x −1/2 ) dx = lim
2
3 t3/2 − 2t 1/2
= − 4 3 − 0=− 4 3
2
t→0 +
3 x3/2 − 2x 1/2 1
t

F.
TX.10
CHAPTER 7 REVIEW ¤ 347
49. Let u =2x +1.Then
∞
−∞
dx
∞
4x 2 +4x +5 =
−∞
1
du 2
u 2 +4 = 1 0
2 −∞
= 1 2
lim 1 tan−1 1
u 0
+ 1
t→−∞ 2 2 t 2
du
u 2 +4 + 1 ∞
2 0
du
u 2 +4
lim
1 tan−1 1
u t
= 1
t→∞ 2 2 0 4
0 − −
π
2 +
1 π − 0 4 2
= π . 4
51. We first make the substitution t = x +1,soln(x 2 +2x +2)=ln (x +1) 2 +1 =ln(t 2 +1). Then we use parts with
u =ln(t 2 +1), dv = dt:
ln(t 2 +1)dt = t ln(t 2 +1)−
t(2t) dt
t 2 +1 = t ln(t2 +1)− 2
t 2
dt
t 2 +1 = t ln(t2 +1)− 2 1 − 1
dt
t 2 +1
= t ln(t 2 +1)− 2t +2arctant + C
=(x +1)ln(x 2 +2x +2)− 2x + 2 arctan(x +1)+K, whereK = C − 2
[Alternatively, we could have integrated by parts immediately with
u =ln(x 2 +2x +2).] Notice from the graph that f =0where F has a
horizontal tangent. Also, F is always increasing, and f ≥ 0.
53. From the graph, it seems as though 2π
cos 2 x sin 3 xdxis equal to 0.
0
To evaluate the integral, we write the integral as
I = 2π
0
cos 2 x (1 − cos 2 x)sinxdxand let u =cosx ⇒
du = − sin xdx. Thus, I = 1
1 u2 (1 − u 2 )(−du) =0.
55.
√
4x2 − 4x − 3 dx = (2x − 1) 2 − 4 dx
39
= 1 2
u =2x − 1,
du =2dx
= √ u 2 − 2 2 1
2 du
u √
u2 − 2
2 2 − 22
2 ln
√ u + u2 − 2 2 + C = 1 u √ 4
u 2 − 4 − ln √ u + u2 − 4 + C
57. Let u =sinx,sothatdu =cosxdx.Then
59. (a)
= 1 4 (2x − 1) √ 4x 2 − 4x − 3 − ln 2x − 1+ √ 4x 2 − 4x − 3 + C
cos x 4+sin 2 xdx= √ 2 2 + u 2 du 21 = u √
22 + u
2 2 + 22
2 ln u + √ 2 2 + u 2 + C
= 1 sin x
4+sin 2 x +2ln sin x +
4+sin 2 x + C
2
d
− 1 √ u
a2 − u
du u
2 − sin −1 + C = 1 √ 1
a2 − u
a u 2 + √ 2 a2 − u − 1
2 1 − u2 /a · 1
2 a
= a 2 − u 2
−1/2 1 a 2 − u 2 √
a2 − u
+1− 1 =
2
u 2 u 2

F.
348 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION
TX.10
(b) Let u = a sin θ ⇒ du = a cos θdθ, a 2 − u 2 = a 2 1 − sin 2 θ = a 2 cos 2 θ.
√
a2 − u 2 a 2 cos 2
θ 1 − sin 2
du =
u 2 a 2 sin 2 θ dθ = θ
sin 2 θ
√
a2 − u
= −
2 u
− sin −1 + C
u
a
dθ = (csc 2 θ − 1) dθ = − cot θ − θ + C
61. For n ≥ 0, ∞
0
x n dx = lim
t→∞
x n+1 /(n +1) t
0 = ∞. Forn

F.
TX.10
CHAPTER 7 REVIEW ¤ 349
71.
(c)Ifwewanttheerrortobelessthan0.00001,wemusthave|E S| ≤ 3.8π5
180n 4 ≤ 0.00001,
so n 4 ≥
3.8π 5
≈ 646,041.6 ⇒ n ≥ 28.35. Sincen must be even for Simpson’s Rule, we must have n ≥ 30
180(0.00001)
to ensure the desired accuracy.
x 3
x 5 +2 ≤ x3
x 5 = 1 x 2 for x in [1, ∞). ∞
convergent by the Comparison Theorem.
73. For x in 0, π 2
1
1
∞
x dx is convergent by (7.8.2) with p =2> 1. Therefore, 2
, 0 ≤ cos 2 x ≤ cos x.Forx in π
2 ,π , cos x ≤ 0 ≤ cos 2 x.Thus,
area = π/2
(cos x − cos 2 x) dx + π
0 π/2 (cos2 x − cos x) dx
= sin x − 1 x − 1 sin 2x π/2
2 4
+ 1
x + 1 sin 2x − sin x π
=
0 2 4
1 − π π/2 4 − 0 + π − π
− 1 2 4
=2
75. Using the formula for disks, the volume is
V = π/2
π [f(x)] 2 dx = π π/2
(cos 2 x) 2 dx = π π/2
1 (1 + cos 0 0 0 2 2x)2 dx
= π 4
π/2
0
(1 + cos 2 2x +2cos2x) dx = π 4
= π 4
3
2 x + 1 2
77. By the Fundamental Theorem of Calculus,
1 sin 4x +2 1
sin 2x π/2
= π 4 2 0 4
π/2
0 1+
1
3π
4
2 (1 + cos 4x)+2cos2x dx
+ 1
8 · 0+0 − 0 = 3π2
16
1
x 3
dx is
x 5 +2
∞
0
f 0 (x) dx = lim
t→∞
t
0 f 0 (x) dx = lim
t→∞
[f(t) − f(0)] = lim
t→∞
f(t) − f(0) = 0 − f(0) = −f(0).
79. Let u =1/x ⇒ x =1/u ⇒ dx = −(1/u 2 ) du.
∞
0
∞
Therefore,
0
ln x
0
1+x dx = 2
∞
ln x
∞
1+x dx = − 2
ln (1/u)
− du 0
− ln u
0
=
1+1/u 2 u 2 ∞ u 2 +1 (−du) = ∞
0
ln x
dx =0.
1+x2
ln u
∞
1+u du = − 2
0
ln u
1+u 2 du

F.
TX.10

F.
TX.10
PROBLEMS PLUS
1.
By symmetry, the problem can be reduced to finding the line x = c such that the shaded area is one-third of the area of the
quarter-circle.An equation of the semicircle is y = √ 49 − x 2 , so we require that c
√
0 49 − x2 dx = 1 · 1
3 4 π(7)2 ⇔
1 x √ 49 − x
2 2 + 49
2 sin−1 (x/7) c
= 49 π [by Formula 30] ⇔ 1 c √ 49 − c
0 12 2 2 + 49 2 sin−1 (c/7) = 49 π. 12
This equation would be difficult to solve exactly, so we plot the left-hand side as a function of c,andfind that the equation
holds for c ≈ 1.85. So the cuts should be made at distances of about 1.85 inches from the center of the pizza.
3. The given integral represents the difference of the shaded areas, which appears to
be 0. It can be calculated by integrating with respect to either x or y,sowefind x
in terms of y for each curve: y = 3√ 1 − x 7 ⇒ x = 7 1 − y 3 and
y = 7√ 1 − x 3 ⇒ x = 3 1 − y 7 ,so
1
3
0 1 − y7 − 7
1 − y 3 dy = 1√ 7
1 − x3 − 3√ 1 − x 7 dx. But this
equation is of the form z = −z. So 1√ 3
0 1 − x7 − 7√ 1 − x 3 dx =0.
0
5. The area A of the remaining part of the circle is given by
a
a2
A =4I =4 − x 2 − b
a2 − x
a 2 dx =4 1 − b a
a2 − x
a
2 dx
0
30
= 4 a (a − b) x
2
= 4 a (a − b)
0+ a2
2
√
a2 − x 2 + a2
2 sin−1 x
a
a
0
π
− 0
2
which is the area of an ellipse with semiaxes a and a − b.
= 4 a 2
a (a − b) π
= πa(a − b),
4
Alternate solution: Subtracting the area of the ellipse from the area of the circle gives us πa 2 − πab = πa (a − b),
as calculated above. (The formula for the area of an ellipse was derived in Example 2 in Section 7.3.)
0
351

F.
352 ¤ CHAPTER 7 PROBLEMS PLUS
TX.10
7. Recall that cos A cos B = 1 2
[cos(A + B)+cos(A − B)]. So
f(x)= π
cos t cos(x − t) dt = 1 π [cos(t + x − t)+cos(t − x + t)] dt = 1 π
0 2 0 2 0
= 1 2 t cos x +
1
sin(2t − x) π
= π cos x + 1 sin(2π − x) − 1 sin(−x)
2 0 2 4 4
= π cos x + 1 sin(−x) − 1 sin(−x) = π cos x
2 4 4 2
The minimum of cos x on this domain is −1, so the minimum value of f(x) is f(π) =− π 2 .
[cos x +cos(2t − x)] dt
9. In accordance with the hint, we let I k = 1(1−x 2 )k
0
dx,andwefind an expression for I k+1 in terms of I k .WeintegrateI k+1
by parts with u =(1− x 2 ) k+1 ⇒ du =(k +1)(1− x 2 ) k (−2x), dv = dx ⇒ v = x, and then split the remaining
integral into identifiable quantities:
I k+1 = x(1 − x 2 ) k+1 1 +2(k +1) 1
0 0 x2 (1 − x 2 ) k dx =(2k +2) 1
(1 − 0 x2 ) k [1 − (1 − x 2 )] dx
=(2k +2)(I k − I k+1 )
So I k+1 [1 + (2k +2)]=(2k +2)I k ⇒ I k+1 =
2k +2
2k +3 I k. Now to complete the proof, we use induction:
I 0 =1= 20 (0!) 2
, so the formula holds for n =0. Now suppose it holds for n = k. Then
1!
2k +2
I k+1 =
2k +3 I 2k +2 2 2k (k!) 2
k = = 2(k +1)22k (k!) 2 2(k +1)
=
2k +3 (2k +1)! (2k +3)(2k +1)! 2k +2 · 2(k +1)22k (k!) 2
(2k + 3)(2k +1)!
=
[2(k +1)] 2 2 2k (k!) 2
(2k + 3)(2k +2)(2k +1)! = 22(k+1) [(k +1)!] 2
[2(k +1)+1]!
So by induction, the formula holds for all integers n ≥ 0.
11. 0

F.
TX.10
CHAPTER 7 PROBLEMS PLUS ¤ 353
13. An equation of the circle with center (0,c) and radius 1 is x 2 +(y − c) 2 =1 2 ,so
an equation of the lower semicircle is y = c − √ 1 − x 2 . At the points of tangency,
the slopes of the line and semicircle must be equal. For x ≥ 0,wemusthave
y 0 =2
⇒
x
√
1 − x
2 =2 ⇒ x =2√ 1 − x 2 ⇒ x 2 =4(1− x 2 ) ⇒
√
5x 2 =4 ⇒ x 2 = 4 ⇒ x = 2 5 5 5 and so y =2 2
√ √
5 5 =
4
5 5.
The slope of the perpendicular line segment is − 1 , so an equation of the line segment is y − √ √
4
2 5 5=−
1
2 x −
2
5 5
⇔
y = − 1 2 x + 1 5
Thus, the shaded area is
2
(2/5)
√
5
0
√ √
5+
4
5 5 ⇔ y = −
1
x + √ 2
5,soc = √ 5 and an equation of the lower semicircle is y = √ 5 − √ 1 − x 2 .
√ √5
5 − 1 − x
2
− 2x dx =2
30 x √
x − 1 − x2 − 1
√
(2/5) 5
2
2 sin−1 x − x 2 0
√
5
=2 2 −
5 · 1
√ − 1 2√5
5 2 sin−1 − 4
− 2(0)
5
=2
1 − 1
2√5 2√5
2 sin−1 =2− sin −1
15. We integrate by parts with u =
integral becomes
∞
where J =
0
1
−1
, dv =sintdt,sodu =
2
and v = − cos t. The
ln(1 + x + t) (1 + x + t)[ln(1 + x + t)]
∞
b
sin tdt
I =
0 ln(1 + x + t) = lim − cos t
b→∞ ln(1 + x + t)
0
= lim
b→∞
∞
at isolated points. So −
0
− cos b
ln(1 + x + b) + 1
∞
ln(1 + x) +
0
b
−
0
cos tdt
(1 + x + t)[ln(1 + x + t)] 2
− cos tdt
(1 + x + t)[ln(1 + x + t)] 2 = 1
ln(1 + x) + J
− cos tdt
2
.Now−1 ≤−cos t ≤ 1 for all t; in fact, the inequality is strict except
(1 + x + t)[ln(1 + x + t)]
dt
∞
(1 + x + t)[ln(1 + x + t)] 2

F.
TX.10

F.
TX.10
8 FURTHER APPLICATIONS OF INTEGRATION
8.1 Arc Length
1. y =2x − 5 ⇒ L = 3
1+(dy/dx)2 dx = 3
1+(2)2 dx = √ 5[3− (−1)] = 4 √ 5.
−1
−1
The arc length can be calculated using the distance formula, since the curve is a line segment, so
L =[distance from (−1, −7) to (3, 1)] = [3 − (−1)] 2 +[1− (−7)] 2 = √ 80 = 4 √ 5
3. y =cosx ⇒ dy/dx = − sin x ⇒ 1+(dy/dx) 2 =1+sin 2 x.SoL =
2π
0 1+sin 2 xdx.
5. x = y + y 3 ⇒ dx/dy =1+3y 2 ⇒ 1+(dx/dy) 2 =1+(1+3y 2 ) 2 =9y 4 +6y 2 +2.
So L = 4
1
9y4 +6y 2 +2dy.
7. y =1+6x 3/2 ⇒ dy/dx =9x 1/2 ⇒ 1+(dy/dx) 2 =1+81x. So
L = 1 √ 82
0 1+81xdx= u 1/2
1
du u =1+81x,
82
= 1
1 81 du =81dx · √
u 2 3/2 81 3
= 2
243 82 82 − 1
1
9. y = x5
6 + 1
10x 3 ⇒ dy
dx = 5 6 x4 − 3 10 x−4
⇒
1+(dy/dx) 2 =1+ 25
36 x8 − 1 + 9
2 100 x−8 = 25
36 x8 + 1 + 9
2 100 x−8 = 5
6 x4 + 3 10 x−42 .So
L =
2 5
1 6 x4 + 3
10 x−42 dx = 2
5
1 6 x4 + 3 x−4 dx = 1
10 6 x5 − 1 x−3 2
= 32
− 1
10 1 6 80 − 1 −
1
6 10
= 31 + 7 = 1261
6 80 240
11. x = 1 √
3 y (y − 3) =
1
3 y3/2 − y 1/2 ⇒ dx/dy = 1 2 y1/2 − 1 2 y−1/2 ⇒
1+(dx/dy) 2 =1+ 1 y − 1 + 1 4 2 4 y−1 = 1 y + 1 + 1 4 2 4 y−1 1
2.So
=
2 y1/2 + 1 2 y−1/2
L = 9
1
= 1 2
1
2 y1/2 + 1 2 y−1/2
dy = 1 2
24 −
8
3
=
1
2
64
3
=
32
3 .
9
2
3 y3/2 +2y 1/2 = 1 2 · 27 + 2 · 3 − 2 · 1+2· 1
2 3 3
1
13. y =ln(secx) ⇒ dy sec x tan x
= =tanx ⇒ 1+
dx sec x
2 dy
=1+tan 2 x =sec 2 x,so
dx
L = π/4
√
sec2 xdx= π/4
|sec x| dx =
π/4
π/4
sec xdx= ln(sec x +tanx)
0
0 0
0
=ln √ 2+1 − ln(1 + 0) = ln √ 2+1 355

F.
356 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
15. y =ln(1− x 2 ) ⇒ y 0 1
=
1 − x · (−2x) ⇒
2
2 dy
4x 2
1+ =1+
dx (1 − x 2 ) = 1 − 2x2 + x 4 +4x 2
= 1+2x2 + x 4
= (1 + x2 ) 2
⇒
2 (1 − x 2 ) 2 (1 − x 2 ) 2 (1 − x 2 ) 2
1+
So L =
2 dy 1+x
2 2
= = 1+x2
dx 1 − x 2 1 − x = −1+ 2
2 1 − x 2 [by division] = −1+ 1
1+x + 1
1 − x
1/2
0
[partial fractions].
−1+ 1
1+x + 1
dx = −x +ln|1+x| − ln |1 − x| 1/2
= − 1
1 − x
0 2 +ln3 − ln 1
2 2 − 0=ln3−
1
. 2
17. y = e x ⇒ y 0 = e x ⇒ 1+(y 0 ) 2 =1+e 2x .So
L =
1
0
e
1+e
2x
dx = du 1+u
2
u
√ 1+e 2
v
=
√
2 v 2 − 1 vdv
1
v = √ 1+u 2 ,so
v 2 =1+u 2 , vdv= udu
u = e x ,so
x =lnu, dx = du/u
√ 1+e 2
=
√
2
=
e
1
√
1+u
2
udu
u 2
1+ 1/2
v − 1 − 1/2
dv
v +1
= v + 1 2 ln v − 1 √ 1+e 2
= √ 1+e
v +1
2 + 1 √
√
2
2 ln 1+e2 − 1
√
1+e2 +1 − √ 2 − 1 √
2 − 1
2 ln √
2+1
= √ 1+e 2 − √ 2+ln √ 1+e 2 − 1 − 1 − ln √ 2 − 1
Or: Use Formula 23 for √ 1+u 2 /u du, or substitute u =tanθ.
19. y = 1 2 x2 ⇒ dy/dx = x ⇒ 1+(dy/dx) 2 =1+x 2 .So
L = 1
√
1+x2 dx =2 1
√
1+x2 dx [by symmetry]
−1
0
=2 √
1
2 2+
1
ln 1+ √ 2 − 0+ 1 ln 1 = √ 2+ln 1+ √ 2
2 2
21 =2 √
x 1+x2 + 1 ln x + √ 1+x 2
2 2 1
0
or substitute
x =tanθ
21. From the figure, the length of the curve is slightly larger than the hypotenuse
of the triangle formed by the points (1, 0), (3, 0),and(3,f(3)) ≈ (3, 15),
where y = f(x) = 2 3 (x2 − 1) 3/2 . This length is about √ 15 2 +2 2 ≈ 15,so
we might estimate the length to be 15.5.
y = 2 3 (x2 − 1) 3/2 ⇒ y 0 =(x 2 − 1) 1/2 (2x) ⇒ 1+(y 0 ) 2 =1+4x 2 (x 2 − 1) = 4x 4 − 4x 2 +1=(2x 2 − 1) 2 ,
so, using the fact that 2x 2 − 1 > 0 for 1 ≤ x ≤ 3,
L = 3
(2x2 − 1)
1
2 dx = 3
2x 2 − 1 dx = 3
1
1 (2x2 − 1) dx = 2
3 x3 − x 3
=(18− 3) − 2
− 1 = 46 =15.3.
1 3 3
23. y = xe −x ⇒ dy/dx = e −x − xe −x = e −x (1 − x) ⇒ 1+(dy/dx) 2 =1+e −2x (1 − x) 2 .Let
f(x) = 1+(dy/dx) 2 = 1+e −2x (1 − x) 2 .ThenL = 5
5 − 0
f(x) dx. Sincen =10, ∆x = = 1 .Now
0 10 2
L ≈ S 10 = 1/2
3 [f(0) + 4f 1
2
+2f(1) + 4f
3
2
+2f(2) + 4f
5
2
+2f(3) + 4f
7
2
+2f(4) + 4f
9
2
+ f(5)]
≈ 5.115840
The value of the integral produced by a calculator is 5.113568 (to six decimal places).

F.
TX.10
SECTION 8.1 ARC LENGTH ¤ 357
25. y =secx ⇒ dy/dx =secx tan x ⇒ L = π/3
0
f(x) dx,wheref(x) = √ 1+sec 2 x tan 2 x.
27. (a)
Since n =10, ∆x = π/3 − 0
10
L ≈ S 10 = π/30
3
= π 30 .Now
π
f(0) + 4f +2f
30
+2f
6π
+4f
30
2π
30
7π
30
+4f
+2f
3π
30
8π
30
+2f
+4f
4π
30
9π
30
The value of the integral produced by a calculator is 1.569259 (to six decimal places).
+4f
5π
30
π
+ f ≈ 1.569619.
3
(b)
Let f(x) =y = x 3√ 4 − x. The polygon with one side is just
L 2 =
the line segment joining the points (0,f(0)) = (0, 0) and
(4,f(4)) = (4, 0), and its length L 1 =4.
The polygon with two sides joins the points (0, 0),
(2,f(2)) = 2, 2 3√ 2 and (4, 0). Its length
(2 − 0) 2 + 2 3√ 2 − 0 2
+
(4 − 2) 2 + 0 − 2 3√ 2 2
=2
√
4+2
8/3
≈ 6.43
Similarly, the inscribed polygon with four sides joins the points (0, 0), 1, 3√ 3 , 2, 2 3√ 2 , (3, 3),and(4, 0),
so its length
0
L 3 =
1+ √ 3
3 2
+
1+ 2 3√ 2 − 3√ 3
2
+ 1+ 3 − 2 3√ 2 2 √ + 1+9≈ 7.50
(c) Using the arc length formula with dy
dx = x 1
(4 − 3 x)−2/3 (−1) + 3√ 12 − 4x
4 − x = , the length of the curve is
3(4 − x)
2/3
4
2 dy
4
2 12 − 4x
L = 1+ dx = 1+
dx.
dx
3(4 − x) 2/3
0
(d) According to a CAS, the length of the curve is L ≈ 7.7988. The actual value is larger than any of the approximations in
part (b). This is always true, since any approximating straight line between two points on the curve is shorter than the
length of the curve between the two points.

F.
358 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
29. y =lnx ⇒ dy/dx =1/x ⇒ 1+(dy/dx) 2 =1+1/x 2 =(x 2 +1)/x 2 ⇒
2
x 2
+1
2
√ 1+x
2 1+x2 L =
dx =
dx 23 = − ln 1 x 2 1 x
1+√ 1+x 2
x
= √ √ 1+ 5
5 − ln
− √ 2+ln 1+ √ 2
2
31. y 2/3 =1− x 2/3 ⇒ y =(1− x 2/3 ) 3/2 ⇒
dy
dx = 3 (1 − 2 x2/3 ) 1/2 − 2 3 x−1/3 = −x −1/3 (1 − x 2/3 ) 1/2 ⇒
2 dy
= x −2/3 (1 − x 2/3 )=x −2/3 − 1. Thus
dx
L =4
1
0 1+(x
−2/3
− 1) dx =4 1
0 x−1/3 dx = 4 lim
1
3
t→0 + 2 x2/3 t
33. y =2x 3/2 ⇒ y 0 =3x 1/2 ⇒ 1+(y 0 ) 2 =1+9x. The arc length function with starting point P 0 (1, 2) is
s(x) = x √
x
1 1+9tdt=
2
(1 + 27 9t)3/2 = 2 (1 + 9x) 3/2 − 10 √
10 .
27
1
=6.
2
1
35. y =sin −1 x + √ 1 − x 2 ⇒ y 0 =
1
√ − x
√ = √ 1 − x
1 − x
2 1 − x
2 1 − x
2
⇒
1+(y 0 ) 2 (1 − x)2
=1+ = 1 − x2 +1− 2x + x 2
= 2 − 2x
1 − x 2 1 − x 2 1 − x = 2(1 − x)
2 (1 + x)(1 − x) = 2
1+x ⇒
2
1+(y0 ) 2 = . Thus, the arc length function with starting point (0, 1) is given by
1+x
x
x
s(x) = 1+[f 0 2
(t)] 2 dt =
1+t dt = √ 2 2 √ 1+t x
=2√ 2 √ 1+x − 1 .
0
0
0
37. The prey hits the ground when y =0 ⇔ 180 − 1 45 x2 =0 ⇔ x 2 =45· 180 ⇒ x = √ 8100 = 90,
since x must be positive. y 0 = − 2 x ⇒ 45 1+(y0 ) 2 =1+ 4 x 2 , so the distance traveled by the prey is
45 2
90
L = 1+ 4 4
45 2 x2 dx =
1+u
2 45
du u = 2
45 x,
2 du = 2
0
21
= 45 2
0
45 dx
1 u √ 1+u
2 2 + 1 ln u + √ 1+u 2 2 4
= √ 45
0 2 2 17 +
1
ln 4+ √ 17 =45 √ 17 + 45 ln 4+ √ 17 ≈ 209.1 m
2 4
39. Thesinewavehasamplitude1 and period 14, since it goes through two periods in a distance of 28 in., so its equation is
y =1sin 2π
14 x =sin π
7 x . The width w of the flat metal sheet needed to make the panel is the arc length of the sine curve
from x =0to x =28.Wesetuptheintegraltoevaluatew using the arc length formula with dy
dx = π 7 cos π
L = 28
0
1+ π
7 cos π
7 x2 dx =2 14
0
so we use a CAS, and find that L ≈ 29.36 inches.
x :
7
1+ π
cos π
7 7 x2 dx. This integral would be very difficult to evaluate exactly,
41. y = x
√
t3 − 1 dt ⇒ dy/dx = √ x
1
3 − 1 [by FTC1] ⇒ 1+(dy/dx) 2 =1+ √ x 3 − 1 2
= x
3
L = 4
√
x3 dx = 4
1
1 x3/2 dx = 2 5
x 5/2 4
1
= 2 5 (32 − 1) = 62 5 =12.4
⇒

F.
TX.10
SECTION 8.2 AREA OF A SURFACE OF REVOLUTION ¤ 359
8.2 Area of a Surface of Revolution
1. y = x 4 ⇒ dy/dx =4x 3 ⇒ ds = 1+(dy/dx) 2 dx = √ 1+16x 6 dx
(a) By (7), an integral for the area of the surface obtained by rotating the curve about the x-axis is
S = 2πy ds = 1
0 2πx4 √ 1+16x 6 dx.
(b) By (8), an integral for the area of the surface obtained by rotating the curve about the y-axis is
S = 2πx ds = 1
0 2πx √ 1+16x 6 dx.
3. y =tan −1 x ⇒ dy
dx = 1
1+x 2 ⇒ ds =
(a) By (7), S = 2πy ds = 1
0 2π tan−1 x
1+
1+
2
dy
1
dx = 1+
dx
(1 + x 2 ) dx. 2
1
(1 + x 2 ) 2 dx.
(b) By (8), S = 2πx ds = 1
2πx 1
1+
0
(1 + x 2 ) 2 dx.
5. y = x 3 ⇒ y 0 =3x 2 .So
S = 2
2πy 1+(y
0 0 ) 2 dx =2π 2
√
0 x3 1+9x 4 dx [u =1+9x 4 , du =36x 3 dx]
= 2π
36
145 √
1 udu=
π
18
7. y = √ 1+4x ⇒ y 0 = 1 2 (1 + 4x)−1/2 (4) =
145 √
2
3 u3/2 = π 27 145 145 − 1
1
2
√ 1+4x
⇒ 1+(y 0 ) 2 =
S = 5
2πy 1+(y
1 0 ) 2 dx =2π
5 √ 5+4x
1 1+4x
1+4x dx =2π 5 √
1 4x +5dx
=2π 25 √
9 u 1 du 4
u =4x +5,
du =4dx
= 2π 4
9. y =sinπx ⇒ y 0 = π cos πx ⇒ 1+(y 0 ) 2 =1+π 2 cos 2 (πx). So
S = 1
2πy 1+(y
0 0 ) 2 dx =2π 1
sin πx
1+π
0 2 cos 2 (πx) dx
1+ 4
1+4x =
5+4x
1+4x .So
25
2
3 u3/2 = π 3 (253/2 − 9 3/2 )= π (125 − 27) = 98 π
3 3
9
u = π cos πx,
du = −π 2 sin πx dx
−π
=2π 1+u
2
− 1
π
π du = 2 π
1+u2 du
2 π −π
= 4 π
1+u2 du 21 = 4 u
1+u2 + 1 2
π 0
π 2 ln u + π
1+u 2 0
= 4 π √
1+π2 + 1 2
π 2
ln π + √ 1+π 2 − 0
=2 √ 1+π 2 + 2 π ln π + √ 1+π 2
11. x = 1 3 (y2 +2) 3/2 ⇒ dx/dy = 1 2 (y2 +2) 1/2 (2y) =y y 2 +2 ⇒ 1+(dx/dy) 2 =1+y 2 (y 2 +2)=(y 2 +1) 2 .
So S =2π 2
1 y(y2 +1)dy =2π 1
4 y4 + 1 y2 2
=2π 4+2− 1 − 1
2 1 4 2 =
21π
. 2
13. y = 3√ x ⇒ x = y 3 ⇒ 1+(dx/dy) 2 =1+9y 4 .So
S =2π 2
x 1+(dx/dy)
1 2 dy =2π 2
y3
1+9y
1 4 dy = 2π 2
36 1
√ √
145 145 − 10 10
= π 27
1+9y4 36y 3 dy = π 18
2
3
1+9y
4 3/2 2
1

F.
360 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
15. x = a 2 − y 2 ⇒ dx/dy = 1 2 (a2 − y 2 ) −1/2 (−2y) =−y/ a 2 − y 2 ⇒
1+
S =
2 dx
=1+
dy
a/2
0
y2
a 2 − y = a2 − y 2
2 a 2 − y + 2
y2
a 2 − y 2 =
2π
a
a/2
a 2 − y 2
a2 − y dy =2π 2
0
a2
a 2 − y 2
⇒
ady =2πa y a/2
0
a
=2πa
2 − 0 = πa 2 .
Note that this is 1 the surface area of a sphere of radius a, and the length of the interval y =0to y = a/2 is 1 the length of the
4 4
interval y = −a to y = a.
17. y =lnx ⇒ dy/dx =1/x ⇒ 1+(dy/dx) 2 =1+1/x 2 ⇒ S = 3
1 2π ln x 1+1/x 2 dx.
Let f(x) =lnx 1+1/x 2 .Sincen =10, ∆x = 3 − 1
10
= 1 5 . Then
S ≈ S 10 =2π · 1/5
3
[f(1) + 4f(1.2) + 2f(1.4) + ···+2f(2.6) + 4f(2.8) + f(3)] ≈ 9.023754.
The value of the integral produced by a calculator is 9.024262 (to six decimal places).
19. y =secx ⇒ dy/dx =secx tan x ⇒ 1+(dy/dx) 2 =1+sec 2 x tan 2 x ⇒
S = π/3
2π sec x √ 1+sec
0 2 x tan 2 xdx.Letf(x) =secx √ 1+sec 2 x tan 2 x.Sincen =10, ∆x = π/3 − 0
10
Then S ≈ S 10 =2π · π/30 π
f(0) + 4f +2f
3
30
2π
+ ···+2f
30
8π
+4f
30
The value of the integral produced by a calculator is 13.516987 (to six decimal places).
9π
π
+ f
30 3
≈ 13.527296.
21. y =1/x ⇒ ds = 1+(dy/dx) 2 dx = 1+(−1/x 2 ) 2 dx = 1+1/x 4 dx ⇒
2
S = 2π · 1
1+ 1 2
√
1 x x dx =2π x4 +1
4
√
u2 +1 dx =2π
1 4 1 x 3 1 u du 2 [u = x 2 , du =2xdx]
2
4
√ √
1+u
2
= π
du 24 1+u
2
= π − +ln
u + 4
1+u
1 u 2
u
2 1
= π − √ 17
+ln 4+ √ 17 + √ 2
− ln 1+ √ 2 = π √ √ √ √
4 1 4ln 17 + 4 − 4ln 2+1 − 17 + 4 2
4
23. y = x 3 and 0 ≤ y ≤ 1 ⇒ y 0 =3x 2 and 0 ≤ x ≤ 1.
S = 1
2πx 1+(3x
0 2 ) 2 dx =2π
3
√
0 1+u
2 1
du 6
u =3x 2 ,
du =6xdx
= π 3
3
0
√
1+u2 du
= π 30 .
21
= [or use CAS] π 1 u √ 1+u
3 2 2 + 1 ln u + √ 1+u 2 2 3
= π 3
√
0 3 2 10 +
1
ln 3+ √ 10 √ √
= π 2 6 3 10 + ln 3+ 10
25. S =2π
∞
1
1
y
1+
2 dy
∞
1
dx =2π 1+ 1 ∞
√
dx
1 x x dx =2π x4 +1
dx. Ratherthantryingto
4 1 x 3
evaluate this integral, note that √ x 4 +1> √ x 4 = x 2 for x>0. Thus, if the area is finite,
∞
√
x4 +1
∞
x 2 ∞
S =2π
dx > 2π
x 3
x dx =2π 1
dx. But we know that this integral diverges, so the area S is
3 x
infinite.
1
1

F.
TX.10
SECTION 8.2 AREA OF A SURFACE OF REVOLUTION ¤ 361
27. Since a>0,thecurve3ay 2 = x(a − x) 2 only has points with x ≥ 0.
[3ay 2 ≥ 0 ⇒ x(a − x) 2 ≥ 0 ⇒ x ≥ 0.]
The curve is symmetric about the x-axis (since the equation is unchanged
when y is replaced by −y). y =0when x =0or a, so the curve’s loop
extends from x =0to x = a.
d
dx (3ay2 )= d
dx [x(a − x)2 ] ⇒ 6ay dy
dx = x · 2(a − x)(−1) + (a − x)2 ⇒ dy
dx
2 dy
= (a − x)2 (a − 3x) 2
= (a −
x)2 (a − 3x) 2 3a the last fraction
·
=
dx
36a 2 y 2 36a 2 x(a − x) 2 is 1/y 2
=
(a − x)[−2x + a − x]
6ay
(a − 3x)2
12ax
⇒
⇒
2 dy
1+ =1+ a2 − 6ax +9x 2
dx
12ax
= 12ax
12ax + a2 − 6ax +9x 2
12ax
= a2 +6ax +9x 2
12ax
=
(a +3x)2
12ax
for x 6= 0.
a
a
(a) S = 2πy ds =2π
x=0
0
a
√
x (a − x)
√
3a
· a +3x √
12ax
dx =2π
a
0
(a − x)(a +3x)
6a
= π (a 2 +2ax − 3x 2 ) dx = π
a 2 x + ax 2 − x 3 a
3a 0
3a
= π 0
3a (a3 + a 3 − a 3 )= π 3a · a3 = πa2
3 .
Note that we have rotated the top half of the loop about the x-axis. This generates the full surface.
(b) We must rotate the full loop about the y-axis, so we get double the area obtained by rotating the top half of the loop:
a
a
S =2· 2π xds=4π x √ a +3x dx =
4π a
x=0
0 12ax 2 √ x 1/2 (a +3x) dx =
2π a
√ (ax 1/2 +3x 3/2 ) dx
3a 0
3a 0
= √ 2π 2
3a 3 ax3/2 + 6 a
5 x5/2 = 2π √
3 2
0
3 √ a 3 a5/2 + 6
5 a5/2 = 2π √
3 2
3 3 + 6 a
5
2 = 2π √
3 28
a 2
3 15
= 56π √ 3 a 2
45
29. (a) x2
a + y2
y (dy/dx)
=1 ⇒ = − x ⇒ dy
2 b2 b 2 a 2 dx = − b2 x
⇒
a 2 y
2 dy
1+ =1+ b4 x 2
dx a 4 y = b4 x 2 + a 4 y 2
= b4 x 2 + a 4 b 2 1 − x 2 /a 2
2 a 4 y 2 a 4 b 2 (1 − x 2 /a 2 )
= a4 + b 2 x 2 − a 2 x 2
= a4 − a 2 − b 2 x 2
a 4 − a 2 x 2 a 2 (a 2 − x 2 )
dx
= a4 b 2 + b 4 x 2 − a 2 b 2 x 2
a 4 b 2 − a 2 b 2 x 2
The ellipsoid’s surface area is twice the area generated by rotating the first-quadrant portion of the ellipse about the x-axis.
Thus,
S =2
a
0
2πy
= 4πb
a 2 a
√
=
0
1+
a 2 −b 2
2 dy
a
b a4 − (a
dx =4π a2 − x 2 − b 2 )x 2
dx
0 a
2 a √ dx = 4πb
a 2 − x 2 a 2
a4 − u 2
du
√
a2 − b 2 u = √ a 2 − b 2 x 30
=
√ √
4πb a a2 − b 2
a 2√ a4 − a
a 2 − b 2 2
2 (a 2 − b 2 )+ a4 a2 − b 2
2 sin−1 a
a
4πb u
a 2√ a4 − u
a 2 − b 2 2
2 + a4
⎡
=2π⎢
⎣ b2 +
0
a4 − (a 2 − b 2 )x 2 dx
2 sin−1 u
a 2 a
a 2 b sin −1 √
a2 − b 2
a √
a2 − b 2
0
√
a 2 −b 2
⎤
⎥
⎦

F.
362 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
(b) x2
a + y2
x (dx/dy)
=1 ⇒ = − y ⇒ dx
2 b2 a 2 b 2 dy = y
−a2 b 2 x
⇒
2 dx
1+ =1+ a4 y 2
dy b 4 x = b4 x 2 + a 4 y 2
= b4 a 2 (1 − y 2 /b 2 )+a 4 y 2
2 b 4 x 2 b 4 a 2 (1 − y 2 /b 2 )
= b4 − b 2 y 2 + a 2 y 2
= b4 − (b 2 − a 2 )y 2
b 4 − b 2 y 2 b 2 (b 2 − y 2 )
= a2 b 4 − a 2 b 2 y 2 + a 4 y 2
a 2 b 4 − a 2 b 2 y 2
The oblate spheroid’s surface area is twice the area generated by rotating the first-quadrant portion of the ellipse about the
y-axis. Thus,
S =2
b
0
= 4πa
b 2
30
=
=
2πx
b
0
1+
2 dx
b
dy =4π
dy
0
b4 − (b 2 − a 2 )y 2 dy = 4πa
b 2
a b4 − (b
b2 − y 2 − a 2 )y 2
b
2 b dy
b 2 − y 2
b
√
b 2 −a 2
4πa u
b √ √
b4 − u 2 b 2 − a 2 2
2 + b4 u
b
2 sin−1 b 2
0
0
√
b4 − u 2
b 2 −a 2
√ √
4πa b
b √ b2 − a 2
b4 − b 2 b 2 − a 2 2
2 (b 2 − a 2 )+ b4 b2 − a 2
2 sin−1 b
du
√
u
b2 − a 2 = √
b 2 − a 2 y
⎡
=2π⎢
⎣ a2 +
Notice that this result can be obtained from the answer in part (a) by interchanging a and b.
31. The analogue of f(x ∗ i ) in the derivation of (4) is now c − f(x ∗ i ), so
S = lim
n
n→∞ i=1
2π[c − f(x ∗ i )] 1+[f 0 (x ∗ i )]2 ∆x = b
a 2π[c − f(x)] 1+[f 0 (x)] 2 dx.
33. For the upper semicircle, f(x) = √ r 2 − x 2 , f 0 (x) =−x/ √ r 2 − x 2 . The surface area generated is
S 1 =
r
=4π
−r
r
2π
r −
r 2 − x 2 1+ x2
0
r 2
√
r2 − x − r dx
2
For the lower semicircle, f(x) =− √ r 2 − x 2 and f 0 (x) =
Thus, the total area is S = S 1 + S 2 =8π
r
0
r r
r 2 − x dx =4π −
r 2 − x 2 2
x
√
r2 − x 2 ,soS 2 =4π
r 2
√ dx =8π
r2 − x 2
0
r
0
r 2 sin −1 x
r
r
0
ab 2 sin −1 √
b2 − a 2
b √
b2 − a 2
r
√
r2 − x 2 dx
r 2
√
r2 − x + r dx.
2
=8πr 2 π
=4π 2 r 2 .
2
yi−1 + yi
35. In the derivation of (4), we computed a typical contribution to the surface area to be 2π |P i−1P i|,
2
the area of a frustum of a cone. When f(x) is not necessarily positive, the approximations y i = f(x i ) ≈ f(x ∗ i ) and
y i−1 = f(x i−1 ) ≈ f(x ∗ i ) must be replaced by y i = |f(x i )| ≈ |f(x ∗ i )| and y i−1 = |f(x i−1 )| ≈ |f(x ∗ i )|. Thus,
2π y i−1 + y i
2
|P i−1 P i | ≈ 2π |f(x ∗ i )| 1+[f 0 (x ∗ i )]2 ∆x. Continuing with the rest of the derivation as before,
we obtain S = b
a 2π |f(x)| 1+[f 0 (x)] 2 dx.
⎤
⎥
⎦

F.
TX.10 SECTION 8.3 APPLICATIONS TO PHYSICS AND ENGINEERING ¤ 363
8.3 Applications to Physics and Engineering
1. The weight density of water is δ =62.5lb/ft 3 .
(a) P = δd ≈ (62.5lb/ft 3 )(3 ft) = 187.5lb/ft 2
(b) F = PA ≈ (187.5lb/ft 2 )(5 ft)(2 ft) = 1875 lb.
(A is the area of the bottom of the tank.)
(c) As in Example 1, the area of the ith strip is 2(∆x) and the pressure is δd = δx i.Thus,
F = 3
0 δx · 2 dx ≈ (62.5)(2) 3
0 xdx=125 1
2 x2 3
0 = 125 9
2
=562.5lb.
In Exercises 3–9, n is the number of subintervals of length ∆x and x ∗ i is a sample point in the ith subinterval [x i−1 ,x i ].
3. Set up a vertical x-axisasshown,withx =0at the water’s surface and x increasing in the
downward direction. Then the area of the ith rectangular strip is 6 ∆x and the pressure on
the strip is δx ∗ i (where δ ≈ 62.5lb/ft 3 ). Thus, the hydrostatic force on the strip is
δx ∗
i · 6 ∆x and the total hydrostatic force ≈ n δx ∗ i · 6 ∆x. The total force
F = lim
n
n→∞ i=1
i=1
δx ∗ i · 6 ∆x = 6
δx · 6 dx =6δ 6
xdx=6δ 1
x2 6
=6δ(18 − 2) = 96δ ≈ 6000 lb
2 2 2 2
5. Set up a vertical x-axis as shown. The base of the triangle shown in the figure
has length 3 2 − (x ∗ i )2 ,sow i =2 9 − (x ∗ i )2 ,andtheareaoftheith
rectangular strip is 2 9 − (x ∗ i )2 ∆x. Theith rectangular strip is (x ∗ i − 1) m
below the surface level of the water, so the pressure on the strip is ρg(x ∗ i − 1).
The hydrostatic force on the strip is ρg(x ∗ i − 1) · 2 9 − (x ∗ i )2 ∆x and the total
force on the plate ≈ n ρg(x ∗ i − 1) · 2 9 − (x ∗ i )2 ∆x. The total force
F =lim n
i=1
i=1
ρg(x ∗ i − 1) · 2 9 − (x ∗ i )2 ∆x =2ρg 3
1 (x − 1) √ 9 − x 2 dx
=2ρg 3
x √ 9 − x
1 2 dx − 2ρg 3
√
1 9 − x2 dx 30 3 x
=2ρg − 1 (9 − √ x
3
3 x2 ) 3/2 − 2ρg 9 − x2 + 9
1 2
2 sin−1 3 1
=2ρg √
0+ 1 3 8 8 − 2ρg 0+
9 · π
2 2 − 1
√
2 8+
9
sin−1
1
2 3
√
= 32
3 2 ρg −
9π
ρg +2√ 2
2 ρg +9 sin −1
1
3 ρg = 38
3
≈ 6.835 · 1000 · 9.8 ≈ 6.7 × 10 4 N
√
2 −
9π
2 +9sin−1 1
3
ρg
Note: If you set up a typical coordinate system with the water level at y = −1,thenF = −1
−3 ρg(−1 − y)2 9 − y 2 dy.

F.
364 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
7. Set up a vertical x-axis as shown. Then the area of the ith rectangular strip is
2 − √ 2
√
x ∗ i ∆x. By similar triangles, wi 3 − x
∗
3 2 = √3 i
,sow i =2− √ 2
x ∗ i .
3
The pressure on the strip is ρgx ∗ i , so the hydrostatic force on the strip is
ρgx ∗ i 2 − √ 2
x ∗
i ∆x and the hydrostatic force on the plate ≈ n
3
The total force
F = lim
n→∞ i=1
n
ρgx ∗ i
2 − √ 2
x ∗ i ∆x =
3
√ 3
0
i=1
ρgx ∗ i
2 − √ 2
x ∗ i ∆x.
3
ρgx 2 − √ 2
x dx = ρg
3
√ 3
= ρg x 2 − 2 √ 3
3 √ 3 x3 = ρg [(3 − 2) − 0] = ρg ≈ 1000 · 9.8 =9.8 × 10 3 N
0
0
2x − 2 √
3
x 2
dx
9. Set up coordinate axes as shown in the figure. The length of the ith strip is
2 25 − (yi ∗)2 and its area is 2 25 − (yi ∗)2 ∆y. The pressure on this strip is
approximately δd i =62.5(7 − yi ∗ ) andsotheforceonthestripisapproximately
62.5(7 − yi ∗ )2 25 − (yi ∗)2 ∆y. The total force
F = lim
n
n→∞ i=1
62.5(7 − y ∗ i )2 25 − (y ∗ i )2 ∆y =125 5
0 (7 − y) 25 − y 2 dy
5
=125 7 25 − y
0 2 dy − 5
y 25 − y
0 2 dy
=125 7 5
5
0 25 − y2 dy − − 1 (25 − 3 y2 ) 3/2 0
=125 7 1
π · 52 + 1 (0 − 125) =125
175π
− 125
4 3 4 3 ≈ 11,972 ≈ 1.2 × 10 4 lb
11. Set up a vertical x-axis as shown. Then the area of the ith rectangular strip is
a
h (2h − w i
x∗ i ) ∆x. By similar triangles, = 2a
2h − x ∗ i
2h ,sowi = a
h (2h − x∗ i ).
The pressure on the strip is δx ∗ i , so the hydrostatic force on the plate
≈ n
δx ∗ i
i=1
a
h (2h − x∗ i ) ∆x. The total force
F = lim
n
δx ∗ i
n→∞ i=1
= aδ
hx 2 − 1
h
x3 h
3
a
h (2h − x∗ i ) ∆x = δ a h aδ
x(2h − x) dx =
h
0
h
2h
3
0 = aδ
h
h 3 − 1 3 h3 = aδ
h
3
= 2 3 δah2
h
0
2hx − x
2 dx

F.
13. By similar triangles,
8
4 √ 3 = w i
x ∗ i
⇒
TX.10 SECTION 8.3 APPLICATIONS TO PHYSICS AND ENGINEERING ¤ 365
w i = 2x∗ i
√ . The area of the ith
3
rectangular strip is 2x∗ i
√ ∆x and the pressure on it is ρg 4 √
3 − x ∗ i .
3
F =
4
√
3
0
=4ρg x 2 4 √ 3
0
ρg 4 √ 2x
3 − x √ dx =8ρg
3
4
√
3
0
xdx− 2ρg
√
4 3
√ x 2 dx
3
− 2ρg
3 √ x
3 4 √ 3
=192ρg − 2ρg
0
3
3 √ 3 64 · 3 √ 3 = 192ρg − 128ρg =64ρg
≈ 64(840)(9.8) ≈ 5.27 × 10 5 N
0
15. (a) The top of the cube has depth d =1m− 20 cm = 80 cm = 0.8m.
(b) The area of a strip is 0.2 ∆x and the pressure on it is ρgx ∗ i .
F = ρgdA ≈ (1000)(9.8)(0.8)(0.2) 2 = 313.6 ≈ 314 N
F = 1
ρgx(0.2) dx =0.2ρg 1
x2 1
=(0.2ρg)(0.18) = 0.036ρg =0.036(1000)(9.8) = 352.8 ≈ 353 N
0.8 2 0.8
17. (a) The area of a strip is 20 ∆x and the pressure on it is δx i .
F = 3
0 δx20 dx =20δ 1
2 x2 3
0 =20δ · 9
2 =90δ
= 90(62.5) = 5625 lb ≈ 5.63 × 10 3 lb
(b) F = 9
0 δx20 dx =20δ 1
2 x2 9
0 =20δ · 81 2 = 810δ = 810(62.5) = 50,625 lb ≈ 5.06 × 104 lb.
(c) For the first 3 ft, the length of the side is constant at 40 ft. For 3

F.
366 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
21. The moment M of the system about the origin is M = 2 m i x i = m 1 x 1 + m 2 x 2 =40· 2+30· 5 = 230.
The mass m of the system is m = 2 m i = m 1 + m 2 =40+30=70.
i=1
ThecenterofmassofthesystemisM/m = 230
70 = 23
7 .
23. m = 3 m i =6+5+10=21.
i=1
M x = 3
m i y i =6(5)+5(−2) + 10(−1) = 10; M y = 3 m i x i = 6(1) + 5(3) + 10(−2) = 1.
i=1
x = My
m = 1 Mx
and y =
21 m = 10
21 , so the center of mass of the system is 1
, 10
21 21 .
25. Since the region in the figure is symmetric about the y-axis, we know
i=1
i=1
that x =0. The region is “bottom-heavy,” so we know that y0.5 and y

F.
31. A = π/4
0
(cos x − sin x) dx = sin x +cosx π/4
0
= √ 2 − 1.
x = A −1 π/4
0
x(cos x − sin x) dx
TX.10 SECTION 8.3 APPLICATIONS TO PHYSICS AND ENGINEERING ¤ 367
= A −1 x(sin x +cosx)+cosx − sin x π/4
0
= A −1 √
1
π
π √ 4 2 − 1 =
4
2 − 1
√ .
2 − 1
[integration by parts]
y = A −1 π/4
0
1
2 (cos2 x − sin 2 x) dx = 1 π/4
cos 2xdx= 1
π/4
2A 0 4A sin 2x = 1
0
4A = 1
4 √ 2 − 1 .
π √
2 − 4
Thus, the centroid is (x, y) =
4 √ 2 − 1 , 1
4 √ 2 − 1 ≈ (0.27, 0.60).
33. From the figure we see that y =0.Now
A = 5
2 √
5
5 − xdx=2 − 2 (5 − 0 3 x)3/2 =2
0+ 2 · 3 53/2 0
so
= 20 3
√
5,
x = 1 5 x√ 5 − x − − √ 5 − x
dx = 1 5 2x √ 5 − xdx
A 0 A 0
0√5
2 5 − u 2
u(−2u) du
= 1 A
= 4 A
u = √ 5 − x, x =5− u 2 ,
u 2 =5− x, dx = −2udu
√ 5
u 2 (5 − u 2 ) du = 4 5
0 A 3 u3 − 1 5 u5√ 5
= 3
0 5 √ 25
√ √
5 3 5 − 5 5 =5− 3=2.
Thus, the centroid is (x, y) =(2, 0).
35. The line has equation y = 3 x. A = 1 (4)(3) = 6,som = ρA = 10(6) = 60.
4 2
M x = ρ 4
0
1 3
2 4 x2 dx =10 4
0
M y = ρ 4
x 3
x dx = 15
0 4 2
9
32 x2 dx = 45 1 x3 4
= 45
16 3 0 16
64
3 =60
4
0 x2 dx = 15 1 x3 4
= 15 64
2 3 0 2 3 = 160
x = M y
m = 160
60 = 8 3 and y = M x
m = 60
60 =1. Thus, the centroid is (x, y) = 8
3 , 1 .
37. A =
=
2
0
2
(2 x − x 2 x
) dx =
4
ln 2 − 8 3
2
ln 2 − x3
3
x = 1 x(2 x − x 2 ) dx = 1 A 0
A
= 1 x2
x
2
A ln 2 −
2x
(ln 2) 2 − x4
4
0
= 1 A
2
0
− 1
ln 2 = 3
ln 2 − 8 3 ≈ 1.661418.
2
0
(x2 x − x 3 ) dx
[use parts]
8
ln 2 − 4
(ln 2) 2 − 4+ 1
(ln 2) 2
= 1 A
8
ln 2 − 3
(ln 2) 2 − 4 ≈ 1 (1.297453) ≈ 0.781
A
[continued]

F.
368 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
y = 1 A
2
0
= 1 A · 1
2
1
2 [(2x ) 2 − (x 2 ) 2 ] dx = 1 2
1
A
2 (22x − x 4 ) dx = 1
0
A · 1
2
16
2ln2 − 32
5 − 1
= 1 2ln2 A
Thus, the centroid is (x, y) ≈ (0.781, 1.330).
2
2x
2
2ln2 − x5
5
0
15
4ln2 − 16
≈ 1 (2.210106) ≈ 1.330
5 A
Since the position of a centroid is independent of density when the density is constant, we will assume for convenience that ρ =1in Exercises 38
and 39.
39. Choose x-andy-axes so that the base (one side of the triangle) lies along
the x-axis with the other vertex along the positive y-axisasshown. From
geometry, we know the medians intersect at a point 2 of the way from each
3
vertex (along the median) to the opposite side. The median from B goes to
the midpoint 1
2 (a + c), 0 of side AC, so the point of intersection of the
medians is 2
3 · 1
2 (a + c), 1 3 b = 1
3 (a + c), 1 3 b .
This can also be verified by finding the equations of two medians, and solving them simultaneously to find their point of
intersection. Now let us compute the location of the centroid of the triangle. The area is A = 1 (c − a)b.
2
x = 1 0
x · b
c
A a (a − x) dx +
a
0
x · b
c (c − x) dx
= 1 A b
a
=
Aa b 1
2 ax2 − 1 0
3 x3 + b 1
a
Ac
2 cx2 − 1 c
3 x3 = b − 1
0
Aa 2 a3 + 1
3 a3
=
0
a
(ax − x 2 ) dx + b c
2
a (c − a) · −a3
6 + 2
c (c − a) · c3 6 = 1
3(c − a) (c2 − a 2 )= a + c
3
+ b Ac
c
0
1
2 c3 − 1 3 c3
cx − x
2
dx
and
y = 1 A
0
a
2
1 b c
2
2 a (a − x) 1 b
dx + dx
0 2 c (c − x)
= 1 b
2
0
c
(a 2 − 2ax + x 2 ) dx + b2
(c 2 − 2cx + x 2 ) dx
A 2a 2 2c 2
b
2
= 1 A
= 1 b
2
A
a
2a 2 a 2 x − ax 2 + 1 3 x3 0
a + b2
2c 2 c 2 x − cx 2 + 1 3 x3 c
0
2a 2
−a 3 + a 3 − 1 3 a3 + b2
2c 2
c 3 − c 3 + 1 3 c3 = 1 A
0
b
2
6 (−a + c) =
2 (c − a)b2 · = b (c − a) b 6 3
a + c
Thus, the centroid is (x, y) =
3 , b
, as claimed.
3
Remarks: Actually the computation of y is all that is needed. By considering each side of the triangle in turn to be the base,
we see that the centroid is 1 3
of the way from each side to the opposite vertex and must therefore be the intersection of the
medians.

F.
TX.10 SECTION 8.3 APPLICATIONS TO PHYSICS AND ENGINEERING ¤ 369
The computation of y in this problem (and many others) can be
simplified by using horizontal rather than vertical approximating rectangles.
If the length of a thin rectangle at coordinate y is (y), then its area is
(y) ∆y,itsmassisρ(y) ∆y, and its moment about the x-axis is
∆M x = ρy(y) ∆y. Thus,
M x = ρy(y) dy and y =
ρy(y) dy
ρA
= 1 y(y) dy
A
In this problem, (y) = c − a
b
(b − y) by similar triangles, so
y = 1 A
b
0
c − a
b
y(b − y) dy = 2 b 2 b
0 (by − y2 ) dy = 2 b 2 1
2 by2 − 1 3 y3 b
0 = 2 b 2 · b3 6 = b 3
Notice that only one integral is needed when this method is used.
41. Divide the lamina into two triangles and one rectangle with respective masses of 2, 2 and 4, so that the total mass is 8. Using
the result of Exercise 39, the triangles have centroids
−1, 2 3 and 1,
2
3 . The centroid of the rectangle (its center) is 0, −
1
2 .
So, using Formulas 5 and 7, we have y = M x
m = 1 3
m i y i = 1 8 2 2
3 +2 2
3 +4 −
1
2 =
1 2
8 3 =
1
12
,andx =0,
m
since the lamina is symmetric about the line x =0. Thus, the centroid is (x, y) = 0, 1 12
.
i=1
43.
b (cx + d) f(x) dx = b
cx f(x) dx + b
df(x) dx = c b
xf(x) dx + d b
f(x) dx = cxA + d b
a a a a a a
= cx b
f(x) dx + d b
f(x) dx =(cx + d) b
a a a
f(x) dx
f(x) dx [by (8)]
45. A cone of height h and radius r can be generated by rotating a right triangle
about one of its legs as shown. By Exercise 39, x = 1 3 r,sobytheTheoremof
Pappus, the volume of the cone is
V = Ad = 1
2 · base · height · (2πx) = 1 2 rh · 2π 1
3 r = 1 3 πr2 h.
47. Suppose the region lies between two curves y = f(x) and y = g(x) where f(x) ≥ g(x), as illustrated in Figure 13.
Choose points x i with a = x 0

F.
370 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
8.4 Applications to Economics and Biology
1. By the Net Change Theorem, C(2000) − C(0) = 2000
0
C 0 (x) dx ⇒
C(2000) = 20,000 + 2000
(5 − 0.008x +0.000009x 2 ) dx =20,000 + 5x − 0.004x 2 +0.000003x 3 2000
0 0
=20,000 + 10,000 − 0.004(4,000,000) + 0.000003(8,000,000,000) = 30,000 − 16,000 + 24,000
=$38,000
3. If the production level is raised from 1200 units to 1600 units, then the increase in cost is
C(1600) − C(1200) = 1600
1200 C0 (x) dx = 1600
1200 (74 + 1.1x − 0.002x2 +0.00004x 3 ) dx
= 74x +0.55x 2 − 0.002
3
x 3 +0.00001x 4 1600
=64,331,733.33 − 20,464,800 = $43,866,933.33
1200
5. p(x) =10 ⇒ 450 =10 ⇒ x +8=45 ⇒ x =37.
x +8
37
37
450
Consumer surplus = [p(x) − 10] dx =
x +8 − 10 dx
0
0
= 450 ln (x +8)− 10x 37
=(450ln45− 370) − 450 ln 8
0
= 450 ln
45
8 − 370 ≈ $407.25
7. P = p S (x) ⇒ 400 = 200 + 0.2x 3/2 ⇒ 200 = 0.2x 3/2 ⇒ 1000 = x 3/2 ⇒ x =1000 2/3 = 100.
Producer surplus = 100
[P − p
0 S(x)] dx = 100
[400 − (200 + 0.2x 3/2 )] dx =
100
200 − 1 0 0
5 x3/2 dx
100
= 200x − 2
25 x5/2 =20,000 − 8,000 = $12,000
0
9. p(x) = 800,000e−x/5000
x +20,000
=16 ⇒ x = x 1 ≈ 3727.04.
Consumer surplus = x 1
[p(x) − 16] dx ≈ $37,753
0
11. f(8) − f(4) = 8
f 0 (t) dt = 8
√ 8 √
4 4 tdt=
2
3 t3/2 = 2 3 16 2 − 8 ≈ $9.75 million
4
13. N =
b
a
Similarly,
x
Ax −k −k+1 b
dx = A
=
−k +1 A
a
1 − k (b1−k − a 1−k ).
b
a
Thus, x = 1 N
x
Ax 1−k 2−k b
dx = A
2 − k
a
b
a
= A
2 − k (b2−k − a 2−k ).
Ax 1−k dx = [A/(2 − k)](b2−k − a 2−k )
[A/(1 − k)](b 1−k − a 1−k ) = (1 − k)(b2−k − a 2−k )
(2 − k)(b 1−k − a 1−k ) .
15. F = πPR4
8ηl
= π(4000)(0.008)4
8(0.027)(2)
≈ 1.19 × 10 −4 cm 3 /s

F.
17. From (3), F =
T
0
A 6
=
c (t) dt 20I ,where
I =
10
0
TX.10
10
te −0.6t 1
dt =
(−0.6) 2 (−0.6t − 1) e−0.6t
6(0.36)
Thus, F =
20(1 − 7e −6 ) = 0.108 ≈ 0.1099 L/s or 6.594 L/min.
1 − 7e−6 0
integrating
by parts
SECTION 8.5 PROBABILITY ¤ 371
= 1
0.36 (−7e−6 +1)
19. AsinExample2,wewillestimatethecardiacoutputusingSimpson’sRulewith∆t =(16− 0)/8 =2.
16
c(t) dt ≈ 2 [c(0) + 4c(2) + 2c(4) + 4c(6) + 2c(8) + 4c(10) + 2c(12) + 4c(14) + c(16)]
0 3
Therefore, F ≈
≈ 2 [0 + 4(6.1) + 2(7.4) + 4(6.7) + 2(5.4) + 4(4.1) + 2(3.0) + 4(2.1) + 1.5]
3
= 2 (109.1) = 72.73mg· s/L
3
A
72.73 = 7 ≈ 0.0962 L/s or 5.77 L/min.
72.73
8.5 Probability
1. (a) 40,000
f(x) dx is the probability that a randomly chosen tire will have a lifetime between 30,000 and 40,000 miles.
30,000
(b) ∞
f(x) dx is the probability that a randomly chosen tire will have a lifetime of at least 25,000 miles.
25,000
3. (a) In general, we must satisfy the two conditions that are mentioned before Example 1—namely, (1) f(x) ≥ 0 for all x,
and (2) ∞
3
f(x) dx =1.For0 ≤ x ≤ 4,wehavef(x) = x √ 16 − x
−∞ 64 2 ≥ 0,sof(x) ≥ 0 for all x. Also,
∞
−∞ f(x) dx = 4
0
= − 1
64
Therefore, f is a probability density function.
(b) P (X

F.
372 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
1
(b) P (−1

F.
17. P (μ − 2σ ≤ X ≤ μ +2σ) =
2
−2
μ+2σ
μ−2σ
1
σ √ 2π e− t2 /2 (σdt)= 1 √
2π
2
−2
TX.10
SECTION 8.5 PROBABILITY ¤ 373
1
σ √ 2π exp (x − μ)2
− dx. Substituting t = x − μ and dt = 1 dx gives us
2σ 2
σ
σ
e − t2 /2 dt ≈ 0.9545.
19. (a) First p(r) = 4 a 3 0
r 2 e −2r/a 0
≥ 0 for r ≥ 0. Next,
∞
−∞
∞
p(r) dr =
0
4
a 3 0
r 2 e −2r/a 0
dr = 4 a 3 0
lim
t→∞
By using parts, tables, or a CAS , we find that x 2 e bx dx =(e bx /b 3 )(b 2 x 2 − 2bx +2).
Next, we use ()(withb = −2/a 0 ) and l’Hospital’s Rule to get
a function to be a probability density function.
(b) Using l’Hospital’s Rule,
4
a 3 0
lim
r→∞
r 2
e 2r/a 0 = 4 a 3 0
To find the maximum of p, we differentiate:
p 0 (r) = 4 a 3 0
lim
r→∞
2r
4
a 3 0
(2/a 0 )e 2r/a 0 = 2 a 2 0
t
0
r 2 e −2r/a 0
dr
()
a
3
0
−8 (−2)
=1.Thissatisfies the second condition for
lim
r→∞
2
(2/a 0 )e 2r/a 0 =0.
r 2 e −2r/a 0
− 2
+ e −2r/a 0
(2r) = 4
e −2r/a 0
(2r) − r
+1
a 0 a 3 0 a 0
p 0 (r) =0 ⇔ r =0or 1= r a 0
⇔ r = a 0 [a 0 ≈ 5.59 × 10 −11 m].
p 0 (r) changes from positive to negative at r = a 0 ,sop(r) has its maximum value at r = a 0 .
(c) It is fairly difficult to find a viewing rectangle, but knowing the maximum
value from part (b) helps.
p(a 0 )= 4 a 2
a
0e −2a 0/a 0
= 4 e −2 ≈ 9,684,098,979
3
0
a 0
With a maximum of nearly 10 billion and a total area under the curve of 1,
we know that the “hump” in the graph must be extremely narrow.
r
(d) P (r) =
0
4
a 3 0
P (4a 0 )= 4 a 3 0
∞
4a0
s 2 e −2s/a 0
ds ⇒ P (4a 0)=
0
e
−2s/a 0
−8/a 3 0
=1− 41e −8 ≈ 0.986
4
s 2 e −2s/a 0
ds. Using() from part (a) [with b = −2/a 0],
a 3 0
4
s 2 + 4 4a0
s +2 = 4 a
3
0
[e −8 (64 + 16 + 2) − 1(2)] = − 1
a 2 0 a 0 0
a 3 0 −8
2 (82e−8 − 2)
t
lim
t→∞
0
(e) μ = rp(r) dr = 4 r 3 e −2r/a 0
dr. Integrating by parts three times or using a CAS, we find that
−∞ a 3 0
x 3 e bx dx = ebx
b 3 x 3 − 3b 2 x 2 +6bx − 6 .Sowithb = − 2 , we use l’Hospital’s Rule, and get
b 4 a 0
μ = 4
− a4 0
a 3 0 16 (−6) = 3 a 2 0.

F.
374 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
8 Review
1. (a) The length of a curve is defined to be the limit of the lengths of the inscribed polygons, as described near Figure 3 in
Section 8.1.
(b) See Equation 8.1.2.
(c) See Equation 8.1.4.
2. (a) S = b
a 2πf(x) 1+[f 0 (x)] 2 dx
(b) If x = g(y), c ≤ y ≤ d,thenS = d
c 2πy 1+[g 0 (y)] 2 dy.
(c) S = b
a 2πx 1+[f 0 (x)] 2 dx or S = d
c 2πg(y) 1+[g 0 (y)] 2 dy
3. Let c(x) be the cross-sectional length of the wall (measured parallel to the surface of the fluid) at depth x. Then the hydrostatic
forceagainstthewallisgivenbyF = b
δxc(x) dx,wherea and b are the lower and upper limits for x at points of the wall
a
and δ is the weight density of the fluid.
4. (a) The center of mass is the point at which the plate balances horizontally.
(b) See Equations 8.3.8.
5. If a plane region R that lies entirely on one side of a line in its plane is rotated about , then the volume of the resulting solid
is the product of the area of R and the distance traveled by the centroid of R.
6. See Figure 3 in Section 8.4, and the discussion which precedes it.
7. (a) See the definition in the first paragraph of the subsection Cardiac Output in Section 8.4.
(b) See the discussion in the second paragraph of the subsection Cardiac Output in Section 8.4.
8. A probability density function f is a function on the domain of a continuous random variable X such that b
f(x) dx
a
measures the probability that X lies between a and b. Such a function f has nonnegative values and satisfies the relation
f(x) dx =1,whereD is the domain of the corresponding random variable X. IfD = R,orifwedefine f(x) =0for real
D
numbers x/∈ D,then ∞
f(x) dx =1.(Ofcourse,toworkwithf in this way, we must assume that the integrals of f exist.)
−∞
9. (a) 130
0
f(x) dx represents the probability that the weight of a randomly chosen female college student is less than
130 pounds.
(b) μ = ∞
−∞ xf(x) dx = ∞
0
xf(x) dx
(c) The median of f is the number m such that ∞
m f(x) dx = 1 2 .
10. See the discussion near Equation 3 in Section 8.5.

F.
TX.10
CHAPTER 8 REVIEW ¤ 375
1. y = 1 6 (x2 +4) 3/2 ⇒ dy/dx = 1 4 (x2 +4) 1/2 (2x) ⇒
1+(dy/dx) 2 =1+
Thus, L = 3
0
1
2 x(x2 +4) 1/2 2
=1+
1
4 x2 (x 2 +4)= 1 4 x4 + x 2 +1= 1
2 x2 +1 2 .
1
2 x2 +1 2 dx = 3
1
0 2 x2 +1 dx = 1
6 x3 + x 3
= 15 . 0 2
3. (a) y = x4
16 + 1
2x 2 = 1
16 x4 + 1 2 x−2 ⇒ dy
dx = 1 4 x3 − x −3
⇒
1+(dy/dx) 2 =1+ 1
4 x3 − x −32 =1+ 1 16 x6 − 1 + 2 x−6 = 1 16 x6 + 1 + 2 x−6 = 1
4 x3 + x −32 .
Thus, L = 2
1
1 4 x3 + x −3 dx = 1
16 x4 − 1 x−2 2
=
1 − 1 2 1 8 − 1 − 1
16 2 =
21
. 16
(b) S = 2
2πx 1
1 4 x3 + x −3 dx =2π 2
1
1 4 x4 + x −2 dx =2π
1
20 x5 − 1 2
x 1
=2π 32
20 − 1 2
−
1
− 1 20
=2π 8
− 1 − 1 +1 5 2 20
=2π 41
20
=
41
10 π
5. y = e −x2 ⇒ dy/dx = −2xe −x2 ⇒ 1+(dy/dx) 2 =1+4x 2 e −2x2 . Letf(x) = 1+4x 2 e −2x2 .Then
L =
3
0
f(x) dx ≈ S 6 =
(3 − 0)/6
3
[f(0) + 4f(0.5) + 2f(1) + 4f(1.5) + 2f(2) + 4f(2.5) + f(3)] ≈ 3.292287
7. y = x
√ √ √
1 t − 1 dt ⇒ dy/dx = x − 1 ⇒ 1+(dy/dx) 2 =1+ x − 1 = √ x.
Thus, L = √
16
16
16
1 xdx= x 1/4 dx =
x 4 5/4 = 4 124
(32 − 1) = .
1 5
5 5
1
9. As in Example 1 of Section 8.3,
a
2 − x = 1 2
⇒ 2a =2− x and w =2(1.5+a) =3+2a =3+2− x =5− x.
Thus, F = 2
ρgx(5 − x) dx = ρg 5
0 2 x2 − 1 x3 2
= ρg
10 − 8 3 0 3 =
22
22
δ [ρg = δ] ≈ · 62.5 ≈ 458 lb.
3 3
11. A = 4
√
0 x −
1
x dx =
2
x = 1 4 x √ x − 1 x dx = 3 A 0 2 4
= 3 4
y = 1 A
4
2
3 x3/2 − 1 4 x2 = 16 − 4= 4 3 3
0
4
0
2
5 x5/2 − 1 6 x3 4
0
= 3 4
64
5 − 64 6
4
0
x 3/2 − 1 2 x2
dx
=
3
4
√ 2
1
2 x − 1 2
x
dx = 3 4
2 4 0
Thus, the centroid is (x, y) = 8
5 , 1 .
64
30 =
8
5
1
2 x −
1
x2
dx = 3 1
4 8 2 x2 − 1 x3 4
= 3
12 0 8
8 −
16
3 =
3 8
8 3 =1
13. An equation of the line passing through (0, 0) and (3, 2) is y = 2 x. A = 1 · 3 · 2=3. Therefore, using Equations 8.3.8,
3 2
x = 1 3 x 2
x
3 0 3
dx = 2 27 x
3 3
=2and y = 1 3
1 2
0 3 0 2 3 x2 dx = 2
81 x
3 3
= 2 . Thus, the centroid is (x, y) =
0 3
2, 2 3 .

F.
376 ¤ CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION
TX.10
15. The centroid of this circle, (1, 0), travels a distance 2π(1) when the lamina is rotated about the y-axis. The area of the circle
is π(1) 2 . So by the Theorem of Pappus, V = A(2πx) =π(1) 2 2π(1) = 2π 2 .
17. x =100 ⇒ P = 2000 − 0.1(100) − 0.01(100) 2 = 1890
19. f(x) =
Consumer surplus = 100
[p(x) − P ] dx = 100 0 0 2000 − 0.1x − 0.01x 2 − 1890 dx
π
20 sin π
10 x if 0 ≤ x ≤ 10
0 if x10
(a) f(x) ≥ 0 for all real numbers x and
∞
−∞ f(x) dx = 10
0
= 110x − 0.05x 2 − 0.01 x3 100
=11,000 − 500 − 10,000 ≈ $7166.67
3 0
3
π
sin π
x dx = π · 10
20 10 20 π − cos π x 10
= 1 (− cos π +cos0)= 1 (1 + 1) = 1
10 0 2 2
Therefore, f is a probability density function.
(b) P (X

F.
TX.10
PROBLEMS PLUS
1. x 2 + y 2 ≤ 4y ⇔ x 2 +(y − 2) 2 ≤ 4,soS is part of a circle, as shown
in the diagram. The area of S is
1
0
4y − y2 dy 113
=
y−2
2
4y − y2 +2cos −1 1
2−y
2
0
√
= − 1
2 3+2cos
−1 1
2 − 2cos −1 1
= − √ 3
+2
π
2 3 − 2(0) =
2π
− √ 3
3 2
[a =2]
Another method (without calculus): Note that θ = ∠CAB = π , so the area is
3
(area of sector OAB) − (area of 4ABC) = 1 2 2
2 π
− 1 (1)√ 3= 2π − √ 3
3 2 3 2
3. (a) The two spherical zones, whose surface areas we will call S 1 and S 2 ,are
generated by rotation about the y-axis of circular arcs, as indicated in the figure.
The arcs are the upper and lower portions of the circle x 2 + y 2 = r 2 that are
obtained when the circle is cut with the line y = d. The portion of the upper arc
in the first quadrant is sufficient to generate the upper spherical zone. That
portion of the arc can be described by the relation x = r 2 − y 2 for
d ≤ y ≤ r.Thus,dx/dy = −y/ r 2 − y 2 and
2
dx
ds = 1+ dy =
dy
From Formula 8.2.8 we have
r
2 dx
r
S 1 = 2πx 1+ dy =
dy
d
d
1+ y2
r 2 − y 2 dy =
r 2
r 2 − y 2 dy =
rdy
r2 − y 2
2π
rdy
r
r 2 − y 2
r2 − y = 2πr dy =2πr(r − d)
2
Similarly, we can compute S 2 = d
2πx 1+(dx/dy)
−r 2 dy = d
2πr dy =2πr(r + d). Note that S −r 1 + S 2 =4πr 2 ,
the surface area of the entire sphere.
d
(b) r =3960mi and d = r (sin 75 ◦ ) ≈ 3825 mi,
so the surface area of the Arctic Ocean is about
2πr(r−d) ≈ 2π(3960)(135) ≈ 3.36×10 6 mi 2 .
377

F.
378 ¤ CHAPTER 8 PROBLEMS PLUS
TX.10
(c) The area on the sphere lies between planes y = y 1 and y = y 2,wherey 2 − y 1 = h. Thus, we compute the surface area on
y2
2 dx
y2
the sphere to be S = 2πx 1+ dy = 2πr dy =2πr(y 2 − y 1)=2πrh.
y 1
dy
y 1
This equals the lateral area of a cylinder of radius r and height h, since such
a cylinder is obtained by rotating the line x = r about the y-axis, so the
surface area of the cylinder between the planes y = y 1 and y = y 2 is
y2
A = 2πx
y 1
=2πry
y 2
1+
2 dx
y2
dy = 2πr 1+0
dy
2 dy
y 1
y=y 1
=2πr(y 2 − y 1)=2πrh
(d) h =2r sin 23.45 ◦ ≈ 3152 mi, so the surface area of the
Torrid Zone is 2πrh ≈ 2π(3960)(3152) ≈ 7.84 × 10 7 mi 2 .
5. (a) Choose a vertical x-axis pointing downward with its origin at the surface. In order to calculate the pressure at depth z,
consider n subintervals of the interval [0,z] by points x i and choose a point x ∗ i ∈ [x i−1 ,x i ] for each i. The thin layer of
water lying between depth x i−1 and depth x i has a density of approximately ρ(x ∗ i ), so the weight of a piece of that layer
with unit cross-sectional area is ρ(x ∗ i )g ∆x. The total weight of a column of water extending from the surface to depth z
(with unit cross-sectional area) would be approximately n ρ(x ∗ i )g ∆x. The estimate becomes exact if we take the limit
as n →∞;weight(orforce)perunitareaatdepthz is W = lim
i=1
n
n→∞ i=1
ρ(x ∗ i )g ∆x. Inotherwords,P (z) = z
0 ρ(x)gdx.
More generally, if we make no assumptions about the location of the origin, then P (z) =P 0 + z
ρ(x)gdx,whereP0 is
0
the pressure at x =0. Differentiating, we get dP/dz = ρ(z)g.
(b)
F = r
P (L + x) · 2 √ r
−r 2 − x 2 dx
=
r
P
−r 0 +
L+x
ρ
0 0 e z/H gdz · 2 √ r 2 − x 2 dx
= P 0
r
−r 2 √ r 2 − x 2 dx + ρ 0 gH r
−r
e (L+x)/H − 1 · 2 √ r 2 − x 2 dx
=(P 0 − ρ 0 gH) r
2 √ r
−r 2 − x 2 dx + ρ 0 gH r
−r e(L+x)/H · 2 √ r 2 − x 2 dx
=(P 0 − ρ 0 gH) πr 2 + ρ 0 gHe L/H r
−r ex/H · 2 √ r 2 − x 2 dx

F.
TX.10
CHAPTER 8 PROBLEMS PLUS ¤ 379
7. To find the height of the pyramid, we use similar triangles. The first figure shows a cross-section of the pyramid passing
through the top and through two opposite corners of the square base. Now |BD| = b, since it is a radius of the sphere, which
has diameter 2b since it is tangent to the oppositesidesofthesquarebase.Also,|AD| = b since 4ADB is isosceles. So the
height is |AB| = √ b 2 + b 2 = √ 2 b.
We first observe that the shared volume is equal to half the volume of the sphere, minus the sum of the four equal volumes
(caps of the sphere) cut off by the triangular faces of the pyramid. See Exercise 6.2.51 for a derivation of the formula for the
volume of a cap of a sphere. To use the formula, we need to find the perpendicular distance h of each triangular face from the
surface of the sphere. We first find the distance d from the center of the sphere to one of the triangular faces. The third figure
shows a cross-section of the pyramid through the top and through the midpoints of opposite sides of the square base. From
similar triangles we find that
d
b = |AB|
|AC| =
√
2 b
b 2 + √ 2 b 2
⇒ d =
√
2 b
2
√
3b
2 = √
6
3 b
So h = b − d = b − √ 6
b = 3 − √ 6
3 3
b. So, using the formula V = πh 2 (r − h/3) from Exercise 6.2.51 with r = b,wefind that
3 −
the volume of each of the caps is π
√ 2
6
b b − 3 − √
6
b = 15 − 6√ 6
· 6+√ 6
πb 3 = 2
− √ 7
3 3 · 3 9 9 3 27 6 πb 3 . So, using our first
observation, the shared volume is V = 1 4 πb3 − 4 2
− √ 7
2 3 3 27 6 πb 3 = √
28
27 6 − 2 πb 3 .
9. We can assume that the cut is made along a vertical line x = b>0, that the
disk’s boundary is the circle x 2 + y 2 =1, and that the center of mass of the
smaller piece (to the right of x = b)is 1
2 , 0 .Wewishtofind b to two
decimal places. We have 1 1
2 = x = x · 2 √ 1 − x
b 2 dx
1 2 √ . Evaluating the
1 − x
b 2 dx
numerator gives us −
1
1 (1 − b x2 ) 1/2 (−2x) dx = − 2 3 − x
2 3/2
1
= − 2 0 − 1 − b 2 3/2
= 2 (1 − b 2 3
3
) 3/2 .
b
Using Formula 30 in the table of integrals, we find that the denominator is
√
x 1 − x2 +sin −1 x 1
= √
0+ π b 2 − b 1 − b 2
+sin −1 b . Thus, we have 1 2
2 = x = (1 − b 2 3
) 3/2
π
− b √ 1 − b
2 2 − sin −1 b ,or,
equivalently, 2 (1 − b 2 ) 3/2 = π − 1 b √ 1 − b
3 4 2 2 − 1 2 sin−1 b. Solving this equation numerically with a calculator or CAS, we
obtain b ≈ 0.138173,orb =0.14 m to two decimal places.

F.
380 ¤ CHAPTER 8 PROBLEMS PLUS
TX.10
11. If h = L,thenP =
area under y = L sin θ
area of rectangle
=
π
0
L sin θdθ
πL
= [− cos θ]π 0
π
= − (−1) + 1
π
= 2 π .
If h = L/2,thenP = area under y = 1 2 L sin θ
area of rectangle
=
π
0
1
L sin θdθ
2
= [− cos θ]π 0
πL
2π
= 2
2π = 1 π .

F.
TX.10
9 DIFFERENTIAL EQUATIONS
9.1 Modeling with Differential Equations
1. y = x − x −1 ⇒ y 0 =1+x −2 . To show that y is a solution of the differential equation, we will substitute the expressions
for y and y 0 in the left-hand side of the equation and show that the left-hand side is equal to the right-hand side.
LHS= xy 0 + y = x(1 + x −2 )+(x − x −1 )=x + x −1 + x − x −1 =2x =RHS
3. (a) y = e rx ⇒ y 0 = re rx ⇒ y 00 = r 2 e rx . Substituting these expressions into the differential equation
2y 00 + y 0 − y =0,weget2r 2 e rx + re rx − e rx =0 ⇒ (2r 2 + r − 1)e rx =0 ⇒
(2r − 1)(r +1)=0 [since e rx is never zero] ⇒ r = 1 or −1.
2
(b) Let r 1 = 1 2 and r 2 = −1, so we need to show that every member of the family of functions y = ae x/2 + be −x is a
solution of the differential equation 2y 00 + y 0 − y =0.
y = ae x/2 + be −x ⇒ y 0 = 1 2 aex/2 − be −x ⇒ y 00 = 1 4 aex/2 + be −x .
LHS = 2y 00 + y 0 1
− y =2
4 aex/2 + be −x 1
+
2 aex/2 − be −x − (ae x/2 + be −x )
= 1 2 aex/2 +2be −x + 1 2 aex/2 − be −x − ae x/2 − be −x
= 1
2 a + 1 2 a − a e x/2 +(2b − b − b)e −x
=0=RHS
5. (a) y =sinx ⇒ y 0 =cosx ⇒ y 00 = − sin x.
LHS = y 00 + y = − sin x +sinx =06= sinx,soy =sinx is not a solution of the differential equation.
(b) y =cosx ⇒ y 0 = − sin x ⇒ y 00 = − cos x.
LHS = y 00 + y = − cos x +cosx =06= sinx,soy =cosx is not a solution of the differential equation.
(c) y = 1 x sin x ⇒ 2 y0 = 1 (x cos x +sinx) ⇒ 2 y00 = 1 (−x sin x +cosx +cosx).
2
LHS = y 00 + y = 1 (−x sin x +2cosx)+ 1 x sin x =cosx 6= sinx, soy = 1 x sin x is not a solution of the
2 2 2
differential equation.
(d) y = − 1 x cos x ⇒ 2 y0 = − 1 (−x sin x +cosx) ⇒ 2 y00 = − 1 2
(−x cos x − sin x − sin x).
LHS = y 00 + y = − 1 (−x cos x − 2sinx)+ 2
− 1 x cos x 2
=sinx =RHS,soy = − 1 2
x cos x is a solution of the
differential equation.
7. (a) Since the derivative y 0 = −y 2 is always negative (or 0 if y =0), the function y must be decreasing (or equal to 0)onany
interval on which it is defined.
(b) y = 1
2
x + C ⇒ 1
y0 = −
(x + C) 2 . LHS = 1
1
y0 = −
(x + C) 2 = − = −y 2 =RHS
x + C
381

F.
382 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
(c) y =0is a solution of y 0 = −y 2 that is not a member of the family in part (b).
(d) If y(x) = 1
x + C ,theny(0) = 1
0+C = 1 C .Sincey(0) = 0.5, 1
C = 1 2
⇒ C =2,soy = 1
x +2 .
9. (a) dP
dt =1.2P 1 − P
. Now dP
4200 dt > 0 ⇒ 1 − P
P
> 0 [assuming that P>0] ⇒
4200 4200 < 1 ⇒
P

F.
TX.10
SECTION 9.2 DIRECTION FIELDS AND EULER’S METHOD ¤ 383
7. (a) y(0) = 1
(b) y(0) = 2
(c) y(0) = −1
9.
x y y 0 =1+y
0 0 1
0 1 2
0 2 3
0 −3 −2
0 −2 −1
Note that for y = −1, y 0 =0. The three solution curves sketched go
through (0, 0), (0, −1),and(0, −2).
11.
x y y 0 = y − 2x
−2 −2 2
−2 2 6
2 2 −2
2 −2 −6
Note that y 0 =0for any point on the line y =2x. The slopes are
positive to the left of the line and negative to the right of the line. The
solution curve in the graph passes through (1, 0).
13.
x y y 0 = y + xy
0 ±2 ±2
1 ±2 ±4
−3 ±2 ∓4
Note that y 0 = y(x +1)=0for any point on y =0or on x = −1.
The slopes are positive when the factors y and x +1have the same
sign and negative when they have opposite signs. The solution curve
in the graph passes through (0, 1).

F.
384 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
15. In Maple, we can use either directionfield (in Maple’s share library) or
DEtools[DEplot] to plot the direction field. To plot the solution, we can
either use the initial-value option in directionfield, or actually solve the
equation.
In Mathematica, we use PlotVectorField for the direction field, and the
Plot[Evaluate[...]] construction to plot the solution, which is
y = 2 arctan e x3 /3 · tan 1 .
2
In Derive, use Direction_Field (in utility file ODE_APPR) to plot the direction field. Then use
DSOLVE1(-xˆ2*SIN(y),1,x,y,0,1) (in utility file ODE1) to solve the equation. Simplify each result.
17. The direction field is for the differential equation y 0 = y 3 − 4y.
L = lim
t→∞
y(t) exists for −2 ≤ c ≤ 2;
L = ±2 for c = ±2 and L =0for −2

F.
TX.10
SECTION 9.2 DIRECTION FIELDS AND EULER’S METHOD ¤ 385
(c) (i) For h =0.4: (exact value) − (approximate value) =e 0.4 − 1.4 ≈ 0.0918
(ii) For h =0.2: (exact value) − (approximate value) =e 0.4 − 1.44 ≈ 0.0518
(iii) For h =0.1: (exact value) − (approximate value) =e 0.4 − 1.4641 ≈ 0.0277
Each time the step size is halved, the error estimate also appears to be halved (approximately).
21. h =0.5, x 0 =1, y 0 =0,andF (x, y) =y − 2x.
Note that x 1 = x 0 + h =1+0.5 =1.5, x 2 =2,andx 3 =2.5.
y 1 = y 0 + hF (x 0 ,y 0 )=0+0.5F (1, 0) = 0.5[0 − 2(1)] = −1.
y 2 = y 1 + hF (x 1 ,y 1 )=−1+0.5F (1.5, −1) = −1+0.5[−1 − 2(1.5)] = −3.
y 3 = y 2 + hF (x 2 ,y 2 )=−3+0.5F (2, −3) = −3+0.5[−3 − 2(2)] = −6.5.
y 4 = y 3 + hF (x 3 ,y 3 )=−6.5+0.5F (2.5, −6.5) = −6.5+0.5[−6.5 − 2(2.5)] = −12.25.
23. h =0.1, x 0 =0, y 0 =1,andF (x, y) =y + xy.
Note that x 1 = x 0 + h =0+0.1 =0.1, x 2 =0.2, x 3 =0.3,andx 4 =0.4.
y 1 = y 0 + hF (x 0,y 0)=1+0.1F (0, 1) = 1 + 0.1[1 + (0)(1)] = 1.1.
y 2 = y 1 + hF (x 1,y 1)=1.1+0.1F (0.1, 1.1) = 1.1+0.1[1.1+(0.1)(1.1)] = 1.221.
y 3 = y 2 + hF (x 2 ,y 2 )=1.221 + 0.1F (0.2, 1.221) = 1.221 + 0.1[1.221 + (0.2)(1.221)] = 1.36752.
y 4 = y 3 + hF (x 3 ,y 3 )=1.36752 + 0.1F (0.3, 1.36752) = 1.36752 + 0.1[1.36752 + (0.3)(1.36752)]
=1.5452976.
y 5 = y 4 + hF (x 4 ,y 4 )=1.5452976 + 0.1F (0.4, 1.5452976)
Thus, y(0.5) ≈ 1.7616.
=1.5452976 + 0.1[1.5452976 + (0.4)(1.5452976)] = 1.761639264.
25. (a) dy/dx +3x 2 y =6x 2 ⇒ y 0 =6x 2 − 3x 2 y. Store this expression in Y 1 and use the following simple program to
evaluate y(1) for each part, using H = h =1and N =1for part (i), H =0.1 and N =10for part (ii), and so forth.
h → H: 0 → X: 3 → Y:
For(I,1,N):Y+ H × Y 1 → Y: X + H → X:
End(loop):
Display Y. [To see all iterations, include this statement in the loop.]
(i) H =1,N=1 ⇒ y(1) = 3
(ii) H =0.1,N=10 ⇒ y(1) ≈ 2.3928
(iii) H =0.01,N= 100 ⇒ y(1) ≈ 2.3701
(iv) H =0.001, N = 1000 ⇒ y(1) ≈ 2.3681
(b) y =2+e −x3 ⇒ y 0 = −3x 2 e −x3
LHS = y 0 +3x 2 y = −3x 2 e −x3 +3x 2 2+e −x3 = −3x 2 e −x3 +6x 2 +3x 2 e −x3 =6x 2 =RHS
y(0) = 2 + e −0 =2+1=3

F.
386 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
(c) The exact value of y(1) is 2+e −13 =2+e −1 .
(i) For h =1: (exact value) − (approximate value) =2+e −1 − 3 ≈−0.6321
(ii) For h =0.1: (exact value) − (approximate value) =2+e −1 − 2.3928 ≈−0.0249
(iii) For h =0.01: (exact value) − (approximate value) =2+e −1 − 2.3701 ≈−0.0022
(iv) For h =0.001: (exact value) − (approximate value) =2+e −1 − 2.3681 ≈−0.0002
In (ii)–(iv), it seems that when the step size is divided by 10, the error estimate is also divided by 10 (approximately).
27. (a) R dQ
dt + 1 C Q = E(t) becomes 5Q0 + 1
0.05 Q =60
or Q 0 +4Q =12.
(b) From the graph, it appears that the limiting value of the
charge Q is about 3.
(c) If Q 0 =0,then4Q =12 ⇒ Q =3is an
equilibrium solution.
(d)
(e) Q 0 +4Q =12 ⇒ Q 0 =12− 4Q. NowQ(0) = 0,sot 0 =0and Q 0 =0.
Q 1 = Q 0 + hF (t 0 ,Q 0 )=0+0.1(12 − 4 · 0) = 1.2
Q 2 = Q 1 + hF (t 1,Q 1)=1.2+0.1(12 − 4 · 1.2) = 1.92
Q 3 = Q 2 + hF (t 2 ,Q 2 )=1.92 + 0.1(12 − 4 · 1.92) = 2.352
Q 4 = Q 3 + hF (t 3 ,Q 3 )=2.352 + 0.1(12 − 4 · 2.352) = 2.6112
Q 5 = Q 4 + hF (t 4,Q 4)=2.6112 + 0.1(12 − 4 · 2.6112) = 2.76672
Thus, Q 5 = Q(0.5) ≈ 2.77 C.
9.3 Separable Equations
1.
dy
dx = y x ⇒ dy
y = dx dy dx
x [y 6= 0] ⇒ y
= x
⇒ ln |y| =ln|x| + C ⇒
|y| = e ln|x|+C = e ln|x| e C = e C |x| ⇒ y = Kx,whereK = ±e C is a constant. (In our derivation, K was nonzero,
but we can restore the excluded case y =0by allowing K to be zero.)
3. (x 2 +1)y 0 = xy ⇒ dy
dx = xy
x 2 +1
⇒
dy
y = xdx
x 2 +1
[y 6= 0] ⇒
dy
y =
xdx
x 2 +1
ln |y| = 1 2 ln(x2 +1)+C [u = x 2 +1, du =2xdx] =ln(x 2 +1) 1/2 +lne C =ln e C√ x 2 +1 ⇒
|y| = e C√ x 2 +1 ⇒ y = K √ x 2 +1,whereK = ±e C is a constant. (In our derivation, K was nonzero, but we can
restore the excluded case y =0by allowing K to be zero.)
⇒

F.
5. (1 + tan y) y 0 = x 2 +1 ⇒ (1 + tan y) dy
dx = x2 +1
1 − − sin y
cos y
7.
9.
⇒
1+ sin y
cos y
dy = (x 2 +1)dx ⇒ y − ln |cos y| = 1 3 x3 + x + C.
Note: The left side is equivalent to y +ln|sec y|.
dy
dt =
te t
y 1+y 2 ⇒ y 1+y 2 dy = te t dt ⇒ y 1+y 2 dy = te t dt ⇒ 1 3
SECTION 9.3 SEPARABLE EQUATIONS ¤ 387
dy =(x 2 +1)dx ⇒
1+y
2 3/2 = te t − e t + C
[where the first integral is evaluated by substitution and the second by parts] ⇒ 1+y 2 =[3(te t − e t + C)] 2/3 ⇒
y = ± [3(te t − e t + C)] 2/3 − 1
du
dt
TX.10
=2+2u + t + tu ⇒
du
dt =(1+u)(2 + t) ⇒
du
1+u = (2 + t)dt [u 6=−1] ⇒
ln |1+u| = 1 2 t2 +2t + C ⇒ |1+u| = e t2 /2+2t + C = Ke t2 /2+2t ,whereK = e C ⇒ 1+u = ±Ke t2 /2+2t
⇒
u = −1 ± Ke t2 /2+2t where K>0. u = −1 is also a solution, so u = −1+Ae t2 /2+2t ,whereA is an arbitrary constant.
11.
dy
dx = x y
⇒ ydy= xdx ⇒ ydy= xdx ⇒ 1 2 y2 = 1 2 x2 + C. y(0) = −3 ⇒
1
2 (−3)2 = 1 2 (0)2 + C ⇒ C = 9 ,so 1 2 2 y2 = 1 2 x2 + 9 ⇒ y 2 = x 2 +9 ⇒ y = − √ x
2 2 +9since y(0) = −3 < 0.
13. x cos x =(2y + e 3y ) y 0 ⇒ x cos xdx =(2y + e 3y ) dy ⇒ (2y + e 3y ) dy = x cos xdx ⇒
y 2 + 1 3 e3y = x sin x +cosx + C
[where the second integral is evaluated using integration by parts].
15.
Now y(0) = 0 ⇒ 0+ 1 3 =0+1+C ⇒ C = − 2 3 . Thus, a solution is y2 + 1 3 e3y = x sin x +cosx − 2 3 .
We cannot solve explicitly for y.
du
dt = 2t +sec2 t
, u(0) = −5.
2u
2udu=
2t +sec 2 t dt ⇒ u 2 = t 2 +tant + C,
where [u(0)] 2 =0 2 +tan0+C ⇒ C =(−5) 2 =25. Therefore, u 2 = t 2 +tant +25,sou = ± √ t 2 +tant +25.
Since u(0) = −5,wemusthaveu = − √ t 2 +tant +25.
17. y 0 tan x = a + y, 0

F.
388 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
21. u = x + y ⇒ d
dx (u) = d
du
(x + y) ⇒
dx
du
1+u = dx [u 6=−1] ⇒
du
1+u =
dx =1+dy dx
dy
du
,but = x + y = u, so
dx dx =1+u
dx ⇒ ln |1+u| = x + C ⇒ |1+u| = e x+C ⇒
1+u = ±e C e x ⇒ u = ±e C e x − 1 ⇒ x + y = ±e C e x − 1 ⇒ y = Ke x − x − 1, whereK = ±e C 6=0.
If u = −1,then−1 =x + y ⇒ y = −x − 1,whichisjusty = Ke x − x − 1 with K =0. Thus, the general solution
⇒
is y = Ke x − x − 1,whereK ∈ R.
23. (a) y 0 =2x 1 − y 2 ⇒ dy
dx =2x 1 − y 2
⇒
dy
=2xdx ⇒ 1 − y
2
dy
= 1 − y
2
2xdx
⇒
sin −1 y = x 2 + C for − π 2 ≤ x2 + C ≤ π 2 .
(b) y(0) = 0 ⇒ sin −1 0=0 2 + C ⇒ C =0,
so sin −1 y = x 2 and y =sin x 2 for
− π/2 ≤ x ≤ π/2.
25.
(c) For 1 − y 2 to be a real number, we must have −1 ≤ y ≤ 1;thatis,−1 ≤ y(0) ≤ 1. Thus, the initial-value problem
y 0 =2x 1 − y 2 , y(0) = 2 does not have a solution.
dy
dx = sin x
sin y , y(0) = π 2 . So sin ydy= sin xdx ⇔
− cos y = − cos x + C ⇔ cos y =cosx − C. From the initial condition,
we need cos π 2
=cos0− C ⇒ 0=1− C ⇒ C =1, so the solution is
cos y =cosx − 1. Note that we cannot take cos −1 of both sides, since that would
unnecessarily restrict the solution to the case where −1 ≤ cos x − 1 ⇔ 0 ≤ cos x,
as cos −1 is defined only on [−1, 1]. Instead we plot the graph using Maple’s
plots[implicitplot] or Mathematica’s Plot[Evaluate[···]].
27. (a)
x y y 0 =1/y
0 0.5 2
0 −0.5 −2
0 1 1
0 −1 −1
0 2 0.5
x y y 0 =1/y
0 −2 −0.5
0 4 0.25
0 3 0.3
0 0.25 4
0 0.3 3

F.
TX.10
SECTION 9.3 SEPARABLE EQUATIONS ¤ 389
(b) y 0 =1/y ⇒ dy/dx =1/y ⇒
(c)
ydy= dx ⇒ ydy= dx ⇒ 1 2 y2 = x + C ⇒
y 2 =2(x + C) or y = ± 2(x + C).
29. The curves x 2 +2y 2 = k 2 form a family of ellipses with major axis on the x-axis. Differentiating gives
d
dx (x2 +2y 2 )= d
dx (k2 ) ⇒ 2x +4yy 0 =0 ⇒ 4yy 0 = −2x ⇒ y 0 = −x . Thus, the slope of the tangent line
2y
at any point (x, y) on one of the ellipses is y 0 = −x , so the orthogonal trajectories
2y
must satisfy y 0 = 2y x
dy
y =2 dx
x
⇔
dy
dx = 2y x
⇔
dy
y =2= dx x
⇔ ln |y| =2ln|x| + C 1 ⇔ ln |y| =ln|x| 2 + C 1 ⇔
|y| = e ln x2 +C 1
⇔ y = ± x 2 · e C 1
= Cx 2 . This is a family of parabolas.
⇔
31. The curves y = k/x form a family of hyperbolas with asymptotes x =0and y =0. Differentiating gives
d
dx (y) = d k
⇒ y 0 = − k ⇒ y 0 = − xy [since y = k/x ⇒ xy = k] ⇒ y 0 = − y . Thus, the slope
dx x
x 2 x 2 x
of the tangent line at any point (x, y) on one of the hyperbolas is y 0 = −y/x,
so the orthogonal trajectories must satisfy y 0 = x/y ⇔ dy
dx = x y
⇔
ydy= xdx ⇔ ydy= xdx ⇔ 1 2 y2 = 1 2 x2 + C 1 ⇔
y 2 = x 2 + C 2 ⇔ x 2 − y 2 = C. This is a family of hyperbolas with
asymptotes y = ±x.
33. From Exercise 9.2.27, dQ
dt =12− 4Q ⇔
dQ
12 − 4Q =
dt ⇔ − 1 ln|12 − 4Q| = t + C ⇔
4
ln|12 − 4Q| = −4t − 4C ⇔ |12 − 4Q| = e −4t−4C ⇔ 12 − 4Q = Ke −4t [K = ±e −4C ] ⇔
4Q =12− Ke −4t ⇔ Q =3− Ae −4t [A = K/4]. Q(0) = 0 ⇔ 0=3− A ⇔ A =3 ⇔
Q(t) =3− 3e −4t .Ast →∞, Q(t) → 3 − 0=3(the limiting value).
35.
dP
dt = k(M − P ) ⇔
dP
P − M = (−k) dt ⇔ ln|P − M| = −kt + C ⇔ |P − M| = e −kt+C ⇔
P − M = Ae −kt [A = ±e C ] ⇔ P = M + Ae −kt . If we assume that performance is at level 0 when t =0,then
P (0) = 0 ⇔ 0=M + A ⇔ A = −M ⇔ P (t) =M − Me −kt . lim
t→∞
P (t) =M − M · 0=M.

F.
390 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
37. (a) If a = b, then dx
dt = k(a − x)(b − x)1/2 becomes dx
dt = k(a − x)3/2 ⇒ (a − x) −3/2 dx = kdt ⇒
(a − x) −3/2 dx = kdt ⇒ 2(a − x) −1/2 2
= kt + C [by substitution] ⇒
kt + C = √ a − x
(b) dx
dt
2 2
= a − x ⇒ x(t) =a −
kt + C
0=a − 4
C 2 ⇒ 4
C 2 = a ⇒ C2 = 4 a
Thus, x(t) =a −
4
(kt +2/ √ a ) 2 .
= k(a − x)(b − x)1/2 ⇒
4
. The initial concentration of HBr is 0, sox(0) = 0
(kt + C) ⇒
2
⇒ C =2/ √ a [C is positive since kt + C =2(a − x) −1/2 > 0].
dx
(a − x) √ b − x = kdt ⇒
dx
(a − x) √ b − x =
From the hint, u = √ b − x ⇒ u 2 = b − x ⇒ 2udu= −dx,so
dx
(a − x) √ b − x = −2udu
[a − (b − u 2 )]u = −2 du
a − b + u = −2 2
17 1
= −2 √ tan −1 u
√ a − b a − b
kdt ().
du
√
a − b
2
+ u
2
⇒
39. (a) dC
dt
So ()becomes
√
−2 b − x
√ tan −1 √ = kt + C. Nowx(0) = 0 ⇒ C = −2
√
b
√ tan −1 √ and we have
a − b a − b a − b a − b
√
−2 b − x
√ tan −1 √ = kt −
a − b a − b
2
√
a − b
tan −1
√
b
√
a − b
2
b
b − x
t(x) =
k √ tan −1
a − b a − b − tan−1 a − b
= r − kC ⇒
dC
dt = −(kC − r) ⇒
.
⇒
dC
kC − r =
2
b
b − x
√ tan −1
a − b a − b − tan−1 a − b
−dt ⇒ (1/k) ln|kC − r| = −t + M 1 ⇒
ln|kC − r| = −kt + M 2 ⇒ |kC − r| = e −kt+M 2
⇒ kC − r = M 3 e −kt ⇒ kC = M 3 e −kt + r ⇒
C(t) =M 4 e −kt + r/k. C(0) = C 0 ⇒ C 0 = M 4 + r/k ⇒ M 4 = C 0 − r/k ⇒
C(t) =(C 0 − r/k)e −kt + r/k.
(b) If C 0

F.
TX.10
SECTION 9.3 SEPARABLE EQUATIONS ¤ 391
43. Let y(t) be the amount of alcohol in the vat after t minutes. Then y(0) = 0.04(500) = 20 gal. The amount of beer in the vat
is 500 gallons at all times, so the percentage at time t (in minutes) is y(t)/500 × 100, and the change in the amount of alcohol
with respect to time t is dy
dt = rate in − rate out =0.06 5 gal
− y(t)
5 gal
=0.3 − y
min 500 min 100 = 30 − y gal
100 min .
Hence,
dy dt
30 − y = 1
and − ln |30 − y| =
100
t + C. Because y(0) = 20, wehave− ln 10 = C, so
100
− ln |30 − y| = 1
100 t − ln 10 ⇒ ln |30 − y| = −t/100 + ln 10 ⇒ ln |30 − y| =lne−t/100 +ln10 ⇒
ln |30 − y| =ln(10e −t/100 ) ⇒ |30 − y| =10e −t/100 .Sincey is continuous, y(0) = 20, and the right-hand side is
never zero, we deduce that 30 − y is always positive. Thus, 30 − y =10e −t/100 ⇒ y =30− 10e −t/100 .The
percentage of alcohol is p(t) =y(t)/500 × 100 = y(t)/5 =6− 2e −t/100 . The percentage of alcohol after one hour is
p(60) = 6 − 2e −60/100 ≈ 4.9.
45. Assume that the raindrop begins at rest, so that v(0) = 0. dm/dt = km and (mv) 0 = gm ⇒ mv 0 + vm 0 = gm ⇒
mv 0 + v(km) =gm ⇒ v 0 + vk = g ⇒ dv
dt = g − kv ⇒ dv
g − kv = dt ⇒
− (1/k)ln|g − kv| = t + C ⇒ ln |g − kv| = −kt − kC ⇒ g − kv = Ae −kt . v(0) = 0 ⇒ A = g.
So kv = g − ge −kt ⇒ v =(g/k)(1 − e −kt ).Sincek>0,ast →∞, e −kt → 0 and therefore, lim
t→∞
v(t) =g/k.
47. (a) The rate of growth of the area is jointly proportional to A(t) and M − A(t); that is, the rate is proportional to the
product of those two quantities. So for some constant k, dA/dt = k √ A (M − A). We are interested in the maximum of
the function dA/dt (when the tissue grows the fastest), so we differentiate, using the Chain Rule and then substituting for
dA/dt from the differential equation:
d dA √A dA
= k (−1) 1 dA
+(M − A) ·
2
dt dt
dt A−1/2 = 1 dA
2
dt
kA−1/2 [−2A +(M − A)]
dt
=
k √
1
2 kA−1/2 A(M − A) [M − 3A] = 1 2 k2 (M − A)(M − 3A)
This is 0 when M − A =0[this situation never actually occurs, since the graph of A(t) is asymptotic to the line y = M,
as in the logistic model] and when M − 3A =0 ⇔ A(t) =M/3. This represents a maximum by the First Derivative
Test, since d dA
goes from positive to negative when A(t) =M/3.
dt dt
Ce √ 2 Mkt − 1
(b) From the CAS, we get A(t) =M
Ce √ .TogetC in terms of the initial area A 0 and the maximum area M,
Mkt
+1
we substitute t =0and A = A 0 = A(0): A 0 = M
2 C − 1
⇔ (C +1) √ A 0 =(C − 1) √ M ⇔
C +1
C √ A 0 + √ A 0 = C √ M − √ M ⇔ √ M + √ A 0 = C √ M − C √ A 0 ⇔
√ √ √M √
√ √ M + A0
M + A0 = C
− A0 ⇔ C = √ √ . [Notice that if A 0 =0,thenC =1.]
M − A0

F.
392 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
9.4 Models for Population Growth
1. (a) dP/dt =0.05P − 0.0005P 2 =0.05P (1 − 0.01P )=0.05P (1 − P/100). Comparing to Equation 4,
dP/dt = kP(1 − P/K), we see that the carrying capacity is K = 100 and the value of k is 0.05.
(b) The slopes close to 0 occur where P is near 0 or 100. The largest slopes appear to be on the line P =50. The solutions
are increasing for 0 100.
(c)
All of the solutions approach P =100as t increases. As in
part (b), the solutions differ since for 0 100 they are decreasing. Also, some
have an IP and some don’t. It appears that the solutions which
have P 0 =20and P 0 =40have inflection points at P =50.
(d) The equilibrium solutions are P =0(trivial solution) and P = 100. The increasing solutions move away from P =0and
all nonzero solutions approach P =100as t →∞.
3. (a) dy 1
dt = ky − y
K
K
K − y(0)
⇒ y(t) =
with A = .WithK =8× 10 7 , k =0.71,and
1+Ae−kt y(0)
y(0) = 2 × 10 7 ,wegetthemodely(t) = 8 × 107
8 × 107
,soy(1) =
1+3e−0.71t 1+3e ≈ 3.23 × −0.71 107 kg.
(b) y(t) =4× 10 7
⇒
8 × 10 7
1+3e −0.71t =4× 107 ⇒ 2=1+3e −0.71t ⇒ e −0.71t = 1 3
⇒
−0.71t =ln 1 3
⇒ t = ln 3 ≈ 1.55 years
0.71
5. (a) We will assume that the difference in the birth and death rates is 20 million/year. Let t =0correspond to the year 1990
anduseaunitof1 billion for all calculations. k ≈ 1 dP
P dt = 1
5.3 (0.02) = 1
265 ,so
dP
1
dt = kP − P
= 1
K 265 P 1 − P
, P in billions
100
(b) A = K − P0 100 − 5.3
= = 947
P 0 5.3 53 ≈ 17.8679. P (t) = K
1+Ae = 100
,soP (10) ≈ 5.49 billion.
−kt e−(1/265)t
1+ 947
53
(c) P (110) ≈ 7.81,andP (510) ≈ 27.72. The predictions are 7.81 billion in the year 2100 and 27.72 billion in 2500.
(d) If K =50,thenP (t) =
1+ 447
53
50
.SoP (10) ≈ 5.48, P (110) ≈ 7.61,andP (510) ≈ 22.41. The predictions
e−(1/265)t
become 5.48 billion in the year 2000, 7.61 billion in 2100, and 22.41 billion in the year 2500.
7. (a) Our assumption is that dy = ky(1 − y),wherey is the fraction of the population that has heard the rumor.
dt

F.
TX.10
SECTION 9.4 MODELS FOR POPULATION GROWTH ¤ 393
(b) Using the logistic equation (4), dP 1
dt = kP − P
, we substitute y = P dP
, P = Ky,and
K
K dt = K dy
dt ,
to obtain K dy
dt
dy
= k(Ky)(1 − y) ⇔ = ky(1 − y), our equation in part (a).
dt
Now the solution to (4) is P (t) =
We use the same substitution to obtain Ky =
K
K − P0
,whereA = .
1+Ae−kt P 0
K
y 0
1+ K − ⇒ y =
Ky0 e
Ky −kt y 0 +(1− y 0 )e . −kt
0
Alternatively, we could use the same steps as outlined in the solution of Equation 4.
(c) Let t be the number of hours since 8 AM. Theny 0 = y(0) = 80
1000 =0.08 and y(4) = 1 2 ,so
1
2 = y(4) = 0.08
0.08 + 0.92e −4k . Thus, 0.08 + 0.92e−4k =0.16, e −4k = 0.08
0.92 = 2
23 ,ande−k = 2
23
1/4
,
so y =
0.08
0.08 + 0.92(2/23) = 2
. Solving this equation for t,weget
t/4 t/4
2 + 23(2/23)
t/4 2
⇒
= 2 − 2y t/4 2
⇒
= 2 23 23y 23 23 · 1 − y
y
t/4 2
2y +23y =2
23
It follows that t 4
When y =0.9, 1 − y
y
the rumor by 3:36 PM.
9. (a) dP 1
dt = kP − P
K
ln[(1 − y)/y]
− 1=
ln 2 ,sot =4
23
= 1 9 ,sot =4
1 − ln 9
ln 2 23
⇒
d2 P
dt 2
1+
= k P − 1 K
= k kP 1 − P K
ln((1 − y)/y)
ln 2
23
.
⇒
2
23
t/4−1
= 1 − y .
y
≈ 7.6 hor7 h 36 min. Thus, 90% of the population will have heard
dP
+ 1 − P dt K
dP
= k dP
dt dt
1 − P K
1 − 2P
= k 2 P
K
− P K +1− P
K
1 − 2P
K
(b) P grows fastest when P 0 has a maximum, that is, when P 00 =0. From part (a), P 00 =0 ⇔ P =0, P = K,
or P = K/2. Since0

F.
394 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
(b) Since k>0, there will be an exponential expansion ⇔ P 0 − m k > 0 ⇔ mkP0.
(d) P 0 =8,000,000, k = α − β =0.016, m = 210,000 ⇒ m>kP 0 (= 128,000), so by part (c), the population was
declining.
15. (a) The term −15 represents a harvesting of fish at a constant rate—in this case, 15 fish/week. This is the rate at which fish
are caught.
(b) (c) From the graph in part (b), it appears that P (t) = 250 and P (t) =750
are the equilibrium solutions. We confirm this analytically by solving the
equation dP/dt =0as follows: 0.08P (1 − P/1000) − 15 = 0
⇒
0.08P − 0.00008P 2 − 15 = 0 ⇒
−0.00008(P 2 − 1000P + 187,500) = 0
⇒
(P − 250)(P − 750) = 0 ⇒ P = 250 or 750.
(d)
For 0

F.
SECTION 9.4 MODELS FOR POPULATION GROWTH ¤ 395
17. (a) dP 1
dt =(kP) − P
1 − m
. Ifm

F.
396 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
(b)
As k increases, the amplitude
increases, but the minimum value
stays the same.
As r increases, the amplitude and
the period decrease.
A change in φ produces slight
adjustments in the phase shift and
amplitude.
P (t) oscillates between P 0 e (k/r)(1+sin φ) and P 0 e (k/r)(−1+sin φ) (the extreme values are attained when rt − φ is an odd
multiple of π ), so lim P (t) does not exist.
2
t→∞
21. By Equation (7), P (t) =
1+tanhu =1+ eu − e −u
e u + e −u
K
1+Ae . By comparison, if c =(lnA)/k and u = 1 k(t − c), then
−kt 2
=
eu + e −u
e u + e + eu − e −u
−u e u + e = 2e u e−u
· −u e u + e−u e = 2
−u 1+e −2u
and e −2u = e −k(t−c) = e kc e −kt = e ln A e −kt = Ae −kt ,so
1
K 1+tanh 1
k(t − c) = K 2 2
2 [1 + tanh u] = K 2 · 2
1+e = K
−2u 1+e = K
= P (t).
−2u 1+Ae−kt 9.5 Linear Equations
1. y 0 +cosx = y ⇒ y 0 +(−1)y = − cos x is linear since it can be put into the standard linear form (1),
y 0 + P (x) y = Q(x).
3. yy 0 + xy = x 2 ⇒ y 0 + x = x 2 /y ⇒ y 0 − x 2 /y = −x is not linear since it cannot be put into the standard linear
form (1), y 0 + P (x) y = Q(x).
5. Comparing the given equation, y 0 +2y =2e x , with the general form, y 0 + P (x)y = Q(x),weseethatP (x) =2and the
integrating factor is I(x) =e P (x)dx = e 2 dx = e 2x . Multiplying the differential equation by I(x) gives
e 2x y 0 +2e 2x y =2e 3x ⇒ (e 2x y) 0 =2e 3x ⇒ e 2x y = 2e 3x dx ⇒ e 2x y = 2 3 e3x + C ⇒ y = 2 3 ex + Ce −2x .
7. xy 0 − 2y = x 2 [divide by x] ⇒ y 0 + − 2
y = x
x
().
I(x) =e P (x) dx = e (−2/x) dx = e −2ln|x| = e ln|x|−2 = e ln(1/x2) =1/x 2 . Multiplying the differential equation ()
by I(x) gives 1 x 2 y0 − 2 x 3 y = 1 x
⇒
1
x 2 y 0
= 1 x
⇒ 1 x 2 y =ln|x| + C ⇒
y = x 2 (ln |x| + C )=x 2 ln |x| + Cx 2 .

F.
TX.10
SECTION 9.5 LINEAR EQUATIONS ¤ 397
9. Since P (x) is the derivative of the coefficient of y 0 [P (x) =1and the coefficient is x], we can write the differential equation
xy 0 + y = √ x in the easily integrable form (xy) 0 = √ x ⇒ xy = 2 3 x3/2 + C ⇒ y = 2 3
√ x + C/x.
11. sin x dy
dx +(cosx) y =sin(x2 ) ⇒ [(sin x) y] 0 =sin(x 2 ) ⇒ (sin x) y =
sin(x 2
sin(x 2 ) dx + C
) dx ⇒ y =
.
sin x
13. (1 + t) du
du
+ u =1+t, t>0 [divide by 1+t] ⇒
dt dt + 1 u =1 (), which has the
1+t
form u 0 + P (t) u = Q(t). The integrating factor is I(t) =e P (t) dt = e [1/(1+t)] dt = e ln(1+t) =1+t.
Multiplying () byI(t) gives us our original equation back. We rewrite it as [(1 + t)u] 0 =1+t. Thus,
(1 + t)u = (1 + t) dt = t + 1 2 t2 + C ⇒ u = t + 1 2 t2 + C
1+t
or u = t2 +2t +2C
.
2(t +1)
15. y 0 = x + y ⇒ y 0 +(−1)y = x. I(x) =e (−1) dx = e −x . Multiplying by e −x gives e −x y 0 − e −x y = xe −x ⇒
(e −x y) 0 = xe −x ⇒ e −x y = xe −x dx = −xe −x − e −x + C [integration by parts with u = x, dv = e −x dx] ⇒
y = −x − 1+Ce x . y(0) = 2 ⇒ −1+C =2 ⇒ C =3,soy = −x − 1+3e x .
17.
dv
dt − 2tv =3t2 e t2 , v (0) = 5.
−t2 dv
e
dt − 2te−t2 v =3t 2
⇒
I(t) =e (−2t) dt = e −t2 . Multiply the differential equation by I(t) to get
0
e −t2 v =3t
2
⇒ e −t2 v = 3t 2 dt = t 3 + C ⇒ v = t 3 e t2 + Ce t2 .
5=v(0) = 0 · 1+C · 1=C,sov = t 3 e t2 +5e t2 .
19. xy 0 = y + x 2 sin x ⇒ y 0 − 1 x y = x sin x. I(x) =e (−1/x) dx = e − ln x = e ln x−1 = 1 x .
Multiplying by 1 x gives 1 x y0 − 1 0 1
x y =sinx ⇒ 2 x y =sinx ⇒ 1 y = − cos x + C ⇒ y = −x cos x + Cx.
x
y(π) =0 ⇒ −π · (−1) + Cπ =0 ⇒ C = −1,soy = −x cos x − x.
21. xy 0 +2y = e x ⇒ y 0 + 2 x y = ex x .
I(x) =e (2/x) dx = e 2ln|x| =
e ln|x| 2
= |x| 2 = x 2 .
Multiplying by I(x) gives x 2 y 0 +2xy = xe x ⇒ (x 2 y) 0 = xe x ⇒
x 2 y = xe x dx =(x − 1)e x + C [by parts] ⇒
y =[(x − 1)e x + C]/x 2 . The graphs for C = −5, −3, −1, 1, 3, 5,and7 are
shown. C =1is a transitional value. For C1, there is a local minimum. As |C| gets larger, the “branches” get
further from the origin.

F.
398 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
23. Setting u = y 1−n , du
dx =(1− n) dy dy
y−n or
dx dx =
becomes un/(1−n)
1 − n
yn du
1 − n dx = un/(1−n)
1 − n
du
. Then the Bernoulli differential equation
dx
du
dx + P (x)u1/(1−n) = Q(x)u n/(1−n) or du +(1− n)P (x)u = Q(x)(1 − n).
dx
25. Here y 0 + 2 x y = y3
x ,son =3, P (x) = 2 2 x and Q(x) = 1 x .Settingu = 2 y−2 , u satisfies u 0 − 4u x = − 2 x . 2
Then I(x) =e (−4/x) dx = x −4 and u = x 4 − 2 2
x dx + C = x 4 6 5x + C = Cx 4 + 2
5 5x .
Thus, y = ± Cx 4 + 2 −1/2
.
5x
27. (a) 2 dI
dI
+10I =40or
dt dt +5I =20. Then the integrating factor is e 5 dt = e 5t . Multiplying the differential equation
by the integrating factor gives e 5t dI
dt +5Ie5t =20e 5t ⇒ (e 5t I) 0 =20e 5t ⇒
I(t) =e −5t 20e 5t dt + C =4+Ce −5t .But0=I(0) = 4 + C,soI(t) =4− 4e −5t .
(b) I(0.1) = 4 − 4e −0.5 ≈ 1.57 A
29. 5 dQ
dt +20Q =60with Q(0) = 0 C. Then the integrating factor is e 4 dt = e 4t , and multiplying the differential
31.
equation by the integrating factor gives e 4t dQ
dt +4e4t Q =12e 4t ⇒ (e 4t Q) 0 =12e 4t ⇒
Q(t) =e −4t 12e 4t dt + C =3+Ce −4t .But0=Q(0) = 3 + C so Q(t) =3(1− e −4t ) isthechargeattimet
and I = dQ/dt =12e −4t is the current at time t.
dP
dt + kP = kM,soI(t) =e kdt = e kt . Multiplying the differential equation
by I(t) gives e kt dP
dt + kPekt = kMe kt ⇒ (e kt P ) 0 = kMe kt ⇒
P (t) =e −kt kMe kt dt + C = M + Ce −kt , k>0. Furthermore,itis
reasonable to assume that 0 ≤ P (0) ≤ M,so−M ≤ C ≤ 0.
33. y(0) = 0 kg. Salt is added at a rate of 0.4 kg
5 L
=2 kg . Since solution is drained from the tank at a rate of
L min min
3 L/min,butsaltsolutionisaddedatarateof5 L/min, the tank, which starts out with 100 L of water, contains (100 + 2t) L
of liquid after t min. Thus, the salt concentration at time t is
y(t)
100 + 2t
kg
3 L
=
L min
3y
100 + 2t
y(t)
100 + 2t
kg
. Salt therefore leaves the tank at a rate of
L
kg
. Combining the rates at which salt enters and leaves the tank, we get
min

F.
dy
dt =2−
3y
dy
. Rewriting this equation as
100 + 2t
I(t) =exp
TX.10
SECTION 9.6 PREDATOR-PREY SYSTEMS ¤ 399
dt + 3
y =2, we see that it is linear.
100 + 2t
=exp 3
ln(100 + 2t) = (100 + 2t) 3/2
2
3 dt
100 + 2t
Multiplying the differential equation by I(t) gives (100 + 2t) 3/2 dy
dt +3(100+2t)1/2 y =2(100+2t) 3/2
[(100 + 2t) 3/2 y] 0 =2(100+2t) 3/2 ⇒ (100 + 2t) 3/2 y = 2 5 (100 + 2t)5/2 + C ⇒
y = 2 (100 + 2t)+C(100 + 5 2t)−3/2 .Now0=y(0) = 2 (100) + C · 5 100−3/2 =40+ 1 C ⇒ C = −40,000,so
1000
2
y = (100 + 2t) − 40,000(100 + 5 2t)−3/2 kg. From this solution (no pun intended), we calculate the salt concentration
at time t to be C(t) =
y(t)
100 + 2t = −40,000
(100 + 2t) + 2 kg
−40,000
. In particular, C(20) = 5/2 5 L 140 + 2 5/2 5 ≈ 0.2275 kg L
and y(20) = 2 5 (140) − 40,000(140)−3/2 ≈ 31.85 kg.
35. (a) dv
dt + c m v = g and I(t) =e (c/m) dt = e (c/m)t , and multiplying the differential equation by
I(t) gives e (c/m)t dv
dt + vce(c/m)t
0
= ge (c/m)t ⇒ e v (c/m)t = ge (c/m)t . Hence,
m
v(t) =e −(c/m)t ge (c/m)t dt + K = mg/c + Ke −(c/m)t . But the object is dropped from rest, so v(0) = 0 and
K = −mg/c. Thus, the velocity at time t is v(t) =(mg/c)[1 − e −(c/m)t ].
(b) lim
t→∞
v(t) =mg/c
(c) s(t) = v(t) dt =(mg/c)[t +(m/c)e −(c/m)t ]+c 1 where c 1 = s(0) − m 2 g/c 2 .
s(0) is the initial position, so s(0) = 0 and s(t) =(mg/c)[t +(m/c)e −(c/m)t ] − m 2 g/c 2 .
⇒
9.6 Predator-Prey Systems
1. (a) dx/dt = −0.05x +0.0001xy. Ify =0,wehavedx/dt = −0.05x, which indicates that in the absence of y, x declines at
a rate proportional to itself. So x represents the predator population and y represents the prey population. The growth of
the prey population, 0.1y (from dy/dt =0.1y − 0.005xy), is restricted only by encounters with predators (the term
−0.005xy). The predator population increases only through the term 0.0001xy; that is, by encounters with the prey and
not through additional food sources.
(b) dy/dt = −0.015y +0.00008xy. Ifx =0,wehavedy/dt = −0.015y, which indicates that in the absence of x, y would
decline at a rate proportional to itself. So y represents the predator population and x represents the prey population. The
growth of the prey population, 0.2x (from dx/dt =0.2x − 0.0002x 2 − 0.006xy =0.2x(1 − 0.001x) − 0.006xy), is
restricted by a carrying capacity of 1000 [from the term 1 − 0.001x =1− x/1000] and by encounters with predators (the
term −0.006xy). The predator population increases only through the term 0.00008xy;thatis,byencounterswiththeprey
and not through additional food sources.

F.
400 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
3. (a) At t =0, there are about 300 rabbits and 100 foxes. At t = t 1, the number
of foxes reaches a minimum of about 20 while the number of rabbits is
about 1000. Att = t 2 , the number of rabbits reaches a maximum of about
2400, while the number of foxes rebounds to 100. Att = t 3, the number of
rabbits decreases to about 1000 and the number of foxes reaches a
maximum of about 315. Ast increases, the number of foxes decreases
greatly to 100, and the number of rabbits decreases to 300 (the initial
populations), and the cycle starts again.
(b)
5.
7.
dW
dR
=
−0.02W +0.00002RW
0.08R − 0.001RW
0.08 − 0.001W
W
dW =
−0.02 + 0.00002R
R
⇔ (0.08 − 0.001W )RdW =(−0.02 + 0.00002R)W dR ⇔
dR
⇔
0.08
W
− 0.001 dW = − 0.02
R
+0.00002 dR
⇔
0.08 ln|W | − 0.001W = −0.02 ln|R| +0.00002R + K ⇔ 0.08 ln W +0.02 ln R =0.001W +0.00002R + K ⇔
ln W 0.08 R 0.02 =0.00002R +0.001W + K ⇔ W 0.08 R 0.02 = e 0.00002R+0.001W +K ⇔
R 0.02 W 0.08 = Ce 0.00002R e 0.001W
⇔
R 0.02 W 0.08
e 0.00002R e 0.001W
dy −ry + bxy
= C. In general, if =
dx kx − axy ,thenC =
xr y k
e bx e ay .
9. (a) Letting W =0gives us dR/dt =0.08R(1 − 0.0002R). dR/dt =0 ⇔ R =0or 5000. SincedR/dt > 0 for
0

F.
TX.10
CHAPTER 9 REVIEW ¤ 401
R = 1
0.0002
=5000 [as in part (a)]. If R = 1000, then0 = 1000[0.08(1 − 0.0002 · 1000) − 0.001W ] ⇔
0=80(1− 0.2) − W ⇔ W =64.
Case (i):
Case (ii):
Case (iii):
W =0, R =0: both populations are zero
W =0, R = 5000: seepart(a)
R =1000, W =64: the predator/prey interaction balances and the populations are stable.
(c) The populations of wolves and rabbits fluctuate around 64 and 1000, respectively, and eventually stabilize at those values.
(d)
9 Review
1. (a) A differential equation is an equation that contains an unknown function and one or more of its derivatives.
(b) The order of a differential equation is the order of the highest derivative that occurs in the equation.
(c) An initial condition is a condition of the form y(t 0 )=y 0 .
2. y 0 = x 2 + y 2 ≥ 0 for all x and y. y 0 =0only at the origin, so there is a horizontal tangent at (0, 0), but nowhere else. The
graph of the solution is increasing on every interval.
3. See the paragraph preceding Example 1 in Section 9.2.
4. See the paragraph next to Figure 14 in Section 9.2.
5. A separable equation is a first-order differential equation in which the expression for dy/dx can be factored as a function of x
times a function of y, thatis,dy/dx = g(x)f(y). We can solve the equation by integrating both sides of the equation
dy/f(y) =g(x)dx and solving for y.
6. A first-order linear differential equation is a differential equation that can be put in the form dy + P (x) y = Q(x),whereP
dx
and Q are continuous functions on a given interval. To solve such an equation, multiply it by the integrating factor
I(x) =e P (x)dx to put it in the form [I(x) y] 0 = I(x) Q(x) and then integrate both sides to get I(x) y = I(x) Q(x) dx,
that is, e P (x) dx y = e P (x) dx Q(x) dx. Solving for y gives us y = e − P (x) dx e P (x) dx Q(x) dx.

F.
402 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
7. (a) dy
dt = ky; the relative growth rate, 1 y
dy
, is constant.
dt
(b) The equation in part (a) is an appropriate model for population growth, assuming that there is enough room and nutrition to
support the growth.
(c) If y(0) = y 0 , then the solution is y(t) =y 0 e kt .
8. (a) dP/dt = kP(1 − P/K),whereK is the carrying capacity.
(b) The equation in part (a) is an appropriate model for population growth, assuming that the population grows at a rate
proportional to the size of the population in the beginning, but eventually levels off and approaches its carrying capacity
because of limited resources.
9. (a) dF/dt = kF − aF S and dS/dt = −rS + bF S.
(b) In the absence of sharks, an ample food supply would support exponential growth of the fish population, that is,
dF/dt = kF,wherek is a positive constant. In the absence of fish, we assume that the shark population would decline at a
rate proportional to itself, that is, dS/dt = −rS,wherer is a positive constant.
1. True. Since y 4 ≥ 0, y 0 = −1 − y 4 < 0 and the solutions are decreasing functions.
3. False. x + y cannot be written in the form g(x)f(y).
5. True. e x y 0 = y ⇒ y 0 = e −x y ⇒ y 0 +(−e −x )y =0, which is of the form y 0 + P (x) y = Q(x),sothe
equation is linear.
7. True. By comparing dy 1
dt =2y − y
with the logistic differential equation (9.4.4), we see that the carrying
5
capacity is 5;thatis, lim
t→∞
y =5.
1. (a) (b) lim
t→∞
y(t) appears to be finite for 0 ≤ c ≤ 4. In fact
lim y(t) =4for c =4, lim y(t) =2for 0

F.
TX.10
CHAPTER 9 REVIEW ¤ 403
3. (a) We estimate that when x =0.3, y =0.8,soy(0.3) ≈ 0.8.
(b) h =0.1, x 0 =0, y 0 =1and F (x, y) =x 2 − y 2 .Soy n = y n−1 +0.1 x 2 n−1 − y 2 n−1
.Thus,
y 1 =1+0.1 0 2 − 1 2 =0.9, y 2 =0.9+0.1 0.1 2 − 0.9 2 =0.82, y 3 =0.82 + 0.1 0.2 2 − 0.82 2 =0.75676.
This is close to our graphical estimate of y(0.3) ≈ 0.8.
(c) The centers of the horizontal line segments of the direction field are located on the lines y = x and y = −x.
When a solution curve crosses one of these lines, it has a local maximum or minimum.
5. y 0 = xe − sin x − y cos x ⇒ y 0 +(cosx) y = xe − sin x (). This is a linear equation and the integrating factor is
I(x) =e cos xdx = e sin x . Multiplying ()bye sin x gives e sin x y 0 + e sin x (cos x) y = x ⇒ (e sin x y) 0 = x ⇒
e sin x y = 1 2 x2 + C ⇒ y = 1
2 x2 + C e − sin x .
7. 2ye y2 y 0 =2x +3 √ y2 dy
x ⇒ 2ye
dx =2x +3√ x ⇒ 2ye y2 dy =
9.
2x +3 √
x dx
2ye
y 2 dy = 2x +3 √ x
dx ⇒ e y2 = x 2 +2x 3/2 + C ⇒ y 2 =ln(x 2 +2x 3/2 + C) ⇒
y = ± ln(x 2 +2x 3/2 + C)
dr
dr
dr
+2tr = r ⇒
dt dt = r − 2tr = r(1 − 2t) ⇒ r = (1 − 2t) dt ⇒ ln |r| = t − t 2 + C ⇒
|r| = e t−t2 +C = ke t−t2 .Sincer(0) = 5, 5=ke 0 = k. Thus,r(t) =5e t−t2 .
11. xy 0 − y = x ln x ⇒ y 0 − 1 x y =lnx. I(x) =e −1
(−1/x) dx = e − ln|x| = e ln|x| = |x| −1 =1/x since the condition
y(1) = 2 implies that we want a solution with x>0. Multiplying the last differential equation by I(x) gives
1
x y0 − 1 x 2 y = 1 x ln x ⇒ 1
x y 0
= 1 x ln x ⇒ 1 x y = ln x
x dx ⇒ 1 x y = 1 2 (ln x)2 + C ⇒
y = 1 2 x(ln x)2 + Cx. Nowy(1) = 2 ⇒ 2=0+C ⇒ C =2,soy = 1 2 x(ln x)2 +2x.
⇒
13.
d
dx (y) = d
dx (kex ) ⇒ y 0 = ke x = y, so the orthogonal trajectories must have y 0 = − 1 y
⇒
dy
dx = − 1 y
⇒
ydy= −dx ⇒ ydy= − dx ⇒ 1 2 y2 = −x + C ⇒ x = C − 1 2 y2 , which are parabolas with a horizontal axis.

F.
404 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
15. (a) Using (4) and (7) in Section 9.4, we see that for dP
dt =0.1P 1 − P
with P (0) = 100, wehavek =0.1,
2000
K =2000, P 0 =100,andA =
P (t) =
2000 − 100
100
2000
2000
and P (20) = ≈ 560.
1+19e−0.1t 1+19e−2 2000
(b) P = 1200 ⇔ 1200 =
⇔ 1+19e −0.1t = 2000
1+19e −0.1t 1200
e −0.1t =
2
3 /19 ⇔ −0.1t =ln
2
⇔ t = −10 ln 2 ≈ 33.5.
57 57
17. (a) dL
dt ∝ L ∞ − L ⇒ dL
dt = k(L ∞ − L)
⇒
=19. Thus, the solution of the initial-value problem is
dL
L ∞ − L =
⇔ 19e −0.1t = 5 3 − 1 ⇔
kdt ⇒ −ln |L ∞ − L| = kt + C ⇒
ln |L ∞ − L| = −kt − C ⇒ |L ∞ − L| = e −kt−C ⇒ L ∞ − L = Ae −kt ⇒ L = L ∞ − Ae −kt .
At t =0, L = L(0) = L ∞ − A ⇒ A = L ∞ − L(0) ⇒ L(t) =L ∞ − [L ∞ − L(0)]e −kt .
(b) L ∞ =53cm, L(0) = 10 cm, and k =0.2 ⇒ L(t) =53− (53 − 10)e −0.2t =53− 43e −0.2t .
19. Let P represent the population and I the number of infected people. The rate of spread dI/dt is jointly proportional to I and
to P − I, so for some constant k, dI
1
dt = kI(P − I) =(kP)I − I
. From Equation 9.4.7 with K = P and k replaced by
P
21.
kP,wehaveI(t) =
P
1+Ae = I 0 P
−kP t I 0 +(P − I 0 )e . −kP t
Now, measuring t in days, we substitute t =7, P = 5000, I 0 = 160 and I(7) = 1200 to find k:
1200 =
e −35,000k =
160 · 5000
2000
⇔ 3=
⇔ 480 + 14,520e −35,000k =2000 ⇔
160 + (5000 − 160)e −5000·7·k 160 + 4840e −35,000k
2000 − 480
14,520
⇔ −35,000k =ln 38
363
I =5000× 80% = 4000,andsolvefort: 4000 =
160 + 4840e −5000kt =200 ⇔ e −5000kt =
t =
−1
5000k ln 1
121 = 1
1
to be infected.
dh
dt = − R h
V k + h
38
ln
7 363
⇒
· ln 1
121
k + h
h
=7· ln 121
ln 363
38
dh =
⇔ k = −1 38
ln ≈ 0.00006448. Next, let
35,000 363
160 · 5000
200
⇔ 1=
⇔
160 + (5000 − 160)e −k·5000·t 160 + 4840e −5000kt
200 − 160
4840
− R
dt
V
⇔ −5000kt =ln 1
121
≈ 14.875. So it takes about 15 days for 80% of the population
⇒
1+ k
dh = − R h V
⇔
1 dt ⇒
h + k ln h = − R V
t + C. This equation gives a relationship between h and t, but it is not possible to isolate h and express it in
terms of t.
23. (a) dx/dt =0.4x(1 − 0.000005x) − 0.002xy, dy/dt = −0.2y +0.000008xy. Ify =0,then
dx/dt =0.4x(1 − 0.000005x),sodx/dt =0 ⇔ x =0or x = 200,000, which shows that the insect population
increases logistically with a carrying capacity of 200,000. Sincedx/dt > 0 for 0

F.
TX.10
CHAPTER 9 REVIEW ¤ 405
(b) x and y are constant ⇒ x 0 =0and y 0 =0 ⇒
0=0.4x(1 − 0.000005x) − 0.002xy
0=−0.2y +0.000008xy
⇒
0=0.4x[(1 − 0.000005x) − 0.005y]
0=y(−0.2+0.000008x)
The second equation is true if y =0or x = 0.2
0.000008
=25,000. Ify =0in the first equation, then either x =0
or x =
1
0.000005
=200,000. Ifx =25,000, then0=0.4(25,000)[(1 − 0.000005 · 25,000) − 0.005y] ⇒
0=10,000[(1 − 0.125) − 0.005y] ⇒ 0 = 8750 − 50y ⇒ y =175.
Case (i):
y =0, x =0: Zero populations
Case (ii): y =0, x = 200,000: In the absence of birds, the insect population is always 200,000.
Case (iii):
x =25,000, y =175: The predator/prey interaction balances and the populations are stable.
(c) The populations of the birds and insects fluctuate
(d)
around 175 and 25,000, respectively, and
eventually stabilize at those values.
25. (a) d 2 y
dx 2 = k
1+
2 dy
.
dx
Setting z = dy dz
,weget
dx dx = k √ 1+z 2
⇒
dz
√ = kdx. Using Formula 25 gives
1+z
2
ln z + √ 1+z 2 = kx + c ⇒ z + √ 1+z 2 = Ce kx [where C = e c ] ⇒ √ 1+z 2 = Ce kx − z ⇒
1+z 2 = C 2 e 2kx − 2Ce kx z + z 2 ⇒ 2Ce kx z = C 2 e 2kx − 1 ⇒ z = C 2 ekx − 1
2C e−kx .Now
dy
dx = C 2 ekx − 1
2C e−kx ⇒ y = C 2k ekx + 1
2Ck e−kx + C 0 . From the diagram in the text, we see that y(0) = a
and y(±b) =h. a = y(0) = C 2k + 1
2Ck + C 0 ⇒ C 0 = a − C 2k − 1
2Ck
⇒
y = C 2k (ekx − 1) + 1
2Ck (e−kx − 1) + a. Fromh = y(±b), wefind h = C 2k (ekb − 1) + 1
2Ck (e−kb − 1) + a
and h = C 2k (e−kb − 1) + 1
2Ck (ekb − 1) + a. Subtracting the second equation from the first, we get
0= C e kb − e −kb
k 2
− 1 e kb − e −kb
= 1 Ck 2 k
C − 1
sinh kb.
C
Now k>0 and b>0, sosinh kb > 0 and C = ±1. IfC =1,then
y = 1
2k (ekx − 1) + 1
2k (e−kx − 1) + a = 1 e kx + e −kx
− 1 k 2 k + a = a + 1 (cosh kx − 1). IfC = −1,
k
then y = − 1
2k (ekx − 1) − 1
2k (e−kx − 1) + a = −1
k
e kx + e −kx
+ 1 2 k + a = a − 1 (cosh kx − 1).
k
Since k>0, cosh kx ≥ 1,andy ≥ a,weconcludethatC =1and y = a + 1 (cosh kx − 1),where
k

F.
406 ¤ CHAPTER 9 DIFFERENTIAL EQUATIONS
TX.10
h = y(b) =a + 1 (cosh kb − 1). Sincecosh(kb) =cosh(−kb), there is no further information to extract from the
k
condition that y(b) =y(−b). However, we could replace a with the expression h − 1 (cosh kb − 1), obtaining
k
y = h + 1 (cosh kx − cosh kb). It would be better still to keep a in the expression for y, and use the expression for h to
k
solve for k in terms of a, b,andh. That would enable us to express y in terms of x and the given parameters a, b, andh.
Sadly, it is not possible to solve for k in closed form. That would have to be done by numerical methods when specific
parameter values are given.
(b) The length of the cable is
L = b
1+(dy/dx)2 dx = b
−b
−b
=2 (1/k)sinhkx =(2/k)sinhkb
b
0
1+sinh 2 kx dx = b
−b cosh kx dx =2 b
0
cosh kx dx

F.
TX.10
PROBLEMS PLUS
1. We use the Fundamental Theorem of Calculus to differentiate the given equation:
[f(x)] 2 =100+
x
[f(t)] 2 +[f 0 (t)] 2 dt ⇒ 2f(x)f 0 (x) =[f(x)] 2 +[f 0 (x)] 2 ⇒
0
[f(x)] 2 +[f 0 (x)] 2 − 2f(x)f 0 (x) =0 ⇒ [f(x) − f 0 (x)] 2 =0 ⇔ f(x) =f 0 (x).Wecansolvethisasaseparable
equation, or else use Theorem 9.4.2 with k =1, which says that the solutions are f(x) =Ce x .Now[f(0)] 2 =100,so
f(0) = C = ±10, and hence f(x) =±10e x are the only functions satisfying the given equation.
3. f 0 f(x + h) − f(x) f(x)[f(h) − 1]
(x) =lim
=lim
[since f(x + h) =f(x)f(h)]
h→0 h
h→0 h
f(h) − 1
f(h) − f(0)
= f(x) lim = f(x) lim
= f(x)f 0 (0) = f(x)
h→0 h
h→0 h − 0
Therefore, f 0 (x) =f(x) for all x and from Theorem 9.4.2 we get f(x) =Ae x .Nowf(0) = 1 ⇒ A =1 ⇒
f(x) =e x .
5. “The area under the graph of f from 0 to x is proportional to the (n +1)st power of f(x)” translates to
x
0 f(t) dt = k[f(x)]n+1 for some constant k. ByFTC1,
d
dx
x
0
f(t) dt = d
k[f(x)]
n+1
dx
f(x) =k(n +1)[f(x)] n f 0 (x) ⇒ 1=k(n +1)[f(x)] n−1 f 0 (x) ⇒ 1=k(n +1)y n−1 dy
dx
k(n +1)y n−1 dy = dx ⇒ k(n +1)y n−1 dy = dx ⇒ k(n +1) 1 n yn = x + C.
Now f(0) = 0 ⇒ 0=0+C ⇒ C =0and then f(1) = 1 ⇒ k(n +1) 1 n =1 ⇒ k = n
n +1 ,
so y n = x and y = f(x) =x 1/n .
⇒
⇒
7. Let y(t) denote the temperature of the peach pie t minutes after 5:00 PM and R the temperature of the room. Newton’s Law of
Cooling gives us dy/dt = k(y − R). Solving for y we get
dy
= kdt ⇒ ln|y − R| = kt + C ⇒
y − R
|y − R| = e kt+C ⇒ y − R = ±e kt · e C ⇒ y = Me kt + R, whereM is a nonzero constant. We are given
temperatures at three times.
Substituting 100 − M for R in (1) and (2) gives us
y(0) = 100 ⇒ 100 = M + R ⇒ R =100− M
y(10) = 80 ⇒ 80 = Me 10k + R (1)
y(20) = 65 ⇒ 65 = Me 20k + R (2)
Dividing (3) by (4) gives us −20
−35 = M e 10k − 1
M(e 20k − 1)
−20 = Me 10k − M (3) and −35 = Me 20k − M (4)
⇒ 4 7 = e10k − 1
e 20k − 1
⇒ 4e 20k − 4=7e 10k − 7 ⇒
407

F.
408 ¤ CHAPTER 9 PROBLEMS PLUS
TX.10
4e 20k − 7e 10k +3=0. This is a quadratic equation in e 10k . 4e 10k − 3 e 10k − 1 =0 ⇒ e 10k = 3 4 or 1 ⇒
10k =ln 3 4
or ln 1 ⇒ k =
1
10 ln 3 4 since k is a nonzero constant of proportionality. Substituting 3 4 for e10k in (3) gives us
−20 = M · 3
4 − M ⇒ −20 = − 1 4 M ⇒ M =80.NowR =100− M so R =20◦ C.
9. (a) While running from (L, 0) to (x, y), the dog travels a distance
s = L
1+(dy/dx)2 dx = − x
1+(dy/dx)2 dx,so
x
L
ds
dx = − 1+(dy/dx) 2 . The dog and rabbit run at the same speed, so the
rabbit’s position when the dog has traveled a distance s is (0,s). Since the
dog runs straight for the rabbit, dy
dx = s − y (see the figure).
0 − x
Thus, s = y − x dy
dx ⇒ ds
dx = dy
dx −
x d 2 y
dx 2 +1dy dx
gives us x d 2 y
dx 2 =
1+
(b) Letting z = dy
dx
ln x =
dz
√
1+z
2
2 dy
, as claimed.
dx
, we obtain the differential equation x
dz
dx = √ 1+z 2 ,or
= −x d2 y
ds
. Equating the two expressions for
dx2 dx
dz
√
1+z
2 = dx x .Integrating:
25
=ln
z +
1+z 2 + C. Whenx = L, z = dy/dx =0,soln L =ln1+C. Therefore,
C =lnL, soln x =ln √ 1+z 2 + z +lnL =ln L √ 1+z 2 + z ⇒ x = L √ 1+z 2 + z ⇒
√
1+z2 = x x
2
L − z ⇒ 2xz
x
2 x 1+z2 = −
L L + z2 ⇒ − 2z − 1=0 ⇒
L L
z = (x/L)2 − 1
2(x/L)
= x2 − L 2
2Lx
= x
2L − L 1
2 x
dy
[for x>0]. Since z =
dx , y = x2
4L − L ln x + C1.
2
Since y =0when x = L, 0= L 4 − L 2 ln L + C1 ⇒ C1 = L 2 ln L − L 4 .Thus,
y = x2
4L − L 2 ln x + L 2 ln L − L 4 = x2 − L 2
− L x
4L 2 ln .
L
(c) As x → 0 + , y →∞, so the dog never catches the rabbit.
11. (a) We are given that V = 1 3 πr2 h, dV/dt =60,000π ft 3 /h, and r =1.5h = 3 2 h.SoV = 1 3 π 3
2 h2 h = 3 4 πh3 ⇒
dV
dt = 3 π · dh
4 3h2 dt = 9 dh
dh
4 πh2 . Therefore,
dt dt = 4(dV/dt) = 240,000π = 80,000 () ⇒
9πh 2 9πh 2 3h 2
3h 2 dh = 80,000 dt ⇒ h 3 =80,000t + C. Whent =0, h =60.Thus,C =60 3 = 216,000, so
h 3 =80,000t + 216,000. Leth =100.Then100 3 =1,000,000 = 80,000t +216,000
⇒
80,000t =784,000 ⇒ t =9.8, so the time required is 9.8 hours.

F.
TX.10
CHAPTER 9 PROBLEMS PLUS ¤ 409
(b) The floor area of the silo is F = π · 200 2 =40,000π ft 2 , and the area of the base of the pile is
A = πr 2 = π 3
2 h2 = 9π 4 h2 .Sotheareaofthefloor which is not covered when h =60is
F − A =40,000π − 8100π =31,900π ≈ 100,217 ft 2 .NowA = 9π 4 h2 ⇒ dA/dt = 9π 4
· 2h (dh/dt),
and from () in part (a) we know that when h =60, dh/dt = 80,000 = 200
3(60) 2 27
ft/h. Therefore,
dA/dt = 9π (2)(60)
200
4 27 =2000π ≈ 6283 ft 2 /h.
(c) At h =90ft, dV/dt =60,000π − 20,000π =40,000π ft 3 /h. From () inpart(a),
dh
dt = 4(dV/dt) = 4(40,000π) = 160,000 ⇒ 9h 2 dh = 160,000 dt ⇒ 3h 3 = 160,000t + C. Whent =0,
9πh 2 9πh 2 9h 2
h =90; therefore, C =3· 729,000 = 2,187,000. So3h 3 =160,000t +2,187,000. At the top, h = 100
⇒
3(100) 3 =160,000t +2,187,000 ⇒ t = 813,000 ≈ 5.1. The pile reaches the top after about 5.1 h.
160,000
13. Let P (a, b) be any point on the curve. If m is the slope of the tangent line at P ,thenm = y 0 (a), and an equation of the
normal line at P is y − b = − 1 m (x − a), or equivalently, y = − 1 m x + b + a .They-intercept is always 6,so
m
b + a m =6 ⇒ a m =6− b ⇒ m = a
dy
. We will solve the equivalent differential equation
6 − b dx = x
6 − y
(6 − y) dy = xdx ⇒ (6 − y) dy = xdx ⇒ 6y − 1 2 y2 = 1 2 x2 + C ⇒ 12y − y 2 = x 2 + K.
⇒
Since (3, 2) is on the curve, 12(2) − 2 2 =3 2 + K ⇒ K =11.Sothecurveisgivenby12y − y 2 = x 2 +11 ⇒
x 2 + y 2 − 12y +36=−11 + 36 ⇒ x 2 +(y − 6) 2 =25, a circle with center (0, 6) and radius 5.

F.
TX.10

F.
TX.10
10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
10.1 Curves Defined by Parametric Equations
1. x =1+ √ t, y = t 2 − 4t, 0 ≤ t ≤ 5
t 0 1 2 3 4 5
x 1 2 1+ √ 2
2.41
1+ √ 3
2.73
3 1+ √ 5
3.24
y 0 −3 −4 −3 0 5
3. x =5sint, y = t 2 , −π ≤ t ≤ π
t −π −π/2 0 π/2 π
x 0 −5 0 5 0
y π 2 π 2 /4 0 π 2 /4
9.87 2.47 2.47
π 2
9.87
5. x =3t − 5, y =2t +1
(a)
t −2 −1 0 1 2 3 4
x −11 −8 −5 −2 1 4 7
y −3 −1 1 3 5 7 9
(b) x =3t − 5 ⇒ 3t = x +5 ⇒ t = 1 (x +5) ⇒
3
y =2· 1
3 (x +5)+1,soy = 2 3 x + 13 3 .
7. x = t 2 − 2, y =5− 2t, −3 ≤ t ≤ 4
(a)
t −3 −2 −1 0 1 2 3 4
x 7 2 −1 −2 −1 2 7 14
y 11 9 7 5 3 1 −1 −3
(b) y =5− 2t ⇒ 2t =5− y ⇒ t = 1 2
(5 − y) ⇒
x = 1
2 (5 − y)2 − 2,sox = 1 4 (5 − y)2 − 2, −3 ≤ y ≤ 11.
411

F.
412 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
9. x = √ t, y =1− t
(a)
t 0 1 2 3 4
x 0 1 1.414 1.732 2
y 1 0 −1 −2 −3
(b) x = √ t ⇒ t = x 2 ⇒ y =1− t =1− x 2 .Sincet ≥ 0, x ≥ 0.
So the curve is the right half of the parabola y =1− x 2 .
11. (a) x =sinθ, y =cosθ, 0 ≤ θ ≤ π. x 2 + y 2 =sin 2 θ +cos 2 θ =1.Since
0 ≤ θ ≤ π,wehavesin θ ≥ 0,sox ≥ 0. Thus, the curve is the right half of
(b)
the circle x 2 + y 2 =1.
13. (a) x =sint, y =csct, 0

F.
TX.10SECTION 10.1 CURVES DEFINED BY PARAMETRIC EQUATIONS ¤ 413
23. We must have 1 ≤ x ≤ 4 and 2 ≤ y ≤ 3. So the graph of the curve must be contained in the rectangle [1, 4] by [2, 3].
25. When t = −1, (x, y) =(0, −1). Ast increases to 0, x decreases to −1 and y
increases to 0. Ast increases from 0 to 1, x increases to 0 and y increases to 1.
As t increases beyond 1,bothx and y increase. For t

F.
414 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
(c) To start at (0, 3) using the original equations, we must have x 1 =0;thatis,2cost =0. Hence, t = π .Soweuse
2
x =2cost, y =1+2sint, π ≤ t ≤ 3π .
2 2
Alternatively, if we want t to start at 0, we could change the equations of the curve. For example, we could use
x = −2sint, y =1+2cost, 0 ≤ t ≤ π.
35. Big circle: It’s centered at (2, 2) with a radius of 2, so by Example 4, parametric equations are
x =2+2cost, y =2+2sint, 0 ≤ t ≤ 2π
Small circles: They are centered at (1, 3) and (3, 3) with a radius of 0.1. By Example 4, parametric equations are
(left) x =1+0.1cost, y =3+0.1sint, 0 ≤ t ≤ 2π
and (right) x =3+0.1cost, y =3+0.1sint, 0 ≤ t ≤ 2π
Semicircle: It’sthelowerhalfofacirclecenteredat(2, 2) with radius 1. By Example 4, parametric equations are
x =2+1cost, y =2+1sint, π ≤ t ≤ 2π
To get all four graphs on the same screen with a typical graphing calculator, we need to change the last t-interval to [0, 2π] in
order to match the others. We can do this by changing t to 0.5t. This change gives us the upper half. There are several ways to
get the lower half—one is to change the “+”toa“−”inthey-assignment, giving us
x =2+1cos(0.5t), y =2− 1sin(0.5t), 0 ≤ t ≤ 2π
37. (a) x = t 3 ⇒ t = x 1/3 ,soy = t 2 = x 2/3 .
We get the entire curve y = x 2/3 traversedinaleftto
right direction.
(b) x = t 6 ⇒ t = x 1/6 ,soy = t 4 = x 4/6 = x 2/3 .
Since x = t 6 ≥ 0, we only get the right half of the
curve y = x 2/3 .
(c) x = e −3t =(e −t ) 3 [so e −t = x 1/3 ],
y = e −2t =(e −t ) 2 =(x 1/3 ) 2 = x 2/3 .
If t0,thenx and y are between 0
and 1. Sincex>0 and y>0, the curve never quite reaches the origin.
39. The case π 2

F.
TX.10SECTION 10.1 CURVES DEFINED BY PARAMETRIC EQUATIONS ¤ 415
41. It is apparent that x = |OQ| and y = |QP | = |ST|. From the diagram,
x = |OQ| = a cos θ and y = |ST| = b sin θ. Thus, the parametric equations are
x = a cos θ and y = b sin θ. To eliminate θ we rearrange: sin θ = y/b
⇒
sin 2 θ =(y/b) 2 and cos θ = x/a ⇒ cos 2 θ =(x/a) 2 . Adding the two
equations: sin 2 θ +cos 2 θ =1=x 2 /a 2 + y 2 /b 2 . Thus, we have an ellipse.
43. C =(2a cot θ, 2a),sothex-coordinate of P is x =2a cot θ. LetB =(0, 2a).
Then ∠OAB isarightangleand∠OBA = θ,so|OA| =2a sin θ and
A =((2a sin θ)cosθ, (2a sin θ)sinθ). Thus,they-coordinate of P
is y =2a sin 2 θ.
45. (a) There are 2 points of intersection:
(−3, 0) and approximately (−2.1, 1.4).
(b) A collision point occurs when x 1 = x 2 and y 1 = y 2 for the same t. So solve the equations:
3sint = −3+cost (1)
2cost =1+sint (2)
From (2), sin t =2cost − 1. Substituting into (1),weget3(2 cos t − 1) = −3+cost ⇒ 5cost =0 () ⇒
cos t =0 ⇒ t = π or 3π . We check that t = 3π satisfies (1) and (2) but t = π 2 2 2 2
does not. So the only collision point
occurs when t = 3π 2
, and this gives the point (−3, 0). [We could check our work by graphing x1 and x2 together as
functions of t and, on another plot, y 1 and y 2 as functions of t. If we do so, we see that the only value of t for which both
pairs of graphs intersect is t = 3π .] 2
(c) The circle is centered at (3, 1) instead of (−3, 1). There are still 2 intersection points: (3, 0) and (2.1, 1.4),butthereare
no collision points, since ()inpart(b)becomes5cost =6 ⇒ cos t = 6 > 1. 5
47. x = t 2 ,y = t 3 − ct.Weuseagraphingdevicetoproducethegraphsforvariousvaluesofc with −π ≤ t ≤ π. Note that all
the members of the family are symmetric about the x-axis. For c0 the graph crosses itself at x = c, so the loop grows larger as c increases.

F.
416 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
49. Note that all the Lissajous figures are symmetric about the x-axis. The parameters a and b simplystretchthegraphinthe
x-andy-directions respectively. For a = b = n =1the graph is simply a circle with radius 1. Forn =2the graph crosses
itself at the origin and there are loops above and below the x-axis. In general, the figures have n − 1 points of intersection,
all of which are on the y-axis, and a total of n closed loops.
a = b =1 n =2 n =3
10.2 Calculus with Parametric Curves
1. x = t sin t, y = t 2 + t ⇒ dy dx
dy
=2t +1, = t cos t +sint,and
dt dt dx = dy/dt
dx/dt = 2t +1
t cos t +sint .
3. x = t 4 +1, y = t 3 + t; t = −1.
dy
dt =3t2 +1, dx
dt =4t3 ,and dy
dx = dy/dt
dx/dt = 3t2 +1
.Whent = −1,
4t 3
(x, y) =(2, −2) and dy/dx = 4 −4
= −1, so an equation of the tangent to the curve at the point corresponding to t = −1
is y − (−2) = (−1)(x − 2),ory = −x.
5. x = e √t , y = t − ln t 2 ; t =1.
dy 2t
=1−
dt t =1− 2 2 t , dx
dt = e√ t
2 √ dy
,and
t dx = dy/dt
dx/dt = 1 − 2/t
e √ t
/ 2 √ t · 2t
2t = √ 2t − 4
√
te
t
.
When t =1, (x, y) =(e, 1) and dy
dx = − 2 e , so an equation of the tangent line is y − 1=−2 e (x − e),ory = −2 e x +3.
7. (a) x =1+lnt, y = t 2 +2; (1, 3).
dy
dt
dx
=2t,
dt = 1 dy
, and
t dx = dy/dt
dx/dt = 2t
1/t =2t2 .
At (1, 3), x =1+lnt =1 ⇒ ln t =0 ⇒ t =1and dy =2,soanequationofthetangentisy − 3=2(x − 1),
dx
or y =2x +1.
(b) x =1+lnt ⇒ x − 1=lnt ⇒ t = e x−1 ,soy =(e x−1 ) 2 +2=e 2x−2 +2and dy
dx =2e2x−2 .
When x =1, dy
dx =2e0 =2, so an equation of the tangent is y =2x +1,asinpart(a).
9. x =6sint, y = t 2 + t; (0, 0).
dy
dx = dy/dt
dx/dt
2t +1
= .Thepoint(0, 0) corresponds to t =0,sothe
6cost
slope of the tangent at that point is 1 6
. An equation of the tangent is therefore
y − 0= 1 6 (x − 0),ory = 1 6 x.

F.
TX.10
SECTION 10.2 CALCULUS WITH PARAMETRIC CURVES ¤ 417
11. x =4+t 2 , y = t 2 + t 3 ⇒ dy
dx = dy/dt
dx/dt
d 2 y
dx 2 = d
dx
=
2t +3t2
2t
=1+ 3 2 t ⇒
dy
= d(dy/dx)/dt = (d/dt) 1+ 3 t 2
= 3/2
dx dx/dt
2t 2t = 3 4t .
The curve is CU when d2 y
dx 2 > 0,thatis,whent>0.
13. x = t − e t , y = t + e −t ⇒
dy
dx = dy/dt
dx/dt = 1 − 1 − 1 e t − 1
d dy
e−t
= e t
1 − e t 1 − e = e t
t 1 − e = t −e−t ⇒ d2 y
dx = dt dx
=
2 dx/dt
The curve is CU when e t < 1 [since e −t > 0] ⇒ t

F.
418 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
23. We graph the curve x = t 4 − 2t 3 − 2t 2 , y = t 3 − t in the viewing rectangle [−2, 1.1] by [−0.5, 0.5]. This rectangle
corresponds approximately to t ∈ [−1, 0.8].
We estimate that the curve has horizontal tangents at about (−1, −0.4) and (−0.17, 0.39) and vertical tangents at
about (0, 0) and (−0.19, 0.37). We calculate dy
dx = dy/dt
dx/dt = 3t 2 − 1
. The horizontal tangents occur when
4t 3 − 6t 2 − 4t
dy/dt =3t 2 − 1=0 ⇔ t = ± 1 √
3
, so both horizontal tangents are shown in our graph. The vertical tangents occur when
dx/dt =2t(2t 2 − 3t − 2) = 0 ⇔ 2t(2t +1)(t − 2) = 0 ⇔ t =0, − 1 or 2. It seems that we have missed one vertical
2
tangent, and indeed if we plot the curve on the t-interval [−1.2, 2.2] we see that there is another vertical tangent at (−8, 6).
25. x =cost, y =sint cos t. dx/dt = − sin t, dy/dt = − sin 2 t +cos 2 t =cos2t.
(x, y) =(0, 0) ⇔ cos t =0 ⇔ t is an odd multiple of π 2 .Whent = π 2 ,
dx/dt = −1 and dy/dt = −1,sody/dx =1. Whent = 3π , dx/dt =1and
2
dy/dt = −1. Sody/dx = −1. Thus, y = x and y = −x are both tangent to the
curve at (0, 0).
27. x = rθ − d sin θ, y = r − d cos θ.
(a) dx
dy
dy
= r − d cos θ, = d sin θ,so
dθ dθ dx = d sin θ
r − d cos θ .
(b) If 0

F.
TX.10
SECTION 10.2 CALCULUS WITH PARAMETRIC CURVES ¤ 419
33. The curve x =1+e t , y = t − t 2 = t(1 − t) intersects the x-axis when y =0,
that is, when t =0and t =1. The corresponding values of x are 2 and 1+e.
The shaded area is given by
x=1+e
x=2
(y T − y B ) dx =
t=1
t=0
35. x = rθ − d sin θ, y = r − d cos θ.
[y(t) − 0] x 0 (t) dt = 1
0 (t − t2 )e t dt
= 1
0 tet dt − 1
0 t2 e t dt = 1
0 tet dt − t 2 e t 1
0 +2 1
0 tet dt [Formula 97 or parts]
=3 1
0 tet dt − (e − 0) = 3 (t − 1)e t 1
− e [Formula 96 or parts]
0
=3[0− (−1)] − e =3− e
A = 2πr
0
ydx= 2π
0 (r − d cos θ)(r − d cos θ) dθ = 2π
0 (r2 − 2dr cos θ + d 2 cos 2 θ) dθ
= r 2 θ − 2dr sin θ + 1 2 d2 θ + 1 2 sin 2θ 2π
0
=2πr 2 + πd 2
37. x = t − t 2 , y = 4 3 t3/2 , 1 ≤ t ≤ 2. dx/dt =1− 2t and dy/dt =2t 1/2 ,so
(dx/dt) 2 +(dy/dt) 2 =(1− 2t) 2 +(2t 1/2 ) 2 =1− 4t +4t 2 +4t =1+4t 2 .
Thus, L = b
(dx/dt)2 +(dy/dt)
a
2 dt = 2
√
1+4t2 dt ≈ 3.1678.
1
39. x = t +cost, y = t − sin t, 0 ≤ t ≤ 2π. dx/dt =1− sin t and dy/dt =1− cos t,so
dx
2
dt + dy
2
dt =(1− sin t) 2 +(1− cos t) 2 =(1− 2sint +sin 2 t)+(1− 2cost +cos 2 t)=3− 2sint − 2cost.
Thus, L = b
(dx/dt)2 +(dy/dt)
a
2 dt = 2π √
0 3 − 2sint − 2costdt≈ 10.0367.
41. x =1+3t 2 , y =4+2t 3 , 0 ≤ t ≤ 1.
dx/dt =6t and dy/dt =6t 2 ,so(dx/dt) 2 +(dy/dt) 2 =36t 2 +36t 4 .
Thus, L = 1
√
36t2 +36t
0
4 dt = 1
6t √ 1+t
0 2 dt
=6 2 √
1 u 1 du [u =1+t 2 , du =2tdt]
2
=3
2
3 u3/2 2
1
=2(2 3/2 − 1) = 2 2 √ 2 − 1
43. x = t
1+t , y =ln(1+t), 0 ≤ t ≤ 2. dx (1 + t) · 1 − t · 1 1 dy
= = and
dt (1 + t) 2 (1 + t)
2
dt = 1
1+t ,
2 2 dx dy 1
so + =
dt dt (1 + t) + 1
4 (1 + t) = 1 1+(1+t)
2
= t2 +2t +2
. Thus,
2 (1 + t) 4 (1 + t) 4
2
√
t2 +2t +2
3
√
u2 +1
L =
dt =
du
0 (1 + t) 2 1 u 2
= − √ 10
3
+ln 3+ √ 10 + √ 2 − ln 1+ √ 2
u = t +1,
du = dt
√
24 u2 +1
= − +ln u + 3
u
u
2 +1
1

F.
420 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
45. x = e t cos t, y = e t sin t, 0 ≤ t ≤ π.
dx
dt
Thus, L = π
0
2
+
dy
2
dt =[e t (cos t − sin t)] 2 +[e t (sin t +cost)] 2
=(e t ) 2 (cos 2 t − 2cost sin t +sin 2 t)
+(e t ) 2 (sin 2 t +2sint cos t +cos 2 t
= e 2t (2 cos 2 t +2sin 2 t)=2e 2t
√
2e
2t
dt = π
√
0 2 e t dt = √ 2 e t π
= √ 2(e π − 1).
0
47. x = e t − t, y =4e t/2 , −8 ≤ t ≤ 3
dx
dt
2 + dy
2
dt =(e t − 1) 2 +(2e t/2 ) 2 = e 2t − 2e t +1+4e t
= e 2t +2e t +1=(e t +1) 2
L = 3
(et +1)
−8
2 dt = 3
−8 (et +1)dt = e t + t 3t
−8
=(e 3 +3)− (e −8 − 8) = e 3 − e −8 +11
49. x = t − e t , y = t + e t , −6 ≤ t ≤ 6.
dx
2
dt + dy
2
dt =(1− e t ) 2 +(1+e t ) 2 =(1− 2e t + e 2t )+(1+2e t + e 2t )=2+2e 2t ,soL = 6
√
−6 2+2e
2t
dt.
Set f(t) = √ 2+2e 2t . Then by Simpson’s Rule with n =6and ∆t = 6−(−6) =2,weget
6
L ≈ 2 [f(−6) + 4f(−4) + 2f(−2) + 4f(0) + 2f(2) + 4f(4) + f(6)] ≈ 612.3053.
3
51. x =sin 2 t, y =cos 2 t, 0 ≤ t ≤ 3π.
(dx/dt) 2 +(dy/dt) 2 =(2sint cos t) 2 +(−2cost sin t) 2 =8sin 2 t cos 2 t =2sin 2 2t ⇒
Distance = 3π
√ √ π/2
0 2 |sin 2t| dt =6 2 sin 2tdt [by symmetry] = −3 √ π/2
2 cos 2t = −3 √ 2(−1 − 1) = 6 √ 2.
0
0
The full curve is traversed as t goes from 0 to π 2
, because the curve is the segment of x + y =1that lies in the first quadrant
(since x, y ≥ 0), and this segment is completely traversed as t goes from 0 to π 2 . Thus, L = π/2
0
sin 2tdt= √ 2,asabove.
53. x = a sin θ, y = b cos θ, 0 ≤ θ ≤ 2π.
dx
2
dt + dy
2
dt =(a cos θ) 2 +(−b sin θ) 2 = a 2 cos 2 θ + b 2 sin 2 θ = a 2 (1 − sin 2 θ)+b 2 sin 2 θ
= a 2 − (a 2 − b 2 )sin 2 θ = a 2 − c 2 sin 2 θ = a 2 1 − c2
a 2 sin2 θ = a 2 (1 − e 2 sin 2 θ)
1 − e2 sin 2 θdθ.
So L =4
π/2
0
a 2 1 − e 2 sin 2 θ dθ [by symmetry] =4a π/2
0
55. (a) x =11cost − 4cos(11t/2), y =11sint − 4sin(11t/2).
Notice that 0 ≤ t ≤ 2π does not give the complete curve because
x(0) 6=x(2π). In fact, we must take t ∈ [0, 4π] in order to obtain the
complete curve, since the first term in each of the parametric equations has
period 2π and the second has period 2π
= 4π
11/2 11
integer multiple of these two numbers is 4π.
, and the least common

F.
TX.10
SECTION 10.2 CALCULUS WITH PARAMETRIC CURVES ¤ 421
(b)WeusetheCAStofind the derivatives dx/dt and dy/dt, and then use Formula 1 to find the arc length. Recent versions
of Maple express the integral 4π
(dx/dt)2 +(dy/dt)
0
2 dt as 88E 2 √ 2 i ,whereE(x) is the elliptic integral
1
√
1 − x2 t
√ 2
dt and i is the imaginary number √ −1.
1 − t
2
0
Some earlier versions of Maple (as well as Mathematica) cannot do the integral exactly, so we use the command
evalf(Int(sqrt(diff(x,t)ˆ2+diff(y,t)ˆ2),t=0..4*Pi)); to estimate the length, and find that the arc
length is approximately 294.03. Derive’sPara_arc_length function in the utility file Int_apps simplifies the
integral to 11
4π
−4cost cos
11t
0
2 − 4sint sin 11t
2 +5dt.
57. x =1+te t , y =(t 2 +1)e t , 0 ≤ t ≤ 1.
dx
dt
2
+
dy
2
dt =(te t + e t ) 2 +[(t 2 +1)e t + e t (2t)] 2 =[e t (t +1)] 2 +[e t (t 2 +2t +1)] 2
= e 2t (t +1) 2 + e 2t (t +1) 4 = e 2t (t +1) 2 [1 + (t +1) 2 ], so
S = 2πy ds = 1
0 2π(t2 +1)e t e 2t (t +1) 2 (t 2 +2t +2)dt = 1
0 2π(t2 +1)e 2t (t +1) √ t 2 +2t +2dt ≈ 103.5999
59. x = t 3 , y = t 2 , 0 ≤ t ≤ 1.
dx
2
dt + dy
2
dt = 3t
2 2 +(2t) 2 =9t 4 +4t 2 .
S =
1
=2π
= π 81
2πy dx
0
13
= 2π
1215
4
dt
2
+
dy
dt
u − 4 √u 1
18
9
du
1
2
dt = 2πt 2 9t 4 +4t 2 dt =2π
0
u =9t 2 +4, t 2 =(u − 4)/9,
du =18tdt,sotdt = 1 18 du
13
13
2
5 u5/2 − 8 3 u3/2 =
3u π · 2 5/2 − 20u 3/2 81 15
4
4
√ 3 · 13
2
13 − 20 · 13 √ 13 − (3 · 32 − 20 · 8) =
1
0
t 2 t 2 (9t 2 +4)dt
= 2π
9 · 18
13
√
2π
1215 247 13 + 64
4
(u 3/2 − 4u 1/2 ) du
61. x = a cos 3 θ, y = a sin 3 θ, 0 ≤ θ ≤ π . dx
2
2 dθ + dy
2
dθ =(−3a cos 2 θ sin θ) 2 +(3a sin 2 θ cos θ) 2 =9a 2 sin 2 θ cos 2 θ.
S = π/2
2π · a sin 3 θ · 3a sin θ cos θdθ=6πa 2 π/2
sin 4 θ cos θdθ= 6 πa2 sin 5 0 0 5
θ π/2
= 6 0 5 πa2
63. x = t + t 3 , y = t − 1 t , 1 ≤ t ≤ 2. dx
2 dt =1+3t2 and dy =1+ 2 dt
t ,so
dx 2 3 dt + dy
2
dt =(1+3t 2 ) 2 +
1+ 2 2
t 3
and S =
2πy ds =
2
1
2π t − 1
(1 + 3t
t 2 ) 2 + 1+ 2 2
dt ≈ 59.101.
2 t 3
65. x =3t 2 , y =2t 3 , 0 ≤ t ≤ 5 ⇒
dx 2
dt + dy
2
dt =(6t) 2 +(6t 2 ) 2 =36t 2 (1 + t 2 ) ⇒
S = 5
2πx (dx/dt)
0 2 +(dy/dt) 2 dt = 5
0 2π(3t2 )6t √ 1+t 2 dt =18π 5
t2√ 1+t
0 2 2tdt
=18π 26
1
(u − 1) √ udu
u =1+t 2 ,
du =2tdt
=18π 26
1 (u3/2 − u 1/2 ) du =18π
=18π 2 · 676 √ 5
26 − 2 · 26 √ 3
26 − 2
− 2
5 3 =
24
π 5
949 √ 26 + 1
26
2
5 u5/2 − 2 3 u3/2
1

F.
422 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
67. If f 0 is continuous and f 0 (t) 6= 0for a ≤ t ≤ b, then either f 0 (t) > 0 for all t in [a, b] or f 0 (t) < 0 for all t in [a, b]. Thus,f
is monotonic (in fact, strictly increasing or strictly decreasing) on [a, b]. It follows that f has an inverse. Set F = g ◦ f −1 ,
that is, define F by F (x) =g(f −1 (x)). Thenx = f(t) ⇒ f −1 (x) =t,soy = g(t) =g(f −1 (x)) = F (x).
dy
69. (a) φ =tan −1 ⇒
dx
d dy
= d ẏ
=
dt dx dt ẋ
t
fact that s =
dx
dt
0
dφ
dt = d
dy
1 d dy
dt tan−1 =
dx 1+(dy/dx) 2 dt dx
ÿẋ − ẍẏ
⇒ dφ
ÿẋ
ẋ 2 dt = 1 − ẍẏ
=
1+(ẏ/ẋ) 2 ẋ 2
. But dy
dx = dy/dt
dx/dt = ẏ
ẋ
⇒
ẋÿ − ẍẏ
. Using the Chain Rule, and the
ẋ 2 + ẏ2 2
+
dy
2
dt dt ⇒
ds = dx
2
dt dt + dy
2
dt = ẋ2 + ẏ 21/2 ,wehavethat
dφ
ds = dφ/dt
ẋÿ − ẍẏ
ds/dt = 1 ẋÿ − ẍẏ
=
ẋ 2 + ẏ 2 (ẋ 2 + ẏ 2 )
1/2
(ẋ 2 + ẏ 2 ) .Soκ = dφ
3/2 ds = ẋÿ − ẍẏ |ẋÿ − ẍẏ|
=
(ẋ 2 + ẏ 2 ) 3/2 (ẋ 2 + ẏ 2 ) . 3/2
(b) x = x and y = f(x) ⇒ ẋ =1, ẍ =0and ẏ = dy
dx , ÿ = d2 y
dx . 2
1 · (d 2 y/dx 2 ) − 0 · (dy/dx)
d 2 y/dx 2
So κ =
=
[1 + (dy/dx) 2 ] 3/2 [1 + (dy/dx) 2 ] . 3/2
71. x = θ − sin θ ⇒ ẋ =1− cos θ ⇒ ẍ =sinθ,andy =1− cos θ ⇒ ẏ =sinθ ⇒ ÿ =cosθ. Therefore,
cos θ − cos 2 θ − sin 2 θ κ =
[(1 − cos θ) 2 +sin 2 θ] = cos θ − (cos 2 θ +sin 2 θ) |cos θ − 1|
3/2 (1 − 2cosθ +cos 2 θ +sin 2 = . The top of the arch is
θ)
3/2 (2 − 2cosθ)
3/2
characterized by a horizontal tangent, and from Example 2(b) in Section 10.2, the tangent is horizontal when θ =(2n − 1)π,
so take n =1and substitute θ = π into the expression for κ: κ =
73. The coordinates of T are (r cos θ, r sin θ). SinceTP was unwound from
arc TA, TP has length rθ. Also∠PTQ = ∠PTR− ∠QT R = 1 2 π − θ,
so P has coordinates x = r cos θ + rθ cos 1
2 π − θ = r(cos θ + θ sin θ),
y = r sin θ − rθ sin 1
2 π − θ = r(sin θ − θ cos θ).
|cos π − 1|
(2 − 2cosπ) 3/2 = |−1 − 1|
[2 − 2(−1)] 3/2 = 1 4 .
10.3 Polar Coordinates
1. (a) 2, π 3
By adding 2π to π 3 , we obtain the point 2, 7π 3
. The direction
opposite π 3 is 4π 3 ,so −2, 4π 3
requirement.
is a point that satisfies the r

F.
(b) 1, − 3π 4
TX.10
SECTION 10.3 POLAR COORDINATES ¤ 423
r>0: 1, − 3π +2π 4
=
1, 5π 4
r0: −(−1), π + π 2
=
1, 3π 2
r

F.
424 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
7. The curves r =1and r =2represent circles with center
O andradii1and2.Theregionintheplanesatisfying
1 ≤ r ≤ 2 consists of both circles and the shaded region
between them in the figure.
9. The region satisfying 0 ≤ r

F.
TX.10
SECTION 10.3 POLAR COORDINATES ¤ 425
29. θ = −π/6 31. r =sinθ ⇔ r 2 = r sin θ ⇔ x 2 + y 2 = y ⇔
x 2 +
y − 1 2
=
1 2
. The reasoning here is the same
2 2
as in Exercise 17. This is a circle of radius 1 2 centered at 0, 1 2
.
33. r =2(1− sin θ). This curve is a cardioid. 35. r = θ, θ ≥ 0
37. r =4sin3θ
39. r =2cos4θ
41. r =1− 2sinθ

F.
426 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
43. r 2 =9sin2θ
45. r =2cos 3
2 θ
47. r =1+2cos2θ
49. For θ =0, π,and2π, r has its minimum value of about 0.5. Forθ = π and 3π , r attains its maximum value of 2.
2 2
We see that the graph has a similar shape for 0 ≤ θ ≤ π and π ≤ θ ≤ 2π.
51. x =(r)cosθ =(4+2secθ)cosθ =4cosθ +2.Now,r →∞ ⇒
(4 + 2 sec θ) →∞ ⇒ θ →
π −
or θ →
3π +
[since we need only
2
2
consider 0 ≤ θ

F.
TX.10
SECTION 10.3 POLAR COORDINATES ¤ 427
53. To show that x =1is an asymptote we must prove lim
r→±∞ x =1.
x =(r)cosθ =(sinθ tan θ)cosθ =sin 2 θ.Now,r →∞ ⇒ sin θ tan θ →∞ ⇒
θ →
π −
,so lim x =
2 r→∞
θ →
π +
2
,so lim x =
r→−∞
lim sin2 θ =1.Also,r →−∞ ⇒ sin θ tan θ →−∞ ⇒
θ→π/2 −
lim sin2 θ =1. Therefore,
θ→π/2 +
lim
r→±∞
x =1 ⇒ x =1is
a vertical asymptote. Also notice that x =sin 2 θ ≥ 0 for all θ,andx =sin 2 θ ≤ 1 for all θ. Andx 6= 1, since the curve is not
defined at odd multiples of π . Therefore, the curve lies entirelywithintheverticalstrip0 ≤ x

F.
428 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
63. r =3cosθ ⇒ x = r cos θ =3cosθ cos θ, y = r sin θ =3cosθ sin θ ⇒
dy
= dθ −3sin2 θ +3cos 2 θ =3cos2θ =0 ⇒ 2θ = π or 3π ⇔ θ = π or 3π .
2 2 4 4
3
So the tangent is horizontal at √
2
, π and − √ 3 4
2
, 3π 3
same as √
4
2
, − π .
4
dx
= −6sinθ cos θ = −3sin2θ =0 ⇒ 2θ =0or π ⇔ θ =0or π . So the tangent is vertical at (3, 0) and
0, π dθ 2 2 .
65. r =1+cosθ ⇒ x = r cos θ =cosθ (1 + cos θ), y = r sin θ =sinθ (1 + cos θ) ⇒
dy
=(1+cosθ) cosθ − dθ sin2 θ =2cos 2 θ +cosθ − 1=(2cosθ − 1)(cos θ +1)=0 ⇒ cos θ = 1 2
or −1 ⇒
θ = π , π,or 5π ⇒ horizontal tangent at 3
, π
3 3 2 3 , (0,π),and 3 , 5π
2 3 .
dx
= −(1 + cos θ)sinθ − cos θ sin θ = − sin θ (1 + 2 cos θ) =0 ⇒ sin θ =0or cos θ = − 1
dθ 2
⇒
θ =0, π, 2π ,or 4π ⇒ vertical tangent at (2, 0), 1
, 2π
3 3 2 3 ,and 1 , 4π
2 3 .
Note that the tangent is horizontal, not vertical when θ = π,since lim
θ→π
dy/dθ
dx/dθ =0.
67. r =2+sinθ ⇒ x = r cos θ =(2+sinθ) cosθ, y = r sin θ =(2+sinθ) sinθ ⇒
dy
dθ
=(2+sinθ) cosθ +sinθ cos θ =cosθ · 2(1 + sin θ) =0 ⇒ cos θ =0or sin θ = −1 ⇒
θ = π 2 or 3π 2
⇒ horizontal tangent at 3, π 2
dx
and
1,
3π
2
=(2+sinθ)(− sin θ)+cosθ cos θ = −2sinθ − dθ sin2 θ +1− sin 2 θ = −2sin 2 θ − 2sinθ +1
sin θ = 2 ± √ 4+8
= 2 ± 2 √ 3
= 1 − √ √
3 1+ 3
< −1 ⇒
−4 −4 −2 −2
.
⇒
θ 1 =sin −1 − 1 + √ 1
2 2 3 and θ2 = π − θ 1 ⇒ vertical tangent at 3
+ √ 1 3,θ1
2 2 and 3 + √ 1 3,θ2
2 2 .
Note that r(θ 1)=2+sin sin −1 − 1 + √ 1
2 2 3 =2−
1
+ √ 1
2 2 3=
3
+ √ 1
2 2 3.
69. r = a sin θ + b cos θ ⇒ r 2 = ar sin θ + br cos θ ⇒ x 2 + y 2 = ay + bx ⇒
x 2 − bx + 1
2 b2 + y 2 − ay + 1
2 a2 = 1
2 b2 + 1
2 a 2
with center 1
2 b, 1 2 a and radius 1 2
√
a2 + b 2 .
⇒
x − 1 2 b2 + y − 1 2 a2 = 1 4 (a2 + b 2 ), and this is a circle
Note for Exercises 71–76: Maple is able to plot polar curves using the polarplot command, or using the coords=polar option in a regular
plot command. In Mathematica, use PolarPlot. In Derive, change to Polar under Options State. If your graphing device cannot
plot polar equations, you must convert to parametric equations. For example, in Exercise 71, x = r cos θ =[1+2sin(θ/2)] cos θ,
y = r sin θ =[1+2sin(θ/2)] sin θ.
71. r =1+2sin(θ/2). The parameter interval is [0, 4π]. 73. r = e sin θ − 2cos(4θ). The parameter interval is [0, 2π].

F.
TX.10
SECTION 10.3 POLAR COORDINATES ¤ 429
75. r =2− 5sin(θ/6). The parameter interval is [−6π, 6π].
77. It appears that the graph of r =1+sin θ − π 6
is the same shape as
the graph of r =1+sinθ, but rotated counterclockwise about the
origin by π 6 . Similarly, the graph of r =1+sin θ − π 3
is rotated by
π
3
. In general, the graph of r = f(θ − α) is the same shape as that of
r = f(θ), but rotated counterclockwise through α about the origin.
That is, for any point (r 0,θ 0) on the curve r = f(θ),thepoint
(r 0,θ 0 + α) is on the curve r = f(θ − α),sincer 0 = f(θ 0)=f((θ 0 + α) − α).
79. (a) r =sinnθ.
n =2 n =3 n =4 n =5
From the graphs, it seems that when n is even, the number of loops in the curve (called a rose) is 2n,andwhenn is odd,
the number of loops is simply n. This is because in the case of n odd, every point on the graph is traversed twice, due to
the fact that
r(θ + π) =sin[n(θ + π)] = sin nθ cos nπ +cosnθ sin nπ =
sin nθ
− sin nθ
(b) The graph of r = |sin nθ| has 2n loops whether n is odd or even, since r(θ + π) =r(θ).
if n is even
if n is odd
n =2 n =3 n =4 n =5

F.
430 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
81. r = 1 − a cos θ . We start with a =0, since in this case the curve is simply the circle r =1.
1+a cos θ
As a increases, the graph moves to the left, and its right side becomes flattened. As a increases through about 0.4, the right
side seems to grow a dimple, which upon closer investigation (with narrower θ-ranges) seems to appear at a ≈ 0.42 [the
actual value is √ 2 − 1]. As a → 1, this dimple becomes more pronounced, and the curve begins to stretch out horizontally,
until at a =1the denominator vanishes at θ = π, and the dimple becomes an actual cusp. For a>1 we must choose our
parameter interval carefully, since r →∞as 1+a cos θ → 0 ⇔ θ → ± cos −1 (−1/a).Asa increases from 1,thecurve
splits into two parts. The left part has a loop, which grows larger as a increases, and the right part grows broader vertically,
and its left tip develops a dimple when a ≈ 2.42 [actually, √ 2+1]. As a increases, the dimple grows more and more
pronounced. If a

F.
83. tan ψ =tan(φ − θ) =
=
dy
dθ − dx
dθ tan θ
dx
dθ + dy =
dθ tan θ
dy
dx − tan θ
1+ dy =
dx tan θ
dr
dr
dθ sin θ + r cos θ − tan θ
dr
dr
dθ cos θ − r sin θ +tanθ
tan φ − tan θ
1+tanφ tan θ =
= r cos2 θ + r sin 2 θ
dr
dθ cos2 θ + dr = r
dr/dθ
dθ sin2 θ
TX.10SECTION 10.4 AREAS AND LENGTHS IN POLAR COORDINATES ¤ 431
dy/dθ
dx/dθ − tan θ
1+ dy/dθ
dx/dθ tan θ
dθ cos θ − r sin θ
dθ sin θ + r cos θ =
r cos θ + r · sin2 θ
cos θ
dr dr
cos θ +
dθ dθ · sin2 θ
cos θ
10.4 Areas and Lengths in Polar Coordinates
1. r = θ 2 , 0 ≤ θ ≤ π 4 . A = π/4
3. r =sinθ, π 3 ≤ θ ≤ 2π 3 .
2π/3
1
A =
2 sin2 θdθ= 1 4
π/3
= 1 4
2π
3 − 1 2
− √ 3
2
0
π/4
π/4
1
2 r2 1
dθ =
2 (θ2 ) 2 1
dθ =
2 θ4 dθ = 1
θ5 π/4
= 1 π
5
= 1
10 0 10 4 10,240 π5
0
0
2π/3
− π 3 + 1 2
π/3
√3
(1 − cos 2θ) dθ = 1 4
2
= 1 4
θ −
1
sin 2θ 2π/3
= 1 2π − 1 sin 4π − π + 1 sin
2π
2 π/3 4 3 2 3 3 2 3
π
+ √
3
= π + √ 3
3 2 12 8
5. r = √ 2π
2π √ 2
2π
1
θ, 0 ≤ θ ≤ 2π. A =
2 r2 1
dθ =
2 θ dθ =
1
θdθ= 1
θ2 2π
= π 2
2 4 0
0
0
0
7. r =4+3sinθ, − π ≤ θ ≤ π .
2 2
π/2
1
A = ((4 + 3 sin 2 θ)2 dθ = 1 2
−π/2
= 1 2 · 2 π/2
=
π/2
0
0
π/2
−π/2
(16 + 24 sin θ +9sin 2 θ) dθ = 1 2
16 + 9 · 1
2 (1 − cos 2θ) dθ [by Theorem 5.5.7(a)]
π/2
−π/2
41
− 9 cos 2θ dθ = 41
θ − 9 sin 2θ π/2
= 41π
− 0 − (0 − 0) = 41π
2 2 2 4 0 4 4
9. The area above the polar axis is bounded by r =3cosθ for θ =0
to θ = π/2 [not π]. By symmetry,
A =2 π/2
0
=9 π/2
0
1
2 r2 dθ = π/2
(3 cos θ) 2 dθ =3 2 π/2
0
1
2 (1 + cos 2θ) dθ = 9 2
cos 2 θdθ
0
θ +
1
sin 2θ π/2
= 9 π +0 − (0 + 0) = 9π 2 0 2 2 4
Also, note that this is a circle with radius 3 2 ,soitsareaisπ 3
2
2
= 9π 4 .
11. The curve goes through the pole when θ = π/4,sowe’llfindtheareafor
0 ≤ θ ≤ π/4 and multiply it by 4.
A =4 π/4
0
1
2 r2 dθ =2 π/4
(4 cos 2θ) dθ
0
=8 π/4
0
cos 2θdθ=4 sin 2θ π/4
0
=4
(16 + 9 sin 2 θ) dθ [by Theorem 5.5.7(b)]

F.
432 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
13. One-sixth of the area lies above the polar axis and is bounded by the curve
r =2cos3θ for θ =0to θ = π/6.
15. A= 2π
0
= 1 2
= 1 2
= 1 2
= 1 2
A =6 π/6
0
= 12 2
1
(2 cos 2 3θ)2 dθ =12 π/6
cos 2 3θdθ
0
π/6
0
(1 + cos 6θ) dθ
=6 θ + 1 6 sin 6θ π/6
0
=6 π
6
= π
1
(1 + 2 sin
2 6θ)2 dθ = 1 2π
2
(1 + 4 sin 6θ 0 +4sin2 6θ) dθ
2π
0 1+4sin6θ +4· 1
(1 − cos 12θ) 2
dθ
2π
(3 + 4 sin 6θ − 2 cos 12θ) dθ
0
3θ −
2
cos 6θ − 1 sin 12θ 2π
3 6 0
(6π −
2
− 0) − 0 − 2 − 0 =3π
3 3
17. Theshadedloopistracedoutfromθ =0to θ = π/2.
A = π/2
0
= 1 π/2
2 0
= 1 4
π
2
1
2 r2 dθ = 1 2
π/2
0
sin 2 2θdθ
1
2 (1 − cos 4θ) dθ = 1 4
=
π
8
θ −
1
4 sin 4θ π/2
0
19. r =0 ⇒ 3cos5θ =0 ⇒ 5θ = π ⇒ θ = π .
2 10
A = π/10 1
(3 cos −π/10 2 5θ)2 dθ = π/10
9cos 2 5θdθ= 9 π/10
(1 + cos 10θ) dθ = 9 0 2 0 2 θ +
1
sin 10θ π/10
= 9π
10 0 20
21. This is a limaçon, with inner loop traced
out between θ = 7π 6
solving r =0].
11π
and [found by
6
3π/2
1
A =2 (1 + 2 sin 2 θ)2 dθ =
7π/6
3π/2
7π/6
= θ − 4cosθ +2θ − sin 2θ 3π/2
= 9π
7π/6 2
1+4sinθ +4sin 2 θ dθ =
3π/2
7π/6
−
7π
+2√ 3 − √
3
= π − 3√ 3
2 2
2
1+4sinθ +4· 1
2 (1 − cos 2θ) dθ
23. 2cosθ =1 ⇒ cos θ = 1 2
⇒ θ = π 3 or 5π 3 .
A =2 π/3
0
= π/3
0
1
[(2 cos 2 θ)2 − 1 2 ] dθ = π/3
(4 cos 2 θ − 1) dθ
0
4
1 (1 + cos 2θ) − 1 dθ = π/3
(1 + 2 cos 2θ) dθ
2 0
= θ +sin2θ π/3
0
= π 3 + √ 3
2

F.
25. To findtheareainsidetheleminiscater 2 =8cos2θ and outside the circle r =2,
we first note that the two curves intersect when r 2 =8cos2θ and r =2,
that is, when cos 2θ = 1 2 .For−π

F.
434 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
35. Thedarkershadedregion(fromθ =0to θ =2π/3)represents 1 of the desired area plus 1 2 2
of the area of the inner loop.
From this area, we’ll subtract 1 2
of the area of the inner loop (the lighter shaded region from θ =2π/3 to θ = π), and then
double that difference to obtain the desired area.
2π/3
1
A =2
1
0 2 2 +cosθ2 dθ − π
1 1
2π/3 2 2 +cosθ2 dθ
= 2π/3
1 +cosθ 0 4 +cos2 θ dθ − π
1 +cosθ 2π/3 4 +cos2 θ dθ
= 2π/3
1 +cosθ + 1 (1 + cos 2θ) 0 4 2
dθ
θ
=
4 +sinθ + θ sin 2θ
+
2 4
π
= + √ 3
+ π − √
3
6 2 3 8
= π 4 + 3 4
√
3=
1
4
− π
1 +cosθ + 1 (1 + cos 2θ) 2π/3 4 2
dθ
2π/3 θ
−
4 +sinθ + θ sin 2θ
+
2 4
0
π +3
√
3
− π
4 + π 2
+
π
+ √ 3
+ π − √ 3
6 2 3 8
π
2π/3
37. The pole is a point of intersection.
1+sinθ =3sinθ ⇒ 1=2sinθ ⇒ sin θ = 1 2
⇒
θ = π or 5π .
6 6
The other two points of intersection are 3
, π
2 6 and 3 , 5π
2 6 .
39. 2sin2θ =1 ⇒ sin 2θ = 1 ⇒ 2θ = π , 5π , 13π 17π
,or .
2 6 6 6 6
By symmetry, the eight points of intersection are given by
(1,θ),whereθ = π 12 , 5π
, 13π
12 12
(−1,θ),whereθ = 7π
12 , 11π
, 19π
12 12
,and
17π
12 ,and
,and
23π
12 .
[There are many ways to describe these points.]
41. The pole is a point of intersection. sin θ =sin2θ =2sinθ cos θ ⇔
sin θ (1 − 2cosθ) =0 ⇔ sin θ =0or cos θ = 1 ⇒
2
√3
θ =0, π, π ,or− π 3 3
⇒ the other intersection points are , π 2 3
√3
and , 2π [by symmetry].
2 3

F.
TX.10SECTION 10.4 AREAS AND LENGTHS IN POLAR COORDINATES ¤ 435
43.
From the first graph, we see that the pole is one point of intersection. By zooming in or using the cursor, we find the θ-values
of the intersection points to be α ≈ 0.88786 ≈ 0.89 and π − α ≈ 2.25. (Thefirst of these values may be more easily
estimated by plotting y =1+sinx and y =2x in rectangular coordinates; see the second graph.) By symmetry, the total
area contained is twice the area contained in the first quadrant, that is,
α
π/2
1
A =2
2 (2θ)2 1
dθ +2 (1 + sin 2 θ)2 dθ =
0
α
45. L =
α
0
4θ 2 dθ +
π/2
= 4
θ3 α
+
3
θ − 2cosθ + 1
θ − 1 sin 2θ π/2
0 2 4
= 4 α 3 α3 + π
+ π 2 4
=3
b
a
π/3
0
α
1+2sinθ +
1
2 (1 − cos 2θ) dθ
−
α − 2cosα +
1
2 α − 1 4 sin 2α ≈ 3.4645
π/3
π/3
r2 +(dr/dθ) 2 dθ = (3 sin θ)2 +(3cosθ) 2 dθ = 9(sin 2 θ +cos 2 θ) dθ
0
dθ =3 θ π/3
0
=3 π
3
= π.
As a check, note that the circumference of a circle with radius 3 is 2π
3
2 2 =3π, and since θ =0to π =
π
traces out 1 of the
3 3
circle (from θ =0to θ = π), 1 (3π) =π.
3
47. L =
=
b
a
2π
0
r2 +(dr/dθ) 2 dθ =
2π
θ 2 (θ 2 +4)dθ =
0
2π
0
(θ 2 ) 2 +(2θ) 2 dθ =
θ θ 2 +4dθ
2π
0
0
θ 4 +4θ 2 dθ
Now let u = θ 2 +4,sothatdu =2θdθ
θdθ= 1 2 du and
2π
0
4π
2 +4
√
θ θ 2 1
+4dθ =
2 udu=
1
u · 2 3/2 4(π 2 +1)
= 1 2 3
3 [43/2 (π 2 +1) 3/2 − 4 3/2 ]= 8 3 [(π2 +1) 3/2 − 1]
4
4
49. The curve r =3sin2θ is completely traced with 0 ≤ θ ≤ 2π. r 2 +
dr 2
dθ =(3sin2θ) 2 +(6cos2θ) 2 ⇒
L =
2π
0 9sin 2 2θ +36cos 2 2θdθ≈ 29.0653

F.
436 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
51. The curve r =sin
θ
2 is completely traced with 0 ≤ θ ≤ 4π.
L =
4π
0
sin 2 θ
2
+
1
4 cos2 θ
2
dθ ≈ 9.6884
r 2 + dr 2
dθ =sin
2 θ
2 + 1 cos
θ 2
2 2
⇒
53. The curve r =cos 4 (θ/4) is completely traced with 0 ≤ θ ≤ 4π.
r 2 +(dr/dθ) 2 =[cos 4 (θ/4)] 2 + 4cos 3 (θ/4) · (− sin(θ/4)) · 1
4
L = 4π
0
=cos 8 (θ/4) + cos 6 (θ/4) sin 2 (θ/4)
=cos 6 (θ/4)[cos 2 (θ/4) + sin 2 (θ/4)] = cos 6 (θ/4)
cos6 (θ/4) dθ = 4π
cos 3 (θ/4) dθ
0
=2 2π
0
cos 3 (θ/4) dθ [since cos 3 (θ/4) ≥ 0 for 0 ≤ θ ≤ 2π] =8 π/2
0
cos 3 udu u = 1 4 θ
68
=8 1
(2 + 3 cos2 u)sinu π/2
= 8 [(2 · 1) − (3 · 0)] = 16 0 3 3
2
55. (a) From (10.2.7),
S = b
a 2πy (dx/dθ) 2 +(dy/dθ) 2 dθ
= b
a 2πy r 2 +(dr/dθ) 2 dθ [from the derivation of Equation 10.4.5]
= b
a 2πr sin θ r 2 +(dr/dθ) 2 dθ
(b) The curve r 2 =cos2θ goes through the pole when cos 2θ =0
2θ = π ⇒ θ = π . We’ll rotate the curve from θ =0to θ = π and double
2 4 4
this value to obtain the total surface area generated.
r 2 =cos2θ ⇒ 2r dr
2 dr
dθ = −2sin2θ ⇒ = sin2 2θ
dθ r 2
S =2
=4π
π/4
0
π/4
0
2π √
cos 2θ sin θ cos 2θ + sin 2 2θ /cos 2θdθ=4π
√
cos 2θ sin θ
1
√
cos 2θ
dθ =4π
π/4
0
⇒
= sin2 2θ
cos 2θ .
π/4
sin θdθ=4π − cos θ π/4
0
0
√
cos 2θ sin θ
cos 2 2θ +sin 2 2θ
cos 2θ
dθ
√2
= −4π − 1 =2π 2 − √
2
2

F.
TX.10
SECTION 10.5 CONIC SECTIONS ¤ 437
10.5 Conic Sections
1. x =2y 2 ⇒ y 2 = 1 x. 4p = 1 ,sop = 1 .Thevertex
2 2 8
3. 4x 2 = −y ⇒ x 2 = − 1 y. 4p = − 1 ,sop = − 1 .
is (0, 0),thefocusis 4 4 16
1
, 0 , and the directrix is x = − 1 . The vertex is (0, 0),thefocusis
0, − 1 8 8 16 ,andthe
directrix is y = 1
16 .
5. (x +2) 2 =8(y − 3). 4p =8,sop =2.Thevertexis
(−2, 3),thefocusis(−2, 5), and the directrix is y =1.
7. y 2 +2y +12x +25=0 ⇒
y 2 +2y +1=−12x − 24 ⇒
(y +1) 2 = −12(x +2). 4p = −12,sop = −3.
The vertex is (−2, −1),thefocusis(−5, −1),andthe
directrix is x =1.
9. The equation has the form y 2 =4px,wherep

F.
438 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
13. 4x 2 + y 2 =16 ⇒ x2
4 + y2
16 =1 ⇒
a = √ 16 = 4, b = √ 4=2,
c = √ a 2 − b 2 = √ 16 − 4=2 √ 3. The ellipse is
centered at (0, 0), with vertices at (0, ±4). Thefoci
are 0, ±2 √ 3 .
15. 9x 2 − 18x +4y 2 =27 ⇔
9(x 2 − 2x +1)+4y 2 =27+9 ⇔
9(x − 1) 2 +4y 2 =36 ⇔
(x − 1)2
4
a =3, b =2, c = √ 5 ⇒ center (1, 0),
vertices (1, ±3),foci 1, ± √ 5
+ y2
9 =1 ⇒
17. The center is (0, 0), a =3,andb =2, so an equation is x2
4 + y2
9 =1. c = √ a 2 − b 2 = √ 5, so the foci are 0, ± √ 5 .
19.
x 2
144 − y2
25 =1 ⇒ a =12, b =5, c = √ 144 + 25 = 13 ⇒
center (0, 0),vertices(±12, 0),foci(±13, 0),asymptotesy = ± 5 12 x.
Note: It is helpful to draw a 2a-by-2b rectangle whose center is the center
of the hyperbola. The asymptotes are the extended diagonals of the
rectangle.
21. y 2 − x 2 =4 ⇔ y2
4 − x2
4 =1 ⇒ a = √ 4=2=b,
c = √ 4+4=2 √ 2 ⇒ center (0, 0),vertices(0, ±2),
foci 0, ±2 √ 2 ,asymptotesy = ±x
23. 4x 2 − y 2 − 24x − 4y +28=0 ⇔
4(x 2 − 6x +9)− (y 2 +4y +4)=−28 + 36 − 4 ⇔
4(x − 3) 2 − (y +2) 2 =4 ⇔
(x − 3)2
1
a = √ 1=1, b = √ 4=2, c = √ 1+4= √ 5
−
(y +2)2
4
⇒
=1 ⇒
center (3, −2), vertices (4, −2) and (2, −2),foci 3 ± √ 5, −2 ,
asymptotes y +2=±2(x − 3).

F.
TX.10
SECTION 10.5 CONIC SECTIONS ¤ 439
25. x 2 = y +1 ⇔ x 2 =1(y +1). This is an equation of a parabola with 4p =1,sop = 1 4
.Thevertexis(0, −1) and the
focus is 0, − 3 4
.
27. x 2 =4y − 2y 2 ⇔ x 2 +2y 2 − 4y =0 ⇔ x 2 +2(y 2 − 2y +1)=2 ⇔ x 2 +2(y − 1) 2 =2 ⇔
x 2
2
+
(y − 1)2
1
=1.Thisisanequationofanellipse with vertices at ± √ 2, 1 . The foci are at ± √ 2 − 1, 1 =(±1, 1).
29. y 2 +2y =4x 2 +3 ⇔ y 2 +2y +1=4x 2 +4 ⇔ (y +1) 2 − 4x 2 =4 ⇔
(y +1)2
4
− x 2 =1.Thisisanequation
of a hyperbola with vertices (0, −1 ± 2) = (0, 1) and (0, −3). Thefociareat 0, −1 ± √ 4+1 = 0, −1 ± √ 5 .
31. The parabola with vertex (0, 0) and focus (0, −2) opens downward and has p = −2, so its equation is x 2 =4py = −8y.
33. Thedistancefromthefocus(−4, 0) to the directrix x =2is 2 − (−4) = 6, so the distance from the focus to the vertex is
1
(6) = 3 and the vertex is (−1, 0). Since the focus is to the left of the vertex, p = −3. An equation is 2 y2 =4p(x +1) ⇒
y 2 = −12(x +1).
35. A parabola with vertical axis and vertex (2, 3) has equation y − 3=a(x − 2) 2 . Since it passes through (1, 5),wehave
5 − 3=a(1 − 2) 2 ⇒ a =2,soanequationisy − 3=2(x − 2) 2 .
37. The ellipse with foci (±2, 0) and vertices (±5, 0) has center (0, 0) and a horizontal major axis, with a =5and c =2,
so b 2 = a 2 − c 2 =25− 4=21. An equation is x2
25 + y2
21 =1.
39. Since the vertices are (0, 0) and (0, 8), the ellipse has center (0, 4) with a vertical axis and a =4. The foci at (0, 2) and (0, 6)
are 2 units from the center, so c =2and b = √ a 2 − c 2 = √ 4 2 − 2 2 = √ 12.Anequationis
x 2 (y − 4)2
+ =1.
12 16
41. An equation of an ellipse with center (−1, 4) and vertex (−1, 0) is
(x − 0)2 (y − 4)2
+ =1 ⇒
b 2 a 2
(x +1)2 (y − 4)2
+ =1.Thefocus(−1, 6) is 2 units
b 2 4 2
from the center, so c =2.Thus,b 2 +2 2 =4 2 ⇒ b 2 =12, and the equation is
(x +1)2
12
+
(y − 4)2
16
=1.
43. An equation of a hyperbola with vertices (±3, 0) is x2
3 2 − y2
b 2 =1.Foci(±5, 0) ⇒ c =5and 32 + b 2 =5 2 ⇒
b 2 =25− 9=16, so the equation is x2
9 − y2
16 =1.
45. The center of a hyperbola with vertices (−3, −4) and (−3, 6) is (−3, 1), soa =5and an equation is
(y − 1) 2 (x +3)2
− =1.Foci(−3, −7) and (−3, 9) ⇒ c =8,so5 2 + b 2 =8 2 ⇒ b 2 =64− 25 = 39 and the
5 2 b 2
equation is
(y − 1)2
25
−
(x +3)2
39
=1.

F.
440 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
47. The center of a hyperbola with vertices (±3, 0) is (0, 0),soa =3andanequationis x2
3 2 − y2
b 2 =1.
Asymptotes y = ±2x ⇒ b a
=2 ⇒ b =2(3)=6and the equation is
x2
9 − y2
36 =1.
49. In Figure 8, we see that the point on the ellipse closest to a focus is the closer vertex (which is a distance
a − c from it) while the farthest point is the other vertex (at a distance of a + c). So for this lunar orbit,
(a − c)+(a + c) =2a = (1728 + 110) + (1728 + 314), ora =1940;and(a + c) − (a − c) =2c = 314 − 110,
or c = 102. Thus,b 2 = a 2 − c 2 =3,753,196,andtheequationis
51. (a) Set up the coordinate system so that A is (−200, 0) and B is (200, 0).
|PA| − |PB| = (1200)(980) = 1,176,000 ft = 2450
11
x 2
3,763,600 + y 2
3,753,196 =1.
1225
mi =2a ⇒ a = ,andc =200so
11
b 2 = c 2 − a 2 = 3,339,375
121
⇒
121x2
1,500,625 − 121y2
3,339,375 =1.
(b) Due north of B ⇒ x = 200 ⇒ (121)(200)2
1,500,625 − 121y2
133,575
=1 ⇒ y = ≈ 248 mi
3,339,375 539
53. The function whose graph is the upper branch of this hyperbola is concave upward. The function is
y = f(x) =a 1+ x2
b = a √
b2 + x 2 b
2 ,soy 0 = a b x(b2 + x 2 ) −1/2 and
y 00 = a
(b 2 + x 2 ) −1/2 − x 2 (b 2 + x 2 ) −3/2 = ab(b 2 + x 2 ) −3/2 > 0 for all x,andsof is concave upward.
b
55. (a) If k>16,thenk − 16 > 0,and x2
k +
y2
=1is an ellipse since it is the sum of two squares on the left side.
k − 16
(b) If 0

F.
TX.10
SECTION 10.5 CONIC SECTIONS ¤ 441
(x 0 − a) 2 = a 2 +4p 2 ⇔ x 0 = a ± a 2 +4p 2 . The slopes of the tangent lines at x = a ± a 2 +4p 2
are a ± a 2 +4p 2
, so the product of the two slopes is
2p
a + a 2 +4p 2
2p
· a − a 2 +4p 2
2p
= a2 − (a 2 +4p 2 )
4p 2
= −4p2
4p 2 = −1,
showing that the tangent lines are perpendicular.
59. For x 2 +4y 2 =4,orx 2 /4+y 2 =1, use the parametrization x =2cost, y =sint, 0 ≤ t ≤ 2π to get
L =4 π/2
0
(dx/dt)2 +(dy/dt) 2 dt =4 π/2
0
4sin 2 t +cos 2 tdt=4
π/2
0 3sin 2 t +1dt
Using Simpson’s Rule with n =10, ∆t = π/2 − 0
10
= π 20 ,andf(t) = 3sin 2 t +1,weget
L ≈ 4 3
π
20
f(0) + 4f
π
20
+2f
2π
20 + ···+2f 8π
20 +4f 9π
20 + f π
2 ≈ 9.69
61.
x 2
a − y2
y2
=1 ⇒ 2 b2 b = x2 − a 2
⇒ y = ± b √
x2 − a 2 a 2 a
2 .
A =2
c
a
b
a
x2 − a 2 dx 39 = 2b
a
x
2
x2 − a 2 − a2
2 ln x +
x2 − a 2
c
a
= b √
c c2 − a
a
2 − a 2 ln √ c + c2 − a 2 + a 2 ln |a|
Since a 2 + b 2 = c 2 ,c 2 − a 2 = b 2 ,and √ c 2 − a 2 = b.
= b cb − a 2 ln(c + b)+a 2 ln a = b cb + a 2 (ln a − ln(b + c))
a
a
= b 2 c/a + ab ln[a/(b + c)], wherec 2 = a 2 + b 2 .
63. Differentiating implicitly, x2
a + y2
2x
=1 ⇒ 2 b2 a + 2yy0 =0 ⇒ y 0 = − b2 x
2 b 2 a 2 y
[y 6= 0]. Thus, the slope of the tangent
line at P is − b2 x 1
a 2 y 1
. The slope of F 1 P is
we have
tan α =
1 −
and tan β =
1 −
Thus, α = β.
y 1
x 1 + c + b2 x 1
a 2 y 1
b 2 x 1 y 1
a 2 y 1 (x 1 + c)
= b2 cx 1 + a 2
cy 1 (cx 1 + a 2 ) = b2
cy 1
− b2 x 1
a 2 y 1
− y 1
x 1 − c
b 2 x 1 y 1
a 2 y 1 (x 1 − c)
y 1
x 1 + c and of F 2P is
y 1
. By the formula in Problem 17 on text page 268,
x 1 − c
= a2 y1 2 + b 2 x 1 (x 1 + c)
= a2 b 2 + b 2 cx 1 using b 2 x 2 1 + a2 y1 2 = a2 b 2 ,
a 2 y 1 (x 1 + c) − b 2 x 1 y 1 c 2 x 1 y 1 + a 2 cy 1 and a 2 − b 2 = c 2
= −a2 y 2 1 − b 2 x 1 (x 1 − c)
a 2 y 1 (x 1 − c) − b 2 x 1 y 1
= −a2 b 2 + b 2 cx 1
c 2 x 1 y 1 − a 2 cy 1
= b2 cx 1 − a 2
cy 1 (cx 1 − a 2 ) = b2
cy 1

F.
442 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
10.6 Conic Sections in Polar Coordinates
1. The directrix y =6is above the focus at the origin, so we use the form with “ + e sin θ” in the denominator. [See Theorem 6
7
ed
and Figure 2(c).] r =
1+e sin θ = · 6 4
1+ 7 sin θ = 42
4+7sinθ
4
3. The directrix x = −5 is to the left of the focus at the origin, so we use the form with “ − e cos θ” in the denominator.
3
ed
r =
1 − e cos θ = · 5 4
1 − 3 cos θ = 15
4 − 3cosθ
4
5. The vertex (4, 3π/2) is 4 units below the focus at the origin, so the directrix is 8 units below the focus (d =8), and we use the
form with “−e sin θ ” in the denominator. e =1for a parabola, so an equation is r =
ed
1 − e sin θ = 1(8)
1 − 1sinθ = 8
1 − sin θ .
7. The directrix r =4secθ (equivalent to r cos θ =4or x =4) is to the right of the focus at the origin, so we will use the form
with “+e cos θ” in the denominator. The distance from the focus to the directrix is d =4,soanequationis
1
ed
r =
1+e cos θ = (4) 2
1+ 1 cos θ · 2
2 = 4
2+cosθ .
2
9. r =
1
1+sinθ = ed
,whered = e =1.
1+e sin θ
(a) Eccentricity = e =1
(b) Since e =1, the conic is a parabola.
(c) Since “+e sin θ ” appears in the denominator, the directrix is above the
focus at the origin. d = |Fl| =1, so an equation of the directrix is y =1.
(d) The vertex is at 1
2 , π 2
, midway between the focus and the directrix.
11. r =
12
4 − sin θ · 1/4
1/4 = 3
1 − 1 sin θ ,wheree = 1 4
and ed =3 ⇒ d =12.
4
(a) Eccentricity = e = 1 4
(b) Since e = 1 < 1, the conic is an ellipse.
4
(c) Since “−e sin θ ” appears in the denominator, the directrix is below the focus
at the origin. d = |Fl| =12,soanequationofthedirectrixisy = −12.
(d) The vertices are
4, π 2 and 12 , 3π
5 2 , so the center is midway between them,
that is, 4
, π
5 2 .

F.
TX.10
SECTION 10.6 CONIC SECTIONS IN POLAR COORDINATES ¤ 443
13. r =
9
6+2cosθ · 1/6
1/6 = 3/2
1+ 1 cos θ ,wheree = 1 and ed = 3 ⇒ d = 9 .
3 2 2
3
(a) Eccentricity = e = 1 3
(b) Since e = 1 3
< 1, the conic is an ellipse.
(c) Since “+e cos θ ” appears in the denominator, the directrix is to the right of
the focus at the origin. d = |Fl| = 9 2 ,soanequationofthedirectrixis
x = 9 2 .
(d) The vertices are 9
8 , 0 and 9
4 ,π , so the center is midway between them,
15. r =
that is, 9
16 ,π .
3
4 − 8cosθ · 1/4
1/4 = 3/4
1 − 2cosθ ,wheree =2and ed = 3 4
⇒ d = 3 . 8
(a) Eccentricity = e =2
(b) Since e =2> 1, the conic is a hyperbola.
(c) Since “−e cos θ ” appears in the denominator, the directrix is to the left of
the focus at the origin. d = |Fl| = 3 8 ,soanequationofthedirectrixis
x = − 3 8 .
(d) The vertices are − 3 4 , 0 and 1
4 ,π , so the center is midway between them,
17. (a) r =
that is, 1
2 ,π .
1
1 − 2sinθ ,wheree =2and ed =1 ⇒ d = 1 . The eccentricity
2
e =2> 1, so the conic is a hyperbola. Since “−e sin θ ” appears in the
denominator, the directrix is below the focus at the origin. d = |Fl| = 1 2 ,
so an equation of the directrix is y = − 1 . The vertices are
−1, π 2 2 and
1 , 3π
3 2 , so the center is midway between them, that is, 2 , 3π
3 2 .
(b) By the discussion that precedes Example 4, the equation
1
is r =
1 − 2sin .
θ − 3π 4

F.
444 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
19. For e

F.
TX.10
CHAPTER 10 REVIEW ¤ 445
the length of the orbit is
L =
2π
0
r2 +(dr/dθ) 2 dθ = a(1 − e 2 )
2π
0
√
1+e2 − 2e cos θ
(1 − e cos θ) 2 dθ ≈ 3.6 × 10 8 km
This seems reasonable, since Mercury’s orbit is nearly circular, and the circumference of a circle of radius a
is 2πa ≈ 3.6 × 10 8 km.
10 Review
1. (a) A parametric curve is a set of points of the form (x, y) =(f(t),g(t)), wheref and g are continuous functions of a
variable t.
(b)Sketchingaparametriccurve,likesketching the graph of a function, is difficult to do in general. We can plot points on the
curve by finding f(t) and g(t) for various values of t, either by hand or with a calculator or computer. Sometimes, when
f and g are given by formulas, we can eliminate t from the equations x = f(t) and y = g(t) to get a Cartesian equation
relating x and y. It may be easier to graph that equation than to work with the original formulas for x and y in terms of t.
2. (a) You can find dy
dy
as a function of t by calculating
dx dx = dy/dt
dx/dt
[if dx/dt 6= 0].
(b) Calculate the area as b
a ydx= β
α g(t) f 0 (t)dt [or α
β g(t) f 0 (t)dt if the leftmost point is (f(β),g(β)) rather
than (f(α),g(α))].
3. (a) L = β
(dx/dt)2 +(dy/dt)
α
2 dt = β
α [f 0
(t)] 2 +[g 0 (t)] 2 dt
(b) S = β
α 2πy (dx/dt) 2 +(dy/dt) 2 dt = β
α 2πg(t) [f 0 (t)] 2 +[g 0 (t)] 2 dt
4. (a) See Figure 5 in Section 10.3.
(b) x = r cos θ, y = r sin θ
(c) To find a polar representation (r, θ) with r ≥ 0 and 0 ≤ θ

F.
446 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
7. (a) An ellipse is a set of points in a plane the sum of whose distances from two fixed points (the foci) is a constant.
(b) x2
a +
y2
2 a 2 − c =1. 2
8. (a) A hyperbola is a set of points in a plane the difference of whose distances from two fixed points (the foci) is a constant.
This difference should be interpreted as the larger distance minus the smaller distance.
(b) x2
a −
y2
2 c 2 − a =1 2
√
c2 − a
(c) y = ±
2
x
a
9. (a) If a conic section has focus F and corresponding directrix l, then the eccentricity e is the fixed ratio |PF| / |Pl| for points
P of the conic section.
(b) e1 for a hyperbola; e =1for a parabola.
(c) x = d: r =
ed
1+e cos θ . x = −d: r = ed
1 − e cos θ . y = d: r = ed
1+e sin θ . y = −d: r = ed
1 − e sin θ .
1. False. Consider the curve defined by x = f(t) =(t − 1) 3 and y = g(t) =(t − 1) 2 .Theng 0 (t) =2(t − 1),sog 0 (1) = 0,
but its graph has a vertical tangent when t =1. Note: The statement is true if f 0 (1) 6= 0when g 0 (1) = 0.
3. False. For example, if f(t) =cost and g(t) =sint for 0 ≤ t ≤ 4π, then the curve is a circle of radius 1, hence its length
is 2π,but 4π
0 [f 0
(t)] 2 +[g 0 (t)] 2 dt = 4π
0 (− sin t)2 +(cost) 2 dt = 4π
1 dt =4π, since as t increases
0
from 0 to 4π, the circle is traversed twice.
5. True. The curve r =1− sin 2θ is unchanged if we rotate it through 180 ◦ about O because
1 − sin 2(θ + π) =1− sin(2θ +2π) =1− sin 2θ. So it’s unchanged if we replace r by −r. (See the discussion
afterExample8inSection10.3.)Inotherwords,it’sthesamecurveasr = −(1 − sin 2θ) =sin2θ − 1.
7. False. The first pair of equations gives the portion of the parabola y = x 2 with x ≥ 0, whereas the second pair of equations
traces out the whole parabola y = x 2 .
9. True. By rotating and translating the parabola, we can assume it has an equation of the form y = cx 2 ,wherec>0.
The tangent at the point a, ca 2 is the line y − ca 2 =2ca(x − a);i.e.,y =2cax − ca 2 . This tangent meets
the parabola at the points x, cx 2 where cx 2 =2cax − ca 2 . This equation is equivalent to x 2 =2ax − a 2
[since c>0]. But x 2 =2ax − a 2 ⇔ x 2 − 2ax + a 2 =0 ⇔ (x − a) 2 =0 ⇔ x = a ⇔
x, cx
2 = a, ca 2 . This shows that each tangent meets the parabola at exactly one point.

F.
TX.10
CHAPTER 10 REVIEW ¤ 447
1. x = t 2 +4t, y =2− t, −4 ≤ t ≤ 1. t =2− y,so
x =(2− y) 2 +4(2− y) =4− 4y + y 2 +8− 4y = y 2 − 8y +12
⇔
x +4=y 2 − 8y +16=(y − 4) 2 .Thisispartofaparabolawithvertex
(−4, 4), opening to the right.
3. y =secθ = 1
cos θ = 1 .Since0 ≤ θ ≤ π/2, 0

F.
448 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
11. r =cos3θ.Thisisa
three-leaved rose. The curve is
traced twice.
13. r =1+cos2θ. Thecurveis
symmetric about the pole and
both the horizontal and vertical
axes.
3
15. r =
⇒ e =2> 1, so the conic is a hyperbola. de =3 ⇒
1+2sinθ
d = 3 2
and the form “+2 sin θ” imply that the directrix is above the focus at
the origin and has equation y = 3 . The vertices are
1, π 2 2 and −3,
3π
2 .
2
17. x + y =2 ⇔ r cos θ + r sin θ =2 ⇔ r(cos θ +sinθ) =2 ⇔ r =
cos θ +sinθ
19. r =(sinθ)/θ. Asθ → ±∞, r → 0.
As θ → 0, r → 1. Inthefirst figure,
there are an infinite number of
x-intercepts at x = πn, n a nonzero
integer. These correspond to pole
points in the second figure.
21. x =lnt, y =1+t 2 ; t =1. dy dx
=2t and
dt dt = 1 dy
,so
t dx = dy/dt
dx/dt = 2t
1/t =2t2 .
When t =1, (x, y) =(0, 2) and dy/dx =2.
23. r = e −θ ⇒ y = r sin θ = e −θ sin θ and x = r cos θ = e −θ cos θ ⇒
dy
dx = dy/dθ dr
dx/dθ = sin θ + r cos θ
dθ
dr
cos θ − r sin θ = −e−θ sin θ + e −θ cos θ
−e −θ cos θ − e −θ sin θ · −eθ sin θ − cos θ
=
−eθ cos θ +sinθ .
dθ
When θ = π, dy
dx = 0 − (−1)
−1+0 = 1 −1 = −1.

F.
TX.10
CHAPTER 10 REVIEW ¤ 449
25. x = t +sint, y = t − cos t ⇒ dy
dx = dy/dt
dx/dt = 1+sint
1+cost
d dy (1 + cos t) cost − (1 + sin t)(− sin t)
d 2 y
dx = dt dx
(1 + cos t)
=
2
2 dx/dt
1+cost
⇒
= cos t +cos2 t +sint +sin 2 t 1+cost +sint
=
(1 + cos t) 3 (1 + cos t) 3
27. We graph the curve x = t 3 − 3t, y = t 2 + t +1for −2.2 ≤ t ≤ 1.2.
By zooming in or using a cursor, we find that the lowest point is about
(1.4, 0.75). Tofind the exact values, we find the t-value at which
dy/dt =2t +1=0 ⇔ t = − 1 2
⇔ (x, y) = 11
8 , 3 4
.
29. x =2a cos t − a cos 2t ⇒ dx
dt
sin t =0or cos t = 1 2
⇒ t =0, π 3 , π,or 5π 3 .
= −2a sin t +2a sin 2t =2a sin t(2 cos t − 1) = 0 ⇔
y =2a sin t − a sin 2t ⇒ dy
dt =2a cos t − 2a cos 2t =2a 1+cost − 2cos 2 t =2a(1 − cos t)(1 + 2 cos t) =0
t =0, 2π 3 ,or 4π 3 .
⇒
Thus the graph has vertical tangents where
t = π , π and 5π , and horizontal tangents where
3 3
t = 2π 3
and 4π 3
. To determine what the slope is
where t =0, we use l’Hospital’s Rule to evaluate
dy/dt
lim =0, so there is a horizontal tangent
t→0 dx/dt
there.
t x y
0 a 0
π
3
3
a √
3
a
2 2
2π
3
− 1 2 a 3 √ 3
2 a
π −3a 0
4π
3
− 1 2 a − 3√ 3
5π
3
2 a
3
a − √ 3
a
2 2
31. The curve r 2 =9cos5θ has 10 “petals.” For instance, for − π ≤ θ ≤ π , there are two petals, one with r>0 and one
10 10
with r

F.
450 ¤ CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES TX.10
35. The curves intersect where 2sinθ =sinθ +cosθ ⇒
sin θ =cosθ ⇒ θ = π 4 , and also at the origin (at which θ = 3π 4
on the second curve).
A = π/4
0
1
(2 sin 2 θ)2 dθ + 3π/4 1
π/4
= π/4
0
(1 − cos 2θ) dθ + 1 2
= θ − 1 2 sin 2θ π/4
0
(sin θ 2 +cosθ)2 dθ
3π/4
(1 + sin 2θ) dθ
π/4
+ 1
θ − 1 cos 2θ 3π/4
= 1 (π − 1)
2 4 π/4 2
37. x =3t 2 , y =2t 3 .
L = 2
0
(dx/dt)2 +(dy/dt) 2 dt = 2
0
(6t)2 +(6t 2 ) 2 dt = 2
√
36t2 +36t
0
4 dt = 2
= 2
0 6 |t| √ 1+t 2 dt =6 2
0 t √ 1+t 2 dt =6 5
1 u1/2 1
2 du u =1+t 2 , du =2tdt
=6· 1 · 2
2 3
5
u 3/2 =2(5 3/2 − 1) = 2 5 √ 5 − 1
1
0
√
36t
2 √ 1+t 2 dt
39. L = 2π
π
24
=
−
r2 +(dr/dθ) 2 dθ =
2π
2π
θ (1/θ)2 +(−1/θ 2 2 +1
)
π
2 dθ =
π θ 2
√ √
θ 2 +1
π2 +1 4π2 +1
θ
= 2√ π 2 +1− √ 4π 2 +1
2π
+ln θ + 2π
θ 2 +1
π
=
√
2π + 4π2 +1
+ln
π + √ π 2 +1
π
−
2π
dθ
√
2π + 4π2 +1
+ln
π + √ π 2 +1
41. x =4 √ t, y = t3 3 + 1
2t 2 , 1 ≤ t ≤ 4
⇒
S = 4
2πy (dx/dt)
1 2 +(dy/dt) 2 dt = 4
2π 1
1 3 t3 + 1 t−2 √ 2
2 2/ t +(t2 − t −3 ) 2 dt
=2π 4
1
1
3 t3 + 1 2 t−2 (t 2 + t −3 ) 2 dt =2π 4
1
1
3 t5 + 5 6 + 1 2 t−5 dt =2π 1
18 t6 + 5 6 t − 1 8 t−4 4
1 = 471,295
1024 π
43. For all c except −1, the curve is asymptotic to the line x =1. For
c0, there is a loop to the left of the origin,
whose size and roundness increase as c increases. Note that the x-intercept
of the curve is always −c.

F.
TX.10
CHAPTER 10 REVIEW ¤ 451
45.
x 2
9 + y2
=1is an ellipse with center (0, 0).
8
a =3, b =2 √ 2, c =1
⇒
foci (±1, 0), vertices (±3, 0).
47. 6y 2 + x − 36y +55=0 ⇔
6(y 2 − 6y +9)=−(x +1) ⇔
(y − 3) 2 = − 1 (x +1), a parabola with vertex (−1, 3),
6
opening to the left, p = − 1 24
⇒ focus − 25
24 , 3 and
directrix x = − 23
24 .
49. The ellipse with foci (±4, 0) and vertices (±5, 0) has center (0, 0) and a horizontal major axis, with a =5and c =4,
so b 2 = a 2 − c 2 =5 2 − 4 2 =9. An equation is x2
25 + y2
9 =1.
51. The center of a hyperbola with foci (0, ±4) is (0, 0),soc =4andanequationis y2
a 2 − x2
b 2 =1.
The asymptote y =3x has slope 3,so a b = 3 1
⇒ a =3b and a 2 + b 2 = c 2 ⇒ (3b) 2 + b 2 =4 2 ⇒
10b 2 =16 ⇒ b 2 = 8 and so 5 a2 =16− 8 = 72 . Thus, an equation is y 2
5 5
72/5 − x2 5y2
=1,or
8/5 72 − 5x2
8 =1.
53. x 2 = −(y − 100) has its vertex at (0, 100), so one of the vertices of the ellipse is (0, 100). Another form of the equation of a
parabola is x 2 =4p(y − 100) so 4p(y − 100) = −(y − 100) ⇒ 4p = −1 ⇒ p = − 1 . Therefore the shared focus is
4
found at
0, 399
4 so 2c =
399
− 0 ⇒ c = 399 and the center of the ellipse is
0, 399
4 8 8 .Soa = 100 −
399
= 401 and
8 8
b 2 = a 2 − c 2 = 4012 − 399 2
=25. So the equation of the ellipse is x2 y −
399 2
8 2 b + 8
=1 ⇒ x2 y −
399 2
2 a 2 25 + 8
401
2
=1,
or x2 (8y − 399)2
+ =1.
25 160,801
55. Directrix x =4 ⇒ d =4,soe = 1 3
⇒ r =
ed
1+e cos θ = 4
3+cosθ .
57. In polar coordinates, an equation for the circle is r =2a sin θ. Thus, the coordinates of Q are x = r cos θ =2a sin θ cos θ
and y = r sin θ =2a sin 2 θ. The coordinates of R are x =2a cot θ and y =2a. SinceP is the midpoint of QR,weusethe
midpoint formula to get x = a(sin θ cos θ +cotθ) and y = a(1 + sin 2 θ).
8

F.
TX.10

F.
TX.10
PROBLEMS PLUS
1. x =
dx
dt
t
1
cos u
u
du, y = t
1
sin u
u
dx
du,sobyFTC1,wehave
dt = cos t and dy
t
dt = sin t . Vertical tangent lines occur when
t
=0 ⇔ cos t =0. The parameter value corresponding to (x, y) =(0, 0) is t =1, so the nearest vertical tangent
occurs when t = π . Therefore, the arc length between these points is
2
L =
π/2
1
2 dx
+
dt
dy
dt
2
dt =
π/2
1
cos 2
t
+ sin2 t
π/2
dt
dt =
t 2 t 2 1 t = ln t π/2
=ln π 1
2
3. In terms of x and y, wehavex = r cos θ =(1+c sin θ)cosθ =cosθ + c sin θ cos θ =cosθ + 1 c sin 2θ and
2
y = r sin θ =(1+c sin θ)sinθ =sinθ + c sin 2 θ.Now−1 ≤ sin θ ≤ 1 ⇒ −1 ≤ sin θ + c sin 2 θ ≤ 1+c ≤ 2,so
−1 ≤ y ≤ 2. Furthermore,y =2when c =1and θ = π ,whiley = −1 for c =0and θ = 3π 2 2
. Therefore, we need a viewing
rectangle with −1 ≤ y ≤ 2.
To find the x-values, look at the equation x =cosθ + 1 2 c sin 2θ and use the fact that sin 2θ ≥ 0 for 0 ≤ θ ≤ π 2 and
sin 2θ ≤ 0 for − π 2
≤ θ ≤ 0. [Because r =1+c sin θ is symmetric about the y-axis, we only need to consider
− π 2 ≤ θ ≤ π 2 .] So for − π 2
≤ θ ≤ 0, x has a maximum value when c =0and then x =cosθ has a maximum value
of 1 at θ =0. Thus, the maximum value of x must occur on 0, π 2
with c =1.Thenx =cosθ +
1
sin 2θ ⇒
2
dx
= − sin θ +cos2θ = − sin θ +1− dθ 2sin2 θ ⇒ dx = −(2 sin θ − 1)(sin θ +1)=0when sin θ = −1 or 1 dθ 2
[but sin θ 6=−1 for 0 ≤ θ ≤ π ]. If sin θ = 1 ,thenθ = π and
2 2 6
x =cos π + 1 sin π = √ √ 3
6 2 3 4 3. Thus, the maximum value of x is
3
4 3, and,
by symmetry, the minimum value is − 3 4
√
3. Therefore, the smallest
viewing rectangle that contains every member of the family of polar curves
r =1+c sin θ,where0 ≤ c ≤ 1,is √ √
− 3 4 3,
3
4 3 × [−1, 2].
3t 1
5. (a) If (a, b) lies on the curve, then there is some parameter value t 1 such that
1+t 3 1
= a and 3t2 1
= b. Ift
1+t 3 1 =0,
1
the point is (0, 0), which lies on the line y = x. Ift 1 6=0, then the point corresponding to t = 1 t 1
is given by
x = 3(1/t 1)
1+(1/t 1 ) = 3t2 1
3 t 3 1 +1 = b, y = 3(1/t 1) 2
1+(1/t 1 ) = 3t 1
= a. So(b, a) also lies on the curve. [Another way to see
3 +1
this is to do part (e) first; the result is immediate.] The curve intersects the line y = x when
t 3 1
3t
1+t 3 = 3t2
1+t 3 ⇒
t = t 2 ⇒ t =0or 1, so the points are (0, 0) and 3
, 3
2 2 .
453

F.
454 ¤ CHAPTER 10 PROBLEMS PLUS
TX.10
(b) dy
dt = (1 + t3 )(6t) − 3t 2 (3t 2 ) 6t − 3t4
=
(1 + t 3 ) 2 (1 + t 3 ) =0when 6t − 2 3t4 =3t(2 − t 3 )=0 ⇒ t =0or t = 3√ 2,sothereare
horizontal tangents at (0, 0) and 3 √ 2, 3√ 4 . Using the symmetry from part (a), we see that there are vertical tangents at
(0, 0) and 3 √ 4, 3√ 2 .
(c) Notice that as t →−1 + ,wehavex →−∞and y →∞.Ast →−1 − ,wehavex →∞and y →−∞.Also
y − (−x − 1) = y + x +1= 3t +3t2 +(1+t 3 ) (t +1)3 (t +1)2
= = → 0 as t →−1. Soy = −x − 1 is a
1+t 3 1+t 3 t 2 − t +1
slant asymptote.
(d) dx
dt = (1 + t3 )(3) − 3t(3t 2 )
= 3 − 6t3
dy
and from part (b) we have
(1 + t 3 ) 2 (1 + t 3 )
2
dt
d dy
Also d2 y
dx = dt dx
= 2(1 + t3 ) 4
2 dx/dt 3(1 − 2t 3 ) > 0 ⇔ t< √ 1 3 3
2 .
=
6t − 3t4
(1 + t 3 ) 2 .Sody dx = dy/dt
dx/dt = t(2 − t3 )
1 − 2t 3 .
So the curve is concave upward there and has a minimum point at (0, 0)
and a maximum point at 3 √ 2, 3√ 4 . Using this together with the
information from parts (a), (b), and (c), we sketch the curve.
3 3t 3t
(e) x 3 + y 3 2
3
=
+
= 27t3 +27t 6
= 27t3 (1 + t 3 )
= 27t3
1+t 3 1+t 3 (1 + t 3 ) 3 (1 + t 3 ) 3 (1 + t 3 ) and 2
3xy =3 3t 3t
2 27t 3
=
1+t 3 1+t 3 (1 + t 3 ) 2 ,sox3 + y 3 =3xy.
(f) We start with the equation from part (e) and substitute x = r cos θ, y = r sin θ. Then x 3 + y 3 =3xy
r 3 cos 3 θ + r 3 sin 3 θ =3r 2 cos θ sin θ. Forr 6= 0,thisgivesr =
by cos 3 θ,weobtainr =
1 sin θ
3
cos θ cos θ
1+ sin3 θ
cos 3 θ
(g) The loop corresponds to θ ∈ 0, π 2
, so its area is
A = π/2
0
r 2
2 dθ = 1 2
π/2
0
9
= lim
b→∞
2 −
1
(1 + 3 u3 ) −1 b
= 3 0 2
=
3secθ tan θ
1+tan 3 θ .
2 3secθ tan θ
dθ = 9 π/2 sec 2 θ tan 2 θ
1+tan 3 θ 2
0
(1 + tan 3 θ) dθ = 9 ∞
2 2 0
3cosθ sin θ
cos 3 θ +sin 3 . Dividing numerator and denominator
θ
⇒
u 2 du
(1 + u 3 ) 2 [let u =tanθ]
(h)Bysymmetry,theareabetweenthefoliumandtheliney = −x − 1 is equal to the enclosed area in the third quadrant,
plus twice the enclosed area in the fourth quadrant. The area in the third quadrant is 1 , and since y = −x − 1 ⇒
2
1
r sin θ = −r cos θ − 1 ⇒ r = −
, the area in the fourth quadrant is
sin θ +cosθ
1 −π/4 2 2
1
3secθ tan θ
−
−
dθ CAS
= 1 2
sin θ +cosθ 1+tan 3 θ
2 . Therefore, the total area is 1 +2
1
2 2 =
3
. 2
−π/2

F.
TX.10
11 INFINITE SEQUENCES AND SERIES
11.1 Sequences
1. (a) A sequence is an ordered list of numbers. It can also be defined as a function whose domain is the set of positive integers.
(b) The terms a n approach 8 as n becomes large. In fact, we can make a n as close to 8 as we like by taking n sufficiently
large.
(c) The terms a n become large as n becomes large. In fact, we can make a n as large as we like by taking n sufficiently large.
3. a n =1− (0.2) n , so the sequence is {0.8, 0.96, 0.992, 0.9984, 0.99968,...}.
5. a n = 3(−1)n
−3
, so the sequence is
n!
1 , 3 2 , −3
6 , 3 24 , −3
120 ,... =
−3, 3 2 , −1 2 , 1 8 , − 1
40 ,... .
7. a 1 =3, a n+1 =2a n − 1. Eachtermisdefined in terms of the preceding term.
9.
a 2 =2a 1 − 1=2(3)− 1=5. a 3 =2a 2 − 1 = 2(5) − 1=9. a 4 =2a 3 − 1=2(9)− 1=17.
a 5 =2a 4 − 1=2(17)− 1=33. The sequence is {3, 5, 9, 17, 33,...}.
1,
1
3 , 1 5 , 1 7 , 1 9 ,... . The denominator of the nth term is the nth positive odd integer, so a n = 1
2n − 1 .
11. {2, 7, 12, 17,...}. Each term is larger than the preceding one by 5,soa n = a 1 + d(n − 1) = 2 + 5(n − 1) = 5n − 3.
13.
1, −
2
3 , 4 9 , − 8 27 ,... . Eachtermis− 2 3 times the preceding one, so a n = − 2 3
n−1
.
15. The first six terms of a n = n
2n +1 are 1 3 , 2 5 , 3 7 , 4 9 , 5
11 , 6
13 . It appears that the sequence is approaching 1 2 .
lim
n→∞
n
2n +1 = lim
n→∞
1
2+1/n = 1 2
17. a n =1− (0.2) n ,so lim
n→∞ a n =1− 0=1by (9).
Converges
19. a n = 3+5n2
n + n 2 = (3 + 5n2 )/n 2
(n + n 2 )/n = 5+3/n2
2 1+1/n
5+0
,soan → =5as n →∞. Converges
1+0
21. Because the natural exponential function is continuous at 0, Theorem 7 enables us to write
lim
n→∞ a n = lim
n→∞ e1/n = e lim n→∞(1/n) = e 0 =1.
Converges
23. If b n = 2nπ ,then lim bn = lim
1+8n n→∞ n→∞
Theorem 7, lim
n→∞ tan 2nπ
1+8n
=tan
(2nπ)/n
(1 + 8n)/n = lim
n→∞
lim
n→∞
2π
1/n +8 = 2π 8 = π 4 .Sincetan is continuous at π 4 ,by
2nπ
=tan π 1+8n 4 =1. Converges 455

F.
456 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
25. a n = (−1)n−1 n
n 2 +1
Theorem 6.
= (−1)n−1
n +1/n ,so0 ≤ |an| = 1
n +1/n ≤ 1 → 0 as n →∞,soan → 0 by the Squeeze Theorem and
n
Converges
27. a n =cos(n/2). This sequence diverges since the terms don’t approach any particular real number as n →∞.
The terms take on values between −1 and 1.
29. a n =
(2n − 1)!
(2n +1)! = (2n − 1)!
(2n +1)(2n)(2n − 1)! = 1
→ 0 as n →∞. Converges
(2n +1)(2n)
31. a n = en + e −n
e 2n − 1 · e−n
e
e n − e −n → 0 as n →∞because 1+e−2n → 1 and e n − e −n →∞.
−n
=
1+e−2n
Converges
33. a n = n 2 e −n = n2
x 2
.Since lim
en x→∞ e x
= H 2x
lim
x→∞ e x
= H 2
lim =0, it follows from Theorem 3 that lim an =0.
x→∞ e Converges
x n→∞
35. 0 ≤ cos2 n
≤ 1
[since 0 ≤ cos 2 1 cos 2
n ≤ 1], so since lim
2 n 2 n n→∞ 2 =0, n
converges to 0 by the Squeeze Theorem.
n 2 n
37. a n = n sin(1/n) = sin(1/n)
sin(1/x) sin t
. Since lim = lim
1/n x→∞ 1/x t→0 + t
that {a n } converges to 1.
39. y =
1+ 2 x
⇒ ln y = x ln 1+ 2
,so
x x
1
− 2 x 2
ln(1 + 2/x) H 1+2/x
lim ln y = lim
= lim
x→∞ x→∞ 1/x x→∞ −1/x 2
lim 1+ 2 x
= lim
x→∞ x x→∞ eln y = e 2 ,sobyTheorem3, lim
n→∞
41. a n =ln(2n 2 +1)− ln(n 2 +1)=ln
= lim
x→∞
[where t =1/x] =1, it follows from Theorem 3
2
1+2/x =2
⇒
1+ 2 n
n
= e 2 . Convergent
2n 2 +1 2+1/n
2
=ln
→ ln 2 as n →∞.
n 2 +1 1+1/n 2
Convergent
43. {0, 1, 0, 0, 1, 0, 0, 0, 1,...} diverges since the sequence takes on only two values, 0 and 1, and never stays arbitrarily close to
either one (or any other value) for n sufficiently large.
45. a n = n!
2 = 1 n 2 · 2
2 · 3 (n − 1)
····· · n
2 2 2 ≥ 1 2 · n
2
[for n>1] = n 4
→∞as n →∞,so{an} diverges.
47. From the graph, it appears that the sequence converges to 1.
{(−2/e) n } converges to 0 by (9), and hence {1+(−2/e) n }
converges to 1+0=1.

F.
TX.10
SECTION 11.1 SEQUENCES ¤ 457
49. From the graph, it appears that the sequence converges to 1 2 .
As n →∞,
3+2n
2
a n =
8n 2 + n = 3/n 2 +2
8+1/n
so lim
n→∞ a n = 1 2 .
⇒
0+2
8+0 =
1
4 = 1 2 ,
n 2 cos n
51. From the graph, it appears that the sequence {a n } =
is
1+n 2
divergent, since it oscillates between 1 and −1 (approximately). To
prove this, suppose that {a n} converges to L. Ifb n =
n2
1+n 2 ,then
a n
{b n} converges to 1,and lim = L
n→∞ b n 1 = L. Buta n
=cosn,so
b n
a n
lim does not exist. This contradiction shows that {a n } diverges.
n→∞ b n
53. From the graph, it appears that the sequence approaches 0.
0 1.
57. If |r| ≥ 1, then{r n } diverges by (9), so {nr n } diverges also, since |nr n | = n |r n | ≥ |r n |.If|r| < 1 then
x
lim
x→∞ xrx = lim
x→∞ r −x
whenever |r| < 1.
H = lim
x→∞
1
= lim
(− ln r) r−x x→∞
r x
=0,so lim
− ln r n→∞ nrn =0, and hence {nr n } converges
59. Since {a n } is a decreasing sequence, a n >a n+1 for all n ≥ 1. Because all of its terms lie between 5 and 8, {a n } is a
bounded sequence. By the Monotonic Sequence Theorem, {a n} is convergent; that is, {a n} has a limit L. L must be less than
8 since {a n } is decreasing, so 5 ≤ L

F.
458 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
63. The terms of a n = n(−1) n alternate in sign, so the sequence is not monotonic. The first five terms are −1, 2, −3, 4,and−5.
Since lim |a n| = lim n = ∞, the sequence is not bounded.
n→∞ n→∞
65. a n = n defines a decreasing sequence since for f(x) =
x
x 2
n 2 +1 x 2 +1 , f 0 +1 (1) − x(2x)
(x) =
(x 2 +1) 2 = 1 − x2
(x 2 +1) 2 ≤ 0
for x ≥ 1. The sequence is bounded since 0 a n and 0 a n ⇒ 1
a n+1
< 1
a n
⇒
− 1
a n+1
> − 1
a n
.Nowa n+2 =3− 1
a n+1
> 3 − 1
a n
= a n+1 ⇔ P n+1 . This proves that {a n } is increasing and bounded
above by 3,so1=a 1 1,soL = 3+√ 5
2
.
71. (a) Let a n be the number of rabbit pairs in the nth month. Clearly a 1 =1=a 2 .Inthenth month, each pair that is
2 or more months old (that is, a n−2 pairs) will produce a new pair to add to the a n−1 pairs already present. Thus,
a n = a n−1 + a n−2 ,sothat{a n } = {f n }, the Fibonacci sequence.
(b) a n = fn+1
⇒ a n−1 = fn fn−1 + fn−2
= =1+ fn−2
1
=1+
=1+ 1 .IfL = lim
f n f n−1 f n−1
f n−1 f n−1 /f n−2 a an,
n−2 n→∞
then L = lim a n−1 and L = lim a n−2,soL must satisfy L =1+ 1
n→∞ n→∞ L ⇒ L2 − L − 1=0 ⇒ L = 1+√ 5
2
[since L must be positive].
n
5
73. (a) From the graph, it appears that the sequence
n!
n 5
converges to 0,thatis, lim
n→∞ n! =0.

F.
TX.10
SECTION 11.1 SEQUENCES ¤ 459
(b)
From the first graph, it seems that the smallest possible value of N corresponding to ε =0.1 is 9,sincen 5 /n! < 0.1
whenever n ≥ 10,but9 5 /9! > 0.1. From the second graph, it seems that for ε =0.001, the smallest possible value for N
is 11 since n 5 /n! < 0.001 whenever n ≥ 12.
75. Theorem 6: If lim |an| =0then lim − |an| =0, and since − |an| ≤ an ≤ |an|,wehavethat lim an =0by the
n→∞ n→∞ n→∞
Squeeze Theorem.
77. To Prove: If lim
n→∞ a n =0and {b n } is bounded, then lim
n→∞ (a nb n )=0.
Proof: Since {b n} is bounded, there is a positive number M such that |b n| ≤ M and hence, |a n||b n| ≤ |a n| M for
all n ≥ 1. Letε>0 be given. Since lim a n =0, there is an integer N such that |a n − 0| < ε if n>N.Then
n→∞ M
|a nb n − 0| = |a nb n| = |a n||b n| ≤ |a n| M = |a n − 0| M< ε M
· M = ε for all n>N.Sinceε was arbitrary,
lim
n→∞ (a nb n )=0.
79. (a) First we show that a>a 1 >b 1 >b.
a 1 − b 1 = a + b − √
ab = 1 a − 2 √ √a √ 2
ab + b = 1 2 2
2 − b > 0 [since a>b] ⇒ a1 >b 1 .Also
a − a 1 = a − 1 (a + b) = 1 (a − b) > 0 and b − b 2 2 1 = b − √ ab = √ √b √
b − a < 0,soa>a 1 >b 1 >b.Inthesame
way we can show that a 1 >a 2 >b 2 >b 1 and so the given assertion is true for n =1. Suppose it is true for n = k,thatis,
a k >a k+1 >b k+1 >b k .Then
a k+2 − b k+2 = 1 (a 2 k+1 + b k+1 ) − a k+1 b k+1 =
a 1 2 k+1 − 2
a k+1 b k+1 + b k+1 = 1 √ak+1
− 2
b
2 k+1 > 0,
a k+1 − a k+2 = a k+1 − 1 2 (a k+1 + b k+1 )= 1 2 (a k+1 − b k+1 ) > 0,and
b k+1 − b k+2 = b k+1 − a k+1 b k+1 = b k+1
bk+1
− √ a k+1
< 0 ⇒ a k+1 >a k+2 >b k+2 >b k+1 ,
so the assertion is true for n = k +1. Thus, it is true for all n by mathematical induction.
(b) From part (a) we have a>a n >a n+1 >b n+1 >b n >b, which shows that both sequences, {a n } and {b n },are
monotonic and bounded. So they are both convergent by the Monotonic Sequence Theorem.
(c) Let lim a n = α and lim b n = β. Then lim a a n + b n
n+1 = lim
n→∞ n→∞ n→∞ n→∞ 2
2α = α + β ⇒ α = β.
⇒
α = α + β
2
⇒

F.
460 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
81. (a) Suppose {p n} converges to p. Thenp n+1 = bp b lim p
n
⇒ lim
a + p n n→∞ pn+1 =
n
n→∞
a + lim p n
n→∞
p 2 + ap = bp ⇒ p(p + a − b) =0 ⇒ p =0or p = b − a.
b
p n
a
(b) p n+1 =
bpn
a + p n
=
1+ pn a
b
< p n since 1+ pn
a a > 1.
⇒ p = bp
a + p
⇒
2 3 n b b b b b b
(c) By part (b), p 1 < p 0 , p 2 < p 1 < p 0 , p 3 < p 2 < p 0 , etc. In general, p n < p 0 ,
a a a
a a
a
n
b
so lim p n ≤ lim · p 0 =0since b

F.
TX.10
SECTION 11.2 SERIES ¤ 461
GRAPH, MORE, FORMT (F3).) Now under E(t)= make the assignments: xt1=t, yt1=12/(-5)ˆt, xt2=t,
yt2=sum seq(yt1,t,1,t,1). (sum and seq are under LIST, OPS (F5), MORE.) Under WIND use
1,10,1,0,10,1,-3,1,1 to obtain a graph similar to the one above. Then use TRACE (F4) to see the values.
5.
n
s n
1 1.55741
2 −0.62763
3 −0.77018
4 0.38764
5 −2.99287
6 −3.28388
7 −2.41243
8 −9.21214
9 −9.66446
10 −9.01610
The series ∞ tan n diverges, since its terms do not approach 0.
n=1
7.
n
s n
1 0.29289
2 0.42265
3 0.50000
4 0.55279
5 0.59175
6 0.62204
7 0.64645
8 0.66667
9 0.68377
10 0.69849
From the graph and the table, it seems that the series converges.
k 1 1 1√1 √n − √ = −
1 1√2
√ + −
n +1 2
n=1
so ∞
n=1
1 √n −
=1−
1
√ k +1
,
1
√ = lim 1 −
n +1 k→∞
1
√
k +1
=1.
1 √
3
+ ···+
1 √k −
1
√
k +1
9. (a) lim a 2n
n = lim
n→∞ n→∞ 3n +1 = 2 3 ,sothesequence {a n} is convergent by (11.1.1).
(b) Since lim a n = 2 6=0,theseries ∞ a
n→∞ 3 n is divergent by the Test for Divergence.
n=1
11. 3+2+ 4 + 8 + ··· is a geometric series with first term a =3andcommonratior = 2 .Since|r| = 2 < 1, theseries
3 9 3 3
converges to
a
= 3 = 3 =9.
1 − r 1−2/3 1/3
13. 3 − 4+ 16 − 64 + ··· is a geometric series with ratio r = − 4 .Since|r| = 4 > 1, the series diverges.
3 9 3 3

F.
462 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
15.
17.
19.
21.
23.
∞
6(0.9) n−1 is a geometric series with first term a =6and ratio r =0.9. Since|r| =0.9 < 1, the series converges to
n=1
a
1 − r = 6
1 − 0.9 = 6
0.1 =60.
∞
n=1
(−3) n−1
converges to
∞
n=0
∞
n=1
∞
n=1
4 n = 1 4
π n
3 n+1 = 1 3
1
2n = 1 2
∞
− 3 n−1
. The latter series is geometric with a =1and ratio r = − 3
4 .Since|r| = 3 < 1, it
4 4
n=1
1
1 − (−3/4) = 4 . Thus, the given series converges to
1 4
7 4 7 =
1
. 7
∞
n=1
∞
n=0
π
3
1
n ,whichdiverges. If
n π
is a geometric series with ratio r = .Since|r| > 1, the series diverges.
3
1
n diverges since each of its partial sums is 1 times the corresponding partial sum of the harmonic series
2
∞
n=1
1
were to converge, then
∞
2n
In general, constant multiples of divergent series are divergent.
∞
k=2
n=1
1
n wouldalsohavetoconvergebyTheorem8(i).
k 2
diverges by the Test for Divergence since lim
k 2 − 1 a k 2
k = lim =16= 0.
k→∞ k→∞ k 2 − 1
25. Converges.
∞
n=1
1+2 n
3 n = ∞
n=1
1
3 n + 2n
3 n
= ∞
n=1
1
3
= 1/3
1 − 1/3 + 2/3
1 − 2/3 = 1 2 +2= 5 2
n n 2
+
3
[sum of two convergent geometric series]
27.
∞
n=1
n√
2=2+
√
2+
3 √ 2+ 4√ 2+··· diverges by the Test for Divergence since
29.
31.
33.
lim a n = lim
n√
2 = lim
n→∞ n→∞ n→∞ 21/n =2 0 =16= 0.
∞
n=1
n 2 +1
ln
diverges by the Test for Divergence since
2n 2 +1
n 2
lim an = lim ln +1
n→∞ n→∞ 2n 2 +1
∞
n=1
∞
n=1
=ln lim
n→∞
n 2 +1
=ln 1
2n 2 2
+1
6=0.
arctan n diverges by the Test for Divergence since lim
n→∞ a n = lim
n→∞ arctan n = π 2 6=0.
1
e = ∞ n 1
is a geometric series with first term a = 1 n n=1 e
e and ratio r = 1 e .Since|r| = 1 < 1, the series converges
e
1/e
to
1 − 1/e = 1/e
1 − 1/e · e
e = 1
e − 1 .ByExample6, ∞
n=1
∞ 1
n=1 e + 1
= ∞ 1
n n(n +1) n=1 e + ∞
n n=1
1
=1. Thus, by Theorem 8(ii),
n(n +1)
1
n(n +1) = 1
e − 1 +1= 1
e − 1 + e − 1
e − 1 =
e
e − 1 .

F.
TX.10
SECTION 11.2 SERIES ¤ 463
35. Using partial fractions, the partial sums of the series ∞
s n = n
=
i=2
1 − 1 3
n=2
2
n 2 − 1 are
2
(i − 1)(i +1) = n 1
i=2 i − 1 − 1
i +1
1
+
2 − 1 1
+
4
3 − 1
+ ···+
5
This sum is a telescoping series and s n =1+ 1 2 − 1
n − 1 − 1 n .
∞ 2
Thus,
n 2 − 1 = lim sn = lim 1+ 1
n→∞ n→∞ 2 − 1
n − 1 − 1
= 3 n 2 .
n=2
37. For the series ∞
n=1
3
n(n +3) , sn = n 3
i(i +3) = n 1
i − 1
i=1
1 −
1
4
+
1
2 − 1 5
+
1
3 − 1 6
+
1
4 − 1 7
Thus,
∞
n=1
39. For the series ∞
i=1
+ ···+
1
n−3 − 1 n
i +3
+
1
n − 3 − 1 1
+
n − 1 n − 2 − 1
n
[using partial fractions]. The latter sum is
1
− 1
n −2 n +1
+
1
− 1
n−1 n+2
+
1
− 1
n n+3
=1+ 1 + 1 − 1 2 3 n +1 n +2 n+3
[telescoping series]
3
n(n +3) = lim n = lim 1+ 1 + 1 − 1 =1+ 1 + 1 = 11 .
n→∞ n→∞
2 3 n +1 n +2 n+3
2 3 6 Converges
n=1
e 1/n − e 1/(n+1)
,
s n = n
e 1/i − e 1/(i+1)
i=1
=(e 1 − e 1/2 )+(e 1/2 − e 1/3 )+···+
e 1/n − e 1/(n+1) = e − e 1/(n+1)
[telescoping series]
∞
Thus,
e 1/n − e 1/(n+1) = lim s n = lim
e − e 1/(n+1) = e − e 0 = e − 1. Converges
n→∞ n→∞
n=1
41. 0.2 = 2 10 + 2
2
+ ··· is a geometric series with a =
102 10 and r = 1 10 .Itconvergesto a
1 − r = 2/10
1 − 1/10 = 2 9 .
43. 3.417 = 3 + 417
10 + 417
3 10 + ···.Now417
6 10 + 417
417
+ ··· is a geometric series with a = 3 106 10 and r = 1
3 10 . 3
It converges to
a
1 − r = 417/103
1 − 1/10 3 = 417/103
999/10 = 417
3 999
.Thus,3.417 = 3 +
417
999 = 3414
999 = 1138
333 .
45. 1.5342 = 1.53 + 42
10 + 42
42
+ ···.Now 4 106 10 + 42
42
+ ··· is a geometric series with a = 4 106 10 and r = 1
4 10 . 2
47.
It converges to
a
1 − r = 42/104
1 − 1/10 2 = 42/104
99/10 2 = 42
9900 .
Thus, 1.5342 = 1.53 + 42
9900 = 153
100 + 42
9900 = 15,147
9900 + 42
9900 = 15,189
9900
∞
n=1
or
5063
3300 .
x n
3 = ∞ x
n x
|x|
is a geometric series with r = , so the series converges ⇔ |r| < 1 ⇔ < 1 ⇔ |x| < 3;
n=1 n 3
3 3
that is, −3

F.
464 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
49.
51.
∞
4 n x n
= ∞ (4x) n is a geometric series with r =4x, so the series converges ⇔ |r| < 1 ⇔ 4 |x| < 1 ⇔
n=0
n=0
|x| < 1 4 . In that case, the sum of the series is 1
1 − 4x .
∞
n=0
cos n x
is a geometric series with first term 1 and ratio r = cos x
|cos x|
,soitconverges ⇔ |r| < 1. But|r| =
2 n 2 2
for all x. Thus, the series converges for all real values of x and the sum of the series is
1
1 − (cos x)/2 = 2
2 − cos x .
53. After defining f, Weuseconvert(f,parfrac); in Maple, Apart in Mathematica, or Expand Rational and
Simplify in Derive to find that the general term is 3n2 +3n +1
= 1 (n 2 + n) 3 n − 1
.Sothenth partial sum is
3 (n +1)
3
s n =
n 1
k − 1
=
1 − 1 1
+
3 (k +1) 3 2 3 2 − 1
1
+ ···+
3 3 3 n − 1
1
=1−
3 (n +1) 3 (n +1) 3
k=1
The series converges to lim
n→∞ s n =1. This can be confirmed by directly computing the sum using sum(f,1..infinity);
(in Maple), Sum[f,{n,1,Infinity}] (in Mathematica), or Calculus Sum (from 1 to ∞)andSimplify (in Derive).
55. For n =1, a 1 =0since s 1 =0.Forn>1,
Also,
∞
n=1
a n = s n − s n−1 = n − 1 (n − 1) − 1 (n − 1)n − (n +1)(n − 2) 2
− = =
n +1 (n − 1) + 1 (n +1)n
n(n +1)
a n = lim s 1 − 1/n
n = lim
n→∞ n→∞ 1+1/n =1.
57. (a) The first step in the chain occurs when the local government spends D dollars. The people who receive it spend a fraction c
of those D dollars, that is, Dc dollars. Those who receive the Dc dollars spend a fraction c of it, that is, Dc 2 dollars.
Continuing in this way, we see that the total spending after n transactions is
≤ 1 2
S n = D + Dc + Dc 2 + ···+ Dc n–1 = D(1 − cn )
1 − c
by (3).
(b) lim S D(1 − c n )
n = lim
= D
n→∞ n→∞ 1 − c 1 − c lim (1 − n→∞ cn )=
D
1 − c
since 0 1 or 1+c0 or c

F.
61. e s n
= e 1+ 1 2 + 1 3 +···+ 1 n = e 1 e 1/2 e 1/3 ···e 1/n > (1 + 1) 1+ 1 2
= 2 3 4
1 2 3
··· n +1
n
= n +1
1+
1
3 ···
1+
1
n
SECTION 11.2 SERIES ¤ 465
[e x > 1+x]
Thus, e s n
>n+1and lim
n→∞ es n
= ∞. Since{s n} is increasing, lim sn = ∞, implying that the harmonic series is
n→∞
divergent.
TX.10
63. Let d n be the diameter of C n . We draw lines from the centers of the C i to
the center of D (or C), and using the Pythagorean Theorem, we can write
1 2 +
1 − 1 2
2 d1 =
1+ 1 2
2 d1 ⇔
1=
1+ 1 2
2 d1 −
1 − 1 2
2 d1 =2d 1 [difference of squares] ⇒ d 1 = 1 . 2
Similarly,
1=
1+ 1 2
2 d2 −
1 − d 1 − 1 2
2 d2 =2d 2 +2d 1 − d 2 1 − d 1d 2
=(2− d 1)(d 1 + d 2)
⇔
d 2 = 1 (1 − d1)2
− d 1 = , 1=
1+ 1 2
2
2 − d 1 2 − d d3 −
1 − d 1 − d 2 − 1 2
2 d3 ⇔ d 3 =
1
d n+1 =
1 −
n
i=1 di 2
2 − n
i=1 d i
[1 − (d1 + d2)]2
, and in general,
2 − (d 1 + d 2 )
. If we actually calculate d 2 and d 3 from the formulas above, we find that they are 1 6 = 1
2 · 3 and
1
12 = 1
3 · 4 respectively, so we suspect that in general, d 1
n = . To prove this, we use induction: Assume that for all
n(n +1)
k ≤ n, d k =
1
k(k +1) = 1 k − 1
k +1 .Then n
i=1
2
1 − n
n +1
formula for d n+1 ,wegetd n+1 = =
2 −
n
n +1
d i =1− 1
n +1 =
1
(n +1) 2
n +2
n +1
=
n
n +1
[telescoping sum]. Substituting this into our
1
, and the induction is complete.
(n +1)(n +2)
Now, we observe that the partial sums n
i=1 d i of the diameters of the circles approach 1 as n →∞;thatis,
∞
a n = ∞
n=1
n=1
1
=1, which is what we wanted to prove.
n(n +1)
65. The series 1 − 1+1− 1+1− 1+··· diverges (geometric series with r = −1) so we cannot say that
0=1− 1+1− 1+1− 1+···.
67. ∞
n=1 ca
n = lim n
n→∞
i=1 ca i = lim c n
n→∞
i=1 a
i = c lim n
n→∞
i=1 a i = c ∞
n=1 a n, which exists by hypothesis.
69. Suppose on the contrary that (a n + b n ) converges. Then (a n + b n ) and a n are convergent series. So by Theorem 8,
[(an + b n ) − a n ] would also be convergent. But [(a n + b n ) − a n ]= b n , a contradiction, since b n is given to be
divergent.
71. The partial sums {s n } form an increasing sequence, since s n − s n−1 = a n > 0 for all n. Also, the sequence {s n } is bounded
since s n ≤ 1000 for all n. So by the Monotonic Sequence Theorem, the sequence of partial sums converges, that is, the series
an is convergent.

F.
466 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
73. (a) At the first step, only the interval 1
, 2
3 3 (length
1
) is removed. At the second step, we remove the intervals 1
, 2
3 9 9 and
7 , 8
9 9 , which have a total length of 2 · 1
2
. At the third step, we remove 2 2 intervals, each of length 1
3
3
3
. In general,
at the nth step we remove 2 n−1 intervals, each of length
1 n
,foralengthof2 n−1 ·
1 n
= 1 2
n−1
. Thus, the total
3
3 3 3
length of all removed intervals is ∞
n=1
the nth step, the leftmost interval that is removed is
1 n
, 2
3
the rightmost interval removed is 1 − 2
3
are 1 3 , 2 3 , 1 9 , 2 9 , 7 9 ,and 8 9 .
1 2
n−1
= 1/3 =1 geometric series with a = 1 and r = 2
3 3
1 − 2/3 3 3 . Notice that at
n
3 ,soweneverremove0,and0 is in the Cantor set. Also,
n
, 1 −
1 n
3 ,so1 is never removed. Some other numbers in the Cantor set
(b) The area removed at the first step is 1 ;atthesecondstep,8 ·
1 2
;atthethirdstep,(8) 2 ·
1 3
. In general, the area
9 9
9
removed at the nth step is (8) n−1
1 n
9
= 1 8
n−1
9 9
, so the total area of all removed squares is
∞ n−1
1 8
= 1/9
9 9 1 − 8/9 =1.
n=1
75. (a) For ∞
n=1
n
(n +1)! , s1 = 1
1 · 2 = 1 2 , s2 = 1 2 + 2
1 · 2 · 3 = 5 6 , s3 = 5 6 + 3
1 · 2 · 3 · 4 = 23
24 ,
s 4 = 23
24 + 4
1 · 2 · 3 · 4 · 5 = 119
120 . The denominators are (n +1)!,soaguesswouldbes n =
(b) For n =1, s 1 = 1 2 = 2! − 1 , so the formula holds for n =1. Assume s k =
2!
s k+1 =
=
(k +1)!− 1
(k +1)!
(k +2)!− 1
(k +2)!
(k +1)!− 1
.Then
(k +1)!
(n +1)!− 1
.
(n +1)!
+ k +1 (k +1)!− 1 k +1 (k +2)!− (k +2)+k +1
= +
=
(k +2)! (k +1)! (k +1)!(k +2) (k +2)!
Thus, the formula is true for n = k +1. So by induction, the guess is correct.
(n +1)!− 1
(c) lim sn = lim
= lim 1 −
n→∞ n→∞ (n +1)! n→∞
1
=1and so ∞
(n +1)!
n=1
n
(n +1)! =1.
11.3 The Integral Test and Estimates of Sums
1. The picture shows that a 2 = 1 2
2 < 1
dx,
1.3 x1.3 a 3 = 1
3 1.3 < 3
2
1
x dx, and so on, so ∞
1.3
1
n=2
1 ∞
n < 1.3
1
1
dx. The
x1.3 integral converges by (7.8.2) with p =1.3 > 1, so the series converges.
3. The function f(x) =1/ 5√ x = x −1/5 is continuous, positive, and decreasing on [1, ∞), so the Integral Test applies.
∞
t
x −1/5 t
dx = lim
1
t→∞ 1 x−1/5 5
dx = lim
t→∞ 4 x4/5 5
= lim
1 t→∞ 4 t4/5 − 5 = ∞,so ∞ 1/ 5√ n diverges.
4
n=1

F.
TX.10SECTION 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS ¤ 467
1
5. The function f(x) = is continuous, positive, and decreasing on [1, ∞), so the Integral Test applies.
(2x +1)
3
∞
1
t
1
dx = lim
dx = lim − 1 t
1
1
= lim −
1 (2x +1)
3 t→∞
1 (2x +1)
3 t→∞ 4 (2x +1) 2 t→∞
1
4(2t +1) + 1
= 1
2 36 36 .
Since this improper integral is convergent, the series ∞
n=1
1
is also convergent by the Integral Test.
(2n +1)
3
7. f(x) =xe −x is continuous and positive on [1, ∞). f 0 (x) =−xe −x + e −x = e −x (1 − x) < 0 for x>1,sof is decreasing
on [1, ∞). Thus, the Integral Test applies.
∞
1
xe −x dx = lim
b→∞
b
1 xe−x dx = lim
b→∞
−xe −x − e −x b
1
[by parts] = lim
b→∞
[−be −b − e −b + e −1 + e −1 ]=2/e
since lim
b→∞
be −b = lim
b→∞
(b/e b ) H = lim
b→∞
(1/e b )=0and lim
b→∞
e −b =0. Thus, ∞
n=1 ne−n converges.
9. The series ∞
n=1
for if it converged, then ∞
1
n is a p-series with p =0.85 ≤ 1,soitdivergesby(1).Therefore,theseries ∞
0.85
n=1
11. 1+ 1 8 + 1 27 + 1 64 + 1
125 + ···= ∞
13. 1+ 1 3 + 1 5 + 1 7 + 1 9 + ··· = ∞
15.
1
would have to converge [by Theorem 8(i) in Section 11.2].
n0.85 n=1
n=1
n=1
1
.Thisisap-series with p =3> 1, so it converges by (1).
n3 1
2n − 1 . The function f(x) = 1
2x − 1 is
continuous, positive, and decreasing on [1, ∞), so the Integral Test applies.
∞
1
diverges.
∞
n=1
1
dx = lim
2x − 1 t→∞
5 − 2 √ n
n 3
=5 ∞
n=1
t
1
2
must also diverge,
n0.85 1
dx = lim 1
2x − 1 ln |2x − 1| t
= 1 lim (ln(2t − 1) − 0) = ∞,sotheseries ∞ 1
t→∞ 2 1 2 t→∞ n=1 2n − 1
1
n − 2 ∞
3 n=1
with p =3> 1 and p =
5
> 1 ∞
2
. Thus,
1
n by Theorem 11.2.8, since ∞
5/2
n=1
5 − 2 √ n
n 3
converges.
n=1
1
n and ∞
3 n=1
1
both converge by (1)
n5/2 17. The function f(x) = 1 is continuous, positive, and decreasing on [1, ∞),sowecanapplytheIntegralTest.
x 2 +4
∞
1
t
t
1
1 x
dx = lim
dx = lim
1 x 2 +4 t→∞
1 x 2 +4 t→∞ 2 tan−1 = 1
t 1
2
1
2 lim tan −1 − tan −1
t→∞ 2
2
= 1 π 1
2 2 2
− tan−1
19.
Therefore, the series ∞
∞
n=1
ln n
n 3
= ∞
n=2
n=1
1
n 2 +4 converges.
ln n ln 1
ln x
since =0. The function f(x) = is continuous and positive on [2, ∞).
n3 1 x 3
f 0 (x) = x3 (1/x) − (ln x)(3x 2 )
= x2 − 3x 2 ln x
= 1 − 3lnx < 0 ⇔ 1 − 3lnx 1
(x 3 ) 2 x 6
x 4 3
⇔

F.
468 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
x>e 1/3 ≈ 1.4,sof is decreasing on [2, ∞), and the Integral Test applies.
∞
2
ln x
x 3
converges.
t
ln x
dx = lim
t→∞
2 x 3
()
dx = lim − ln x
t→∞ 2x − 1 t
= lim − 1
2 4x 2 t→∞
1
4t (2 ln t +1)+ 1
()
= 1 2 4 4 ,sotheseries ∞ ln n
n=2 n 3
(): u =lnx, dv = x −3 dx ⇒ du =(1/x) dx, v = − 1 2 x−2 ,so
ln x
dx = − 1 2 x−2 ln x − − 1 2 x−2 (1/x) dx = − 1 2 x−2 ln x + 1 2
x 3
(): lim −
t→∞
2lnt +1 H 2/t
= − lim
4t 2
t→∞ 8t = − 1 4
lim 1
t→∞ t =0. 2
x −3 dx = − 1 2 x−2 ln x − 1 4 x−2 + C.
21. f(x) = 1
x ln x is continuous and positive on [2, ∞), and also decreasing since f 0 (x) =− 1+lnx < 0 for x>2, so we can
x 2 (ln x)
2
∞
1
use the Integral Test.
dx = lim
2 x ln x [ln(ln t→∞ x)]t 2 = lim [ln(ln t) − ln(ln 2)] = ∞,sotheseries ∞ 1
t→∞ n=2 n ln n diverges.
23. The function f(x) =e 1/x /x 2 is continuous, positive, and decreasing on [1, ∞), so the Integral Test applies.
[g(x) =e 1/x is decreasing and dividing by x 2 doesn’t change that fact.]
∞
t
e 1/x
f(x) dx = lim dx = lim
−e 1/x t
= − lim (e 1/t
− e) =−(1 − e) =e − 1, sotheseries ∞ e 1/n
1
t→∞
1 x 2 t→∞
1 t→∞ n=1 n 2
converges.
25. The function f(x) = 1 is continuous, positive, and decreasing on [1, ∞), so the Integral Test applies. We use partial
x 3 + x
fractions to evaluate the integral:
∞
1
so the series ∞
n=1
1
dx = lim
x 3 + x t→∞
t
1
= lim ln
t→∞
1
n 3 + n converges.
1
x −
x
dx = lim ln x − 1 t
1+x 2 t→∞ 2 ln(1 + x2 ) = lim ln
t→∞
1
t
√ − ln 1
√ = lim ln
1+t
2 2 t→∞
27. We have already shown (in Exercise 21) that when p =1the series ∞
n=2
1
1+1/t
2 + 1 2 ln 2
= 1 2 ln 2
t
x
√
1+x
2
1
1
diverges, so assume that p 6= 1.
n(ln n)
p
1
f(x) =
x(ln x) is continuous and positive on [2, ∞),andf 0 (x) =−
p +lnx
p x 2 (ln x) < 0 if p+1 x>e−p ,sothatf is eventually
decreasing and we can use the Integral Test.
∞
2
1
(ln x)
1−p
dx = lim
x(ln x)
p t→∞ 1 − p
t
2
(ln t)
1−p
[for p 6= 1] = lim
−
t→∞ 1 − p
This limit exists whenever 1 − p1, so the series converges for p>1.
(ln 2)1−p
1 − p

F.
TX.10SECTION 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS ¤ 469
29. Clearly the series cannot converge if p ≥− 1 2
, because then lim n(1 +
n→∞ n2 ) p 6=0. So assume p1000.
35. f(x) =1/(2x +1) 6 is continuous, positive, and decreasing on [1, ∞), so the Integral Test applies. Using (2),
37.
R n ≤
∞
n
(2x +1) −6 dx = lim
t→∞
t
−1
1
=
. To be correct to five decimal places, we want
10(2x +1) 5 n
10(2n +1)
5
1
10(2n +1) ≤ 5
√ ⇔ (2n +1) 5 ≥ 20,000 ⇔ n ≥ 1 5
5 10 6 2 20,000 − 1 ≈ 3.12,sousen =4.
s 4 =
4
n=1
1
(2n +1) = 1 6 3 + 1 6 5 + 1 6 7 + 1 ≈ 0.001 446 ≈ 0.00145.
6 96 ∞
n −1.001 = ∞
n=1
R n ≤
∞
n
n=1
1
is a convergent p-series with p =1.001 > 1. Using(2),weget
n1.001 x
x −1.001 −0.001 t
dx = lim
= −1000 lim
t→∞ −0.001
n
t→∞
1
x 0.001 t
n
= −1000 − 1
= 1000
n 0.001 n . 0.001
We want R n < 0.000 000 005 ⇔ 1000
n 0.001 < 5 × 10−9 ⇔ n 0.001 > 1000
5 × 10 −9 ⇔
n> 2 × 10 11 1000 =2 1000 × 10 11,000 ≈ 1.07 × 10 301 × 10 11,000 =1.07 × 10 11,301 .

F.
470 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
39. (a) From the figure, a 2 + a 3 + ···+ a n ≤ n
f(x) dx,sowith
1
f(x) = 1 x , 1 2 + 1 3 + 1 4 + ···+ 1 n ≤ n
1
1
dx =lnn.
x
Thus, s n =1+ 1 2 + 1 3 + 1 4 + ···+ 1 n ≤ 1+lnn.
(b) By part (a), s 10 6 ≤ 1+ln10 6 ≈ 14.82 < 15 and
s 10 9 ≤ 1+ln10 9 ≈ 21.72 < 22.
41. b ln n = e ln b ln n
= e
ln n ln b
= n
ln b = 1 .Thisisap-series, which converges for all b such that − ln b>1
n ⇔
− ln b
ln b
n
n √ n = √ 1
for all n ≥ 1,so ∞
n
p-series with p = 1 2 ≤ 1.
n=1
9 n
n
3+10 < 9n 9
n 10 = for all n ≥ 1.
n 10
converges by the Comparison Test.
cos 2 n
n 2 +1 ≤ 1
n 2 +1 < 1 n 2 ,sotheseries ∞
n=1
n=1
n
2n 3 +1 converges by comparison with ∞
n +1
n √ n diverges by comparison with ∞
∞
n=1
n=1
n=1
1
, which converges
n2 1
√
n
, which diverges because it is a
9
n
is a convergent geometric series |r| = 9 < 1
,so ∞ 9 n
10
10
n=1 3+10 n
cos 2 n
n 2 +1 convergesbycomparisonwiththep-series ∞
n − 1
n − 1
11. is positive for n>1 and
n 4n n 4 < n
n n 4 = 1 n 1 n 4 =
,so ∞ n 4
geometric series ∞ n 1
.
n=1 4
13.
arctan n
n 1.2 < π/2
n 1.2 for all n ≥ 1,so ∞
n=1
constant times a p-series with p =1.2 > 1.
n=1
n − 1
n 4 n
arctan n
n 1.2 converges by comparison with π 2
n=1
1
n 2 [p =2> 1].
converges by comparison with the convergent
∞
n=1
1
, which converges because it is a
n1.2 15.
2+(−1) n
n √ n
≤ 3
n √ n ,and ∞
n=1
3
n √ n converges because it is a constant multiple of the convergent p-series ∞
n=1
1
n √ n
p =
3
2 > 1 , so the given series converges by the Comparison Test.

F.
TX.10
SECTION 11.4 THE COMPARISON TESTS ¤ 471
17. Use the Limit Comparison Test with a n =
a n
lim = lim
n→∞ b n n→∞
∞
n=1
1
√
n2 +1 .
n
√
n2 +1 = lim
n→∞
1
√
n2 +1 and bn = 1 n :
1
1+(1/n2 ) =1> 0. Since the harmonic series ∞
19. Use the Limit Comparison Test with a n = 1+4n
1+3 n and b n = 4n
3 n :
a n
lim = lim
n→∞ b n n→∞
1+4 n
1+3 n 1+4 n
4 n = lim
n→∞ 1+3 · 3n
n 4 = lim 1+4 n
·
n n→∞ 4 n
3 n
Since the geometric series b n =
4 n
diverges, so does ∞ 3
n=1
1+4 n
1+3 > 1+4n
n 3 n +3 > 4n
n 2(3 n ) = 1 n 4
or use the Test for Divergence.
2 3
21. Use the Limit Comparison Test with a n =
√ n +2
1
and
2n 2 bn =
+ n +1 n : 3/2
a n n √ 3/2 n +2
lim = lim
n→∞ b n n→∞ 2n 2 + n +1 = lim (n √ 3/2 n +2)/(n √ 3/2 n )
n→∞
Since ∞
n=1
(2n 2 + n +1)/n 2 = lim
n→∞
1
n is a convergent p-series p = 3 > 1
,theseries ∞ 3/2 2
n=1
3 n
1
1+3 = lim
n n→∞ 4 +1 ·
n
1
diverges, so does
n
1
=1> 0
1
3 +1 n
1+4 n
. Alternatively, use the Comparison Test with
1+3n n=1
23. Use the Limit Comparison Test with a n = 5+2n
(1 + n 2 ) 2 and bn = 1 n 3 :
a n n 3 (5 + 2n)
lim = lim
n→∞ b n n→∞ (1 + n 2 ) = lim 5n 3 +2n 4
2 n→∞ (1 + n 2 ) · 1/n 4
2 1/(n 2 ) = lim
2 n→∞
p-series [p =3> 1], the series ∞
25. If a n =
so ∞
n=1
n=1
5+2n
also converges.
(1 + n 2 )
2
1+n + n2
√
1+n2 + n and 6 bn = 1 a n
,then lim = lim
n n→∞ b n n→∞
1+2/n
2+1/n +1/n 2 = √
1
2 = 1 2 > 0.
√ n +2
also converges.
2n 2 + n +1
5
+2 n
1
+1 2 =2> 0. Since ∞
n 2 n=1
n + n 2 + n 3
√
1+n2 + n 6 = lim
n→∞
1+n + n 2
√
1+n2 + n diverges by the Limit Comparison Test with the divergent harmonic series ∞
6
27. Use the Limit Comparison Test with a n =
∞
e −n
= ∞
n=1
n=1
1+ 1 2
e −n and b n = e −n a n
: lim
n n→∞
1
e is a convergent geometric series |r| = 1 < 1
,theseries ∞ n e
29. Clearly n! =n(n − 1)(n − 2) ···(3)(2) ≥ 2 · 2 · 2 ·····2 · 2=2 n−1 ,so 1 n! ≤ 1
2 n−1 . ∞
series |r| = 1 2 < 1 ,so ∞
n=1
1
converges by the Comparison Test.
n!
n=1
b n
1
is a convergent
n3 1/n 2 +1/n +1
=1> 0,
1/n6 +1/n 4 +1
n=1
1
n .
= lim 1+ 1 2
=1> 0. Since
n→∞ n
1+ 1 n
2
e −n also converges.
n=1
1
is a convergent geometric
2n−1

F.
472 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES TX.10
1
31. Use the Limit Comparison Test with a n =sin and b n = 1 n n .Then a n and b n are series with positive terms and
33.
35.
a n
lim
n→∞ b n
sin(1/n) sin θ
= lim =lim =1> 0. Since ∞ b n is the divergent harmonic series,
n→∞ 1/n θ→0 θ
n=1
∞
sin (1/n) also diverges.
n=1
[Note that we could also use l’Hospital’s Rule to evaluate the limit:
−1/x
sin(1/x) H cos(1/x) ·
2
lim = lim
= lim
x→∞ 1/x x→∞ −1/x cos 1 2
x→∞ x =cos0=1.]
10
n=1
1
√
n4 +1 = 1 √
2
+ 1 √
17
+ 1 √
82
+ ···+
R 10 ≤ T 10 ≤
10
n=1
∞
10
1
√ 10,001
≈ 1.24856. Now
1
dx = lim − 1 t
= lim
− 1 x2 t→∞ x t→∞
10
t + 1
= 1
10 10 =0.1.
1
1+2 = 1 n 3 + 1 5 + 1 9 + ···+ 1
1025 ≈ 0.76352. Now 1
1+2 < 1 , so the error is
n 2n R 10 ≤ T 10 =
∞
n=11
1
2 n = 1/211
1 − 1/2
37. Since d n
10 n ≤ 9
10 n for each n,andsince ∞
[geometric series] ≈ 0.00098.
n=1
will always converge by the Comparison Test.
1
√
n4 +1 < 1 √
n
4 = 1 n 2 ,sotheerroris
9
10 n is a convergent geometric series |r| = 1
10 < 1 , 0.d 1 d 2 d 3 ...= ∞
39. Since a n converges, lim an =0,sothereexistsN such that |an − 0| < 1 for all n>N ⇒ 0 ≤ an < 1 for
n→∞
all n>N ⇒ 0 ≤ a 2 n ≤ a n . Since a n converges, so does a 2 n by the Comparison Test.
a n
41. (a) Since lim = ∞, there is an integer N such that an
> 1 whenever n>N.(TakeM =1in Definition 11.1.5.)
n→∞ b n
b n
Then a n >b n whenever n>Nand since b n is divergent, a n is also divergent by the Comparison Test.
(b) (i) If a n = 1
ln n and b n = 1 a n
for n ≥ 2,then lim = lim
n n→∞ b n n→∞
so by part (a),
(ii) If a n = ln n
n
∞
n=2
1
is divergent.
ln n
and b n = 1 n ,then ∞
so ∞ a n diverges by part (a).
n=1
n=1
n
ln n = lim
x→∞
x
ln x
H
= lim
x→∞
n=1
1
1/x = lim
x→∞ x = ∞,
d n
10 n
a n
b n is the divergent harmonic series and lim = lim ln n = lim ln x = ∞,
n→∞ b n n→∞ x→∞
a n
43. lim nan = lim
n→∞ n→∞ 1/n , so we apply the Limit Comparison Test with bn = 1 .Since lim nan > 0 we know that either both
n n→∞
series converge or both series diverge, and we also know that
divergent.
∞
n=1
1
n diverges [p-series with p =1]. Therefore, a n must be

F.
TX.10
SECTION 11.5 ALTERNATING SERIES ¤ 473
45. Yes. Since a n is a convergent series with positive terms, lim
n→∞ an =0by Theorem 11.2.6, and b n = sin(a n) is a
b n sin(a n)
series with positive terms (for large enough n). We have lim = lim =1> 0 by Theorem 3.3.2. Thus, b n
n→∞ a n n→∞ a n
is also convergent by the Limit Comparison Test.
11.5 Alternating Series
1. (a) An alternating series is a series whose terms are alternately positive and negative.
(b) An alternating series ∞ (−1) n−1 b n converges if 0 0, {b n} is decreasing, and lim b n =0,sothe
n→∞
5.
7.
n=1
series converges by the Alternating Series Test.
∞
a n = ∞
n=1
n=1
(−1) n−1 1
2n +1 = ∞
n=1
series converges by the Alternating Series Test.
∞
a n = ∞ (−1) n 3n − 1
2n +1 = ∞ (−1) n b n.Now lim
n=1
n=1
n=1
(−1) n−1 b n .Nowb n = 1
2n +1 > 0, {b n} is decreasing, and lim
n→∞ b n =0,sothe
n→∞
(in fact the limit does not exist), the series diverges by the Test for Divergence.
3 − 1/n
bn = lim
n→∞ 2+1/n = 3 6=0.Since lim an 6= 0
2 n→∞
9. b n = n
10 > 0 for n ≥ 1. {b n} is decreasing for n ≥ 1 since
n
x
0 10 x (1) − x · 10 x ln 10
= = 10x (1 − x ln 10) 1 − x ln 10
= < 0 for 1 − x ln 10 < 0 ⇒ x ln 10 > 1 ⇒
10 x (10 x ) 2 (10 x ) 2 10 x
x> 1 ≈ 0.4. Also, lim bn = lim
ln 10 n→∞ n→∞
converges by the Alternating Series Test.
n
10 n = lim
x→∞
x
10 x H = lim
x→∞
x
10 x ln 10 =0. Thus, the series ∞
n=1
(−1) n
n
10 n
11. b n = n2
> 0 for n ≥ 1.
n 3 +4
x
2
13.
x 3 +4
lim
n→∞ b n = lim
n→∞
∞
(−1) n n
n=2
{bn} is decreasing for n ≥ 2 since
0
= (x3 + 4)(2x) − x 2 (3x 2 )
(x 3 +4) 2 = x(2x3 +8− 3x 3 )
ln n . lim
n→∞
= x(8 − x3 )
< 0 for x>2. Also,
(x 3 +4) 2 (x 3 +4)
2
1/n
1+4/n =0.Thus,theseries ∞
(−1) n+1 n 2
converges by the Alternating Series Test.
3 n 3 +4
n
ln n = lim
x→∞
x
ln x
H
= lim
x→∞
n=1
1
= ∞, so the series diverges by the Test for Divergence.
1/x

F.
474 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
15.
17.
19.
∞
n=1
cos nπ
n 3/4
= ∞
n=1
Alternating Series Test.
∞
n=1
π
(−1) n sin . b n =sin
n
(−1) n
n . b 3/4 n = 1 is decreasing and positive and lim
n3/4 π
n
converges by the Alternating Series Test.
π
> 0 for n ≥ 2 and sin ≥ sin
n
n→∞
π
n +1
1
=0, so the series converges by the
n3/4
π
,and lim sin =sin0=0,sotheseries
n→∞ n
n n
n! = n · n ·····n
1 · 2 ·····n ≥ n ⇒ lim n n
n→∞ n! = ∞ ⇒ lim (−1) n n n
does not exist. So the series diverges by the Test for
n→∞ n!
Divergence.
21.
n a n s n
1 1 1
2 −0.35355 0.64645
3 0.19245 0.83890
4 −0.125 0.71390
5 0.08944 0.80334
6 −0.06804 0.73530
7 0.05399 0.78929
8 −0.04419 0.74510
9 0.03704 0.78214
10 −0.03162 0.75051
By the Alternating Series Estimation Theorem, the error in the approximation
∞
n=1
(−1) n−1
n 3/2
decimal places, rounded up).
≈ 0.75051 is |s − s 10 | ≤ b 11 =1/(11) 3/2 ≈ 0.0275 (to four
23. The series ∞
n=1
(−1) n+1
satisfies (i) of the Alternating Series Test because
n 6
1
(n +1) < 1 and (ii) lim
6 n6 n→∞
1
n 6 =0,sothe
series is convergent. Now b 5 = 1 5 =0.000064 > 0.00005 and b 6 6 = 1 ≈ 0.00002 < 0.00005,sobytheAlternatingSeries
66 Estimation Theorem, n =5. (That is, since the 6th term is less than the desired error, we need to add the first 5 terms to get the
sum to the desired accuracy.)
25. The series ∞
n=0
(−1) n
10 n n! satisfies (i) of the Alternating Series Test because 1
10 n+1 (n +1)! < 1
1
and (ii) lim
10 n n! n→∞ 10 n n! =0,
sotheseriesisconvergent.Nowb 3 = 1
10 3 3! ≈ 0.000 167 > 0.000 005 and b 4 = 1 =0.000 004 < 0.000 005,soby
10 4 4!
the Alternating Series Estimation Theorem, n =4(since the series starts with n =0,notn =1). (That is, since the 5th term
is less than the desired error, we need to add the first 4 terms to get the sum to the desired accuracy.)
27. b 7 = 1 7 5 = 1
16,807
∞
n=1
(−1) n+1
n 5 ≈ s 6 = 6
≈ 0.000 059 5, so
n=1
(−1) n+1
n 5 =1− 1 32 + 1
243 − 1
1024 + 1
3125 − 1
7776 ≈ 0.972 080. Addingb 7 to s 6 does not change
the fourth decimal place of s 6 , so the sum of the series, correct to four decimal places, is 0.9721.

F.
SECTION TX.10 11.6 ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS ¤ 475
29. b 7 = 72
=0.000 004 9, so
107 31.
∞
n=1
(−1) n−1 n 2
10 n ≈ s 6 = 6
n=1
(−1) n−1 n 2
= 1
10 − 4 + 9 − 16 + 25 − 36
n 10 100 1000 10,000 100,000 1,000,000
=0.067 614. Addingb7 to s6
does not change the fourth decimal place of s 6 , so the sum of the series, correct to four decimal places, is 0.0676.
∞
n=1
(−1) n−1
n
underestimate, since ∞
=1− 1 2 + 1 3 − 1 4 + ···+ 1
49 − 1
50 + 1
51 − 1 + ···. The 50th partial sum of this series is an
52
n=1
(−1) n−1
The result can be seen geometrically in Figure 1.
n
1
= s 50 +
51 52
− 1 1
+
53 − 1
+ ···, and the terms in parentheses are all positive.
54
33. Clearly b n = 1 is decreasing and eventually positive and lim
n + p b n =0for any p. So the series converges (by the
n→∞
Alternating Series Test) for any p for which every b n is defined, that is, n + p 6= 0for n ≥ 1,orp is not a negative integer.
35.
b2n = 1/(2n) 2 clearlyconverges(bycomparisonwiththep-series for p =2). So suppose that (−1) n−1 b n
converges. Then by Theorem 11.2.8(ii), so does (−1) n−1 b n + b n
=2
1+
1
3 + 1 5 + ··· =2 1
2n − 1 .Butthis
diverges by comparison with the harmonic series, a contradiction. Therefore, (−1) n−1 b n must diverge. The Alternating
Series Test does not apply since {b n} is not decreasing.
11.6 Absolute Convergence and the Ratio and Root Tests
1. (a) Since lim
(b) Since lim
n→∞ an+1
a n
n→∞ an+1
a n
therefore convergent).
(c) Since lim
n=0
n→∞ an+1
a n
=8> 1, part (b) of the Ratio Test tells us that the series a n is divergent.
=0.8 < 1, part (a) of the Ratio Test tells us that the series a n is absolutely convergent (and
=1, the Ratio Test fails and the series a n might converge or it might diverge.
∞ (−10) n
3.
. Using the Ratio Test, lim
n!
n→∞ a n+1 = lim
n→∞ (−10)n+1 n! · = lim
(n +1)! (−10) n n→∞ −10
n +1 =0< 1,sotheseriesis
5.
absolutely convergent.
∞
n=1
(−1) n+1
4√ n
is conditionally convergent.
a n
converges by the Alternating Series Test, but
∞
n=1
1
4√ n
is a divergent p-series p = 1 4 ≤ 1 ,sothegivenseries

F.
476 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
7. lim
k→∞
a
k+1 (k +1) 2
k+1
3
= lim
k→∞
∞
n=1
a k
k
2 k
= lim
k→∞
3
k +1
k
2
3
1
= 2
3 lim 1+ 1
= 2
k→∞
3
k
(1) = 2 < 1,sotheseries
3
k 2
3
k
is absolutely convergent by the Ratio Test. Since the terms of this series are positive, absolute convergence is the
same as convergence.
9. lim
n→∞
a n+1
a n
(1.1)
n+1
= lim
n→∞ (n +1) 4 ·
=(1.1)(1) = 1.1 > 1,
n 4
= lim
(1.1) n n→∞
TX.10
(1.1)n 4
4
=(1.1) lim
(n +1) n→∞
1
(n +1) 4
n 4
1
=(1.1) lim
n→∞ (1 + 1/n) 4
so the series ∞ (−1) n (1.1) n
diverges by the Ratio Test.
n=1
11. Since 0 ≤ e1/n
n 3
n 4
≤ e n 3 = e 1
n 3
and ∞
n=1
1
n is a convergent p-series [p =3> 1], ∞
3
n=1
e 1/n
n 3
converges, and so
∞
n=1
13. lim
n→∞
(−1) n e 1/n
is absolutely convergent.
n 3
a n+1
a n
= lim
n→∞
10 n+1 (n +1)42n+1
·
(n +2)42n+3 10 n
10
= lim
n→∞ 4 · n +1
= 5 2 n +2 8 < 1,sotheseries ∞
n=1
10 n
(n +1)4 2n+1
is absolutely convergent by the Ratio Test. Since the terms of this series are positive, absolute convergence is the same as
convergence.
15.
(−1)n arctan n
< π/2
n 2 n ,sosince ∞
2
17.
19.
n=1
converges absolutely by the Comparison Test.
∞
n=2
(−1) n
ln n
π/2
n 2 = π 2
∞
n=1
converges by the Alternating Series Test since lim
n→∞
1
ln n > 1 n , and since ∞
∞
n=2
(−1) n
ln n
|cos (nπ/3)|
n!
Comparison Test.
n=2
1
n is the divergent (partial) harmonic series, ∞
is conditionally convergent.
≤ 1 n! and ∞
n=1
1
n converges (p =2> 1), the given series ∞
2
n=1
(−1) n arctan n
n 2
1
1
ln n =0and is decreasing. Now ln n1 (byEquation3.6.6),sotheseries ∞ 1+ 1 n
2
n→∞
n→∞ n n→∞ n
n=1 n
diverges by the Root Test.

F.
SECTION TX.10 11.6 ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS ¤ 477
25. UsetheRatioTestwiththeseries
1 − 1 · 3 + 1 · 3 · 5 − 1 · 3 · 5 · 7 + ···+(−1) n−1 1 · 3 · 5 ·····(2n − 1)
+ ···= ∞ (−1) n−1 1 · 3 · 5 ·····(2n − 1)
.
3! 5! 7!
(2n − 1)!
n=1
(2n − 1)!
lim
n→∞ a n+1 = lim
a n n→∞ (−1)n · 1 · 3 · 5 ·····(2n − 1)[2(n +1)− 1]
(2n − 1)!
·
[2(n +1)− 1]!
(−1) n−1 · 1 · 3 · 5 ·····(2n − 1)
= lim
(−1)(2n +1)(2n − 1)!
n→∞ (2n +1)(2n)(2n − 1)! = lim 1
=0< 1,
n→∞ 2n
27.
so the given series is absolutely convergent and therefore convergent.
∞
n=1
2 · 4 · 6 ·····(2n)
n!
= ∞
n=1
Divergence since lim
n→∞ 2n = ∞.
29. Bytherecursivedefinition, lim
31. (a) lim
n→∞
1/(n +1)3
1/n 3
= lim
n→∞
(b) lim
(n +1)
n→∞ · 2n 2 n+1 n = lim
(c) lim
√ (−3)n · n +1
n→∞
(d) lim
n→∞
33. (a) lim
(2 · 1) · (2 · 2) · (2 · 3) ·····(2 · n)
n!
n→∞ an+1
a n
n→∞
= lim
n→∞
n 3
(n +1) 3 = lim
n→∞
n +1
2n
√
n
= 3 lim
(−3) n−1 n→∞
= ∞
n=1
2 n n!
n!
= ∞ 2 n , which diverges by the Test for
n=1
5n +1
4n +3 = 5 > 1, so the series diverges by the Ratio Test.
4
1 = lim
n→∞ 2 + 1
2n
n
n +1
√
n +1
1+(n +1) · 1+n2 √ = lim
2 n n→∞
n→∞ an+1
a n
series ∞
n=0
1
3
=1. Inconclusive
(1 + 1/n)
1+ 1 n ·
= lim
x n+1
n→∞ (n +1)! · n! = lim
x
x n n→∞ n +1 x n
n!
converges for all x.
= 1 . Conclusive (convergent)
2
1
=3 lim
n→∞ 1+1/n =3.
Conclusive (divergent)
1/n 2 +1
1/n 2 +(1+1/n) 2 =1. Inconclusive
= |x| lim
n→∞
x n
(b) Since the series of part (a) always converges, we must have lim =0by Theorem 11.2.6.
n→∞ n!
35. (a) s 5 = 5
n=1
1
n2 = 1 n 2 + 1 8 + 1
24 + 1 64 + 1
160 = 661 ≈ 0.68854. Nowtheratios
960
r n = a n+1 n2 n
=
a n (n +1)2 = n
form an increasing sequence, since
n+1 2(n +1)
1
= |x|·0=0< 1,sobytheRatioTestthe
n +1
r n+1 − r n = n +1
2(n +2) − n
2(n +1) = (n +1)2 − n(n +2) 1
=
> 0. So by Exercise 34(b), the error
2(n +1)(n +2) 2(n +1)(n +2)
a 6
in using s 5 is R 5 ≤
= 1/ 6 · 2 6
1 − lim
n→∞ rn 1 − 1/2 = 1
192 ≈ 0.00521.
(b) The error in using s n as an approximation to the sum is R n = an+1
1 − 1 2
=
2
(n +1)2 n+1 .WewantR n < 0.00005 ⇔
1
(n +1)2 n < 0.00005 ⇔ (n +1)2n > 20,000. Tofind such an n we can use trial and error or a graph. We calculate
(11 + 1)2 11 =24,576,sos 11 = 11
n=1
1
≈ 0.693109 is within 0.00005 of the actual sum.
n2n

F.
478 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
37. (i) Following the hint, we get that |a n| 1 for n ≥ N. Thus,
lim
n→∞ an 6= 0,so ∞
n=1
an diverges by the Test for Divergence.
(iii) Consider ∞
n=1
1
n [diverges] and ∞
n=1
1
n 2
n
[converges]. For each sum, lim |an| =1, so the Root Test is inconclusive.
n→∞
39. (a) Since a n is absolutely convergent, and since a
+
n
≤ |an| and a
−
n
≤ |an| (because a + n and a − n each equal
either a n or 0), we conclude by the Comparison Test that both a + n and a − n must be absolutely convergent.
Or: Use Theorem 11.2.8.
(b) We will show by contradiction that both a + n and a − n must diverge. For suppose that a + n converged. Then so
would a + n − 1 2 an
by Theorem 11.2.8. But
a
+
n − 1 2 an
=
1
2 (an + |an|) − 1 2 an
=
1
2
|an|, which
diverges because a n is only conditionally convergent. Hence, a + n can’t converge. Similarly, neither can a − n .
11.7 Strategy for Testing Series
1.
1
n +3 < 1 n 1 n 3 = for all n ≥ 1.
n 3
converges by the Comparison Test.
∞
n=1
1
3
n
is a convergent geometric series |r| = 1 3 < 1 ,so ∞
n=1
1
n +3 n
n
n
3. lim |an| = lim =1,so lim an = lim
n→∞ n→∞ n +2 n→∞ n→∞ (−1)n n +2 does not exist. Thus, the series ∞ (−1) n n
diverges by
n +2
the Test for Divergence.
5. lim
n→∞ an+1
a n
∞
n=1
n 2 2 n−1
(−5) n
= lim
n→∞
(n +1)2 2 n (−5) n 2(n +1) 2
· = lim
= 2
(−5) n+1 n 2 2 n−1 n→∞ 5n 2 5 lim 1+ 1 2
= 2
n→∞ n 5 (1) = 2 5 < 1,sotheseries
converges by the Ratio Test.
1
7. Let f(x) =
x √ .Thenf is positive, continuous, and decreasing on [2, ∞), so we can apply the Integral Test.
ln x
1 u =lnx,
Since
x √ ln x dx = u −1/2 du =2u 1/2 + C =2 √ ln x + C, wefind
du = dx/x
∞
dx
t
2 x √ ln x = lim dx
t→∞
2 x √ ln x = lim 2 √ t
ln x = lim 2 √ ln t − 2 √
ln 2 = ∞. Since the integral diverges, the
t→∞
2 t→∞
given series ∞ 1
n=2 n √ ln n diverges.
n=1

F.
TX.10
SECTION 11.7 STRATEGY FOR TESTING SERIES ¤ 479
9.
∞
k 2 e −k
= ∞
k=1
a k
k=1
lim
k→∞
a k+1 = lim
(k +1)2
k→∞
e k+1
k 2
. Using the Ratio Test, we get
ek · ek
k 2
= lim
k→∞
k +1
k
2
· 1 =1 2 · 1
e e = 1 < 1, so the series converges.
e
11. b n = 1
n ln n > 0 for n ≥ 2, {b n} is decreasing, and lim b
n =0,sothegivenseries ∞ (−1) n+1
converges by the
n→∞ n=2 n ln n
Alternating Series Test.
13. lim
n→∞ a n+1 = lim
n→∞ 3n+1 (n +1) 2
·
(n +1)!
a n
convergesbytheRatioTest.
a n
n!
3 n n 2
= lim
n→∞
15. lim
n→∞ a n+1 = lim
(n +1)!
n→∞ 2 · 5 · 8 ·····(3n +2)[3(n +1)+2]
so the series ∞
n=0
n!
2 · 5 · 8 ·····(3n +2) convergesbytheRatioTest.
3(n +1) 2 n +1
=3 lim =0< 1,sotheseries ∞ 3 n n 2
(n +1)n2 n→∞ n 2
n=1 n!
· 2 · 5 · 8 ·····(3n +2)
n!
n=1
= lim
n→∞
n +1
3n +5 = 1 3 < 1,
17. lim
n→∞ 21/n =2 0 =1,so lim
n→∞ (−1)n 2 1/n
does not exist and the series ∞ (−1) n 2 1/n diverges by the Test for Divergence.
19. Let f(x) = ln √ x .Thenf 0 (x) = 2 − ln x < 0 when ln x>2 or x>e 2 ,so ln √ n is decreasing for n>e 2 .
x 2x 3/2
n
21.
ln n
By l’Hospital’s Rule, lim √ = lim
n→∞ n
n→∞
Alternating Series Test.
∞
n=1
(−2) 2n
n n
1/n
1/ 2 √ 2
= lim √ =0,sotheseries ∞ (−1) n ln n
√ converges by the
n→∞
n
n n=1 n
=
n=1 ∞ n 4
n
4
. lim |an | = lim =0< 1, so the given series is absolutely convergent by the Root Test.
n n→∞
n→∞ n
1
23. Using the Limit Comparison Test with a n =tan and b n =
n
1 n ,wehave
a n tan(1/n) tan(1/x) H sec 2 (1/x) · (−1/x 2 )
lim = lim = lim = lim
= lim
n→∞ b n n→∞ 1/n x→∞ 1/x x→∞ −1/x 2
x→∞ sec2 (1/x) =1 2 =1> 0. Since
∞
∞
b n is the divergent harmonic series, a n is also divergent.
n=1
25. Use the Ratio Test. lim
27.
converges.
n→∞ an+1
a n
= lim
n→∞
n=1
(n +1)!
e (n+1)2
∞
ln x
dx = lim − ln x
2 x2 t→∞ x − 1 t
x
1
k ln k
(k +1) < k ln k = ln k
3 k 3 k ,thegivenseries ∞
2 k=1
· en2
n!
= lim
n→∞
(n +1)n! · e n2
e n2 +2n+1
n!
[using integration by parts] H =1.So ∞
n=1
ln n
n 2
k ln k
3
converges by the Comparison Test.
(k +1)
n +1
= lim
n→∞ e =0< 1,so ∞ n!
2n+1 n=1 e n2
converges by the Integral Test, and since

F.
480 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
29.
∞
a n = ∞ (−1) n 1
cosh n = ∞ (−1) n b n .Nowb n = 1
cosh n > 0, {b n} is decreasing, and lim b n =0,sotheseries
n→∞
n=1
n=1
n=1
converges by the Alternating Series Test.
Or: Write
1
cosh n = 2
e n + e −n < 2
e n and ∞
n=1
1
e is a convergent geometric series, so ∞
n
Comparison Test. So ∞ (−1) n 1
is absolutely convergent and therefore convergent.
cosh n
31. lim
k→∞ a k = lim
k→∞
Thus,
∞
k=1
n=1
5 k
3 k +4 k =[divide by 4k ] lim
k→∞
5 k
diverges by the Test for Divergence.
3 k +4k n=1
1
is convergent by the
cosh n
(5/4) k
k k 3 5
= ∞ since lim =0and lim = ∞.
(3/4) k +1 k→∞
4
k→∞
4
33. Let a n = sin(1/n) √ and b n = 1
n n √ a n sin(1/n)
.Then lim = lim =1> 0, so ∞ sin(1/n)
√ converges by limit
n n→∞ b n n→∞ 1/n
n=1 n
comparison with the convergent p-series ∞ 1
[p =3/2 > 1].
n 3/2
n
35. lim |an| = lim
n→∞
n→∞
n
n +1
converges by the Root Test.
n=1
n
2 /n
= lim
n→∞
1
[(n +1)/n] n =
1
lim (1 + = 1
n→∞ 1/n)n e < 1,sotheseries ∞
n=1
n
37. lim |an| = lim
n→∞
n→∞ (21/n − 1) = 1 − 1=0< 1,sotheseries ∞ √ n
n
2 − 1 converges by the Root Test.
n=1
n
n +1
n
2
11.8 Power Series
1. A power series is a series of the form ∞
n=0 c nx n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + ···,wherex isavariableandthec n ’s are
constants called the coefficients of the series.
More generally, a series of the form ∞
n=0 c n(x − a) n = c 0 + c 1 (x − a)+c 2 (x − a) 2 + ··· is called a power series in
(x − a) or a power series centered at a or a power series about a,wherea is a constant.
3. If a n = xn
√ n
,then lim
n→∞
a n+1 = lim √ · n +1
a n
By the Ratio Test, the series ∞
n=1
n→∞ xn+1
endpoints, that is, x = ±1. Whenx =1,theseries ∞
the series ∞
n=1
5. If a n = (−1)n−1 x n
,then
a n
√ n
= lim
x n n→∞
x
√ √ = lim
n +1/ n n→∞
|x|
1+1/n
= |x|.
x n
√ n
converges when |x| < 1, so the radius of convergence R =1. Now we’ll check the
n=1
1
√ diverges because it is a p-series with p = 1 ≤ 1. Whenx = −1,
n
2
(−1) n
√ n
converges by the Alternating Series Test. Thus, the interval of convergence is I =[−1, 1).
n 3
lim
n→∞ a
n+1 = lim
n→∞ (−1)n x n+1 n 3 3
·
= lim
(n +1) 3 (−1) n−1 x n n→∞ (−1)xn3
n
= lim
|x|
=1 3 ·|x| = |x|. Bythe
(n +1) 3 n→∞ n +1

F.
TX.10
SECTION 11.8 POWER SERIES ¤ 481
Ratio Test, the series ∞
n=1
(−1) n−1 x n
converges when |x| < 1, so the radius of convergence R =1.Nowwe’llcheckthe
endpoints, that is, x = ±1. Whenx =1,theseries ∞
the series ∞
n=1
n 3
(−1) n−1 (−1) n
= − ∞
n 3
n=1
Thus, the interval of convergence is I =[−1, 1].
a n
n=1
(−1) n−1
7. If a n = xn
,then lim
n! n→∞ a n+1 = lim
x n+1
n→∞ (n +1)! · n! = lim
x
x n n→∞ n +1 So, by the Ratio Test, R = ∞ and I =(−∞, ∞).
9. If a n =(−1) n n 2 x n
2 ,then
n lim
n→∞ a n+1 = lim
n→∞ (n +1)2 x n+1 2 n
· = lim
2 n+1 n 2 x n n→∞
a n
n 3 converges by the Alternating Series Test. When x = −1,
1
converges because it is a constant multiple of a convergent p-series [p =3> 1].
n3
= |x| lim
n→∞
1
= |x|·0=0< 1 for all real x.
n +1
x(n +1)2 |x|
= lim 1+ 1 2
= |x|
2n 2 n→∞ 2 n 2 (1)2 = 1 2
|x|. Bythe
Ratio Test, the series ∞ (−1) n n 2 x n
2 converges when 1 |x| < 1 ⇔ |x| < 2, so the radius of convergence is R =2.
n 2
n=1
When x = ±2, both series ∞ (−1) n n 2 (±2) n
= ∞ (∓1) n n 2 diverge by the Test for Divergence since
n=1 2 n n=1
lim (∓1) n n 2 = ∞. Thus, the interval of convergence is I =(−2, 2).
n→∞
11. a n = (−2)n x n
4√ ,so lim
n
n→∞ a n+1
a n
2 n+1 |x| n+1
= lim
n→∞ 4√ · n +1
4√ n
2 n |x| n = lim
n
2 |x| 4
n→∞ n +1 =2|x|,sobytheRatioTest,the
series converges when 2 |x| < 1 ⇔ |x| < 1 2 ,soR = 1 2 .Whenx = − 1 2 , we get the divergent p-series ∞
p =
1
4 ≤ 1 .Whenx = 1 2 ,wegettheseries ∞
Thus, I = − 1 2 , 1 2
.
13. If a n =(−1) n x n
,then lim
4 n ln n
n=2
n→∞ an+1
a n
n=2
n=1
= lim
n→∞
(−1) n
4√ n
, which converges by the Alternating Series Test.
x n+1
4 n+1 ln(n +1) · 4n ln n
x n
= |x|
4 lim
n→∞
n=1
1
4√ n
ln n
ln(n +1) = |x|
4 · 1
[by l’Hospital’s Rule] = |x|
|x|
. By the Ratio Test, the series converges when < 1 ⇔ |x| < 4, soR =4.When
4 4
∞
x = −4, (−1) n x n
4 n ln n = ∞ [(−1)(−4)] n
= ∞ 1
4 n ln n ln n .Sinceln n 1 n and ∞ 1
is the
n
divergent harmonic series (without the n =1term),
n=2
∞
n=2
n=2
1
is divergent by the Comparison Test. When x =4,
ln n
∞
(−1) n x n
4 n ln n = ∞ (−1) n 1
, which converges by the Alternating Series Test. Thus, I =(−4, 4].
ln n
n=2
n=2

F.
482 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES TX.10
(x − 2)n
15. If a n = ,then lim
n 2 +1 n→∞ a n+1 = lim
(x − 2)n+1
n→∞ (n +1) 2 +1 ·
Ratio Test, the series ∞
n=0
a n
(x − 2) n
n 2 +1
n 2 +1
(x − 2) n
= |x − 2| lim
n→∞
n 2 +1
= |x − 2|. Bythe
(n +1) 2 +1
converges when |x − 2| < 1 [R =1] ⇔ −1

F.
TX.10
SECTION 11.8 POWER SERIES ¤ 483
x n
27. If a n =
1 · 3 · 5 · ··· ·(2n − 1) ,then
lim
n→∞ a n+1 = lim
x n+1
n→∞ 1 · 3 · 5 ·····(2n − 1)(2n +1)
a n
Ratio Test, the series ∞
n=1
1 · 3 · 5 ·····(2n − 1)
· = lim
x n
n→∞
|x|
=0< 1. Thus, by the
2n +1
x n
converges for all real x and we have R = ∞ and I =(−∞, ∞).
1 · 3 · 5 ·····(2n − 1)
29. (a) We are given that the power series ∞
n=0 c nx n is convergent for x =4. So by Theorem 3, it must converge for at least
−4

F.
484 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
37. s 2n−1 =1+2x + x 2 +2x 3 + x 4 +2x 5 + ···+ x 2n−2 +2x 2n−1
=1(1+2x)+x 2 (1 + 2x)+x 4 (1 + 2x)+···+ x 2n−2 (1 + 2x) =(1+2x)(1 + x 2 + x 4 + ···+ x 2n−2 )
=(1+2x) 1 − x2n
[by (11.2.3)] with r = x 2 ] → 1+2x as n →∞ [by (11.2.4)], when |x| < 1.
1 − x 2 1 − x2 Also s 2n = s 2n−1 + x 2n → 1+2x
1 − x since 2 x2n → 0 for |x| < 1. Therefore, s n → 1+2x since s2n and s2n−1 both
1 − x2 approach 1+2x as n →∞. Thus, the interval of convergence is (−1, 1) and f(x) =1+2x
1 − x2 1 − x . 2
39. We use the Root Test on the series
c n x n n
.Weneed lim |cn x n n
| = |x| lim |cn | = c |x| < 1 for convergence, or
n→∞
n→∞
|x| < 1/c,soR =1/c.
41. For 2

F.
SECTION TX.10 11.9 REPRESENTATIONS OF FUNCTIONS AS POWER SERIES ¤ 485
11. f(x) =
3
x 2 − x − 2 = 3
(x − 2)(x +1) = A
x − 2 + B
x +1
⇒
3=A(x +1)+B(x − 2). Letx =2to get A =1and
x = −1 to get B = −1. Thus
3
x 2 − x − 2 = 1
x − 2 − 1
x +1 = 1
1
−2 1 − (x/2)
= ∞ − 1 n
1
− 1(−1) n x n
= ∞
2 2
n=0
−
n=0
1
1 − (−x) = − 1 2
(−1) n+1 − 1
2 n+1
x n
∞ x
n ∞ − (−x) n
n=0 2 n=0
We represented f as the sum of two geometric series; the first converges for x ∈ (−2, 2) and the second converges for (−1, 1).
Thus, the sum converges for x ∈ (−1, 1) = I.
1
13. (a) f(x) =
(1 + x) 2 = d −1
= − d ∞
(−1) n x n [from Exercise 3]
dx 1+x dx n=0
= ∞ (−1) n+1 nx n−1
[from Theorem 2(i)] = ∞ (−1) n (n +1)x n with R =1.
n=1
n=0
In the last step, note that we decreased the initial value of the summation variable n by 1, andthenincreased each
occurrence of n in the term by 1 [also note that (−1) n+2 =(−1) n ].
1
(b) f(x) =
(1 + x) 3 = − 1
d 1
2 dx (1 + x) 2 = − 1
d ∞
(−1) n (n +1)x n [from part (a)]
2 dx n=0
∞
∞
= − 1 (−1) n (n +1)nx n−1 = 1 (−1) n (n +2)(n +1)x n with R =1.
2
2
n=1
n=0
x 2
(c) f(x) =
(1 + x) = 1
3 x2 ·
(1 + x) = 3 x2 · 1
2
= 1 ∞
(−1) n (n +2)(n +1)x n+2
2
n=0
∞
(−1) n (n +2)(n +1)x n
n=0
[from part (b)]
To write the power series with x n rather than x n+2 ,wewilldecrease each occurrence of n in the term by 2 and increase
the initial value of the summation variable by 2. Thisgivesus 1 2
15. f(x) =ln(5− x) =−
17.
dx
5 − x = −1 5
∞
(−1) n (n)(n − 1)x n with R =1.
n=2
dx
1 − x/5 = −1 ∞ x
n
dx = C −
5 n=0 1 5
5
Putting x =0,wegetC =ln5. The series converges for |x/5| < 1 ⇔ |x| < 5,soR =5.
1
2 − x = 1
2(1 − x/2) = 1 ∞ x
2 n=0
2
1
(x − 2) = d 1
= d
∞ 2 dx 2 − x dx
f(x) =
x 3
(x − 2) 2 = ∞
x3
n=0
n
=
∞
n=0
n=0
n +1
2 n+2 xn = ∞
1
2 n+1 xn for x < 1 ⇔ |x| < 2. Now
2
1
2 n+1 xn = ∞
n=0
n=1
n +1
2 n+2 xn+3 or ∞
n
2 n+1 xn−1 = ∞
n=3
n=0
∞
n=0
n +1
2 n+2 xn .So
x n+1
5 n (n +1) = C − ∞
n=1
n − 2
2 n−1 xn for |x| < 2. Thus,R =2and I =(−2, 2).
x n
n 5 n

F.
486 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
x
19. f(x) =
x 2 +16 = x
1
= x ∞
16 1 − (−x 2 /16) 16
n=0
TX.10
− x2
16
n
= x 16
∞
(−1) n 1
16 n x2n = ∞ (−1) n 1
16 n+1 x2n+1 .
The series converges when −x 2 /16 < 1 ⇔ x 2 < 16 ⇔ |x| < 4, soR =4. The partial sums are s 1 = x 16 ,
n=0
n=0
s 2 = s 1 − x3
16 2 , s 3 = s 2 + x5
16 3 , s 4 = s 3 − x7
16 4 , s 5 = s 4 + x9
16 5 , ....Notethats 1 corresponds to the first term of the infinite
sum, regardless of the value of the summation variable and the value of the exponent.
As n increases, s n (x) approximates f better on the interval of convergence, which is (−4, 4).
1+x
21. f(x) =ln
1 − x
∞
= (−1) n x n
+ ∞
n=0
=ln(1+x) − ln(1 − x) =
n=0
= (2 + 2x 2 +2x 4 + ···) dx =
dx
1+x +
dx
1 − x =
dx
1 − (−x) +
dx
1 − x
x n dx = [(1 − x + x 2 − x 3 + x 4 − ···)+(1+x + x 2 + x 3 + x 4 + ···)] dx
∞
n=0
But f(0) = ln 1 1 =0,soC =0and we have f(x) = ∞
2x 2n dx = C + ∞
n=0
n=0
2x 2n+1
2n +1
2x 2n+1
2n +1 with R =1.Ifx = ±1, thenf(x) =±2 ∞
which both diverge by the Limit Comparison Test with b n = 1 n . The partial sums are s 1 = 2x 1 , s 2 = s 1 + 2x3
3 ,
s 3 = s 2 + 2x5
5 , ....
n=0
1
2n +1 ,
As n increases, s n(x) approximates f better on the interval of convergence, which is (−1, 1).

F.
t
23.
1 − t = t · 1
8 1 − t = t ∞ (t 8 ) n
= ∞ 8
n=0
t 8n+1
n=0
SECTION TX.10 11.9 REPRESENTATIONS OF FUNCTIONS AS POWER SERIES ¤ 487
⇒
t
1 − t dt = C + ∞
8
n=0
t 8n+2
8n +2 . Theseriesfor 1
1 − t 8 converges
when t
8 < 1 ⇔ |t| < 1,soR =1for that series and also the series for t/(1 − t 8 ). By Theorem 2, the series for
t
dt also has R =1.
1 − t8 25. By Example 7, tan −1
x = ∞ (−1) n x 2n+1
with R =1,so
n=0 2n +1
x − tan −1 x = x −
x ···
− x3
3 + x5
5 − x7
7 + = x3
3 − x5
5 + x7
7 − ··· = ∞ (−1) n+1 x 2n+1
2n +1 and
x − tan −1 x
x 3 =
∞
(−1) n+1 x 2n−2
2n +1 ,so
n=1
x − tan −1 x
dx = C + ∞ (−1) n+1 x 2n−1
x 3
(2n +1)(2n − 1) = C + ∞ (−1) n+1 x 2n−1
. By Theorem 2, R =1.
4n 2 − 1
n=1
1
27.
1+x = 1
5 1 − (−x 5 ) = ∞ −x
5 n
= ∞ (−1) n x 5n ⇒
n=0
n=0
1
1+x dx = ∞
(−1) n x 5n
dx = C + ∞ (−1) n x 5n+1
5 5n +1 . Thus,
0.2
I =
0
n=0
n=0
1
··· 0.2
1+x dx = x − x6
5 6 + x11
11 −
0
=0.2 − (0.2)6
6
n=1
+ (0.2)11
11
n=1
− ···. The series is alternating, so if we use
the first two terms, the error is at most (0.2) 11 /11 ≈ 1.9 × 10 −9 .SoI ≈ 0.2 − (0.2) 6 /6 ≈ 0.199989 to six decimal places.
29. We substitute 3x for x in Example 7, and find that
x arctan(3x) dx = x ∞ (−1) n (3x) 2n+1 ∞
n=0 2n +1 dx =
(−1) n 3 2n+1 x 2n+2
dx = C + ∞ (−1) n 3 2n+1 x 2n+3
n=0 2n +1
n=0 (2n +1)(2n +3)
So
0.1
0
3x
3
x arctan(3x) dx =
1 · 3 − 33 x 5
3 · 5 + 35 x 7
5 · 7 − 37 x ··· 9
0.1
7 · 9 + 0
The series is alternating, so if we use three terms, the error is at most
0.1
0
= 1
10 3 − 9
5 × 10 5 + 243
35 × 10 7 − 2187
63 × 10 9 + ···.
2187
63 × 10 9 ≈ 3.5 × 10−8 .So
x arctan(3x) dx ≈ 1
10 − 9
3 5 × 10 + 243 ≈ 0.000 983 to six decimal places.
5 35 × 107 31. Using the result of Example 6, ln(1 − x) =− ∞
n=1
x n
,withx = −0.1, wehave
n
ln 1.1 =ln[1− (−0.1)] = 0.1 − 0.01
2 + 0.001 − 0.0001 + 0.00001 − ···. The series is alternating, so if we use only
3 4 5
the first four terms, the error is at most 0.00001
5
=0.000002. Soln 1.1 ≈ 0.1 − 0.01
2 + 0.001 − 0.0001 ≈ 0.09531.
3 4

F.
488 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
33. (a) J 0 (x) = ∞
(b)
1
0
n=0
(−1) n x 2n
2 2n (n!) 2 ,J0 0(x) = ∞
n=1
x 2 J 00
0 (x)+xJ 0 0(x)+x 2 J 0(x)= ∞
J 0 (x) dx =
1
0
∞
n=0
n=1
= ∞
n=1
= ∞
n=1
(−1) n 2nx 2n−1
2 2n (n!) 2 ,andJ 00
0 (x) = ∞
(−1) n 2n(2n − 1)x 2n
2 2n (n!) 2 + ∞
n=1
(−1) n 2n(2n − 1)x 2n
2 2n (n!) 2 + ∞
n=1
(−1) n 2n(2n − 1)x 2n
2 2n (n!) 2 + ∞
n=1
n=1
= ∞ 2n(2n − 1) + 2n − 2
(−1) n 2 n 2
2 2n (n!) 2
n=1
(−1) n 2n(2n − 1)x 2n−2
2 2n (n!) 2 ,so
(−1) n 2nx 2n
2 2n (n!) 2 + ∞
n=0
(−1) n 2nx 2n
2 2n (n!) 2 + ∞
n=1
(−1) n 2nx 2n
2 2n (n!) 2 + ∞
x 2n
= ∞ 4n
(−1) n 2 − 2n +2n − 4n 2
x 2n =0
2 2n (n!) 2
n=1
(−1) n x 2n
2 2n (n!) 2
dx =
1
0
1 ···
− x2
4 + x4
64 − x6
2304 + dx
=
x ··· 1
− x3
3 · 4 + x5
5 · 64 − x7
7 · 2304 + =1− 1
0
12 + 1
320 − 1
16,128 + ···
n=1
(−1) n x 2n+2
2 2n (n!) 2
(−1) n−1 x 2n
2 2n−2 [(n − 1)!] 2
(−1) n (−1) −1 2 2 n 2 x 2n
2 2n (n!) 2
Since
1
0
1
≈ 0.000062, it follows from The Alternating Series Estimation Theorem that, correct to three decimal places,
16,128
J0(x) dx ≈ 1 −
1
12 + 1
320 ≈ 0.920.
35. (a) f(x) = ∞
n=0
x n
n!
⇒
f 0 (x) = ∞
n=1
nx n−1
n!
= ∞
n=1
x n−1
(n − 1)! = ∞
n=0
x n
n! = f(x)
(b) By Theorem 9.4.2, the only solution to the differential equation df(x)/dx = f(x) is f(x) =Ke x ,butf(0) = 1,
so K =1and f(x) =e x .
Or: We could solve the equation df(x)/dx = f(x) as a separable differential equation.
37. If a n = xn
,thenbytheRatioTest, lim
n2 n→∞ a
n+1 = lim
x n+1
a n n→∞ (n +1) · n2
= |x| lim
2 x n n→∞
∞
convergence, so R =1.Whenx = ±1,
xn
= ∞
n 2
n=1
n=1
n
n +1
2
= |x| < 1 for
1
which is a convergent p-series (p =2> 1), so the interval of
n2 convergence for f is [−1, 1]. By Theorem 2, the radii of convergence of f 0 and f 00 are both 1, so we need only check the
endpoints. f(x) = ∞
n=1
x n
⇒ f 0
(x) = ∞
n 2
n=1
nx n−1
n 2
= ∞
n=0
x n
, and this series diverges for x =1(harmonic series)
n +1
and converges for x = −1 (Alternating Series Test), so the interval of convergence is [−1, 1).
f 00 (x) = ∞
n
at both 1 and −1 (Test for Divergence) since lim =16= 0, so its interval of convergence is (−1, 1).
n→∞ n +1
n=1
nx n−1
n +1 diverges

F.
TX.10
SECTION 11.10 TAYLOR AND MACLAURIN SERIES ¤ 489
39. By Example 7, tan −1
x = ∞ (−1) n x 2n+1
for |x| < 1. In particular, for x =
1 2n +1
n=0
have π
√ 2n+1
1√3
6 =tan−1 = ∞ 1/ 3
(−1) n 2n +1
π =
6 √
3
∞
n=0
n=0
(−1) n
(2n +1)3 =2√
3 ∞ n
n=0
(−1) n
(2n +1)3 n .
√
3
,we
=
n=0(−1) ∞ n 1 n 1 1 √3
3 2n +1 ,so
11.10 Taylor and Maclaurin Series
1. UsingTheorem5with ∞
n=0
b n(x − 5) n , b n = f (n) (a)
n!
,sob 8 = f (8) (5)
.
8!
3. Since f (n) (0) = (n +1)!, Equation 7 gives the Maclaurin series
∞
n=0
lim
n→∞
f (n) (0)
x n
= ∞
n!
n=0
= lim
a n+1
a n
n→∞
(n +1)!
n!
radius of convergence R =1.
x n
= ∞ (n +1)x n . Applying the Ratio Test with a n =(n +1)x n gives us
n=0
(n +2)xn+1 n +2
= |x| lim = |x|·1=|x|. For convergence, we must have |x| < 1,sothe
(n +1)x n n→∞ n +1
5.
n f (n) (x) f (n) (0)
0 (1 − x) −2 1
1 2(1 − x) −3 2
2 6(1 − x) −4 6
3 24(1 − x) −5 24
4 120(1 − x) −6 120
.
.
.
(1 − x) −2 = f(0) + f 0 (0)x + f 00 (0)
2!
a n
x 2 + f 000 (0)
3!
=1+2x + 6 2 x2 + 24 6 x3 + 120
24 x4 + ···
x 3 + f (4) (0)
x 4 + ···
4!
=1+2x +3x 2 +4x 3 +5x 4
+ ···= ∞ (n +1)x n
lim
n→∞ a
n+1 = lim
(n +2)xn+1 n +2
n→∞ = |x| lim = |x| (1) = |x| < 1
(n +1)x n n→∞ n +1
for convergence, so R =1.
n=0
7.
n f (n) (x) f (n) (0)
0 sin πx 0
1 π cos πx π
2 −π 2 sin πx 0
3 −π 3 cos πx −π 3
4 π 4 sin πx 0
5 π 5 cos πx π 5
.
.
.
sin πx = f(0) + f 0 (0)x + f 00 (0)
2!
n→∞
+ f (4) (0)
4!
x 2 + f 000 (0)
x 3
3!
x 4 + f (5) (0)
x 5 + ···
5!
=0+πx +0− π3
3! x3 +0+ π5
5! x5 + ···
= πx − π3
3! x3 + π5
5! x5 − π7
7! x7 + ···
= ∞
a n
n=0
(−1) n π 2n+1
(2n +1)! x2n+1
lim
a n+1 = lim
n→∞ π2n+3 x 2n+3
·
(2n +3)!
=0< 1 for all x,soR = ∞.
(2n +1)!
π 2n+1 x 2n+1
= lim
n→∞
π 2 x 2
(2n +3)(2n +2)

F.
490 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
9.
n f (n) (x) f (n) (0)
0 e 5x 1
1 5e 5x 5
2 5 2 e 5x 25
3 5 3 e 5x 125
4 5 4 e 5x 625
.
.
.
e 5x = ∞
n=0
lim
n→∞ an+1
a n
for all x,soR = ∞.
f (n) (0)
x n
= ∞ 5 n
n!
n=0 n! xn .
5 n+1 |x| n+1
= lim
·
n→∞ (n +1)!
n!
5 n |x| n = lim
n→∞
5 |x|
n +1 =0< 1
11.
n f (n) (x) f (n) (0)
0 sinh x 0
1 cosh x 1
2 sinh x 0
3 cosh x 1
4 sinh x 0
.
.
.
f (n) (0) =
0 if n is even
1 if n is odd
Use the Ratio Test to find R. Ifa n =
lim
n→∞ an+1
a n
= lim
n→∞
x 2n+3
(2n +3)!
so sinh x = ∞
n=0
x2n+1
(2n +1)! ,then
· (2n +1)!
x 2n+1
=0< 1 for all x,soR = ∞.
x 2n+1
(2n +1)! .
= x 2 · lim
n→∞
1
(2n + 3)(2n +2)
13.
n f (n) (x) f (n) (1)
0 x 4 − 3x 2 +1 −1
1 4x 3 − 6x −2
2 12x 2 − 6 6
3 24x 24
4 24 24
5 0 0
6 0 0
.
.
.
f (n) (x) =0for n ≥ 5,sof has a finite series expansion about a =1.
f(x)=x 4 − 3x 2 +1=
4
n=0
f (n) (1)
(x − 1) n
n!
= −1
0! (x − 1)0 + −2
1! (x − 1)1 + 6 2! (x − 1)2 + 24
3! (x − 1)3 + 24 (x − 1)4
4!
= −1 − 2(x − 1) + 3(x − 1) 2 +4(x − 1) 3 +(x − 1) 4
A finite series converges for all x,soR = ∞.
15. f(x) =e x ⇒ f (n) (x) =e x ,sof (n) (3) = e 3 and e x = ∞
lim
n→∞ an+1
a n
= lim
n→∞
e3 (x − 3) n+1
·
(n +1)!
n=0
e 3
n! (x − 3)n .Ifa n = e3
n! (x − 3)n ,then
n! |x − 3|
= lim =0< 1 for all x,soR = ∞.
e 3 (x − 3) n n→∞ n +1
17.
n f (n) (x) f (n) (π)
0 cos x −1
1 − sin x 0
2 − cos x 1
3 sin x 0
4 cos x −1
.
.
.
cos x = ∞
k=0
= ∞
n=0
lim
n→∞ an+1
a n
for all x,soR = ∞.
f (k) (π)
(x − π) k (x − π)2 (x − π)4 (x − π)6
= −1+ − + − ···
k!
2! 4! 6!
(−1) n+1 (x − π) 2n
(2n)!
|x − π|
2n+2
= lim
·
n→∞ (2n +2)!
(2n)!
|x − π| 2n = lim
n→∞
|x − π| 2
(2n + 2)(2n +1) =0< 1

F.
TX.10
SECTION 11.10 TAYLOR AND MACLAURIN SERIES ¤ 491
19.
n f (n) (x) f (n) (9)
0 x −1/2 1 3
1 − 1 2 x−3/2 − 1 · 1
2 3 3
3
2
4 x−5/2 − 1 · − 3
2 2 · 1
3 5
3 − 15 8 x−7/2 − 1 · − 3
2 2 · −
5
2 · 1
3 7
.
.
.
lim
n→∞ an+1
a n
1
√ = 1 x 3 − 1
3 (x − 9) 2
(x − 9) +
2 · 33 2 2 · 3 5 2!
− 3 · 5 (x − 9) 3
+ ···
2 3 · 3 7 3!
= 1 3 + ∞
n=1
1 · 3 · 5 · ··· ·(2n − 1)[2(n +1)− 1] |x − 9|
n+1
= lim
·
n→∞
2 n+1 · 3 [2(n+1)+1] · (n +1)!
(2n +1)|x − 9|
= lim
= 1 |x − 9| < 1
n→∞ 2 · 3 2 (n +1) 9
(−1) n 1 · 3 · 5 · ··· ·(2n − 1)
2 n · 3 2n+1 · n!
(x − 9) n .
2 n · 3 2n+1 · n!
1 · 3 · 5 ·····(2n − 1) |x − 9| n
for convergence, so |x − 9| < 9 and R =9.
21. If f(x) =sinπx,thenf (n+1) (x) =±π n+1 sin πx or ±π n+1
cos πx. In each case, f (n+1) (x) ≤ π n+1 ,sobyFormula9
with a =0and M = π n+1 , |R n (x)| ≤
πn+1
(n +1)! |x|n+1 = |πx|n+1
(n +1)! . Thus, |R n(x)| → 0 as n →∞by Equation 10.
So lim Rn(x) =0and, by Theorem 8, the series in Exercise 7 represents sin πx for all x.
n→∞
23. If f(x) =sinhx, thenforalln, f (n+1) (x) =coshx or sinh x. Since|sinh x| < |cosh x| =coshx for all x,wehave
f (n+1) (x) ≤ cosh x for all n. Ifd is any positive number and |x| ≤ d, then
Formula 9 with a =0and M =coshd,wehave|R n(x)| ≤
f (n+1) (x) ≤ cosh x ≤ cosh d,soby
cosh d
(n +1)! |x|n+1 . It follows that |R n(x)| → 0 as n →∞for
|x| ≤ d (by Equation 10). But d was an arbitrary positive number. So by Theorem 8, the series represents sinh x for all x.
25. The general binomial series in (17) is
(1 + x) k
= ∞ k
x n =1+kx +
n
n=0
(1 + x) 1/2
= ∞ 1
2
n
n=0
k(k − 1)
x 2 +
2!
x n =1+ 1
2
k(k − 1)(k − 2)
x 3 + ···.
3!
1
x +
2 −
1
2
x 2 +
2!
1
2 −
1
2
3!
=1+ x 2 − x2
2 2 · 2! + 1 · 3 · x3
− 1 · 3 · 5 · x4
+ ···
2 3 · 3! 2 4 · 4!
=1+ x 2 + ∞
n=2
(−1) n−1 1 · 3 · 5 ·····(2n − 3)x n
2 n · n!
−
3
2
x 3 + ···
for |x| < 1, soR =1.

F.
492 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES TX.10
1
27.
(2 + x) 3 = 1
[2(1 + x/2)] 3 = 1
1+ x −3 1 ∞ −3 x
n.
= The binomial coefficient is
8 2 8 n=0 n 2
Thus,
1
(2 + x) 3 = 1 8
−3
=
n
∞
n=0
(−3)(−4)(−5) ·····(−3 − n +1)
n!
= (−1)n · 2 · 3 · 4 · 5 ·····(n +1)(n +2)
2 · n!
=
(−3)(−4)(−5) ·····[−(n +2)]
n!
= (−1)n (n +1)(n +2)
2
(−1) n (n +1)(n +2) x n
2 2 = ∞ (−1) n (n +1)(n +2)x n
for
n n=0 2 n+4
x < 1 ⇔ |x| < 2,soR =2.
2
29. sin x = ∞ (−1) n x 2n+1
(2n +1)!
n=0
⇒
f(x) =sin(πx) = ∞ (−1) n (πx) 2n+1
(2n +1)! = ∞ (−1) n π 2n+1
(2n +1)! x2n+1 , R = ∞.
n=0
n=0
31. e x = ∞
n=0
R = ∞.
x n
n!
⇒
e 2x = ∞
n=0
(2x) n
n!
= ∞
n=0
2 n x n
,sof(x) =e x + e 2x
= ∞ 1
n!
n=0 n! xn + ∞ 2 n
n=0 n! xn = ∞ 2 n +1
x n ,
n=0 n!
33. cos x = ∞ (−1) n x 2n
(2n)!
n=0
⇒ cos
1
x2
= ∞ 1
(−1) n x2 2n
2
2
(2n)!
n=0
f(x) =x cos 1
x2
= ∞ (−1) n 1
2
2 2n (2n)! x4n+1 , R = ∞.
n=0
= ∞ (−1) n
n=0
x 4n
2 2n (2n)! ,so
35. We must write the binomial in the form (1+ expression), so we’ll factor out a 4.
x
√ = x
4+x
2 4(1 + x2 /4) =
x
2 1+x 2 /4 = x 2
= x 1+
− 1 x 2 −
1
2
2
4 + 2 −
3
2 x
2
2
+
2! 4
= x 2 + x 2
∞
(−1) n 1 · 3 · 5 ·····(2n − 1)
2 n · 4 n · n!
n=1
−1/2 1+ x2
= x 4 2
x 2n
−
1
2
−
3
2
3!
−
5
2
x
∞ − 1 2
2
n 4
n=0
x
2
3
+ ···
= x 2 + ∞ (−1) n 1 · 3 · 5 ·····(2n − 1)
x 2n+1 and x2
n!2 3n+1 4 < 1 ⇔ |x| < 1 ⇔ |x| < 2, soR =2.
2
n=1
37. sin 2 x = 1 2 (1 − cos 2x) =1 2
R = ∞
1 − ∞
n=0
(−1) n (2x) 2n
= 1
1 − 1 − ∞
(2n)! 2
n=1
4
n
(−1) n (2x) 2n
= ∞
(2n)!
n=1
(−1) n+1 2 2n−1 x 2n
,
(2n)!
39. cos x = ∞ (−1) n x 2n
(2n)!
n=0
⇒
f(x) =cos x 2 = ∞
n=0
(−1) n x 2 2n
(2n)!
= ∞
n=0
(−1) n x 4n
, R = ∞
(2n)!
Notice that, as n increases, T n (x)
becomes a better approximation to f(x).

F.
TX.10
SECTION 11.10 TAYLOR AND MACLAURIN SERIES ¤ 493
41. e x (11)
= ∞
n=0
x n
n! ,soe−x = ∞
n=0
(−x) n
n!
f(x)=xe −x
= ∞ (−1) n 1
n=0
= ∞ (−1) n x n
n! ,so
n=0
n! xn+1
= x − x 2 + 1 2 x3 − 1 6 x4 + 1 24 x5 − 1
120 x6 + ···
= ∞ (−1) n−1 x n
(n − 1)!
n=1
The series for e x converges for all x, so the same is true of the series
for f(x);thatis,R = ∞. From the graphs of f and the first few Taylor
polynomials, we see that T n(x) provides a closer fittof(x) near 0 as n increases.
43. e x = ∞
n=0
x n
n! ,soe−0.2 = ∞
n=0
(−0.2) n
n!
=1− 0.2+ 1 2! (0.2)2 − 1 3! (0.2)3 + 1 4! (0.2)4 − 1 5! (0.2)5 + 1 6! (0.2)6 − ···.
But 1 6! (0.2)6 =8.8 × 10 −8 , so by the Alternating Series Estimation Theorem, e −0.2 ≈
5
n=0
(−0.2) n
n!
≈ 0.81873,correctto
five decimal places.
45. (a) 1/ √ 1 − x 2 = 1+ −x 2−1/2 =1+
−
1
2 −x
2 −
1
2 −
3
2
+
−x
2 2 +
2!
=1+ ∞ 1 · 3 · 5 ·····(2n − 1)
x 2n
2 n · n!
(b) sin −1 x =
= x + ∞
n=1
1
√ dx = C + x + ∞
1 − x
2
n=1
n=1
1 · 3 · 5 ·····(2n − 1)
x 2n+1
(2n +1)2 n · n!
1 · 3 · 5 ·····(2n − 1)
x 2n+1
(2n +1)2 n · n!
since 0=sin −1 0=C.
−
1
2
−
3
2
3!
−
5
2
−x
2 3 + ···
47. cos x (16)
= ∞ (−1) n x 2n
(2n)!
n=0
x cos(x 3
)= ∞ (−1) n x 6n+1
(2n)!
n=0
⇒
cos(x 3 )= ∞ (−1) n (x 3 ) 2n
= ∞ (−1) n x 6n
⇒
n=0 (2n)! n=0 (2n)!
⇒ x cos(x 3
) dx = C + ∞ (−1) n x 6n+2
,withR = ∞.
(6n + 2)(2n)!
n=0
49. cos x (16)
= ∞ (−1) n x 2n
(2n)!
n=0
⇒
cos x − 1= ∞ (−1) n x 2n
(2n)!
n=1
⇒ cos x − 1
x
= ∞ (−1) n x 2n−1
(2n)!
n=1
⇒
cos x − 1
dx = C + ∞ (−1) n x 2n
,withR = ∞.
x
2n · (2n)!
51. By Exercise 47,
1
0
n=1
x cos(x 3
) dx = C + ∞ (−1) n x 6n+2
(6n +2)(2n)! ,so
n=0
∞
x cos(x 3
) dx = (−1) n x 6n+2 1
= ∞ (−1) n
n=0 (6n + 2)(2n)!
0 n=0 (6n + 2)(2n)! = 1 2 − 1
8 · 2! + 1
14 · 4! − 1
20 · 6! + ···,but
1
20 · 6! = 1
14,400 ≈ 0.000 069, so 1
Alternating Series Estimation Theorem.
0
x cos(x 3 ) dx ≈ 1 2 − 1 16 + 1 ≈ 0.440 (correct to three decimal places) by the
336

F.
494 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES TX.10
√
53. 1+x4 =(1+x 4 ) 1/2
= ∞ 1/2
1+x4
(x 4 ) n
,so
dx = C + ∞ 1/2 x 4n+1
and hence, since 0.4 < 1,
n
n 4n +1
we have
I =
0.4
0
=(1) (0.4)1
0!
n=0
1+x4 dx = ∞ 1/2 (0.4) 4n+1
n 4n +1
+
=0.4+ (0.4)5
10
n=0
1
2 (0.4) 5
+
1! 5
− (0.4)9
72
1
2
−
1
2 (0.4) 9
+
2! 9
+ (0.4)13
208
1
2
− 5(0.4)17
2176
−
1
2
−
3
2
3!
+ ···
n=0
(0.4) 13
13
+
1
2
−
1
2
−
3
2
4!
−
5
2
(0.4) 17
17
+ ···
Now (0.4)9
72
≈ 3.6 × 10 −6 < 5 × 10 −6 ,sobytheAlternatingSeriesEstimationTheorem,I ≈ 0.4+ (0.4)5
10
(correct to five decimal places).
x − tan −1 x x − x − 1 3
55. lim
= lim
x3 + 1 5 x5 − 1 7 x7 1
+ ···
3
=lim
x3 − 1 5 x5 + 1 7 x7 − ···
x→0 x 3
x→0 x 3
x→0 x 3
= lim
x→0
1
3 − 1 5 x2 + 1 7 x4 − ··· = 1 3
since power series are continuous functions.
≈ 0.40102
57. lim
x→0
sin x − x + 1 6 x3
x 5
since power series are continuous functions.
x −
1
3!
=lim
x3 + 1 5! x5 − 1 7! x7 + ··· − x + 1 6 x3
x→0 x 5
1
5!
=lim
x5 − 1
7! x7 + ···
···
1
=lim
x→0 x 5
x→0 5! − x2
7! + x4
9! − = 1 5! = 1
120
59. From Equation 11, we have e −x2 =1− x2
1! + x4
2! − x6
x2
+ ··· and we know that cos x =1−
3! 2! + x4
− ··· from
4!
61.
Equation 16. Therefore, e −x2 cos x = 1 − x 2 + 1 2 x4 − ··· 1 − 1 2 x2 + 1 24 x4 − ···.
Writing only the terms with
degree ≤ 4,wegete −x2 cos x =1− 1 2 x2 + 1 24 x4 − x 2 + 1 2 x4 + 1 2 x4 + ···=1− 3 2 x2 + 25
24 x4 + ···.
x (15) x
=
sin x x − 1 6 x3 + 1
120 x5 − ···.
x − 1 6 x3 + 1
120 x5 − ··· x
1+ 1 6 x2 + 7
360 x4 + ···
x − 1 6 x3 + 1
120 x5 − ···
1
6 x3 − 1
120 x5 + ···
1
6 x3 − 1 36 x5 + ···
7
360 x5 + ···
7
360 x5 + ···
x
From the long division above,
sin x =1+1 6 x2 + 7
360 x4 + ···.
∞
63. (−1) n x 4n
= ∞ −x
4 n
= e −x4 ,by(11).
n! n!
n=0
n=0
···

F.
TX.10
SECTION 11.10 TAYLOR AND MACLAURIN SERIES ¤ 495
65.
∞
n=0
(−1) n π 2n+1
4 2n+1 (2n +1)! = ∞
n=0
(−1) n
π 2n+1
4
=sin π 4
(2n +1)!
= √ 1
2
, by (15).
67. 3+ 9 2! + 27
3! + 81 31
+ ···=
4! 1! + 32
2! + 33
3! + 34
4! + ···= ∞ 3 n
n=1 n! = ∞ 3 n
n=0 n! − 1=e3 − 1, by (11).
69. Assume that |f 000 (x)| ≤ M, sof 000 (x) ≤ M for a ≤ x ≤ a + d. Now x
a f 000 (t) dt ≤ x
a Mdt
f 00 (x) − f 00 (a) ≤ M(x − a) ⇒ f 00 (x) ≤ f 00 (a)+M(x − a). Thus, x
a f 00 (t) dt ≤ x
a [f 00 (a)+M(t − a)] dt
f 0 (x) − f 0 (a) ≤ f 00 (a)(x − a)+ 1 M(x − 2 a)2 ⇒ f 0 (x) ≤ f 0 (a)+f 00 (a)(x − a)+ 1 M(x − 2 a)2 ⇒
x f 0 (t) dt ≤ x
a a f 0 (a)+f 00 (a)(t − a)+ 1 M(t − a)2 dt ⇒
2
f(x) − f(a) ≤ f 0 (a)(x − a) + 1 2 f 00 (a)(x − a) 2 + 1 6 M(x − a)3 .So
⇒
⇒
f(x) − f(a) − f 0 (a)(x − a) − 1 2 f 00 (a)(x − a) 2 ≤ 1 6 M(x − a)3 .But
R 2 (x) =f(x) − T 2 (x) =f(x) − f(a) − f 0 (a)(x − a) − 1 2 f 00 (a)(x − a) 2 ,soR 2 (x) ≤ 1 6 M(x − a)3 .
A similar argument using f 000 (x) ≥−M shows that R 2 (x) ≥− 1 6 M(x − a)3 .So|R 2 (x 2 )| ≤ 1 6 M |x − a|3 .
Although we have assumed that x>a, a similar calculation shows that this inequality is also true if x

F.
496 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
11.11 Applications of Taylor Polynomials
1. (a)
n f (n) (x) f (n) (0) T n (x)
0 cos x 1 1
1 − sin x 0 1
2 − cos x −1 1 − 1 2 x2
3 sin x 0 1 − 1 2 x2
4 cos x 1 1 − 1 2 x2 + 1
24 x4
5 − sin x 0 1 − 1 2 x2 + 1
24 x4
6 − cos x −1 1 − 1 2 x2 + 1
24 x4 − 1
720 x6
(b)
x f T 0 = T 1 T 2 = T 3 T 4 = T 5 T 6
π
4
0.7071 1 0.6916 0.7074 0.7071
π
2
0 1 −0.2337 0.0200 −0.0009
π −1 1 −3.9348 0.1239 −1.2114
(c) As n increases, T n (x) is a good approximation to f(x) on a larger and larger interval.
3.
n f (n) (x) f (n) (2)
0 1/x
1
2
T 3(x)=
1 −1/x 2 − 1 4
2 2/x 3 1 4
3 −6/x 4 − 3 8
3
n=0
f (n) (2)
n!
(x − 2) n
1 1
1
3
2
=
0! − 4
1! (x − 2) + 4
2! (x − 2)2 8
− (x − 2)3
3!
= 1 − 1 (x − 2) + 1 (x − 2 4 8 2)2 − 1 (x − 2)3
16
5.
n f (n) (x) f (n) (π/2)
0 cos x 0
1 − sin x −1
2 − cos x 0
3 sin x 1
T 3(x)=
3
n=0
= − x − π 2
f (n) (π/2)
n!
+
1
6
x −
π n
2
x −
π 3
2

F.
TX.10
SECTION 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS ¤ 497
7.
n f (n) (x) f (n) (0)
0 arcsin x 0
1
1
√
1 − x
2
1
2
3
x
(1 − x 2 ) 3/2 0
2x 2 +1
(1 − x 2 ) 5/2 1
T 3(x) =
3
n=0
f (n) (0)
n!
x n = x + x3
6
9.
n f (n) (x) f (n) (0)
0 xe −2x 0
1 (1 − 2x)e −2x 1
2 4(x − 1)e −2x −4
3 4(3 − 2x)e −2x 12
T 3 (x) =
3
n=0
f (n) (0)
x n = 0 1
n! · 1+ 1 1 x1 + −4
2 x2 + 12 6 x3 = x − 2x 2 +2x 3
11. You may be able to simply find the Taylor polynomials for
f(x) =cotx using your CAS. We will list the values of f (n) (π/4)
for n =0to n =5.
n 0 1 2 3 4 5
f (n) (π/4) 1 −2 4 −16 80 −512
T 5 (x)=
5
n=0
f (n) (π/4)
x −
π n
n!
4
=1− 2 x − π 4
+2
x −
π
4
2
− 8 3
x −
π
4
3
+ 10
3
x −
π
4
4
− 64
15
x −
π 5
4
For n =2to n =5, T n(x) is the polynomial consisting of all the terms up to and including the x − π 4
n
term.
13.
(a) f(x) = √ x ≈ T 2 (x)=2+ 1 1/32
(x − 4) − (x − 4) 2
n f (n) (x) f (n) 4 2!
(4)
3
3
8 x−5/2 on [4, 4.2], we can take M = |f 000 (4)| = 3 8 4−5/2 = 3 256
0
√ x 2
=2+ 1 1
(x − 4) − (x − 4)2
4 64
1
1
2 x−1/2 1 4
(b) |R 2 (x)| ≤ M 3! |x − 4|3 ,where|f 000 (x)| ≤ M. Now 4 ≤ x ≤ 4.2 ⇒
2 − 1 4 x−3/2 − 1 32
|x − 4| ≤ 0.2 ⇒ |x − 4| 3 ≤ 0.008. Sincef 000 (x) is decreasing
|R 2 (x)| ≤ 3/256 (0.008) = 0.008 =0.000 015 625.
6
512

F.
498 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
(c)
From the graph of |R 2 (x)| = | √ x − T 2 (x)|,itseemsthatthe
error is less than 1.52 × 10 −5 on [4, 4.2].
15.
(c)
(a) f(x) =x 2/3 ≈ T 3 (x)=1+ 2 2/9
(x − 1) −
n f (n) (x) f (n) 3
2! (x − 1)2 + 8/27 (x − 1) 3
3!
(1)
=1+ 2
0 x 2/3 1
(x − 1) − 1 (x − 3 9 1)2 + 4 81
(x − 1)3
2
1
3 x−1/3 2 (b) |R
3
3(x)| ≤ M 4! |x − 1|4 ,where f (4) (x) ≤ M. Now0.8 ≤ x ≤ 1.2 ⇒
2 − 2 9 x−4/3 − 2 9
|x − 1| ≤ 0.2 ⇒ |x − 1| 4
≤ 0.0016. Sincef (4) (x) is decreasing
8
3
27 x−7/3 8 27
4 − 56
on [0.8, 1.2],wecantakeM = f (4) (0.8) = 56
81 x−10/3 81 (0.8)−10/3 ,so
|R 3 (x)| ≤
56
81 (0.8)−10/3
24
(0.0016) ≈ 0.000 096 97.
From the graph of |R 3(x)| = x 2/3
− T 3(x) , it seems that the
errorislessthan0.000 053 3 on [0.8, 1.2].
17.
n f (n) (x) f (n) (0)
0 sec x 1
1 sec x tan x 0
2 sec x (2 sec 2 x − 1) 1
3 sec x tan x (6 sec 2 x − 1)
(a) f(x) =secx ≈ T 2 (x) =1+ 1 2 x2
(b) |R 2(x)| ≤ M 3! |x|3 ,where f (3) (x) ≤ M. Now−0.2 ≤ x ≤ 0.2 ⇒ |x| ≤ 0.2 ⇒ |x| 3 ≤ (0.2) 3 .
f (3) (x) is an odd function and it is increasing on [0, 0.2] since sec x and tan x are increasing on [0, 0.2],
so f (3) (x) ≤ f (3) (0.2) ≈ 1.085 158 892. Thus,|R 2 (x)| ≤ f (3) (0.2)
(0.2) 3 ≈ 0.001 447.
3!
(c)
From the graph of |R 2(x)| = |sec x − T 2(x)|, it seems that the
error is less than 0.000 339 on [−0.2, 0.2].

F.
TX.10
SECTION 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS ¤ 499
19.
n f (n) (x) f (n) (0)
0 e x2 1
(a) f(x) =e x2 ≈ T 3(x) =1+ 2 2! x2 =1+x 2
(b) |R 3 (x)| ≤ M
4! |x|4 ,wheref (4) (x) ≤ M. Now0 ≤ x ≤ 0.1
⇒
1 e x2 (2x) 0
2 e x2 (2 + 4x 2 ) 2
3 e x2 (12x +8x 3 ) 0
4 e x2 (12 + 48x 2 +16x 4 )
x 4 ≤ (0.1) 4 , and letting x =0.1 gives
|R 3(x)| ≤ e0.01 (12 + 0.48 + 0.0016)
(0.1) 4 ≈ 0.00006.
24
(c)
From the graph of |R 3 (x)| = e x2 − T 3 (x) , it appears that the
error is less than 0.000 051 on [0, 0.1].
21.
n f (n) (x) f (n) (0)
0 x sin x 0
1 sin x + x cos x 0
2 2cosx − x sin x 2
3 −3sinx − x cos x 0
4 −4cosx + x sin x −4
5 5sinx + x cos x
(a) f(x) =x sin x ≈ T 4 (x) = 2 2! (x − 0)2 + −4
4! (x − 0)4 = x 2 − 1 6 x4
(b) |R 4 (x)| ≤ M 5! |x|5 ,where f (5) (x) ≤ M. Now−1 ≤ x ≤ 1
|x| ≤ 1, and a graph of f (5) (x) shows that f (5) (x) ≤ 5 for −1 ≤ x ≤ 1.
Thus, we can take M =5and get |R 4 (x)| ≤ 5 5! · 15 = 1
24 =0.0416.
⇒
(c)
From the graph of |R 4(x)| = |x sin x − T 4(x)|, it seems that the
error is less than 0.0082 on [−1, 1].
23. From Exercise 5, cos x = −
x − π 2 +
1
6 x −
π 3
2
+ R 3(x), where|R 3(x)| ≤ M x −
π 4
2
with
4!
f (4) (x) = |cos x| ≤ M =1.Nowx =80 ◦ =(90 ◦ − 10 ◦ )= π
− π
2 18 =
4π
radians, so the error is
9
R 4π
3
≤ 1 π
4
≈ 0.000 039, which means our estimate would not be accurate to five decimal places. However,
9 24 18
T 3 = T 4 ,sowecanuse R 4
4π
9
cos 80 ◦ ≈− − π 18
+
1
6 −
π 3
18
≈ 0.17365.
25. All derivatives of e x are e x ,so|R n(x)| ≤
≤ 1
120
π
18
5
≈ 0.000 001. Therefore, to five decimal places,
e x
(n +1)! |x|n+1 ,where0

F.
500 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
R 3(0.1) < 0.0000046. Thus, by adding the four terms of the Maclaurin series for e x corresponding to n =0, 1, 2,and3,
we can estimate e 0.1 to within 0.00001. (In fact, this sum is 1.10516 and e 0.1 ≈ 1.10517.)
27. sin x = x − 1 3! x3 + 1 5! x5 − ···. By the Alternating Series
Estimation Theorem, the error in the approximation
sin x = x − 1 3! x3 is less than
1
5! x5 < 0.01 ⇔
x
5 < 120(0.01) ⇔ |x| < (1.2) 1/5 ≈ 1.037. The curves
y = x − 1 6 x3 and y =sinx − 0.01 intersect at x ≈ 1.043,so
the graph confirms our estimate. Since both the sine function
and the given approximation are odd functions, we need to check the estimate only for x>0. Thus, the desired range of
values for x is −1.037

F.
TX.10
SECTION 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS ¤ 501
35. (a) If the water is deep, then 2πd/L is large, and we know that tanh x → 1 as x →∞. So we can approximate
tanh(2πd/L) ≈ 1,andsov 2 ≈ gL/(2π) ⇔ v ≈ gL/(2π).
(b) From the table, the first term in the Maclaurin series of
tanh x is x,soifthewaterisshallow,wecanapproximate
tanh 2πd
L
≈ 2πd
L ,andsov2 ≈ gL
2π · 2πd
L
⇔
v ≈ √ gd.
n f (n) (x) f (n) (0)
0 tanh x 0
1 sech 2 x 1
2 −2sech 2 x tanh x 0
3 2sech 2 x (3 tanh 2 x − 1) −2
(c) Since tanh x is an odd function, its Maclaurin series is alternating, so the error in the approximation
tanh 2πd
L
≈ 2πd
L is less than the first neglected term, which is |f 000 3
(0)| 2πd
= 1 3 2πd
.
3! L 3 L
If L>10d, then 1 3 2πd
< 1 3
1
2π · = π3
3 L 3 10 375 , so the error in the approximation v2 = gd is less
than gL
2π · π 3
375 ≈ 0.0132gL.
37. (a) L is the length of the arc subtended by the angle θ,soL = Rθ ⇒
θ = L/R. Nowsec θ =(R + C)/R ⇒ R sec θ = R + C ⇒
C = R sec θ − R = R sec(L/R) − R.
(b) Extending the result in Exercise 17, we have f (4) (x) =secx (18 sec 2 x tan 2 x +6sec 4 x − sec 2 x − tan 2 x),
so f (4) (0) = 5, andsec x ≈ T 4(x) =1+ 1 2 x2 + 5 24 x4 . By part (a),
C ≈ R 1+ 1 2 L
+ 5 4 L
− R = R + 1 2 R 24 R
2 R · L2
R + 5 2 24 R · L4
R − R = L2
4 2R + 5L4
24R . 3
(c) Taking L =100km and R = 6370 km, the formula in part (a) says that
C = R sec(L/R) − R = 6370 sec(100/6370) − 6370 ≈ 0.785 009 965 44 km.
The formula in part (b) says that C ≈ L2
2R + 5L4
24R = 1002
3 2 · 6370 + 5 · 1004 ≈ 0.785 009 957 36 km.
24 · 63703 The difference between these two results is only 0.000 000 008 08 km, or 0.000 008 08 m!
39. Using f(x) =T n (x)+R n (x) with n =1and x = r,wehavef(r) =T 1 (r)+R 1 (r),whereT 1 is the first-degree Taylor
polynomial of f at a. Because a = x n , f(r) =f(x n )+f 0 (x n )(r − x n )+R 1 (r). Butr is a root of f,sof(r) =0
and we have 0=f(x n )+f 0 (x n )(r − x n )+R 1 (r). Takingthefirst two terms to the left side gives us
f 0 (x n )(x n − r) − f(x n )=R 1 (r). Dividing by f 0 (x n ),wegetx n − r − f(xn)
f 0 (x = R1(r) . By the formula for Newton’s
n) f 0 (x n) method, the left side of the preceding equation is x n+1 − r, so|x n+1 − r| =
R 1(r)
f 0 (x n ) . Taylor’s Inequality gives us
|R 1 (r)| ≤ |f 00 (r)|
2!
|x n+1 − r| ≤ M 2K |x n − r| 2 .
|r − x n | 2 . Combining this inequality with the facts |f 00 (x)| ≤ M and |f 0 (x)| ≥ K gives us

F.
502 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
11 Review
1. (a) See Definition 11.1.1.
(b) See Definition 11.2.2.
(c) The terms of the sequence {a n} approach 3 as n becomes large.
(d) By adding sufficiently many terms of the series, we can make the partial sums as close to 3 as we like.
2. (a) See Definition 11.1.11.
(b) A sequence is monotonic if it is either increasing or decreasing.
(c) By Theorem 11.1.12, every bounded, monotonic sequence is convergent.
3. (a) See (4) in Section 11.2.
(b) The p-series ∞
n=1
1
is convergent if p>1.
np 4. If a n =3,then lim
n→∞ a n =0and lim
n→∞ s n =3.
5. (a) Test for Divergence: If lim
n→∞ a n does not exist or if lim
n→∞ a n 6= 0,thentheseries ∞
n=1 a n is divergent.
(b) Integral Test: Suppose f is a continuous, positive, decreasing function on [1, ∞) and let a n = f(n). Then the series
∞
n=1 a n is convergent if and only if the improper integral ∞
f(x) dx is convergent. In other words:
1
(i) If ∞
f(x) dx is convergent, then ∞
1 n=1 a n is convergent.
(ii) If ∞
f(x) dx is divergent, then ∞
1 n=1
an is divergent.
(c) Comparison Test: Suppose that a n and b n are series with positive terms.
(i) If b n is convergent and a n ≤ b n for all n,then a n is also convergent.
(ii) If b n is divergent and a n ≥ b n for all n,then a n is also divergent.
(d) Limit Comparison Test: Suppose that a n and b n are series with positive terms. If lim
n→∞ (a n/b n )=c,wherec is a
finite number and c>0, then either both series converge or both diverge.
(e) Alternating Series Test: If the alternating series ∞
n=1 (−1)n−1 b n = b 1 − b 2 + b 3 − b 4 + b 5 − b 6 + ··· [b n > 0]
satisfies (i) b n+1 ≤ b n for all n and (ii) lim b n =0, then the series is convergent.
n→∞
(f ) Ratio Test:
(i) If lim
n→∞ an+1
= L1 or lim
a n
n→∞ a
n+1
= ∞, then the series ∞ a n is divergent.
a n
n=1
(iii) If lim =1, the Ratio Test is inconclusive; that is, no conclusion can be drawn about the convergence or
n→∞ an+1
a n
divergence of a n.

F.
TX.10
CHAPTER 11 REVIEW ¤ 503
(g) Root Test:
(i) If lim
n→∞
(ii) If lim
n→∞
(iii) If lim
n→∞
n |an| = L1 or lim |an | = ∞, then the series ∞
n→∞
n |an | =1, the Root Test is inconclusive..
n=1 a n is divergent.
6. (a) A series a n is called absolutely convergent if the series of absolute values |a n | is convergent.
(b) If a series a n is absolutely convergent, then it is convergent.
(c) A series a n is called conditionally convergent if it is convergent but not absolutely convergent.
7. (a) Use (3) in Section 11.3.
(b) See Example 5 in Section 11.4.
(c) By adding terms until you reach the desired accuracy given by the Alternating Series Estimation Theorem on page 712.
∞
8. (a) c n (x − a) n
n=0
(b) Given the power series ∞ c n (x − a) n , the radius of convergence is:
n=0
(i) 0 if the series converges only when x = a
(ii) ∞ if the series converges for all x,or
(iii) a positive number R such that the series converges if |x − a| R.
(c) The interval of convergence of a power series is the interval that consists of all values of x for which the series converges.
Corresponding to the cases in part (b), the interval of convergence is: (i) the single point {a}, (ii) all real numbers, that is,
the real number line (−∞, ∞), or (iii) an interval with endpoints a − R and a + R which can contain neither, either, or
both of the endpoints. In this case, we must test the series for convergence at each endpoint to determine the interval of
convergence.
9. (a), (b) See Theorem 11.9.2.
10. (a) T n(x) = n
(b)
∞
n=0
i=0
f (n) (a)
n!
f (i) (a)
i!
(x − a) n
(x − a) i
(c)
∞
n=0
f (n) (0)
n!
x n
[a =0in part (b)]
(d) See Theorem 11.10.8.
(e) See Taylor’s Inequality (11.10.9).
11. (a)–(e) See Table 1 on page 743.
12. See the binomial series (11.10.17) for the expansion. The radius of convergence for the binomial series is 1.

F.
504 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
1. False. See Note 2 after Theorem 11.2.6.
3. True. If lim
n→∞ a n = L,thengivenanyε>0,wecanfind a positive integer N such that |a n − L| N.
If n>N,then2n +1>Nand |a 2n+1 − L|

F.
TX.10
CHAPTER 11 REVIEW ¤ 505
11.
n
n 3 +1 < n n = 1 3 n ,so ∞
2
13. lim
n→∞
n=1
n
n 3 +1 converges by the Comparison Test with the convergent p-series ∞
n=1
1
[ p =2> 1].
n2 a
n+1 (n +1)
3
= lim
· 5n
= lim 1+ 1 3
· 1
a n n→∞ 5 n+1 n 3 n→∞ n 5 = 1 5 < 1,so ∞ n 3
n=1 5 convergesbytheRatioTest.
n
1
15. Let f(x) =
x √ .Thenf is continuous, positive, and decreasing on [2, ∞), so the Integral Test applies.
ln x
∞
2
so the series ∞
n=2
t
f(x) dx = lim
t→∞
2
1
x √ ln x dx u =lnx, du = 1
x dx
= lim 2 √ ln t − 2 √ ln 2
t→∞
1
n √ ln n diverges.
= ∞,
= lim
t→∞
ln t
ln 2
u −1/2 du = lim 2 √ ln t
u
t→∞ ln 2
17. |a n | =
cos 3n 1
≤
1+(1.2) n 1+(1.2) < 1 n 5 n (1.2) =
,so ∞ |a n n | converges by comparison with the convergent geometric
6 n=1
series ∞ 5
n
6 r =
5
< 1 ∞
6
. It follows that a n converges (by Theorem 3 in Section 11.6).
19. lim
n→∞
n=1
a n+1
a n
n=1
1 · 3 · 5 ·····(2n − 1)(2n +1)
= lim
·
n→∞ 5 n+1 (n +1)!
convergesbytheRatioTest.
5 n n!
1 · 3 · 5 ·····(2n − 1) = lim
n→∞
2n +1
5(n +1) = 2 5 < 1,sotheseries
√ √
n
21. b n = > 0, {bn} is decreasing, and lim
n +1 n→∞ bn =0,sotheseries ∞ n
(−1) n−1 converges by the Alternating
n=1 n +1
Series Test.
23. Consider the series of absolute values:
∞
n −1/3 is a p-series with p = 1 ≤ 1 and is therefore divergent. But if we apply the
3
n=1
Alternating Series Test, we see that b n = 3√ 1 > 0, {b n } is decreasing, and lim b
n =0,sotheseries ∞ (−1) n−1 n −1/3
n n→∞
converges. Thus,
∞
(−1) n−1 n −1/3 is conditionally convergent.
n=1
25.
a
n+1 =
a n
(−1)n+1 (n +2)3 n+1 2 2n+1
·
= n +2
2 2n+3 (−1) n (n +1)3 n n +1 · 3
4 = 1+(2/n)
1+(1/n) · 3
4 → 3 < 1 as n →∞,sobytheRatio
4
Test,
∞
n=1
(−1) n (n +1)3 n
2 2n+1 is absolutely convergent.
n=1
27.
∞
n=1
(−3) n−1
2 3n = ∞
n=1
= 1 8 ·
(−3) n−1
(2 3 ) n = ∞
8
11 = 1
11
n=1
(−3) n−1
8 n = 1 8
∞
n=1
(−3) n−1
8 n−1 = 1 8
∞
n=1
− 3 n−1
= 1
1
8 8 1 − (−3/8)

F.
506 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
29.
∞
[tan −1 (n +1)− tan −1 n]= lim
n=1
31. 1 − e + e2
2! − e3
3! + e4
4! − ···= ∞
n→∞ s n
= lim
n→∞ [(tan−1 2 − tan −1 1) + (tan −1 3 − tan −1 2) + ···+(tan −1 (n +1)− tan −1 n)]
= lim
n→∞ [tan−1 (n +1)− tan −1 1] = π 2 − π 4 = π 4
n=0
(−1) n e n
n! = ∞
n=0
(−e) n
n!
= e −e since e x = ∞
33. cosh x = 1 2 (ex + e −x )= 1 ∞
x n
2 n=0 n! + ∞ (−x) n
n=0 n!
= 1 1+x ···
+ x2
2
2! + x3
3! + x4
4! + +
1 ···
− x + x2
2! − x3
3! + x4
4! −
= 1 2+2· ···
x2 x4
+2·
2 2! 4! + =1+ 1
2 x2 + ∞ x 2n
n=2 (2n)! ≥ 1+ 1 2 x2 for all x
35.
37.
∞
n=1
(−1) n+1
n 5 =1− 1
32 + 1
243 − 1
1024 + 1
3125 − 1
7776 + 1
16,807 − 1
32,768 + ···.
Since b 8 = 1 8 5 = 1
32,768 < 0.000031, ∞
∞
n=1
1
2+5 ≈ 8
n
R 8 = ∞
n=9
n=1
1
2+5 < ∞
n
n=1
(−1) n+1
n 5 ≈ 7
n=1
(−1) n+1
n 5 ≈ 0.9721.
1
2+5 ≈ 0.18976224. To estimate the error, note that 1
n
n=9
39. Use the Limit Comparison Test. lim
n=0
x n
n!
for all x.
2+5 < 1 , so the remainder term is
n 5n 1
5 n = 1/59
1 − 1/5 =6.4 × 10−7 geometric series with a = 1
5 9 and r = 1 5
.
n→∞
n +1
n
a n
an
n +1
= lim = lim 1+ 1
=1> 0.
n→∞ n n→∞ n
Since |a n | is convergent, so is n +1
a n , by the Limit Comparison Test.
n
41. lim
n→∞
a n+1
a n
|x +2|
n+1
= lim
n→∞ (n +1)4 · n+1
n 4 n
|x +2| n = lim
n→∞
n
n +1
|x +2| < 4 ⇔ −4

F.
TX.10
CHAPTER 11 REVIEW ¤ 507
45.
n f (n) (x) f (n)
π
6
0 sin x
1 cos x
1
2
√
3
2
47.
49.
2 − sin x − 1 2
3 − cos x − √ 3
4 sin x
.
.
2
1
2
π
π
sin x = f + f 0 x − π
+
6 6 6
= 1 2
= 1 2
.
1 − 1
x − π 2! 6
∞
n=0
1
1+x = 1
1 − (−x) = ∞
1
1 − x = ∞
n=0
2 1
+ x − π 4! 6
(−1) n 1
x − π √
2n 3
+
(2n)! 6 2
n=0
(−x) n = ∞
π
π
π
f 00
6 x − π 2 f (3)
+ 6 x − π 3 f (4)
+ 6
2! 6 3! 6 4!
√ 4 3
− ···
+ x − π
− 1
x − π 3
+ ···
2 6 3! 6
n=0
∞
n=0
x n for |x| < 1 ⇒ ln (1 − x) =−
ln (1 − 0) = C − 0 ⇒ C =0 ⇒ ln (1 − x) =− ∞
(−1) n 1
x − π 2n+1
(2n +1)! 6
(−1) n x n for |x| < 1 ⇒ x2
1+x = ∞
dx
∞
1 − x = −
n=0
n=0
x n+1
n +1 = ∞
n=1
n=0
x n dx = C − ∞
−x n
n
x − π 6
4
+ ···
(−1) n x n+2 with R =1.
n=0
with R =1.
x n+1
n +1 .
51. sin x = ∞
n=0
(−1) n x 2n+1
(2n +1)!
⇒
sin(x 4 )= ∞
n=0
(−1) n (x 4 ) 2n+1
(2n +1)!
= ∞
n=0
(−1) n x 8n+4
(2n +1)!
for all x, so the radius of
convergence is ∞.
53. f(x) =
= 1 2
1
4√ 16 − x
=
1+
1
=
16(1 − x/16)
4
− 1
− x
+
4 16
−
1
4
1
4√ 16 1 −
1
x = 1
1/4 2 1 −
1
x −1/4
16
16
−
5
4
2!
− x 16
= 1 2 + ∞ 1 · 5 · 9 ·····(4n − 3)
x n = 1
n=1 2 · 4 n · n! · 16 n 2 + ∞
n=1
for − x < 1 ⇔ |x| < 16,soR =16.
16
2
+
−
1
4
−
5
4
3!
−
9
4
1 · 5 · 9 ·····(4n − 3)
2 6n+1 n!
− x 3
+ ···
16
x n
55. e x = ∞
n=0
x n
n! ,so ex x = 1 x
∞
n=0
x n
n!
= ∞
n=0
x n−1
n!
= x −1 + ∞
n=1
x n−1
n!
= 1 x + ∞
n=1
x n−1
n!
and
e
x
x dx = C +ln|x| + ∞
n=1
x n
n · n! .

F.
508 ¤ CHAPTER 11 INFINITE SEQUENCES AND SERIES
TX.10
57. (a)
(b)
(d)
n f (n) (x) f (n) (1)
0 x 1/2 1
1
1
2 x−1/2 1 2
2 − 1 4 x−3/2 − 1 4
3
3
8 x−5/2 3 8
4 − 15
16 x−7/2 − 15
16
.
.
.
√
x ≈ T3 (x)=1+ 1/2
1!
(x − 1) − 1/4
2!
(x − 1) 2 + 3/8
3!
=1+ 1 (x − 1) − 1 (x − 2 8 1)2 + 1 (x − 1)3
16
(x − 1) 3
(c) |R 3 (x)| ≤ M 4! |x −
1|4 ,wheref (4) (x) ≤ M with
f (4) (x) =− 15
16 x−7/2 .Now0.9 ≤ x ≤ 1.1 ⇒
−0.1 ≤ x − 1 ≤ 0.1 ⇒ (x − 1) 4 ≤ (0.1) 4 ,
and letting x =0.9 gives M =
|R 3(x)| ≤
15
,so
16(0.9)
7/2
15
16(0.9) 7/2 4! (0.1)4 ≈ 0.000 005 648
≈ 0.000 006 = 6 × 10 −6
From the graph of |R 3 (x)| = | √ x − T 3 (x)|,itappears
that the error is less than 5 × 10 −6 on [0.9, 1.1].
59. sin x = ∞
sin x − x
x 3
(−1) n x 2n+1
(2n +1)! = x − x3
3! + x5
5! − x7
x3
+ ···,sosin x − x = −
7! 3! + x5
5! − x7
+ ··· and
7!
= − 1
3! + x2
5! − x4
7! + ···.Thus,lim sin x − x
=lim − 1 ···
x→0 x 3 x→0 6 + x2
120 − x4
5040 + = − 1 6 .
n=0
61. f(x) = ∞ c n x n
⇒ f(−x) = ∞ c n(−x) n
= ∞ (−1) n c n x n
n=0
n=0
n=0
(a) If f is an odd function, then f(−x) =−f(x) ⇒ ∞
(−1) n c n x n = ∞ −c n x n . The coefficients of any power series
n=0
are uniquely determined (by Theorem 11.10.5), so (−1) n c n = −c n .
If n is even, then (−1) n =1,soc n = −c n ⇒ 2c n =0 ⇒ c n =0.Thus,allevencoefficients are 0, thatis,
c 0 = c 2 = c 4 = ···=0.
(b) If f is even, then f(−x) =f(x) ⇒ ∞ (−1) n c n x n
= ∞ c n x n ⇒ (−1) n c n = c n .
n=0
If n is odd, then (−1) n = −1,so−c n = c n ⇒ 2c n =0 ⇒ c n =0. Thus, all odd coefficients are 0,
that is, c 1 = c 3 = c 5 = ···=0.
n=0
n=0

F.
TX.10
PROBLEMS PLUS
1. Itwouldbefartoomuchworktocompute15 derivatives of f. The key idea is to remember that f (n) (0) occurs in the
coefficient of x n in the Maclaurin series of f. We start with the Maclaurin series for sin: sin x = x − x3
3! + x5
5! − ···.
Then sin(x 3 )=x 3 − x9
3! + x15
− ···, and so the coefficient of x 15 is f (15) (0)
= 1 5!
15! 5! . Therefore,
f (15) (0) = 15!
5!
=6· 7 · 8 · 9 · 10 · 11 · 12 · 13 · 14 · 15 = 10,897,286,400.
3. (a) From Formula 14a in Appendix D, with x = y = θ, wegettan 2θ = 2tanθ
1 − tan 2 θ ,socot 2θ = 1 − tan2 θ
2tanθ
2cot2θ = 1 − tan2 θ
tan θ
tan 1 2 x =cot1 2 x − 2cotx.
=cotθ − tan θ. Replacing θ by 1 2 x,weget2cotx =cot1 2 x − tan 1 2 x,or
⇒
(b) From part (a) with
x
2 n−1 in place of x, tan x 2 n =cot x 2 n − 2cot x
2 n−1 ,sothenth partial sum of ∞
s n = tan(x/2) + tan(x/4) + tan(x/8) + ···+ tan(x/2n )
2 4 8
2
n
cot(x/2)
cot(x/4)
=
− cot x + − cot(x/2) cot(x/8)
+
2
4 2
8
+
− cot(x/4)
+ ···
4
n=1
1
2 n tan x 2 n is
cot(x/2 n )
− cot(x/2n−1 )
= − cot x + cot(x/2n )
[telescoping sum]
2 n 2 n−1 2 n
Now cot(x/2n )
= cos(x/2n )
2 n 2 n sin(x/2 n ) = cos(x/2n ) x/2 n
·
x sin(x/2 n ) → 1 x · 1= 1 x as n →∞since x/2n → 0
for x 6= 0. Therefore, if x 6= 0and x 6=kπ where k is any integer, then
∞ 1
2 tan x
n 2 = lim s n n = lim − cot x + 1
n→∞ n→∞
2 cot x
= − cot x + 1 n 2 n x
n=1
If x =0, then all terms in the series are 0,sothesumis0.
5. (a) At each stage, each side is replaced by four shorter sides, each of length
1
of the side length at the preceding stage. Writing s 3 0 and 0 for the
number of sides and the length of the side of the initial triangle, we
generate the table at right. In general, we have s n =3· 4 n and
n =
1 n
3
, so the length of the perimeter at the nth stage of construction
is p n = s n n =3· 4 n ·
1 n
=3·
4 n
.
3
3
s 0 =3 0 =1
s 1 =3· 4 1 =1/3
s 2 =3· 4 2 2 =1/3 2
s 3 =3· 4 3 3 =1/3 3
.
.
(b) p n =
4n
3 n−1 =4 4
3
n−1
. Since 4 3 > 1, pn →∞as n →∞. 509

F.
510 ¤ CHAPTER 11 PROBLEMS PLUS
TX.10
(c) The area of each of the small triangles added at a given stage is one-ninth of the area of the triangle added at the preceding
stage. Let a be the area of the original triangle. Then the area a n of each of the small triangles added at stage n is
a n = a ·
1
9 = a
n 9 . Since a small triangle is added to each side at every stage, it follows that the total area A n n added to the
figure at the nth stage is A n = s n−1 · a n =3· 4 n−1 a ·
9 = a · 4n−1
. Then the total area enclosed by the snowflake
n 32n−1 curve is A = a + A 1 + A 2 + A 3 + ···= a + a · 1
3 + a · 4
3 + a · 42
3 3 + a · 43
+ ···.Afterthefirst term, this is a
5 37 geometric series with common ratio 4 a/3
,soA = a +
9 1 − 4 9
= a + a 3 · 9
5 = 8a . But the area of the original equilateral
5
triangle with side 1 is a = 1 2 · 1 · sin π √
3
3 = 4 . So the area enclosed by the snowflake curve is 8 √
3
5 · 4 = 2 √ 3
5 .
7. (a) Let a =arctanx and b =arctany. Then, from Formula 14b in Appendix D,
tan(a − b) =
tan a − tan b tan(arctan x) − tan(arctan y)
=
1+tana tan b 1 + tan(arctan x) tan(arctan y) = x − y
1+xy
Now arctan x − arctan y = a − b =arctan(tan(a − b)) = arctan x − y
1+xy since −π 2

F.
TX.10
CHAPTER 11 PROBLEMS PLUS ¤ 511
(f) From part (c) we have π =16arctan 1 − 4arctan 1
5 239
, so from parts (d) and (e) we have
16(0.197395560) − 4(0.004184077)

F.
512 ¤ CHAPTER 11 PROBLEMS PLUS
TX.10
(b) The total volume is
π ∞
0
e −x/5 sin 2 xdx= ∞
n=1
V n = 250π
101
∞
[e −(n−1)π/5 − e −nπ/5 ]= 250π [telescoping sum].
101
n=1
Another method: If the volume in part (a) has been written as V n = 250π
101 e−nπ/5 (e π/5 − 1), then we recognize ∞
as a geometric series with a = 250π
101 (1 − e−π/5 ) and r = e −π/5 .
15. If L is the length of a side of the equilateral triangle, then the area is A = 1 2 L · √3
2 L = √ 3
4 L2 and so L 2 = 4
Let r be the radius of one of the circles. When there are n rows of circles, the figure shows that
L = √ 3 r + r +(n − 2)(2r)+r + √ 3 r = r 2n − 2+2 √ 3 ,sor =
The number of circles is 1+2+···+ n =
A n =
=
n(n +1)
πr 2 n(n +1)
= π
2
2
n(n +1)
2
A n
A = n(n +1) π
√ 2
n + 3 − 1 2 √ 3
=
π
L
2 n + √ 3 − 1 .
n(n +1)
, and so the total area of the circles is
2
L 2
4 n + √ 3 − 1 2
4A/ √ 3
4 n + √ 3 − 1 2 = n(n +1) πA
√ 2
n + 3 − 1 2 √ 3
1+1/n π
√ 2
1+ 3 − 1 /n 2 √ 3 → π
2 √ 3
as n →∞
⇒
√
3
A.
V n
n=1
17. As in Section 11.9 we have to integrate the function x x by integrating series. Writing x x =(e ln x ) x = e x ln x and using the
Maclaurin series for e x ,wehavex x =(e ln x ) x = e x ln x
= ∞ (x ln x) n
= ∞ x n (ln x) n
. As with power series, we can
n!
n!
integrate this series term-by-term:
1
with u =(lnx) n , dv = x n dx,sodu =
1
0
x n (ln x) n dx = lim
0
t→0 + 1
t
=0− n
n +1
x x dx = ∞
n=0
n(ln x)n−1
x
1
0
n=0
x n (ln x) n
dx = ∞
n!
dx and v = xn+1
n +1 :
n=0
n=0
1
1
n!
x
x n (ln x) n n+1
1
dx = lim
t→0 + n +1 (ln x)n t
1
0
x n (ln x) n−1 dx
0
x n (ln x) n dx. We integrate by parts
1
− lim
t→0 + t
(where l’Hospital’s Rule was used to help evaluate the first limit). Further integration by parts gives
1
0
1
0
1
0
x n (ln x) k dx = −
k
n +1
1
x n (ln x) n dx = (−1)n n!
(n +1) n 1
x x dx = ∞
n=0
1
n!
1
0
0
0
x n (ln x) k−1 dx and, combining these steps, we get
x n (ln x) n dx = ∞
x n dx = (−1)n n!
(n +1) n+1 ⇒
n=0
1 (−1) n n!
n! (n +1) = ∞
n+1
n=0
(−1) n
(n +1) = ∞
n+1
n=1
n
n +1 xn (ln x) n−1 dx
(−1) n−1
n n .

F.
TX.10
CHAPTER 11 PROBLEMS PLUS ¤ 513
19. Let f(x) = ∞ c m x m and g(x) =e f(x)
= ∞ d n x n .Theng 0
(x) = ∞ nd n x n−1 ,sond n occurs as the coefficient
m=0
n=0
of x n−1 .Butalso
∞
g 0 (x) =e f(x) f 0
∞
(x) = d n x n
mc m x m−1
n=0
m=1
= d 0 + d 1x + d 2x 2 + ···+ d n−1x n−1 + ··· c 1 +2c 2x +3c 3x 2 + ···+ nc nx n−1 + ···
so the coefficient of x n−1
is c 1d n−1 +2c 2d n−2 +3c 3d n−3 + ···+ nc nd 0 = n
ic id n−i. Therefore, nd n = n ic id n−i.
21. Call the series S. We group the terms according to the number of digits in their denominators:
S = 1
+ 1 + ···+ 1 +
1 + 1
1 2 8 9
+ ···+
1 + 1
11 99
+ ···+
1 + ···
111 999
g 1 g 2 g 3
Now in the group g n ,sincewehave9 choices for each of the n digits in the denominator, there are 9 n terms.
Furthermore, each term in g n is less than
Now ∞
n=1
S = ∞
g n < ∞
n=1
n=0
1
[except for the first term in g
10 n−1 1 ]. So g n < 9 n 1 · =9
9 n−1
.
10 n−1 10
9
9 n−1
is a geometric series with a =9and r = 9 < 1. Therefore, by the Comparison Test,
10
10
n=1
9 9
10
n−1
=
9
=90.
1 − 9/10
i=1
i=1
23. u =1+ x3
3! + x6
6! + x9
x4
+ ···, v = x +
9! 4! + x7
7! + x10
x2
+ ···, w =
10! 2! + x5
5! + x8
8! + ···.
Use the Ratio Test to show that the series for u, v, andw have positive radii of convergence (∞ in each case), so
Theorem 11.9.2 applies, and hence, we may differentiate each of these series:
du
dx = 3x2 + 6x5 + 9x8 + ···= x2
3! 6! 9! 2! + x5
5! + x8
8! + ···= w
Similarly, dv
dx =1+x3 3! + x6
6! + x9
dw
+ ···= u,and
9! dx = x + x4
4! + x7
7! + x10
+ ···= v.
10!
So u 0 = w, v 0 = u,andw 0 = v. Now differentiate the left hand side of the desired equation:
d
dx (u3 + v 3 + w 3 − 3uvw) =3u 2 u 0 +3v 2 v 0 +3w 2 w 0 − 3(u 0 vw + uv 0 w + uvw 0 )
=3u 2 w +3v 2 u +3w 2 v − 3(vw 2 + u 2 w + uv 2 )=0 ⇒
u 3 + v 3 + w 3 − 3uvw = C. Tofind the value of the constant C, weputx =0in the last equation and get
1 3 +0 3 +0 3 − 3(1 · 0 · 0) = C ⇒ C =1,sou 3 + v 3 + w 3 − 3uvw =1.

F.
TX.10

F.
TX.10
13 VECTORS AND THE GEOMETRY OF SPACE ET 12
13.1 Three-Dimensional Coordinate Systems ET 12.1
1. We start at the origin, which has coordinates (0, 0, 0). Firstwemove4 units along the positive x-axis, affecting only the
x-coordinate, bringing us to the point (4, 0, 0). Wethenmove3 units straight downward, in the negative z-direction. Thus
only the z-coordinate is affected, and we arrive at (4, 0, −3).
3. The distance from a point to the xz-plane is the absolute value of the y-coordinate of the point. Q(−5, −1, 4) has the
y-coordinate with the smallest absolute value, so Q is the point closest to the xz-plane. R(0, 3, 8) must lie in the yz-plane
since the distance from R to the yz-plane, given by the x-coordinate of R,is0.
5. The equation x + y =2represents the set of all points in
R 3 whose x-andy-coordinates have a sum of 2,or
equivalently where y =2− x. This is the set
{(x, 2 − x, z) | x ∈ R,z ∈ R} which is a vertical plane
that intersects the xy-plane in the line y =2− x, z =0.
7. We can find the lengths of the sides of the triangle by using the distance formula between pairs of vertices:
|PQ| = (7 − 3) 2 +[0− (−2)] 2 +[1− (−3)] 2 = √ 16 + 4 + 16 = 6
|QR| = (1 − 7) 2 +(2− 0) 2 +(1− 1) 2 = √ 36 + 4 + 0 = √ 40 = 2 √ 10
|RP | = (3 − 1) 2 +(−2 − 2) 2 +(−3 − 1) 2 = √ 4+16+16=6
The longest side is QR, but the Pythagorean Theorem is not satisfied: |PQ| 2 + |RP | 2 6=|QR| 2 . Thus PQRis not a right
triangle. PQRis isosceles, as two sides have the same length.
9. (a) First we find the distances between points:
|AB| = (3 − 2) 2 +(7− 4) 2 +(−2 − 2) 2 = √ 26
|BC| = (1 − 3) 2 +(3− 7) 2 +[3− (−2)] 2 = √ 45 = 3 √ 5
|AC| = (1 − 2) 2 +(3− 4) 2 +(3− 2) 2 = √ 3
In order for the points to lie on a straight line, the sum of the two shortest distances must be equal to the longest distance.
Since √ 26 + √ 3 6= 3 √ 5, the three points do not lie on a straight line.
105

F.
106 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ETTX.10
CHAPTER 12
(b) First we find the distances between points:
|DE| = (1 − 0) 2 +[−2 − (−5)] 2 +(4− 5) 2 = √ 11
|EF| = (3 − 1) 2 +[4− (−2)] 2 +(2− 4) 2 = √ 44 = 2 √ 11
|DF| = (3 − 0) 2 +[4− (−5)] 2 +(2− 5) 2 = √ 99 = 3 √ 11
Since |DE| + |EF| = |DF|, the three points lie on a straight line.
11. An equation of the sphere with center (1, −4, 3) and radius 5 is (x − 1) 2 +[y − (−4)] 2 +(z − 3) 2 =5 2 or
(x − 1) 2 +(y +4) 2 +(z − 3) 2 =25. The intersection of this sphere with the xz-plane is the set of points on the sphere
whose y-coordinate is 0. Putting y =0into the equation, we have (x − 1) 2 +4 2 +(z − 3) 2 =25,y=0or
(x − 1) 2 +(z − 3) 2 =9,y=0, which represents a circle in the xz-plane with center (1, 0, 3) and radius 3.
13. The radius of the sphere is the distance between (4, 3, −1) and (3, 8, 1): r = (3 − 4) 2 +(8− 3) 2 +[1− (−1)] 2 = √ 30.
Thus, an equation of the sphere is (x − 3) 2 +(y − 8) 2 +(z − 1) 2 =30.
15. Completing squares in the equation x 2 + y 2 + z 2 − 6x +4y − 2z =11gives
(x 2 − 6x +9)+(y 2 +4y +4)+(z 2 − 2z +1)=11+9+4+1 ⇒ (x − 3) 2 +(y +2) 2 +(z − 1) 2 =25,whichwe
recognize as an equation of a sphere with center (3, −2, 1) and radius 5.
17. Completing squares in the equation 2x 2 − 8x +2y 2 +2z 2 +24z =1gives
2(x 2 − 4x +4)+2y 2 +2(z 2 +12z +36)=1+8+72 ⇒ 2(x − 2) 2 +2y 2 +2(z +6) 2 =81 ⇒
(x − 2) 2 + y 2 +(z +6) 2 = 81 , which we recognize as an equation of a sphere with center (2, 0, −6) and radius
2
81
=9/√ 2.
2
19. (a) If the midpoint of the line segment from P 1(x 1,y 1,z 1) to P 2(x 2,y 2,z 2) is Q =
x1 + x 2
2
, y 1 + y 2
2
, z 1 + z
2
,
2
then the distances |P 1 Q| and |QP 2 | are equal, and each is half of |P 1 P 2 |. We verify that this is the case:
|P 1 P 2 | = (x 2 − x 1 ) 2 +(y 2 − y 1 ) 2 +(z 2 − z 1 ) 2
|P 1Q| = 1
2 (x1 + x2) − 2 x1 + 1
2 (y1 + y2) − 2 y1 + 1
2 (z1 + z2) − 2
z1
= 1
2 x2 − 1 2
2 x1 + 1
2 y2 − 1 2
2 y1 + 1
2 z2 − 1 2
2 z1
= 1
2
(x2 − x
2 1) 2 +(y 2 − y 1) 2 +(z 2 − z 1) 2
= 1 (x
2 2 − x 1) 2 +(y 2 − y 1) 2 +(z 2 − z 1) 2
= 1 |P1P2|
2
x2
|QP 2 | = − 1 (x 2 1 + x 2 ) 2 + y 2 − 1 (y 2 1 + y 2 ) 2 + z 2 − 1 (z 2 1 + z 2 ) 2
= 1 x 2 2 − 1 x 2
2 1 + 1
y 2 2 − 1 y 2
2 1 +
1
z 2 2 − 1 z 2
2 1
= 1
2
(x2 − x
2
1 ) 2 +(y 2 − y 1 ) 2 +(z 2 − z 1 ) 2
(x 2 − x 1 ) 2 +(y 2 − y 1 ) 2 +(z 2 − z 1 ) 2 = 1 |P 2 1P 2 |
= 1 2
So Q is indeed the midpoint of P 1 P 2 .

F.
SECTION 13.1 TX.10 THREE-DIMENSIONAL COORDINATE SYSTEMS ET SECTION 12.1 ¤ 107
(b) By part (a), the midpoints of sides AB, BC and CA are P 1
−
1
2 , 1, 4 , P 2
1,
1
2 , 5 and P 3
5
2 , 3 2 , 4 . (Recall that a median
of a triangle is a line segment from a vertex to the midpoint of the opposite side.) Then the lengths of the medians are:
|AP 2 | = 0 2 +
1
− 2 22 +(5− 3) 2 9
= +4= 25
= 5
4 4 2
|BP 3 | = 5
2 +22 +
3 2
+(4− 5) 2 81
= + 9 +1= 94
= √ 1
2 4 4 4 2 94
−
1
|CP 1 | = − 2 42 +(1− 1) 2 +(4− 5) 2 81
= +1= √ 1
4 2 85
21. (a) Since the sphere touches the xy-plane, its radius is the distance from its center, (2, −3, 6),tothexy-plane, namely 6.
Therefore r =6and an equation of the sphere is (x − 2) 2 +(y +3) 2 +(z − 6) 2 =6 2 =36.
(b) The radius of this sphere is the distance from its center (2, −3, 6) to the yz-plane, which is 2. Therefore, an equation is
(x − 2) 2 +(y +3) 2 +(z − 6) 2 =4.
(c) Here the radius is the distance from the center (2, −3, 6) to the xz-plane, which is 3. Therefore, an equation is
(x − 2) 2 +(y +3) 2 +(z − 6) 2 =9.
23. The equation y = −4 represents a plane parallel to the xz-plane and 4 units to the left of it.
25. The inequality x>3 represents a half-space consisting of all points in front of the plane x =3.
27. The inequality 0 ≤ z ≤ 6 represents all points on or between the horizontal planes z =0(the xy-plane) and z =6.
29. The inequality x 2 + y 2 + z 2 ≤ 3 is equivalent to x 2 + y 2 + z 2 ≤ √ 3, so the region consists of those points whose distance
from the origin is at most √ 3. This is the set of all points on or inside the sphere with radius √ 3 and center (0, 0, 0).
31. Here x 2 + z 2 ≤ 9 or equivalently √ x 2 + z 2 ≤ 3 which describes the set of all points in R 3 whose distance from the y-axis is
at most 3. Thus, the inequality represents the region consisting of all points on or insideacircularcylinderofradius3 with
axis the y-axis.
33. This describes all points whose x-coordinate is between 0 and 5, that is, 0

F.
108 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ETTX.10
CHAPTER 12
39. We need to find a set of points P (x, y, z) |AP | = |BP|
.
(x +1)2 +(y − 5) 2 +(z − 3) 2 = (x − 6) 2 +(y − 2) 2 +(z +2) 2
⇒
(x +1) 2 +(y − 5) + (z − 3) 2 =(x − 6) 2 +(y − 2) 2 +(z +2) 2 ⇒
x 2 +2x +1+y 2 − 10y +25+z 2 − 6z +9=x 2 − 12x +36+y 2 − 4y +4+z 2 +4z +4 ⇒ 14x − 6y − 10z =9.
Thus the set of points is a plane perpendicular to the line segment joining A and B (since this plane must contain the
perpendicular bisector of the line segment AB).
13.2 Vectors ET 12.2
1. (a) The cost of a theater ticket is a scalar, because it has only magnitude.
(b) The current in a river is a vector, because it has both magnitude (the speed of the current) and direction at any given
location.
(c) If we assume that the initial path is linear, the initial flight path from Houston to Dallas is a vector, because it has both
magnitude (distance) and direction.
(d) The population of the world is a scalar, because it has only magnitude.
3. Vectors are equal when they share the same length and direction (but not necessarily location). Using the symmetry of the
parallelogram as a guide, we see that −→
AB = −→
DC,
−→
DA =
−→
CB,
−−→
DE =
−→
EB,and
−→
EA =
−→
CE.
5. (a) (b) (c) (d)
7. a = h−2 − 2, 1 − 3i = h−4, −2i 9. a = h2 − (−1), 2 − 3i = h3, −1i
11. a = h2 − 0, 3 − 3, −1 − 1i = h2, 0, −2i 13. h−1, 4i + h6, −2i = h−1+6, 4+(−2)i = h5, 2i

F.
TX.10
SECTION 13.2 VECTORS ET SECTION 12.2 ¤ 109
15. h0, 1, 2i + h0, 0, −3i = h0+0, 1+0, 2+(−3)i
= h0, 1, −1i
17. a + b = h5+(−3) , −12 + (−6)i = h2, −18i
2a +3b = h10, −24i + h−9, −18i = h1, −42i
|a| = 5 2 +(−12) 2 = √ 169 = 13
|a − b| = |h5 − (−3), −12 − (−6)i| = |h8, −6i| = 8 2 +(−6) 2 = √ 100 = 10
19. a + b =(i +2j − 3 k)+(−2 i − j +5k) =− i + j +2k
2a +3b =2(i +2j − 3 k)+3(−2 i − j +5k) =2i +4j − 6 k − 6 i − 3 j +15k = − 4 i + j +9k
|a| = 1 2 +2 2 +(−3) 2 = √ 14
|a − b| = |(i +2j − 3 k) − (−2 i − j +5k)| = |3 i +3j − 8 k| = 3 2 +3 2 +(−8) 2 = √ 82
21. |−3 i +7j| = (−3) 2 +7 2 = √ 58,sou = 1 √
58
(−3 i +7j) =− 3 √
58
i + 7 √
58
j.
23. The vector 8 i − j +4k has length |8 i − j +4k| = 8 2 +(−1) 2 +4 2 = √ 81 = 9, so by Equation 4 the unit vector with
the same direction is 1 9 (8 i − j +4k) = 8 9 i − 1 9 j + 4 9 k.
25. From the figure, we see that the x-component of v is
v 1 = |v| cos(π/3) = 4 · 1 =2and the y-component is
2
v 2 = |v| sin(π/3) = 4 · √3
v = hv 1 ,v 2 i = 2, 2 √ 3 .
2 =2√ 3. Thus
27. The velocity vector v makes an angle of 40 ◦ with the horizontal and
has magnitude equal to the speed at which the football was thrown.
From the figure, we see that the horizontal component of v is
|v| cos 40 ◦ =60cos40 ◦ ≈ 45.96 ft/s and the vertical component is
|v| sin 40 ◦ =60sin40 ◦ ≈ 38.57 ft/s.
29. The given force vectors can be expressed in terms of their horizontal and vertical components as −300 i and
200 cos 60 ◦ i +200sin60 ◦ j = 200 √3
1
2 i +200 j = 100 i + 100 √ 2
3 j. TheresultantforceF is the sum of these two
vectors: F =(−300 + 100) i + 0 + 100 √ 3 j = −200 i + 100 √ 3 j. Thenwehave

F.
110 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ET TX.10 CHAPTER 12
|F| ≈ (−200) 2 + 100 √ 3 2 √ √
= 70,000 = 100 7 ≈ 264.6 N. Let θ be the angle F makes with the positive x-axis.
Then tan θ = 100 √ √
3 3
−200 = − 2
θ =tan −1 −
√
3
2
and the terminal point of F lies in the second quadrant, so
+180 ◦ ≈−40.9 ◦ + 180 ◦ =139.1 ◦ .
31. With respect to the water’s surface, the woman’s velocity is the vector sum of the velocity of the ship with respect to the water,
and the woman’s velocity with respect to the ship. If we let north be the positive y-direction, then
v = h0, 22i + h−3, 0i = h−3, 22i. The woman’s speed is |v| = √ 9 + 484 ≈ 22.2mi/h. Thevectorv makes an angle θ
with the east, where θ =tan −1 22
≈ 98 ◦ . Therefore, the woman’s direction is about N(98 − 90) ◦ W=N8 ◦ W.
33. Let T 1 and T 2 represent the tension vectors in each side of the
clothesline as shown in the figure. T 1 and T 2 have equal vertical
−3
components and opposite horizontal components, so we can write
T 1 = −a i + b j and T 2 = a i + b j [a, b > 0]. By similar triangles, b a = 0.08
4
⇒
a =50b. The force due to gravity
acting on the shirt has magnitude 0.8g ≈ (0.8)(9.8) = 7.84 N, hence we have w = −7.84 j. The resultant T 1 + T 2
of the tensile forces counterbalances w, soT 1 + T 2 = −w ⇒ (−a i + b j)+(a i + b j) =7.84 j ⇒
(−50b i + b j)+(50b i + b j) =2b j =7.84 j ⇒ b = 7.84
2
=3.92 and a =50b = 196. Thus the tensions are
T 1 = −a i + b j = −196 i +3.92 j and T 2 = a i + b j = 196 i +3.92 j.
Alternatively, we can find the value of θ and proceed as in Example 7.
35. The slope of the tangent line to the graph of y = x 2 at the point (2, 4) is
dy
dx =2x
=4
x=2 x=2
and a parallel vector is i +4j which has length |i +4j| = √ 1 2 +4 2 = √ 17, so unit vectors parallel to the tangent line are
± 1 √
17
(i +4j).
37. BytheTriangleLaw, −→
AB + −→ −→ −→ −→ −→ −→ −→ −→ −→ −→
BC = AC. ThenAB + BC + CA = AC + CA,butAC + CA = AC + − −→
AC
So −→
AB + −→
BC +
−→
CA = 0.
39. (a), (b) (c) From the sketch, we estimate that s ≈ 1.3 and t ≈ 1.6.
(d) c = s a + t b ⇔ 7=3s +2t and 1=2s − t.
Solving these equations gives s = 9 7 and t = 11 7 .
= 0.
41. |r − r 0 | is the distance between the points (x, y, z) and (x 0 ,y 0 ,z 0 ), so the set of points is a sphere with radius 1 and
center (x 0 ,y 0 ,z 0 ).
Alternate method: |r − r 0| =1 ⇔ (x − x 0) 2 +(y − y 0) 2 +(z − z 0) 2 =1 ⇔
(x − x 0 ) 2 +(y − y 0 ) 2 +(z − z 0 ) 2 =1, which is the equation of a sphere with radius 1 and center (x 0 ,y 0 ,z 0 ).

F.
TX.10
SECTION 13.3 THE DOT PRODUCT ET SECTION 12.3 ¤ 111
43. a +(b + c) =ha 1,a 2i +(hb 1,b 2i + hc 1,c 2i) =ha 1,a 2i + hb 1 + c 1,b 2 + c 2i
= ha 1 + b 1 + c 1 ,a 2 + b 2 + c 2 i = h(a 1 + b 1 )+c 1 , (a 2 + b 2 )+c 2 i
= ha 1 + b 1 ,a 2 + b 2 i + hc 1 ,c 2 i =(ha 1 ,a 2 i + hb 1 ,b 2 i)+hc 1 ,c 2 i
=(a + b)+c
45. Consider triangle ABC, whereD and E are the midpoints of AB and BC. We know that −→
AB + −→
BC =
−→
AC (1) and
−−→
DB + −→ −−→
−−→ −→
BE = DE (2). However, DB =
1
AB, and −→ −→
BE =
1
BC. Substituting these expressions for DB −−→
and −→
BE into
2
−→ −→
(2)gives 1 AB + 1 BC = −−→
DE. Comparing this with (1)gives −−→ −→
DE = 1 AC. Therefore −→
AC and −−→
DE are parallel and
2 2
2
−−→
DE = 1 2
−→
AC.
2
13.3 The Dot Product ET 12.3
1. (a) a · b is a scalar, and the dot product is defined only for vectors, so (a · b) · c has no meaning.
(b) (a · b) c is a scalar multiple of a vector, so it does have meaning.
(c) Both |a| and b · c are scalars, so |a| (b · c) is an ordinary product of real numbers, and has meaning.
(d) Both a and b + c are vectors, so the dot product a · (b + c) has meaning.
(e) a · b is a scalar, but c is a vector, and so the two quantities cannot be added and a · b + c has no meaning.
(f ) |a| is a scalar, and the dot product is defined only for vectors, so |a|·(b + c) has no meaning.
3. a · b = −2, 1 3
·h−5, 12i =(−2)(−5) +
1
3
(12) = 10 + 4 = 14
5. a · b = 4, 1, 1 4
·h6, −3, −8i = (4)(6) + (1)(−3) +
1
4
(−8) = 19
7. a · b =(i − 2 j +3k) · (5 i +9k) = (1)(5) + (−2)(0) + (3)(9) = 32
9. a · b = |a||b| cos θ =(6)(5)cos 2π 3 =30 − 1 2
= −15
11. u, v, and w are all unit vectors, so the triangle is an equilateral triangle. Thus the angle between u and v is 60 ◦ and
u · v = |u||v| cos 60 ◦ = (1)(1)
1
2 =
1
. If w is moved so it has the same initial point as u, we can see that the angle
2
between them is 120 ◦ and we have u · w = |u||w| cos 120 ◦ =(1)(1)
− 1 2 = −
1
. 2
13. (a) i · j = h1, 0, 0i·h0, 1, 0i = (1)(0) + (0)(1) + (0)(0) = 0. Similarly, j · k = (0)(0) + (1)(0) + (0)(1) = 0 and
k · i = (0)(1) + (0)(0) + (1)(0) = 0.
Another method: Because i, j,andk are mutually perpendicular, the cosine factor in each dot product (see Theorem 3)
is cos π 2 =0.
(b) By Property 1 of the dot product, i · i = |i| 2 =1 2 =1since i is a unit vector. Similarly, j · j = |j| 2 =1and
k · k = |k| 2 =1.
15. |a| = (−8) 2 +6 2 =10, |b| =
√7 2
+32 =4,anda · b =(−8) √ 7 + (6)(3) = 18 − 8 √ 7. From Corollary 6,
we have cos θ = a · b
|a||b| = 18 − 8 √ 7
10 · 4
= 9 − 4 √ √
7
9 − 4 7
. So the angle between a and b is θ =cos −1 ≈ 95 ◦ .
20
20

F.
112 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ET TX.10 CHAPTER 12
17. |a| = 3 2 +(−1) 2 +5 2 = √ 35, |b| = (−2) 2 +4 2 +3 2 = √ 29,anda · b =(3)(−2) + (−1)(4) + (5)(3) = 5. Then
cos θ = a · b
|a||b| = 5
√ √ = 5
√ and the angle between a and b is θ =cos −1 √
5
35 · 29 1015
1015
≈ 81 ◦ .
19. |a| = √ 0 2 +1 2 +1 2 = √ 2, |b| = 1 2 +2 2 +(−3) 2 = √ 14, anda · b = (0)(1) + (1)(2) + (1)(−3) = −1.
Then cos θ = a · b
|a||b| = −1
√ √ =
−1
2 · 14 2 √ 7 and θ =cos−1 − 1
2 √ ≈ 101 ◦ .
7
21. Let a, b,andc be the angles at vertices A, B,andC respectively.
Then a is the angle between vectors −→
AB and −→
AC, b is the angle
between vectors −→
BA and −→
BC,andc is the angle between vectors
−→
CA and −→
CB.
−→
AB · −→
AC
Thus cos a =
−→
AB
−→
h2, 6i·h−2, 4i
= √ AC
22 +6 2 (−2) 2 +4 = 1
√ √ (−4+24)= 20
√
2
√ =
2 40 20 800 2 and
√ 2
a =cos −1 =45 ◦ . Similarly, cos b =
2
−→ −→
BA · BC
−→
BA
−→ = BC
h−2, −6i·h−4, −2i
√ 4+36
√ 16 + 4
=
1
√ √ (8 + 12) = 20
√
2
√ =
40 20 800 2
√ 2
so b =cos −1 =45 ◦ and c =180 ◦ − (45 ◦ + 45 ◦ )=90 ◦ .
2
−→
2 −→
2 −→
2
BC − AB − AC
Alternate solution: Apply the Law of Cosines three times as follows: cos a =
2 −→
AB
−→
,
AC
−→
AC 2 − −→
AB 2 − −→
2
BC −→
AB 2 − −→
AC 2 − −→
2
BC
cos b =
2 −→
AB
−→ ,andcos c =
BC 2 −→
AC
−→ . BC
23. (a) a · b =(−5)(6) + (3)(−8) + (7)(2) = −40 6= 0,soa and b are not orthogonal. Also, since a is not a scalar multiple
25.
of b, a and b are not parallel.
(b) a · b =(4)(−3) + (6)(2) = 0,soa and b are orthogonal (and not parallel).
(c) a · b =(−1)(3) + (2)(4) + (5)(−1) = 0,soa and b are orthogonal (and not parallel).
(d) Because a = − 2 b, a and b are parallel.
3
−→
QP = h−1, −3, 2i, −→
QR = h4, −2, −1i,and −→ −→
−→ −→
QP · QR = −4+6− 2=0. Thus QP and QR are orthogonal, so the angle of
thetriangleatvertexQ is a right angle.
27. Let a = a 1 i + a 2 j + a 3 k be a vector orthogonal to both i + j and i + k. Thena · (i + j) =0 ⇔ a 1 + a 2 =0and
a · (i + k) =0 ⇔ a 1 + a 3 =0,so a 1 = −a 2 = −a 3 .Furthermorea istobeaunitvector,so1=a 2 1 + a 2 2 + a 2 3 =3a 2 1
implies a 1 = ± 1 √
3
.Thusa = 1 √
3
i − 1 √
3
j − 1 √
3
k and a = − 1 √
3
i + 1 √
3
j + 1 √
3
k are two such unit vectors.

F.
TX.10
SECTION 13.3 THE DOT PRODUCT ET SECTION 12.3 ¤ 113
29. Since |h3, 4, 5i| = √ 9+16+25= √ 50 = 5 √ 2, using Equations 8 and 9 we have cos α = 3
5 √ , cos β = 4
2 5 √ ,and 2
cos γ = 5
5 √ = √ 1
2 2
. The direction angles are given by α =cos −1 3
5 √ ≈ 65 ◦ , β =cos −1 4
2
5 √ ≈ 56 ◦ ,and
2
γ =cos −1 √
1
2
=45 ◦ .
31. Since |2 i +3j − 6 k| = √ 4+9+36= √ 49 = 7, Equations 8 and 9 give cos α = 2 7 , cos β = 3 7
α =cos −1
2
7 ≈ 73 ◦ , β =cos −1
3
7 ≈ 65 ◦ ,andγ =cos −1
− 6 7 ≈ 149 ◦ .
,andcos γ =
−6
7 ,while
33. |hc, c, ci| = √ c 2 + c 2 + c 2 = √ 3c [since c>0], so cos α =cosβ =cosγ = c √
3c
=
1 √
3
and
α = β = γ =cos −1 √
1
3
≈ 55 ◦ .
35. |a| = 3 2 +(−4) 2 =5. The scalar projection of b onto a is comp a b = a · b
|a|
a · b a
projection of b onto a is proj a b =
|a| |a| =3· 1 h3, −4i = 9
, − 12
5 5 5 .
37. |a| = √ 9+36+4=7so the scalar projection of b onto a is comp ab = a · b
|a|
= 3 · 5+(−4) · 0
5
projection of b onto a is proj ab = 9 a
7 |a| = 9 · 1 h3, 6, −2i = 9 h3, 6, −2i = 27
, 54 , − 18
7 7 49 49 49 49 .
39. |a| = √ 4+1+16= √ 21 so the scalar projection of b onto a is comp a b = a · b
|a|
=3and the vector
= 1 (3 + 12 − 6) = 9 7 7
. The vector
= 0 − 1+2 √
21
= 1 √
21
while the vector
projection of b onto a is proj a b = 1 √
21
a
|a| = 1 √
21
· 2 i − j +4k √
21
= 1 21 (2 i − j +4k) = 2 21 i − 1 21 j + 4 21 k.
41. (orth a b) · a =(b − proj a b) · a = b · a − (proj a b) · a = b · a − a · b
|a| 2
So they are orthogonal by (7).
43. comp a b = a · b
|a|
a · a = b · a − a · b
|a| 2 |a|2 = b · a − a · b =0.
=2 ⇔ a · b =2|a| =2 √ 10. Ifb = hb 1 ,b 2 ,b 3 i,thenweneed3b 1 +0b 2 − 1b 3 =2 √ 10.
One possible solution is obtained by taking b 1 =0, b 2 =0, b 3 = −2 √ 10. In general, b = s, t, 3s − 2 √ 10 , s, t ∈ R.
45. The displacement vector is D =(6− 0) i +(12− 10) j +(20− 8) k =6i +2j +12k so by Equation 12 the work done is
W = F · D =(8i − 6 j +9k) · (6 i +2j +12k) =48− 12 + 108 = 144 joules.
47. Here |D| =80ft, |F| =30lb, and θ =40 ◦ . Thus
W = F · D = |F||D| cos θ = (30)(80) cos 40 ◦ = 2400 cos 40 ◦ ≈ 1839 ft-lb.
49. First note that n = ha, bi is perpendicular to the line, because if Q 1 =(a 1 ,b 1 ) and Q 2 =(a 2 ,b 2 ) lie on the line, then
n · −−−→
Q 1 Q 2 = aa 2 − aa 1 + bb 2 − bb 1 =0,sinceaa 2 + bb 2 = −c = aa 1 + bb 1 from the equation of the line.
Let P 2 =(x 2 ,y 2 ) lie on the line. Then the distance from P 1 to the line is the absolute value of the scalar projection

F.
114 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ET TX.10 CHAPTER 12
of −−→
P 1 P 2 onto n.
−−→
|n ·hx2 − x1,y2 − y1i| |ax2 − ax1 + by2 − by1| |ax1 + by1 + c|
comp n P 1 P 2 = = √ = √
|n|
a2 + b 2 a2 + b 2
since ax 2 + by 2 = −c. The required distance is
|3 · −2+−4 · 3+5|
√
32 +4 2 = 13 5 .
51. For convenience, consider the unit cube positioned so that its back left corner is at the origin, and its edges lie along the
coordinate axes. The diagonal of the cube that begins at the origin and ends at (1, 1, 1) has vector representation h1, 1, 1i.
The angle θ between this vector and the vector of the edge which also begins at the origin and runs along the x-axis [that is,
h1, 1, 1i·h1, 0, 0i
h1, 0, 0i]isgivenbycos θ =
|h1, 1, 1i| |h1, 0, 0i| = √ 1
⇒ θ =cos −1 √
1
3
3
≈ 55 ◦ .
53. Consider the H — C — H combination consisting of the sole carbon atom and the two hydrogen atoms that are at (1, 0, 0) and
(0, 1, 0) (or any H — C — H combination, for that matter). Vector representations of the line segments emanating from the
carbon atom and extending to these two hydrogen atoms are 1 − 1 2 , 0 − 1 2 , 0 − 1 2
=
1
2 , − 1 2 , − 1 2
and
0 −
1
, 1 − 1 , 0 − 1
2 2 2 = −
1
, 1 , − 1
2 2 2 . The bond angle, θ, is therefore given by
1 , − 1 , −
1
2 2 2 · −
1
2
cos θ =
, 1 , − 1 2 2
1
, − 1 , − 1 2 2 2 −
1
, 1 , − 1
= − 1 − 1 + 1 4 4 4
= − 1 3
2 2 2
55. Let a = ha 1,a 2,a 3i and = hb 1,b 2,b 3i.
Property 2: a · b = ha 1,a 2,a 3i·hb 1,b 2,b 3i = a 1b 1 + a 2b 2 + a 3b 3
3
4
= b 1a 1 + b 2a 2 + b 3a 3 = hb 1,b 2,b 3i·ha 1,a 2,a 3i = b · a
3
4
⇒ θ =cos −1 − 1 3
≈ 109.5 ◦ .
Property 4: (c a) · b = hca 1,ca 2,ca 3i·hb 1,b 2,b 3i =(ca 1)b 1 +(ca 2)b 2 +(ca 3)b 3
= c (a 1 b 1 + a 2 b 2 + a 3 b 3 )=c (a · b) =a 1 (cb 1 )+a 2 (cb 2 )+a 3 (cb 3 )
= ha 1 ,a 2 ,a 3 i·hcb 1 ,cb 2 ,cb 3 i = a · (c b)
Property 5: 0 · a = h0, 0, 0i·ha 1,a 2,a 3i =(0)(a 1)+(0)(a 2)+(0)(a 3)=0
57. |a · b| = |a||b| cos θ = |a||b||cos θ|. Since|cos θ| ≤ 1, |a · b| = |a||b||cos θ| ≤ |a||b|.
Note: We have equality in the case of cos θ = ±1,soθ =0or θ = π, thus equality when a and b are parallel.
59. (a) The Parallelogram Law states that the sum of the squares of the
lengths of the diagonals of a parallelogram equals the sum of the
squares of its (four) sides.
(b) |a + b| 2 =(a + b) · (a + b) =|a| 2 +2(a · b)+|b| 2 and |a − b| 2 =(a − b) · (a − b) =|a| 2 − 2(a · b)+|b| 2 .
Adding these two equations gives |a + b| 2 + |a − b| 2 =2|a| 2 +2|b| 2 .

F.
TX.10
SECTION 13.4 THE CROSS PRODUCT ET SECTION 12.4 ¤ 115
13.4 The Cross Product ET 12.4
i j k
0 −2
1. a × b =
6 0 −2
=
0 8 0
8 0 i − 6 −2
0 0 j + 6 0
0 8 k
=[0− (−16)] i − (0 − 0) j +(48− 0) k =16i +48k
Now (a × b) · a = h16, 0, 48i·h6, 0, −2i =96+0− 96 = 0 and (a × b) · b = h16, 0, 48i·h0, 8, 0i =0+0+0=0,so
a × b is orthogonal to both a and b.
i j k
3 −2
3. a × b =
1 3 −2
=
−1 0 5
0 5 i − 1 −2
−1 5 j + 1 3
−1 0 k
=(15− 0) i − (5 − 2) j +[0− (−3)] k =15i − 3 j +3k
Since (a × b) · a =(15i − 3 j +3k) · (i +3j − 2 k) =15− 9 − 6=0, a × b is orthogonal to a.
Since (a × b) · b =(15i − 3 j +3k) · (−i +5k) =−15 + 0 + 15 = 0, a × b is orthogonal to b.
i j k
−1
5. a × b =
1 −1 −1
=
1
1
1
2
1
2
−1
i − 1
1
1
2
2
−1
j + 1
1
1
2
−1
k
2
1
= − 1 2 − (−1) i − 1
2 − (− 1 2 ) j + 1 − (− 1 2 ) k = 1 2 i − j + 3 2 k
Now (a × b) · a = 1
2 i − j + 3 2 k · (i − j − k) = 1 2 +1− 3 2 =0and
(a × b) · b = 1
i − j + 3 k 2 2 · 1
i + j + 1 k 2 2
= 1 − 1+ 3 4 4
=0,soa × b is orthogonal to both a and b.
i j k 7. a × b =
t t 2 t 3
t 2 t 3 t t 3 t t 2
=
i −
j +
2t 3t 2 1 3t 2 1 2t k
1 2t 3t 2
=(3t 4 − 2t 4 ) i − (3t 3 − t 3 ) j +(2t 2 − t 2 ) k = t 4 i − 2t 3 j + t 2 k
Since (a × b) · a = t 4 , −2t 3 ,t 2 · t,
t 2 ,t 3 = t 5 − 2t 5 + t 5 =0, a × b is orthogonal to a.
Since (a × b) · b = t 4 , −2t 3 ,t 2 · 1, 2t, 3t 2 = t 4 − 4t 4 +3t 4 =0, a × b is orthogonal to b.
9. According to the discussion preceding Theorem 8, i × j = k,so(i × j) × k = k × k = 0 [by Example 2].
11. (j − k) × (k − i) =(j − k) × k +(j − k) × (−i) by Property 3 of Theorem 8
= j × k +(−k) × k + j × (−i)+(−k) × (−i) by Property 4 of Theorem 8
=(j × k)+(−1)(k × k)+(−1)(j × i)+(−1) 2 (k × i) by Property 2 of Theorem 8
= i +(−1) 0 +(−1)(−k)+j = i + j + k by Example 2 and the
discussion preceding Theorem 8

F.
116 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ET TX.10 CHAPTER 12
13. (a) Since b × c is a vector, the dot product a · (b × c) is meaningful and is a scalar.
(b) b · c is a scalar, so a × (b · c) is meaningless, as the cross product is defined only for two vectors.
(c) Since b × c is a vector, the cross product a × (b × c) is meaningful and results in another vector.
(d) a · b is a scalar, so the cross product (a · b) × c is meaningless.
(e) Since (a · b) and (c · d) are both scalars, the cross product (a · b) × (c · d) is meaningless.
(f ) a × b and c × d are both vectors, so the dot product (a × b) · (c × d) is meaningful and is a scalar.
15. If we sketch u and v starting from the same initial point, we see that
the angle between them is 30 ◦ .UsingTheorem6,wehave
|u × v| = |u||v| sin 30 ◦ =(6)(8) 1
2
=24.
By the right-hand rule, u × v is directed into the page.
i j k
2 1
17. a × b =
1 2 1
=
0 1 3
1 3 i − 1 1
0 3 j + 1 2
k =(6− 1) i − (3 − 0) j +(1− 0) k =5i − 3 j + k
0 1 i j k
1 3
b × a =
0 1 3
=
1 2 1
2 1 i − 0 3
1 1 j + 0 1
k =(1− 6) i − (0 − 3) j +(0− 1) k = −5 i +3j − k
1 2 Notice a × b = −b × a here, as we know is always true by Theorem 8.
19. We know that the cross product of two vectors is orthogonal to both. So we calculate
i j k
−1 1
h1, −1, 1i×h0, 4, 4i =
1 −1 1
=
0 4 4
4 4 i − 1 1
0 4 j + 1 −1
k = −8 i − 4 j +4k.
0 4 h−8, −4, 4i
So two unit vectors orthogonal to both are ± √ = ± 64 + 16 + 16
and
√
2
6
,
√
1
6
, − √ 1
6
.
21. Let a = ha 1 ,a 2 ,a 3 i.Then
i j k 0 0
0 × a =
0 0 0 =
i −
a 2 a 3
a 1 a 2 a 3
0 0 j +
a 1 a 3
0 0 k = 0,
a 1 a 2
i j k
a 2 a 3
a × 0 =
a 1 a 2 a 3
=
0 0 0
0 0 i − a 1 a 3
0 0 j + a 1 a 2
0 0 k = 0.
23. a × b = ha 2b 3 − a 3b 2,a 3b 1 − a 1b 3,a 1b 2 − a 2b 1i
= h(−1)(b 2a 3 − b 3a 2) , (−1)(b 3a 1 − b 1a 3) , (−1)(b 1a 2 − b 2a 1)i
= − hb 2 a 3 − b 3 a 2 ,b 3 a 1 − b 1 a 3 ,b 1 a 2 − b 2 a 1 i = −b × a
h−8, −4, 4i
4 √ ,thatis, − √ 2
6
6
, − √ 1
6
,
√
1
6

F.
TX.10
SECTION 13.4 THE CROSS PRODUCT ET SECTION 12.4 ¤ 117
25. a × (b + c) =a ×hb 1 + c 1,b 2 + c 2,b 3 + c 3i
= ha 2 (b 3 + c 3 ) − a 3 (b 2 + c 2 ) , a 3 (b 1 + c 1 ) − a 1 (b 3 + c 3 ) , a 1 (b 2 + c 2 ) − a 2 (b 1 + c 1 )i
= ha 2 b 3 + a 2 c 3 − a 3 b 2 − a 3 c 2 , a 3 b 1 + a 3 c 1 − a 1 b 3 − a 1 c 3 , a 1 b 2 + a 1 c 2 − a 2 b 1 − a 2 c 1 i
= h(a 2 b 3 − a 3 b 2 )+(a 2 c 3 − a 3 c 2 ) , (a 3 b 1 − a 1 b 3 )+(a 3 c 1 − a 1 c 3 ) , (a 1 b 2 − a 2 b 1 )+(a 1 c 2 − a 2 c 1 )i
= ha 2b 3 − a 3b 2,a 3b 1 − a 1b 3,a 1b 2 − a 2b 1i + ha 2c 3 − a 3c 2,a 3c 1 − a 1c 3,a 1c 2 − a 2c 1i
=(a × b)+(a × c)
27. By plotting the vertices, we can see that the parallelogram is determined by the
vectors −→
AB = h2, 3i and −→
AD = h4, −2i. We know that the area of the parallelogram
determined by two vectors is equal to the length of the cross product of these vectors.
In order to compute the cross product, we consider the vector −→
AB as the threedimensional
vector h2, 3, 0i (and similarly for −→
AD), and then the area of
parallelogram ABCD is
−→
AB × −→
i j k
AD = 2 3 0
= |(0) i − (0) j +(−4 − 12) k| = |−16 k| =16
4 −2 0
29. (a) Because the plane through P , Q, andR contains the vectors −→ −→
PQand PR, a vector orthogonal to both of these vectors
(such as their cross product) is also orthogonal to the plane. Here −→
−→
PQ = h−1, 2, 0i and PR = h−1, 0, 3i,so
−→
PQ× −→
PR = h(2)(3) − (0)(0), (0)(−1) − (−1)(3), (−1)(0) − (2)(−1)i = h6, 3, 2i
Therefore, h6, 3, 2i (or any scalar multiple thereof) is orthogonal to the plane through P , Q,andR.
(b) Note that the area of the triangle determined by P , Q,andR is equal to half of the area of the parallelogram determined by
the three points. From part (a), the area of the parallelogram is −→ −→ √ PQ× PR = |h6, 3, 2i| = 36 + 9 + 4 = 7, so the area
of the triangle is 1 2 (7) = 7 2 .
31. (a) −→
PQ = h4, 3, −2i and
−→
PR = h5, 5, 1i, so a vector orthogonal to the plane through P , Q, andR is
−→
PQ× −→
PR = h(3)(1) − (−2)(5), (−2)(5) − (4)(1), (4)(5) − (3)(5)i = h13, −14, 5i [or any scalar mutiple thereof].
(b) The area of the parallelogram determined by −→ −→
PQ and PR is
−→ −→
PQ× PR = |h13, −14, 5i| = 132 +(−14) 2 +5 2 = √ √
390, so the area of triangle PQRis 1 2 390.
33. We know that the volume of the parallelepiped determined by a, b,andc is the magnitude of their scalar triple product, which
6 3 −1
1 2
is a · (b × c) =
0 1 2
=6
4 −2 5
−2 5 − 3 0 2
4 5 + (−1) 0 1
=6(5+4)− 3(0 − 8) − (0 − 4) = 82.
4 −2
Thus the volume of the parallelepiped is 82 cubic units.

F.
118 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ET TX.10 CHAPTER 12
35. a = −→
−→
−→
PQ = h2, 1, 1i, b = PR = h1, −1, 2i,andc = PS = h0, −2, 3i.
2 1 1
−1 2
a · (b × c) =
1 −1 2
=2
0 −2 3
−2 3 − 1 1 2
0 3 + 1 1 −1
=2− 3 − 2=−3,
0 −2
so the volume of the parallelepiped is 3 cubic units.
1 5 −2
−1 0
37. u · (v × w) =
3 −1 0
=1
5 9 −4
9 −4 − 5 3 0
5 −4 +(−2) 3 −1
=4+60− 64 = 0, which says that the volume
5 9 of the parallelepiped determined by u, v and w is 0, and thus these three vectors are coplanar.
39. The magnitude of the torque is |τ | = |r × F| = |r||F| sin θ =(0.18 m)(60 N)sin(70+10) ◦ =10.8sin80 ◦ ≈ 10.6 N·m.
41. Using the notation of the text, r = h0, 0.3, 0i and F has direction h0, 3, −4i. Theangleθ between them can be determined by
cos θ =
h0, 0.3, 0i·h0, 3, −4i
|h0, 0.3, 0i| |h0, 3, −4i|
⇒ cos θ = 0.9
(0.3)(5)
⇒ cos θ =0.6 ⇒ θ ≈ 53.1 ◦ .Then|τ | = |r||F| sin θ ⇒
100 = 0.3 |F| sin 53.1 ◦ ⇒ |F| ≈ 417 N.
43. (a) The distance between a point and a line is the length of the perpendicular
from the point to the line, here −→
PS = d. But referring to triangle PQS,
d = −→
PS = −→ −→
QP sin θ = |b| sin θ. Butθ is the angle between QP = b
and −→
|a × b|
QR = a. Thus by Theorem 6, sin θ =
|a||b|
and so d = |b| sin θ =
(b) a = −→
QR = h−1, −2, −1i and b = −→
QP = h1, −5, −7i. Then
|b||a × b|
|a||b|
=
|a × b|
.
|a|
a × b = h(−2)(−7) − (−1)(−5), (−1)(1) − (−1)(−7), (−1)(−5) − (−2)(1)i = h9, −8, 7i.
Thus the distance is d =
|a × b|
|a|
√
= √ 1
6 81 + 64 + 49 =
194 97
6
= . 3
45. (a − b) × (a + b) =(a − b) × a +(a − b) × b by Property 3 of Theorem 8
= a × a +(−b) × a + a × b +(−b) × b by Property 4 of Theorem 8
=(a × a) − (b × a)+(a × b) − (b × b) by Property 2 of Theorem 8 (with c = −1)
= 0 − (b × a)+(a × b) − 0 by Example 2
=(a × b)+(a × b) by Property 1 of Theorem 8
=2(a × b)
47. a × (b × c)+b × (c × a)+c × (a × b)
=[(a · c)b − (a · b)c]+[(b · a)c − (b · c)a]+[(c · b)a − (c · a)b] by Exercise 46
=(a · c)b − (a · b)c +(a · b)c − (b · c)a +(b · c)a − (a · c)b = 0

F.
TX.10 SECTION 13.5 EQUATIONS OF LINES AND PLANES ET SECTION 12.5 ¤ 119
49. (a) No. If a · b = a · c,thena · (b − c) =0,soa is perpendicular to b − c, which can happen if b 6=c. For example,
let a = h1, 1, 1i, b = h1, 0, 0i and c = h0, 1, 0i.
(b) No. If a × b = a × c then a × (b − c) =0, which implies that a is parallel to b − c, which of course can happen
if b 6=c.
(c) Yes. Since a · c = a · b, a is perpendicular to b − c, by part (a). From part (b), a is also parallel to b − c. Thussince
a 6=0 but is both parallel and perpendicular to b − c,wehaveb − c = 0,sob = c.
13.5 Equations of Lines and Planes ET 12.5
1. (a) True; each of the first two lines has a direction vector parallel to the direction vector of the third line, so these vectors are
each scalar multiples of the third direction vector. Then the first two direction vectors are also scalar multiples of each
other, so these vectors, and hence the two lines, are parallel.
(b) False; for example, the x-andy-axes are both perpendicular to the z-axis, yet the x-andy-axes are not parallel.
(c) True; each of the first two planes has a normal vector parallel to the normal vector of the third plane, so these two normal
vectors are parallel to each other and the planes are parallel.
(d) False; for example, the xy-andyz-planes are not parallel, yet they are both perpendicular to the xz-plane.
(e) False; the x-andy-axes are not parallel, yet they are both parallel to the plane z =1.
(f ) True; if each line is perpendicular to a plane, then the lines’ direction vectors are both parallel to a normal vector for the
plane. Thus, the direction vectors are parallel to each other and the lines are parallel.
(g) False; the planes y =1and z =1are not parallel, yet they are both parallel to the x-axis.
(h) True; if each plane is perpendicular to a line, then any normal vector for each plane is parallel to a direction vector for the
line. Thus, the normal vectors are parallel to each other and the planes are parallel.
(i) True; see Figure 9 and the accompanying discussion.
( j) False; they can be skew, as in Example 3.
(k) True. Consider any normal vector for the plane and any direction vector for the line. If the normal vector is perpendicular
to the direction vector, the line and plane are parallel. Otherwise, the vectors meet at an angle θ, 0 ◦ ≤ θ

F.
120 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ETTX.10
CHAPTER 12
9. v = 2 − 0, 1 − 1 2 , −3 − 1 = 2, 1 2 , −4 , and letting P 0 =(2, 1, −3), parametric equations are x =2+2t, y =1+ 1 2 t,
z = −3 − 4t, while symmetric equations are x − 2
2
= y − 1
1/2 = z +3
−4 or x − 2 =2y − 2= z +3
2
−4 .
11. Thelinehasdirectionv = h1, 2, 1i. LettingP 0 =(1, −1, 1), parametric equations are x =1+t, y = −1+2t, z =1+t
and symmetric equations are x − 1= y +1
2
= z − 1.
13. Direction vectors of the lines are v 1 = h−2 − (−4), 0 − (−6), −3 − 1i = h2, 6, −4i and
v 2 = h5 − 10, 3 − 18, 14 − 4i = h−5, −15, 10i, and since v 2 = − 5 2
v1, the direction vectors and thus the lines are parallel.
15. (a) The line passes through the point (1, −5, 6) and a direction vector for the line is h−1, 2, −3i, so symmetric equations for
thelineare x − 1
−1 = y +5 = z − 6
2 −3 .
(b) The line intersects the xy-plane when z =0, so we need x − 1
−1 = y +5 = 0 − 6
2 −3 or x − 1 =2 ⇒ x = −1,
−1
y +5
2
=2 ⇒ y = −1. Thus the point of intersection with the xy-plane is (−1, −1, 0). Similarly for the yz-plane,
we need x =0 ⇒ 1= y +5
2
= z − 6
−3
the xz-plane, we need y =0 ⇒ x − 1
−1 = 5 2 = z − 6
−3
at − 3 2 , 0, − 3 2
.
⇒
y = −3, z =3. Thus the line intersects the yz-plane at (0, −3, 3). For
⇒
x = − 3 , z = − 3 . So the line intersects the xz-plane
2 2
17. From Equation 4, the line segment from r 0 =2i − j +4k to r 1 =4i +6j + k is
r(t) =(1− t) r 0 + t r 1 =(1− t)(2 i − j +4k)+t(4 i +6j + k) =(2i − j +4k)+t(2 i +7j − 3 k), 0 ≤ t ≤ 1.
19. Since the direction vectors are v 1 = h−6, 9, −3i and v 2 = h2, −3, 1i,wehavev 1 = −3v 2 so the lines are parallel.
21. Since the direction vectors h1, 2, 3i and h−4, −3, 2i are not scalar multiples of each other, the lines are not parallel, so we
check to see if the lines intersect. The parametric equations of the lines are L 1 : x = t, y =1+2t, z =2+3t and L 2 :
x =3− 4s, y =2− 3s, z =1+2s. For the lines to intersect, we must be able to find one value of t and one value of s that
produce the same point from the respective parametric equations. Thus we need to satisfy the following three equations:
t =3− 4s, 1+2t =2− 3s, 2+3t =1+2s. Solving the first two equations we get t = −1, s =1and checking, we see
that these values don’t satisfy the third equation. Thus the lines aren’t parallel and don’t intersect, so they must be skew lines.
23. Since the plane is perpendicular to the vector h−2, 1, 5i,wecantakeh−2, 1, 5i as a normal vector to the plane.
(6, 3, 2) is a point on the plane, so setting a = −2, b =1, c =5and x 0 =6, y 0 =3, z 0 =2in Equation 7 gives
−2(x − 6) + 1(y − 3) + 5(z − 2) = 0 or −2x + y +5z =1to be an equation of the plane.
25. i + j − k = h1, 1, −1i is a normal vector to the plane and (1, −1, 1) is a point on the plane, so setting a =1, b =1, c = −1,
x 0 =1, y 0 = −1, z 0 =1in Equation 7 gives 1(x − 1) + 1[y − (−1)] − 1(z − 1) = 0 or x + y − z = −1 to be an equation
of the plane.
27. Since the two planes are parallel, they will have the same normal vectors. So we can take n = h2, −1, 3i, and an equation of
the plane is 2(x − 0) − 1(y − 0) + 3(z − 0) = 0 or 2x − y +3z =0.

F.
TX.10 SECTION 13.5 EQUATIONS OF LINES AND PLANES ET SECTION 12.5 ¤ 121
29. Since the two planes are parallel, they will have the same normal vectors. So we can take n = h3, 0, −7i, and an equation of
the plane is 3(x − 4) + 0[y − (−2)] − 7(z − 3) = 0 or 3x − 7z = −9.
31. Here the vectors a = h1 − 0, 0 − 1, 1 − 1i = h1, −1, 0i and b = h1 − 0, 1 − 1, 0 − 1i = h1, 0, −1i lie in the plane, so
a × b is a normal vector to the plane. Thus, we can take n = a × b = h1 − 0, 0+1, 0+1i = h1, 1, 1i. IfP 0 is the point
(0, 1, 1), an equation of the plane is 1(x − 0) + 1(y − 1) + 1(z − 1) = 0 or x + y + z =2.
33. Here the vectors a = h8 − 3, 2 − (−1), 4 − 2i = h5, 3, 2i and b = h−1 − 3, −2 − (−1), −3 − 2i = h−4, −1, −5i lie in
the plane, so a normal vector to the plane is n = a × b = h−15 + 2, −8+25, −5+12i = h−13, 17, 7i andanequationof
the plane is −13(x − 3) + 17[y − (−1)] + 7(z − 2) = 0 or −13x +17y +7z = −42.
35. If we first find two nonparallel vectors in the plane, their cross product will be a normal vector to the plane. Since the given
line lies in the plane, its direction vector a = h−2, 5, 4i is one vector in the plane. We can verify that the given point (6, 0, −2)
does not lie on this line, so to find another nonparallel vector b which lies in the plane, we can pick any point on the line and
find a vector connecting the points. If we put t =0, we see that (4, 3, 7) is on the line, so
b = h6 − 4, 0 − 3, −2 − 7i = h2, −3, −9i and n = a × b = h−45 + 12, 8 − 18, 6 − 10i = h−33, −10, −4i. Thus,an
equation of the plane is −33(x − 6) − 10(y − 0) − 4[z − (−2)] = 0 or 33x +10y +4z = 190.
37. A direction vector for the line of intersection is a = n 1 × n 2 = h1, 1, −1i×h2, −1, 3i = h2, −5, −3i,anda is parallel to the
desired plane. Another vector parallel to the plane is the vector connecting any point on the line of intersection to the given
point (−1, 2, 1) in the plane. Setting x =0, the equations of the planes reduce to y − z =2and −y +3z =1with
simultaneous solution y = 7 and z = 3 .Soapointonthelineis 2 2
0, 7 , 3
2 2 and another vector parallel to the plane is
−1, −
3
, − 1
2 2 . Then a normal vector to the plane is n = h2, −5, −3i× −1, −
3
, − 1
2 2 = h−2, 4, −8i and an equation of
the plane is −2(x +1)+4(y − 2) − 8(z − 1) = 0 or x − 2y +4z = −1.
39. To find the x-intercept we set y = z =0in the equation 2x +5y + z =10
and obtain 2x =10 ⇒ x =5so the x-intercept is (5, 0, 0). When
x = z =0we get 5y =10 ⇒ y =2,sothey-intercept is (0, 2, 0).
Setting x = y =0gives z =10,sothez-intercept is (0, 0, 10) and we
graph the portion of the plane that lies in the first octant.
41. Setting y = z =0in the equation 6x − 3y +4z =6gives 6x =6 ⇒
x =1,whenx = z =0we have −3y =6 ⇒ y = −2,andx = y =0
implies 4z =6 ⇒ z = 3 2
, so the intercepts are (1, 0, 0), (0, −2, 0),and
(0, 0, 3 ).Thefigure shows the portion of the plane cut off by the coordinate
2
planes.

F.
122 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ETTX.10
CHAPTER 12
43. Substitute the parametric equations of the line into the equation of the plane: (3 − t) − (2 + t)+2(5t) =9 ⇒
8t =8 ⇒ t =1. Therefore, the point of intersection of the line and the plane is given by x =3− 1=2, y =2+1=3,
and z =5(1)=5, that is, the point (2, 3, 5).
45. Parametric equations for the line are x = t, y =1+t, z = 1 2
t and substituting into the equation of the plane gives
4(t) − (1 + t)+3 1
t 2
=8 ⇒ 9 t =9 ⇒ t =2. Thus x =2, y =1+2=3, z = 1 2 2
(2) = 1 and the point of
intersection is (2, 3, 1).
47. Setting x =0,weseethat(0, 1, 0) satisfies the equations of both planes, so that they do in fact have a line of intersection.
v = n 1 × n 2 = h1, 1, 1i×h1, 0, 1i = h1, 0, −1i is the direction of this line. Therefore, direction numbers of the intersecting
line are 1, 0, −1.
49. Normal vectors for the planes are n 1 = h1, 4, −3i and n 2 = h−3, 6, 7i, so the normals (and thus the planes) aren’t parallel.
But n 1 · n 2 = −3+24− 21 = 0, so the normals (and thus the planes) are perpendicular.
51. Normal vectors for the planes are n 1 = h1, 1, 1i and n 2 = h1, −1, 1i. The normals are not parallel, so neither are the planes.
Furthermore, n 1 · n 2 =1− 1+1=16= 0, so the planes aren’t perpendicular. The angle between them is given by
cos θ = n 1 · n 2
|n 1 ||n 2 | = 1 √
3
√
3
= 1 3
⇒ θ =cos −1 1
3
≈ 70.5 ◦ .
53. The normals are n 1 = h1, −4, 2i and n 2 = h2, −8, 4i. Sincen 2 =2n 1 , the normals (and thus the planes) are parallel.
55. (a) To find a point on the line of intersection, set one of the variables equal to a constant, say z =0. (This will fail if the line of
intersection does not cross the xy-plane;inthatcase,trysettingx or y equal to 0.) The equations of the two planes reduce
to x + y =1and x +2y =1. Solving these two equations gives x =1, y =0. Thus a point on the line is (1, 0, 0).
Avectorv in the direction of this intersecting line is perpendicular to the normal vectors of both planes, so we can take
v = n 1 × n 2 = h1, 1, 1i×h1, 2, 2i = h2 − 2, 1 − 2, 2 − 1i = h0, −1, 1i. By Equations 2, parametric equations for the
line are x =1, y = −t, z = t.
(b) The angle between the planes satisfies cos θ = n 1 · n 2
|n 1 ||n 2 | = 1+2+2 √ √ = 5
5
3 9 3 √ 3 . Therefore θ =cos−1 3 √ 3
57. Setting z =0, the equations of the two planes become 5x − 2y =1and 4x + y =6. Solving these two equations gives
x =1, y =2soapointonthelineofintersectionis(1, 2, 0). Avectorv in the direction of this intersecting line is
≈ 15.8 ◦ .
perpendicular to the normal vectors of both planes. So we can use v = n 1 × n 2 = h5, −2, −2i×h4, 1, 1i = h0, −13, 13i or
equivalently we can take v = h0, −1, 1i, and symmetric equations for the line are x =1, y − 2
−1
= z or x =1, y − 2=−z.
1
59. The distance from a point (x, y, z) to (1, 0, −2) is d 1 = (x − 1) 2 + y 2 +(z +2) 2 and the distance from (x, y, z) to
(3, 4, 0) is (x − 3) 2 +(y − 4) 2 + z 2 . The plane consists of all points (x, y, z) where d 1 = d 2 ⇒ d 2 1 = d 2 2 ⇔
(x − 1) 2 + y 2 +(z +2) 2 =(x − 3) 2 +(y − 4) 2 + z 2 ⇔
x 2 − 2x + y 2 + z 2 +4z +5=x 2 − 6x + y 2 − 8y + z 2 +25 ⇔ 4x +8y +4z =20so an equation for the plane is
4x +8y +4z =20or equivalently x +2y + z =5.
Alternatively, you can argue that the segment joining points (1, 0, −2) and (3, 4, 0) is perpendicular to the plane and the plane
includes the midpoint of the segment.

F.
TX.10 SECTION 13.5 EQUATIONS OF LINES AND PLANES ET SECTION 12.5 ¤ 123
61. The plane contains the points (a, 0, 0), (0,b,0) and (0, 0,c). Thus the vectors a = h−a, b, 0i and b = h−a, 0,ci lie in the
plane, and n = a × b = hbc − 0, 0+ac, 0+abi = hbc, ac, abi is a normal vector to the plane. The equation of the plane is
therefore bcx + acy + abz = abc +0+0or bcx + acy + abz = abc. Notice that if a 6= 0, b 6= 0and c 6= 0then we can
rewrite the equation as x a + y b + z c
=1. This is a good equation to remember!
63. Two vectors which are perpendicular to the required line are the normal of the given plane, h1, 1, 1i, and a direction vector for
the given line, h1, −1, 2i. So a direction vector for the required line is h1, 1, 1i×h1, −1, 2i = h3, −1, −2i. Thus L is given
by hx, y, zi = h0, 1, 2i + th3, −1, −2i, or in parametric form, x =3t, y =1− t, z =2− 2t.
65. Let P i have normal vector n i .Thenn 1 = h4, −2, 6i, n 2 = h4, −2, −2i, n 3 = h−6, 3, −9i, n 4 = h2, −1, −1i. Now
n 1 = − 2 3 n 3,son 1 and n 3 are parallel, and hence P 1 and P 3 are parallel; similarly P 2 and P 4 are parallel because n 2 =2n 4 .
However, n 1 and n 2 are not parallel. 0, 0, 1 2
lies on P1,butnotonP 3, so they are not the same plane, but both P 2 and P 4
contain the point (0, 0, −3),sothesetwoplanesareidentical.
67. Let Q =(1, 3, 4) and R =(2, 1, 1), points on the line corresponding to t =0and t =1.Let
P =(4, 1, −2). Thena = −→
QR = h1, −2, −3i, b = −→
QP = h3, −2, −6i. The distance is
d =
|a × b|
|a|
=
|h1, −2, −3i×h3, −2, −6i|
|h1, −2, −3i|
=
|h6, −3, 4i|
|h1, −2, −3i| =
62 +(−3) 2 +4 2
12 +(−2) 2 +(−3) 2 = √
61
√
14
=
69. By Equation 9, the distance is D = |ax 1 + by 1 + cz 1 + d| |3(1) + 2(−2) + 6(4) − 5|
√ = √ = √ |18| = 18
a2 + b 2 + c 2 32 +2 2 +6 2 49 7 .
64
14 .
71. Put y = z =0in the equation of the first plane to get the point (2, 0, 0) on the plane. Because the planes are parallel, the
distance D between them is the distance from (2, 0, 0) to the second plane. By Equation 9,
D =
|4(2) − 6(0) + 2(0) − 3|
42 +(−6) 2 +(2) 2 = 5 √
56
= 5
2 √ 14 or 5 √ 14
28 .
73. The distance between two parallel planes is the same as the distance between a point on one of the planes and the other plane.
Let P 0 =(x 0,y 0,z 0) be a point on the plane given by ax + by + cz + d 1 =0.Thenax 0 + by 0 + cz 0 + d 1 =0and the
distance between P 0 and the plane given by ax + by + cz + d 2 =0is, from Equation 9,
D = |ax 0 + by 0 + cz 0 + d 2 |
√
a2 + b 2 + c 2 = |−d 1 + d 2 |
√
a2 + b 2 + c 2 = |d 1 − d 2 |
√
a2 + b 2 + c 2 .
75. L 1 : x = y = z ⇒ x = y (1). L 2 : x +1=y/2 =z/3 ⇒ x +1=y/2 (2). The solution of (1)and(2)is
x = y = −2. However,whenx = −2, x = z ⇒ z = −2,butx +1=z/3 ⇒ z = −3, a contradiction. Hence the
lines do not intersect. For L 1, v 1 = h1, 1, 1i,andforL 2, v 2 = h1, 2, 3i, so the lines are not parallel. Thus the lines are skew
lines. If two lines are skew, they can be viewed as lying in two parallel planes and so the distance between the skew lines
would be the same as the distance between these parallel planes. The common normal vector to the planes must be
perpendicular to both h1, 1, 1i and h1, 2, 3i, the direction vectors of the two lines. So set

F.
124 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ETTX.10
CHAPTER 12
n = h1, 1, 1i×h1, 2, 3i = h3 − 2, −3+1, 2 − 1i = h1, −2, 1i. From above, we know that (−2, −2, −2) and (−2, −2, −3)
are points of L 1 and L 2 respectively. So in the notation of Equation 8, 1(−2) − 2(−2) + 1(−2) + d 1 =0 ⇒ d 1 =0and
1(−2) − 2(−2) + 1(−3) + d 2 =0 ⇒ d 2 =1.
By Exercise 73, the distance between these two skew lines is D =
|0 − 1|
√ 1+4+1
= 1 √
6
.
Alternate solution (without reference to planes): A vector which is perpendicular to both of the lines is
n = h1, 1, 1i×h1, 2, 3i = h1, −2, 1i.Pickanypointoneachofthelines,say(−2, −2, −2) and (−2, −2, −3), and form the
vector b = h0, 0, 1i connecting the two points. The distance between the two skew lines is the absolute value of the scalar
projection of b along n,thatis,D =
|n · b|
|n|
=
|1 · 0 − 2 · 0+1· 1|
√ 1+4+1
= 1 √
6
.
77. If a 6= 0,thenax + by + cz + d =0 ⇒ a(x + d/a)+b(y − 0) + c(z − 0) = 0 which by (7) is the scalar equation of the
plane through the point (−d/a, 0, 0) with normal vector ha, b, ci. Similarly, if b 6= 0(or if c 6= 0) the equation of the plane can
be rewritten as a(x − 0) + b(y + d/b)+c(z − 0) = 0 [or as a(x − 0) + b(y − 0) + c(z + d/c) =0] which by (7) is the
scalar equation of a plane through the point (0, −d/b, 0) [or the point (0, 0, −d/c)] with normal vector ha, b, ci.
13.6 Cylinders and Quadric Surfaces ET 12.6
1. (a) In R 2 , the equation y = x 2 represents a parabola.
(b) In R 3 , the equation y = x 2 doesn’t involve z,soany
horizontal plane with equation z = k intersects the graph
in a curve with equation y = x 2 . Thus, the surface is a
parabolic cylinder, made up of infinitely many shifted
copies of the same parabola. The rulings are parallel to
the z-axis.
(c) In R 3 , the equation z = y 2 also represents a parabolic
cylinder. Since x doesn’t appear, the graph is formed by
moving the parabola z = y 2 in the direction of the x-axis.
Thus, the rulings of the cylinder are parallel to the x-axis.

F.
SECTION TX.1013.6 CYLINDERS AND QUADRIC SURFACES ET SECTION 12.6 ¤ 125
3. Since x is missing from the equation, the vertical traces
y 2 +4z 2 =4, x = k, are copies of the same ellipse in the
plane x = k. Thus,thesurfacey 2 +4z 2 =4is an elliptic
cylinder with rulings parallel to the x-axis.
5. Since z is missing, each horizontal trace x = y 2 , z = k,
is a copy of the same parabola in the plane z = k. Thus,
the surface x − y 2 =0is a parabolic cylinder with rulings
parallel to the z-axis.
7. Since y is missing, each vertical trace z =cosx, y = k is a copy of a cosine curve in the plane y = k. Thus, the surface
z =cosx is a cylindrical surface with rulings parallel to the y-axis.
9. (a) The traces of x 2 + y 2 − z 2 =1in x = k are y 2 − z 2 =1− k 2 , a family of hyperbolas. (Note that the hyperbolas are
oriented differently for −1

F.
126 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ETTX.10
CHAPTER 12
11. For x = y 2 +4z 2 , the traces in x = k are y 2 +4z 2 = k. Whenk>0 we
have a family of ellipses. When k =0we have just a point at the origin, and
the trace is empty for k 1,andthetracesinz = k are 4y 2 − x 2 =4+k 2 ,afamily
of hyperbolas. Thus the surface is a hyperboloid of two sheets with
axis the y-axis.
17. 36x 2 + y 2 +36z 2 =36. The traces in x = k are y 2 +36z 2 = 36(1 − k 2 ),
a family of ellipses for |k| < 1. (The traces are a single point for |k| =1
and are empty for |k| > 1.) The traces in y = k are the circles
36x 2 +36z 2 =36− k 2 ⇔ x 2 + z 2 =1− 1 36 k2 , |k| < 6,andthe
traces in z = k are the ellipses 36x 2 + y 2 = 36(1 − k 2 ), |k| < 1. The
graph is an ellipsoid centered at the origin with intercepts x = ±1, y = ±6,
z = ±1.
19. y = z 2 − x 2 . The traces in x = k are the parabolas y = z 2 − k 2 ;
the traces in y = k are k = z 2 − x 2 , which are hyperbolas (note the hyperbolas
are oriented differently for k>0 than for k

F.
SECTION TX.1013.6 CYLINDERS AND QUADRIC SURFACES ET SECTION 12.6 ¤ 127
23. This is the equation of a hyperboloid of one sheet, with a = b = c =1. Sincethecoefficient of y 2 is negative, the axis of the
hyperboloid is the y-axis, hence the correct graph is II.
25. There are no real values of x and z that satisfy this equation for y0 in an ellipse. Notice that y occurs to the first power whereas x and z
occur to the second power. So the surface is an elliptic paraboloid with axis the y-axis. Its graph is VI.
27. This surface is a cylinder because the variable y is missing from the equation. The intersection of the surface and the xz-plane
is an ellipse. So the graph is VIII.
29. z 2 =4x 2 +9y 2 +36or −4x 2 − 9y 2 + z 2 =36or
− x2
9 − y2
4 + z2
=1represents a hyperboloid of two
36
sheets with axis the z-axis.
31. x =2y 2 +3z 2 or x = y2
1/2 + z2
1/3 or x 6 = y2
3 + z2
2
represents an elliptic paraboloid with vertex (0, 0, 0) and
axis the x-axis.
33. Completing squares in y and z gives
4x 2 +(y − 2) 2 +4(z − 3) 2 =4or
x 2 (y − 2)2
+ +(z − 3) 2 =1, an ellipsoid with
4
center (0, 2, 3).
35. Completing squares in all three variables gives
(x − 2) 2 − (y +1) 2 +(z − 1) 2 =0or
(y +1) 2 =(x − 2) 2 +(z − 1) 2 , a circular cone with
center (2, −1, 1) and axis the horizontal line x =2,
z =1.
37. Solving the equation for z we get z = ± 1+4x 2 + y 2 , so we plot separately z = 1+4x 2 + y 2 and
z = − 1+4x 2 + y 2 .
To restrict the z-range as in the second graph, we can use the option view = -4..4 in Maple’s plot3d command, or
PlotRange -> {-4,4}inMathematica’sPlot3D command.

F.
128 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ETTX.10
CHAPTER 12
39. Solving the equation for z we get z = ± 4x 2 + y 2 , so we plot separately z = 4x 2 + y 2 and z = − 4x 2 + y 2 .
41.
43. The surface is a paraboloid of revolution (circular paraboloid) with vertex at the origin, axis the y-axis and opens to the right.
Thus the trace in the yz-plane is also a parabola: y = z 2 , x =0. The equation is y = x 2 + z 2 .
45. Let P =(x, y, z) be an arbitrary point equidistant from(−1, 0, 0) and the plane x =1. Then the distance from P to
(−1, 0, 0) is (x +1) 2 + y 2 + z 2 and the distance from P to the plane x =1is |x − 1| / √ 1 2 = |x − 1|
(by Equation 13.5.9 [ ET 12.5.9]). So |x − 1| = (x +1) 2 + y 2 + z 2 ⇔ (x − 1) 2 =(x +1) 2 + y 2 + z 2 ⇔
x 2 − 2x +1=x 2 +2x +1+y 2 + z 2 ⇔ −4x = y 2 + z 2 . Thus the collection of all such points P is a circular
paraboloid with vertex at the origin, axis the x-axis, which opens in the negative direction.
47. (a) An equation for an ellipsoid centered at the origin with intercepts x = ±a, y = ±b,andz = ±c is x2
a 2 + y2
b 2 + z2
c 2 =1.
Here the poles of the model intersect the z-axis at z = ±6356.523 and the equator intersects the x-andy-axes at
x = ±6378.137, y = ±6378.137,soanequationis
(b) Traces in z = k are the circles
x 2 + y 2 =(6378.137) 2 −
x 2
(6378.137) + y 2
2 (6378.137) + z 2
2 (6356.523) =1 2
x 2
(6378.137) + y 2
2 (6378.137) =1− k 2
2
2 6378.137
k 2 .
6356.523
(6356.523) 2 ⇔

F.
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CHAPTER 13 REVIEW ET CHAPTER 12 ¤ 129
(c) To identify the traces in y = mx we substitute y = mx into the equation of the ellipsoid:
As expected, this is a family of ellipses.
x 2
(6378.137) 2 + (mx)2
(6378.137) 2 + z 2
(6356.523) 2 =1
(1 + m 2 )x 2
(6378.137) 2 + z 2
(6356.523) 2 =1
x 2
(6378.137) 2 /(1 + m 2 ) + z 2
(6356.523) =1 2
49. If (a, b, c) satisfies z = y 2 − x 2 ,thenc = b 2 − a 2 . L 1: x = a + t, y = b + t, z = c +2(b − a)t,
L 2 : x = a + t, y = b − t, z = c − 2(b + a)t. Substitute the parametric equations of L 1 into the equation
of the hyperbolic paraboloid in order to find the points of intersection: z = y 2 − x 2
⇒
c +2(b − a)t =(b + t) 2 − (a + t) 2 = b 2 − a 2 +2(b − a)t ⇒ c = b 2 − a 2 .Asthisistrueforallvaluesoft,
L 1 lies on z = y 2 − x 2 . Performing similar operations with L 2 gives: z = y 2 − x 2
⇒
c − 2(b + a)t =(b − t) 2 − (a + t) 2 = b 2 − a 2 − 2(b + a)t ⇒ c = b 2 − a 2 . This tells us that all of L 2 also lies on
z = y 2 − x 2 .
51. The curve of intersection looks like a bent ellipse. The projection
of this curve onto the xy-plane is the set of points (x, y, 0) which
satisfy x 2 + y 2 =1− y 2 ⇔ x 2 +2y 2 =1 ⇔
x 2 +
y 2
1/
√
2
2
=1.Thisisanequationofanellipse.
13 Review ET 12
1. A scalar is a real number, while a vector is a quantity that has both a real-valued magnitude and a direction.
2. To add two vectors geometrically, we can use either the Triangle Law or the Parallelogram Law, as illustrated in Figures 3
and 4 in Section 13.2 [ ET 12.2]. Algebraically, we add the corresponding components of the vectors.
3. For c>0, c a is a vector with the same direction as a and length c times the length of a. Ifc

F.
130 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ETTX.10
CHAPTER 12
6. The dot product can be used to find the angle between two vectors and the scalar projection of one vector onto another. In
particular, the dot product can determine if two vectors are orthogonal. Also, the dot product can be used to determine the
work done moving an object given the force and displacement vectors.
7. See the boxed equations on page 819 [ ET 783] as well as Figures 4 and 5 and the accompanying discussion on pages 818–19
[ ET 782–83] .
8. See Theorem 13.4.6 [ ET 12.4.6] and the preceding discussion; use either (1) or (4) in Section 13.4 [ ET 12.4].
9. The cross product can be used to create a vector orthogonal to two given vectors as well as to determine if two vectors are
parallel. The cross product can also be used to find the area of a parallelogram determined by two vectors. In addition, the
cross product can be used to determine torque if the force and position vectors are known.
10. (a) The area of the parallelogram determined by a and b is the length of the cross product: |a × b|.
(b) The volume of the parallelepiped determined by a, b,andc is the magnitude of their scalar triple product: |a · (b × c)|.
11. If an equation of the plane is known, it can be written as ax + by + cz + d =0. A normal vector, which is perpendicular to the
plane, is ha, b, ci (or any scalar multiple of ha, b, ci). If an equation is not known, we can use points on the plane to find two
non-parallel vectors which lie in the plane. The cross product of these vectors is a vector perpendicular to the plane.
12. The angle between two intersecting planes is defined as the acute angle between their normal vectors. We can find this angle
using Corollary 13.3.6 [ ET 12.3.6].
13. See (1), (2), and (3) in Section 13.5 [ ET 12.5].
14. See (5), (6), and (7) in Section 13.5 [ ET 12.5].
15. (a) Two (nonzero) vectors are parallel if and only if one is a scalar multiple of the other. In addition, two nonzero vectors are
parallel if and only if their cross product is 0.
(b) Two vectors are perpendicular if and only if their dot product is 0.
(c) Two planes are parallel if and only if their normal vectors are parallel.
16. (a) Determine the vectors −→
PQ = ha1 ,a 2 ,a 3 i and −→
PR = hb 1 ,b 2 ,b 3 i. If there is a scalar t such that
ha 1 ,a 2 ,a 3 i = t hb 1 ,b 2 ,b 3 i, then the vectors are parallel and the points must all lie on the same line.
Alternatively, if −→
PQ×
−→
PR = 0,then
−→
PQand
−→
PRare parallel, so P , Q,andR are collinear.
Thirdly, an algebraic method is to determine an equation of the line joining two of the points, and then check whether or
not the third point satisfies this equation.
(b) Find the vectors −→
PQ = a,
−→
PR = b,
−→
PS = c. a × b is normal to the plane formed by P , Q and R,andsoS lies on this
plane if a × b and c are orthogonal, that is, if (a × b) · c =0. (Or use the reasoning in Example 5 in Section 13.4
[ ET 12.4].)
Alternatively, find an equation for the plane determined by three of the points and check whether or not the fourth point
satisfies this equation.

F.
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CHAPTER 13 REVIEW ET CHAPTER 12 ¤ 131
17. (a) See Exercise 13.4.43 [ ET 12.4.43].
(b) See Example 8 in Section 13.5 [ ET 12.5].
(c) See Example 10 in Section 13.5 [ ET 12.5].
18. The traces of a surface are the curves of intersection of the surface with planes parallel to the coordinate planes. We can find
the trace in the plane x = k (parallel to the yz-plane) by setting x = k and determining the curve represented by the resulting
equation. Traces in the planes y = k (parallel to the xz-plane) and z = k (parallel to the xy-plane) are found similarly.
19. SeeTable1inSection13.6[ET12.6].
1. True, by Theorem 13.3.2 [ ET 12.3.2], property 2.
3. True. If θ istheanglebetweenu and v, then by Theorem 13.4.6 [ ET 12.4.6],
|u × v| = |u||v| sin θ = |v||u| sin θ = |v × u|.
(Or, by Theorem 13.4.8 [ ET 12.4.8], |u × v| = |−v × u| = |−1||v × u| = |v × u|.)
5. Theorem 13.4.8 [ ET 12.4.8], property 2 tells us that this is true.
7. This is true by Theorem 13.4.8 [ ET 12.4.8], property 5.
9. This is true because u × v is orthogonal to u (see Theorem 13.4.5 [ ET 12.4.5]), and the dot product of two orthogonal vectors
is 0.
11. If |u| =1, |v| =1and θ is the angle between these two vectors (so 0 ≤ θ ≤ π), then by Theorem 13.4.6 [ ET 12.4.6],
|u × v| = |u||v| sin θ =sinθ, which is equal to 1 ifandonlyifθ = π (that is, if and only if the two vectors are orthogonal).
2
Therefore, the assertion that the cross product of two unit vectors is a unit vector is false.
13. This is false. In R 2 , x 2 + y 2 =1represents a circle, but (x, y, z) | x 2 + y 2 =1 represents a three-dimensional surface,
namely, a circular cylinder with axis the z-axis.
15. False. For example, i · j = 0 but i 6=0 and j 6=0.
17. This is true. If u and v are both nonzero, then by (7) in Section 13.3 [ET 12.3], u · v =0implies that u and v are orthogonal.
But u × v = 0 implies that u and v are parallel (see Corollary 13.4.7 [ET 12.4.7]). Two nonzero vectors can’t be both parallel
and orthogonal, so at least one of u, v must be 0.
1. (a) The radius of the sphere is the distance between the points (−1, 2, 1) and (6, −2, 3), namely,
[6 − (−1)]2 +(−2 − 2) 2 +(3− 1) 2 = √ 69. By the formula for an equation of a sphere (see page 804 [ET 768]),
an equation of the sphere with center (−1, 2, 1) and radius √ 69 is (x +1) 2 +(y − 2) 2 +(z − 1) 2 =69.

F.
132 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ETTX.10
CHAPTER 12
(b) The intersection of this sphere with the yz-plane is the set of points on the sphere whose x-coordinate is 0. Putting x =0
into the equation, we have (y − 2) 2 +(z − 1) 2 =68,x=0which represents a circle in the yz-plane with center (0, 2, 1)
and radius √ 68.
(c) Completing squares gives (x − 4) 2 +(y +1) 2 +(z +3) 2 = −1+16+1+9=25. Thus the sphere is centered at
(4, −1, −3) and has radius 5.
3. u · v = |u||v| cos 45 ◦ =(2)(3) √ 2
2 =3√ 2. |u × v| = |u||v| sin 45 ◦ =(2)(3) √ 2
2 =3√ 2.
By the right-hand rule, u × v is directed out of the page.
5. For the two vectors to be orthogonal, we need h3, 2,xi·h2x, 4,xi =0 ⇔ (3)(2x) + (2)(4) + (x)(x) =0 ⇔
x 2 +6x +8=0 ⇔ (x +2)(x +4)=0 ⇔ x = −2 or x = −4.
7. (a) (u × v) · w = u · (v × w) =2
(b) u · (w × v) =u · [− (v × w)] = −u · (v × w) =−2
(c) v · (u × w) =(v × u) · w = − (u × v) · w = −2
(d) (u × v) · v = u · (v × v) =u · 0 =0
9. For simplicity, consider a unit cube positioned with its back left corner at the origin. Vector representations of the diagonals
11.
joining the points (0, 0, 0) to (1, 1, 1) and (1, 0, 0) to (0, 1, 1) are h1, 1, 1i and h−1, 1, 1i. Letθ be the angle between these
two vectors. h1, 1, 1i·h−1, 1, 1i = −1+1+1=1=|h1, 1, 1i| |h−1, 1, 1i| cos θ =3cosθ ⇒ cos θ = 1 3
⇒
θ =cos −1 1
3
≈ 71 ◦ .
−→
AB = h1, 0, −1i, −→
AC = h0, 4, 3i,so
(a) a vector perpendicular to the plane is −→
AB × −→
AC = h0+4, −(3 + 0), 4 − 0i = h4, −3, 4i.
(b) 1 2
−→
AB × −→
√
AC = 1 √
2 16 + 9 + 16 = 41
. 2
13. Let F 1 be the magnitude of the force directed 20 ◦ away from the direction of shore, and let F 2 be the magnitude of the other
force. Separating these forces into components parallel to the direction of the resultant force and perpendicular to it gives
F 1 cos 20 ◦ + F 2 cos 30 ◦ =255 (1),andF 1 sin 20 ◦ − F 2 sin 30 ◦ sin 30 ◦
=0 ⇒ F 1 = F 2 (2). Substituting (2)
sin 20◦ into (1) gives F 2 (sin 30 ◦ cot 20 ◦ +cos30 ◦ )=255 ⇒ F 2 ≈ 114 N. Substituting this into (2) gives F 1 ≈ 166 N.
15. The line has direction v = h−3, 2, 3i. LettingP 0 =(4, −1, 2), parametric equations are
x =4− 3t, y = −1+2t, z =2+3t.
17. A direction vector for the line is a normal vector for the plane, n = h2, −1, 5i, and parametric equations for the line are
x = −2+2t, y =2− t, z =4+5t.

F.
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CHAPTER 13 REVIEW ET CHAPTER 12 ¤ 133
19. Here the vectors a = h4 − 3, 0 − (−1) , 2 − 1i = h1, 1, 1i and b = h6 − 3, 3 − (−1), 1 − 1i = h3, 4, 0i lie in the plane,
so n = a × b = h−4, 3, 1i is a normal vector to the plane and an equation of the plane is
−4(x − 3) + 3(y − (−1)) + 1(z − 1) = 0 or −4x +3y + z = −14.
21. Substitution of the parametric equations into the equation of the plane gives 2x − y + z =2(2− t) − (1 + 3t)+4t =2 ⇒
−t +3=2 ⇒ t =1.Whent =1, the parametric equations give x =2− 1=1, y =1+3=4and z =4. Therefore,
the point of intersection is (1, 4, 4).
23. Since the direction vectors h2, 3, 4i and h6, −1, 2i aren’t parallel, neither are the lines. For the lines to intersect, the three
equations 1+2t = −1+6s, 2+3t =3− s, 3+4t = −5+2s must be satisfied simultaneously. Solving the first two
equations gives t = 1 , s = 2 and checking we see these values don’t satisfy the third equation. Thus the lines aren’t parallel
5 5
andtheydon’tintersect,sotheymustbeskew.
25. n 1 = h1, 0, −1i and n 2 = h0, 1, 2i. Settingz =0, it is easy to see that (1, 3, 0) is a point on the line of intersection of
x − z =1and y +2z =3. The direction of this line is v 1 = n 1 × n 2 = h1, −2, 1i. A second vector parallel to the desired
plane is v 2 = h1, 1, −2i, since it is perpendicular to x + y − 2z =1. Therefore, the normal of the plane in question is
n = v 1 × v 2 = h4 − 1, 1+2, 1+2i =3h1, 1, 1i. Taking(x 0 ,y 0 ,z 0 )=(1, 3, 0), the equation we are looking for is
(x − 1) + (y − 3) + z =0 ⇔ x + y + z =4.
27. By Exercise 13.5.73 [ ET 12.5.73], D =
|2 − 24|
√
26
= 22 √
26
.
29. The equation x = z represents a plane perpendicular to
the xz-plane and intersecting the xz-plane in the line
x = z, y =0.
31. The equation x 2 = y 2 +4z 2 represents a (right elliptical)
cone with vertex at the origin and axis the x-axis.
33. An equivalent equation is −x 2 + y2
4 − z2 =1,a
hyperboloid of two sheets with axis the y-axis. For
|y| > 2, traces parallel to the xz-plane are circles.
35. Completing the square in y gives
4x 2 +4(y − 1) 2 + z 2 =4or x 2 +(y − 1) 2 + z2
4 =1,
an ellipsoid centered at (0, 1, 0).

F.
134 ¤ CHAPTER 13 VECTORSANDTHEGEOMETRYOFSPACE ETTX.10
CHAPTER 12
37. 4x 2 + y 2 =16 ⇔ x2
4 + y2
x2
=1. The equation of the ellipsoid is
16 4 + y2
16 + z2
=1, since the horizontal trace in the
c 2
plane z =0must be the original ellipse. The traces of the ellipsoid in the yz-plane must be circles since the surface is obtained
by rotation about the x-axis. Therefore, c 2 =16and the equation of the ellipsoid is x2
4 + y2
16 + z2
16 =1 ⇔
4x 2 + y 2 + z 2 =16.

F.
TX.10
PROBLEMS PLUS
1. Since three-dimensional situations are often difficult to visualize and work with, let
us first try to find an analogous problem in two dimensions. The analogue of a cube
is a square and the analogue of a sphere is a circle. Thus a similar problem in two
dimensions is the following: if five circles with the same radius r are contained in a
square of side 1 m so that the circles touch each other and four of the circles touch
two sides of the square, find r.
The diagonal of the square is √ 2. The diagonal is also 4r +2x. Butx is the diagonal of a smaller square of side r. Therefore
x = √ 2 r ⇒ √ 2=4r +2x =4r +2 √ 2 r = 4+2 √ 2 r ⇒ r = √ 2
4+2 √ 2 .
Let’s use these ideas to solve the original three-dimensional problem. The diagonal of the cube is √ 1 2 +1 2 +1 2 = √ 3.
The diagonal of the cube is also 4r +2x where x is the diagonal of a smaller cube with edge r. Therefore
x = √ r 2 + r 2 + r 2 = √ 3 r ⇒ √ 3=4r +2x =4r +2 √ 3 r = 4+2 √ 3 √
3
r.Thusr =
4+2 √ 3 = 2 √ 3 − 3
.
2
The radius of each ball is √ 3 − 3 2
m.
3. (a) We findthelineofintersectionL as in Example 13.5.7(b) [ ET 12.5.7(b)]. Observe that the point (−1,c,c) lies on both
planes. Now since L lies in both planes, it is perpendicular to both of the normal vectors n 1 and n 2 , and thus parallel to
i j k
their cross product n 1 × n 2 =
c 1 1
= 2c, −c 2 +1, −c 2 − 1 . So symmetric equations of L can be written as
1 −c c
x +1
−2c = y − c
c 2 − 1 = z − c , provided that c 6= 0, ±1.
c 2 +1
If c =0, then the two planes are given by y + z =0and x = −1, so symmetric equations of L are x = −1, y = −z. If
c = −1, then the two planes are given by −x + y + z = −1 and x + y + z = −1, and they intersect in the line x =0,
y = −z − 1. Ifc =1, then the two planes are given by x + y + z =1and x − y + z =1, and they intersect in the line
y =0, x =1− z.
(b) If we set z = t in the symmetric equations and solve for x and y separately, we get x +1=
y − c = (t − c)(c2 − 1)
c 2 +1
⇒ x = −2ct +(c2 − 1)
c 2 +1
(t − c)(−2c)
,
c 2 +1
, y = (c2 − 1)t +2c
. Eliminating c from these equations, we
c 2 +1
have x 2 + y 2 = t 2 +1. So the curve traced out by L in the plane z = t is a circle with center at (0, 0,t) and
radius √ t 2 +1.
(c) The area of a horizontal cross-section of the solid is A(z) =π(z 2 +1),soV = 1
0 A(z)dz = π 1
3 z3 + z 1
0 = 4π 3 . 135

F.
136 ¤ PROBLEMS PLUS
TX.10
5. (a) When θ = θ s, the block is not moving, so the sum of the forces on the block
must be 0, thus N + F + W = 0. This relationship is illustrated
geometrically in the figure. Since the vectors form a right triangle, we have
tan(θ s)= |F|
|N| = μ sn
n = μ s.
(b) We place the block at the origin and sketch the force vectors acting on the block, including the additional horizontal force
H, with initial points at the origin. We then rotate this system so that F lies along the positive x-axis and the inclined plane
is parallel to the x-axis.
|F| is maximal, so |F| = μ s n for θ>θ s. Then the vectors, in terms of components parallel and perpendicular to the
inclined plane, are
N = n j
W =(−mg sin θ) i +(−mg cos θ) j
F =(μ s n) i
H =(h min cos θ) i +(−h min sin θ) j
Equating components, we have
μ s n − mg sin θ + h min cos θ =0 ⇒ h min cos θ + μ s n = mg sin θ (1)
n − mg cos θ − h min sin θ =0 ⇒ h min sin θ + mg cos θ = n (2)
(c) Since (2) is solved for n, we substitute into (1):
h min cos θ + μ s (h min sin θ + mg cos θ)=mg sin θ
⇒
h min cos θ + h min μ s sin θ = mg sin θ − mgμ s cos θ
⇒
sin θ − μs cos θ tan θ − μs
h min = mg
= mg
cos θ + μ s sin θ 1+μ s tan θ
tan θ − tan θs
From part (a) we know μ s =tanθ s , so this becomes h min = mg
and using a trigonometric identity,
1+tanθ s tan θ
this is mg tan(θ − θ s ) as desired.
Note for θ = θ s, h min = mg tan 0 = 0, which makes sense since the block is at rest for θ s, thus no additional force H
is necessary to prevent it from moving. As θ increases, the factor tan(θ − θ s), and hence the value of h min,increases
slowly for small values of θ − θ s but much more rapidly as θ − θ s becomes significant. This seems reasonable, as the

F.
TX.10
PROBLEMS PLUS ¤ 137
steeper the inclined plane, the less the horizontal components of the various forces affect the movement of the block, so we
would need a much larger magnitude of horizontal force to keep the block motionless. If we allow θ → 90 ◦ , corresponding
to the inclined plane being placed vertically, the value of h min is quite large; this is to be expected, as it takes a great
amount of horizontal force to keep an object from moving vertically. In fact, without friction (so θ s =0), we would have
θ → 90 ◦ ⇒ h min →∞, and it would be impossible to keep the block from slipping.
(d) Since h max is the largest value of h that keeps the block from slipping, the force of friction is keeping the block from
moving up the inclined plane; thus, F is directed down the plane. Our system of forces is similar to that in part (b), then,
except that we have F = −(μ s n) i. (Note that |F| is again maximal.) Following our procedure in parts (b) and (c), we
equate components:
−μ s n − mg sin θ + h max cos θ =0 ⇒ h max cos θ − μ s n = mg sin θ
n − mg cos θ − h max sin θ =0 ⇒ h max sin θ + mg cos θ = n
Then substituting,
h max cos θ − μ s (h max sin θ + mg cos θ) =mg sin θ
⇒
h max cos θ − h max μ s sin θ = mg sin θ + mgμ s cos θ
⇒
sin θ + μs cos θ tan θ + μs
h max = mg
= mg
cos θ − μ s sin θ 1 − μ s tan θ
tan θ +tanθs
= mg
= mg tan(θ + θ s )
1 − tan θ s tan θ
We would expect h max to increase as θ increases, with similar behavior as we established for h min ,butwithh max values
always larger than h min .Wecanseethatthisisthecaseifwegraphh max as a function of θ,asthecurveisthegraphof
h min translated 2θ s to the left, so the equation does seem reasonable. Notice that the equation predicts h max →∞as
θ → (90 ◦ − θ s ). In fact, as h max increases, the normal force increases as well. When (90 ◦ − θ s ) ≤ θ ≤ 90 ◦ ,the
horizontal force is completely counteracted by the sum of the normal and frictional forces, so no part of the horizontal
force contributes to moving the block up the plane no matter how large its magnitude.

F.
TX.10

F.
TX.10
14 VECTOR FUNCTIONS ET 13
14.1 Vector Functions and Space Curves ET 13.1
1. The component functions √ 4 − t 2 , e −3t ,andln(t +1)are all defined when 4 − t 2 ≥ 0 ⇒ −2 ≤ t ≤ 2 and
t +1> 0 ⇒ t>−1, so the domain of r is (−1, 2].
ln t
3. lim cos t =cos0=1, lim sin t =sin0=0, lim t ln t = lim
t→0 + t→0 + t→0 + t→0 + 1/t = lim 1/t
t→0 + −1/t = lim −t =0
2 t→0 +
[by l’Hospital’s Rule]. Thus lim hcos t, sin t, t ln ti = lim cos t, lim sin t, lim t ln t = h1, 0, 0i.
t→0 + t→0 + t→0 + t→0 +
5. lim e −3t = e 0 t 2
=1, lim
t→0 t→0 sin 2 t =lim
t→0
1
sin 2 t
t 2 =
lim
t→0
1
sin 2 =
t
t 2
and lim
t→0
cos 2t =cos0=1. Thus the given limit equals i + j + k.
lim
t→0
1
sin t
t
7. The corresponding parametric equations for this curve are x =sint, y = t.
2
= 1 1 2 =1
We can make a table of values, or we can eliminate the parameter: t = y
⇒
x =siny,withy ∈ R. By comparing different values of t,wefind the direction in
which t increases as indicated in the graph.
9. The corresponding parametric equations are x = t, y =cos2t, z =sin2t.
Note that y 2 + z 2 =cos 2 2t +sin 2 2t =1, so the curve lies on the circular
cylinder y 2 + z 2 =1.Sincex = t, the curve is a helix.
11. The corresponding parametric equations are x =1, y =cost, z =2sint.
Eliminating the parameter in y and z gives y 2 +(z/2) 2 =cos 2 t +sin 2 t =1
or y 2 + z 2 /4=1.Sincex =1, the curve is an ellipse centered at (1, 0, 0) in
the plane x =1.
139

F.
140 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
13. The parametric equations are x = t 2 , y = t 4 , z = t 6 . These are positive
for t 6= 0and 0 when t =0.Sothecurveliesentirelyinthefirst quadrant.
Theprojectionofthegraphontothexy-plane is y = x 2 , y>0, a half parabola.
On the xz-plane z = x 3 , z>0, a half cubic, and the yz-plane, y 3 = z 2 .
15. Taking r 0 = h0, 0, 0i and r 1 = h1, 2, 3i, we have from Equation 13.5.4 [ET 12.5.4]
r(t) =(1− t) r 0 + t r 1 =(1− t) h0, 0, 0i + t h1, 2, 3i, 0 ≤ t ≤ 1 or r(t) =ht, 2t, 3ti, 0 ≤ t ≤ 1.
Parametric equations are x = t, y =2t, z =3t, 0 ≤ t ≤ 1.
17. Taking r 0 = h1, −1, 2i and r 1 = h4, 1, 7i,wehave
r(t) =(1− t) r 0 + t r 1 =(1− t) h1, −1, 2i + t h4, 1, 7i, 0 ≤ t ≤ 1 or r(t) =h1+3t, −1+2t, 2+5ti, 0 ≤ t ≤ 1.
Parametric equations are x =1+3t, y = −1+2t, z =2+5t, 0 ≤ t ≤ 1.
19. x =cos4t, y = t, z =sin4t. At any point (x, y, z) on the curve, x 2 + z 2 =cos 2 4t +sin 2 4t =1.Sothecurveliesona
circular cylinder with axis the y-axis. Since y = t, this is a helix. So the graph is VI.
21. x = t, y =1/(1 + t 2 ), z = t 2 . Note that y and z arepositiveforallt. The curve passes through (0, 1, 0) when t =0.
As t →∞, (x, y, z) → (∞, 0, ∞),andast →−∞, (x, y, z) → (−∞, 0, ∞). So the graph is IV.
23. x =cost, y =sint, z =sin5t. x 2 + y 2 =cos 2 t +sin 2 t =1, so the curve lies on a circular cylinder with axis the
z-axis. Each of x, y and z is periodic, and at t =0and t =2π the curve passes through the same point, so the curve repeats
itself and the graph is V.
25. If x = t cos t, y = t sin t, z = t,thenx 2 + y 2 = t 2 cos 2 t + t 2 sin 2 t = t 2 = z 2 ,
so the curve lies on the cone z 2 = x 2 + y 2 .Sincez = t, the curve is a spiral on
this cone.
27. Parametric equations for the curve are x = t, y =0, z =2t − t 2 . Substituting into the equation of the paraboloid gives
2t − t 2 = t 2 ⇒ 2t =2t 2 ⇒ t =0, 1. Sincer(0) = 0 and r(1) = i + k, the points of intersection
are (0, 0, 0) and (1, 0, 1).
29. r(t) =hcos t sin 2t, sin t sin 2t, cos 2ti.
We include both a regular plot and a plot
showing a tube of radius 0.08 around the
curve.

F.
SECTION TX.10 14.1 VECTOR FUNCTIONS AND SPACE CURVES ET SECTION 13.1 ¤ 141
31. r(t) =ht, t sin t, t cos ti
33. x =(1+cos16t)cost, y =(1+cos16t)sint, z = 1 + cos 16t. Atany
point on the graph,
x 2 + y 2 =(1+cos16t) 2 cos 2 t +(1+cos16t) 2 sin 2 t
=(1+cos16t) 2 = z 2 , so the graph lies on the cone x 2 + y 2 = z 2 .
From the graph at left, we see that this curve looks like the projection of a
leaved two-dimensional curve onto a cone.
35. If t = −1,thenx =1, y =4, z =0, so the curve passes through the point (1, 4, 0). Ift =3,thenx =9, y = −8, z =28,
so the curve passes through the point (9, −8, 28). For the point (4, 7, −6) to be on the curve, we require y =1− 3t =7
⇒
t = −2. But then z =1+(−2) 3 = −7 6=−6,so(4, 7, −6) is not on the curve.
37. Both equations are solved for z, so we can substitute to eliminate z: x 2 + y 2 =1+y ⇒ x 2 + y 2 =1+2y + y 2 ⇒
x 2 =1+2y ⇒ y = 1 2 (x2 − 1). We can form parametric equations for the curve C of intersection by choosing a
parameter x = t,theny = 1 2 (t2 − 1) and z =1+y =1+ 1 2 (t2 − 1) = 1 2 (t2 +1). Thus a vector function representing C
is r(t) =t i + 1 2 (t2 − 1) j + 1 2 (t2 +1)k.
39. The projection of the curve C of intersection onto the
xy-plane is the circle x 2 + y 2 =4,z =0. Then we can write
x =2cost, y =2sint, 0 ≤ t ≤ 2π. SinceC also lies on
the surface z = x 2 ,wehavez = x 2 =(2cost) 2 =4cos 2 t.
Then parametric equations for C are x =2cost, y =2sint,
z =4cos 2 t, 0 ≤ t ≤ 2π.
41. For the particles to collide, we require r 1(t) =r 2(t) ⇔ t 2 , 7t − 12,t 2 = 4t − 3,t 2 , 5t − 6 . Equating components
gives t 2 =4t − 3, 7t − 12 = t 2 ,andt 2 =5t − 6. Fromthefirst equation, t 2 − 4t +3=0 ⇔ (t − 3)(t − 1) = 0 so t =1
or t =3. t =1does not satisfy the other two equations, but t =3does. The particles collide when t =3,atthe
point (9, 9, 9).
43. (a) lim u(t) + lim v(t) = lim u1(t), lim u2(t), lim u3(t) + lim v1(t), lim v2(t), lim v3(t) and the limits of these
t→a t→a t→a t→a t→a t→a t→a t→a
component functions must each exist since the vector functions both possess limits as t → a. Then adding the two vectors

F.
142 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
andusingtheadditionpropertyoflimits for real-valued functions, we have that
lim u(t) + lim v(t)= lim u1(t)+lim v1(t), lim u2(t)+lim
t→a t→a t→a t→a t→a t→a
=
lim
t→a
v2(t), lim u3(t)+limv3(t)
t→a t→a
[u1(t)+v1(t)] , lim [u2(t)+v2(t)] , lim [u3(t)+v3(t)]
t→a t→a
=lim
t→a
hu 1 (t)+v 1 (t),u 2 (t)+v 2 (t),u 3 (t)+v 3 (t)i
=lim[u(t)+v(t)]
t→a
(b) lim cu(t) =limhcu 1 (t),cu 2 (t),cu 3 (t)i = lim cu
t→a t→a
1(t), lim cu 2 (t), lim
t→a t→a
= c lim u
t→a
1(t),clim u
t→a
2(t),clim u
t→a
3(t) = c lim
t→a
= c lim hu
t→a
1(t),u 2(t),u 3(t)i = c lim u(t)
t→a
(c) lim u(t) · lim v(t) = lim u1(t), lim u2(t), lim u3(t) ·
t→a t→a t→a t→a t→a
lim
t→a
= lim u1(t) lim v1(t) + lim u2(t) lim v2(t)
t→a t→a t→a t→a
=lim
t→a
u 1(t)v 1(t)+lim
t→a
u 2(t)v 2(t)+lim
t→a
u 3(t)v 3(t)
cu 3 (t)
t→a
u1(t), lim u2(t), lim u3(t)
t→a t→a
v1(t), lim v2(t), lim v3(t)
t→a t→a
+ lim
[using (1) backward]
u3(t) lim v3(t)
t→a t→a
=lim[u t→a 1(t)v 1(t)+u 2(t)v 2(t)+u 3(t)v 3(t)] = lim [u(t) · v(t)]
t→a
(d) lim u(t) × lim v(t) = lim u
t→a t→a
1(t), lim u 2 (t), lim u 3 (t) × lim v
t→a t→a t→a
1(t), lim v 2 (t), lim v 3 (t)
t→a t→a t→a
= lim u 2(t) lim v 3(t) − lim u 3(t) lim v 2(t) ,
t→a t→a t→a t→a
lim u 3(t) lim v 1(t) − lim u 1(t) lim v 3(t) ,
t→a t→a t→a t→a
lim u 1(t) lim v 2(t) − lim u 2(t) lim v 1(t)
t→a t→a t→a t→a
=
lim
t→a
[u2(t)v3(t) − u3(t)v2(t)] , lim [u3(t)v1(t) − u1(t)v3(t)] ,
t→a
lim [u1(t)v2(t) − u2(t)v1(t)]
t→a
=lim
t→a
hu 2(t)v 3(t) − u 3(t)v 2(t),u 3 (t) v 1(t) − u 1(t)v 3(t),u 1(t)v 2(t) − u 2(t)v 1(t)i
=lim
t→a
[u(t) × v(t)]
45. Let r(t) =hf (t) ,g(t) ,h(t)i and b = hb 1,b 2,b 3i. Iflim r(t) =b,thenlim r(t) exists,soby(1),
t→a t→a
b =limr(t) = lim f(t), lim g(t), lim h(t) .Bythedefinition of equal vectors we have lim f(t) =b
t→a t→a t→a t→a t→a 1, lim g(t) =b 2
t→a
and lim
t→a
h(t) =b 3. But these are limits of real-valued functions, so by the definition of limits, for every ε>0 there exists
δ 1 > 0, δ 2 > 0, δ 3 > 0 so that if 0 < |t − a|

F.
SECTION 14.2 DERIVATIVES TX.10 AND INTEGRALS OF VECTOR FUNCTIONS ET SECTION 13.2 ¤ 143
Thus for every ε>0 there exists δ>0 such that if 0 < |t − a| 0 such
that if 0 < |t − a|

F.
144 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
3. Since (x +2) 2 = t 2 = y − 1 ⇒
y =(x +2) 2 − 1,thecurveisa
parabola.
(a), (c)
(b) r 0 (t) =h1, 2ti,
r 0 (−1) = h1, −2i
5. x =sint, y =2cost so
x 2 +(y/2) 2 =1and the curve is
an ellipse.
(a), (c) (b) r 0 (t) =cost i − 2sint j,
π
√
2
r 0 =
4 2 i − √ 2 j
7. Since y = e 3t =(e t ) 3 = x 3 ,the
curve is part of a cubic cuve. Note
that here, x>0.
(a), (c) (b) r 0 (t) =e t i +3e 3t j,
r 0 (0) = i +3j
d
9. r 0 (t)=
dt [t sin t] , d dt
t
2 , d dt [t cos 2t]
= ht cos t +sint, 2t, t(− sin 2t) · 2+cos2ti
= ht cos t +sint, 2t, cos 2t − 2t sin 2ti
11. r(t) =i − j + e 4t k ⇒ r 0 (t) =0i +0j +4e 4t k =4e 4t k
13. r(t) =e t2 i − j +ln(1+3t) k ⇒ r 0 (t) =2te t2 i + 3
1+3t k
15. r 0 (t) =0 + b +2t c = b +2t c by Formulas 1 and 3 of Theorem 3.
17. r 0 (t) = −te −t + e −t , 2/(1 + t 2 ), 2e t ⇒ r 0 (0) = h1, 2, 2i. So|r 0 (0)| = √ 1 2 +2 2 +2 2 = √ 9=3and
T(0) = r0 (0)
|r 0 (0)| = 1 h1, 2, 2i = 1
2
3 3 3 3 .
19. r 0 (t) =− sin t i +3j +4cos2t k ⇒ r 0 (0) = 3 j +4k. Thus
T(0) = r0 (0)
|r 0 (0)| = 1
√
02 +3 2 +4 (3 j +4k) = 1 (3 j +4k) = 3 j + 4 k.
2 5 5 5
21. r(t) = t, t 2 ,t 3 ⇒ r 0 (t) = 1, 2t, 3t 2 .Thenr 0 (1) = h1, 2, 3i and |r 0 (1)| = √ 1 2 +2 2 +3 2 = √ 14, so
T(1) = r0 (1)
|r 0 (1)| = √ 1
1
14
h1, 2, 3i = √
2
14
, √
3
14
, √
14
. r 00 (t) =h0, 2, 6ti,so
i j k
r 0 (t) × r 00 (t) =
1 2t 3t 2
2t 3t 2
=
0 2 6t
2 6t i − 1 3t 2
0 6t j + 1 2t
0 2 k
=(12t 2 − 6t 2 ) i − (6t − 0) j +(2− 0) k = 6t 2 , −6t, 2

F.
SECTION 14.2 DERIVATIVES TX.10 AND INTEGRALS OF VECTOR FUNCTIONS ET SECTION 13.2 ¤ 145
23. The vector equation for the curve is r(t) = 1+2 √ t, t 3 − t, t 3 + t ,sor 0 (t) = 1/ √ t, 3t 2 − 1, 3t 2 +1 .Thepoint
(3, 0, 2) corresponds to t =1, so the tangent vector there is r 0 (1) = h1, 2, 4i. Thus, the tangent line goes through the point
(3, 0, 2) and is parallel to the vector h1, 2, 4i. Parametric equations are x =3+t, y =2t, z =2+4t.
25. The vector equation for the curve is r(t) = e −t cos t, e −t sin t, e −t ,so
r 0 (t) = e −t (− sin t)+(cost)(−e −t ), e −t cos t +(sint)(−e −t ), (−e −t )
= −e −t (cos t +sint),e −t (cos t − sin t), −e −t
The point (1, 0, 1) corresponds to t =0, so the tangent vector there is
r 0 (0) = −e 0 (cos 0 + sin 0),e 0 (cos 0 − sin 0), −e 0 = h−1, 1, −1i. Thus, the tangent line is parallel to the vector
h−1, 1, −1i and parametric equations are x =1+(−1)t =1− t, y =0+1· t = t, z =1+(−1)t =1− t.
27. r(t) = t, e −t , 2t − t 2 ⇒ r 0 (t) = 1, −e −t , 2 − 2t .At(0, 1, 0),
t =0and r 0 (0) = h1, −1, 2i. Thus, parametric equations of the tangent
line are x = t, y =1− t, z =2t.
29. r(t) =ht cos t, t, t sin ti ⇒ r 0 (t) =hcos t − t sin t, 1,tcos t +sinti.
At (−π, π, 0), t = π and r 0 (π) =h−1, 1, −πi. Thus, parametric equations
of the tangent line are x = −π − t, y = π + t, z = −πt.
31. The angle of intersection of the two curves is the angle between the two tangent vectors to the curves at the point of
intersection. Since r 0 1(t) = 1, 2t, 3t 2 and t =0at (0, 0, 0), r 0 1(0) = h1, 0, 0i is a tangent vector to r 1 at (0, 0, 0). Similarly,
r 0 2(t) =hcos t, 2cos2t, 1i and since r 2 (0) = h0, 0, 0i, r 0 2 (0) = h1, 2, 1i is a tangent vector to r 2 at (0, 0, 0). Ifθ is the angle
between these two tangent vectors, then cos θ = √ 1 √
1 6
h1, 0, 0i·h1, 2, 1i = √ 1
6
and θ =cos −1 √
1
6
≈ 66 ◦ .
1
33.
0 (16t3 i − 9t 2 j +25t 4 k) dt =
π/2
1
1
1
dt
0 16t3 i −
0 9t2 dt j +
0 25t4 dt k
= 4t 4 1
i − 3t 3 1
j + 5t 5 1
k =4i − 3 j +5k
0 0 0
35. (3 sin 2 t cos t i +3sint cos 2 t j +2sint cos t k) dt
0
π/2
= 3sin 2 π/2
t cos tdt i + 3sint cos 2 π/2
tdt j + 2sint cos tdt k
0 0 0
= sin 3 t π/2
0
i + − cos 3 t π/2
0
j+ sin 2 t π/2
0
k =(1− 0) i +(0+1)j +(1− 0) k = i + j + k

F.
146 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
37.
e t i +2t j +lnt k dt = e t dt i + 2tdt j + ln tdt k
= e t i + t 2 j +(t ln t − t) k + C, whereC is a vector constant of integration.
39. r 0 (t) =2t i +3t 2 j + √ t k ⇒ r(t) =t 2 i + t 3 j + 2 3 t3/2 k + C, whereC is a constant vector.
But i + j = r (1) = i + j + 2 k + C. ThusC = − 2 k and r(t) 3 3 =t2 i + t 3 2
j +
3 t3/2 − 2 k.
3
For Exercises 41–44, let u(t) =hu 1 (t),u 2 (t),u 3 (t)i and v(t) =hv 1 (t),v 2 (t),v 3 (t)i. In each of these exercises, the procedure is to apply
Theorem 2 so that the corresponding properties of derivatives of real-valued functions can be used.
41.
d
dt [u(t)+v(t)] = d dt hu 1(t)+v 1 (t),u 2 (t)+v 2 (t),u 3 (t)+v 3 (t)i
d
=
dt [u1(t)+v1 (t)] , d dt [u2(t)+v2(t)] , d
dt [u3(t)+v3(t)]
= hu 0 1(t)+v 0 1(t),u 0 2(t)+v 0 2(t),u 0 3(t)+v 0 3(t)i
= hu 0 1(t),u 0 2 (t) ,u 0 3(t)i + hv 0 1(t),v 0 2(t),v 0 3(t)i = u 0 (t)+v 0 (t)
43.
d
dt [u(t) × v(t)] = d dt hu 2(t)v 3 (t) − u 3 (t)v 2 (t),u 3 (t)v 1 (t) − u 1 (t)v 3 (t),u 1 (t)v 2 (t) − u 2 (t)v 1 (t)i
= hu 0 2v 3 (t)+u 2 (t)v 0 3(t) − u 0 3(t)v 2 (t) − u 3 (t)v 0 2(t),
u 0 3(t)v 1(t)+u 3(t)v 0 1 (t) − u 0 1(t)v 3(t) − u 1(t)v 0 3(t),
45.
u 0 1(t)v 2 (t)+u 1 (t)v 0 2(t) − u 0 2(t)v 1 (t) − u 2 (t)v 0 1(t)i
= hu 0 2(t)v 3 (t) − u 0 3(t)v 2 (t) ,u 0 3(t)v 1 (t) − u 0 1(t)v 3 (t),u 0 1(t)v 2 (t) − u 0 2(t)v 1 (t)i
+ hu 2(t)v 0 3(t) − u 3(t)v 0 2(t),u 3(t)v 0 1 (t) − u 1(t)v 0 3(t),u 1(t)v 0 2(t) − u 2(t)v 0 1(t)i
= u 0 (t) × v(t)+u(t) × v 0 (t)
Alternate solution: Let r(t) =u(t) × v(t). Then
r(t + h) − r(t) =[u(t + h) × v(t + h)] − [u(t) × v(t)]
=[u(t + h) × v(t + h)] − [u(t) × v(t)] + [u(t + h) × v(t)] − [u(t + h) × v(t)]
= u(t + h) × [v(t + h) − v(t)] + [u(t + h) − u(t)] × v(t)
(Be careful of the order of the cross product.) Dividing through by h and taking the limit as h → 0 we have
r 0 u(t + h) × [v(t + h) − v(t)] [u(t + h) − u(t)] × v(t)
(t) = lim
+ lim
= u(t) × v 0 (t)+u 0 (t) × v(t)
h→0 h
h→0 h
by Exercise 14.1.43(a) [ET 13.1.43(a)] and Definition 1.
d
dt [u(t) · v(t)] = u0 (t) · v(t)+u(t) · v 0 (t) [by Formula 4 of Theorem 3]
= hcos t, − sin t, 1i·ht, cos t, sin ti + hsin t, cos t, ti·h1, − sin t, cos ti
= t cos t − cos t sin t +sint +sint − cos t sin t + t cos t
=2t cos t +2sint − 2cost sin t

F.
47.
49.
TX.10 SECTION 14.3 ARCLENGTHANDCURVATURE ET SECTION 13.3 ¤ 147
d
dt [r(t) × r0 (t)] = r 0 (t) × r 0 (t)+r(t) × r 00 (t) by Formula 5 of Theorem 3. But r 0 (t) × r 0 (t) =0 (by Example 2 in
Section 13.4 [ET 12.4]). Thus,
d
dt [r(t) × r0 (t)] = r(t) × r 00 (t).
d
dt |r(t)| = d dt [r(t) · r(t)]1/2 = 1 [r(t) · 2 r(t)]−1/2 [2r(t) · r 0 (t)] = 1
|r(t)| r(t) · r0 (t)
51. Since u(t) =r(t) · [r 0 (t) × r 00 (t)],
u 0 (t) = r 0 (t) · [r 0 (t) × r 00 (t)] + r(t) · d
dt [r0 (t) × r 00 (t)]
=0+r(t) · [r 00 (t) × r 00 (t)+r 0 (t) × r 000 (t)]
[since r 0 (t) ⊥ r 0 (t) × r 00 (t)]
= r(t) · [r 0 (t) × r 000 (t)] [since r 00 (t) × r 00 (t) =0]
14.3 Arc Length and Curvature ET 13.3
1. r(t) =h2sint, 5t, 2costi ⇒ r 0 (t) =h2cost, 5, −2sinti ⇒ |r 0 (t)| = (2 cos t) 2 +5 2 +(−2sint) 2 = √ 29.
Then using Formula 3, we have L = 10
−10 |r0 (t)| dt = 10
√ √ 10
−10 29 dt = 29 t =20√ 29.
−10
3. r(t) = √ 2 t i + e t j + e −t k ⇒ r 0 (t) = √ 2 i + e t j − e −t k ⇒
|r 0 (t)| =
√2 2
+(et ) 2 +(−e −t ) 2 = √ 2+e 2t + e −2t = (e t + e −t ) 2 = e t + e −t [since e t + e −t > 0].
Then L = 1
0 |r0 (t)| dt = 1
0 (et + e −t ) dt = e t − e −t 1
0 = e − e−1 .
5. r(t) =i + t 2 j + t 3 k ⇒ r 0 (t) =2t j +3t 2 k ⇒ |r 0 (t)| = √ 4t 2 +9t 4 = t √ 4+9t 2 [since t ≥ 0].
Then L = 1
0 |r0 (t)| dt = 1
t √ 1
4+9t
0 2 dt = 1 · 2 (4 + 18 3 9t2 ) 3/2 = 1 27 (133/2 − 4 3/2 )= 1
27 (133/2 − 8).
0
2
+12 +(2t) 2 1
=
4t +1+4t2 ,so
7. r(t) = √ t, t, t 2 1
⇒ r 0 (t) =
2 √ t , 1, 2t ⇒ |r 0 (t)| =
L = 4
1 |r0 (t)| dt =
4 1
1 4t +1+4t2 dt ≈ 15.3841.
1
2 √ t
9. r(t) =hsin t, cos t, tan ti ⇒ r 0 (t) = cos t, − sin t, sec 2 t ⇒
|r 0 (t)| = cos 2 t +(− sin t) 2 +(sec 2 t) 2 = √ 1+sec 4 t and L = π/4
|r 0 (t)| dt = π/4
√
1+sec4 tdt≈ 1.2780.
0 0
11. The projection of the curve C onto the xy-plane is the curve x 2 =2y or y = 1 2 x2 , z =0. Then we can choose the parameter
x = t ⇒ y = 1 2 t2 .SinceC also lies on the surface 3z = xy,wehavez = 1 3 xy = 1 3 (t)( 1 2 t2 )= 1 6 t3 . Then parametric
equations for C are x = t, y = 1 2 t2 , z = 1 6 t3 and the corresponding vector equation is r(t) = t, 1 2 t2 , 1 t3 . The origin
6
corresponds to t =0and the point (6, 18, 36) corresponds to t =6,so
L = 6
0 |r0 (t)| dt = 6
1,t, 1 t2 dt =
6
1
0
2 0
2 + t 2 + 1
2 t22 dt =
6
1+t
0
2 + 1 4 t4 dt
= 6
0
(1 + 1 2 t2 ) 2 dt = 6
0 (1 + 1 2 t2 ) dt = t + 1 6 t3 6
0 =6+36=42

F.
148 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
13. r(t) =2t i +(1− 3t) j +(5+4t) k ⇒ r 0 (t) =2i − 3 j +4k and ds = dt |r0 (t)| = √ 4+9+16= √ 29. Then
√ √
29 du = 29 t. Therefore, t = √
1
29
s, and substituting for t in the original equation, we
s = s(t) = t
0 |r0 (u)| du = t
0
have r(t(s)) = 2 √
29
s i +
1 − 3 √
29
s
j + 5+ √ 4
29
s k.
15. Here r(t) =h3sint, 4t, 3costi,sor 0 (t) =h3cost, 4, −3sinti and |r 0 (t)| = 9cos 2 t +16+9sin 2 t = √ 25 = 5.
The point (0, 0, 3) corresponds to t =0, so the arc length function beginning at (0, 0, 3) and measuring in the positive
direction is given by s(t) = t
0 |r0 (u)| du = t
5 du =5t. s(t) =5 ⇒ 5t =5 ⇒ t =1, thus your location after
0
moving 5 units along the curve is (3 sin 1, 4, 3 cos 1).
17. (a) r(t) =h2sint, 5t, 2costi ⇒ r 0 (t) =h2cost, 5, −2sinti ⇒ |r 0 (t)| = 4cos 2 t +25+4sin 2 t = √ 29.
Then T(t) = r0 (t)
|r 0 (t)| = 1 √
29
h2cost, 5, −2sinti or
√
2
29
cos t,
√
5
29
, − √ 2
29
sin t .
T 0 (t) = 1 √
29
h−2sint, 0, −2costi ⇒ |T 0 (t)| = 1 √
29
4sin 2 t +0+4cos 2 t = 2 √
29
. Thus
N(t) = T0 (t)
|T 0 (t)| = 1/√ 29
2/ √ h−2sint, 0, −2costi = h− sin t, 0, − cos ti.
29
(b) κ(t) = |T0 (t)|
|r 0 (t)|
= 2/√ 29
√
29
= 2 29
19. (a) r(t) = √ 2 t, e t ,e −t ⇒ r 0 (t) = √ 2,e t , −e −t ⇒ |r 0 (t)| = √ 2+e 2t + e −2t = (e t + e −t ) 2 = e t + e −t .
Then
T(t) = r0 (t)
|r 0 (t)| = 1 √
2,e t , −e −t 1 √
=
2e t ,e 2t , −1
after multiplying by et
and
e t + e −t e 2t +1
e t
T 0 1 √
(t)=
2e t , 2e 2t , 0 2e 2t √
−
2e t
e 2t +1
(e 2t +1) 2 ,e 2t , −1
1
=
(e 2t +1) √ 2e t , 2e 2t , 0 − 2e √ 2t 2e t ,e 2t , −1 1 √
=
2e
t
1 − e 2t , 2e 2t , 2e 2t
(e 2t +1) 2 (e 2t +1) 2
Then
|T 0 1 1
(t)| =
2e
2t
(1 − 2e
(e 2t +1) 2t + e 4t )+4e 4t +4e 4t =
2e 2 (e 2t +1) 2 2t
(1 + 2e 2t + e 4t )
Therefore
(b) κ(t) = |T0 (t)|
|r 0 (t)|
=
1
2e
(e 2t +1) 2t (1 + e 2t ) 2 =
2
√
2e t (1 + e 2t )
(e 2t +1) 2 =
√
2 e
t
e 2t +1
N(t)= T0 (t)
|T 0 (t)| = e2t +1 1 √
√ 2 e t (1 − e 2t ), 2e 2t , 2e 2t
2 e
t (e 2t +1) 2
1 √
= √ 2 e t (1 − e 2t ), 2e 2t , 2e 2t 1
= 1 − e 2t , √ 2 e t , √ 2 e t
2 et (e 2t +1)
e 2t +1
=
√
2 e
t
e 2t +1 ·
1
e t + e −t =
√
2 e
t
e 3t +2e t + e −t =
√
2 e
2t
e 4t +2e 2t +1 =
√
2 e
2t
(e 2t +1) 2

F.
TX.10 SECTION 14.3 ARCLENGTHANDCURVATURE ET SECTION 13.3 ¤ 149
21. r(t) =t 2 i + t k ⇒ r 0 (t) =2t i + k, r 00 (t) =2i, |r 0 (t)| = (2t) 2 +0 2 +1 2 = √ 4t 2 +1, r 0 (t) × r 00 (t) =2j,
|r 0 (t) × r 00 (t)| =2.Thenκ(t) = |r0 (t) × r 00 (t)| 2
|r 0 (t)| 3 = √
4t2 +1 3 = 2
(4t 2 +1) . 3/2
23. r(t) =3t i +4sint j +4cost k ⇒ r 0 (t) =3i +4cost j − 4sint k, r 00 (t) =−4sint j − 4cost k,
|r 0 (t)| = 9+16cos 2 t +16sin 2 t = √ 9+16=5, r 0 (t) × r 00 (t) =−16 i +12cost j − 12 sin t k,
|r 0 (t) × r 00 (t)| = 256 + 144 cos 2 t +144sin 2 t = √ 400 = 20. Thenκ(t) = |r0 (t) × r 00 (t)|
|r 0 (t)| 3 = 20
5 3 = 4 25 .
25. r(t) = t, t 2 ,t 3 ⇒ r 0 (t) = 1, 2t, 3t 2 . The point (1, 1, 1) corresponds to t =1,andr 0 (1) = h1, 2, 3i ⇒
|r 0 (1)| = √ 1+4+9= √ 14. r 00 (t) =h0, 2, 6ti ⇒ r 00 (1) = h0, 2, 6i. r 0 (1) × r 00 (1) = h6, −6, 2i, so
|r 0 (1) × r 00 (1)| = √ 36 + 36 + 4 = √ √
76. Thenκ(1) = |r0 (1) × r 00 (1)| 76
|r 0 (1)| 3 = √ 3 = 1
19
14 7 14 .
27. f(x) =2x − x 2 , f 0 (x) =2− 2x, f 00 (x) =−2,
κ(x) =
|f 00 (x)|
[1 + (f 0 (x)) 2 ] 3/2 = |−2|
[1 + (2 − 2x) 2 ] 3/2 = 2
(4x 2 − 8x +5) 3/2
29. f(x) =4x 5/2 , f 0 (x) =10x 3/2 , f 00 (x) =15x 1/2 ,
15x
|f 00 1/2
(x)|
κ(x) =
[1 + (f 0 (x)) 2 ] =
3/2 [1 + (10x 3/2 ) 2 ] = 15 √ x
3/2 (1 + 100x 3 ) 3/2
31. Since y 0 = y 00 = e x , the curvature is κ(x) =
|y 00 (x)|
[1 + (y 0 (x)) 2 ] = e x
3/2 (1 + e 2x ) = 3/2 ex (1 + e 2x ) −3/2 .
To find the maximum curvature, we first find the critical numbers of κ(x):
κ 0 (x) =e x (1 + e 2x ) −3/2 + e x − 3 2
(1 + e 2x ) −5/2 (2e 2x )=e x 1+e 2x − 3e 2x
(1 + e 2x ) 5/2 = e x 1 − 2e 2x
(1 + e 2x ) 5/2 .
κ 0 (x) =0when 1 − 2e 2x =0,soe 2x = 1 or x = − 1 ln 2. Andsince1 − 2 2 2e2x > 0 for x− 1 ln 2, the maximum curvature is attained at the point − 1 ln 2 2 2,e(− ln 2)/2 = − 1 ln 2, √2
1
.
2
Since lim
x→∞ ex (1 + e 2x ) −3/2 =0,κ(x) approaches 0 as x →∞.
33. (a) C appears to be changing direction more quickly at P than Q, so we would expect the curvature to be greater at P .
(b) First we sketch approximate osculating circles at P and Q. Usingthe
axes scale as a guide, we measure the radius of the osculating circle
at P to be approximately 0.8 units, thus ρ = 1 κ
⇒
κ = 1 ρ ≈ 1 ≈ 1.3. Similarly, we estimate the radius of the
0.8
osculating circle at Q to be 1.4 units, so κ = 1 ρ ≈ 1
1.4 ≈ 0.7.

F.
150 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
35. y = x −2 ⇒ y 0 = −2x −3 , y 00 =6x −4 ,and
κ(x) =
|y 00 |
6x −4
[1 + (y 0 ) 2 ] = 3/2 [1 + (−2x −3 ) 2 ] = 6
3/2 x 4 (1 + 4x −6 ) . 3/2
The appearance of the two humps in this graph is perhaps a little surprising, but it is
explained by the fact that y = x −2 increases asymptotically at the origin from both
directions, and so its graph has very little bend there. (Note that κ(0) is undefined.)
37. Notice that the curve b has two inflection points at which the graph appears almost straight. We would expect the curvature to
be 0 or nearly 0 at these values, but the curve a isn’t near 0 there. Thus, a must be the graph of y = f(x) rather than the graph
of curvature, and b is the graph of y = κ(x).
39. Using a CAS, we find (after simplifying)
κ(t) = 6 √ 4cos 2 t − 12 cos t +13
.(Tocomputecross
(17 − 12 cos t) 3/2
products in Maple, use the VectorCalculus package and
the CrossProduct(a,b) command; in Mathematica, use
Cross[a,b].) Curvature is largest at integer multiples of 2π.
41. x = e t cos t ⇒ ẋ = e t (cos t − sin t) ⇒ ẍ = e t (− sin t − cos t)+e t (cos t − sin t) =−2e t sin t,
43.
y = e t sin t ⇒ ẏ = e t (cos t +sint) ⇒ ÿ = e t (− sin t +cost)+e t (cos t +sint) =2e t cos t. Then
κ(t)=
=
|ẋÿ − ẏẍ|
[ẋ 2 + ẏ 2 ] = e t (cos t − sin t)(2e t cos t) − e t (cos t +sint)(−2e t sin t) 3/2 ([e t (cos t − sin t)] 2 +[e t (cos t +sint)] 2 ) 3/2
2e 2t (cos 2 t − sin t cos t +sint cos t +sin 2 t)
e
2t
(cos 2 t − 2cost sin t +sin 2 t +cos 2 t +2cost sin t +sin 2 t) 3/2 =
2e 2t (1) 2e2t
=
[e 2t 3/2
(1 + 1)] e 3t (2) = √ 1
3/2 2 e
t
1,
2
, 1 corresponds to t =1. T(t) = r0 (t) 2t, 2t 2
3
|r 0 (t)| = , 1 2t, 2t 2
√
4t2 +4t 4 +1 = , 1
2t 2 +1 ,soT(1) = 2
, 2 , 1
3 3 3 .
T 0 (t) =−4t(2t 2 +1) −2 2t, 2t 2 , 1 +(2t 2 +1) −1 h2, 4t, 0i (by Formula 3 of Theorem 14.2 [ET 13.2])
=(2t 2 +1) −2 −8t 2 +4t 2 +2, −8t 3 +8t 3 +4t, −4t =2(2t 2 +1) −2 1 − 2t 2 , 2t, −2t
N(t) = T0 (t)
|T 0 (t)| = 2(2t 2 +1) −2 1 − 2t 2 , 2t, −2t
1 − 2t 2
2(2t 2 +1) −2 (1 − 2t 2 ) 2 +(2t) 2 +(−2t) = , 2t, −2t
1 − 2t 2
√ 2 1 − 4t2 +4t 4 +8t = , 2t, −2t
2 1+2t 2
N(1) = − 1 3 , 2 3 , − 2 3
and B(1) = T(1) × N(1) = −
4
− 2 , − − 4 + 1
9 9 9 9 ,
4
+ 2
9 9 = −
2
, 1 , 2
3 3 3 .
45. (0,π,−2) corresponds to t = π. r(t) =h2sin3t, t, 2cos3ti ⇒
T(t) = r0 (t)
|r 0 (t)| = h6cos3t, 1, −6sin3ti
36 cos2 3t +1+36sin 2 3t = 1 √
37
h6cos3t, 1, −6sin3ti.
T(π) = 1 √
37
h−6, 1, 0i is a normal vector for the normal plane, and so h−6, 1, 0i is also normal. Thus an equation for the

F.
plane is −6(x − 0) + 1(y − π)+0(z +2)=0or y − 6x = π.
TX.10 SECTION 14.3 ARCLENGTHANDCURVATURE ET SECTION 13.3 ¤ 151
T 0 (t) = 1 √
37
h−18 sin 3t, 0, −18 cos 3ti ⇒ |T 0 (t)| =
182 sin 2 3t +18 2 cos 2 3t
√
37
= 18 √
37
⇒
N(t) = T0 (t)
= h− sin 3t, 0, − cos 3ti. SoN(π) =h0, 0, 1i and B(π) =
|T 0 √
1
(t)| 37
h−6, 1, 0i×h0, 0, 1i = √ 1
37
h1, 6, 0i.
Since B(π) is a normal to the osculating plane, so is h1, 6, 0i.
An equation for the plane is 1(x − 0) + 6(y − π)+0(z +2)=0or x +6y =6π.
47. The ellipse 9x 2 +4y 2 =36is given by the parametric equations x =2cost, y =3sint, so using the result from
Exercise 40,
|ẋÿ − ẍẏ| |(−2sint)(−3sint) − (3 cos t)(−2cost)|
6
κ(t) =
=
[ẋ 2 + ẏ 2 ]
3/2
(4 sin 2 =
t +9cos 2 t) 3/2 (4 sin 2 t +9cos 2 t) . 3/2
At (2, 0), t =0.Nowκ(0) = 6 = 2 , so the radius of the osculating circle is
27 9
1/κ(0) = 9 and its center is 2
− 5 , 0 2
. Its equation is therefore
x + 5 2
2
+ y 2 = 81 . 4
At (0, 3), t = π ,andκ
π
2 2 =
6
= 3 . So the radius of the osculating circle is 4 and
8 4 3
its center is 0, 5 3
. Hence its equation is x 2 + y − 5 3
2
= 16 9 .
49. The tangent vector is normal to the normal plane, and the vector h6, 6, −8i is normal to the given plane.
But T(t) k r 0 (t) and h6, 6, −8i kh3, 3, −4i, so we need to find t such that r 0 (t) kh3, 3, −4i.
r(t) = t 3 , 3t, t 4 ⇒ r 0 (t) = 3t 2 , 3, 4t 3 kh3, 3, −4i when t = −1. So the planes are parallel at the point (−1, −3, 1).
dT
51. κ =
dT
ds = dT/dt
ds/dt = |dT/dt| and N = dT/dt
ds/dt |dT/dt| ,soκN = dt dT
dt
dT
dt ds = dT/dt
ds/dt = dT by the Chain Rule.
ds
dt
53. (a) |B| =1 ⇒ B · B =1 ⇒ d ds
(b) B = T × N
⇒
(B · B) =0 ⇒ 2
dB
ds
· B =0 ⇒
dB
ds ⊥ B
dB
ds = d ds (T × N) = d dt (T × N) 1
ds/dt = d dt (T × N) 1
|r 0 (t)| =[(T0 × N)+(T × N 0 1
)]
|r 0 (t)|
=
T 0 × T0
+(T × N 0 1
)
|T 0 |
|r 0 (t)| = T × N0
|r 0 (t)|
⇒
dB
ds ⊥ T
(c) B = T × N ⇒ T ⊥ N, B ⊥ T and B ⊥ N. SoB, T and N form an orthogonal set of vectors in the threedimensional
space R 3 . From parts (a) and (b), dB/ds is perpendicular to both B and T, sodB/ds is parallel to N.
Therefore, dB/ds = −τ(s)N,whereτ(s) is a scalar.
(d) Since B = T × N, T ⊥ N and both T and N are unit vectors, B is a unit vector mutually perpendicular to both T and
N. Foraplanecurve,T and N always lie in the plane of the curve, so that B is a constant unit vector always
perpendicular to the plane. Thus dB/ds = 0,butdB/ds = −τ(s)N and N 6=0,soτ(s) =0.

F.
152 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
55. (a) r 0 = s 0 T ⇒ r 00 = s 00 T + s 0 T 0 = s 00 T + s 0 dT
ds s0 = s 00 T + κ(s 0 ) 2 N by the first Serret-Frenet formula.
(b) Using part (a), we have
r 0 × r 00 =(s 0 T) × [s 00 T + κ(s 0 ) 2 N]
(c) Using part (a), we have
=[(s 0 T) × (s 00 T)] + (s 0 T) × (κ(s 0 ) 2 N) (by Property 3 of Theorem 13.4.8 [ ET 12.4.8])
=(s 0 s 00 )(T × T)+κ(s 0 ) 3 (T × N) =0 + κ(s 0 ) 3 B = κ(s 0 ) 3 B
r 000 =[s 00 T + κ(s 0 ) 2 N] 0 = s 000 T + s 00 T 0 + κ 0 (s 0 ) 2 N +2κs 0 s 00 N + κ(s 0 ) 2 N 0
= s 000 T + s 00 dT
ds s0 + κ 0 (s 0 ) 2 N +2κs 0 s 00 N + κ(s 0 ) 2 dN
ds s0
= s 000 T + s 00 s 0 κ N + κ 0 (s 0 ) 2 N +2κs 0 s 00 N + κ(s 0 ) 3 (−κ T + τ B) [by the second formula]
=[s 000 − κ 2 (s 0 ) 3 ] T +[3κs 0 s 00 + κ 0 (s 0 ) 2 ] N + κτ(s 0 ) 3 B
(d) Using parts (b) and (c) and the facts that B · T =0, B · N =0,andB · B =1,weget
(r 0 × r 00 ) · r 000
|r 0 × r 00 | 2 = κ(s0 ) 3 B · [s 000 − κ 2 (s 0 ) 3 ] T +[3κs 0 s 00 + κ 0 (s 0 ) 2 ] N + κτ(s 0 ) 3 B
|κ(s 0 ) 3 B| 2 = κ(s0 ) 3 κτ(s 0 ) 3
[κ(s 0 ) 3 ] 2 = τ.
57. r = t, 1 2 t2 , 1 t3 ⇒ r 0 3
= 1,t,t 2 , r 00 = h0, 1, 2ti, r 000 = h0, 0, 2i ⇒ r 0 × r 00 = t 2 , −2t, 1 ⇒
τ = (r0 × r 00 ) · r 000 t 2 , −2t, 1 ·h0, 0, 2i 2
|r 0 × r 00 | 2 =
=
t 4 +4t 2 +1 t 4 +4t 2 +1
59. For one helix, the vector equation is r(t) =h10 cos t, 10 sin t, 34t/(2π)i (measuring in angstroms), because the radius of each
helix is 10 angstroms, and z increases by 34 angstroms for each increase of 2π in t. Using the arc length formula, letting t go
from 0 to 2.9 × 10 8 × 2π,wefind the approximate length of each helix to be
L = 2.9×10 8 ×2π
|r 0 (t)| dt =
2.9×10 8 ×2π
(−10 sin t)
0 0
2 +(10cost) 2 +
34 2
2π
dt = 100 +
2.9×10
8 ×2π
34 2
2π
=2.9 × 10 8 × 2π 100 +
34 2
≈ 2.07 × 10 10 Å — more than two meters!
2π
14.4 Motion in Space: Velocity and Acceleration ET 13.4
1. (a) If r(t) =x(t) i + y (t) j + z(t) k is the position vector of the particle at time t, then the average velocity over the time
interval [0, 1] is
v ave =
r(1) − r(0)
1 − 0
intervals we have
=
[0.5, 1] : v ave =
[1, 2] : v ave =
[1, 1.5] : v ave =
(4.5 i +6.0 j +3.0 k) − (2.7 i +9.8 j +3.7 k)
1
r(1) − r(0.5)
1 − 0.5
r(2) − r(1)
2 − 1
r(1.5) − r(1)
1.5 − 1
=
=
=
(4.5 i +6.0 j +3.0 k) − (3.5 i +7.2 j +3.3 k)
0.5
(7.3 i +7.8 j +2.7 k) − (4.5 i +6.0 j +3.0 k)
1
(5.9 i +6.4 j +2.8 k) − (4.5 i +6.0 j +3.0 k)
0.5
=1.8 i − 3.8 j − 0.7 k. Similarly, over the other
=2.0 i − 2.4 j − 0.6 k
=2.8 i +1.8 j − 0.3 k
=2.8 i +0.8 j − 0.4 k

F.
SECTION 14.4 TX.10 MOTION IN SPACE: VELOCITY AND ACCELERATION ET SECTION 13.4 ¤ 153
(b) We can estimate the velocity at t =1by averaging the average velocities over the time intervals [0.5, 1] and [1, 1.5]:
v(1) ≈ 1 [(2 i − 2.4 j − 0.6 k)+(2.8 i +0.8 j − 0.4 k)] = 2.4 i − 0.8 j − 0.5 k. Then the speed is
2
|v(1)| ≈ (2.4) 2 +(−0.8) 2 +(−0.5) 2 ≈ 2.58.
3. r(t) = − 1 2 t2 ,t ⇒ At t =2:
v(t) =r 0 (t) =h−t, 1i
a(t) =r 00 (t) =h−1, 0i
v(2) = h−2, 1i
a(2) = h−1, 0i
|v(t)| = √ t 2 +1
5. (t) =3cost i +2sint j ⇒ At t = π/3:
v(t) =−3sint i +2cost j
a(t) =−3cost i − 2sint j
v
π
3 = −
3 √ 3
a π
3
i + j
2
= −
3
i − √ 3 j
2
|v(t)| = 9sin 2 t +4cos 2 t = 4+5sin 2 t
Notice that x 2 /9+y 2 /4=sin 2 t +cos 2 t =1, so the path is an ellipse.
7. r(t) =t i + t 2 j +2k ⇒ At t =1:
v(t) =i +2t j v(1) = i +2j
a(t) =2j
a(1) = 2 j
|v(t)| = √ 1+4t 2
Here x = t, y = t 2 ⇒ y = x 2 and z =2, so the path of the particle is a
parabola in the plane z =2.
9. r(t) = t 2 +1,t 3 ,t 2 − 1 ⇒ v(t) =r 0 (t) = 2t, 3t 2 , 2t , a(t) =v 0 (t) =h2, 6t, 2i,
|v(t)| = (2t) 2 +(3t 2 ) 2 +(2t) 2 = √ 9t 4 +8t 2 = |t| √ 9t 2 +8.
11. r(t) = √ 2 t i + e t j + e −t k ⇒ v(t) =r 0 (t) = √ 2 i + e t j − e −t k, a(t) =v 0 (t) =e t j + e −t k,
|v(t)| = √ 2+e 2t + e −2t = (e t + e −t ) 2 = e t + e −t .
13. r(t) =e t hcos t, sin t, ti ⇒
v(t) =r 0 (t) =e t hcos t, sin t, ti + e t h− sin t, cos t, 1i = e t hcos t − sin t, sin t +cost, t +1i
a(t) =v 0 (t) =e t hcos t − sin t − sin t − cos t, sin t +cost +cost − sin t, t +1+1i
= e t h−2sint, 2cost, t +2i
|v(t)| = e t cos 2 t +sin 2 t − 2cost sin t +sin 2 t +cos 2 t +2sint cos t + t 2 +2t +1
= e t√ t 2 +2t +3

F.
154 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
15. a(t) =i +2j ⇒ v(t) = a(t) dt = (i +2j) dt = t i +2t j + C and k = v (0) = C,
so C = k and v(t) =t i +2t j + k. r(t) = v(t) dt = (t i +2t j + k) dt = 1 2 t2 i + t 2 j + t k + D.
But i = r (0) = D,soD = i and r(t) = 1
2 t2 +1 i + t 2 j + t k.
17. (a) a(t) =2t i +sint j +cos2t k ⇒
(b)
v(t) = (2t i +sint j +cos2t k) dt = t 2 i − cos t j + 1 2
sin 2t k + C
and i = v (0) = −j + C,soC = i + j
and v(t) = t 2 +1 i +(1− cos t) j + 1 sin 2t k.
2
r(t) = [ t 2 +1 i +(1− cos t) j + 1 sin 2t k]dt
2
= 1
3 t3 + t i +(t − sin t) j − 1 cos 2t k + D
4
But j = r (0) = − 1 4 k + D,soD = j + 1 4 k and r(t) = 1
3 t3 + t i +(t − sin t +1)j + 1
4 − 1 4 cos 2t k.
19. r(t) = t 2 , 5t, t 2 − 16t ⇒ v(t) =h2t, 5, 2t − 16i, |v(t)| = √ 4t 2 +25+4t 2 − 64t + 256 = √ 8t 2 − 64t + 281
and d dt |v(t)| = 1 2 (8t2 − 64t + 281) −1/2 (16t − 64). This is zero if and only if the numerator is zero, that is,
16t − 64 = 0 or t =4.Since d dt |v(t)| < 0 for t 0 for t>4, the minimum speed of √ 153 is attained
at t =4units of time.
21. |F(t)| =20N in the direction of the positive z-axis, so F(t) =20k. Alsom =4kg, r(0) = 0 and v(0) = i − j.
Since 20k = F(t) =4a(t), a(t) =5k. Thenv(t) =5t k + c 1 where c 1 = i − j so v(t) =i − j +5t k and the
speed is |v(t)| = √ 1+1+25t 2 = √ 25t 2 +2.Alsor(t) =t i − t j + 5 2 t2 k + c 2 and 0 = r(0), soc 2 = 0
and r(t) =t i − t j + 5 2 t2 k.
23. |v(0)| = 500 m/s and since the angle of elevation is 30 ◦ , the direction of the velocity is 1 2
√
3 i + j
.Thus
v(0) = 250 √ 3 i + j and if we set up the axes so the projectile starts at the origin, then r(0) = 0. Ignoring air resistance, the
only force is that due to gravity, so F(t) =−mg j where g ≈ 9.8 m/s 2 .Thusa(t) =−g j and v(t) =−gt j + c 1 .But
250 √ 3 i + j = v(0) = c 1 ,sov(t) =250 √ 3 i + (250 − gt) j and r(t) = 250 √ 3 t i + 250t − 1 2 gt2 j + c 2 where
0 = r(0) = c 2 .Thusr(t) =250 √ 3 t i + 250t − 1 2 gt2 j.
(a) Setting 250t − 1 2 gt2 =0gives t =0or t = 500
g
≈ 51.0 s. So the range is 250 √ 3 · 500
g
≈ 22 km.
(b) 0= d dt
250t −
1
2 gt2 = 250 − gt implies that the maximum height is attained when t = 250/g ≈ 25.5 s.
Thus, the maximum height is (250)(250/g) − g(250/g) 2 1 2 = (250)2 /(2g) ≈ 3.2 km.
(c) From part (a), impact occurs at t = 500/g ≈ 51.0. Thus, the velocity at impact is
v(500/g) =250 √ 3 i +[250− g(500/g)] j =250 √ 3 i − 250 j andthespeedis|v(500/g)| = 250 √ 3+1=500m/s.

F.
SECTION 14.4 TX.10 MOTION IN SPACE: VELOCITY AND ACCELERATION ET SECTION 13.4 ¤ 155
25. As in Example 5, r(t) =(v 0 cos 45 ◦ )t i + (v 0 sin 45 ◦ )t − 1 gt2 √ √
j = 1 v0
2 2 2 t i + v0 2 t − gt
2
j . Then the ball
lands at t = v √ √
0 2
√
s. Now since it lands 90 maway,90 = 1 v 0 2
g
2 v0 2 or v0 2 =90g and the initial velocity
g
is v 0 = √ 90g ≈ 30 m/s.
27. Let α be the angle of elevation. Then v 0 = 150 m/s and from Example 5, the horizontal distance traveled by the projectile is
d = v2 0 sin 2α
. Thus 1502 sin 2α
=800 ⇒ sin 2α = 800g
g
g
150 ≈ 0.3484 ⇒ 2α ≈ 2 20.4◦ or 180 − 20.4 =159.6 ◦ .
Two angles of elevation then are α ≈ 10.2 ◦ and α ≈ 79.8 ◦ .
29. Place the catapult at the origin and assume the catapult is 100 meters from the city, so the city lies between (100, 0)
and (600, 0). The initial speed is v 0 =80m/sandletθ be the angle the catapult is set at. As in Example 5, the trajectory of
the catapulted rock is given by r (t) =(80cosθ)t i + (80 sin θ)t − 4.9t 2 j. The top of the near city wall is at (100, 15),
which the rock will hit when (80 cos θ) t =100 ⇒ t = 5
4cosθ and (80 sin θ)t − 4.9t2 =15
⇒
80 sin θ ·
2
5
5
4cosθ − 4.9 =15 ⇒ 100 tan θ − 7.65625 sec 2 θ =15. Replacing sec 2 θ with tan 2 θ +1gives
4cosθ
7.65625 tan 2 θ − 100 tan θ +22.62625 = 0. Using the quadratic formula, we have tan θ ≈ 0.230324, 12.8309 ⇒
θ ≈ 13.0 ◦ , 85.5 ◦ .Sofor13.0 ◦

F.
156 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
reaches the east bank after 8 s, and it is located at r(8) = 5(8)i + 3
4 (8)2 − 1 16 (8)3 j =40i +16j. Thus the boat is 16 m
downstream.
(b) Let α be the angle north of east that the boat heads. Then the velocity of the boat in still water is given by
5(cos α) i +5(sinα) j. Att seconds, the boat is 5(cos α)t meters from the west bank, at which point the velocity
of the water is
3
[5(cos α)t][40 − 5(cos α)t] j. The resultant velocity of the boat is given by
400
v(t) =5(cosα) i + 5sinα + 3
400 (5t cos α)(40 − 5t cos α) j =(5cosα) i + 5sinα + 3 2 t cos α − 3 16 t2 cos 2 α j.
Integrating, r(t) =(5t cos α) i + 5t sin α + 3 4 t2 cos α − 1 16 t3 cos 2 α j (where we have again placed
the origin at A). The boat will reach the east bank when 5t cos α =40 ⇒ t = 40
5cosα = 8
cos α .
In order to land at point B(40, 0) we need 5t sin α + 3 4 t2 cos α − 1 16 t3 cos 2 α =0
2 3 8
8
8
5 sin α + 3 cos α − 1 cos 2 α =0 ⇒ 1 (40 sin α +48− 32) = 0
4
16
cos α
cos α
cos α
cos α ⇒
⇒
40 sin α +16=0 ⇒ sin α = − 2 . Thus α =sin−1
− 2 5 5 ≈−23.6 ◦ , so the boat should head 23.6 ◦ south of
east (upstream). The path does seem realistic. The boat initially heads
upstream to counteract the effect of the current. Near the center of the river,
the current is stronger and the boat is pushed downstream. When the boat
nears the eastern bank, the current is slower and the boat is able to progress
upstream to arrive at point B.
33. r(t) =(3t − t 3 ) i +3t 2 j ⇒ r 0 (t) =(3− 3t 2 ) i +6t j,
|r 0 (t)| = (3 − 3t 2 ) 2 +(6t) 2 = √ 9+18t 2 +9t 4 = (3 − 3t 2 ) 2 =3+3t 2 ,
r 00 (t) =−6t i +6j, r 0 (t) × r 00 (t) =(18+18t 2 ) k. Then Equation 9 gives
a T = r0 (t) · r 00 (t)
|r 0 (t)|
a T = v 0 = d 3+3t
2
=6t
dt
= (3 − 3t2 )(−6t)+(6t)(6) 18t +18t3
= = 18t(1 + t2 )
3+3t 2 3+3t 2 3(1 + t 2 )
and Equation 10 gives a N = |r0 (t) × r 00 (t)|
=
|r 0 (t)|
=6t
or by Equation 8,
18 + 18t2
3+3t 2 = 18(1 + t2 )
3(1 + t 2 ) =6.
35. r(t) =cost i +sint j + t k ⇒ r 0 (t) =− sin t i +cost j + k, |r 0 (t)| = sin 2 t +cos 2 t +1= √ 2,
r 00 (t) =− cos t i − sin t j, r 0 (t) × r 00 (t) =sint i − cos t j + k.
Then a T = r0 (t) · r 00 (t)
|r 0 (t)|
=
sin t cos t − sin t cos t
√
2
=0and a N = |r0 (t) × r 00 (t)|
|r 0 (t)|
=
√
sin 2 t +cos 2 t +1 2
√ = √ =1.
2 2
37. r(t) =e t i + √ 2 t j + e −t k ⇒ r 0 (t) =e t i + √ 2 j − e −t k, |r(t)| = √ e 2t +2+e −2t = (e t + e −t ) 2 = e t + e −t ,
r 00 (t) =e t i + e −t k.Thena T = e2t − e −2t
= (et + e −t )(e t − e −t )
= e t − e −t =2sinht
e t + e −t e t + e −t
√ 2e −t i − 2 j − √ 2e t k 2(e
−2t
+2+e
and a N =
=
2t )
= √ 2 et + e −t
e t + e −t e t + e −t
e t + e = √ 2.
−t

F.
TX.10
CHAPTER 14 REVIEW ET CHAPTER 13 ¤ 157
39. The tangential component of a is the length of the projection of a onto T,sowesketch
the scalar projection of a in the tangential direction to the curve and estimate its length to
be 4.5 (using the fact that a has length 10 as a guide). Similarly, the normal component of
a is the length of the projection of a onto N,sowesketchthescalarprojectionofa in the
normal direction to the curve and estimate its length to be 9.0. Thus a T ≈ 4.5 cm/s 2 and
a N ≈ 9.0 cm/s 2 .
41. If the engines are turned off at time t, then the spacecraft will continue to travel in the direction of v(t), so we need a t such
that for some scalar s>0, r(t)+s v(t) =h6, 4, 9i. v(t) =r 0 (t) =i + 1 t j + 8t
(t 2 +1) k ⇒
2
r(t)+s v(t) = 3+t + s, 2+lnt + s t , 7 − 4
t 2 +1 + 8st
⇒ 3+t + s =6 ⇒ s =3− t,
(t 2 +1) 2
so 7 − 4 8(3 − t)t
+
t 2 +1 (t 2 +1) =9 ⇔ 24t − 12t2 − 4
=2 ⇔ t 4 +8t 2 − 12t +3=0.
2 (t 2 +1) 2
It is easily seen that t =1is a root of this polynomial. Also 2+ln1+ 3 − 1
1
=4,sot =1is the desired solution.
14 Review ET 13
1. A vector function is a function whose domain is a set of real numbers and whose range is a set of vectors. To find the derivative
or integral, we can differentiate or integrate each component of the vector function.
2. Thetipofthemovingvectorr(t) of a continuous vector function traces out a space curve.
3. The tangent vector to a smooth curve at a point P with position vector r(t) is the vector r 0 (t). The tangent line at P is the line
through P parallel to the tangent vector r 0 (t). The unit tangent vector is T(t) = r0 (t)
|r 0 (t)| .
4. (a) – (f ) See Theorem 14.2.3 [ET 13.2.3].
5. Use Formula 14.3.2 [ ET 13.3.2], or equivalently, 14.3.3 [ ET 13.3.3].
6. (a) The curvature of a curve is κ =
dT
ds where T is the unit tangent vector.
(b) κ(t) =
T0 (t)
r 0 (t)
(c) κ(t) = |r0 (t) × r 00 (t)|
|f 00 (x)|
|r 0 (t)| 3 (d) κ(x) =
[1 + (f 0 (x)) 2 ] 3/2
7. (a) The unit normal vector: N(t) = T0 (t)
. The binormal vector: B(t) =T(t) × N(t).
|T 0 (t)|
(b) See the discussion preceding Example 7 in Section 14.3 [ ET 13.3].

F.
158 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
8. (a) If r(t) is the position vector of the particle on the space curve, the velocity v(t) =r 0 (t), the speed is given by |v(t)|,
and the acceleration a(t) =v 0 (t) =r 00 (t).
(b) a = a T T + a N N where a T = v 0 and a N = κv 2 .
9. See the statement of Kepler’s Laws on page 880 [ET page 844].
1. True. If we reparametrize the curve by replacing u = t 3 ,wehaver(u) =u i +2u j +3u k, which is a line through the origin
with direction vector i +2j +3k.
3. False. By Formula 5 of Theorem 14.2.3[ ET 13.2.3],
d
dt [u(t) × v(t)] = u0 (t) × v(t)+u(t) × v 0 (t).
5. False. κ is the magnitude of the rate of change of the unit tangent vector T with respect to arc length s,notwithrespecttot.
7. True. At an inflection point where f is twice continuously differentiable we must have f 00 (x) =0, and by Equation 14.3.11
[ET 13.3.11], the curvature is 0 there.
9. False. If r(t) is the position of a moving particle at time t and |r(t)| =1then the particle lies on the unit circle or the unit
sphere, but this does not mean that the speed |r 0 (t)| must be constant. As a counterexample, let r(t) = t, √ 1 − t 2 ,then
r 0 (t) = 1, −t/ √ 1 − t 2 and |r(t)| = √ t 2 +1− t 2 =1but |r 0 (t)| = 1+t 2 /(1 − t 2 )=1/ √ 1 − t 2 which is not
constant.
11. True. See the discussion preceding Example 7 in Section 14.3 [ ET 13.3].
1. (a) The corresponding parametric equations for the curve are x = t,
y =cosπt, z =sinπt. Sincey 2 + z 2 =1, the curve is contained in a
circular cylinder with axis the x-axis. Since x = t, the curve is a helix.
(b) r(t) =t i +cosπt j +sinπt k ⇒
r 0 (t) =i − π sin πt j + π cos πt k ⇒
r 00 (t) =−π 2 cos πt j − π 2 sin πt k
3. The projection of the curve C of intersection onto the xy-plane is the circle x 2 + y 2 =16,z =0.Sowecanwrite
x =4cost, y =4sint, 0 ≤ t ≤ 2π. From the equation of the plane, we have z =5− x =5− 4cost, so parametric
equations for C are x =4cost, y =4sint, z =5− 4cost, 0 ≤ t ≤ 2π, and the corresponding vector function is
r(t) =4cost i +4sint j +(5− 4cost) k, 0 ≤ t ≤ 2π.

F.
5.
1
1
1
0 (t2 i + t cos πt j +sinπt k) dt = dt
0 t2 i + t cos πt dt 1
j + sin πt dt k
0 0
=
1
t3 1
i + 3 0
where we integrated by parts in the y-component.
TX.10
t
sin πt 1
− 1
π 0 0
CHAPTER 14 REVIEW ET CHAPTER 13 ¤ 159
1
sin πt dt j + − 1 cos πt 1
k
π π 0
= 1 3 i + 1
π 2 cos πt 1
0 j + 2 π k = 1 3 i − 2
π 2 j + 2 π k
7. r(t) = t 2 ,t 3 ,t 4 ⇒ r 0 (t) = 2t, 3t 2 , 4t 3 ⇒ |r 0 (t)| = √ 4t 2 +9t 4 +16t 6 and
L = 3
0 |r0 (t)| dt = 3
0
√
4t2 +9t 4 +16t 6 dt. Using Simpson’s Rule with f(t) = √ 4t 2 +9t 4 +16t 6 and n =6we
have ∆t = 3−0 = 1 and
6 2
L ≈ ∆t
3 f(0) + 4f 1
2 +2f(1) + 4f 3
2 +2f(2) + 4f 5
2 + f(3)
= 1 6
√0+0+0+4· 4
1 2
+9
1 4
+16
1 6
+2· 4(1)
2 2
2
2 +9(1) 4 + 16(1) 6
≈ 86.631
+4·
+4·
4 3
2
4 5
2
2
+9 3
2
2
+9 5
2
4
+16
3 6
+2· 4(2)
2
2 +9(2) 4 +16(2) 6
4
+16
5 6
+
4(3)
2
2 +9(3) 4 +16(3) 6
9. The angle of intersection of the two curves, θ, is the angle between their respective tangents at the point of intersection.
For both curves the point (1, 0, 0) occurs when t =0.
r 0 1(t) =− sin t i +cost j + k ⇒ r 0 1(0) = j + k and r 0 2(t) = i +2t j +3t 2 k ⇒ r 0 2(0) = i.
r 0 1(0) · r 0 2(0) = (j + k) · i =0. Therefore, the curves intersect in a right angle, that is, θ = π 2 .
11. (a) T(t) = r0 (t)
|r 0 (t)| =
t 2 ,t,1
|ht 2 ,t,1i| =
t 2 ,t,1
√
t4 + t 2 +1
(b) T 0 (t) =− 1 2 (t4 + t 2 +1) −3/2 (4t 3 +2t) t 2 ,t,1 +(t 4 + t 2 +1) −1/2 h2t, 1, 0i
−2t 3 − t
=
t 2 ,t,1 1
+
h2t, 1, 0i
(t 4 + t 2 +1) 3/2 (t 4 + t 2 +1)
1/2
−2t 5 − t 3 , −2t 4 − t 2 , −2t 3 − t + 2t 5 +2t 3 +2t, t 4 + t 2 +1, 0
=
|T 0 (t)| =
(c) κ(t) = |T0 (t)|
|r 0 (t)|
(t 4 + t 2 +1) 3/2 =
√ √
4t2 + t 8 − 2t 4 +1+4t 6 +4t 4 + t 2 t8 +4t
=
6 +2t 4 +5t 2
and N(t) =
(t 4 + t 2 +1) 3/2 (t 4 + t 2 +1) 3/2
√
t8 +4t
=
6 +2t 4 +5t 2
(t 4 + t 2 +1) 2
13. y 0 =4x 3 , y 00 =12x 2 and κ(x) =
|y 00 |
[1 + (y 0 ) 2 ] = 12x 2
12
,soκ(1) = 3/2 (1 + 16x 6 )
3/2
17 . 3/2
2t, −t 4 +1, −2t 3 − t
(t 4 + t 2 +1) 3/2
2t, 1 − t 4 , −2t 3 − t
√
t8 +4t 6 +2t 4 +5t 2 .
15. r(t) =hsin 2t, t, cos 2ti ⇒ r 0 (t) =h2cos2t, 1, −2sin2ti ⇒ T(t) = 1 √
5
h2cos2t, 1, −2sin2ti
⇒
T 0 (t) = 1 √
5
h−4sin2t, 0, −4cos2ti ⇒ N(t) =h− sin 2t, 0, − cos 2ti. SoN = N(π) =h0, 0, −1i and

F.
160 ¤ CHAPTER 14 VECTOR FUNCTIONS ET CHAPTER 13
TX.10
B = T × N = 1 √
5
h−1, 2, 0i. So a normal to the osculating plane is h−1, 2, 0i and an equation is
−1(x − 0) + 2(y − π)+0(z − 1) = 0 or x − 2y +2π =0.
17. r(t) =t ln t i + t j + e −t k, v(t) =r 0 (t) =(1+lnt) i + j − e −t k,
|v (t)| = (1 + ln t) 2 +1 2 +(−e −t ) 2 = 2+2lnt +(lnt) 2 + e −2t , a(t) =v 0 (t) = 1 t i + e−t k
19. We set up the axes so that the shot leaves the athlete’s hand 7 ft above the origin. Then we are given r(0) = 7j,
|v(0)| =43ft/s, and v(0) has direction given by a 45 ◦ angle of elevation. Then a unit vector in the direction of v(0) is
√
1
2
(i + j) ⇒ v(0) = √ 43
2
(i + j). Assuming air resistance is negligible, the only external force is due to gravity, so as
in Example 14.4.5 [ET 13.4.5] we have a = −g j where here g ≈ 32 ft/s 2 .Sincev 0 (t) =a(t), weintegrate,giving
v(t) =−gt j + C where C = v(0) = √ 43
2
(i + j) ⇒ v (t) = √ 43 43
2
i + √
2
− gt j. Sincer 0 (t) =v(t) we integrate
again, so r(t) = 43 √
2
t i +
(a) At 2 seconds, the shot is at r(2) = 43 √
2
(2) i +
43
√
2
t − 1 2 gt2
j + D. ButD = r(0) = 7 j ⇒ r(t) = 43 √
2
t i +
the ground, at a horizontal distance of 60.8 ft from the athlete.
√
43
2
t − 1 2 gt2 +7 j.
√
43
2
(2) − 1 2 g(2)2 +7 j ≈ 60.8 i +3.8 j, so the shot is about 3.8 ft above
(b) The shot reaches its maximum height when the vertical component of velocity is 0:
√
43
2
− gt =0
⇒
t = 43 √
2 g
≈ 0.95 s. Then r(0.95) ≈ 28.9 i +21.4 j, so the maximum height is approximately 21.4 ft.
(c) The shot hits the ground when the vertical component of r(t) is 0,so 43 √
2
t − 1 2 gt2 +7=0
⇒
−16t 2 + 43 √
2
t +7=0 ⇒ t ≈ 2.11 s. r(2.11) ≈ 64.2 i − 0.08 j, thus the shot lands approximately 64.2 ft from the
athlete.
21. (a) Instead of proceeding directly, we use Formula 3 of Theorem 14.2.3 [ ET 13.2.3]: r(t) =t R(t) ⇒
v = r 0 (t) =R(t)+t R 0 (t) =cosωt i +sinωt j + t v d .
(b) Using the same method as in part (a) and starting with v = R(t)+t R 0 (t), wehave
a = v 0 = R 0 (t)+R 0 (t)+t R 00 (t) =2R 0 (t)+t R 00 (t) =2v d + t a d .
(c)Herewehaver(t) =e −t cos ωt i + e −t sin ωt j = e −t R(t). So, as in parts (a) and (b),
v = r 0 (t) =e −t R 0 (t) − e −t R(t) =e −t [R 0 (t) − R(t)]
⇒
a = v 0 = e −t [R 00 (t) − R 0 (t)] − e −t [R 0 (t) − R(t)] = e −t [R 00 (t) − 2 R 0 (t)+R(t)]
= e −t a d − 2e −t v d + e −t R
Thus, the Coriolis acceleration (the sum of the “extra” terms not involving a d )is−2e −t v d + e −t R.

F.
TX.10
PROBLEMS PLUS
1. (a) r(t) =R cos ωt i + R sin ωt j ⇒ v = r 0 (t) =−ωR sin ωt i + ωR cos ωt j, sor = R(cos ωt i +sinωt j) and
v = ωR(− sin ωt i +cosωt j). v · r = ωR 2 (− cos ωt sin ωt +sinωt cos ωt) =0,sov ⊥ r. Sincer points along a
radius of the circle, and v ⊥ r, v is tangent to the circle. Because it is a velocity vector, v points in the direction of motion.
(b) In (a), we wrote v in the form ωR u,whereu is the unit vector − sin ωt i +cosωt j. Clearly |v| = ωR |u| = ωR. At
speed ωR, the particle completes one revolution, a distance 2πR,intimeT = 2πR
ωR = 2π ω .
(c) a = dv
dt = −ω2 R cos ωt i − ω 2 R sin ωt j = −ω 2 R(cos ωt i +sinωt j),soa = −ω 2 r. This shows that a is proportional
to r and points in the opposite direction (toward the origin). Also, |a| = ω 2 |r| = ω 2 R.
(d) By Newton’s Second Law (see Section 14.4 [ET 13.4]), F = ma,so|F| = m |a| = mRω 2 = m (ωR)2
R
= m |v|2
R .
3. (a) The projectile reaches maximum height when 0= dy
dt = d dt [(v 0 sin α)t − 1 2 gt2 ]=v 0 sin α − gt; thatis,when
t = v
0 sin α
v0 sin α
and y =(v 0 sin α)
− 1 2
g
g 2 g v0 sin α
= v2 0 sin 2 α
. This is the maximum height attained when
g
2g
the projectile is fired with an angle of elevation α. This maximum height is largest when α = π . In that case, sin α =1
2
and the maximum height is v2 0
2g .
(b) Let R = v0
2 g. We are asked to consider the parabola x 2 +2Ry − R 2 =0which can be rewritten as y = − 1
2R x2 + R 2 .
The points on or inside this parabola are those for which −R ≤ x ≤ R and 0 ≤ y ≤ −1
2R x2 + R 2 .
fired at angle of elevation α, the points (x, y) along its path satisfy the relations x =(v 0 cos α) t and
y =(v 0 sin α)t − 1 2 gt2 ,where0 ≤ t ≤ (2v 0 sin α)/g (as in Example 14.4.5 [ET 13.4.5]). Thus
|x| ≤
v0 cos α 2v0 sin α =
g v2 0
g sin 2α ≤
v2 0
g = |R|. This shows that −R ≤ x ≤ R.
For t in the specified range, we also have y = t v 0 sin α − 1 2 gt = 1 2 gt 2v0 sin α
g
− t ≥ 0 and
When the projectile is
x
y =(v 0 sin α)
v 0 cos α − g 2 x
g
1
= (tan α) x −
2 v 0 cos α
2v0 2 cos2 α x2 = −
2R cos 2 α x2 + (tan α) x. Thus
−1
y −
2R x2 + R
−1
=
2 2R cos 2 α x2 + 1
2R x2 +(tanα) x − R 2
= x2
1 − 1
+ (tan α) x − R 2R cos 2 α
2 = x2 (1 − sec 2 α)+2R (tan α) x − R 2
2R
= −(tan2 α) x 2 +2R (tan α) x − R 2
2R
=
− [(tan α) x − R]2
2R
We have shown that every target that can be hit by the projectile lies on or inside the parabola y = − 1
2R x2 + R 2 . 161
≤ 0

F.
162 ¤ PROBLEMS PLUS
TX.10
Now let (a, b) be any point on or inside the parabola y = − 1
2R x2 + R 1
.Then−R ≤ a ≤ R and 0 ≤ b ≤−
2 2R a2 + R 2 .
We seek an angle α such that (a, b) lies in the path of the projectile; that is, we wish to find an angle α such that
1
b = −
2R cos 2 α a2 +(tanα) a or equivalently b = −1
2R (tan2 α +1)a 2 + (tan α) a. Rearranging this equation we get
a 2
a
2
2R tan2 α − a tan α +
2R + b =0or a 2 (tan α) 2 − 2aR(tan α)+(a 2 +2bR) =0 (∗) . This quadratic equation
for tan α has real solutions exactly when the discriminant is nonnegative. Now B 2 − 4AC ≥ 0
⇔
(−2aR) 2 − 4a 2 (a 2 +2bR) ≥ 0 ⇔ 4a 2 (R 2 − a 2 − 2bR) ≥ 0 ⇔ −a 2 − 2bR + R 2 ≥ 0 ⇔
b ≤ 1
2R (R2 − a 2 ) ⇔ b ≤ −1
2R a2 + R . This condition is satisfied since (a, b) is on or inside the parabola
2
y = − 1
2R x2 + R . It follows that (a, b) lies in the path of the projectile when tan α satisfies (∗), that is, when
2
tan α = 2aR ± 4a 2 (R 2 − a 2 − 2bR)
= R ± √ R 2 − 2bR − a 2
.
2a 2 a
(c)
If the gun is pointed at a target with height h at a distance D downrange, then
tan α = h/D. When the projectile reaches a distance D downrange (remember
we are assuming that it doesn’t hit the ground first), we have D = x =(v 0 cos α)t,
so t =
D
v 0 cos α and y =(v0 sin α)t − 1 2 gt2 = D tan α −
gD2
2v0 2 cos2 α .
Meanwhile, the target, whose x-coordinate is also D, has fallen from height h to height
h − 1 2 gt2 = D tan α −
gD2 . Thus the projectile hits the target.
2v0 2 cos2 α
5. (a) a = −g j ⇒ v = v 0 − gt j =2i − gt j ⇒ s = s 0 +2t i − 1 2 gt2 j =3.5 j +2t i − 1 2 gt2 j ⇒
s =2t i + 3.5 − 1 2 gt2 j. Therefore y =0when t = 7/g seconds. At that instant, the ball is 2 7/g ≈ 0.94 ft to the
right of the table top. Its coordinates (relative to an origin on the floor directly under the table’s edge) are (0.94, 0). At
impact, the velocity is v =2i − √ 7g j,sothespeedis|v| = √ 4+7g ≈ 15 ft/s.
7
(b) The slope of the curve when t =
g
and θ ≈ 7.6 ◦ .
dy
is
dx = dy/dt
dx/dt = −gt
2 = −g 7/g
2
= −√ √
7g
7g
.Thuscot θ =
2
2
(c) From (a), |v| = √ 4+7g. So the ball rebounds with speed 0.8 √ 4+7g ≈ 12.08 ft/s at angle of inclination
90 ◦ − θ ≈ 82.3886 ◦ . By Example 14.4.5 [ET 13.4.5], the horizontal distance traveled between bounces is
d = v2 0 sin 2α
,wherev 0 ≈ 12.08 ft/sandα ≈ 82.3886 ◦ . Therefore, d ≈ 1.197 ft. So the ball strikes the floor at
g
about 2 7/g +1.197 ≈ 2.13 ft to the right of the table’s edge.

F.
TX.10
PROBLEMS PLUS ¤ 163
7. The trajectory of the projectile is given by r(t) =(v cos α)t i + (v sin α)t − 1 2 gt2 j,so
v(t) =r 0 (t) =v cos α i +(v sin α − gt) j and
|v(t)| = (v cos α) 2 +(v sin α − gt) 2 =
v 2 − (2vg sin α) t + g 2 t 2 =
g 2 t 2 − 2v
v2
(sin α) t +
g g 2
= g t − v
2
g sin α + v2
g − v2
2 g 2 sin2 α = g t − v 2
g sin α + v2
g 2 cos2 α
The projectile hits the ground when (v sin α)t − 1 2 gt2 =0 ⇒ t = 2v g
sin α, so the distance traveled by the projectile is
(2v/g)sinα
(2v/g)sinα
L(α)=
|v(t)| dt =
g t − v 2
0
0
g sin α + v2
g 2 cos2 αdt
⎡
= g⎣ t − (v/g)sinα t − v 2 2 v
2
g sin α +
g cos α
+ [(v/g)cosα]2
2
⎛
⎞⎤
ln⎝t − v g sin α + t − v 2 2 v
g sin α +
g cos α ⎠⎦
(2v/g)sinα
[using Formula 21 in the Table of Integrals]
⎡ ⎛
= g ⎣ v 2 2
⎞
2 v
v v v
2 g sin α g sin α +
g cos α +
g cos α ln⎝ v 2 2 v
g sin α + g sin α +
g cos α ⎠
= g 2
= v2
g
⎛
+ v 2 2
⎞⎤
2 v v v v
g sin α g sin α +
g cos α −
g cos α ln⎝− v 2 2 v
g sin α + g sin α +
g cos α ⎠⎦
v
g sin α · v
g + v2 v
g 2 cos2 α ln
g sin α + v
+ v g g sin α · v
g − v2
g 2 cos2 α ln − v g sin α + v
g
v2
(v/g)sinα + v/g
sin α +
2g cos2 α ln
= v2 v2 1+sinα
sin α +
− (v/g)sinα + v/g g 2g cos2 α ln
1 − sin α
0
We want to maximize L(α) for 0 ≤ α ≤ π/2.
L 0 (α)= v2
g
= v2
g
= v2
g
v2
cos α +
2g
cos α +
v2
2g
cos 2 α · 1 − sin α
1+sinα ·
cos 2 α ·
cos α +
v2
g cos α
1 − sin α ln
2cosα
1+sinα
(1 − sin α) 2 − 2cosα sin α ln 1 − sin α
2
1+sinα
cos α − 2cosα sin α ln 1 − sin α
1+sinα
1 − sin α
L(α) has critical points for 0

F.
TX.10

F.
TX.10
15 PARTIAL DERIVATIVES ET 14
15.1 Functions of Several Variables ET 14.1
1. (a) From Table 1, f(−15, 40) = −27, which means that if the temperature is −15 ◦ C and the wind speed is 40 km/h, then the
air would feel equivalent to approximately −27 ◦ C without wind.
(b) The question is asking: when the temperature is −20 ◦ C, what wind speed gives a wind-chill index of −30 ◦ C?From
Table 1, the speed is 20 km/h.
(c) The question is asking: when the wind speed is 20 km/h, what temperature gives a wind-chill index of −49 ◦ C?From
Table 1, the temperature is −35 ◦ C.
(d) The function W = f(−5,v) means that we fix T at −5 and allow v to vary, resulting in a function of one variable. In
other words, the function gives wind-chill index values for different wind speeds when the temperature is −5 ◦ C.From
Table 1 (look at the row corresponding to T = −5), the function decreases and appears to approach a constant value as v
increases.
(e) The function W = f(T,50) means that we fix v at 50 and allow T to vary, again giving a function of one variable. In
other words, the function gives wind-chill index values for different temperatures when the wind speed is 50 km/h.From
Table 1 (look at the column corresponding to v =50), the function increases almost linearly as T increases.
3. If the amounts of labor and capital are both doubled, we replace L, K in the function with 2L, 2K,giving
P (2L, 2K) =1.01(2L) 0.75 (2K) 0.25 =1.01(2 0.75 )(2 0.25 )L 0.75 K 0.25 =(2 1 )1.01L 0.75 K 0.25 =2P (L, K)
Thus, the production is doubled. It is also true for the general case P (L, K) =bL α K 1 − α :
P (2L, 2K) =b(2L) α (2K) 1−α = b(2 α )(2 1−α )L α K 1−α =(2 α+1−α )bL α K 1−α =2P (L, K).
5. (a) According to Table 4, f(40, 15) = 25, which means that if a 40-knot wind has been blowing in the open sea for 15 hours,
it will create waves with estimated heights of 25 feet.
(b) h = f(30,t) means we fix v at 30 and allow t to vary, resulting in a function of one variable. Thus here, h = f(30,t)
gives the wave heights produced by 30-knot winds blowing for t hours. From the table (look at the row corresponding to
v =30), the function increases but at a declining rate as t increases. In fact, the function values appear to be approaching a
limiting value of approximately 19, which suggests that 30-knot winds cannot produce waves higher than about 19 feet.
(c) h = f(v, 30) means we fix t at 30, again giving a function of one variable. So, h = f(v, 30) gives the wave heights
produced by winds of speed v blowing for 30 hours. From the table (look at the column corresponding to t =30), the
function appears to increase at an increasing rate, with no apparent limiting value. This suggests that faster winds (lasting
30 hours) always create higher waves.
7. (a) f(2, 0) = 2 2 e 3(2)(0) =4(1)=4
(b) Since both x 2 and the exponential function are defined everywhere, x 2 e 3xy is defined for all choices of values for x and y.
Thus the domain of f is R 2 .
165

F.
166 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
(c) Because the range of g(x, y) =3xy is R, and the range of e x is (0, ∞), the range of e g(x,y) = e 3xy is (0, ∞).
The range of x 2 is [0, ∞), so the range of the product x 2 e 3xy is [0, ∞).
√
6−2
9. (a) f(2, −1, 6) = e
2 −(−1) 2 = e √1 = e.
√
z−x
(b) e
2 −y 2 is defined when z − x 2 − y 2 ≥ 0 ⇒ z ≥ x 2 + y 2 . Thus the domain of f is (x, y, z) | z ≥ x 2 + y 2 .
(c) Since √
z − x 2 − y 2 z−x
≥ 0,wehavee
2 −y 2 ≥ 1. Thus the range of f is [1, ∞).
√
11. x + y is defined only when x + y ≥ 0,ory ≥−x. So
the domain of f is {(x, y) | y ≥−x}.
13. ln(9 − x 2 − 9y 2 ) is definedonlywhen
9 − x 2 − 9y 2 > 0,or 1 9 x2 + y 2 < 1. So the domain of f
is (x, y) 1
9 x2 + y 2 < 1 , the interior of an ellipse.
√
15. 1 − x2 is defined only when 1 − x 2 ≥ 0,orx 2 ≤ 1
17. y − x 2 is defined only when y − x 2 ≥ 0,ory ≥ x 2 .
⇔ −1 ≤ x ≤ 1,and 1 − y 2 is defined only when
1 − y 2 ≥ 0,ory 2 ≤ 1 ⇔ −1 ≤ y ≤ 1. Thusthe
domain of f is {(x, y) | −1 ≤ x ≤ 1, − 1 ≤ y ≤ 1}.
In addition, f is not defined if 1 − x 2 =0
x = ±1. Thus the domain of f is
(x, y) | y ≥ x 2 ,x6=±1 .
⇒
19. We need 1 − x 2 − y 2 − z 2 ≥ 0 or x 2 + y 2 + z 2 ≤ 1,
21. z =3, a horizontal plane through the point (0, 0, 3).
so D = (x, y, z) | x 2 + y 2 + z 2 ≤ 1 (the points inside
or on the sphere of radius 1, center the origin).

F.
SECTION TX.1015.1 FUNCTIONS OF SEVERAL VARIABLES ET SECTION 14.1 ¤ 167
23. z =10− 4x − 5y or 4x +5y + z =10,aplanewith
intercepts 2.5, 2,and10.
25. z = y 2 +1, a parabolic cylinder
27. z =4x 2 + y 2 +1,anellipticparaboloidwithvertex
at (0, 0, 1).
29. z = x 2 + y 2 so x 2 + y 2 = z 2 and z ≥ 0,thetophalf
of a right circular cone.
31. The point (−3, 3) lies between the level curves with z-values 50 and 60. Since the point is a little closer to the level curve with
z =60,weestimatethatf(−3, 3) ≈ 56. Thepoint(3, −2) appears to be just about halfway between the level curves with
z-values 30 and 40,soweestimatef(3, −2) ≈ 35. The graph rises as we approach the origin, gradually from above, steeply
from below.
33. Near A, the level curves are very close together, indicating that the terrain is quite steep. At B, the level curves are much
farther apart, so we would expect the terrain to be much less steep than near A, perhaps almost flat.
35. 37.

F.
168 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
39. The level curves are (y − 2x) 2 = k or y =2x ± √ k,
k ≥ 0, a family of pairs of parallel lines.
TX.10
41. The level curves are y − ln x = k or y =lnx + k.
43. The level curves are ye x = k or y = ke −x ,afamilyof
exponential curves.
45. The level curves are y 2 − x 2 = k or y 2 − x 2 = k 2 ,
k ≥ 0. Whenk =0the level curve is the pair of lines
y = ±x. Fork>0, the level curves are hyperbolas with
axis the y-axis.
47. The contour map consists of the level curves k = x 2 +9y 2 ,afamilyof
ellipses with major axis the x-axis. (Or, if k =0, the origin.)
The graph of f (x, y) is the surface z = x 2 +9y 2 , an elliptic paraboloid.
If we visualize lifting each ellipse k = x 2 +9y 2 of the contour map to the plane
z = k, we have horizontal traces that indicate the shape of the graph of f.
49. The isothermals are given by k = 100/(1 + x 2 +2y 2 ) or
x 2 +2y 2 =(100− k)/k [0

F.
SECTION TX.1015.1 FUNCTIONS OF SEVERAL VARIABLES ET SECTION 14.1 ¤ 169
51. f(x, y) =e −x2 + e −2y2
53. f(x, y) =xy 2 − x 3
The traces parallel to the yz-plane (such as the left-front trace in the graph above) are parabolas; those parallel to the xz-plane
(such as the right-front trace) are cubic curves. The surface is called a monkey saddle because a monkey sitting on the surface
near the origin has places for both legs and tail to rest.
55. (a) C (b) II
Reasons: This function is periodic in both x and y, and the function is the same when x is interchanged with y,soitsgraphis
symmetric about the plane y = x. In addition, the function is 0 along the x-andy-axes. These conditions are satisfied only by
C and II.
57. (a) F (b) I
Reasons: This function is periodic in both x and y but is constant along the lines y = x + k, a condition satisfied only by F and
I.
59. (a) B (b) VI
Reasons: This function is 0 along the lines x = ±1 and y = ±1. The only contour map in which this could occur is VI. Also
note that the trace in the xz-plane is the parabola z =1− x 2 and the trace in the yz-plane is the parabola z =1− y 2 ,sothe
graph is B.
61. k = x +3y +5z is a family of parallel planes with normal vector h1, 3, 5i.
63. k = x 2 − y 2 + z 2 are the equations of the level surfaces. For k =0, the surface is a right circular cone with vertex the origin
and axis the y-axis. For k>0, we have a family of hyperboloids of one sheet with axis the y-axis. For k

F.
170 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
67. f(x, y) =3x − x 4 − 4y 2 − 10xy
Three-dimensional view
Front view
It does appear that the function has a maximum value, at the higher of the two “hilltops.” From the front view graph, the
maximum value appears to be approximately 15. Both hilltops could be considered local maximum points, as the values of f
there are larger than at the neighboring points. There does not appear to be any local minimum point; although the valley shape
between the two peaks looks like a minimum of some kind, some neighboring points have lower function values.
69. f(x, y) = x + y .Asbothx and y become large, the function values
x 2 + y2 appear to approach 0, regardless of which direction is considered. As
(x, y) approaches the origin, the graph exhibits asymptotic behavior.
From some directions, f(x, y) →∞, while in others f(x, y) →−∞.
(These are the vertical spikes visible in the graph.) If the graph is
examined carefully, however, one can see that f(x, y) approaches 0
along the line y = −x.
71. f (x, y) =e cx2 +y 2 .First,ifc =0, the graph is the cylindrical surface
z = e y2 (whose level curves are parallel lines). When c>0,theverticaltrace
above the y-axis remains fixed while the sides of the surface in the x-direction
“curl” upward, giving the graph a shape resembling an elliptic paraboloid. The
level curves of the surface are ellipses centered at the origin.
c =0
For 0

F.
For c =1the level curves are circles centered at the origin.
SECTION TX.1015.1 FUNCTIONS OF SEVERAL VARIABLES ET SECTION 14.1 ¤ 171
c =1(level curves in increments of 1)
When c>1, the level curves are ellipses with major axis the y-axis, and the eccentricity increases as c increases.
c =2(level curves in increments of 4)
For values of c

F.
172 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
73. z = x 2 + y 2 + cxy. Whenc0 thegraphshavethesameshape,butarereflected in the plane x =0, because
x 2 + y 2 + cxy =(−x) 2 + y 2 +(−c)(−x)y. That is, the value of z is the same for c at (x, y) as it is for −c at (−x, y).
c = −1, z =2 c =0 c =10
So the surface is an elliptic paraboloid for 0

F.
Year x =ln(L/K) y =ln(P/K)
1899 0 0
1900 −0.02 −0.06
1901 −0.04 −0.02
1902 −0.04 0
1903 −0.07 −0.05
1904 −0.13 −0.12
1905 −0.18 −0.04
1906 −0.20 −0.07
1907 −0.23 −0.15
1908 −0.41 −0.38
1909 −0.33 −0.24
1910 −0.35 −0.27
TX.10
SECTION 15.2 LIMITS AND CONTINUITY ET SECTION 14.2 ¤ 173
Year x =ln(L/K) y =ln(P/K)
1911 −0.38 −0.34
1912 −0.38 −0.24
1913 −0.41 −0.25
1914 −0.47 −0.37
1915 −0.53 −0.34
1916 −0.49 −0.28
1917 −0.53 −0.39
1918 −0.60 −0.50
1919 −0.68 −0.57
1920 −0.74 −0.57
1921 −1.05 −0.85
1922 −0.98 −0.59
After entering the (x, y) pairs into a calculator or CAS, the resulting least squares regression line through the points is
approximately y =0.75136x +0.01053,whichweroundtoy =0.75x +0.01.
(c) Comparing the regression line from part (b) to the equation y =lnb + αx with x =ln(L/K) and y =ln(P/K),wehave
α =0.75 and ln b =0.01 ⇒ b = e 0.01 ≈ 1.01. Thus, the Cobb-Douglas production function is
P = bL α K 1−α =1.01L 0.75 K 0.25 .
15.2 Limits and Continuity ET 14.2
1. In general, we can’t say anything about f(3, 1)! lim f(x, y) =6means that the values of f(x, y) approach 6 as
(x,y)→(3,1)
(x, y) approaches, but is not equal to, (3, 1). Iff is continuous, we know that
lim f(x, y) =f(3, 1) = 6.
(x,y)→(3,1)
lim
(x,y)→(a,b)
f(x, y) =f(a, b), so
3. We make a table of values of
f(x, y) = x2 y 3 + x 3 y 2 − 5
for a set
2 − xy
of (x, y) points near the origin.
As the table shows, the values of f(x, y) seem to approach −2.5 as (x, y) approaches the origin from a variety of different
directions. This suggests that
lim
(x,y)→(0,0)
f(x, y) =−2.5. Sincef is a rational function, it is continuous on its domain. f is
defined at (0, 0), so we can use direct substitution to establish that lim f(x, y) 0 3 +0 3 0 2 − 5
(x,y)→(0,0) =02 = − 5 2 − 0 · 0 2 , verifying
our guess.

F.
174 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
5. f(x, y) =5x 3 − x 2 y 2 is a polynomial, and hence continuous, so lim
(x,y)→(1,2) f(x, y) =f(1, 2) = 5(1)3 − (1) 2 (2) 2 =1.
7. f(x, y) = 4 − xy is a rational function and hence continuous on its domain.
x 2 +3y2 4 − (2)(1)
(2, 1) is in the domain of f,sof is continuous there and lim f(x, y) =f(2, 1) =
(x,y)→(2,1) (2) 2 +3(1) = 2 2 7 .
9. f(x, y) =y 4 / x 4 +3y 4 . First approach (0, 0) along the x-axis. Then f(x, 0) = 0/x 4 =0for x 6= 0,sof(x, y) → 0.
Now approach (0, 0) along the y-axis. Then for y 6= 0, f(0,y)=y 4 /3y 4 =1/3,sof(x, y) → 1/3. Sincef has two different
limits along two different lines, the limit does not exist.
11. f(x, y) =(xy cos y)/(3x 2 + y 2 ). Onthex-axis, f (x, 0) = 0 for x 6= 0,sof(x, y) → 0 as (x, y) → (0, 0) along the
x-axis. Approaching (0, 0) along the line y = x, f(x, x) =(x 2 cos x)/4x 2 = 1 cos x for x 6= 0,sof(x, y) → 1 4 4
along this
line. Thus the limit does not exist.
13. f(x, y) =
xy
x2 + y 2 .
We can see that the limit along any line through (0, 0) is 0, as well as along other paths through
(0, 0) such as x = y 2 and y = x 2 . So we suspect that the limit exists and equals 0; we use the Squeeze Theorem to prove our
xy
assertion. 0 ≤
≤ |x| since |y| ≤ x
2 + y 2 ,and|x| → 0 as (x, y) → (0, 0). So lim f(x, y) =0.
x2 + y 2 (x,y)→(0,0)
15. Let f(x, y) = x2 ye y
.Thenf(x, 0) = 0 for x 6= 0,sof(x, y) → 0 as (x, y) → (0, 0) along the x-axis. Approaching
x 4 +4y2 (0, 0) along the y-axis or the line y = x also gives a limit of 0. Butf x, x 2 = x2 x 2 e x2
x 4 +4(x 2 ) = x4 e x2
2 5x 4
= ex2
5
for x 6= 0,so
f(x, y) → e 0 /5= 1 5 as (x, y) → (0, 0) along the parabola y = x2 . Thus the limit doesn’t exist.
17. lim
(x,y)→(0,0)
x 2 + y 2
x2 + y 2 +1− 1 =
lim
(x,y)→(0,0)
= lim
(x,y)→(0,0)
x 2 + y 2
x2 + y 2 +1− 1 ·
x2 + y 2 +1+1
x2 + y 2 +1+1
x 2 + y 2
x2 + y +1+1
2
x 2 + y 2 = lim
(x,y)→(0,0)
19. e −xy and sin(πz/2) are each compositions of continuous functions, and hence continuous, so their product
f(x, y, z) =e −xy sin(πz/2) is a continuous function. Then
lim
(x,y,z)→(3,0,1) f(x, y, z) =f (3, 0, 1) = e−(3)(0) sin(π · 1/2) = 1.
x2 + y +1+1
2 =2
21. f(x, y, z) = xy + yz2 + xz 2
x 2 + y 2 + z 4 . Thenf(x, 0, 0) = 0/x2 =0for x 6= 0,soas(x, y, z) → (0, 0, 0) along the x-axis,
f(x, y, z) → 0. Butf(x, x, 0) = x 2 /(2x 2 )= 1 for x 6= 0,soas(x, y, z) → (0, 0, 0) along the line y = x, z =0,
2
TX.10
f(x, y, z) → 1 . Thus the limit doesn’t exist.
2

F.
TX.10
SECTION 15.2 LIMITS AND CONTINUITY ET SECTION 14.2 ¤ 175
23. From the ridges on the graph, we see that as (x, y) → (0, 0) along the
lines under the two ridges, f(x, y) approaches different values. So the
limit does not exist.
25. h(x, y) =g(f(x, y)) = (2x +3y − 6) 2 + √ 2x +3y − 6. Sincef is a polynomial, it is continuous on R 2 and g is
continuous on its domain {t | t ≥ 0}. Thus h is continuous on its domain.
D = {(x, y) | 2x +3y − 6 ≥ 0} = (x, y) | y ≥− 2 3 x +2 , which consists of all points on or above the line y = − 2 3 x +2.
27. From the graph, it appears that f is discontinuous along the line y = x.
If we consider f(x, y) =e 1/(x−y) as a composition of functions,
g(x, y) =1/(x − y) is a rational function and therefore continuous except
where x − y =0 ⇒ y = x. Since the function h(t) =e t is continuous
everywhere, the composition h(g(x, y)) = e 1/(x−y) = f(x, y) is
continuous except along the line y = x, as we suspected.
29. The functions sin(xy) and e x − y 2 are continuous everywhere, so F (x, y) = sin(xy) is continuous except where
e x − y2 e x − y 2 =0 ⇒ y 2 = e x ⇒ y = ± √ e x = ±e 1 2 x .ThusF is continuous on its domain (x, y) | y 6=±e x/2 .
31. F (x, y) = arctan x +
y = g(f(x, y)) where f(x, y) =x + y, continuous on its domain {(x, y) | y ≥ 0}, and
g(t) = arctan t is continuous everywhere. Thus F is continuous on its domain {(x, y) | y ≥ 0}.
33. G(x, y) =ln x 2 + y 2 − 4 = g(f(x, y)) where f(x, y) =x 2 + y 2 − 4, continuous on R 2 ,andg(t) =lnt, continuous on
its domain {t | t>0}. ThusG is continuous on its domain (x, y) | x 2 + y 2 − 4 > 0 = (x, y) | x 2 + y 2 > 4 ,the
exterior of the circle x 2 + y 2 =4.
35.
y is continuous on its domain {y | y ≥ 0} and x 2 − y 2 + z 2 is continuous everywhere, so f(x, y, z) =
continuous for y ≥ 0 and x 2 − y 2 + z 2 6=0 ⇒ y 2 6=x 2 + z 2 ,thatis, (x, y, z) | y ≥ 0, y 6= √ x 2 + z 2 .
⎧
⎪⎨ x 2 y 3
if (x, y) 6= (0, 0)
37. f(x, y) = 2x 2 + y 2
⎪⎩
1 if (x, y) =(0, 0)
y
x 2 − y 2 + z 2 is
The first piece of f is a rational function defined everywhere except at the
origin, so f is continuous on R 2 except possibly at the origin. Since x 2 ≤ 2x 2 + y 2 ,wehave x 2 y 3 /(2x 2 + y 2 ) ≤
y 3 .We
know that y 3 → 0 as (x, y) → (0, 0). So, by the Squeeze Theorem,
lim
(x,y)→(0,0)
f(x, y) =
lim
(x,y)→(0,0)
x 2 y 3
2x 2 + y 2 =0.
But f(0, 0) = 1,sof is discontinuous at (0, 0). Therefore, f is continuous on the set {(x, y) | (x, y) 6= (0, 0)}.

F.
176 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
x 3 + y 3
39. lim
(x,y)→(0,0) x 2 + y = lim (r cos θ) 3 +(r sin θ) 3
= lim
2 r→0 + r 2
r→0 +(r cos3 θ + r sin 3 θ)=0
41. lim
(x,y)→(0,0)
e −x2 −y 2 − 1 e −r2 − 1 e −r2 (−2r)
= lim = lim
x 2 + y 2 r→0 + r 2 r→0 + 2r
[using l’Hospital’s Rule]
= lim
r→0 + −e−r2 = −e 0 = −1
⎧
⎨ sin(xy)
if (x, y) 6= (0, 0)
43. f(x, y) = xy
⎩
1 if (x, y) =(0, 0)
From the graph, it appears that f is continuous everywhere. We know
xy is continuous on R 2 and sin t is continuous everywhere, so
sin(xy) is continuous on R 2 and sin(xy) is continuous on R 2
xy
except possibly where xy =0. To show that f is continuous at those points, consider any point (a, b) in R 2 where ab =0.
Because xy is continuous, xy → ab =0as (x, y) → (a, b). Ifwelett = xy, thent → 0 as (x, y) → (a, b) and
sin(xy) sin(t)
lim
=lim
(x,y)→(a,b) xy t→0 t
on R 2 .
=1by Equation 3.4.2 [ET 3.3.2]. Thus
lim
(x,y)→(a,b)
f(x, y) =f(a, b) and f is continuous
45. Since |x − a| 2 = |x| 2 + |a| 2 − 2 |x||a| cos θ ≥ |x| 2 + |a| 2 − 2 |x||a| =(|x| − |a|) 2 ,wehave |x| − |a|
≤ |x − a|. Let
>0 be given and set δ = . Thenif0 < |x − a|

F.
TX.10
SECTION 15.3 PARTIAL DERIVATIVES ET SECTION 14.3 ¤ 177
temperature increases from the morning to the afternoon as the sun warms it, we would expect f t(158, 21, 9) to be
positive.
f(−15 + h, 30) − f(−15, 30)
3. (a) By Definition 4, f T (−15, 30) = lim
, which we can approximate by considering h =5
h→0 h
and h = −5 and using the values given in the table:
f T (−15, 30) ≈
f T (−15, 30) ≈
f(−10, 30) − f(−15, 30)
5
f(−20, 30) − f(−15, 30)
−5
=
=
−20 − (−26)
5
−33 − (−26)
−5
= 6 5 =1.2,
= −7 =1.4. Averaging these values, we estimate
−5
f T (−15, 30) to be approximately 1.3. Thus, when the actual temperature is −15 ◦ C and the wind speed is 30 km/h, the
apparent temperature rises by about 1.3 ◦ C for every degree that the actual temperature rises.
f(−15, 30 + h) − f(−15, 30)
Similarly, f v(−15, 30) = lim
which we can approximate by considering h =10and
h→0 h
h = −10: f v (−15, 30) ≈
f v (−15, 30) ≈
f(−15, 40) − f(−15, 30)
10
f(−15, 20) − f(−15, 30)
−10
=
=
−24 − (−26)
−10
−27 − (−26)
10
= −1
10 = −0.1,
= 2 = −0.2. Averaging these values, we estimate
−10
f v (−15, 30) to be approximately −0.15. Thus, when the actual temperature is −15 ◦ C and the wind speed is 30 km/h, the
apparent temperature decreases by about 0.15 ◦ C for every km/h that the wind speed increases.
(b) For a fixed wind speed v, the values of the wind-chill index W increase as temperature T increases (look at a column of
the table), so ∂W
∂T
(look at a row of the table), so ∂W
∂v
is positive. For a fixed temperature T ,thevaluesofW decrease (or remain constant) as v increases
is negative (or perhaps 0).
(c) For fixed values of T , the function values f(T,v) appear to become constant (or nearly constant) as v increases, so the
corresponding rate of change is 0 or near 0 as v increases. This suggests that lim (∂W/∂v) =0.
v→∞
5. (a) If we start at (1, 2) andmoveinthepositivex-direction, the graph of f increases. Thus f x (1, 2) is positive.
(b) If we start at (1, 2) andmoveinthepositivey-direction, the graph of f decreases. Thus f y (1, 2) is negative.
7. (a) f xx = ∂
∂x (f x), sof xx is the rate of change of f x in the x-direction. f x is negative at (−1, 2) and if we move in the
positive x-direction, the surface becomes less steep. Thus the values of f x are increasing and f xx(−1, 2) is positive.
(b) f yy is the rate of change of f y in the y-direction. f y is negative at (−1, 2) andifwemoveinthepositivey-direction, the
surface becomes steeper. Thus the values of f y are decreasing, and f yy (−1, 2) is negative.
9. Firstofall,ifwestartatthepoint(3, −3) and move in the positive y-direction, we see that both b and c decrease, while a
increases. Both b and c have a low point at about (3, −1.5),whilea is 0 at this point. So a is definitely the graph of f y,and
one of b and c is the graph of f. To see which is which, we start at the point (−3, −1.5) and move in the positive x-direction.
b traces out a line with negative slope, while c traces out a parabola opening downward. This tells us that b is the x-derivative
of c. Soc is the graph of f, b is the graph of f x,anda is the graph of f y.

F.
178 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
11. f(x, y) =16− 4x 2 − y 2 ⇒ f x(x, y) =−8x and f y(x, y) =−2y ⇒ f x(1, 2) = −8 and f y(1, 2) = −4. The graph
of f is the paraboloid z =16− 4x 2 − y 2 and the vertical plane y =2intersects it in the parabola z =12− 4x 2 , y =2
(the curve C 1 in the first figure). The slope of the tangent line
to this parabola at (1, 2, 8) is f x(1, 2) = −8. Similarly the
plane x =1intersects the paraboloid in the parabola
z =12− y 2 , x =1(the curve C 2 in the second figure) and
the slope of the tangent line at (1, 2, 8) is f y(1, 2) = −4.
13. f(x, y) =x 2 + y 2 + x 2 y ⇒ f x =2x +2xy, f y =2y + x 2
Note that the traces of f in planes parallel to the xz-plane are parabolas which open downward for y−1, and the traces of f x in these planes are straight lines, which have negative slopes for y−1. The traces of f in planes parallel to the yz-plane are parabolas which always open upward, and the traces of f y in
these planes are straight lines with positive slopes.
15. f(x, y) =y 5 − 3xy ⇒ f x (x, y) =0− 3y = −3y, f y (x, y) =5y 4 − 3x
17. f(x, t) =e −t cos πx ⇒ f x(x, t) =e −t (− sin πx)(π) =−πe −t sin πx, f t(x, t) =e −t (−1) cos πx = −e −t cos πx
19. z =(2x +3y) 10 ⇒ ∂z
∂x =10(2x +3y)9 · 2=20(2x +3y) 9 , ∂z
∂y = 10(2x +3y)9 · 3=30(2x +3y) 9
21. f(x, y) = x − y
x + y
f y(x, y) =
⇒
f x (x, y) =
(1)(x + y) − (x − y)(1)
(x + y) 2 =
(−1)(x + y) − (x − y)(1)
(x + y) 2 = − 2x
(x + y) 2
23. w =sinα cos β ⇒ ∂w
∂α =cosα cos β, ∂w
= − sin α sin β
∂β
25. f(r, s) =r ln(r 2 + s 2 ) ⇒ f r (r, s) =r ·
f s (r, s) =r ·
2s
r 2 + s +0= 2rs
2 r 2 + s 2
2y
(x + y) 2 ,
2r
r 2 + s 2 +ln(r2 + s 2 ) · 1=
2r2
r 2 + s 2 +ln(r2 + s 2 ),

F.
27. u = te w/t ⇒ ∂u
∂t = t · ew/t (−wt −2 )+e w/t · 1=e w/t − w t ew/t = e
1 w/t − w
,
t
SECTION 15.3 PARTIAL DERIVATIVES ET SECTION 14.3 ¤ 179
∂u
∂w = tew/t · 1
t = ew/t
29. f(x, y, z) =xz − 5x 2 y 3 z 4 ⇒ f x (x, y, z) =z − 10xy 3 z 4 , f y (x, y, z) =−15x 2 y 2 z 4 , f z (x, y, z) =x − 20x 2 y 3 z 3
31. w =ln(x +2y +3z) ⇒ ∂w
∂x = 1
x +2y +3z , ∂w
∂y = 2
x +2y +3z , ∂w
∂z = 3
x +2y +3z
33. u = xy sin −1 (yz) ⇒ ∂u
∂x = y sin−1 (yz),
∂u
∂z = xy · 1
(y) = xy 2
1 − (yz)
2 1 − y2 z 2
35. f(x, y, z, t) =xyz 2 tan(yt) ⇒ f x (x, y, z, t) =yz 2 tan(yt),
f y (x, y, z, t) =xyz 2 · sec 2 (yt) · t + xz 2 tan(yt) =xyz 2 t sec 2 (yt)+xz 2 tan(yt),
f z (x, y, z, t) =2xyz tan(yt), f t (x, y, z, t) =xyz 2 sec 2 (yt) · y = xy 2 z 2 sec 2 (yt)
∂u
∂y = xy · 1
xyz
1 − (yz)
2 (z)+sin−1 (yz) · x =
1 − y2 z + x 2 sin−1 (yz),
37. u = x 2 1 + x2 2 + ···+ x2 n. For each i =1, ..., n, u xi = 1 2
x
2
1 + x 2 2 + ···+ x 2 n −1/2
(2x i )=
39. f(x, y) =ln
x +
x 2 + y 2
f x (x, y) =
so f x (3, 4) =
41. f(x, y, z) =
so f y (2, 1, −1) =
⇒
1
x + 1+ 1
x 2 + y 2 2 (x2 + y 2 ) −1/2 (2x) =
1
3+ √ 1+
3 2 +4 2
y
x + y + z
43. f(x, y) =xy 2 − x 3 y ⇒
⇒
3
√
32 +4 2
= 1 8
f y (x, y, z) =
2+(−1)
(2+1+(−1)) 2 = 1 4 .
1
x + 1+
x 2 + y 2
1+
3
5 =
1
. 5
1(x + y + z) − y(1)
(x + y + z) 2 =
x + z
(x + y + z) 2 ,
x
x2 + y 2 ,
f(x + h, y) − f(x, y) (x + h)y 2 − (x + h) 3 y − (xy 2 − x 3 y)
f x (x, y) =lim
= lim
h→0 h
h→0 h
h(y 2 − 3x 2 y − 3xyh − yh 2 )
=lim
=lim(y 2 − 3x 2 y − 3xyh − yh 2 )=y 2 − 3x 2 y
h→0 h
h→0
x i
x
2
1 + x 2 2 + ···+ .
x2 n
f(x, y + h) − f(x, y) x(y + h) 2 − x 3 (y + h) − (xy 2 − x 3 y) h(2xy + xh − x 3 )
f y (x, y) =lim
= lim
=lim
h→0 h
h→0 h
h→0 h
=lim
h→0
(2xy + xh − x 3 )=2xy − x 3
45. x 2 + y 2 + z 2 =3xyz ⇒ ∂
∂x (x2 + y 2 + z 2 )= ∂
∂z
(3xyz) ⇒ 2x +0+2z
∂x ∂x =3y x ∂z
∂x + z · 1
2z ∂z ∂z
∂z
∂z 3yz − 2x
− 3xy =3yz − 2x ⇔ (2z − 3xy) =3yz − 2x,so =
∂x ∂x ∂x ∂x 2z − 3xy .
∂
∂y (x2 + y 2 + z 2 )= ∂
∂z
(3xyz) ⇒ 0+2y +2z
∂y ∂y =3x y ∂z
∂y + z · 1
(2z − 3xy) ∂z
∂z 3xz − 2y
=3xz − 2y,so =
∂y ∂y
2z − 3xy . TX.10
⇔
⇔
2z ∂z ∂z
− 3xy =3xz − 2y
∂y ∂y ⇔

F.
180 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
47. x − z =arctan(yz) ⇒ ∂
∂x
1=
∂
∂z
(x − z) = (arctan(yz)) ⇒ 1 −
∂x ∂x = 1
1+(yz) · y ∂z
2 ∂x
y ∂z
y +1+y 2
1+y 2 z +1 2 ∂x ⇔ 1= z 2 ∂z
1+y 2 z 2 ∂x ,so∂z ∂x = 1+y2 z 2
1+y + y 2 z . 2
∂
∂y (x − z) = ∂
∂z
(arctan(yz)) ⇒ 0 −
∂y ∂y = 1
1+(yz) · y ∂z
2 ∂y + z · 1
z
−
1+y 2 z = y ∂z
2 1+y 2 z +1 z y +1+y 2
⇔ −
2 ∂y 1+y 2 z = z 2 ∂z
2 1+y 2 z 2 ∂y
49. (a) z = f(x)+g(y) ⇒ ∂z
∂x = f 0 (x),
(b) z = f(x + y).
∂z
∂y = g0 (y)
Let u = x + y. Then ∂z
∂x = df ∂u
du ∂x = df
du (1) = f 0 (u) =f 0 (x + y),
∂z
∂y = df ∂u
du ∂y = df
du (1) = f 0 (u) =f 0 (x + y).
⇔
⇔
⇔
∂z
∂y = − z
1+y + y 2 z . 2
51. f(x, y) =x 3 y 5 +2x 4 y ⇒ f x(x, y) =3x 2 y 5 +8x 3 y, f y(x, y) =5x 3 y 4 +2x 4 .Thenf xx(x, y) =6xy 5 +24x 2 y,
f xy (x, y) =15x 2 y 4 +8x 3 , f yx (x, y) =15x 2 y 4 +8x 3 ,andf yy (x, y) =20x 3 y 3 .
53. w = √ u 2 + v 2 ⇒ w u = 1 2 (u2 + v 2 ) −1/2 · 2u =
u
√
u2 + v , w 2 v = 1 2 (u2 + v 2 ) −1/2 v
· 2v = √
u2 + v .Then 2
w uu = 1 · √u 2 + v 2 − u · 1
2 (u2 + v 2 ) −1/2 √
(2u) u2 + v
√
u2 + v 2 2
=
2 − u 2 / √ u 2 + v 2
= u2 + v 2 − u 2
u 2 + v 2 (u 2 + v 2 ) = v 2
3/2 (u 2 + v 2 ) , 3/2
w uv = u
− 1 2 u 2 + v 2−3/2 uv
(2v) =−
(u 2 + v 2 ) , w 3/2 vu = v
− 1 2 u 2 + v 2−3/2 uv
(2u) =−
(u 2 + v 2 ) , 3/2
w vv = 1 · √u 2 + v 2 − v · 1
2 (u2 + v 2 ) −1/2 √
(2v) u2 + v
√
u2 + v 2 2
=
2 − v 2 / √ u 2 + v 2
= u2 + v 2 − v 2
u 2 + v 2 (u 2 + v 2 ) = u 2
3/2 (u 2 + v 2 ) . 3/2
55. z =arctan x + y
1 − xy
⇒
z x =
1+
=
z y =
1+
1
x+y
1−xy
2 ·
1+y 2
(1 + x 2 )(1 + y 2 ) = 1
1+x 2 ,
1
x+y
1−xy
2 ·
(1)(1 − xy) − (x + y)(−y) 1+y 2
=
(1 − xy) 2 (1 − xy) 2 +(x + y) = 1+y 2
2 1+x 2 + y 2 + x 2 y 2
(1)(1 − xy) − (x + y)(−x) 1+x 2
=
(1 − xy) 2 (1 − xy) 2 +(x + y) = 1+x 2
2 (1 + x 2 )(1 + y 2 ) = 1
1+y . 2
Then z xx = −(1 + x 2 ) −2 2x
· 2x = −
(1 + x 2 ) , 2 zxy =0, zyx =0, zyy = −(1 + y2 ) −2 2y
· 2y = −
(1 + y 2 ) . 2
57. u = x sin(x +2y) ⇒ u x = x · cos(x +2y)(1) + sin(x +2y) · 1=x cos(x +2y)+sin(x +2y),
u xy = x(− sin(x +2y)(2)) + cos(x +2y)(2) = 2 cos(x +2y) − 2x sin(x +2y),
u y = x cos (x +2y)(2)=2x cos(x +2y),
u yx =2x · (− sin(x +2y)(1)) + cos (x +2y) · 2=2cos(x +2y) − 2x sin(x +2y). Thus u xy = u yx .

F.
TX.10
SECTION 15.3 PARTIAL DERIVATIVES ET SECTION 14.3 ¤ 181
59. u =ln x 2 + y 2 =ln(x 2 + y 2 ) 1/2 = 1 2 ln(x2 + y 2 ) ⇒ u x = 1 1
2 x 2 + y · 2x = x
2 x 2 + y , 2
u xy = x(−1)(x 2 + y 2 ) −2 (2y) =−
2xy
(x 2 + y 2 ) and u 2 y = 1 1
2 x 2 + y · 2y =
2
u yx = y(−1)(x 2 + y 2 ) −2 (2x) =−
2xy
(x 2 + y 2 ) 2 . Thus u xy = u yx .
y
x 2 + y 2 ,
61. f(x, y) =3xy 4 + x 3 y 2 ⇒ f x =3y 4 +3x 2 y 2 , f xx =6xy 2 , f xxy =12xy and
f y =12xy 3 +2x 3 y, f yy =36xy 2 +2x 3 , f yyy =72xy.
63. f(x, y, z) =cos(4x +3y +2z) ⇒
f x = − sin(4x +3y +2z)(4) = −4sin(4x +3y +2z), f xy = −4cos(4x +3y +2z)(3) = −12 cos(4x +3y +2z),
f xyz = −12(− sin(4x +3y +2z))(2) = 24 sin(4x +3y +2z) and
f y = − sin(4x +3y +2z)(3) = −3sin(4x +3y +2z),
f yz = −3cos(4x +3y +2z)(2) = −6cos(4x +3y +2z), f yzz = −6(− sin(4x +3y +2z))(2) = 12 sin(4x +3y +2z).
65. u = e rθ sin θ ⇒ ∂u
∂θ = erθ cos θ +sinθ · e rθ (r) =e rθ (cos θ + r sin θ),
∂ 2 u
∂r ∂θ = erθ (sin θ)+(cosθ + r sin θ) e rθ (θ) =e rθ (sin θ + θ cos θ + rθ sin θ),
∂ 3 u
∂r 2 ∂θ = erθ (θ sin θ)+(sinθ + θ cos θ + rθ sin θ) · e rθ (θ) =θe rθ (2 sin θ + θ cos θ + rθ sin θ).
67. w = x
y +2z = x(y +2z)−1 ⇒ ∂w
∂x =(y +2z)−1 ,
∂ 2 w
∂y ∂x = −(y +2z)−2 (1) = −(y +2z) −2 ,
∂ 3 w
∂z ∂y ∂x = −(−2)(y +2z)−3 (2) = 4(y +2z) −3 4
=
and ∂w
(y +2z) 3 ∂y = x(−1)(y +2z)−2 (1) = −x(y +2z) −2 ,
∂ 2 w
∂x∂y = −(y +2z)−2 ,
∂ 3 w
∂x 2 ∂y =0.
f(3 + h, 2) − f(3, 2)
69. By Definition 4, f x (3, 2) = lim
which we can approximate by considering h =0.5 and h = −0.5:
h→0 h
f x (3, 2) ≈
f(3.5, 2) − f(3, 2)
0.5
=
22.4 − 17.5
0.5
=9.8, f x (3, 2) ≈
f(2.5, 2) − f(3, 2)
−0.5
=
10.2 − 17.5
−0.5
=14.6. Averaging
f(3 + h, 2.2) − f(3, 2.2)
these values, we estimate f x (3, 2) to be approximately 12.2. Similarly, f x (3, 2.2) = lim
which
h→0 h
we can approximate by considering h =0.5 and h = −0.5: f x(3, 2.2) ≈
f x(3, 2.2) ≈
f(2.5, 2.2) − f(3, 2.2)
−0.5
=
9.3 − 15.9
−0.5
To estimate f xy(3, 2), wefirstneedanestimateforf x(3, 1.8):
f x (3, 1.8) ≈
f(3.5, 1.8) − f(3, 1.8)
0.5
=
20.0 − 18.1
0.5
f(3.5, 2.2) − f(3, 2.2)
0.5
=
26.1 − 15.9
0.5
=13.2. Averaging these values, we have f x(3, 2.2) ≈ 16.8.
=3.8, f x (3, 1.8) ≈
f(2.5, 1.8) − f(3, 1.8)
−0.5
=
12.5 − 18.1
−0.5
Averaging these values, we get f x (3, 1.8) ≈ 7.5. Nowf xy (x, y) = ∂ ∂y [f x(x, y)] and f x (x, y) is itself a function of two
=20.4,
=11.2.

F.
182 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
variables, so Definition 4 says that f xy (x, y) = ∂
∂y [f f x(x, y + h) − f x(x, y)
x(x, y)] = lim
h→0 h
f x(3, 2+h) − f x(3, 2)
f xy(3, 2) = lim
.
h→0 h
We can estimate this value using our previous work with h =0.2 and h = −0.2:
f xy (3, 2) ≈ f x(3, 2.2) − f x (3, 2)
0.2
=
16.8 − 12.2
0.2
=23, f xy (3, 2) ≈ f x(3, 1.8) − f x (3, 2)
−0.2
Averaging these values, we estimate f xy (3, 2) to be approximately 23.25.
=
⇒
7.5 − 12.2
−0.2
71. u = e −α2 k 2t sin kx ⇒ u x = ke −α2 k 2t cos kx, u xx = −k 2 e −α2 k 2t sin kx,andu t = −α 2 k 2 e −α2 k 2t sin kx.
Thus α 2 u xx = u t .
73. u =
1
x2 + y 2 + z 2 ⇒ u x = − 1 2
(x 2 + y 2 + z 2 ) −3/2 (2x) =−x(x 2 + y 2 + z 2 ) −3/2 and
u xx = −(x 2 + y 2 + z 2 ) −3/2 − x − 3 2
(x 2 + y 2 + z 2 ) −5/2 (2x) = 2x2 − y 2 − z 2
(x 2 + y 2 + z 2 ) 5/2 .
By symmetry, u yy = 2y2 − x 2 − z 2
(x 2 + y 2 + z 2 ) 5/2 and u zz = 2z2 − x 2 − y 2
(x 2 + y 2 + z 2 ) 5/2 .
Thus u xx + u yy + u zz = 2x2 − y 2 − z 2 +2y 2 − x 2 − z 2 +2z 2 − x 2 − y 2
(x 2 + y 2 + z 2 ) 5/2 =0.
75. Let v = x + at, w = x − at. Thenu t = ∂[f(v)+g(w)]
∂t
u tt = ∂[af 0 (v) − ag 0 (w)]
∂t
=
df (v) ∂v
dv
∂t + dg(w)
dw
=23.5.
∂w
∂t = af 0 (v) − ag 0 (w) and
= a[af 00 (v)+ag 00 (w)] = a 2 [f 00 (v)+g 00 (w)]. Similarly, by using the Chain Rule we have
u x = f 0 (v)+g 0 (w) and u xx = f 00 (v)+g 00 (w). Thus u tt = a 2 u xx .
77. z =ln(e x + e y ) ⇒ ∂z
∂x =
∂ 2 z
∂x 2 = ex (e x + e y ) − e x (e x )
(e x + e y ) 2 =
ex ∂z
and
e x + ey ∂y =
∂ 2 z
∂y = ey (e x + e y ) − e y (e y ) e x+y
=
2 (e x + e y ) 2 (e x + e y ) . Thus 2
∂ 2 z ∂ 2 z ∂ 2 2
∂x 2 ∂y − z
=
2 ∂x∂y
ey ∂z
,so
e x + ey ∂x + ∂z
∂y =
ex
e x + e y +
ey
e x + e y
e x+y
(e x + e y ) 2 , ∂ 2 z
∂x∂y = 0 − ey (e x )
(e x + e y ) 2 = − ex+y
(e x + e y ) 2 , and
e x+y
(e x + e y ) 2 ·
e x+y
(e x + e y ) − 2
= ex + e y
e x + e y =1.
2
−
ex+y
= (ex+y ) 2
(e x + e y ) 2 (e x + e y ) − (ex+y ) 2
4 (e x + e y ) =0 4
79. If we fix K = K 0 ,P(L, K 0 ) is a function of a single variable L,and dP
dL = α P is a separable differential equation. Then
L
dP
dP
P = αdL L ⇒ P = α dL ⇒ ln |P | = α ln |L| + C (K0), whereC(K0) can depend on K0. Then
L
|P | = e α ln|L| + C(K 0) , and since P>0 and L>0,wehaveP = e α ln L e C(K 0) = e C(K 0) e ln Lα = C 1 (K 0 )L α where
C 1(K 0)=e C(K 0) .
81. By the Chain Rule, taking the partial derivative of both sides with respect to R 1 gives
∂R −1
∂R
∂R
∂R 1
= ∂ [(1/R1)+(1/R2)+(1/R3)]
∂R 1
or −R −2 ∂R
= −R −2 ∂R
1 .Thus = R2
.
∂R 1 ∂R 1
R 2 1

F.
TX.10
SECTION 15.3 PARTIAL DERIVATIVES ET SECTION 14.3 ¤ 183
83. By Exercise 82, PV = mRT ⇒ P = mRT
V
Since T = PV
∂P
,wehaveT
mR ∂T
∂P
,so
∂T = mR
V
∂V
∂T = PV
mR · mR
V · mR
P
= mR.
.Also,PV = mRT ⇒ V =
mRT
P
and ∂V
∂T = mR
P .
85.
∂K
∂m = 1 2 v2 ,
∂K
∂v = mv, ∂ 2 K
∂v 2
∂K
= m. Thus
∂m · ∂2 K
∂v = 1 2 2 v2 m = K.
87. f x (x, y) =x +4y ⇒ f xy (x, y) =4and f y (x, y) =3x − y ⇒ f yx (x, y) =3.Sincef xy and f yx are continuous
everywhere but f xy(x, y) 6=f yx(x, y), Clairaut’s Theorem implies that such a function f(x, y) does not exist.
89. By the geometry of partial derivatives, the slope of the tangent line is f x(1, 2). By implicit differentiation of
4x 2 +2y 2 + z 2 =16,weget8x +2z (∂z/∂x) =0 ⇒ ∂z/∂x = −4x/z, sowhenx =1and z =2we have
∂z/∂x = −2. Sotheslopeisf x (1, 2) = −2. Thus the tangent line is given by z − 2=−2(x − 1), y =2.Takingthe
parameter to be t = x − 1, we can write parametric equations for this line: x =1+t, y =2, z =2− 2t.
91. By Clairaut’s Theorem, f xyy =(f xy ) y
=(f yx ) y
= f yxy =(f y ) xy
=(f y ) yx
= f yyx .
93. Let g(x) =f(x, 0) = x(x 2 ) −3/2 e 0 = x |x| −3 .Butweareusingthepoint(1, 0), sonear(1, 0), g(x) =x −2 .Then
g 0 (x) =−2x −3 and g 0 (1) = −2,sousing(1)wehavef x (1, 0) = g 0 (1) = −2.
95. (a) (b) For (x, y) 6= (0, 0),
f x(x, y)= (3x2 y − y 3 )(x 2 + y 2 ) − (x 3 y − xy 3 )(2x)
(x 2 + y 2 ) 2
= x4 y +4x 2 y 3 − y 5
(x 2 + y 2 ) 2
andbysymmetryf y(x, y) = x5 − 4x 3 y 2 − xy 4
(x 2 + y 2 ) 2 .
f(h, 0) − f(0, 0) (0/h 2 ) − 0
f(0,h) − f(0, 0)
(c) f x (0, 0) = lim
= lim
=0and f y (0, 0) = lim
=0.
h→0 h
h→0 h
h→0 h
(d) By (3), f xy (0, 0) = ∂f x
∂y =lim f x (0,h) − f x (0, 0) (−h 5 − 0)/h 4
= lim
h→0 h
h→0 h
f yx(0, 0) = ∂f y
∂x =lim f y(h, 0) − f y(0, 0) h 5 /h 4
= lim =1.
h→0 h
h→0 h
(e) For (x, y) 6= (0, 0), we use a CAS to compute
f xy (x, y) = x6 +9x 4 y 2 − 9x 2 y 4 − y 6
(x 2 + y 2 ) 3
Now as (x, y) → (0, 0) along the x-axis, f xy(x, y) → 1 while as
(x, y) → (0, 0) along the y-axis, f xy (x, y) →−1. Thus f xy isn’t
continuous at (0, 0) and Clairaut’s Theorem doesn’t apply, so there is
no contradiction. The graphs of f xy and f yx are identical except at the
origin, where we observe the discontinuity.
= −1 while by (2),

F.
184 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
15.4 Tangent Planes and Linear Approximations ET 14.4
1. z = f(x, y) =4x 2 − y 2 +2y ⇒ f x(x, y) =8x, f y(x, y) =−2y +2,sof x(−1, 2) = −8, f y(−1, 2) = −2.
By Equation 2, an equation of the tangent plane is z − 4=f x (−1, 2)[x − (−1)] + f y (−1, 2)(y − 2)
⇒
z − 4=−8(x +1)− 2(y − 2) or z = −8x − 2y.
3. z = f(x, y) = xy ⇒ f x(x, y) = 1 2 (xy)−1/2 · y = 1 2
y/x, fy(x, y) = 1 2 (xy)−1/2 · x = 1 2
x/y,sofx(1, 1) = 1 2
and f y(1, 1) = 1 . Thus an equation of the tangent plane is z − 1=fx(1, 1)(x − 1) + fy(1, 1)(y − 1) ⇒
2
z − 1= 1 (x − 1) + 1 (y − 1) or x + y − 2z =0.
2 2
5. z = f(x, y) =y cos(x − y) ⇒ f x = y(− sin(x − y)(1)) = −y sin(x − y),
f y = y(− sin(x − y)(−1)) + cos(x − y) =y sin(x − y)+cos(x − y), sof x (2, 2) = −2sin(0) = 0,
f y(2, 2) = 2 sin(0) + cos(0) = 1 and an equation of the tangent plane is z − 2=0(x − 2) + 1(y − 2) or z = y.
6. z = f(x, y) =e x2 −y 2 ⇒ f x (x, y) =2xe x2 −y 2 , f y (x, y) =−2ye x2 −y 2 ,sof x (1, −1) = 2, f y (1, −1) = 2.
By Equation 2, an equation of the tangent plane is z − 1=f x(1, −1)(x − 1) + f y(1, −1)[y − (−1)]
⇒
z − 1=2(x − 1) + 2(y +1)or z =2x +2y +1.
7. z = f(x, y) =x 2 + xy +3y 2 ,sof x (x, y) =2x + y ⇒ f x (1, 1) = 3, f y (x, y) =x +6y ⇒ f y (1, 1) = 7 and an
equation of the tangent plane is z − 5=3(x − 1) + 7(y − 1) or z =3x +7y − 5. After zooming in, the surface and the
tangent plane become almost indistinguishable. (Here, the tangent plane is below the surface.) If we zoom in farther, the
surface and the tangent plane will appear to coincide.
9. f(x, y) =
xy sin (x − y)
sin (x − y)+xy cos (x − y)
. A CASgives
1+x 2 fx (x, y) =y − 2x2 y sin (x − y)
+ y2 1+x 2 + y 2 (1 + x 2 + y 2 ) 2 and
x sin (x − y) − xy cos (x − y)
f y (x, y) = − 2xy2 sin (x − y)
1+x 2 + y 2 (1 + x 2 + y 2 2
. We use the CAS to evaluate these at (1, 1), andthen
)
substitute the results into Equation 2 to compute an equation of the tangent plane: z = 1 x − 1 y. The surface and tangent
3 3
planeareshowninthefirst graph below. After zooming in, the surface and the tangent plane become almost indistinguishable,

F.
SECTION 15.4TX.10
TANGENT PLANES AND LINEAR APPROXIMATIONS ET SECTION 14.4 ¤ 185
as shown in the second graph. (Here, the tangent plane is shown with fewer traces than the surface.) If we zoom in farther, the
surface and the tangent plane will appear to coincide.
11. f(x, y) =x y. The partial derivatives are f x (x, y) = y and f y (x, y) = x
2 y ,sof x(1, 4) = 2 and f y (1, 4) = 1 4 .Both
f x and f y are continuous functions for y>0, so by Theorem 8, f is differentiable at (1, 4). By Equation 3, the linearization of
f at (1, 4) is given by L(x, y) =f(1, 4) + f x (1, 4)(x − 1) + f y (1, 4)(y − 4) = 2 + 2(x − 1) + 1 4 (y − 4) = 2x + 1 4 y − 1.
13. f(x, y) = x
x + y .
The partial derivatives are f x(x, y) =
1(x + y) − x(1)
(x + y) 2 = y/(x + y) 2 and
f y (x, y) =x(−1)(x + y) −2 · 1=−x/(x + y) 2 ,sof x (2, 1) = 1 9 and f y(2, 1) = − 2 9 .Bothf x and f y are continuous
functions for y 6=−x, sof is differentiable at (2, 1) by Theorem 8. The linearization of f at (2, 1) is given by
L (x, y) =f(2, 1) + f x(2, 1)(x − 2) + f y(2, 1)(y − 1) = 2 3 + 1 9 (x − 2) − 2 9 (y − 1) = 1 9 x − 2 9 y + 2 3 .
15. f(x, y) =e −xy cos y. The partial derivatives are f x (x, y) =e −xy (−y)cosy = −ye −xy cos y and
f y(x, y) =e −xy (− sin y)+(cosy)e −xy (−x) =−e −xy (sin y + x cos y),sof x(π, 0) = 0 and f y(π, 0) = −π.
Both f x and f y are continuous functions, so f is differentiable at (π, 0), and the linearization of f at (π, 0) is
L(x, y) =f(π, 0) + f x(π, 0)(x − π)+f y(π, 0)(y − 0) = 1 + 0(x − π) − π(y − 0) = 1 − πy.
17. Let f(x, y) =
2x +3
4y +1 .Thenf x(x, y) = 2
4y +1 and f y(x, y) =(2x +3)(−1)(4y +1) −2 −8x − 12
(4) =
(4y +1) .Bothf 2 x and f y
are continuous functions for y 6=− 1 4 , so by Theorem 8, f is differentiable at (0, 0). Wehavef x(0, 0) = 2, f y (0, 0) = −12
and the linear approximation of f at (0, 0) is f(x, y) ≈ f(0, 0) + f x (0, 0)(x − 0) + f y (0, 0)(y − 0) = 3 + 2x − 12y.
19. f(x, y) = x
20 − x 2 − 7y 2 ⇒ f x (x, y) =− 20 − x2 − 7y and f 7y
y(x, y) =− 2 20 − x2 − 7y ,
2
so f x (2, 1) = − 2 and f 3 y(2, 1) = − 7 . Then the linear approximation of f at (2, 1) is given by
3
f (x, y) ≈ f(2, 1) + f x(2, 1)(x − 2) + f y(2, 1)(y − 1) = 3 − 2 3 (x − 2) − 7 3 (y − 1) = − 2 3 x − 7 3 y + 20
3 .
Thus f(1.95, 1.08) ≈− 2 (1.95) − 7 20
(1.08) + =2.84¯6.
3 3 3
21. f(x, y, z) = x 2 + y 2 + z 2 ⇒ f x(x, y, z) =
f z (x, y, z) =
x
y
, fy(x, y, z) =
x2 + y 2 + z2 x2 + y 2 + z ,and
2
z
x2 + y 2 + z ,sof x(3, 2, 6) = 3 , f
2 7 y(3, 2, 6) = 2 , f 7 z(3, 2, 6) = 6 . Then the linear approximation of f at
7

F.
186 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
(3, 2, 6) is given by
f(x, y, z) ≈ f(3, 2, 6) + f x(3, 2, 6)(x − 3) + f y(3, 2, 6)(y − 2) + f z(3, 2, 6)(z − 6)
=7+ 3 7 (x − 3) + 2 7 (y − 2) + 6 7 (z − 6) = 3 7 x + 2 7 y + 6 7 z
Thus (3.02) 2 +(1.97) 2 +(5.99) 2 = f(3.02, 1.97, 5.99) ≈ 3 (3.02) + 2 (1.97) + 6 7 7 7
(5.99) ≈ 6.9914.
23. From the table, f(94, 80) = 127. Toestimatef T (94, 80) and f H (94, 80) we follow the procedure used in Section 15.3
f(94 + h, 80) − f(94, 80)
[ET 14.3]. Since f T (94, 80) = lim
, we approximate this quantity with h = ±2 and use the
h→0 h
values given in the table:
f T (94, 80) ≈
f(96, 80) − f(94, 80)
2
=
135 − 127
2
=4, f T (94, 80) ≈
f(92, 80) − f(94, 80)
−2
=
119 − 127
−2
f(94, 80 + h) − f(94, 80)
Averaging these values gives f T (94, 80) ≈ 4. Similarly, f H (94, 80) = lim
,soweuseh = ±5:
h→0 h
f H(94, 80) ≈
f(94, 85) − f(94, 80)
5
=
132 − 127
5
=1, f H(94, 80) ≈
Averaging these values gives f H (94, 80) ≈ 1. The linear approximation, then, is
f(94, 75) − f(94, 80)
−5
f(T,H) ≈ f(94, 80) + f T (94, 80)(T − 94) + f H (94, 80)(H − 80)
≈ 127 + 4(T − 94) + 1(H − 80) [or 4T + H − 329]
=
122 − 127
−5
Thus when T =95and H =78, f(95, 78) ≈ 127 + 4(95 − 94) + 1(78 − 80) = 129, so we estimate the heat index to be
approximately 129 ◦ F.
25. z = x 3 ln(y 2 ) ⇒ dz = ∂z ∂z
dx +
∂x ∂y dy =3x2 ln(y 2 ) dx + x 3 1 ·
y 2(2y) dy =3x2 ln(y 2 ) dx + 2x3
y dy
27. m = p 5 q 3 ⇒ dm = ∂m ∂m
dp +
∂p ∂q dq =5p4 q 3 dp +3p 5 q 2 dq
29. R = αβ 2 cos γ ⇒ dR = ∂R ∂R ∂R
dα + dβ +
∂α ∂β ∂γ dγ = β2 cos γdα+2αβ cos γdβ− αβ 2 sin γdγ
31. dx = ∆x =0.05, dy = ∆y =0.1, z =5x 2 + y 2 , z x =10x, z y =2y. Thus when x =1and y =2,
dz = z x (1, 2) dx + z y (1, 2) dy = (10)(0.05) + (4)(0.1) = 0.9 while
∆z = f(1.05, 2.1) − f(1, 2) = 5(1.05) 2 +(2.1) 2 − 5 − 4=0.9225.
33. dA = ∂A ∂A
dx + dy = ydx+ xdyand |∆x| ≤ 0.1, |∆y| ≤ 0.1. Weusedx =0.1, dy =0.1 with x =30, y =24;
∂x ∂y
then the maximum error in the area is about dA =24(0.1) + 30(0.1) = 5.4 cm 2 .
35. ThevolumeofacanisV = πr 2 h and ∆V ≈ dV is an estimate of the amount of tin. Here dV =2πrh dr + πr 2 dh,soput
dr =0.04, dh =0.08 (0.04 on top, 0.04 on bottom) and then ∆V ≈ dV =2π(48)(0.04) + π(16)(0.08) ≈ 16.08 cm 3 .
Thus the amount of tin is about 16 cm 3 .
=4
=1

F.
TX.10
SECTION 15.5 THE CHAIN RULE ET SECTION 14.5 ¤ 187
37. TheareaoftherectangleisA = xy,and∆A ≈ dA is an estimate of the area of paint in the stripe. Here dA = ydx+ xdy,
so with dx = dy = 3+3
12 = 1 2 , ∆A ≈ dA = (100) 1
2
+ (200)
1
2
= 150 ft 2 . Thus there are approximately 150 ft 2 of paint
in the stripe.
39. First we find ∂R implicitly by taking partial derivatives of both sides with respect to R 1 :
∂R 1
∂ 1
= ∂ [(1/R 1)+(1/R 2 )+(1/R 3 )]
⇒ −R −2 ∂R
= −R −2
1 ⇒ ∂R = R2
.Thenbysymmetry,
∂R 1 R
∂R 1 ∂R 1 ∂R 1
∂R
= R2
,
∂R 2
R 2 2
∂R
= R2
.WhenR 1 =25, R 2 =40and R 3 =50,
∂R 3
R 2 3
1
R = 17
200
R 2 1
⇔ R = 200
17 Ω.
Since the possible error for each R i is 0.5%, the maximum error of R is attained by setting ∆R i =0.005R i .So
∆R ≈ dR = ∂R ∆R 1 + ∂R ∆R 2 + ∂R
1
∆R 3 =(0.005)R 2 + 1 + 1
=(0.005)R = 1 ≈ 0.059 Ω.
17
∂R 1 ∂R 2 ∂R 3 R 1 R 2 R 3 41. The errors in measurement are at most 2%,so
∆w
w ≤ 0.02 and ∆h
h ≤ 0.02. The relative error in the calculated surface
area is
∆S
S
≈ dS S = 0.1091(0.425w0.425−1 )h 0.725 dw +0.1091w 0.425 (0.725h 0.725−1 ) dh
=0.425 dw 0.1091w 0.425 h 0.725 w +0.725 dh h
To estimate the maximum relative error, we use dw
w = ∆w
w =0.02 and dh
h = ∆h
h =0.02 ⇒
dS
S
=0.425 (0.02) + 0.725 (0.02) = 0.023. Thus the maximum percentage error is approximately 2.3%.
43. ∆z = f(a + ∆x, b + ∆y) − f(a, b) =(a + ∆x) 2 +(b + ∆y) 2 − (a 2 + b 2 )
= a 2 +2a ∆x +(∆x) 2 + b 2 +2b ∆y +(∆y) 2 − a 2 − b 2 =2a ∆x +(∆x) 2 +2b ∆y +(∆y) 2
But f x (a, b) =2a and f y (a, b) =2b and so ∆z = f x (a, b) ∆x + f y (a, b) ∆y + ∆x ∆x + ∆y ∆y,whichisDefinition 7
with ε 1 = ∆x and ε 2 = ∆y. Hence f is differentiable.
45. To show that f is continuous at (a, b) we need to show that lim f(x, y) =f(a, b) or
(x,y)→(a,b)
equivalently
lim
(∆x,∆y)→(0,0)
f(a + ∆x, b + ∆y) =f(a, b). Sincef is differentiable at (a, b),
f(a + ∆x, b + ∆y) − f(a, b) =∆z = f x (a, b) ∆x + f y (a, b) ∆y + ε 1 ∆x + ε 2 ∆y, where 1 and 2 → 0 as
(∆x, ∆y) → (0, 0). Thusf(a + ∆x, b + ∆y) =f(a, b)+f x(a, b) ∆x + f y(a, b) ∆y + ε 1 ∆x + ε 2 ∆y. Taking the limit of
both sides as (∆x, ∆y) → (0, 0) gives
lim
(∆x,∆y)→(0,0)
f(a + ∆x, b + ∆y) =f(a, b). Thus f is continuous at (a, b).
15.5 The Chain Rule ET 14.5
1. z = x 2 + y 2 + xy, x =sint, y = e t ⇒ dz
dt = ∂z dx
∂x dt + ∂z dy
=(2x + y)cost +(2y + x)et
∂y dt
3. z = 1+x 2 + y 2 , x =lnt, y =cost ⇒
dz
dt = ∂z dx
∂x dt + ∂z dy
∂y dt = 1 (1 + 2 x2 + y 2 ) −1/2 (2x) · 1
t + 1 (1 + 2 x2 + y 2 ) −1/2 1
x
(2y)(− sin t) =
1+x2 + y 2 t − y sin t

F.
188 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
5. w = xe y/z , x = t 2 , y =1− t, z =1+2t ⇒
dw
dt = ∂w dx
∂x dt + ∂w dy
∂y dt + ∂w
dz
1
∂z dt = ey/z · 2t + xe y/z · (−1) + xe y/z − y
· 2=e y/z 2t − x z
z 2 z − 2xy
z 2
7. z = x 2 y 3 , x = s cos t, y = s sin t ⇒
9. z =sinθ cos φ, θ = st 2 , φ = s 2 t ⇒
∂z
∂s = ∂z ∂x
∂x ∂s + ∂z ∂y
∂y ∂s =2xy3 cos t +3x 2 y 2 sin t
∂z
∂t = ∂z ∂x
∂x ∂t + ∂z ∂y
∂y ∂t =(2xy3 )(−s sin t)+(3x 2 y 2 )(s cos t) =−2sxy 3 sin t +3sx 2 y 2 cos t
∂z
∂s = ∂z ∂θ
∂θ ∂s + ∂z ∂φ
∂φ ∂s =(cosθ cos φ)(t2 )+(− sin θ sin φ)(2st) =t 2 cos θ cos φ − 2st sin θ sin φ
∂z
∂t = ∂z ∂θ
∂θ ∂t + ∂z ∂φ
∂φ ∂t =(cosθ cos φ)(2st)+(− sin θ sin φ)(s2 )=2st cos θ cos φ − s 2 sin θ sin φ
11. z = e r cos θ, r = st, θ = √ s 2 + t 2 ⇒
∂z
∂s = ∂z ∂r
∂r ∂s + ∂z ∂θ
∂θ ∂s = er cos θ · t + e r (− sin θ) · 1
2 (s2 + t 2 ) −1/2 (2s) =te r cos θ − e r sin θ ·
= e r t cos θ −
s
√
s2 + t sin θ 2
∂z
∂t = ∂z ∂r
∂r ∂t + ∂z ∂θ
∂θ ∂t = er cos θ · s + e r (− sin θ) · 1
2 (s2 + t 2 ) −1/2 (2t) =se r cos θ − e r sin θ ·
= e r s cos θ −
t
√
s2 + t sin θ 2
s
√
s2 + t 2
t
√
s2 + t 2
13. When t =3, x = g(3) = 2 and y = h(3) = 7. BytheChainRule(2),
dz
dt = ∂f dx
∂x dt + ∂f dy
∂y dt = fx(2, 7)g0 (3) + f y(2, 7) h 0 (3) = (6)(5) + (−8)(−4) = 62.
15. g(u, v) =f(x(u, v),y(u, v)) where x = e u +sinv, y = e u +cosv ⇒
∂x
∂u = eu ,
∂x
∂v =cosv, ∂y
∂u = eu ,
∂y
∂g
= − sin v.BytheChainRule(3),
∂v ∂u = ∂f
∂x
∂x
∂u + ∂f
∂y
∂y
∂u .Then
g u(0, 0) = f x(x(0, 0),y(0, 0)) x u(0, 0) + f y(x(0, 0),y(0, 0)) y u(0, 0) = f x(1, 2)(e 0 )+f y(1, 2)(e 0 ) = 2(1) + 5(1) = 7.
Similarly, ∂g
∂v = ∂f ∂x
∂x ∂v + ∂f ∂y
∂y ∂v .Then
g v(0, 0) = f x(x(0, 0),y(0, 0)) x v(0, 0) + f y(x(0, 0),y(0, 0)) y v(0, 0) = f x(1, 2)(cos 0) + f y(1, 2)(− sin 0)
=2(1)+5(0)=2
17. u = f(x, y), x = x(r, s, t), y = y(r, s, t) ⇒
∂u
∂r = ∂u ∂x
∂x ∂r + ∂u ∂y
∂y ∂r , ∂u
∂s = ∂u
∂x
∂x
∂s + ∂u
∂y
∂y
∂s , ∂u
∂t = ∂u ∂x
∂x ∂t + ∂u
∂y
∂y
∂t

F.
TX.10
SECTION 15.5 THE CHAIN RULE ET SECTION 14.5 ¤ 189
19. w = f(r, s, t), r = r(x, y), s = s(x, y), t = t(x, y) ⇒
∂w
∂x = ∂w ∂r
∂r ∂x + ∂w ∂s
∂s ∂x + ∂w ∂t
∂t ∂x ,
∂w
∂y = ∂w ∂r
∂r ∂y + ∂w ∂s
∂s ∂y + ∂w ∂t
∂t ∂y
21. z = x 2 + xy 3 , x = uv 2 + w 3 , y = u + ve w ⇒
∂z
∂u = ∂z ∂x
∂x ∂u + ∂z ∂y
∂y ∂u =(2x + y3 )(v 2 )+(3xy 2 )(1),
∂z
∂v = ∂z ∂x
∂x ∂v + ∂z ∂y
∂y ∂v =(2x + y3 )(2uv)+(3xy 2 )(e w ),
∂z
∂w = ∂z ∂x
∂x ∂w + ∂z ∂y
∂y ∂w =(2x + y3 )(3w 2 )+(3xy 2 )(ve w ).
When u =2, v =1,andw =0,wehavex =2, y =3,
so ∂z
∂u = (31)(1) + (54)(1) = 85, ∂z
∂v = (31) (4) + (54)(1) = 178, ∂z
= (31)(0) + (54)(1) = 54.
∂w
23. R =ln(u 2 + v 2 + w 2 ), u = x +2y, v =2x − y, w =2xy ⇒
∂R
∂x = ∂R ∂u
∂u ∂x + ∂R ∂v
∂v ∂x + ∂R
∂w
=
2u +4v +4wy
u 2 + v 2 + w 2 ,
∂R
∂y = ∂R ∂u
∂u ∂y + ∂R ∂v
∂v ∂y + ∂R
∂w
=
4u − 2v +4wx
u 2 + v 2 + w 2 .
∂w
∂x =
∂w
∂y =
2u
u 2 + v 2 + w 2 (1) +
2u
u 2 + v 2 + w 2 (2) +
2v
u 2 + v 2 + w 2 (2) +
2v
u 2 + v 2 + w 2 (−1) +
When x = y =1we have u =3, v =1,andw =2,so ∂R
∂x = 9 ∂R
and
7 ∂y = 9 7 .
25. u = x 2 + yz, x = pr cos θ, y = pr sin θ, z = p + r ⇒
2w
u 2 + v 2 + w 2 (2y)
2w
u 2 + v 2 + w 2 (2x)
∂u
∂p = ∂u ∂x
∂x ∂p + ∂u ∂y
∂y ∂p + ∂u ∂z
=(2x)(r cos θ)+(z)(r sin θ)+(y)(1) = 2xr cos θ + zr sin θ + y,
∂z ∂p
∂u
∂r = ∂u ∂x
∂x ∂r + ∂u ∂y
∂y ∂r + ∂u ∂z
=(2x)(p cos θ)+(z)(p sin θ)+(y)(1) = 2xp cos θ + zp sin θ + y,
∂z ∂r
∂u
∂θ = ∂u ∂x
∂x ∂θ + ∂u ∂y
∂y ∂θ + ∂u ∂z
=(2x)(−pr sin θ)+(z)(pr cos θ)+(y)(0) = −2xpr sin θ + zpr cos θ.
∂z ∂θ
When p =2, r =3,andθ =0we have x =6, y =0,andz =5,so ∂u ∂u ∂u
=36, =24,and
∂p ∂r ∂θ =30.
27.
xy =1+x 2 y,soletF (x, y) =(xy) 1/2 − 1 − x 2 y =0.ThenbyEquation6
dy
dx = − Fx
F y
1
2
= −
(xy)−1/2 (y) − 2xy
1
2 (xy)−1/2 (x) − x = − y − 4xy xy
2 x − 2x 2 xy = 4(xy)3/2 − y
x − 2x 2 xy .
29. cos(x − y) =xe y ,soletF (x, y) =cos(x − y) − xe y =0.
Then dy
dx = −F x − sin(x − y) − ey sin(x − y)+ey
= − =
F y − sin(x − y)(−1) − xey sin(x − y) − xe . y

F.
190 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
31. x 2 + y 2 + z 2 =3xyz, soletF (x, y, z) =x 2 + y 2 + z 2 − 3xyz =0. Then by Equations 7
∂z
∂x = −Fx F z
2x − 3yz 3yz − 2x
= − =
2z − 3xy 2z − 3xy
and
∂z
∂y = −Fy F z
33. x − z =arctan(yz),soletF (x, y, z) =x − z − arctan(yz) =0.Then
∂z
∂x = −Fx F z
∂z
∂y = −F y
F z
2y − 3xz 3xz − 2y
= − =
2z − 3xy 2z − 3xy .
1
= −
= 1+y2 z 2
1
−1 −
1+(yz) (y) 1+y + y 2 z 2
2
1
−
1+(yz) (z)
z
= −
2 1+y
= −
2 z 2
z
1
−1 −
1+(yz) (y) 1+y 2 z 2 = −
+ y 1+y + y 2 z 2
2 1+y 2 z 2
35. Since x and y are each functions of t, T (x, y) is a function of t,sobytheChainRule, dT
dt = ∂T dx
∂x dt + ∂T dy
∂y dt .After
3 seconds, x = √ 1+t = √ 1+3=2, y =2+ 1 t dx
3 =2+1 (3) = 3,
3
dt = 1
2 √ 1+t = 1
2 √ 1+3 = 1 dy
,and
4 dt = 1 3 .
Then dT
dt
dx
dy
= Tx(2, 3) + Ty(2, 3)
dt dt =4
1
4 +3 1
3 =2. Thus the temperature is rising at a rate of 2 ◦ C/s.
37. C =1449.2+4.6T − 0.055T 2 +0.00029T 3 +0.016D, so ∂C
∂T =4.6 − 0.11T +0.00087T 2 and ∂C
∂D =0.016.
According to the graph, the diver is experiencing a temperature of approximately 12.5 ◦ C at t =20minutes, so
∂C
∂T =4.6 − 0.11(12.5) + 0.00087(12.5)2 ≈ 3.36. By sketching tangent lines at t =20to the graphs given, we estimate
dD
dt ≈ 1 dT
and
2 dt ≈−1 dC
. Then, by the Chain Rule,
10 dt = ∂C dT
∂T dt + ∂C dD
∂D dt ≈ (3.36)
− 1 10 +(0.016) 1
2 ≈−0.33.
Thus the speed of sound experienced by the diver is decreasing at a rate of approximately 0.33 m/sperminute.
39. (a) V = wh,sobytheChainRule,
dV
dt = ∂V d
∂ dt + ∂V dw
∂w dt + ∂V dh d dw dh
= wh + h + w
∂h dt dt dt dt =2· 2 · 2+1· 2 · 2+1· 2 · (−3) = 6 m3 /s.
(b) S =2(w + h + wh),sobytheChainRule,
dS
dt = ∂S d
∂ dt + ∂S dw
∂w dt + ∂S dh
d
dw
dh
=2(w + h) +2( + h) +2( + w)
∂h dt dt dt dt
=2(2+2)2+2(1+2)2+2(1+2)(−3) = 10 m 2 /s
(c) L 2 = 2 + w 2 + h 2 ⇒ 2L dL
dt
dL/dt =0m/s.
d dw dh
=2 +2w +2h = 2(1)(2) + 2(2)(2) + 2(2)(−3) = 0
dt dt dt ⇒
41.
dP dT
=0.05,
dt dt =0.15, V =8.31 T dV
and
P dt = 8.31 dT
P dt − 8.31 T dP
. Thus whenP =20and T = 320,
P 2 dt
dV 0.15
dt =8.31 20 − (0.05)(320)
≈−0.27 L/s.
400
43. Let x be the length of the first side of the triangle and y the length of the second side. The area A of the triangle is given by
A = 1 xy sin θ where θ is the angle between the two sides. Thus A is a function of x, y,andθ,andx, y,andθ are each in
2

F.
TX.10
SECTION 15.5 THE CHAIN RULE ET SECTION 14.5 ¤ 191
turn functions of time t. We are given that dx
dt
dA
dt = ∂A dx
∂x dt + ∂A dy
∂y dt + ∂A dθ
∂θ dt
and θ = π/6 we have
⇒
0= 1 2 (30) sin π 6
=3,
dy
dt
= −2, and because A is constant,
dA
dt
=0. By the Chain Rule,
dA
dt = 1 dx
2
y sin θ ·
dt + 1 dy
2
x sin θ ·
dt + 1 dθ
2
xy cos θ · .Whenx =20, y =30,
dt
(3) +
1
(20)
2
sin π 6 (−2) +
1
(20)(30)
2
cos π dθ
6
dt
=45· 1
2 − 20 · 1
2 + 300 · √
3
2 · dθ
dt = 25 2
+150√ 3 dθ
dt
Solving for dθ dθ
gives
dt dt = −25/2
150 √ 3 = − 1
12 √ , so the angle between the sides is decreasing at a rate of
3
1/ 12 √ 3 ≈ 0.048 rad/s.
45. (a)BytheChainRule, ∂z
∂r = ∂z ∂z
cos θ +
∂x ∂y sin θ, ∂z
∂θ = ∂z
∂z
(−r sin θ)+ r cos θ.
∂x ∂y
2 2 ∂z ∂z
(b) = cos 2 θ +2 ∂z
2
∂z
∂z
∂r ∂x
∂x ∂y cos θ sin θ + sin 2 θ,
∂y
2 2 ∂z ∂z
= r 2 sin 2 θ − 2 ∂z ∂z
∂θ ∂x
∂x ∂y r2 cos θ sin θ +
2 ∂z
+ 1
2 2
2
∂z ∂z ∂z
= + (cos 2 θ +sin 2 θ)=
∂r r 2 ∂θ ∂x ∂y
47. Let u = x − y. Then ∂z
∂x = dz ∂u
du ∂x = dz
du
2 ∂z
r 2 cos 2 θ.Thus
∂y
2 ∂z
+
∂x
∂z
and
∂y = dz (−1). Thus∂z
du ∂x + ∂z
∂y =0.
2 ∂z
.
∂y
49. Let u = x + at, v = x − at. Thenz = f(u)+g(v),so∂z/∂u = f 0 (u) and ∂z/∂v = g 0 (v).
51.
Thus ∂z
∂t = ∂z ∂u
∂u ∂t + ∂z ∂v
∂v ∂t = af 0 (u) − ag 0 (v) and
∂ 2 z
∂t 2
= a ∂ df
∂t [f 0 (u) − g 0 0 (u) ∂u
(v)] = a
du ∂t − dg0 (v)
dv
∂v
= a 2 f 00 (u)+a 2 g 00 (v).
∂t
Similarly ∂z
∂x = f 0 (u)+g 0 (v) and ∂2 z
∂x = f 00 (u)+g 00 (v). Thus ∂2 z
2 ∂t = ∂ 2 z
2 a2
∂x . 2
∂z
∂s = ∂z ∂z
2s + 2r. Then
∂x ∂y
∂ 2 z
∂r ∂s = ∂ ∂r
∂z
∂x 2s + ∂ ∂z
∂r ∂y 2r
= ∂2 z ∂x
∂x 2 ∂r 2s + ∂ ∂z ∂y ∂z ∂
2s +
∂y ∂x ∂r ∂x ∂r 2s + ∂2 z ∂y
∂y 2 ∂r 2r + ∂ ∂z ∂x ∂z
2r +
∂x ∂y ∂r ∂y 2
=4rs ∂2 z
∂x 2 +
By the continuity of the partials,
∂2 z
∂y ∂x 4s2 +0+4rs ∂2 z
∂y + 2
∂2 z
∂x∂y 4r2 +2 ∂z
∂y
∂ 2 z
∂r∂s =4rs ∂2 z
∂x 2 +4rs ∂2 z
∂y 2 +(4r2 +4s 2 ) ∂2 z
∂x∂y +2∂z ∂y .

F.
192 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
53.
∂z
∂r = ∂z ∂z ∂z
cos θ + sin θ and
∂x ∂y ∂θ = − ∂z ∂z
r sin θ + r cos θ. Then
∂x ∂y
and
∂ 2 z
∂r 2 =cosθ ∂ 2 z
∂x 2 cos θ +
∂2 z
∂y ∂x sin θ +sinθ
=cos 2 θ ∂2 z
+2cosθ sin θ
∂2 z
∂x2 ∂x∂y +sin2 θ ∂2 z
∂y 2
∂ 2 z
∂z
∂ 2
2
= −r cos θ
∂θ ∂x +(−r sin θ) z
∂x
2
(−r sin θ)+
∂2
∂ 2 z
∂y 2 sin θ +
z
∂y ∂x r cos θ
∂2 z
∂x∂y cos θ
−r sin θ ∂z
∂ 2
∂y + r cos θ z
∂y r cos θ +
∂2 z
2 ∂x∂y (−r sin θ)
= −r cos θ ∂z ∂z
− r sin θ
∂x ∂y + r2 sin 2 θ ∂2 z
∂x − 2 2r2 cos θ sin θ
∂2 z
∂x∂y + r2 cos 2 θ ∂2 z
∂y 2
Thus
∂ 2 z
∂r + 1 ∂ 2 z
2 r 2 ∂θ 2 + 1 ∂z
r ∂r =(cos2 θ +sin 2 θ) ∂2 z
− 1 r
= ∂2 z
∂x + ∂2 z
as desired.
2 ∂y2 ∂x + sin 2 θ +cos 2 θ ∂ 2 z
2 ∂y 2
∂z
cos θ
∂x − 1 ∂z
sin θ
r ∂y + 1 r
cos θ ∂z
∂x
∂z
+sinθ
∂y
55. (a) Since f is a polynomial, it has continuous second-order partial derivatives, and
f(tx, ty) =(tx) 2 (ty)+2(tx)(ty) 2 +5(ty) 3 = t 3 x 2 y +2t 3 xy 2 +5t 3 y 3 = t 3 (x 2 y +2xy 2 +5y 3 )=t 3 f (x, y).
Thus, f is homogeneous of degree 3.
(b) Differentiating both sides of f(tx, ty) =t n f(x, y) with respect to t using the Chain Rule, we get
∂
∂t f(tx, ty) = ∂ ∂t [tn f(x, y)]
⇔
∂
∂(tx)
f(tx, ty) · + ∂
∂(ty)
f(tx, ty) · = x ∂
∂
f(tx, ty)+y
∂(tx) ∂t ∂(ty) ∂t ∂(tx) ∂(ty) f(tx, ty) =ntn−1 f(x, y).
Setting t =1: x ∂
∂x f(x, y)+y ∂ f(x, y) =nf(x, y).
∂y
57. Differentiating both sides of f(tx, ty) =t n f(x, y) with respect to x using the Chain Rule, we get
∂
∂
f(tx, ty) =
∂x ∂x [tn f(x, y)]
⇔
∂
∂ (tx)
f(tx, ty) ·
∂ (tx) ∂x + ∂
∂ (ty)
f(tx, ty) ·
∂ (ty) ∂x = ∂
tn
∂x f(x, y) ⇔ tfx(tx, ty) =tn f x(x, y).
Thus f x(tx, ty) =t n−1 f x(x, y).
15.6 Directional Derivatives and the Gradient Vector ET 14.6
1. We can approximate the directional derivative of the pressure function at K in the direction of S by the average rate of change
of pressure between the points where the red line intersects the contour lines closest to K (extendtheredlineslightlyatthe
left). In the direction of S, the pressure changes from 1000 millibars to 996 millibars and we estimate the distance between
these two points to be approximately 50 km (using the fact that the distance from K to S is 300 km). Then the rate of change of
pressure in the direction given is approximately
996 − 1000
50
= −0.08 millibar/km.

F.
SECTION 15.6 DIRECTIONAL TX.10 DERIVATIVES AND THE GRADIENT VECTOR ET SECTION 14.6 ¤ 193
1
3. D u f(−20, 30) = ∇f (−20, 30) · u = f T (−20, 30) √
2
+ f v(−20, 30)
√
1
2
.
f(−20 + h, 30) − f(−20, 30)
f T (−20, 30) = lim
, so we can approximate f T (−20, 30) by considering h = ±5 and
h→0 h
using the values given in the table: f T (−20, 30) ≈
f T (−20, 30) ≈
f(−25, 30) − f(−20, 30)
−5
=
f(−15, 30) − f(−20, 30)
5
−39 − (−33)
−5
=
−26 − (−33)
5
=1.4,
=1.2. Averaging these values gives f T (−20, 30) ≈ 1.3.
f(−20, 30 + h) − f(−20, 30)
Similarly, f v (−20, 30) = lim
, so we can approximate f v (−20, 30) with h = ±10:
h→0 h
f v (−20, 30) ≈
f(−20, 40) − f(−20, 30)
10
=
−34 − (−33)
10
= −0.1,
f(−20, 20) − f(−20, 30) −30 − (−33)
f v(−20, 30) ≈ = = −0.3. Averaging these values gives f v(−20, 30) ≈−0.2.
−10
−10
1
Then D u f(−20, 30) ≈ 1.3 √
1
2
+(−0.2) √
2
≈ 0.778.
5. f(x, y) =ye −x ⇒ f x(x, y) =−ye −x and f y(x, y) =e −x .Ifu is a unit vector in the direction of θ =2π/3, then
from Equation 6, D u f(0, 4) = f x(0, 4) cos
2π
3 + fy(0, 4) sin √
2π
3 = −4 · −
1
2 +1· 3
2 =2+√ 3
. 2
7. f(x, y) =sin(2x +3y)
(a) ∇f(x, y) = ∂f
∂x i + ∂f j =[cos(2x +3y) · 2] i +[cos(2x +3y) · 3] j =2cos(2x +3y) i +3cos(2x +3y) j
∂y
(b) ∇f(−6, 4) = (2 cos 0)i + (3 cos 0)j =2i +3j
√ √ √
(c) By Equation 9, D u f(−6, 4) = ∇f(−6, 4) · u =(2i +3j) · 1
2 3 i − j =
1
2 2 3 − 3 = 3 −
3
. 2
9. f(x, y, z) =xe 2yz
(a) ∇f(x, y, z) =hf x(x, y, z),f y(x, y, z),f z(x, y, z)i = e 2yz , 2xze 2yz , 2xye 2yz
(b) ∇f(3, 0, 2) = h1, 12, 0i
2
(c) By Equation 14, D u f(3, 0, 2) = ∇f(3, 0, 2) · u = h1, 12, 0i· , − 2 , 1
3 3 3 =
2
− 24 +0=− 22 .
3 3 3
11. f(x, y) =1+2x
y ⇒ ∇f(x, y) = 2
y, 2x · 1
2 y−1/2 = 2 y, x/
y , ∇f(3, 4) =
4, 3 2 ,andaunitvectorin
the direction of v is u =
1
42 +(−3) h4, −3i = 4
, − 3 ,soDu 2 5 5
f(3, 4) = ∇f(3, 4) · u =
4, 3 2 · 4 , − 3
5 5 =
23
. 10
13. g(p, q) =p 4 − p 2 q 3 ⇒ ∇g(p, q) = 4p 3 − 2pq 3 i + −3p 2 q 2 j, ∇g(2, 1) = 28 i − 12 j, and a unit
vector in the direction of v is u =
D u g(2, 1) = ∇g(2, 1) · u =(28i − 12 j) ·
1
(i +3j) = √
1
√1 2 +32 10
(i +3j), so
√
1
10
(i +3j) = √ 1
10
(28 − 36) = − √ 8
10
or − 4√ 10
.
5
15. f(x, y, z) =xe y + ye z + ze x ⇒ ∇f(x, y, z) =he y + ze x ,xe y + e z ,ye z + e x i, ∇f(0, 0, 0) = h1, 1, 1i,andaunit
vector in the direction of v is u =
D u f(0, 0, 0) = ∇f(0, 0, 0) · u = h1, 1, 1i·
√
1
25+1+4
h5, 1, −2i = √ 1
30
h5, 1, −2i, so
√
1
30
h5, 1, −2i = √ 4
30
.

F.
194 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
17. g(x, y, z) =(x +2y +3z) 3/2 ⇒
3
∇g(x, y, z) = (x +2y 2 +3z)1/2 (1), 3 (x +2y 2 +3z)1/2 (2), 3 (x +2y 2 +3z)1/2 (3)
= 3
√ √
2 x +2y +3z, 3 x +2y +3z,
9
√
2 x +2y +3z , ∇g(1, 1, 2) = 9 , 9, 27
2 2 ,
and a unit vector in the direction of v =2j − k is u = 2 √
5
j − 1 √
5
k,so
D u g(1, 1, 2) = 9
, 9, 27
2 2 ·
2
0, √
5
, − √ 1 5
= √ 18
5
− 27
2 √ = 9
5 2 √ . 5
19. f(x, y) = xy ⇒ ∇f(x, y) =
1
2 (xy)−1/2 (y), 1 y
2 (xy)−1/2 (x) =
2 xy , x
2 ,so∇f(2, 8) =
1, 1 4 .
xy
The unit vector in the direction of −→
PQ = h5 − 2, 4 − 8i = h3, −4i is u =
3 , − 4
5 5 ,so
D u f(2, 8) = ∇f(2, 8) · u =
1, 1 4 · 3 , − 4
5 5 =
2
. 5
21. f(x, y) =y 2 /x = y 2 x −1 ⇒ ∇f(x, y) = −y 2 x −2 , 2yx −1 = −y 2 /x 2 , 2y/x .
∇f(2, 4) = h−4, 4i, or equivalently h−1, 1i, is the direction of maximum rate of change, and the maximum rate
is |∇f(2, 4)| = √ 16 + 16 = 4 √ 2.
23. f(x, y) =sin(xy) ⇒ ∇f(x, y) =hy cos(xy),xcos(xy)i, ∇f(1, 0) = h0, 1i. Thus the maximum rate of change is
|∇f(1, 0)| =1in the direction h0, 1i.
25. f(x, y, z) = x 2 + y 2 + z 2 ⇒
1
∇f(x, y, z)=
2 (x2 + y 2 + z 2 ) −1/2 · 2x, 1 2 (x2 + y 2 + z 2 ) −1/2 · 2y, 1 2 (x2 + y 2 + z 2 ) −1/2 · 2z
x
=
x2 + y 2 + z , y
2 x2 + y 2 + z , z
,
2 x2 + y 2 + z 2
∇f(3, 6, −2) =
√
3
49
,
|∇f(3, 6, −2)| =
3
7
√
6
49
, √ −2
49
= 3
, 6 , − 2
7 7 7 . Thus the maximum rate of change is
2
+
6 2
+
− 2 2 9+36+4
=
=1in the direction 3
, 6 , − 2
7 7 49 7 7 7 or equivalently h3, 6, −2i.
27. (a) As in the proof of Theorem 15, D u f = |∇f| cos θ. Since the minimum value of cos θ is −1 occurring when θ = π,the
minimum value of D u f is − |∇f| occurring when θ = π, that is when u is in the opposite direction of ∇f
(assuming ∇f 6=0).
(b) f(x, y) =x 4 y − x 2 y 3 ⇒ ∇f(x, y) = 4x 3 y − 2xy 3 ,x 4 − 3x 2 y 2 ,sof decreases fastest at the point (2, −3) in the
direction −∇f(2, −3) = − h12, −92i = h−12, 92i.
29. The direction of fastest change is ∇f(x, y) =(2x − 2) i +(2y − 4) j,soweneedtofind all points (x, y) where ∇f(x, y) is
parallel to i + j ⇔ (2x − 2) i +(2y − 4) j = k (i + j) ⇔ k =2x − 2 and k =2y − 4. Then2x − 2=2y − 4 ⇒
y = x +1, so the direction of fastest change is i + j at all points on the line y = x +1.

F.
31. T =
SECTION 15.6 DIRECTIONAL TX.10 DERIVATIVES AND THE GRADIENT VECTOR ET SECTION 14.6 ¤ 195
k
x2 + y 2 + z and 120 = T (1, 2, 2) = k so k =360.
2 3
h1, −1, 1i
(a) u = √ ,
3
D u T (1, 2, 2) = ∇T (1, 2, 2)·u = −360 x 2 + y 2 + z 2
−3/2
hx, y, zi ·u = − 40 h1, 2, 2i· √3
1
h1, −1, 1i = − 40
3
(1,2,2)
3 √ 3
(b) From (a), ∇T = −360 x 2 + y 2 + z 2 −3/2 hx, y, zi,andsincehx, y, zi is the position vector of the point (x, y, z),the
vector − hx, y, zi, and thus ∇T , always points toward the origin.
33. ∇V (x, y, z) =h10x − 3y + yz, xz − 3x, xyi, ∇V (3, 4, 5) = h38, 6, 12i
(a) D u V (3, 4, 5) = h38, 6, 12i·
√
1
3
h1, 1, −1i = √ 32
3
(b) ∇V (3, 4, 5) = h38, 6, 12i,orequivalently,h19, 3, 6i.
(c) |∇V (3, 4, 5)| = √ 38 2 +6 2 +12 2 = √ 1624 = 2 √ 406
35. A unit vector in the direction of −→
AB is i and a unit vector in the direction of −→
AC is j. ThusD −→
AB
f(1, 3) = f x(1, 3) = 3 and
D −→
AC
f(1, 3) = f y (1, 3) = 26. Therefore ∇f(1, 3) = hf x (1, 3),f y (1, 3)i = h3, 26i, and by definition,
D −→ f(1, 3) = ∇f · u where u is a unit vector in the direction of −→
AD, whichis
5 , 12
AD 13 13 . Therefore,
D −→
5
f (1, 3) = h3, 26i· , 12
AD 13 13 =3· 5
12
+26· = 327 .
13 13 13
∂(au + bv)
37. (a) ∇(au + bv) =
,
∂x
(b) ∇(uv) = v ∂u
= a ∇u + b ∇v
∂x + u ∂v
∂x
∂u
u
v
(c) ∇ = ∂x − u ∂v
∂x ,
v v 2
∂(au + bv)
= a ∂u
∂y
∂x + b ∂v
∂x
∂u
,a
∂y + b ∂v ∂u
= a
∂y ∂x , ∂u ∂v
+ b
∂y ∂x , ∂v
∂y
∂u
,v
∂y + u ∂v ∂u
= v
∂y ∂x , ∂u ∂v
+ u
∂y ∂x , ∂v
= v ∇u + u ∇v
∂y
v ∂u
∂y − u ∂v
∂u
v
∂y ∂x , ∂u
∂y
=
v 2
∂v
− u
∂(u
(d) ∇u n n )
=
∂x , ∂(un )
= nu n−1 ∂u ∂u
∂y
∂x ,nun−1 = nu n−1 ∇u
∂y
v 2
∂x , ∂v
∂y
=
v ∇u − u ∇v
v 2
39. Let F (x, y, z) =2(x − 2) 2 +(y − 1) 2 +(z − 3) 2 .Then2(x − 2) 2 +(y − 1) 2 +(z − 3) 2 =10isalevelsurfaceofF .
F x(x, y, z) =4(x − 2) ⇒ F x(3, 3, 5) = 4, F y(x, y, z) =2(y − 1) ⇒ F y(3, 3, 5) = 4,and
F z(x, y, z) =2(z − 3) ⇒ F z(3, 3, 5) = 4.
(a) Equation 19 gives an equation of the tangent plane at (3, 3, 5) as 4(x − 3) + 4(y − 3) + 4(z − 5) = 0
4x +4y +4z =44or equivalently x + y + z =11.
(b) By Equation 20, the normal line has symmetric equations x − 3
4
= y − 3
4
= z − 5
4
or equivalently
x − 3=y − 3=z − 5. Corresponding parametric equations are x =3+t, y =3+t, z =5+t.
⇔

F.
196 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14 TX.10
41. Let F (x, y, z) =x 2 − 2y 2 + z 2 + yz. Thenx 2 − 2y 2 + z 2 + yz =2isalevelsurfaceofF
and ∇F (x, y, z) =h2x, −4y + z, 2z + yi.
(a) ∇F (2, 1, −1) = h4, −5, −1i is a normal vector for the tangent plane at (2, 1, −1),soanequationofthetangentplane
is 4(x − 2) − 5(y − 1) − 1(z +1)=0or 4x − 5y − z =4.
(b) The normal line has direction h4, −5, −1i, so parametric equations are x =2+4t, y =1− 5t, z = −1 − t, and
symmetric equations are x − 2 = y − 1
4 −5 = z +1
−1 .
43. F (x, y, z) =−z + xe y cos z ⇒ ∇F (x, y, z) =he y cos z, xe y cos z, −1 − xe y sin zi and ∇F (1, 0, 0) = h1, 1, −1i.
(a) 1(x − 1) + 1(y − 0) − 1(z − 0) = 0 or x + y − z =1
(b) x − 1=y = −z
45. F (x, y, z) =xy + yz + zx, ∇F (x, y, z) =hy + z, x + z, y + xi, ∇F (1, 1, 1) = h2, 2, 2i,soanequationofthetangent
plane is 2x +2y +2z =6or x + y + z =3, and the normal line is given by x − 1=y − 1=z − 1 or x = y = z. Tograph
thesurfacewesolveforz: z = 3 − xy
x + y .
47. f(x, y) =xy ⇒ ∇f(x, y) =hy, xi, ∇f(3, 2) = h2, 3i. ∇f(3, 2)
is perpendicular to the tangent line, so the tangent line has equation
∇f(3, 2) ·hx − 3,y− 2i =0 ⇒ h2, 3i·hx − 3,x− 2i =0 ⇒
2(x − 3) + 3(y − 2) = 0 or 2x +3y =12.
2x0
49. ∇F (x 0 ,y 0 ,z 0 )=
a , 2y0
2 b , 2z0 . Thus an equation of the tangent plane at (x 2 c 2 0 ,y 0 ,z 0 ) is
2x 0
a x + 2y 0
2 b y + 2z
0 x
2
2 c z =2 0
2 a + y2 0
2 b + z2 0
=2(1)=2since (x 2 c 2 0 ,y 0 ,z 0 ) is a point on the ellipsoid. Hence
x 0
a x + y0
2 b y + z0
2 c z =1is an equation of the tangent plane. 2

F.
SECTION 15.6 DIRECTIONAL TX.10 DERIVATIVES AND THE GRADIENT VECTOR ET SECTION 14.6 ¤ 197
2x0
51. ∇F (x 0,y 0,z 0)=
a , 2y 0
2 b , −1
, so an equation of the tangent plane is 2x 0
2 c
a x + 2y 0
2 b y − 1 2 c z = 2x2 0
a + 2y2 0
2 b − z 0
2 c
or 2x 0
a x + 2y 0
2 b y = z x
2
2 c +2 0
a + y2 0
− z 0
2 b 2 c .Butz 0
c = x2 0
a + y2 0
, so the equation can be written as
2 b2 2x 0
a x + 2y 0
2 b y = z + z 0
.
2 c
53. The hyperboloid x 2 − y 2 − z 2 =1is a level surface of F (x, y, z) =x 2 − y 2 − z 2 and ∇F (x, y, z) =h2x, −2y, −2zi is a
normal vector to the surface and hence a normal vector for the tangent plane at (x, y, z). The tangent plane is parallel to the
plane z = x + y or x + y − z =0if and only if the corresponding normal vectors are parallel, so we need a point (x 0 ,y 0 ,z 0 )
on the hyperboloid where h2x 0, −2y 0, −2z 0i = c h1, 1, −1i or equivalently hx 0, −y 0, −z 0i = k h1, 1, −1i for some k 6= 0.
Then we must have x 0 = k, y 0 = −k, z 0 = k and substituting into the equation of the hyperboloid gives
k 2 − (−k) 2 − k 2 =1 ⇔ −k 2 =1, an impossibility. Thus there is no such point on the hyperboloid.
55. Let (x 0 ,y 0 ,z 0 ) be a point on the cone [other than (0, 0, 0)]. Then an equation of the tangent plane to the cone at this point is
2x 0 x +2y 0 y − 2z 0 z =2 x 2 0 + y 2 0 − z 2 0 .Butx
2
0 + y 2 0 = z 2 0 so the tangent plane is given by x 0 x + y 0 y − z 0 z =0,aplane
which always contains the origin.
57. Let (x 0,y 0,z 0) be a point on the surface. Then an equation of the tangent plane at the point is
x
2 √ + y
x 0 2 + z
√
y 0 2 √ x0 + y 0 + √ z 0
=
.But √ x 0 + y 0 + √ z 0 = √ c, so the equation is
z 0 2
x
√
x0
+
y
y0
+
z √
z0
= √ c.Thex-, y-, and z-intercepts are √ cx 0 , cy 0 and √ cz 0 respectively. (The x-intercept is found
by setting y = z =0and solving the resulting equation for x,andthey-andz-intercepts are found similarly.) So the sum of
the intercepts is √ √x0
c + y 0 + √
z 0 = c, a constant.
59. If f(x, y, z) =z − x 2 − y 2 and g(x, y, z) =4x 2 + y 2 + z 2 , then the tangent line is perpendicular to both ∇f and ∇g
at (−1, 1, 2). The vector v = ∇f × ∇g will therefore be parallel to the tangent line.
We have ∇f(x, y, z) =h−2x, −2y, 1i ⇒ ∇f(−1, 1, 2) = h2, −2, 1i,and∇g(x, y, z) =h8x, 2y, 2zi ⇒
i j k
∇g(−1, 1, 2) = h−8, 2, 4i. Hence v = ∇f × ∇g =
2 −2 1
= −10 i − 16 j − 12 k.
−8 2 4
Parametric equations are: x = −1 − 10t, y =1− 16t, z =2− 12t.
61. (a) The direction of the normal line of F is given by ∇F ,andthatofG by ∇G. Assuming that
∇F 6= 06=∇G, the two normal lines are perpendicular at P if ∇F · ∇G =0at P
⇔
h∂F/∂x,∂F/∂y,∂F/∂zi·h∂G/∂x, ∂G/∂y, ∂G/∂zi =0at P ⇔ F x G x + F y G y + F z G z =0at P .
(b) Here F = x 2 + y 2 − z 2 and G = x 2 + y 2 + z 2 − r 2 ,so
∇F · ∇G = h2x, 2y, −2zi·h2x, 2y, 2zi =4x 2 +4y 2 − 4z 2 =4F =0, since the point (x, y, z) lies on the graph of

F.
198 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
F =0. To see that this is true without using calculus, note that G =0is the equation of a sphere centered at the origin and
F =0is the equation of a right circular cone with vertex at the origin (which is generated by lines through the origin).
At any point of intersection, the sphere’s normal line (which passes through the origin) lies on the cone, and thus is
perpendicular to the cone’s normal line. So the surfaces with equations F =0and G =0are everywhere orthogonal.
63. Let u = ha, bi and v = hc, di. Then we know that at the given point, D u f = ∇f · u = af x + bf y and
D v f = ∇f · v = cf x + df y . But these are just two linear equations in the two unknowns f x and f y , and since u and v
are not parallel, we can solve the equations to find ∇f = hf x,f yi at the given point. In fact,
dDu f − bD v f aDv f − cDu f
∇f =
, .
ad − bc ad − bc
15.7 Maximum and Minimum Values ET 14.7
1. (a) First we compute D(1, 1) = f xx (1, 1) f yy (1, 1) − [f xy (1, 1)] 2 = (4)(2) − (1) 2 =7.SinceD(1, 1) > 0 and
f xx (1, 1) > 0, f has a local minimum at (1, 1) by the Second Derivatives Test.
(b) D(1, 1) = f xx(1, 1) f yy(1, 1) − [f xy(1, 1)] 2 =(4)(2)− (3) 2 = −1. SinceD(1, 1) < 0, f has a saddle point at (1, 1)
by the Second Derivatives Test.
3. In the figure, a point at approximately (1, 1) is enclosed by level curves which are oval in shape and indicate that as we move
away from the point in any direction the values of f are increasing. Hence we would expect a local minimum at or near (1, 1).
The level curves near (0, 0) resemble hyperbolas, and as we move away from the origin, the values of f increase in some
directions and decrease in others, so we would expect to find a saddle point there.
To verify our predictions, we have f(x, y) =4+x 3 + y 3 − 3xy ⇒ f x(x, y) =3x 2 − 3y, f y(x, y) =3y 2 − 3x. We
have critical points where these partial derivatives are equal to 0: 3x 2 − 3y =0, 3y 2 − 3x =0. Substituting y = x 2 from the
first equation into the second equation gives 3(x 2 ) 2 − 3x =0 ⇒ 3x(x 3 − 1) = 0 ⇒ x =0or x =1.Thenwehave
two critical points, (0, 0) and (1, 1). The second partial derivatives are f xx (x, y) =6x, f xy (x, y) =−3,andf yy (x, y) =6y,
so D(x, y) =f xx(x, y) f yy(x, y) − [f xy(x, y)] 2 =(6x)(6y) − (−3) 2 =36xy − 9. ThenD(0, 0) = 36(0)(0) − 9=−9,
and D(1, 1) = 36(1)(1) − 9=27.SinceD(0, 0) < 0, f has a saddle point at (0, 0) by the Second Derivatives Test. Since
D(1, 1) > 0 and f xx (1, 1) > 0, f has a local minimum at (1, 1).
5. f(x, y) =9− 2x +4y − x 2 − 4y 2 ⇒ f x = −2 − 2x, f y =4− 8y,
f xx = −2, f xy =0, f yy = −8. Thenf x =0and f y =0imply
x = −1 and y = 1 2 , and the only critical point is −1, 1 2
.
D(x, y) =f xx f yy − (f xy ) 2 =(−2)(−8) − 0 2 =16, and since
D
−1, 1 2 =16> 0 and fxx −1,
1
2 = −2 < 0, f −1,
1
2 =11is a
local maximum by the Second Derivatives Test.

F.
TX.10 SECTION 15.7 MAXIMUM AND MINIMUM VALUES ET SECTION 14.7 ¤ 199
7. f(x, y) =x 4 + y 4 − 4xy +2 ⇒ f x =4x 3 − 4y, f y =4y 3 − 4x,
f xx =12x 2 , f xy = −4, f yy =12y 2 .Thenf x =0implies y = x 3 ,
and substitution into f y =0 ⇒ x = y 3 gives x 9 − x =0 ⇒
x(x 8 − 1) = 0 ⇒ x =0or x = ±1. Thus the critical points are (0, 0),
(1, 1),and(−1, −1). NowD(0, 0) = 0 · 0 − (−4) 2 = −16 < 0,
so (0, 0) is a saddle point. D(1, 1) = (12)(12) − (−4) 2 > 0 and
f xx (1, 1) = 12 > 0, sof(1, 1) = 0 is a local minimum.
D(−1, −1) = (12)(12) − (−4) 2 > 0 and
f xx =(−1, −1) = 12 > 0,sof(−1, −1) = 0 is also a local minimum.
9. f(x, y) =(1+xy)(x + y) =x + y + x 2 y + xy 2 ⇒
f x =1+2xy + y 2 , f y =1+x 2 +2xy, f xx =2y, f xy =2x +2y,
f yy =2x. Thenf x =0implies 1+2xy + y 2 =0and f y =0implies
1+x 2 +2xy =0. Subtracting the second equation from the first gives
y 2 − x 2 =0 ⇒ y = ±x,butify = x then 1+2xy + y 2 =0 ⇒
1+3x 2 =0which has no real solution. If y = −x then
1+2xy + y 2 =0 ⇒ 1 − x 2 =0 ⇒ x = ±1, so critical points are (1, −1) and (−1, 1).
D(1, −1) = (−2)(2) − 0 < 0 and D(−1, 1) = (2)(−2) − 0 < 0,so(−1, 1) and (1, −1) are saddle points.
11. f(x, y) =x 3 − 12xy +8y 3 ⇒ f x =3x 2 − 12y, f y = −12x +24y 2 ,
f xx =6x, f xy = −12, f yy =48y. Thenf x =0implies x 2 =4y and
f y =0implies x =2y 2 . Substituting the second equation into the first
gives (2y 2 ) 2 =4y ⇒ 4y 4 =4y ⇒ 4y(y 3 − 1) = 0 ⇒ y =0or
y =1.Ify =0then x =0and if y =1then x =2, so the critical points
are (0, 0) and (2, 1).
D(0, 0) = (0)(0) − (−12) 2 = −144 < 0, so(0, 0) is a saddle point.
D(2, 1) = (12)(48) − (−12) 2 =432> 0 and f xx(2, 1) = 12 > 0 so f(2, 1) = −8 is a local minimum.
13. f(x, y) =e x cos y ⇒ f x = e x cos y, f y = −e x sin y.
Now f x =0implies cos y =0or y = π + nπ for n an integer.
2
But sin π
2 + nπ 6=0,sotherearenocriticalpoints.

F.
200 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
15. f(x, y) =(x 2 + y 2 )e y2 −x 2 ⇒
f x =(x 2 + y 2 )e y2 −x 2 (−2x)+2xe y2 −x 2 =2xe y2 −x 2 (1 − x 2 − y 2 ),
f y =(x 2 + y 2 )e y2 −x 2 (2y)+2ye y2 −x 2 =2ye y2 −x 2 (1 + x 2 + y 2 ),
f xx =2xe y2 −x 2 (−2x)+(1− x 2 − y 2 ) 2x −2xe y2 −x 2 +2e y2 −x 2 =2e y2 −x 2 ((1 − x 2 − y 2 )(1 − 2x 2 ) − 2x 2 ),
f xy =2xe y2 −x 2 (−2y)+2x(2y)e y2 −x 2 (1 − x 2 − y 2 )=−4xye y2 −x 2 (x 2 + y 2 ),
f yy =2ye y2 −x 2 (2y)+(1+x 2 + y 2 ) 2y 2ye y2 −x 2 +2e y2 −x 2 =2e y2 −x 2 ((1 + x 2 + y 2 )(1 + 2y 2 )+2y 2 ).
f y =0implies y =0, and substituting into f x =0gives
2xe −x2 (1 − x 2 )=0 ⇒ x =0or x = ±1. Thus the critical points are
(0, 0) and (±1, 0). NowD(0, 0) = (2)(2) − 0 > 0 and f xx (0, 0) = 2 > 0,
so f(0, 0) = 0 is a local minimum. D(±1, 0) = (−4e −1 )(4e −1 ) − 0 < 0
so (±1, 0) are saddle points.
17. f(x, y) =y 2 − 2y cos x ⇒ f x =2y sin x, f y =2y − 2cosx,
f xx =2y cos x, f xy =2sinx, f yy =2.Thenf x =0implies y =0or
sin x =0 ⇒ x =0, π,or2π for −1 ≤ x ≤ 7. Substituting y =0into
f y =0gives cos x =0 ⇒ x = π or 3π , substituting x =0or x =2π
2 2
into f y =0gives y =1, and substituting x = π into f y =0gives y = −1.
Thus the critical points are (0, 1), π
2 , 0 , (π, −1), 3π
2 , 0 ,and(2π, 1).
D π
2 , 0 = D 3π
2 , 0 = −4 < 0 so π
2 , 0 and 3π
2 , 0 aresaddlepoints.D(0, 1) = D(π, −1) = D(2π, 1) = 4 > 0 and
f xx (0, 1) = f xx (π, −1) = f xx (2π, 1) = 2 > 0,sof(0, 1) = f(π, −1) = f(2π, 1) = −1 are local minima.
19. f(x, y) =x 2 +4y 2 − 4xy +2 ⇒ f x =2x − 4y, f y =8y − 4x, f xx =2, f xy = −4, f yy =8.Thenf x =0
and f y =0each implies y = 1 x, so all points of the form x
2 0 , 1 x 2 0
are critical points and for each of these we have
D x 0 , 1 x 2 0 = (2)(8) − (−4) 2 =0. The Second Derivatives Test gives no information, but
f(x, y) =x 2 +4y 2 − 4xy +2=(x − 2y) 2 +2≥ 2 with equality if and only if y = 1 x. Thus f
2
x 0, 1 2 x0 =2are all local
(and absolute) minima.

F.
TX.10 SECTION 15.7 MAXIMUM AND MINIMUM VALUES ET SECTION 14.7 ¤ 201
21. f(x, y) =x 2 + y 2 + x −2 y −2
From the graphs, there appear to be local minima of about f(1, ±1) = f(−1, ±1) ≈ 3 (and no local maxima or saddle
points). f x =2x − 2x −3 y −2 , f y =2y − 2x −2 y −3 , f xx =2+6x −4 y −2 , f xy =4x −3 y −3 , f yy =2+6x −2 y −4 .Then
f x =0implies 2x 4 y 2 − 2=0or x 4 y 2 =1or y 2 = x −4 . Note that neither x nor y can be zero. Now f y =0implies
2x 2 y 4 − 2=0,andwithy 2 = x −4 this implies 2x −6 − 2=0or x 6 =1. Thus x = ±1 and if x =1, y = ±1;ifx = −1,
y = ±1. So the critical points are (1, 1), (1, −1),(−1, 1) and (−1, −1). NowD(1, ±1) = D(−1, ±1) = 64 − 16 > 0 and
f xx > 0 always, so f(1, ±1) = f(−1, ±1) = 3 are local minima.
23. f(x, y) =sinx +siny +sin(x + y), 0 ≤ x ≤ 2π, 0 ≤ y ≤ 2π
From the graphs it appears that f has a local maximum at about (1, 1) with value approximately 2.6, a local minimum
at about (5, 5) with value approximately −2.6, and a saddle point at about (3, 3).
f x =cosx +cos(x + y), f y =cosy +cos(x + y), f xx = − sin x − sin(x + y), f yy = − sin y − sin(x + y),
f xy = − sin(x + y). Setting f x =0and f y =0andsubtractinggivescos x − cos y =0or cos x =cosy. Thusx = y
or x =2π − y. Ifx = y, f x =0becomes cos x +cos2x =0or 2cos 2 x +cosx − 1=0,aquadraticincos x. Thus
cos x = −1 or 1 and x = π, π ,or 5π , yielding the critical points (π, π), π
, π
2 3 3 3 3 and 5π , 5π
3 3 . Similarly if
x =2π − y, f x =0becomes (cos x)+1=0and the resulting critical point is (π, π). Now
D(x, y) =sinx sin y +sinx sin(x + y)+siny sin(x + y). SoD(π, π) =0and the Second Derivatives Test doesn’t apply.
However, along the line y = x we have f(x, x) =2sinx +sin2x =2sinx +2sinx cos x =2sinx(1 + cos x), and
f(x, x) > 0 for 0

F.
202 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
positive as well as points where f is negative, so the graph crosses its tangent plane (z =0)there and (π, π) isasaddlepoint.
D π
, π
3 3 =
9
> 0 and f π
4 xx , π
3 3 < 0 so f π , π
3 3 =
3 √ 3
is a local maximum while D 5π
, 5π
2 3 3 =
9
> 0 and
4
f 5π xx , 5π
3 3 > 0,sof 5π , 5π
3 3 = −
3 √ 3
is a local minimum.
2
25. f(x, y) =x 4 − 5x 2 + y 2 +3x +2 ⇒ f x(x, y) =4x 3 − 10x +3and f y(x, y) =2y. f y =0 ⇒ y =0, and the graph
of f x shows that the roots of f x =0are approximately x = −1.714, 0.312 and 1.402. (Alternatively, we could have used a
calculator or a CAS to find these roots.) So to three decimal places, the critical points are (−1.714, 0), (1.402, 0), and
(0.312, 0). Nowsincef xx =12x 2 − 10, f xy =0, f yy =2,andD =24x 2 − 20, wehaveD(−1.714, 0) > 0,
f xx (−1.714, 0) > 0, D(1.402, 0) > 0, f xx (1.402, 0) > 0,andD(0.312, 0) < 0. Therefore f(−1.714, 0) ≈−9.200 and
f(1.402, 0) ≈ 0.242 are local minima, and (0.312, 0) is a saddle point. The lowest point on the graph is approximately
(−1.714, 0, −9.200).
27. f(x, y) =2x +4x 2 − y 2 +2xy 2 − x 4 − y 4 ⇒ f x(x, y) =2+8x +2y 2 − 4x 3 , f y(x, y) =−2y +4xy − 4y 3 .
Now f y =0 ⇔ 2y(2y 2 − 2x +1)=0 ⇔ y =0or y 2 = x − 1 .Thefirst of these implies that f 2 x = −4x 3 +8x +2,
and the second implies that f x =2+8x +2
x − 1 2 − 4x 3 = −4x 3 +10x +1. From the graphs, we see that the first
possibility for f x has roots at approximately −1.267, −0.259,and1.526, and the second has a root at approximately 1.629
(the negative roots do not give critical points, since y 2 = x − 1 must be positive). So to three decimal places, f has critical
2
points at (−1.267, 0), (−0.259, 0), (1.526, 0), and(1.629, ±1.063). Nowsincef xx =8− 12x 2 , f xy =4y,
f yy =4x − 12y 2 ,andD =(8− 12x 2 )(4x − 12y 2 ) − 16y 2 ,wehaveD(−1.267, 0) > 0, f xx (−1.267, 0) > 0,
D(−0.259, 0) < 0, D(1.526, 0) < 0, D(1.629, ±1.063) > 0,andf xx (1.629, ±1.063) < 0. Therefore, to three decimal
places, f(−1.267, 0) ≈ 1.310 and f(1.629, ±1.063) ≈ 8.105 are local maxima, and (−0.259, 0) and (1.526, 0) are saddle
points. The highest points on the graph are approximately (1.629, ±1.063, 8.105).

F.
TX.10 SECTION 15.7 MAXIMUM AND MINIMUM VALUES ET SECTION 14.7 ¤ 203
29. Since f is a polynomial it is continuous on D,soanabsolutemaximumandminimumexist.Heref x =4, f y = −5 so
there are no critical points inside D. Thus the absolute extrema must both occur on the boundary. Along L 1 : x =0and
f(0,y)=1− 5y for 0 ≤ y ≤ 3, a decreasing function in y, so the maximum value is f (0, 0) = 1 and the minimum value
is f(0, 3) = −14. Along L 2 : y =0and f(x, 0) = 1 + 4x for 0 ≤ x ≤ 2,an
increasing function in x, so the minimum value is f(0, 0) = 1 and the maximum
value is f(2, 0) = 9. Along L 3 : y = − 3 2 x +3and f x, − 3 2 x +3 = 23 2 x − 14
for 0 ≤ x ≤ 2, an increasing function in x, so the minimum value is
f(0, 3) = −14 and the maximum value is f(2, 0) = 9. Thus the absolute
maximum of f on D is f(2, 0) = 9 and the absolute minimum is f(0, 3) = −14.
31. f x (x, y) =2x +2xy, f y (x, y) =2y + x 2 , and setting f x = f y =0
gives (0, 0) as the only critical point in D,withf(0, 0) = 4.
On L 1: y = −1, f(x, −1) = 5,aconstant.
On L 2 : x =1, f(1,y)=y 2 + y +5, a quadratic in y which attains its
maximum at (1, 1), f(1, 1) = 7 and its minimum at
1, − 1 2 , f 1, −
1
2 =
19
. 4
On L 3: f(x, 1) = 2x 2 +5which attains its maximum at (−1, 1) and (1, 1)
with f(±1, 1) = 7 and its minimum at (0, 1), f(0, 1) = 5.
On L 4: f(−1,y)=y 2 + y +5with maximum at (−1, 1), f(−1, 1) = 7 and minimum at −1, − 1 2
, f −1, −
1
2 =
19
. 4
Thus the absolute maximum is attained at both (±1, 1) with f(±1, 1) = 7 and the absolute minimum on D is attained at
(0, 0) with f(0, 0) = 4.
33. f(x, y) =x 4 + y 4 − 4xy +2is a polynomial and hence continuous on D,so
it has an absolute maximum and minimum on D.InExercise7,wefoundthe
critical points of f;only(1, 1) with f(1, 1) = 0 is inside D. OnL 1 : y =0,
f(x, 0) = x 4 +2, 0 ≤ x ≤ 3, a polynomial in x which attains its maximum
at x =3, f(3, 0) = 83, and its minimum at x =0, f(0, 0) = 2.
On L 2 : x =3, f(3,y)=y 4 − 12y +83, 0 ≤ y ≤ 2, a polynomial in y
which attains its minimum at y = 3√ 3, f 3, 3√ 3 =83− 9 3√ 3 ≈ 70.0,anditsmaximumaty =0, f(3, 0) = 83.
On L 3 : y =2, f(x, 2) = x 4 − 8x +18, 0 ≤ x ≤ 3, a polynomial in x which attains its minimum at x = 3√ 2,
f 3 √ 2, 2 =18− 6 3√ 2 ≈ 10.4, and its maximum at x =3,f(3, 2) = 75. OnL 4 : x =0, f(0,y)=y 4 +2, 0 ≤ y ≤ 2,a
polynomial in y which attains its maximum at y =2, f(0, 2) = 18, and its minimum at y =0, f(0, 0) = 2. Thus the absolute
maximum of f on D is f(3, 0) = 83 and the absolute minimum is f(1, 1) = 0.

F.
204 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
35. f x(x, y) =6x 2 and f y(x, y) =4y 3 .Andsof x =0and f y =0only occur when x = y =0. Hence, the only critical point
inside the disk is at x = y =0where f(0, 0) = 0. Now on the circle x 2 + y 2 =1, y 2 =1− x 2 so let
g(x) =f(x, y) =2x 3 +(1− x 2 ) 2 = x 4 +2x 3 − 2x 2 +1,−1 ≤ x ≤ 1. Theng 0 (x) =4x 3 +6x 2 − 4x =0 ⇒ x =0,
−2, or 1 . f(0, ±1) = g (0) = 1, f 1
, ± √
3
= g
1
2 2 2
2 =
13
,and(−2, −3) is not in D. Checking the endpoints, we get
16
f(−1, 0) = g(−1) = −2 and f(1, 0) = g(1) = 2. Thus the absolute maximum and minimum of f on D are f(1, 0) = 2 and
f(−1, 0) = −2.
Another method: On the boundary x 2 + y 2 =1we can write x =cosθ, y =sinθ,sof(cos θ, sin θ) =2cos 3 θ +sin 4 θ,
0 ≤ θ ≤ 2π.
37. f(x, y) =−(x 2 − 1) 2 − (x 2 y − x − 1) 2 ⇒ f x(x, y) =−2(x 2 − 1)(2x) − 2(x 2 y − x − 1)(2xy − 1) and
f y (x, y) =−2(x 2 y − x − 1)x 2 .Settingf y (x, y) =0gives either x =0or x 2 y − x − 1=0.
There are no critical points for x =0,sincef x (0,y)=−2,sowesetx 2 y − x − 1=0 ⇔ y = x +1 [x 6= 0],
x 2
so f x x, x +1
= −2(x 2 − 1)(2x) − 2 x 2 x +1
− x − 1 2x x +1
− 1 = −4x(x 2 − 1). Therefore
x 2
x 2 x 2
f x (x, y) =f y (x, y) =0at the points (1, 2) and (−1, 0). To classify these critical points, we calculate
f xx(x, y) =−12x 2 − 12x 2 y 2 +12xy +4y +2, f yy(x, y) =−2x 4 ,
and f xy (x, y) =−8x 3 y +6x 2 +4x. In order to use the Second
Derivatives Test we calculate
D(−1, 0) = f xx(−1, 0) f yy(−1, 0) − [f xy(−1, 0)] 2 =16> 0,
f xx(−1, 0) = −10 < 0, D(1, 2) = 16 > 0,andf xx(1, 2) = −26 < 0,so
both (−1, 0) and (1, 2) give local maxima.
39. Let d be the distance from (2, 1, −1) to any point (x, y, z) on the plane x + y − z =1,so
d = (x − 2) 2 +(y − 1) 2 +(z +1) 2 where z = x + y − 1, and we minimize
d 2 = f(x, y) =(x − 2) 2 +(y − 1) 2 +(x + y) 2 .Thenf x (x, y) =2(x − 2) + 2(x + y) =4x +2y − 4,
f y(x, y) =2(y − 1) + 2(x + y) =2x +4y − 2. Solving 4x +2y − 4=0and 2x +4y − 2=0simultaneously gives x =1,
y =0. An absolute minimum exists (since there is a minimum distance from the point to the plane) and it must occur at a
critical point, so the shortest distance occurs for x =1, y =0for which d = (1 − 2) 2 +(0− 1) 2 +(0+1) 2 = √ 3.
41. Let d be the distance from the point (4, 2, 0) to any point (x, y, z) on the cone, so d = (x − 4) 2 +(y − 2) 2 + z 2 where
z 2 = x 2 + y 2 , and we minimize d 2 =(x − 4) 2 +(y − 2) 2 + x 2 + y 2 = f (x, y). Then
f x (x, y) =2(x − 4) + 2x =4x − 8, f y (x, y) =2(y − 2) + 2y =4y − 4, and the critical points occur when f x =0
⇒
x =2, f y =0 ⇒ y =1. Thus the only critical point is (2, 1). An absolute minimum exists (since there is a minimum
distance from the cone to the point) which must occur at a critical point, so the points on the cone closest
to (4, 2, 0) are 2, 1, ± √ 5 .

F.
TX.10 SECTION 15.7 MAXIMUM AND MINIMUM VALUES ET SECTION 14.7 ¤ 205
43. x + y + z =100, so maximize f(x, y) =xy(100 − x − y). f x =100y − 2xy − y 2 , f y = 100x − x 2 − 2xy,
f xx = −2y, f yy = −2x, f xy =100− 2x − 2y. Thenf x =0implies y =0or y = 100 − 2x. Substituting y =0into
f y =0gives x =0or x = 100 and substituting y =100− 2x into f y =0gives 3x 2 − 100x =0so x =0or 100
Thus the critical points are (0, 0), (100, 0), (0, 100) and 100
, 100
3 3
D(0, 0) = D(100, 0) = D(0, 100) = −10,000 while D 100
, 100
3 3
(100, 0) and (0, 100) are saddle points whereas f 100
, 100
3 3
.
=
10,000
and f 100
3 xx , 100
3 3 = −
200
3 .
3
< 0. Thus (0, 0),
is a local maximum. Thus the numbers are x = y = z =
100
3 .
45. Center the sphere at the origin so that its equation is x 2 + y 2 + z 2 = r 2 , and orient the inscribed rectangular box so that its
edges are parallel to the coordinate axes. Any vertex of the box satisfies x 2 + y 2 + z 2 = r 2 ,sotake(x, y, z) to be the vertex
in the first octant. Then the box has length 2x, width2y, and height 2z =2 r 2 − x 2 − y 2 with volume given by
V (x, y) =(2x)(2y)
2
r 2 − x 2 − y 2 =8xy r 2 − x 2 − y 2 for 0 0. ThenV x = 1 4 cy − 2xy − y2 and V y = 1 4 cx − x2 − 2xy,
so V x =0=V y when 2x + y = 1 4 c and x +2y = 1 4 c. Solving, we get x = 1 12 c, y = 1
12 c and z = 1 4 c − x − y = 1
12 c.From
the geometrical nature of the problem, this critical point must give an absolute maximum. Thus the box is a cube with edge
length 1 12 c.
51. Let the dimensions be x, y and z, then minimize xy +2(xz + yz) if xyz =32,000 cm 3 .Then
f(x, y) =xy +[64,000(x + y)/xy] =xy +64,000(x −1 + y −1 ), f x = y − 64,000x −2 , f y = x − 64,000y −2 .
And f x =0implies y =64,000/x 2 ; substituting into f y =0implies x 3 =64,000 or x =40and then y =40.Now
D(x, y) = [(2)(64,000)] 2 x −3 y −3 − 1 > 0 for (40, 40) and f xx (40, 40) > 0 so this is indeed a minimum. Thus the
dimensions of the box are x = y =40cm, z =20cm.

F.
206 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
53. Let x, y, z be the dimensions of the rectangular box. Then the volume of the box is xyz and
L = x 2 + y 2 + z 2 ⇒ L 2 = x 2 + y 2 + z 2 ⇒ z = L 2 − x 2 − y 2 .
Substituting, we have volume V (x, y) =xy L 2 − x 2 − y 2 , (x, y > 0).
V x = xy · 1
2 (L2 − x 2 − y 2 ) −1/2 (−2x)+y L 2 − x 2 − y 2 = y x 2 y
L 2 − x 2 − y 2 −
L2 − x 2 − y ,
2
V y = x L 2 − x 2 − y 2 −
xy 2
L2 − x 2 − y 2 . V x =0implies y(L 2 − x 2 − y 2 )=x 2 y ⇒ y(L 2 − 2x 2 − y 2 )=0 ⇒
2x 2 + y 2 = L 2 (since y>0), and V y =0implies x(L 2 − x 2 − y 2 )=xy 2 ⇒ x(L 2 − x 2 − 2y 2 )=0 ⇒
x 2 +2y 2 = L 2 (since x>0). Substituting y 2 = L 2 − 2x 2 into x 2 +2y 2 = L 2 gives x 2 +2L 2 − 4x 2 = L 2 ⇒
3x 2 = L 2 ⇒ x = L/ √
3 (since x>0)andtheny = L 2 − 2 L/ √ 3 2 √
= L/ 3. So the only critical point is
L/
√
3,L/
√
3
which, from the geometrical nature of the problem, must give an absolute maximum. Thus the maximum
volume is V L/ √ 3,L/ √ 3 = L/ √ 3 2 L 2 − L/ √ 3 2
−
L/
√
3
2
= L 3 / 3 √ 3 cubic units.
55. Note that here the variables are m and b,andf(m, b) = n [y i − (mx i + b)] 2
.Thenf m = n −2x i [y i − (mx i + b)] = 0
implies
n
xi y i − mx 2 i − bx i =0or
i =1
n
y i = m n x i +
i =1
Now f mm =
i =1
n
i =1
n
i =1
2x 2 i , f bb =
i =1
n
x i y i = m n x 2
i + b n x i and f b =
i =1
i =1
i =1
n
i =1
n
b = m x i + nb. Thus we have the two desired equations.
i =1
n
i =1
n
D(m, b) =4n x 2 n
i − 4
i =1
equations do indeed minimize
i =1
n
i =1
i =1
2=2n and f mb = n 2x i. And f mm(m, b) > 0 always and
2
n
x i =4 n
d 2 i .
i =1
x 2 i
i =1
−
−2[y i − (mx i + b)] = 0 implies
n
2
x i > 0 always so the solutions of these two
i =1
15.8 Lagrange Multipliers ET 14.8
1. At the extreme values of f, the level curves of f just touch the curve g(x, y) =8with a common tangent line. (See Figure 1
and the accompanying discussion.) We can observe several such occurrences on the contour map, but the level curve
f(x, y) =c with the largest value of c which still intersects the curve g(x, y) =8is approximately c =59, and the smallest
value of c corresponding to a level curve which intersects g(x, y) =8appears to be c =30.Thusweestimatethemaximum
value of f subject to the constraint g(x, y) =8to be about 59 and the minimum to be 30.
3. f(x, y) =x 2 + y 2 , g(x, y) =xy =1,and∇f = λ∇g ⇒ h2x, 2yi = hλy, λxi,so2x = λy, 2y = λx, andxy =1.
From the last equation, x 6= 0and y 6= 0,so2x = λy ⇒ λ =2x/y. Substituting, we have 2y =(2x/y) x ⇒
y 2 = x 2 ⇒ y = ±x. Butxy =1,sox = y = ±1 and the possible points for the extreme values of f are (1, 1) and
(−1, −1). Here there is no maximum value, since the constraint xy =1allows x or y to become arbitrarily large, and hence
f(x, y) =x 2 + y 2 canbemadearbitrarilylarge.Theminimumvalueisf(1, 1) = f(−1, −1) = 2.

F.
TX.10
SECTION 15.8 LAGRANGE MULTIPLIERS ET SECTION 14.8 ¤ 207
5. f(x, y) =x 2 y, g(x, y) =x 2 +2y 2 =6 ⇒ ∇f = 2xy, x 2 , λ∇g = h2λx, 4λyi. Then2xy =2λx implies x =0or
λ = y. Ifx =0,thenx 2 =4λy implies λ =0or y =0.However,ify =0then g(x, y) =0, a contradiction. So λ =0and
then g(x, y) =6 ⇒ y = ± √ 3.Ifλ = y,thenx 2 =4λy implies x 2 =4y 2 ,andsog(x, y) =6 ⇒
4y 2 +2y 2 =6 ⇒ y 2 =1 ⇒ y = ±1. Thus f has possible extreme values at the points 0, ± √ 3 , (±2, 1), and
(±2, −1). After evaluating f at these points, we find the maximum value to be f(±2, 1) = 4 and the minimum to be
f(±2, −1) = −4.
7. f(x, y, z) =2x +6y +10z, g(x, y, z) =x 2 + y 2 + z 2 =35 ⇒ ∇f = h2, 6, 10i, λ∇g = h2λx, 2λy, 2λzi. Then
2λx =2, 2λy =6, 2λz =10imply x = 1 λ , y = 3 λ ,andz = 5 2 2 2
1 3 5
λ .But35 = x2 + y 2 + z 2 = + +
λ λ λ
⇒
35 = 35
λ 2 ⇒ λ = ±1,sof has possible extreme values at the points (1, 3, 5), (−1, −3, −5). The maximum value of f on
x 2 + y 2 + z 2 =35is f(1, 3, 5) = 70, and the minimum is f(−1, −3, −5) = −70.
9. f(x, y, z) =xyz, g(x, y, z) =x 2 +2y 2 +3z 2 =6 ⇒ ∇f = hyz, xz, xyi, λ∇g = h2λx, 4λy, 6λzi. Ifλ =0then at
least one of the coordinates is 0, in which case f(x, y, z) =0. (None of these ends up giving a maximum or minimum.)
If λ 6= 0,then∇f = λ∇g implies λ =(yz)/(2x) =(xz)/(4y) =(xy)/(6z) or x 2 =2y 2 and z 2 = 2 3 y2 . Thus
√2,
x 2 +2y 2 +3z 2 =6implies 6y 2 =6or y = ±1. Thus the possible remaining points are ±1,
2
,
3
√2,
±1, −
2
, − √
2
2, ±1, , − √
2
2, ±1, − . The maximum value of f on the ellipsoid is √ 2
3
3
3
3
,occurringwhen
all coordinates are positive or exactly two are negative and the minimum is − 2 √
3
occurring when 1 or 3 of the coordinates are
negative.
11. f(x, y, z) =x 2 + y 2 + z 2 , g(x, y, z) =x 4 + y 4 + z 4 =1 ⇒ ∇f = h2x, 2y, 2zi, λ∇g = 4λx 3 , 4λy 3 , 4λz 3 .
Case 1: If x 6= 0, y 6= 0and z 6= 0,then∇f = λ∇g implies λ =1/(2x 2 )=1/(2y 2 )=1/(2z 2 ) or x 2 = y 2 = z 2 and
3x 4 =1or x = ± √ 1 giving the points ± 1 4 4
3 √ , 1 1
3
4√ ,
3
4√
, ± √ 1 , − 1
3
4 4
3 √ , 1
3
4√
, ± √ 1 , 1
3
4
3
4√ , − √ 1 , ± 1
3
4 4
3
√ , − 1 4
3 √ , − 1 4
3 √ 3
all with an f-value of √ 3.
Case 2: If one of the variables equals zero and the other two are not zero, then the squares of the two nonzero coordinates
are equal with common value 1 √
2
and corresponding f value of √ 2.
Case 3: If exactly two of the variables are zero, then the third variable has value ±1 with the corresponding f value of 1.
Thus on x 4 + y 4 + z 4 =1, the maximum value of f is √ 3 and the minimum value is 1.
13. f(x, y, z, t) =x + y + z + t, g(x, y, z, t) =x 2 + y 2 + z 2 + t 2 =1 ⇒ h1, 1, 1, 1i = h2λx, 2λy, 2λz, 2λti, so
λ =1/(2x) =1/(2y) =1/(2z) =1/(2t) and x = y = z = t. Butx 2 + y 2 + z 2 + t 2 =1, so the possible points are
±
1
, ± 1 , ± 1 , ± 1
2 2 2 2 . Thus the maximum value of f is f 1 , 1 , 1 , 1
2 2 2 2 =2and the minimum value is
f − 1 , − 1 , − 1 , − 1
2 2 2 2 = −2.

F.
208 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
15. f(x, y, z) =x +2y, g(x, y, z) =x + y + z =1, h(x, y, z) =y 2 + z 2 =4 ⇒ ∇f = h1, 2, 0i, λ∇g = hλ, λ, λi
and μ∇h = h0, 2μy, 2μzi. Then1=λ, 2=λ +2μy and 0=λ +2μz so μy = 1 = −μz or y =1/(2μ), z = −1/(2μ).
2
Thus x + y + z =1implies x =1and y 2 + z 2 =4implies μ = ± 1
2 √ . Then the possible points are 1, ± √ 2, ∓ √ 2
2
and the maximum value is f 1, √ 2, − √ 2 =1+2 √ 2 and the minimum value is f 1, − √ 2, √ 2 =1− 2 √ 2.
17. f(x, y, z) =yz + xy, g(x, y, z) =xy =1, h(x, y, z) =y 2 + z 2 =1 ⇒ ∇f = hy, x + z, yi, λ∇g = hλy, λx, 0i,
μ∇h = h0, 2μy, 2μzi. Theny = λy implies λ =1[y 6= 0since g(x, y, z) =1], x + z = λx +2μy and y =2μz. Thus
μ = z/(2y) =y/(2y) or y 2 = z 2 ,andsoy 2 + z 2 =1implies y = ± 1 √
2
, z = ± 1 √
2
.Thenxy =1implies x = ± √ 2 and
the possible points are ± √
2, ± √ 1 1
2
, √
2
, ± √ 2, ± √ 1
2
, − √ 1
2
. Hence the maximum of f subject to the constraints is
f ± √
2, ± √ 1
2
, ± √ 1
2
= 3 and the minimum is f ± √
2, ± √ 1
2 2
, ∓√ 1
2
= 1 . 2
Note: Since xy =1is one of the constraints we could have solved the problem by solving f(y, z) =yz +1subject to
y 2 + z 2 =1.
19. f(x, y) =e −xy . For the interior of the region, we find the critical points: f x = −ye −xy , f y = −xe −xy , so the only
critical point is (0, 0), andf(0, 0) = 1. For the boundary, we use Lagrange multipliers. g(x, y) =x 2 +4y 2 =1
⇒
λ∇g = h2λx, 8λyi, sosetting∇f = λ∇g we get −ye −xy =2λx and −xe −xy =8λy. Thefirst of these gives
e −xy = −2λx/y, and then the second gives −x(−2λx/y) =8λy ⇒ x 2 =4y 2 . Solving this last equation with the
constraint x 2 +4y 2 =1gives x = ± √ 1
2
and y = ± 1
2 √ .Nowf ± √ 1
2 2
, ∓ 1
2 √ = e 1/4 ≈ 1.284 and
2
f ± √ 1
2
, ± 1
2 √ = e −1/4 ≈ 0.779. The former are the maxima on the region and the latter are the minima.
2
21. (a) f(x, y) =x, g(x, y) =y 2 + x 4 − x 3 =0 ⇒ ∇f = h1, 0i = λ∇g = λ 4x 3 − 3x 2 , 2y .Then
1=λ(4x 3 − 3x 2 ) (1) and 0=2λy (2). Wehaveλ 6= 0from (1),so(2) gives y =0. Then, from the constraint equation,
x 4 − x 3 =0 ⇒ x 3 (x − 1) = 0 ⇒ x =0or x =1.Butx =0contradicts (1), so the only possible extreme value
subject to the constraint is f(1, 0) = 1. (The question remains whether this is indeed the minimum of f.)
(b) The constraint is y 2 + x 4 − x 3 =0 ⇔ y 2 = x 3 − x 4 . The left side is non-negative, so we must have x 3 − x 4 ≥ 0
which is true only for 0 ≤ x ≤ 1. Therefore the minimum possible value for f(x, y) =x is 0 which occurs for x = y =0.
However, λ∇g(0, 0) = λ h0 − 0, 0i = h0, 0i and ∇f(0, 0) = h1, 0i,so∇f(0, 0) 6=λ∇g(0, 0) for all values of λ.
(c) Here ∇g(0, 0) = 0 but the method of Lagrange multipliers requires that ∇g 6=0 everywhere on the constraint curve.
23. P (L, K) =bL α K 1−α , g(L, K) =mL + nK = p ⇒ ∇P = αbL α−1 K 1−α , (1 − α)bL α K −α , λ∇g = hλm, λni.
Then αb(K/L) 1−α = λm and (1 − α)b(L/K) α = λn and mL + nK = p,soαb(K/L) 1−α /m =(1− α)b(L/K) α /n or
nα/[m(1 − α)] = (L/K) α (L/K) 1−α or L = Knα/[m(1 − α)]. Substituting into mL + nK = p gives K =(1− α)p/n
and L = αp/m for the maximum production.

F.
TX.10
SECTION 15.8 LAGRANGE MULTIPLIERS ET SECTION 14.8 ¤ 209
25. Let the sides of the rectangle be x and y. Thenf(x, y) =xy, g(x, y) =2x +2y = p ⇒ ∇f(x, y) =hy, xi,
λ∇g = h2λ, 2λi. Thenλ = 1 2 y = 1 2 x implies x = y and the rectangle with maximum area is a square with side length 1 4 p.
27. Let f(x, y, z) =d 2 =(x − 2) 2 +(y − 1) 2 +(z +1) 2 , then we want to minimize f subject to the constraint
g(x, y, z) =x + y − z =1. ∇f = λ∇g ⇒ h2(x − 2), 2(y − 1), 2(z +1)i = λ h1, 1, −1i, sox =(λ +4)/2,
y =(λ +2)/2, z = −(λ +2)/2. Substituting into the constraint equation gives λ +4
2
+ λ +2
2
+ λ +2
2
=1 ⇒
3λ +8=2 ⇒ λ = −2,sox =1, y =0,andz =0. This must correspond to a minimum, so the shortest distance is
d = (1 − 2) 2 +(0− 1) 2 +(0+1) 2 = √ 3.
29. Let f(x, y, z) =d 2 =(x − 4) 2 +(y − 2) 2 + z 2 . Then we want to minimize f subject to the constraint
g (x, y, z) =x 2 + y 2 − z 2 =0. ∇f = λ∇g ⇒ h2(x − 4) , 2(y − 2) , 2zi = h2λx, 2λy, −2λzi,sox − 4=λx,
y − 2=λy,andz = −λz. From the last equation we have z + λz =0 ⇒ z (1 + λ) =0,soeitherz =0or λ = −1.
But from the constraint equation we have z =0 ⇒ x 2 + y 2 =0 ⇒ x = y =0which is not possible from the first two
equations. So λ = −1 and x − 4=λx ⇒ x =2, y − 2=λy ⇒ y =1,andx 2 + y 2 − z 2 =0 ⇒
4+1− z 2 =0 ⇒ z = ± √ 5. This must correspond to a minimum, so the points on the cone closest to (4, 2, 0)
are 2, 1, ± √ 5 .
31. f(x, y, z) =xyz, g(x, y, z) =x + y + z =100 ⇒ ∇f = hyz, xz, xyi = λ∇g = hλ, λ, λi. Thenλ = yz = xz = xy
implies x = y = z = 100
3 .
33. If the dimensions are 2x, 2y, and2z, thenmaximizef(x, y, z) =(2x)(2y)(2z) =8xyz subject to
g(x, y, z) =x 2 + y 2 + z 2 = r 2 (x>0, y>0, z>0). Then ∇f = λ∇g ⇒ h8yz, 8xz, 8xyi = λ h2x, 2y, 2zi ⇒
8yz =2λx, 8xz =2λy,and8xy =2λz,soλ = 4yz
x
= 4xz
y
= 4xy
z .Thisgivesx2 z = y 2 z ⇒ x 2 = y 2 (since z 6= 0)
and xy 2 = xz 2 ⇒ z 2 = y 2 ,sox 2 = y 2 = z 2 ⇒ x = y = z, and substituting into the constraint
equation gives 3x 2 = r 2 ⇒ x = r/ √ 3=y = z. Thus the largest volume of such a box is
r
f √
r
3
, √
r
3
, √
r
3
=8 √ √
r
√
r
3 3 3
= 8
3 √ 3 r3 .
35. f(x, y, z) =xyz, g(x, y, z) =x +2y +3z =6 ⇒ ∇f = hyz, xz, xyi = λ∇g = hλ, 2λ, 3λi.
Then λ = yz = 1 xz = 1 xy implies x =2y, z = 2 y.But2y +2y +2y =6so y =1, x =2, z = 2 and the volume
2 3 3 3
is V = 4 3 .
37. f(x, y, z) =xyz, g(x, y, z) =4(x + y + z) =c ⇒ ∇f = hyz, xz, xyi, λ∇g = h4λ, 4λ, 4λi. Thus
4λ = yz = xz = xy or x = y = z = 1 c are the dimensions giving the maximum volume.
12
39. If the dimensions of the box are given by x, y, andz, thenweneedtofind the maximum value of f(x, y, z) =xyz
[x, y, z > 0] subject to the constraint L = x 2 + y 2 + z 2 or g(x, y, z) =x 2 + y 2 + z 2 = L 2 . ∇f = λ∇g ⇒
hyz, xz, xyi = λh2x, 2y, 2zi,soyz =2λx ⇒ λ = yz
xz
xy
, xz =2λy ⇒ λ = ,andxy =2λz ⇒ λ =
2x 2y 2z . Thus

F.
210 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
λ = yz
2x = xz
2y
⇒ x 2 = y 2 [since z 6= 0] ⇒ x = y and λ = yz
2x = xy
2z
⇒ x = z [since y 6= 0].
Substituting into the constraint equation gives x 2 + x 2 + x 2 = L 2 ⇒ x 2 = L 2 /3 ⇒ x = L/ √ 3=y = z and the
maximum volume is L/ √ 3 3
= L 3 / 3 √ 3 .
41. We need to find the extreme values of f(x, y, z) =x 2 + y 2 + z 2 subject to the two constraints g(x, y, z) =x + y +2z =2
and h(x, y, z) =x 2 + y 2 − z =0.
∇f = h2x, 2y, 2zi, λ∇g = hλ, λ, 2λi and μ∇h = h2μx, 2μy, −μi. Thusweneed
2x = λ +2μx (1), 2y = λ +2μy (2), 2z =2λ − μ (3), x + y +2z =2 (4),andx 2 + y 2 − z =0 (5).
From (1) and (2), 2(x − y) =2μ(x − y),soifx 6=y, μ =1. Putting this in (3) gives 2z =2λ − 1 or λ = z + 1 , but putting
2
μ =1into (1) says λ =0. Hence z + 1 2 =0or z = − 1 2 .Then(4) and (5) become x + y − 3=0and x2 + y 2 + 1 2 =0.The
last equation cannot be true, so this case gives no solution. So we must have x = y. Then(4) and (5) become 2x +2z =2and
2x 2 − z =0which imply z =1− x and z =2x 2 .Thus2x 2 =1− x or 2x 2 + x − 1=(2x − 1)(x +1)=0so x = 1 or 2
x = −1. The two points to check are 1
, 1 , 1
2 2 2 and (−1, −1, 2): f 1 , 1 , 1
2 2 2 =
3
and f(−1, −1, 2) = 6. Thus 1
, 1 , 1
4 2 2 2 is
the point on the ellipse nearest the origin and (−1, −1, 2) is the one farthest from the origin.
43. f(x, y, z) =ye x−z , g(x, y, z) =9x 2 +4y 2 +36z 2 =36, h(x, y, z) =xy + yz =1. ∇f = λ∇g + μ∇h ⇒
ye x−z ,e x−z , −ye x−z = λh18x, 8y, 72zi + μhy, x + z, yi, soye x−z =18λx + μy, e x−z =8λy + μ(x + z),
−ye x−z =72λz + μy, 9x 2 +4y 2 +36z 2 =36, xy + yz =1. Using a CAS to solve these 5 equations simultaneously for x,
y, z, λ,andμ (in Maple, use the allvalues command), we get 4 real-valued solutions:
x ≈ 0.222444, y ≈−2.157012, z ≈−0.686049, λ ≈−0.200401, μ ≈ 2.108584
x ≈−1.951921, y ≈−0.545867, z ≈ 0.119973, λ ≈ 0.003141, μ ≈−0.076238
x ≈ 0.155142, y ≈ 0.904622, z ≈ 0.950293, λ ≈−0.012447, μ ≈ 0.489938
x ≈ 1.138731, y ≈ 1.768057, z ≈−0.573138, λ ≈ 0.317141, μ ≈ 1.862675
Substituting these values into f gives f(0.222444, −2.157012, −0.686049) ≈−5.3506,
f(−1.951921, −0.545867, 0.119973) ≈−0.0688, f(0.155142, 0.904622, 0.950293) ≈ 0.4084,
f(1.138731, 1.768057, −0.573138) ≈ 9.7938. Thus the maximum is approximately 9.7938, and the mininum is
approximately −5.3506.
45. (a) We wish to maximize f(x 1,x 2, ..., x n)= n√ x 1x 2 ···x n subject to g(x 1,x 2, ..., x n)=x 1 + x 2 + ···+ x n = c
and x i > 0.
1
∇f = (x n 1x 2 ···x n ) n 1 −1 (x 2 ···x n ) , 1 (x n 1x 2 ···x n ) n 1 −1 (x 1 x 3 ···x n ) , ..., 1 (x n 1x 2 ···x n ) n 1 −1 (x 1 ···x n−1 )
and λ∇g = hλ, λ, ..., λi, so we need to solve the system of equations
1
(x n 1x 2 ···x n ) n 1 −1 (x 2 ···x n )=λ ⇒ x 1/n
1 x 1/n
2 ···x 1/n
n = nλx 1
1
n (x1x2 ···xn) n 1 −1 (x 1x 3 ···x n)=λ ⇒ x 1/n
1 x 1/n
2 ···x 1/n
n = nλx 2
1
n (x1x2 ···xn) n 1 −1 (x 1 ···x n−1) =λ ⇒ x 1/n
1 x 1/n
2 ···x 1/n
n = nλx n
.

F.
TX.10
CHAPTER 15 REVIEW ET CHAPTER 14 ¤ 211
This implies nλx 1 = nλx 2 = ···= nλx n.Noteλ 6= 0, otherwise we can’t have all x i > 0. Thusx 1 = x 2 = ···= x n.
But x 1 + x 2 + ···+ x n = c ⇒ nx 1 = c ⇒ x 1 = c n = x 2 = x 3 = ···= x n . Then the only point where f can
c
have an extreme value is
n , c n n
, ..., c . Since we can choose values for (x 1,x 2,...,x n) that make f as close to
zero (but not equal) as we like, f has no minimum value. Thus the maximum value is
c
f
n , c n n
, ..., c c
= n n · c
n ····· c
n = c n .
(b) From part (a), c n is the maximum value of f. Thus f(x1,x2, ..., xn) = n√ x 1x 2 ···x n ≤ c n .But
x 1 + x 2 + ···+ x n = c,so n√ x1 + x2 + ···+ xn
x 1x 2 ···x n ≤ . These two means are equal when f attains its
n
maximum value c c
n , but this can occur only at the point n , c n , ..., c
we found in part (a). So the means are equal only
n
when x 1 = x 2 = x 3 = ···= x n = c n .
15 Review ET 14
1. (a) A function f of two variables is a rule that assigns to each ordered pair (x, y) of real numbers in its domain a unique real
number denoted by f(x, y).
(b) One way to visualize a function of two variables is by graphing it, resulting in the surface z = f(x, y). Another method for
visualizing a function of two variables is a contour map. The contour map consists of level curves of the function which are
horizontal traces of the graph of the function projected onto the xy-plane. Also, we can use an arrow diagram such as
Figure1inSection15.1[ET14.1].
2. Afunctionf of three variables is a rule that assigns to each ordered triple (x, y, z) in its domain a unique real number
f(x, y, z). We can visualize a function of three variables by examining its level surfaces f(x, y, z) =k,wherek is a constant.
3. lim f(x, y) =L means the values of f(x, y) approach the number L as the point (x, y) approaches the point (a, b)
(x,y)→(a,b)
along any path that is within the domain of f. We can show that a limit at a point does not exist by finding two different paths
approaching the point along which f(x, y) has different limits.
4. (a) See Definition 15.2.4 [ET 14.2.4].
(b) If f is continuous on R 2 , its graph will appear as a surface without holes or breaks.
5. (a) See (2) and (3) in Section 15.3 [ET 14.3].
(b) See “Interpretations of Partial Derivatives” on page 917 [ET 881].
(c) To find f x ,regardy as a constant and differentiate f(x, y) with respect to x. Tofind f y ,regardx as a constant and
differentiate f(x, y) with respect to y.
6. See the statement of Clairaut’s Theorem on page 921 [ET 885].

F.
212 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
7. (a)See(2)inSection15.4[ET14.4]
(b) See (19) and the preceding discussion in Section 15.6 [ET 14.6].
8. See (3) and (4) and the accompanying discussion in Section 15.4 [ET 14.4]. We can interpret the linearization of f at (a, b)
geometrically as the linear function whosegraphisthetangentplanetothegraphoff at (a, b). Thus it is the linear function
which best approximates f near (a, b).
9. (a) See Definition 15.4.7 [ET 14.4.7].
(b) Use Theorem 15.4.8 [ET 14.4.8].
10. See (10) and the associated discussion in Section 15.4 [ET 14.4].
11. See (2) and (3) in Section 15.5 [ET 14.5].
12. See (7) and the preceding discussion in Section 15.5 [ET 14.5].
13. (a) See Definition 15.6.2 [ET 14.6.2]. We can interpret it as the rate of change of f at (x 0 ,y 0 ) in the direction of u.
Geometrically, if P is the point (x 0,y 0,f(x 0,y 0)) on the graph of f and C is the curve of intersection of the graph of f
with the vertical plane that passes through P in the direction u, the directional derivative of f at (x 0,y 0) in the direction of
u is the slope of the tangent line to C at P . (See Figure 5 in Section 15.6 [ET 14.6].)
(b) See Theorem 15.6.3 [ET 14.6.3].
14. (a) See (8) and (13) in Section 15.6 [ET 14.6].
(b) D u f(x, y) =∇f(x, y) · u or D u f(x, y, z) =∇f(x, y, z) · u
(c) The gradient vector of a function points in the direction of maximum rate of increase of the function. On a graph of the
function, the gradient points in the direction of steepest ascent.
15. (a) f has a local maximum at (a, b) if f(x, y) ≤ f(a, b) when (x, y) is near (a, b).
(b) f has an absolute maximum at (a, b) if f(x, y) ≤ f(a, b) for all points (x, y) in the domain of f.
(c) f has a local minimum at (a, b) if f(x, y) ≥ f(a, b) when (x, y) is near (a, b).
(d) f has an absolute minimum at (a, b) if f(x, y) ≥ f(a, b) for all points (x, y) in the domain of f.
(e) f has a saddle point at (a, b) if f(a, b) is a local maximum in one direction but a local minimum in another.
16. (a) By Theorem 15.7.2 [ET 14.7.2], if f has a local maximum at (a, b) and the first-order partial derivatives of f exist there,
then f x (a, b) =0and f y (a, b) =0.
(b) A critical point of f is a point (a, b) such that f x (a, b) =0and f y (a, b) =0or one of these partial derivatives does not
exist.
17. See (3) in Section 15.7 [ET 14.7]
18. (a) See Figure 11 and the accompanying discussion in Section 15.7 [ET 14.7].
(b) See Theorem 15.7.8 [ ET 14.7.8].
(c) See the procedure outlined in (9) in Section 15.7 [ET 14.7].
19. See the discussion beginning on page 970 [ET 934]; see “Two Constraints” on page 974 [ET 938].

F.
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CHAPTER 15 REVIEW ET CHAPTER 14 ¤ 213
f(a, b + h) − f(a, b)
1. True. f y(a, b) = lim
from Equation 15.3.3 [ET 14.3.3]. Let h = y − b. Ash → 0, y → b. Thenby
h→0 h
f(a, y) − f(a, b)
substituting, we get f y(a, b) =lim
.
y→b y − b
3. False. f xy = ∂2 f
∂y ∂x .
5. False. See Example 15.2.3 [ET 14.2.3].
7. True. If f has a local minimum and f is differentiable at (a, b) then by Theorem 15.7.2 [ET 14.7.2], f x (a, b) =0and
f y (a, b) =0,so∇f(a, b) =hf x (a, b),f y (a, b)i = h0, 0i = 0.
9. False. ∇f(x, y) =h0, 1/yi.
11. True. ∇f = hcos x, cos yi, so|∇f| = cos 2 x +cos 2 y.But|cos θ| ≤ 1, so|∇f| ≤ √ 2.Now
D u f (x, y) =∇f · u = |∇f||u| cos θ,butu is a unit vector, so |D u f(x, y)| ≤ √ 2 · 1 · 1= √ 2.
1. ln(x + y +1)is defined only when x + y +1> 0 ⇒ y>−x − 1,
so the domain of f is {(x, y) | y>−x − 1}, all those points above the
line y = −x − 1.
3. z = f(x, y) =1− y 2 , a parabolic cylinder 5. The level curves are 4x 2 + y 2 = k or 4x 2 + y 2 = k 2 ,
k ≥ 0, a family of ellipses.

F.
214 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
7.
9. f is a rational function, so it is continuous on its domain. Since f is defined at (1, 1), we use direct substitution to evaluate the
limit:
lim
(x,y)→(1,1)
2xy
x 2 +2y 2 = 2(1)(1)
1 2 +2(1) 2 = 2 3 .
T (6 + h, 4) − T (6, 4)
11. (a) T x (6, 4) = lim
, so we can approximate T x (6, 4) by considering h = ±2 and
h→0 h
using the values given in the table: T x (6, 4) ≈
T x(6, 4) ≈
T (4, 4) − T (6, 4)
−2
=
72 − 80
−2
T (8, 4) − T (6, 4)
2
=
86 − 80
2
=3,
=4. Averaging these values, we estimate T x(6, 4) to be approximately
3.5 ◦ T (6, 4+h) − T (6, 4)
C/m. Similarly, T y (6, 4) = lim
, which we can approximate with h = ±2:
h→0 h
T y (6, 4) ≈
T (6, 6) − T (6, 4)
2
=
75 − 80
2
= −2.5, T y (6, 4) ≈
values, we estimate T y (6, 4) to be approximately −3.0 ◦ C/m.
(b) Here u =
√
1
2
,
T (6, 2) − T (6, 4)
−2
=
87 − 80
−2
= −3.5. Averagingthese
√
1
2
, so by Equation 15.6.9 [ ET 14.6.9], D u T (6, 4) = ∇T (6, 4) · u = T x(6, 4) √ 1
2
+ T y(6, 4) √ 1 2
.
Using our estimates from part (a), we have D u T (6, 4) ≈ (3.5) √ 1
2
+(−3.0) √ 1
2
= 1
2 √ ≈ 0.35. This means that as we
2
move through the point (6, 4) in the direction of u, the temperature increases at a rate of approximately 0.35 ◦ C/m.
T 6+h √ 1
2
, 4+h √ 1
2
− T (6, 4)
Alternatively, we can use Definition 15.6.2 [ ET 14.6.2]: D u T (6, 4) = lim
,
h→0 h
which we can estimate with h = ±2 √ 2.ThenD u T (6, 4) ≈
D u T (6, 4) ≈
(c) T xy(x, y) = ∂ ∂y
T (4, 2) − T (6, 4)
−2 √ 2
=
[Tx(x, y)] = lim
h→0
T (8, 6) − T (6, 4)
2 √ 2
=
80 − 80
2 √ 2
=0,
74 − 80
−2 √ 2 = 3 √
2
. Averaging these values, we have D u T (6, 4) ≈ 3
2 √ 2 ≈ 1.1◦ C/m.
T x(x, y + h) − T x(x, y)
T x(6, 4+h) − T x(6, 4)
,soT xy(6, 4) = lim
which we can
h
h→0 h
estimate with h = ±2. WehaveT x(6, 4) ≈ 3.5 from part (a), but we will also need values for T x(6, 6) and T x(6, 2). Ifwe
use h = ±2 and the values given in the table, we have
T x(6, 6) ≈
T (8, 6) − T (6, 6)
2
=
80 − 75
2
=2.5, T x(6, 6) ≈
Averaging these values, we estimate T x(6, 6) ≈ 3.0. Similarly,
T (4, 6) − T (6, 6)
−2
=
68 − 75
−2
=3.5.

F.
TX.10
CHAPTER 15 REVIEW ET CHAPTER 14 ¤ 215
T x (6, 2) ≈
T (8, 2) − Tx(6, 2)
2
=
90 − 87
2
=1.5, T x (6, 2) ≈
T (4, 2) − T (6, 2)
−2
Averaging these values, we estimate T x (6, 2) ≈ 4.0. Finally, we estimate T xy (6, 4):
T xy(6, 4) ≈ T x(6, 6) − T x (6, 4)
2
=
3.0 − 3.5
2
Averaging these values, we have T xy (6, 4) ≈−0.25.
13. f(x, y) = 2x + y 2 ⇒ f x = 1 2 (2x + y2 ) −1/2 (2) =
=
74 − 87
−2
= −0.25, T xy(6, 4) ≈ T x(6, 2) − T x (6, 4)
−2
=
=6.5.
4.0 − 3.5
−2
1
, 2x + y
2 fy = 1 (2x + 2 y2 ) −1/2 y
(2y) =
2x + y
2
= −0.25.
15. g(u, v) =u tan −1 v ⇒ g u =tan −1 v, g v = u
1+v 2
17. T (p, q, r) =p ln(q + e r ) ⇒ T p =ln(q + e r ), T q = p
q + e , T r r =
per
q + e r
19. f(x, y) =4x 3 − xy 2 ⇒ f x =12x 2 − y 2 , f y = −2xy, f xx =24x, f yy = −2x, f xy = f yx = −2y
21. f(x, y, z) =x k y l z m ⇒ f x = kx k−1 y l z m , f y = lx k y l−1 z m , f z = mx k y l z m−1 , f xx = k(k − 1)x k−2 y l z m ,
f yy = l(l − 1)x k y l−2 z m , f zz = m(m − 1)x k y l z m−2 , f xy = f yx = klx k−1 y l−1 z m , f xz = f zx = kmx k−1 y l z m−1 ,
f yz = f zy = lmx k y l−1 z m−1
23. z = xy + xe y/x ⇒ ∂z
∂x = y − y x ey/x + e y/x ,
x ∂z
∂x + y ∂z
∂y = x
y − y x ey/x + e y/x
+ y
∂z
∂y = x + ey/x and
x + e y/x = xy − ye y/x + xe y/x + xy + ye y/x = xy + xy + xe y/x = xy + z.
25. (a) z x =6x +2 ⇒ z x(1, −2) = 8 and z y = −2y ⇒ z y(1, −2) = 4, so an equation of the tangent plane is
z − 1=8(x − 1) + 4(y +2)or z =8x +4y +1.
(b) A normal vector to the tangent plane (and the surface) at (1, −2, 1) is h8, 4, −1i. Then parametric equations for the normal
line there are x =1+8t, y = −2+4t, z =1− t, and symmetric equations are x − 1
8
= y +2
4
= z − 1
−1 .
27. (a) Let F (x, y, z) =x 2 +2y 2 − 3z 2 .ThenF x =2x, F y =4y, F z = −6z, soF x (2, −1, 1) = 4, F y (2, −1, 1) = −4,
F z (2, −1, 1) = −6. From Equation 15.6.19 [ET 14.6.19], an equation of the tangent plane is
4(x − 2) − 4(y +1)− 6(z − 1) = 0 or, equivalently, 2x − 2y − 3z =3.
(b) From Equations 15.6.20 [ET 14.6.20], symmetric equations for the normal line are x − 2
4
= y +1
−4 = z − 1
−6 .
29. (a) r(u, v) =(u + v) i + u 2 j + v 2 k and the point (3, 4, 1) corresponds to u =2, v =1.Thenr u = i +2u j ⇒
r u(2, 1) = i +4j and r v = i +2v k ⇒ r v(2, 1) = i +2j. A normal vector to the surface at (3, 4, 1) is
r u × r v =8i − 2 j − 4 k, so an equation of the tangent plane there is 8(x − 3) − 2(y − 4) − 4(z − 1) = 0 or equivalently
4x − y − 2z =6.
(b) A direction vector for the normal line through (3, 4, 1) is 8 i − 2 j − 4 k, so a vector equation is
r(t) =(3i +4j + k)+t (8 i − 2 j − 4 k), and the corresponding parametric equations are x =3+8t, y =4− 2t,
z =1− 4t.

F.
216 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
31. The hyperboloid is a level surface of the function F (x, y, z) =x 2 +4y 2 − z 2 , so a normal vector to the surface at (x 0,y 0,z 0)
is ∇F (x 0, y 0 ,z 0 )=h2x 0 , 8y 0 , −2z 0 i. A normal vector for the plane 2x +2y + z =5is h2, 2, 1i. For the planes to be
parallel, we need the normal vectors to be parallel, so h2x 0 , 8y 0 , −2z 0 i = k h2, 2, 1i,orx 0 = k , y 0 = 1 4 k,andz 0 = − 1 2 k.
But x 2 0 +4y 2 0 − z 2 0 =4 ⇒ k 2 + 1 4 k2 − 1 4 k2 =4 ⇒ k 2 =4 ⇒ k = ±2. So there are two such points:
2,
1
2 , −1 and −2, − 1 2 , 1 .
33. f(x, y, z) =x 3 y 2 + z 2 ⇒ f x (x, y, z) =3x 2 y 2 + z 2 , f y (x, y, z) =
yx 3
y2 + z , f zx 3
z(x, y, z) = 2 y2 + z ,
2
so f(2, 3, 4) = 8(5) = 40, f x(2, 3, 4) = 3(4) √ 25 = 60, f y(2, 3, 4) = 3(8)
linear approximation of f at (2, 3, 4) is
√
25
= 24
5
,andfz(2, 3, 4) =
4(8)
f(x, y, z) ≈ f(2, 3, 4) + f x (2, 3, 4)(x − 2) + f y (2, 3, 4)(y − 3) + f z (2, 3, 4)(z − 4)
=40+60(x − 2) + 24 5
Then (1.98) 3 (3.01) 2 +(3.97) 2 = f(1.98, 3.01, 3.97) ≈ 60(1.98) + 24 5
(y − 3) +
32
5 (z − 4) = 60x + 24 5 y + 32
5 z − 120
√
25
= 32
5
32
(3.01) + (3.97) − 120 = 38.656.
5
. Then the
35.
du
dp = ∂u dx
∂x dp + ∂u dy
∂y dp + ∂u dz
∂z dp =2xy3 (1 + 6p)+3x 2 y 2 (pe p + e p )+4z 3 (p cos p +sinp)
37. By the Chain Rule, ∂z
∂s = ∂z ∂x
∂x ∂s + ∂z ∂y
.Whens =1and t =2, x = g(1, 2) = 3 and y = h(1, 2) = 6, so
∂y ∂s
39.
41.
∂z
∂s = f x(3, 6)g s (1, 2) + f y (3, 6) h s (1, 2) = (7)(−1) + (8)(−5) = −47. Similarly, ∂z
∂t = ∂z ∂x
∂x ∂t + ∂z ∂y
∂y ∂t ,so
∂z
∂t = f x(3, 6)g t (1, 2) + f y (3, 6) h t (1, 2) = (7)(4) + (8)(10) = 108.
∂z
∂x =2xf 0 (x 2 − y 2 ),
∂z
∂y =1− 2yf0 (x 2 − y 2 )
y ∂z
∂x + x ∂z
∂y =2xyf 0 (x 2 − y 2 )+x − 2xyf 0 (x 2 − y 2 )=x.
∂z
∂x = ∂z
∂u y + ∂z −y
∂v x and 2
∂ 2 z
∂x = y ∂
2 ∂x
∂z
+ 2y ∂z
∂u x 3 ∂v + −y
x 2
∂
∂x
= 2y ∂z
x 3 ∂v + ∂ 2 z
y2
∂u − 2y2 ∂ 2 z
2 x 2 ∂u∂v + y2 ∂ 2 z
x 4 ∂v 2
Also ∂z
∂y = x ∂z
∂u + 1 x
∂ 2 z
∂y 2 = x ∂
∂y
∂z
∂u
∂z
∂v and
+ 1 ∂
x ∂y
∂z ∂ 2 z
= x
∂v ∂u x + 2
where f 0 df
=
.Then
d(x 2 − y 2 )
∂z
= 2y
∂z ∂ 2
∂v x 3 ∂v + y z
∂u y + 2
1
∂v ∂u x
∂2 z
+ 1 ∂ 2 z 1
x ∂v 2 x +
∂2 z −y
+ −y
∂ 2 z −y
∂v ∂u x 2 x 2 ∂v 2 x + ∂2 z
2 ∂u∂v y
∂2 z
∂u∂v x = x 2 ∂ 2 z
∂u +2 ∂2 z
2 ∂u∂v + 1 ∂ 2 z
x 2 ∂v 2

F.
TX.10
CHAPTER 15 REVIEW ET CHAPTER 14 ¤ 217
Thus
x 2 ∂ 2 z
∂x − ∂ 2 z
2 y2
∂y = 2y ∂z
2 x ∂v + x2 y 2 ∂ 2 z
∂u − 2 2y2
= 2y x
∂z
∂v − 4y2
since y = xv = uv
y or y2 = uv.
43. ∇f =
z 2 √ ye
x √y , xz2 e x√ y
2 √ y , 2zex√ y
∂ 2 z ∂z
=2v
∂u∂v ∂v − 4uv
= ze x√ y
z √ y,
∂ 2 z
∂u∂v + y2
x 2 ∂ 2 z
∂v 2 − x2 y 2 ∂ 2 z
∂u 2 − 2y2
xz
2 √ y , 2
∂2 z
∂u∂v
45. ∇f = h1/ √ x, −2yi, ∇f(1, 5) = h1, −10i, u = 1 5 h3, −4i. ThenD u f(1, 5) = 43
5 .
47. ∇f = 2xy, x 2 +1/ 2 √ y , |∇f(2, 1)| = 4, 9 2
direction 4, 9 2
.
∂ 2 z
∂u∂v − y2 ∂ 2 z
x 2 ∂v 2
. Thus the maximum rate of change of f at (2, 1) is
√
145
2
in the
49. First we draw a line passing through Homestead and the eye of the hurricane. We can approximate the directional derivative at
Homestead in the direction of the eye of the hurricane by the average rate of change of wind speed between the points where
this line intersects the contour lines closest to Homestead. In the direction of the eye of the hurricane, the wind speed changes
from 45 to 50 knots. We estimate the distance between these two points to be approximately 8 miles, so the rate of change of
wind speed in the direction given is approximately
50 − 45
8
= 5 8
=0.625 knot/mi.
51. f(x, y) =x 2 − xy + y 2 +9x − 6y +10 ⇒ f x =2x − y +9,
f y = −x +2y − 6, f xx =2=f yy , f xy = −1. Thenf x =0and f y =0
imply y =1, x = −4. Thus the only critical point is (−4, 1) and
f xx(−4, 1) > 0, D(−4, 1) = 3 > 0,sof(−4, 1) = −11 is a local minimum.
53. f(x, y) =3xy − x 2 y − xy 2 ⇒ f x =3y − 2xy − y 2 ,
f y =3x − x 2 − 2xy, f xx = −2y, f yy = −2x, f xy =3− 2x − 2y. Then
f x =0implies y(3 − 2x − y) =0so y =0or y =3− 2x. Substituting into
f y =0implies x(3 − x) =0or 3x(−1+x) =0. Hence the critical points are
(0, 0), (3, 0), (0, 3) and (1, 1). D(0, 0) = D(3, 0) = D(0, 3) = −9 < 0 so
(0, 0), (3, 0),and(0, 3) are saddle points. D(1, 1) = 3 > 0 and
f xx (1, 1) = −2 < 0,sof(1, 1) = 1 is a local maximum.
55. First solve inside D. Heref x =4y 2 − 2xy 2 − y 3 , f y =8xy − 2x 2 y − 3xy 2 .
Then f x =0implies y =0or y =4− 2x,buty =0isn’t inside D. Substituting
y =4− 2x into f y =0implies x =0, x =2or x =1,butx =0isn’t inside D,
and when x =2, y =0but (2, 0) isn’t inside D. Thus the only critical point inside
D is (1, 2) and f(1, 2) = 4. Secondly we consider the boundary of D.
On L 1 : f(x, 0) = 0 and so f =0on L 1 .OnL 2 : x = −y +6and

F.
218 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
f(−y +6,y)=y 2 (6 − y)(−2) = −2(6y 2 − y 3 ) which has critical pointsat y =0and y =4.Thenf(6, 0) = 0 while
f(2, 4) = −64. OnL 3 : f(0,y)=0,sof =0on L 3 .ThusonD the absolute maximum of f is f(1, 2) = 4 while the
absolute minimum is f(2, 4) = −64.
57. f(x, y) =x 3 − 3x + y 4 − 2y 2
From the graphs, it appears that f has a local maximum f(−1, 0) ≈ 2, local minima f(1, ±1) ≈−3, and saddle points at
(−1, ±1) and (1, 0).
To find the exact quantities, we calculate f x =3x 2 − 3=0 ⇔ x = ±1 and f y =4y 3 − 4y =0 ⇔
y =0, ±1, giving the critical points estimated above. Also f xx =6x, f xy =0, f yy =12y 2 − 4, sousingtheSecond
Derivatives Test, D(−1, 0) = 24 > 0 and f xx(−1, 0) = −6 < 0 indicating a local maximum f(−1, 0) = 2;
D(1, ±1) = 48 > 0 and f xx(1, ±1) = 6 > 0 indicating local minima f(1, ±1) = −3; andD(−1, ±1) = −48 and
D(1, 0) = −24, indicating saddle points.
59. f(x, y) =x 2 y, g(x, y) =x 2 + y 2 =1 ⇒ ∇f = 2xy, x 2 = λ∇g = h2λx, 2λyi. Then2xy =2λx and x 2 =2λy
imply λ = x 2 /(2y) and λ = y if x 6= 0and y 6= 0.Hencex 2 =2y 2 .Thenx 2 + y 2 =1implies 3y 2 =1so y = ± 1 √
3
and
2
x = ± .[Noteifx =0then 3 x2 2
=2λy implies y =0and f (0, 0) = 0.] Thus the possible points are ± , ± √ 1
3 3
and
2
the absolute maxima are f ± , √3
1
= 2
3 3 √ while the absolute minima are f 2
± , − √ 1
3 3 3
= − 2
3 √ . 3
61. f(x, y, z) =xyz, g(x, y, z) =x 2 + y 2 + z 2 =3. ∇f = λ∇g ⇒ hyz, xz, xyi = λh2x, 2y, 2zi. Ifanyofx, y,orz is
zero, then x = y = z =0which contradicts x 2 + y 2 + z 2 =3.Thenλ = yz
2x = xz
2y = xy
2z
⇒ 2y 2 z =2x 2 z ⇒
y 2 = x 2 ,andsimilarly2yz 2 =2x 2 y ⇒ z 2 = x 2 . Substituting into the constraint equation gives x 2 + x 2 + x 2 =3 ⇒
x 2 =1=y 2 = z 2 . Thus the possible points are (1, 1, ±1), (1, −1, ±1), (−1, 1, ±1), (−1, −1, ±1). The absolute maximum
is f(1, 1, 1) = f(1, −1, −1) = f(−1, 1, −1) = f(−1, −1, 1) = 1 and the absolute
minimum is f(1, 1, −1) = f(1, −1, 1) = f(−1, 1, 1) = f(−1, −1, −1) = −1.
63. f(x, y, z) =x 2 + y 2 + z 2 , g(x, y, z) =xy 2 z 3 =2 ⇒ ∇f = h2x, 2y, 2zi = λ∇g = λy 2 z 3 , 2λxyz 3 , 3λxy 2 z 2 .
Since xy 2 z 3 =2, x 6= 0, y 6= 0and z 6= 0,so2x = λy 2 z 3 (1), 1=λxz 3 (2), 2=3λxy 2 z (3). Then(2) and (3) imply
1
xz = 2
3 3xy 2 z or y2 = 2 3 z2 2
2x
so y = ±z . Similarly (1) and (3) imply
3
y 2 z = 2
3 3xy 2 z or 3x2 = z 2 so x = ± √ 1
3
z.But

F.
TX.10
CHAPTER 15 REVIEW ET CHAPTER 14 ¤ 219
xy 2 z 3 =2so x and z must have the same sign, that is, x = 1 √
3
z.Thusg(x, y, z) =2implies 1 √
3
z 2
3 z2 z 3 =2or
z = ±3 1/4 and the possible points are (±3 −1/4 , 3 −1/4√ 2, ±3 1/4 ), (±3 −1/4 , −3 −1/4√ 2, ±3 1/4 ). However at each of these
points f takes on the same value, 2 √ 3.But(2, 1, 1) also satisfies g(x, y, z) =2and f(2, 1, 1) = 6 > 2 √ 3. Thus f has an
absolute minimum value of 2 √ 3 and no absolute maximum subject to the constraint xy 2 z 3 =2.
Alternate solution: g(x, y, z) =xy 2 z 3 =2implies y 2 = 2
xz 3 , so minimize f(x, z) =x2 + 2
xz 3 + z2 .Then
f x =2x − 2
x 2 z 3 , f z = − 6
xz 4 +2z, f xx =2+ 4
x 3 z 3 , f zz = 24
xz 5 +2and f xz = 6
x 2 z 4 .Nowf x =0implies
2x 3 z 3 − 2=0or z =1/x. Substituting into f y =0implies −6x 3 +2x −1 =0or x =
± √ 1 , ± 4√
3 .ThenD ± 1 4 4
3 √ , ± 4√
3
3
=(2+4) 2+ 24
3
1 4 √ 3 ,sothetwocriticalpointsare
2
− √
6
3
> 0 and fxx
± √ 1 , ± 4√
3 =6> 0, so each point
4
3
is a minimum. Finally, y 2 = 2
xz , so the four points closest to the origin are ± 1 3 √ , √ 2
4
3
4√ , ± 4√
3 , ± √ 1 , − √ 2
3
4
3
4√ , ± 4√
3 .
3
65. The area of the triangle is 1 2
ca sin θ and the area of the rectangle is bc. Thus, the
area of the whole object is f(a, b, c) = 1 ca sin θ + bc. The perimeter of the object
2
is g(a, b, c) =2a +2b + c = P . To simplify sin θ in terms of a, b,andc notice
that a 2 sin 2 θ + 1
2 c2 = a 2 ⇒ sin θ = 1 √
4a2 − c
2a
2 .Thus
f(a, b, c) = c 4
√
4a2 − c 2 + bc. (Instead of using θ, we could just have used the
Pythagorean Theorem.) As a result, by Lagrange’s method, we must find a, b, c,andλ by solving ∇f = λ∇g which gives the
following equations: ca(4a 2 − c 2 ) −1/2 =2λ (1), c =2λ (2),
1
4 (4a2 − c 2 ) 1/2 − 1 4 c2 (4a 2 − c 2 ) −1/2 + b = λ (3),and
2a +2b + c = P (4). From(2), λ = 1 2 c and so (1) produces ca(4a2 − c 2 ) −1/2 = c ⇒ (4a 2 − c 2 ) 1/2 = a ⇒
4a 2 − c 2 = a 2 ⇒ c = √ 3 a (5). Similarly, since 4a 2 − c 21/2 = a and λ = 1 c, (3) gives a 2
4 − c2
4a + b = c , so from
2
(5), a 4 − 3a 4 + b = √
3 a
2
⇒
− a 2 − √
3 a
2
= −b ⇒ b = a 2
1+
√
3
(6). Substituting (5) and (6) into (4) we get:
2a + a 1+ √ 3 + √ 3 a = P ⇒ 3a +2 √ P
3 a = P ⇒ a =
3+2 √ 3 = 2 √ 3 − 3
P and thus
3
√ √
2 3 − 3 1+ 3
b =
P = 3 − √ 3
P and c = 2 − √ 3 P .
6
6

F.
TX.10

F.
TX.10
PROBLEMS PLUS
1. The areas of the smaller rectangles are A 1 = xy, A 2 =(L − x)y,
A 3 =(L − x)(W − y), A 4 = x(W − y). For0 ≤ x ≤ L, 0 ≤ y ≤ W ,let
f(x, y) =A 2 1 + A 2 2 + A 2 3 + A 2 4
= x 2 y 2 +(L − x) 2 y 2 +(L − x) 2 (W − y) 2 + x 2 (W − y) 2
=[x 2 +(L − x) 2 ][y 2 +(W − y) 2 ]
Then we need to find the maximum and minimum values of f(x, y). Here
f x (x, y) =[2x − 2(L − x)][y 2 +(W − y) 2 ]=0 ⇒ 4x − 2L =0or x = 1 2 L,and
f y (x, y) =[x 2 +(L − x) 2 ][2y − 2(W − y)] = 0 ⇒ 4y − 2W =0or y = W/2. Also
f xx =4[y 2 +(W − y) 2 ], f yy =4[x 2 +(L − x) 2 ],andf xy =(4x − 2L)(4y − 2W ). Then
D =16[y 2 +(W − y) 2 ][x 2 +(L − x) 2 ] − (4x − 2L) 2 (4y − 2W ) 2 .Thuswhenx = 1 L and y = 1 W , D>0 and
2 2
f xx =2W 2 > 0. Thus a minimum of f occurs at 1
L, 1 W 2 2
and this minimum value is f 1
L, 1 W 2 2
= 1 4 L2 W 2 .
There are no other critical points, so the maximum must occur on the boundary. Now along the width of the rectangle let
g(y) =f(0,y)=f(L, y) =L 2 [y 2 +(W − y) 2 ], 0 ≤ y ≤ W .Theng 0 (y) =L 2 [2y − 2(W − y)] = 0 ⇔ y = 1 W . 2
And g
1
2 =
1
2 L2 W 2 . Checking the endpoints, we get g(0) = g(W )=L 2 W 2 . Along the length of the rectangle let
h(x) =f(x, 0) = f(x, W )=W 2 [x 2 +(L − x) 2 ], 0 ≤ x ≤ L. Bysymmetryh 0 (x) =0 ⇔ x = 1 L and 2
h 1
L = 1 2 2 L2 W 2 . At the endpoints we have h(0) = h(L) =L 2 W 2 . Therefore L 2 W 2 is the maximum value of f.
This maximum value of f occurs when the “cutting” lines correspond to sides of the rectangle.
3. (a) The area of a trapezoid is 1 2 h(b 1 + b 2 ),whereh is the height (the distance between the two parallel sides) and b 1 , b 2 are
the lengths of the bases (the parallel sides). From the figure in the text, we see that h = x sin θ, b 1 = w − 2x, and
b 2 = w − 2x +2x cos θ. Therefore the cross-sectional area of the rain gutter is
A(x, θ) = 1 2
x sin θ [(w − 2x)+(w − 2x +2x cos θ)] = (x sin θ)(w − 2x + x cos θ)
= wx sin θ − 2x 2 sin θ + x 2 sin θ cos θ, 0

F.
222 ¤ PROBLEMS PLUS
TX.10
Since x>0,wemusthavex = 1 3 w,inwhichcasecos θ = 1 2 ,soθ = π 3 , sin θ = √ 3
2 , k = √ 3
6 w, b1 = 1 3 w, b2 = 2 3 w,
and A = √ 3
12 w2 . As in Example 15.7.6 [ET 14.7.6], we can argue from the physical nature of this problem that we have
found a local maximum of A. Now checking the boundary of A,let
g(θ) =A(w/2,θ)= 1 2 w2 sin θ − 1 2 w2 sin θ + 1 4 w2 sin θ cos θ = 1 8 w2 sin 2θ, 0

F.
TX.10
16 MULTIPLE INTEGRALS ET 15
16.1 Double Integrals over Rectangles ET 15.1
1. (a) The subrectangles are shown in the figure.
The surface is the graph of f(x, y) =xy and ∆A =4, so we estimate
V ≈
3 2
f(x i ,y j ) ∆A
(b) V ≈
i =1 j =1
= f(2, 2) ∆A + f(2, 4) ∆A + f(4, 2) ∆A + f(4, 4) ∆A + f(6, 2) ∆A + f(6, 4) ∆A
= 4(4) + 8(4) + 8(4) + 16(4) + 12(4) + 24(4) = 288
3
i =1 j =1
2
f
x i, y j ∆A = f(1, 1) ∆A + f(1, 3) ∆A + f(3, 1) ∆A + f(3, 3) ∆A + f(5, 1) ∆A + f(5, 3) ∆A
=1(4)+3(4)+3(4)+9(4)+5(4)+15(4)=144
3. (a) The subrectangles are shown in the figure. Since ∆A = π 2 /4, we estimate
R sin(x + y) dA ≈ 2 2
f x ∗ ij,yij ∗ ∆A
i=1 j=1
(b) R sin(x + y) dA ≈ 2
= f(0, 0) ∆A + f
0, π 2 ∆A + f π , 0 2
∆A + f π
, π
2 2 ∆A
π
=0
2 π
+1
2 π
+1
2 π
+0
2
= π2 ≈ 4.935
4
4
4
4 2
i=1 j=1
2
f(x i , y j ) ∆A
= f π
, π
4 4 ∆A + f π , 3π 4 4
=1
π 2
4
+0
π 2
4
+0
∆A + f
3π
4 , π 4
π 2
4
+(−1)
π 2
=0
4
∆A + f
3π
4 , 3π 4
∆A
5. (a) Each subrectangle and its midpoint are shown in the figure. The area of each
subrectangle is ∆A =2,soweevaluatef at each midpoint and estimate
R f(x, y) dA ≈ 2 2
f
x i , y j ∆A
i =1j =1
= f(1.5, 1) ∆A + f(1.5, 3) ∆A
+ f(2.5, 1) ∆A + f(2.5, 3) ∆A
=1(2)+(−8)(2) + 5(2) + (−1)(2) = −6
223

F.
224 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
(b) The subrectangles are shown in the figure. In each subrectangle, the sample point
farthest from the origin is the upper right corner, and the area of each subrectangle
is ∆A = 1 2 .Thusweestimate
R f(x, y) dA ≈ 4
i =1j =1
4
f(x i,y j) ∆A
= f(1.5, 1) ∆A + f(1.5, 2) ∆A + f(1.5, 3) ∆A + f(1.5, 4) ∆A
+ f(2, 1) ∆A + f(2, 2) ∆A + f(2, 3) ∆A + f(2, 4) ∆A
+ f(2.5, 1) ∆A + f(2.5, 2) ∆A + f(2.5, 3) ∆A + f(2.5, 4) ∆A
+ f(3, 1) ∆A + f(3, 2) ∆A + f(3, 3) ∆A + f(3, 4) ∆A
=1
1
2 +(−4) 1
2 +(−8) 1
2 +(−6) 1
2 +3 1
2 +0 1
2 +(−5) 1
2 +(−8) 1
2
+5
1
2 +3 1
2 +(−1) 1
2 +(−4) 1
2 +8 1
2 +6 1
2 +3 1
2 +0 1
2
= −3.5
7. The values of f(x, y) = 52 − x 2 − y 2 get smaller as we move farther from the origin, so on any of the subrectangles in the
problem, the function will have its largest value at the lower left corner of the subrectangle and its smallest value at the upper
right corner, and any other value will lie between these two. So using these subrectangles we have U

F.
TX.10
SECTION 16.2 ITERATED INTEGRALS ET SECTION 15.2 ¤ 225
15. To calculate the estimates using a programmable calculator, we can use an algorithm
similartothatofExercise5.1.7[ET 5.1.7]. In Maple, we can define the function
f(x, y) = √ 1+xe −y (calling it f), load the student package, and then use the
command
middlesum(middlesum(f,x=0..1,m),
y=0..1,m);
to get the estimate with n = m 2 squares of equal size. Mathematica has no special
Riemann sum command, but we can define f andthenusenestedSum commands to
calculate the estimates.
n estimate
1 1.141606
4 1.143191
16 1.143535
64 1.143617
256 1.143637
1024 1.143642
17. If we divide R into mn subrectangles, R kdA≈ m
But f m
x ∗ ij,yij
∗ = k always and
i =1
points,
m
n
i =1 j =1
R kdA=
lim
i =1 j =1
n
f
x ∗ ij,yij ∗ ∆A for any choice of sample points x
∗
ij ,yij ∗ .
n
∆A = area of R =(b − a)(d − c). Thus, no matter how we choose the sample
j =1
f m
x ∗ ij,yij
∗ ∆A = k
i =1
m
n
m,n→∞ i =1 j =1
n
∆A = k(b − a)(d − c) and so
j =1
f x ∗ ij,yij ∗ ∆A = lim k m
m,n→∞ i =1
n
∆A =
j =1
lim
m,n→∞
k(b − a)(d − c) =k(b − a)(d − c).
16.2 Iterated Integrals ET 15.2
1.
3.
5.
7.
9.
x=5
5
0 12x2 y 3 dx =
12 x3
3 y3 =4x 3 y 3 x=5
x=0 =4(5)3 y 3 − 4(0) 3 y 3 = 500y 3 ,
x=0
1
0 12x2 y 3 dy =
12x 2 y 4 y=1
=3x 2 y 4 y=1
4
y=0 =3x2 (1) 4 − 3x 2 (0) 4 =3x 2
3
1
2
0
2
0
y=0
1 (1 + 4xy) dx dy = 3
0 1 x +2x 2 y x=1
dy = 3
(1 + 2y) dy = y + y 2 3
=(3+9)− (1 + 1) = 10
x=0 1 1
π/2
0
x sin ydydx= 2
0 xdx π/2
0
sin ydy [as in Example 5] =
2
1 1 (2x + (2x + y) 9
0 y)8 dx dy =
2 9
4
2
1
1
= 1 18
x
y + y x
dy dx =
0
2
0
x=1
x=0
[(2 + y) 9 − (0 + y) 9 ] dy = 1 18
x
2
2
2 π/2
− cos y =(2− 0)(0 + 1) = 2
0
0
dy [substitute u =2x + y ⇒ dx = 1 2 du]
(2 + y)
10
= 1
180 [(410 − 2 10 ) − (2 10 − 0 10 )] = 1,046,528
180
= 261,632
45
4
1
x ln |y| + 1 x · 1 y=2
2 y2 dx =
y=1
4
1
10
2
− y10
10
0
=8ln2+ 3 2 ln 4 − 1 2 ln 2 = 15 2 ln 2 + 3 ln 41/2 = 21
2 ln 2
x ln 2 + 3
dx = 1
2
2x
x2 ln 2 + 3 ln |x| 4
2 1

F.
226 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
11.
13.
1
0
2
0
1 (u − 0 v)5 du dv = 1
1 (u − v)6 u=1
dv = 1 1
0 6 u=0 6 0 (1 − v) 6 − (0 − v) 6 dv
= 1 1
6 0 (1 − v) 6 − v 6
dv = 1 6 −
1
(1 − 7 v)7 − 1 v7 1
7 0
= − 1 [(0 + 1) − (1 + 0)] = 0
42
π r 0 sin2 θdθdr = 2
rdr π
0 0 sin2 θdθ [as in Example 5] = 2
rdr π
0 0
= 1
2 r2 2
0· 1
2
θ −
1
2 sin 2θ π
0
=2· 1
2
[(π − 0) − (0 − 0)] = π
1
(1 − cos 2θ) dθ
2
1
=(2− 0) ·
2 π −
1
sin 2π 2
− 0 − 1 2
sin 0
15.
17.
19.
21.
R (6x2 y 3 − 5y 4 ) dA = 3
0
R
xy 2 1
x 2 +1 dA =
= 1 2
0
1
0 (6x2 y 3 − 5y 4 ) dy dx = 3
3
0 2 x2 y 4 − y 5 y=1
dx = 3
3
y=0 0 2 x2 − 1 dx
= 1
2 x3 − x 3
0 = 27 2 − 3= 21 2
3
−3
π/6
π/3
x sin(x + y) dy dx
0 0
= π/6
0
xy 2 1
x 2 +1 dy dx =
1
(ln 2 − ln 1) · (27 + 27) = 9 ln 2
3
−x cos(x + y)
y = π/3
= x sin x − sin x + π 3
y =0
π/6
− π/6
0 0
= π 6
1
2 − 1 − − cos x +cos x + π 3
R xyex2y dA= 2
0
1
0 xyex2y dx dy = 2
0
0
x
3
x 2 +1 dx
−3
dx = π/6
0 x cos x − x cos x +
π
3 dx
sin x − sin x +
π
π/6
= 1 2 [(e2 − 2) − (1 − 0)] = 1 2 (e2 − 3)
0
1 3 1 1
y 2 dy =
2 ln(x2 +1)
0
3 y3 −3
3
dx [by integrating by parts separately for each term]
= − π − − √ 3
+0− −1+ 1 = √ 3−1
− π 12 2 2
2 12
x=1
1
2 ex2 y
dy = 1 2
2 0 (ey − 1) dy = 1 2 e y − y 2
x=0
0
23. z = f(x, y) =4− x − 2y ≥ 0 for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. Sothesolid
is the region in the first octant which lies below the plane z =4− x − 2y
and above [0, 1] × [0, 1].
25. V = (12 − 3x − 2y) dA = 3
1 (12 − 3x − 2y) dx dy = 3
R −2 0 −2 12x −
3
2 x2 − 2xy x=1
= 3
21 − 2y dy = 21
y − y2 3
= 95
−2 2 2 −2 2
27. V = 2
1
−2 −1 1 −
1
4 x2 − 1 y2 dx dy =4 2 1
9 0 0 1 −
1
4 x2 − 1 y2 dx dy
9
=4 2
0
x −
1
12 x3 − 1 9 y2 x x =1
dy =4 2
x =0 0
x=0 dy
11
− 1 y2 dy =4 11
y − 1 y3 2 83
=4· = 166
12 9 12 27 0 54 27

F.
SECTION 16.3 TX.10 DOUBLE INTEGRALS OVER GENERAL REGIONS ET SECTION 15.3 ¤ 227
29. Here we need the volume of the solid lying under the surface z = x sec 2 y and above the rectangle R =[0, 2] × [0,π/4] in the
xy-plane.
V = 2 π/4
x sec 2 ydydx= 2
xdx π/4
sec 2 ydy= 1
x2 2 π/4
0 0 0 0 2 0 tan y
0
=(2− 0)(tan π − tan 0) = 2(1 − 0) = 2
4
31. The solid lies below the surface z =2+x 2 +(y − 2) 2 and above the plane z =1for −1 ≤ x ≤ 1, 0 ≤ y ≤ 4. Thevolume
of the solid is the difference in volumes between the solid that lies under z =2+x 2 +(y − 2) 2 over the rectangle
R =[−1, 1] × [0, 4] andthesolidthatliesunderz =1over R.
V = 4 1 [2 + 0 −1 x2 +(y − 2) 2 ] dx dy − 4
0
= 4
0
= 4
0
1
−1 (1) dx dy = 4
0
(2 +
1
3 +(y − 2)2 ) − (−2 − 1 3 − (y − 2)2 ) dy − [x] 1 −1 [y]4 0
14
3 +2(y − 2)2 dy − [1 − (−1)][4 − 0] = 14
3 y + 2 3 (y − 2)3 4
= 56
3 + 16 3
− 0 −
16
3 − 8=
88
64
− 8=
3 3
33. In Maple, we can calculate the integral by defining the integrand as f
and then using the command int(int(f,x=0..1),y=0..1);.
In Mathematica, we can use the command
Integrate[f,{x,0,1},{y,0,1}]
We find that R x5 y 3 e xy dA =21e − 57 ≈ 0.0839. Wecanuseplot3d
(in Maple) or Plot3D (in Mathematica) to graph the function.
35. R is the rectangle [−1, 1] × [0, 5]. Thus, A(R) =2· 5=10and
2x +
1
3 x3 + x(y − 2) 2 x =1
x = −1 dy − 1
−1 dx 4
0 dy
0 − (2)(4)
f ave = 1
A(R)
f(x, y) dA = 1 5
1
R 10 0 −1 x2 ydxdy= 1 5
1
10 0 3 x3 y x =1
dy = 1 5
x = −1 10 0
2
ydy= 1 1 y2 5
= 5 .
3 10 3 0 6
37. Let f(x, y) = x − y
(x + y) .ThenaCASgives 1 1 f(x, y) dy dx = 1 and 1 1 f(x, y) dx dy = − 1 .
3 0 0 2 0 0 2
To explain the seeming violation of Fubini’s Theorem, note that f has an infinite discontinuity at (0, 0) and thus does not
satisfy the conditions of Fubini’s Theorem. In fact, both iterated integrals involve improper integrals which diverge at their
lower limits of integration.
16.3 Double Integrals over General Regions ET 15.3
1.
3.
5.
4
0
1
0
√ y
xy 2 dx dy = 4
1
0 0 2 x2 y 2 x= √ y
dy = 4 1
x=0 0 2 y2 [(
y ) 2 − 0 2 ]dy = 1 4
2 0 y3 dy = 1 1 y4 4
= 1
2 4 0 2
(64 − 0) = 32
x
(1 + 2y)dy dx = 1
x 2 0 y + y
2 y=x
dx = 1
y=x 2 0 x + x 2 − x 2 − (x 2 ) 2 dx
π/2
cos θ
e sin θ dr dθ = π/2
0 0 0
= 1
0 (x − x4 )dx = 1
2 x2 − 1 5 x5 1
0 = 1 2 − 1 5 − 0+0= 3 10
re
sin θ r=cos θ
r=0
dθ = π/2
(cos θ) e sin θ dθ = e sin θ π/2
= e sin(π/2) − e 0 = e − 1
0 0

F.
228 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
7.
9.
11.
D y2 dA = 1
y
−1 −y−2 y2 dx dy = 1
−1 xy
2 x=y
dy = 1
x=−y−2 −1 y2 [y − (−y − 2)] dy
= 1
−1 (2y3 +2y 2 )dy = 1
2 y4 + 2 y3 1
= 1 + 2 − 1 + 2 = 4
3 −1 2 3 2 3 3
D xdA= π
sin x
xdydx= π
0 0 0 [xy]y=sin x
y=0
dx =
π
x sin xdx 0
= −x cos x +sinx π
= −π cos π +sinπ +0− sin 0 = π
0
D y2 e xy dA = 4
0
=
y
0 y2 e xy dx dy = 4
0 ye
xy x=y
dy =
4
ye y2 − y dy
x=0 0
4
1
2 ey2 − 1 2 y2 = 1 2 e16 − 8 − 1 +0= 1 2 2 e16 − 17
2
0
integrate by parts
with u = x, dv =sinxdx
13.
1
x
2
0
0
x cos ydydx= 1
0
y = x
2
x sin y dx = 1
x sin y =0 0 x2 dx = − 1 cos x2 1
= 1 (1 − cos 1)
2 0 2
15. 2
1
2y−1
2−y
y 3 dx dy =
2
1
xy 3 x=2y−1
dy =
x=2−y
2
= 2
1 (3y4 − 3y 3 ) dy = 3
5 y5 − 3 4 y4 2
1
= 96 5 − 12 − 3 5 + 3 4 = 147
20
1
[(2y − 1) − (2 − y)] y 3 dy
17. 2
√ 4−x 2
(2x − y) dy dx
−2 −
√4−x 2 √
2
=
2xy − 1 y2 y= 4−x 2
2
√4−x dx
2
−2
y=−
= 2
√
−2 2x 4 − x2 − 1 2 4 − x
2
+2x √
4 − x 2 + 1 2 4 − x
2
dx
= 2
4x √
4 − x
−2 2 dx = − 4 3 4 − x
2 3/2
2
=0
−2
[Or, note that 4x √ 4 − x 2 is an odd function, so 2
−2 4x √ 4 − x 2 dx =0.]
19. V = 1
x
(x +2y) dy dx
0 x 4
= 1
0 xy + y
2 y=x
dx = 1
y=x 4 0 (2x2 − x 5 − x 8 ) dx
= 2
3 x3 − 1 6 x6 − 1 9 x9 1
0 = 2 3 − 1 6 − 1 9 = 7 18
21. V = 2 7 − 3y
xy dx dy = 2
1
1 1 1 2 x2 y x =7− 3y
dy
x =1
= 1 2
2
1 (48y − 42y2 +9y 3 ) dy
= 1 2
24y 2 − 14y 3 + 9 y4 2
= 31
4 1 8

F.
23. V = 2 3 − 3
2
x
(6 − 3x − 2y) dy dx
0 0
SECTION 16.3 TX.10 DOUBLE INTEGRALS OVER GENERAL REGIONS ET SECTION 15.3 ¤ 229
= 2
0
6y − 3xy − y
2 y =3− 3 2 x
dx
= 2
0 6(3 −
3
= 2
0
y =0
x) − 3x(3 − 3 x) − (3 − 3 x)2 dx
2 2 2
9
4 x2 − 9x +9 dx = 3
4 x3 − 9 2 x2 +9x 2
=6− 0=6
0
25.
V = 2
−2
4
x 2 x 2 dy dx
= 2
−2 x2 y y=4
y=x 2 dx = 2
−2 (4x2 − x 4 ) dx
= 4
3 x3 − 1 x5 2
= 32 − 32 + 32 − 32 = 128
5 −2 3 5 3 5 15
27.
1
√ 1 − x 2 1
y
2
V =
ydydx=
0 0
0 2
1
1 − x 2
=
dx = 1
0 2
2 x −
1
x3 1
= 1
3 0 3
y =
√1 − x 2
y =0
dx
29. From the graph, it appears that the two curves intersect at x =0and
at x ≈ 1.213. Thus the desired integral is
D xdA≈ 1.213 3x − x
2
xdydx= 1.213
0 x 4 0
xy
y =3x − x
2
y = x 4
= 1.213
(3x 2 − x 3 − x 5 ) dx = x 3 − 1 0 4 x4 − 1 x6 1.213
6 0
≈ 0.713
31. The two bounding curves y =1− x 2 and y = x 2 − 1 intersect at (±1, 0) with 1 − x 2 ≥ x 2 − 1 on [−1, 1]. Withinthis
region, the plane z =2x +2y +10is above the plane z =2− x − y,so
V = 1
1−x
2
(2x +2y +10)dy dx − 1
−1 x 2 −1 −1
= 1
−1
1−x
2
x 2 −1
1−x
2
x 2 −1
(2x +2y +10− (2 − x − y)) dy dx
(2 − x − y) dy dx
= 1
1−x
2
(3x +3y +8)dy dx =
y=1−x
2
1
3xy + 3 −1 x 2 −1 −1
2 y2 +8y dx
y=x 2 −1
= 1
−1 3x(1 − x 2 )+ 3 (1 − 2 x2 ) 2 +8(1− x 2 ) − 3x(x 2 − 1) − 3 2 (x2 − 1) 2 − 8(x 2 − 1) dx
= 1
−1 (−6x3 − 16x 2 +6x +16)dx = − 3 2 x4 − 16
3 x3 +3x 2 +16x 1
−1
= − 3 2 − 16
3 +3+16+ 3 2 − 16 3 − 3+16= 64 3
dx

F.
230 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
33. The solid lies below the plane z =1− x − y
or x + y + z =1and above the region
D = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x}
in the xy-plane. The solid is a tetrahedron.
35. The two bounding curves y = x 3 − x and y = x 2 + x intersect at the origin and at x =2,withx 2 + x>x 3 − x on (0, 2).
Using a CAS, we find that the volume is
V =
2
x
2 + x
0
x 3 − x
zdydx=
2
x
2 + x
0
x 3 − x
(x 3 y 4 + xy 2 ) dy dx = 13,984,735,616
14,549,535
37. The two surfaces intersect in the circle x 2 + y 2 =1, z =0and the region of integration is the disk D: x 2 + y 2 ≤ 1.
1
√
Using a CAS, the volume is (1 − x 2 − y 2 1−x 2
) dA =
(1 − x
D
−1 −
√1−x 2 − y 2 ) dy dx = π
2 2 .
39. Because the region of integration is
D = {(x, y) | 0 ≤ y ≤ √ x, 0 ≤ x ≤ 4} = (x, y) | y 2 ≤ x ≤ 4, 0 ≤ y ≤ 2
we have 4
√ x
f(x, y) dy dx = f(x, y) dA = 2
4
f(x, y) dx dy.
0 0 D 0 y 2
41. Because the region of integration is
D = (x, y) | − 9 − y 2 ≤ x ≤
9 − y 2 , 0 ≤ y ≤ 3
= (x, y) | 0 ≤ y ≤ √ 9 − x 2 , −3 ≤ x ≤ 3
f(x, y) dx dy =
we have
3
√ 9−y 2
f(x, y) dA
0 −
√9−y 2 D
3
√ 9−x 2
=
f(x, y) dy dx
−3 0
43. Because the region of integration is
D = {(x, y) | 0 ≤ y ≤ ln x, 1 ≤ x ≤ 2} = {(x, y) | e y ≤ x ≤ 2, 0 ≤ y ≤ ln 2}
45.
we have
2
ln x
ln 2
2
f(x, y) dy dx = f(x, y) dA =
1 0
D
0
f(x, y) dx dy
e y
1
3
3
x/3
3 y=x/3
e x2 dx dy = e x2 dy dx = e x2 y dx
y=0
0
3y
0
0
0
=
3
0
x
3
e x2 dx = 1 6 ex2 3
0
= e9 − 1
6

F.
SECTION 16.3 TX.10 DOUBLE INTEGRALS OVER GENERAL REGIONS ET SECTION 15.3 ¤ 231
47. 4
0
2
√ x
1
y 3 +1 dy dx = 2
0
2
=
0
y
2
0
1
dx dy
y 3 +1
1 x=y
2
x
2
dy = y 3 +1
x=0
0
y 2
y 3 +1 dy
= 1 ln y 3 +1 2
= 1 (ln 9 − ln 1) = 1 ln 9
3 3 3
0
49.
1
π/2
0
arcsin y
= π/2
0
cos x 1+cos 2 xdxdy
sin x
0
cos x √ 1+cos 2 xdydx
= π/2
0
cos x √ 1+cos 2 x y y=sin x
y=0
= π/2
0
cos x √ 1+cos 2 x sin xdx
dx
= 0
−u √
1+u
1 2 du = − 1 3 1+u
2 3/2
0
1
√ √
= 1 3 8 − 1 =
1
3 2 2 − 1
Let u =cosx, du = − sin xdx,
dx = du/(− sin x)
51. D = {(x, y) | 0 ≤ x ≤ 1, − x +1≤ y ≤ 1} ∪ {(x, y) | −1 ≤ x ≤ 0, x +1≤ y ≤ 1}
D
∪ {(x, y) | 0 ≤ x ≤ 1, − 1 ≤ y ≤ x − 1} ∪ {(x, y) | −1 ≤ x ≤ 0, − 1 ≤ y ≤−x − 1}, all type I.
x 2 dA =
=4
1
1
0 1 − x
1
1
0
1 − x
0
1
x 2 dy dx + x 2 dy dx +
−1 x +1
1
x − 1
0
−1
0
x 2 dy dx +
−1
−x − 1
x 2 dy dx [by symmetry of the regions and because f(x, y) =x 2 ≥ 0]
−1
x 2 dy dx
=4 1
0 x3 dx =4 1
1
4 0
53. Here Q = (x, y) | x 2 + y 2 ≤ 1 ,x≥ 0,y ≥ 0 ,and0 ≤ (x 2 + y 2 ) 2 ≤
1 2
4 4
⇒ − 1
16 ≤−(x2 + y 2 ) 2 ≤ 0 so
e −1/16 ≤ e −(x2 +y 2 ) 2 ≤ e 0 =1since e t is an increasing function. We have A(Q) = 1 π
1 2
= π , so by Property 11,
4 2 16
e −1/16 A(Q) ≤ Q e−(x2 +y 2 ) 2 dA ≤ 1 · A(Q) ⇒ π 16 e−1/16 ≤ Q e−(x2 +y 2 ) 2 dA ≤ π or we can say
16
0.1844 < Q e−(x2 +y 2 ) 2 dA < 0.1964. (We have rounded the lower bound down and the upper bound up to preserve the
inequalities.)
55. The average value of a function f of two variables defined on a rectangle R was
defined in Section 16.1 [ET 15.1] as f ave = 1 f(x, y)dA. Extending
A(R) R
this definition to general regions D,wehavef ave = 1 f(x, y)dA.
A(D)
Here D = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 3x},soA(D) = 1 (1)(3) = 3 and
2 2
f ave = 1
1 1
3x
f(x, y)dA = xy dy dx
A(D)
D 3/2 0 0
= 2 1
1 xy2 y=3x
dx = 1 1
3 0 2 y=0 3 0 9x3 dx = 3 x4 1
= 3
4 0 4
57. Since m ≤ f(x, y) ≤ M, mdA≤ f(x, y) dA ≤ MdAby (8) ⇒
D D D
m 1 dA ≤ f(x, y) dA ≤ M 1 dA by (7) ⇒ mA(D) ≤ f(x, y) dA ≤ MA(D) by (10).
D D D D
D

F.
232 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15 TX.10
59.
D (x2 tan x + y 3 +4)dA = D x2 tan xdA+ D y3 dA + 4 dA. D Butx2 tan x is an odd function of x and D is
symmetric with respect to the y-axis, so D x2 tan xdA=0. Similarly, y 3 isanoddfunctionofy and D is symmetric with
respect to the x-axis, so D y3 dA =0.Thus
D (x2 tan x + y 3 +4)dA =4 dA =4(areaofD) =4· π√ 2 2
D =8π
61. Since 1 − x 2 − y 2 ≥ 0, we can interpret
D 1 − x2 − y 2 dA as the volume of the solid that lies below the graph of
z = 1 − x 2 − y 2 and above the region D in the xy-plane. z = 1 − x 2 − y 2 is equivalent to x 2 + y 2 + z 2 =1, z ≥ 0
which meets the xy-plane in the circle x 2 + y 2 =1, the boundary of D. Thus, the solid is an upper hemisphere of radius 1
which has volume 1 4 π (1)3 = 2 π.
2 3 3
16.4 Double Integrals in Polar Coordinates ET 15.4
1. The region R is more easily described by polar coordinates: R =
(r, θ) | 0 ≤ r ≤ 4, 0 ≤ θ ≤ 3π 2 .
Thus f(x, y) dA = 3π/2 4
f(r cos θ, r sin θ) rdrdθ.
R 0 0
3. The region R is more easily described by rectangular coordinates: R = (x, y) | −1 ≤ x ≤ 1, 0 ≤ y ≤ 1 2 x + 1 2
.
Thus f(x, y) dA = 1
(x+1)/2
f(x, y) dy dx.
R −1 0
5. The integral 2π
7
π 4
rdrdθrepresents the area of the region
R = {(r, θ) | 4 ≤ r ≤ 7, π ≤ θ ≤ 2π},thelowerhalfofaring.
2π
7
rdrdθ= 2π
7
dθ rdr π 4
π
4
= θ 2π 1 r2 7
= π · 1
33π
(49 − 16) =
π 2 4 2 2
7. The disk D can be described in polar coordinates as D = {(r, θ) | 0 ≤ r ≤ 3, 0 ≤ θ ≤ 2π}. Then
D xy dA = 2π
3
(r cos θ)(r sin θ) rdrdθ= 2π
3
sin θ cos θdθ
0 0 0 0 r3 dr = 1
2 sin2 θ 2π 1 r4 3
=0.
0 4 0
9.
R cos(x2 + y 2 ) dA = π
3
0 0 cos(r2 ) rdrdθ= π
dθ
3 r 0 0 cos(r2 ) dr
= θ π 1
0 2 sin(r2 ) 3
= π · 1 (sin 9 − sin 0) = π sin 9
0 2 2
11.
D e−x2 −y 2 dA = π/2
2
π/2
−π/2 0 e−r2 rdrdθ= dθ 2
−π/2 0 re−r2 dr
= θ π/2
−π/2
− 1 2 e−r2 2
= π
− 1 2 (e −4 − e 0 )= π (1 − 2 e−4 )
0
13. R is the region shown in the figure, and can be described
by R = {(r, θ) | 0 ≤ θ ≤ π/4, 1 ≤ r ≤ 2}. Thus
R arctan(y/x) dA = π/4
2
arctan(tan θ) rdrdθsince y/x =tanθ.
0 1
Also, arctan(tan θ) =θ for 0 ≤ θ ≤ π/4, so the integral becomes
π/4
2 θrdrdθ= π/4
θdθ 2
rdr= 1
θ2 π/4 1 r2 2
= π2 · 3 = 3
0 1 0 1 2 0 2 1 32 2 64 π2 .

F.
SECTION 16.4 TX.10 DOUBLE INTEGRALS IN POLAR COORDINATES ET SECTION 15.4 ¤ 233
15. One loop is given by the region
D = {(r, θ) |−π/6 ≤ θ ≤ π/6, 0 ≤ r ≤ cos 3θ },sotheareais
π/6
cos 3θ
π/6
r=cos 3θ
1
dA =
rdrdθ=
D
−π/6 0
−π/6 2 r2 dθ
π/6
1
=
−π/6 2 cos2 3θdθ=2
θ + 1 π/6
6 sin 6θ
π/6
0
1
2
r=0
1+cos6θ
2
dθ
= 1 2
17. By symmetry,
0
= π 12
A =2 π/4
0
sin θ
rdrdθ=2 π/4
0 0
1 r2 r=sin θ
dθ
2
r=0
= π/4
sin 2 θdθ= π/4 1
(1 − cos 2θ) dθ
0 0 2
= 1 2 θ −
1
sin 2θ π/4
2 0
= 1 π − 1 sin π − 0+ 1 sin 0 = 1 (π − 2)
2 4 2 2 2 8
19. V = x 2 + y 2 ≤4
x2 + y 2 dA = 2π 2
√
r2 rdrdθ= 2π
dθ 2
0 0 0 0 r2 dr = θ 2π 1 r3 2
=2π
8
0 3 0 3 =
16
π 3
21. The hyperboloid of two sheets −x 2 − y 2 + z 2 =1intersects the plane z =2when −x 2 − y 2 +4=1or x 2 + y 2 =3.Sothe
solid region lies above the surface z = 1+x 2 + y 2 and below the plane z =2for x 2 + y 2 ≤ 3, and its volume is
V =
x 2 + y 2 ≤ 3
2 − 1+x 2 + y 2
dA =
2π
√ 3
0
0
(2 − 1+r 2 ) rdrdθ
= 2π
dθ √ 3
(2r − r √ 1+r
0 0 2 )dr = θ
2π
r √ 2 − 1 (1 + 0
3 r2 ) 3/2 3
0
=2π 3 − 8 − 0+ 1
3 3 =
4
π 3
23. By symmetry,
2π
a 2π
V =2
a2 − x 2 − y 2 dA =2 a2 − r rdrdθ=2 2 dθ
0 0
0
x 2 + y 2 ≤ a 2
=2 θ
2π
a
− 1 0 3 (a2 − r 2 ) 3/2 =2(2π) 0+ 1 a3 = 4π 3 3 a3
0
a
0
r a 2 − r 2 dr
25. The cone z = x 2 + y 2 intersects the sphere x 2 + y 2 + z 2 =1when x 2 + y 2 +
x2 + y 2 2
=1or x 2 + y 2 = 1 2 .So
V =
x 2 + y 2 ≤ 1/2
= 2π
dθ 1/ √ 2
0 0
1 − x2 − y 2 − x 2 + y 2
dA =
2π
√
1/ 2
0
0
1 − r2 − r
rdrdθ
√
r 1 − r2 − r 2 dr = θ
√
2π
1/ 2
− 1 (1 − 0 3 r2 ) 3/2 − 1 3 r3 =2π √
− 1 √2
1
− 1 = π 3 3 2 − 2
0

F.
234 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
27. The given solid is the region inside the cylinder x 2 + y 2 =4between the surfaces z = 64 − 4x 2 − 4y 2
and z = − 64 − 4x 2 − 4y 2 .So
V =
64 − 4x2 − 4y 2 −
−
64 − 4x 2 − 4y 2 dA =
2 64 − 4x 2 − 4y 2 dA
x 2 + y 2 ≤ 4
=4 2π
0
x 2 +y 2 ≤ 4
2
√
0 16 − r2 rdrdθ=4 2π
dθ 2
r √ 16 − r
0 0 2 dr =4 θ 2π
0
=8π √
− 1 3 (12 3/2 − 16 2/3 )= 8π 3 64 − 24 3
− 1 3 (16 − r2 ) 3/2 2
0
29.
3
−3
√ 9−x 2
sin(x 2 + y 2 )dy dx =
0
π
3
0
0
sin r 2 rdrdθ
= π
0 dθ 3
0 r sin r 2 dr =[θ] π 0
−
1
2 cos r 2 3
0
= π − 1 2
(cos 9 − 1) =
π
(1 − cos 9)
2
31.
π/4
√ 2
(r cos θ + r sin θ) rdrdθ = π/4
(cos θ +sinθ) dθ √ 2
r 2 dr
0 0 0 0
=[sinθ − cos θ] π/4 1
3 r3√ 2
=
√2
0
2 − √ 2
2 − 0+1
· 1
3
0
2
√
2 − 0
=
2 √ 2
3
33. The surface of the water in the pool is a circular disk D with radius 20 ft. If we place D on coordinate axes with the origin at
35.
the center of D and define f(x, y) to be the depth of the water at (x, y), then the volume of water in the pool is the volume of
the solid that lies above D = (x, y) | x 2 + y 2 ≤ 400 and below the graph of f(x, y). We can associate north with the
positive y-direction, so we are given that the depth is constant in the x-direction and the depth increases linearly in the
y-direction from f(0, −20) = 2 to f(0, 20) = 7. The trace in the yz-plane is a line segment from (0, −20, 2) to (0, 20, 7).
The slope of this line is
7 − 2
= 1 , so an equation of the line is z − 7= 1 (y − 20) ⇒ z = 1 y + 9
20 − (−20) 8 8 8 2
.Sincef(x, y) is
independent of x, f(x, y) = 1 y + 9 . Thus the volume is given by 8 2
f(x, y) dA, which is most conveniently evaluated
D
using polar coordinates. Then D = {(r, θ) | 0 ≤ r ≤ 20, 0 ≤ θ ≤ 2π} and substituting x = r cos θ, y = r sin θ the integral
becomes
2π
0
20
0
1 r sin θ + 9 2π
8 2 rdrdθ= 1
0 24 r3 sin θ + 9 r2 r =20
4 r =0
Thus the pool contains 1800π ≈ 5655 ft 3 of water.
1
1/ √ 2
x
√1 − x 2 xy dy dx +
=
π/4
2
= 15
4
0 1
π/4
0
√ 2 x
1
r 3 cos θ sin θdrdθ=
0
= − 1000 cos θ + 900θ 2π
= 1800π
3 0
2
xy dy dx +
π/4
sin θ cos θdθ= 15 sin 2 θ
4 2
0
√ 4 − x 2
xy dy dx
√
2 0
r
4
r =2
4 cos θ sin θ dθ
π/4
0
= 15
16
r =1
dθ = 2π
1000
sin θ +900 dθ
0 3

F.
SECTION TX.10 16.5 APPLICATIONS OF DOUBLE INTEGRALS ET SECTION 15.5 ¤ 235
37. (a) We integrate by parts with u = x and dv = xe −x2 dx. Thendu = dx and v = − 1 2 e−x2 ,so
∞
0
x 2 e −x2 t
dx = lim
t→∞ 0 x2 e −x2 dx = lim − 1 xe−x2 t
+
t 1
t→∞ 2 0 2 e−x2 dx
0
= lim
− 1 te−t2
+ 1 ∞
t→∞ 2 2
e −x2 dx =0+ 1 ∞
0 2
e −x2 dx [by l’Hospital’s Rule]
0
= 1 ∞
4 −∞ e−x2 dx [since e −x2 is an even function]
= 1 √
4 π [by Exercise 36(c)]
(b) Let u = √ x.Thenu 2 = x ⇒ dx =2udu ⇒
∞ √
0 xe −x t √
dx = lim
t→∞ 0 xe −x √ t
dx = lim ue −u2 2udu=2 ∞
u 2 e −u2 du =2 1
√
t→∞ 0 0 4 π
[by part(a)] = 1 2
√ π.
16.5 Applications of Double Integrals ET 15.5
1. Q = D σ(x, y) dA = 3
1
= 3
1
2 (2xy + 0 y2 ) dy dx = 3
1 xy 2 + 1 y3 y=2
dx
3 y=0
4x +
8
3 dx = 2x 2 + 8 x 3
3 1 =16+16 = 64 C
3 3
3. m = ρ(x, y) dA = 2
1
D 0 −1 xy2 dy dx = 2
xdx 1
0 −1 y2 dy = 1
x2 2 1 y3 1
=2· 2 = 4 ,
2 0 3 −1 3 3
x = 1 xρ(x, y) dA = 3 2
1
m D 4 0 −1 x2 y 2 dy dx = 3 2
4 0 x2 dx 1
−1 y2 dy = 3 1 x3 2 1 y3 1
= 3 · 8 · 2 = 4 ,
4 3 0 3 −1 4 3 3 3
y = 1 yρ(x, y) dA = 3 2
1
m D 4 0 −1 xy3 dy dx = 3 2 xdx 1
4 0 −1 y3 dy = 3 1 x2 2 1 y4 1
= 3 · 2 · 0=0.
4 2 0 4 −1 4
Hence, (x, y) = 4
3 , 0 .
5. m = 2 3−x
(x + y) dy dx = 2
0 x/2 0 xy +
1
y2 y=3−x
dx = 2
2 y=x/2 0 x 3 −
3
x 2
+ 1 (3 − 2 x)2 − 1 x2 dx
8
= 2
0 −
9
8 x2 + 9 2 dx = −
9 1 x3 + 9 x 2
=6,
8 3 2 0
M y = 2 3−x
0 x/2 (x2 + xy) dy dx = 2
0 x 2 y + 1 xy2 y=3−x
dx = 2
9 x − 9 x3 dx = 9 ,
2 y=x/2 0 2 8 2
M x = 2 3−y
(xy + 0 x/2 y2 ) dy dx = 2
1
0 2 xy2 + 1 y3 y=3−x
dx = 2
3 y=x/2 0 9 −
9
x 2
dx =9.
My
Hence m =6, (x, y) =
m , M
x 3
=
m 4 , 3 .
2
7. m = 1
e
x
ydydx= 1
1 y2 y=e x
dx = 1 1
0 0 0 2 y=0 2 0 e2x dx = 1 e2x 1
= 1
4 0 4 (e2 − 1),
M y = 1
e
x
xy dy dx = 1 1
0 0 2 0 xe2x dx = 1 1
2 2 xe2x − 1 e2x 1
= 1
4 0 8 (e2 +1),
M x = 1
e
x
y 2 dy dx = 1
1 y3 y=e x
dx = 1 1
0 0 0 3 y=0 3 0 e3x dx = 1 1 e3x 1
= 1
3 3 0 9 (e3 − 1).
1
Hence m = 1 4 (e2 8
− 1), (x, y) =
(e2 1
+1)
1
4 (e2 − 1) , 9 (e3 − 1)
1
4 (e2 − 1)
e 2 +1
=
2(e 2 − 1) , 4(e3 − 1)
.
9(e 2 − 1)

F.
236 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
9. Note that sin(πx/L) ≥ 0 for 0 ≤ x ≤ L.
m = L
0
sin(πx/L)
ydydx= L 1
0 0 2 sin2 (πx/L) dx = 1 2
M y = L sin(πx/L)
x · ydydx= 1 L
0 0 2 0
= 1 · x 1
x − L sin(2πx/L) L
− 1 L
2 2 4π 0 2 0
= 1 4 L2 − 1 2
L
1
4 x2 + cos(2πx/L) L2
= 1 4π 2 4 L2 − 1 2
0
1 x − L sin(2πx/L) L
= 1 L,
2 4π 0 4
x sin2 (πx/L) dx
1
2 x − L 4π sin(2πx/L) dx
integrate by parts with
u = x, dv =sin 2 (πx/L) dx
1
4 L2 + L2
4π 2 − L2
4π 2
= 1 8 L2 ,
M x = L sin(πx/L)
y · ydydx= L
1
0 0 0 3 sin3 (πx/L) dx = 1 L
3 0 1 − cos 2 (πx/L) sin(πx/L) dx
[substitute u =cos(πx/L)] ⇒ du = − π sin(πx/L)]
L
= 1 3 −
L
π cos(πx/L) −
1
3 cos3 (πx/L) L
= − L
0 3π −1+
1
− 1+ 1
3 3 =
4
L. 9π
Hence m = L L 2
4 , (x, y) = /8
L/4 , 4L/(9π)
=
L/4
L
2 , 16
.
9π
11. ρ(x, y) =ky = kr sin θ, m = π/2 1
0 0 kr2 sin θdrdθ = 1 k π/2
3
sin θdθ = 1 k 0 3
− cos θ π/2
= 1 k,
0 3
M y = π/2 1
0 0 kr3 sin θ cos θdrdθ= 1 k π/2
4
sin θ cos θdθ= 1 k 0 8
− cos 2θ π/2
= 1 k,
0 8
M x = π/2
0
1
0 kr3 sin 2 θdrdθ= 1 k π/2
sin 2 θdθ= 1 k θ +sin2θ π/2
= π k.
4 0 8 0 16
Hence (x, y) = 3
8 , 3π
16
.
13. ρ(x, y) =k x 2 + y 2 = kr,
m = ρ(x, y)dA = π 2
kr · rdrdθ
D 0 1
= k π
0 dθ 2
1 r2 dr = k(π) 1
3 r3 2
1 = 7 3 πk,
M y = xρ(x, y)dA = π 2 (r cos θ)(kr) rdrdθ= k π
cos θdθ 2
D 0 1 0 1 r3 dr
= k sin θ π 1 r4 2
= k(0)
15
0 4 1 4 =0
[this is to be expected as the region and density
function are symmetric about the y-axis]
Hence (x, y) =
M x = yρ(x, y)dA = π 2 (r sin θ)(kr) rdrdθ= k π
sin θdθ 2
D 0 1 0
= k − cos θ π 1 r4 2
= k(1 + 1)
15
0 4 1 4 =
15
k. 2
0, 15k/2
7πk/3
= 0, 45
14π
.
15. Placing the vertex opposite the hypotenuse at (0, 0), ρ(x, y) =k(x 2 + y 2 ).Then
m = a
0
a − x
By symmetry,
k x 2 + y 2 dy dx = k a
0 0
Hence (x, y) = 2
5 a, 2 5 a .
1 r3 dr
ax 2 − x 3 + 1 3 (a − x)3 dx = k 1
3 ax3 − 1 4 x4 − 1 12 (a − x)4 a
0 = 1 6 ka4 .
M y = M x = a a − x
ky(x 2 + y 2 ) dy dx = k a
1 (a − 0 0 0 2 x)2 x 2 + 1 (a − x)4 dx
4
= k 1
6 a2 x 3 − 1 4 ax4 + 1
10 x5 − 1 20 (a − x)5 a
0 = 1 15 ka5

F.
SECTION TX.10 16.5 APPLICATIONS OF DOUBLE INTEGRALS ET SECTION 15.5 ¤ 237
17. I x = D y2 ρ(x, y)dA = 1
e
x
y 2 · ydydx= 1
1 y4 y=e x
dx = 1 1
0 0 0 4 y=0 4 0 e4x dx = 1 1 e4x 1
= 1
4 4 0 16 (e4 − 1),
I y = D x2 ρ(x, y) dA = 1
e
x
x 2 ydydx= 1
x2 1
y2 y=e x
dx = 1 1
0 0 0 2 y=0 2 0 x2 e 2x dx
= 1 1
2 2 x2 − 1 x + 1
2 4 e
2x 1
[integrate by parts twice] = 1 0
8 (e2 − 1),
and I 0 = I x + I y = 1 16 (e4 − 1) + 1 8 (e2 − 1) = 1 16 (e4 +2e 2 − 3).
19. As in Exercise 15, we place the vertex opposite the hypotenuse at (0, 0) and the equal sides along the positive axes.
I x = a a−x
y 2 k(x 2 + y 2 ) dy dx = k a a−x
(x 2 y 2 + y 4 ) dy dx = k a 1
0 0 0 0 0 3 x2 y 3 + 1 y5 y=a−x
5
dx
y=0
= k a 1
0 3 x2 (a − x) 3 + 1 (a − x)5 dx = k
1 1
5 3 3 a3 x 3 − 3 4 a2 x 4 + 3 5 ax5 − 1 x6 − 1 (a − x)6 a
= 7
6 30 0 180 ka6 ,
I y = a a−x
0
= k a
0
x 2 k(x 2 + y 2 ) dy dx = k a a−x
(x 4 + x 2 y 2 ) dy dx = k a 0 0 0 0 x 4 y + 1 3 x2 y 3 y=a−x
dx
y=0
x 4 (a − x)+ 1 3 x2 (a − x) 3 dx = k 1
5 ax5 − 1 6 x6 + 1 3
and I 0 = I x + I y = 7 90 ka6 .
1
3 a3 x 3 − 3 4 a2 x 4 + 3 5 ax5 − 1 6 x6 a
0 = 7
180 ka6 ,
21. Using a CAS, we find m = ρ(x, y) dA = π
sin x
xy dy dx = π2
D 0 0
8 .Then
x = 1
xρ(x, y) dA =
8 π
sin x
x 2 ydydx = 2π m D
π 2 0 0
3 − 1 π and
y = 1
yρ(x, y) dA = 8 π
sin x
xy 2 dy dx = 16
2π
m
π 2 9π ,so(x, y) = 3 − 1 π , 16
.
9π
D
0
0
The moments of inertia are I x = D y2 ρ(x, y) dA = π
sin x
xy 3 dy dx = 3π2
0 0
64 ,
I y = D x2 ρ(x, y) dA = π
sin x
x 3 ydydx= π2
0 0
16 (π2 − 3),andI 0 = I x + I y = π2
64 (4π2 − 9).
23. I x = D y2 ρ(x, y)dA = h b
0 0 ρy2 dx dy = ρ b
dx h
0 0 y2 dy = ρ x b 1 y3 h
= ρb 1
h3 = 1 0 3 0 3 3 ρbh3 ,
I y = D x2 ρ(x, y)dA = h b
0 0 ρx2 dx dy = ρ b
0 x2 dx h
dy = ρ 1
0 3 x3 b
0 [y]h 0 = 1 3 ρb3 h,
and m = ρ (area of rectangle) = ρbh since the lamina is homogeneous. Hence x 2 = Iy m = 1
3 ρb3 h
ρbh = b2 3
⇒ x = b √
3
and y 2 = Ix 1
m = 3 ρbh3
ρbh
= h2
3
⇒ y = h √
3
.
25. In polar coordinates, the region is D = (r, θ) | 0 ≤ r ≤ a, 0 ≤ θ ≤ π 2
,so
I x = D y2 ρdA= π/2 a ρ(r sin 0 0 θ)2 rdrdθ= ρ π/2
sin 2 dθ a
0
= ρ 1
θ − 1 sin 2θ π/2 1 r4 a
= ρ
π 1 a4 = 1 2 4 0 4 0 4 4 16 ρa4 π,
0 r3 dr
I y = D x2 ρdA= π/2 a ρ(r cos 0 0 θ)2 rdrdθ= ρ π/2
cos 2 dθ a
0
= ρ 1
θ + 1 sin 2θ π/2 1 r4 a
= ρ
π 1 a4 = 1 2 4 0 4 0 4 4 16 ρa4 π,
0 r3 dr
and m = ρ · A(D) =ρ · 1
4 πa2 since the lamina is homogeneous. Hence x 2 = y 2 =
1
16 ρa4 π
1
4 ρa2 π = a2
4
⇒ x = y = a 2 .

F.
238 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
27. (a) f(x, y) is a joint density function, so we know R 2 f(x, y) dA =1.Sincef(x, y) =0outside the
rectangle [0, 1] × [0, 2], we can say
Then 2C =1 ⇒ C = 1 2 .
(b) P (X ≤ 1,Y ≤ 1) = 1
−∞
R 2 f(x, y) dA = ∞
−∞
= 1
0
∞ f(x, y) dy dx = 1 2
Cx(1 + y) dy dx
−∞ 0 0
= C 1
0 x y + 1 2 y2 y=2
y=0 dx = C 1
0 4xdx= C 2x 2 1
0 =2C
1 f(x, y) dy dx = 1 1
−∞ 0 0
1
x y + 1 y2 y =1
dx = 1 1
x 3
2 2 y =0 0 2 2
1
x(1 + y) dy dx
2
dx =
3 1 x2 1
= 3 or 0.375
4 2 0 8
(c) P (X + Y ≤ 1) = P ((X, Y ) ∈ D) where D is the triangular region shown in
the figure. Thus
P (X + Y ≤ 1) = f(x, y) dA = 1
D 0
= 1
0
= 1 1
4 0
1 − x
0
1
x(1 + y) dy dx
2
1
x 2
y + 1 y2 y =1−x
2
dx = 1 1
x 1
y =0 0 2
2 x2 − 2x + 3 2
dx
x 3 − 4x 2 +3x
dx = 1 x 4
− 4 x3
4 4 3 +3x2 2
= 5
48 ≈ 0.1042
29. (a) f(x, y) ≥ 0,sof is a joint density function if R 2 f(x, y) dA =1. Here, f(x, y) =0outside the first quadrant, so
R 2 f(x, y) dA = ∞
0
∞
0
0.1e −(0.5x +0.2y) dy dx =0.1 ∞
0
=0.1 lim
t→∞
t
0 e−0.5x dx lim
t→∞
t
0 e−0.2y dy =0.1 lim
t→∞
−2e
−0.5x t
0 lim
t→∞
∞
0
1
0
e −0.5x e −0.2y dy dx =0.1 ∞
e −0.5x dx ∞
e −0.2y dy
0 0
−5e
−0.2y t
0
=0.1 lim
t→∞
−2(e −0.5t − 1) lim
t→∞
−5(e −0.2t − 1) =(0.1) · (−2)(0 − 1) · (−5)(0 − 1) = 1
Thus f(x, y) is a joint density function.
(b) (i) No restriction is placed on X,so
P (Y ≥ 1) = ∞
−∞
∞
1
(ii) P (X ≤ 2,Y ≤ 4) = 2
−∞
f(x, y) dy dx = ∞
0
=0.1 ∞
e −0.5x dx ∞
e −0.2y dy =0.1 lim
0 1
=0.1 lim
t→∞
−2e
−0.5x t
0 lim
t→∞
(c) The expected value of X is given by
∞
1
0.1e −(0.5x+0.2y) dy dx
t
t
t→∞ 0 e−0.5x dx lim
t→∞ 1 e−0.2y dy
−5e
−0.2y t
=0.1 lim
1 t→∞
(0.1) · (−2)(0 − 1) · (−5)(0 − e −0.2 )=e −0.2 ≈ 0.8187
4 f(x, y) dy dx = 2 4
−∞ 0 0 0.1e−(0.5x+0.2y) dy dx
=0.1 2
0 e−0.5x dx 4
0 e−0.2y dy =0.1 −2e −0.5x 2
0
=(0.1) · (−2)(e −1 − 1) · (−5)(e −0.8 − 1)
−2(e −0.5t − 1) lim
t→∞
−5(e −0.2t − e −0.2 )
−5e
−0.2y 4
0
=(e −1 − 1)(e −0.8 − 1) = 1 + e −1.8 − e −0.8 − e −1 ≈ 0.3481
μ 1 = R 2 xf(x, y) dA = ∞
0
∞
x0.1e −(0.5x+0.2y) dy dx
0
=0.1 ∞
0
xe −0.5x dx ∞
0
e −0.2y dy =0.1 lim
t→∞
t
0 xe−0.5x dx lim
t→∞
t
0 e−0.2y dy
To evaluate the first integral, we integrate by parts with u = x and dv = e −0.5x dx (or we can use Formula 96

F.
SECTION TX.10 16.5 APPLICATIONS OF DOUBLE INTEGRALS ET SECTION 15.5 ¤ 239
in the Table of Integrals): xe −0.5x dx = −2xe −0.5x − −2e −0.5x dx = −2xe −0.5x − 4e −0.5x = −2(x +2)e −0.5x .
Thus
μ 1 =0.1 lim
−2(x +2)e
−0.5x t
lim
t→∞ 0 −5e
−0.2y t
t→∞
0
=0.1 lim(−2) (t +2)e −0.5t − 2 lim
t→∞
=0.1(−2)
lim
t→∞
t +2
e 0.5t
− 2
(−5)(−1) = 2
t→∞
(−5) e −0.2t − 1
[by l’Hospital’s Rule]
The expected value of Y is given by
μ 2 = yf(x, y) dA = ∞
∞
y0.1e −(0.5+0.2y) dy dx
R 2 0 0
=0.1 ∞
0
e −0.5x dx ∞
0
ye −0.2y dy =0.1 lim
t→∞
t
0 e−0.5x dx lim
t→∞
t
0 ye−0.2y dy
To evaluate the second integral, we integrate by parts with u = y and dv = e −0.2y dy (oragainwecanuseFormula96in
the Table of Integrals) which gives ye −0.2y dy = −5ye −0.2y + 5e −0.2y dy = −5(y +5)e −0.2y .Then
μ 2 =0.1 lim
−2e
−0.5x t
lim
t→∞ 0 −5(y +5)e
−0.2y t
t→∞
0
=0.1 lim −2(e −0.5t − 1)
lim −5 (t +5)e −0.2t − 5
t→∞ t→∞
t +5
=0.1(−2)(−1) · (−5) lim
t→∞ e − 5 =5 [by l’Hospital’s Rule]
0.2t
31. (a) The random variables X and Y are normally distributed with μ 1 =45, μ 2 =20, σ 1 =0.5,andσ 2 =0.1.
1
The individual density functions for X and Y , then, are f 1 (x) =
0.5 √ 2π e−(x−45)2 /0.5 and
1
f 2 (y) =
0.1 √ 2π e−(y−20)2 /0.02 .SinceX and Y are independent, the joint density function is the product
f(x, y) =f 1 (x)f 2 (y) =
Then P (40 ≤ X ≤ 50, 20 ≤ Y ≤ 25) = 50
40
1
0.5 √ 2π e−(x−45)2 /0.5 1
0.1 √ 2π e−(y−20)2 /0.02 = 10
π e−2(x−45)2 −50(y−20) 2 .
25
20
f(x, y) dy dx =
10
π
50
25
40 20 e−2(x−45)2 −50(y−20) 2 dy dx.
Using a CAS or calculator to evaluate the integral, we get P (40 ≤ X ≤ 50, 20 ≤ Y ≤ 25) ≈ 0.500.
(b) P (4(X − 45) 2 +100(Y − 20) 2 ≤ 2) = 10
D π e−2(x−45)2 −50(y−20) 2 dA,whereD is the region enclosed by the ellipse
4(x − 45) 2 + 100(y − 20) 2 =2. Solving for y gives y =20± 1 10 2 − 4(x − 45)2 , the upper and lower halves of the
ellipse, and these two halves meet where y =20 [since the ellipse is centered at (45, 20)] ⇒ 4(x − 45) 2 =2 ⇒
x =45± 1 √
2
.Thus
D
√
45+1/ 2
20+ 1
10
−50(y−20) 2
π e−2(x−45)2 dA = 10
10
√2 − 4(x−45) 2
e −2(x−45)2 −50(y−20) 2 dy dx.
π
45−1/ √ 2 20−
10
1 √2 − 4(x−45) 2
Using a CAS or calculator to evaluate the integral, we get P (4(X − 45) 2 +100(Y − 20) 2 ≤ 2) ≈ 0.632.
33. (a) If f(P, A) is the probability that an individual at A will be infected by an individual at P ,andkdAis the number of
infected individuals in an element of area dA, thenf(P, A)kdAis the number of infections that should result from
exposure of the individual at A to infected people in the element of area dA. IntegrationoverD gives the number of
infections of the person at A due to all the infected people in D. In rectangular coordinates (with the origin at the city’s

F.
240 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
center), the exposure of a person at A is
E = kf(P, A) dA = k
D
(b) If A =(0, 0),then
E = k
= k
D
2π
10
0
D
1 − 1
x2 + y
20
2 dx dy
0
=2πk 50 − 50 3
1 − r
20
20 − d(P, A)
20
r
2
10
rdrdθ=2πk
2 − r3
60
0
=
200
3
πk ≈ 209k
dA = k 1 −
D
(x − x0) 2 +(y − y 0) 2
dx dy
20
For A at the edge of the city, it is convenient to use a polar coordinate system centered at A. Then the polar equation for
the circular boundary of the city becomes r =20cosθ instead of r =10, and the distance from A to a point P in the city
is again r (see the figure). So
E = k
π/2
20 cos θ
−π/2
0
1 − r
π/2
r
2
r=20 cos θ
rdrdθ= k
20
−π/2 2 − r3
dθ
60
r=0
1 + 1 cos 2θ − 2 2 2 3
= k π/2
−π/2 200 cos 2 θ − 400
3 cos3 θ dθ =200k π/2
−π/2
1 − sin 2 θ cos θ dθ
=200k 1
θ + 1 sin 2θ − 2 sin θ + 2 · 1
2 4 3 3 3 sin3 θ π/2
=200k π
+0− 2 + 2 + π +0− 2 +
2
−π/2 4 3 9 4 3 9
=200k π
2 − 8 9
≈ 136k
Therefore the risk of infection is much lower at the edge of the city than in the middle, so it is better to live at the edge.
16.6 Triple Integrals ET 15.6
1.
3.
5.
B xyz2 dV = 1 3
0 0
1
z
0 0
3
1
0 0
= 1
0
x+z
6xz dy dx dz = 1
z
0 0 0
2
−1 xyz2 dy dz dx = 1 3
0 0
1
2 xy2 z 2 y=2
dz dx = 1 3
y=−1 0 0
1 xz3 z=3
dx = 1 27
27
xdx= x2 1
= 27
2 z=0 0 2 4 0 4
y=x+z
6xyz
= 1
0
√ 1−z 2
0
ze y dx dz dy = 3
3
2 xz2 dz dx
dx dz = 1
z
6xz(x + z) dx dz
y=0
0 0
2x 3 z +3x 2 z 2 x=z
dz = 1
x=0 0 (2z4 +3z 4 ) dz = 1
0 5z4 dz = z 5 1
√
= 3
0
xze
y x= 1−z 2
dz dy = 3
1 zey√ 1 − z
x=0
0 0 2 dz dy
z=1
− 1 (1 − 3 z2 ) 3/2 e y dy = 3
0
1
0 0
z=0
1
3 ey dy = 1 ey 3
= 1
3 0 3 (e3 − 1)
0 =1
7.
π/2
y
0 0
x cos(x + y + z)dz dx dy = π/2
y
z=x
0 0 0 sin(x + y + z) dx dy
z=0
= π/2
0
= π/2
0
= π/2
0
y
0
[sin(2x + y) − sin(x + y)] dx dy
−
1
2 cos(2x + y)+cos(x + y) x=y
x=0 dy
−
1
2 cos 3y +cos2y + 1 2 cos y − cos y dy
= − 1 6 sin 3y + 1 2 sin 2y − 1 2 sin y π/2
0
= 1 6 − 1 2 = − 1 3
9.
E 2xdV = 2
√ 4−y 2
0 0
x 2 y x=
= 2
0
y 2xdzdxdy = 2
√ 4−y 2
0
0
√
4−y 2
dy = 2
(4 − x=0 0 y2 )ydy= 2y 2 − 1 y4 2
4
0
z=y
2xz dx dy = 2
√ 4−y 2
2xy dx dy
z=0 0 0
0 =4

F.
TX.10
SECTION 16.6 TRIPLE INTEGRALS ET SECTION 15.6 ¤ 241
11. Here E = {(x, y, z) | 0 ≤ x ≤ 1, 0 ≤ y ≤ √ x, 0 ≤ z ≤ 1+x + y},so
E 6xy dV = 1
0
= 1
0
√ x
1+x+y
6xy dz dy dx = 1
0 0 0
√ x
0
6xyz
z=1+x+y
z=0
dy dx = 1
√ x
0
3xy 2 +3x 2 y 2 +2xy 3 y= √ x
y=0
dx = 1
0 (3x2 +3x 3 +2x 5/2 ) dx =
6xy(1 + x + y) dy dx
0
1
x 3 + 3 4 x4 + 4 7 x7/2 = 65
28
0
13. E is the region below the parabolic cylinder z =1− y 2 and above the
square [−1, 1] × [−1, 1] in the xy-plane.
E x2 e y dV = 1
1
1−y
2
x 2 e y dz dy dx
−1 −1 0
= 1
1
−1 −1 x2 e y (1 − y 2 ) dy dx
= 1
−1 x2 dx 1
−1 (ey − y 2 e y ) dy
= 1
x3 1
3 −1 e y − (y 2 − 2y +2)e y 1
−1
integrate by
parts twice
= 1 3 (2)[e − e − e−1 +5e −1 ]= 8 3e
15. Here T = {(x, y, z) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x, 0 ≤ z ≤ 1 − x − y},so
T x2 dV = 1
1−x
1−x−y
x 2 dz dy dx = 1
1−x
0 0 0 0
= 1
0
= 1
0
= 1
0
1−x
(x 2 − x 3 − x 2 y)dy dx = 1
0 0
x 2 (1 − x) − x 3 (1 − x) − 1 2 x2 (1 − x) 2 dx
1
2 x4 − x 3 + 1 2 x2 dx = 1
10 x5 − 1 4 x4 + 1 6 x3 1
0
= 1
10 − 1 4 + 1 6 = 1 60
x 2 (1 − x − y)dy dx
0
x 2 y − x 3 y − 1 2 x2 y 2 y=1−x
dx
y=0
17. The projection E on the yz-plane is the disk y 2 + z 2 ≤ 1. Using polar
19. The plane 2x + y + z =4intersects the xy-plane when
2x + y +0=4 ⇒ y =4− 2x,so
E = {(x, y, z) | 0 ≤ x ≤ 2, 0 ≤ y ≤ 4 − 2x, 0 ≤ z ≤ 4 − 2x − y} and
V = 2
0
= 2
0
= 2
0
4−2x
0
4−2x−y
dz dy dx = 2
4−2x
(4 − 2x − y) dy dx
0 0 0
4y − 2xy −
1
y2 y=4−2x
dx
2 y=0
4(4 − 2x) − 2x(4 − 2x) −
1
(4 − 2x)2 dx
2
= 2
0 (2x2 − 8x +8)dx = 2
3 x3 − 4x 2 +8x 2
= 16
0 3
coordinates y = r cos θ and z = r sin θ,weget
xdV =
4
xdx dA = 1 E D 4y 2 +4z 2 2 D 4 2 − (4y 2 +4z 2 ) 2 dA
=8 2π 1 (1 − 0 0 r4 ) rdrdθ=8 2π
dθ 1
(r − 0 0 r5 ) dr
=8(2π) 1
2 r2 − 1 r6 1
= 16π
6 0 3

F.
242 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
3
21. V =
−3
√ 9−x 2 5−y
√
− 9−x 2 1
3
√ 9−x 2
3
dz dy dx =
(5 − y − 1) dy dx =
4y −
−3 −
√9−x 1 y2 y=
2 2
−3
= 3
8 √ 9 − x
−3 2 dx =8 √
x
2 9 − x2 + 9 sin−1
x 3
2 3 −3
=8 9
2 sin−1 (1) − 9 2 sin−1 (−1) =36 π
2 − − π 2
=36π
Alternatively, use polar coordinates to evaluate the double integral:
3
√ 9−x 2
2π
3
(4 − y) dy dx = (4 − r sin θ) rdrdθ
−3 −
√9−x 2 0 0
= 2π
0 2r 2 − 1 3 r3 sin θ r=3
dθ r=0
= 2π
(18 − 9sinθ) dθ
0
=18θ +9cosθ
2π
0
=36π
23. (a) The wedge can be described as the region
D = (x, y, z) | y 2 + z 2 ≤ 1, 0 ≤ x ≤ 1, 0 ≤ y ≤ x
=
(x, y, z) | 0 ≤ x ≤ 1, 0 ≤ y ≤ x, 0 ≤ z ≤
1 − y 2
So the integral expressing the volume of the wedge is
D dV = 1 x
√ 1 − y 2
dz dy dx.
0 0 0
(b) A CAS gives 1 x
√ 1 − y 2
dz dy dx = π − 1 .
0 0 0 4 3
(Or use Formulas 30 and 87 from the Table of Integrals.)
25. Here f(x, y, z) =
using trigonometric substitution or
Formula 30 in the Table of Integrals
1
and ∆V =2· 4 · 2=16, so the Midpoint Rule gives
ln(1 + x + y + z)
B f(x, y, z) dV ≈ l
m
n
i=1 j=1 k=1
TX.10
f x i , y j , z k
∆V
√
y=−
= 16[f(1, 2, 1) + f(1, 2, 3) + f(1, 6, 1) + f(1, 6, 3)
+ f(3, 2, 1) + f(3, 2, 3) + f(3, 6, 1) + f(3, 6, 3)]
9−x 2
√9−x 2 dx
=16 1
ln 5 + 1
ln 7 + 1
ln 9 + 1
ln 11 + 1
ln 7 + 1
ln 9 + 1
ln 11 + 1
ln 13
≈ 60.533
27. E = {(x, y, z) | 0 ≤ x ≤ 1, 0 ≤ z ≤ 1 − x, 0 ≤ y ≤ 2 − 2z},
the solid bounded by the three coordinate planes and the planes
z =1− x, y =2− 2z.

F.
TX.10
SECTION 16.6 TRIPLE INTEGRALS ET SECTION 15.6 ¤ 243
29.
If D 1 , D 2 , D 3 are the projections of E on the xy-, yz-, and xz-planes, then
Therefore
Then
D 1 = (x, y) | −2 ≤ x ≤ 2, 0 ≤ y ≤ 4 − x 2 = (x, y) | 0 ≤ y ≤ 4, − √ 4 − y ≤ x ≤ √ 4 − y
D 2 = (y, z) | 0 ≤ y ≤ 4, − 1 √
2 4 − y ≤ z ≤
1
√
2 4 − y = (y, z) | −1 ≤ z ≤ 1, 0 ≤ y ≤ 4 − 4z
2
D 3 = (x, z) | x 2 +4z 2 ≤ 4
E = (x, y, z) | −2 ≤ x ≤ 2, 0 ≤ y ≤ 4 − x 2 , − 1 2 4 − x2 − y ≤ z ≤ 1 2 4 − x2 − y
= (x, y, z) | 0 ≤ y ≤ 4, − √ 4 − y ≤ x ≤ √
4 − y, − 1 2 4 − x2 − y ≤ z ≤ 1 2 4 − x2 − y
=
(x, y, z) | −1 ≤ z ≤ 1, 0 ≤ y ≤ 4 − 4z 2 , − 4 − y − 4z 2 ≤ x ≤
4 − y − 4z 2
= (x, y, z) | 0 ≤ y ≤ 4, − 1 √
2 4 − y ≤ z ≤
1
√
2 4 − y, − 4 − y − 4z2 ≤ x ≤
4 − y − 4z 2
√ √
= (x, y, z) | −2 ≤ x ≤ 2, − 1 2 4 − x2 ≤ z ≤ 1 2 4 − x2 , 0 ≤ y ≤ 4 − x 2 − 4z 2
=
(x, y, z) | −1 ≤ z ≤ 1, − √ 4 − 4z 2 ≤ x ≤ √
4 − 4z 2 , 0 ≤ y ≤ 4 − x 2 − 4z 2
E f(x, y, z) dV = 2
4−x
2
−2 0
= 1
4−4z
2
−1 0
√ √
4−x2−y/2
f(x, y, z) dz dy dx = 4
4−x 2 −y/2 0
−
= 2
√ √
4−x2 /2
−2
−
√ √
4−y−4z2
f(x, y, z) dx dy dz = 4
− 4−y−4z 2 0
4−x 2 /2
4−x
2 −4z
2
0
f(x, y, z) dy dz dx = 1
−1
√ 4−y
− √ 4−y
√ 4−y/2
− √ 4−y/2
√
√
4−4z2
−
√ 4−x 2 −y/2
−
√4−x 2 −y/2
√ 4−y−4z 2
−
√4−y−4z
4−4z 2 4−x
2 −4z
2
f(x, y, z) dz dx dy
2
f(x, y, z) dx dz dy
0
f(x, y, z) dy dx dz

F.
244 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
31.
If D 1 , D 2 ,andD 3 are the projections of E on the xy-, yz-, and xz-planes, then
D 1 =
D 2 =
(x, y) | −2 ≤ x ≤ 2,x 2 ≤ y ≤ 4 =
(y, z) | 0 ≤ y ≤ 4, 0 ≤ z ≤ 2 − 1 y =
2
(x, y) | 0 ≤ y ≤ 4, − y ≤ x ≤
y ,
(y, z) | 0 ≤ z ≤ 2, 0 ≤ y ≤ 4 − 2z ,and
D 3 =
(x, z) | −2 ≤ x ≤ 2, 0 ≤ z ≤ 2 − 1 2 x2 = (x, z) | 0 ≤ z ≤ 2, − √ 4 − 2z ≤ x ≤ √
4 − 2z
Therefore E =
(x, y, z) | −2 ≤ x ≤ 2, x 2 ≤ y ≤ 4, 0 ≤ z ≤ 2 − 1 y 2
=
(x, y, z) | 0 ≤ y ≤ 4, − y ≤ x ≤
y, 0 ≤ z ≤ 2 − 1 y 2
= (x, y, z) | 0 ≤ y ≤ 4, 0 ≤ z ≤ 2 − 1 y, − y ≤ x ≤
y
2
= (x, y, z) | 0 ≤ z ≤ 2, 0 ≤ y ≤ 4 − 2z, − y ≤ x ≤
y
= (x, y, z) | −2 ≤ x ≤ 2, 0 ≤ z ≤ 2 − 1 2 x2 , x 2 ≤ y ≤ 4 − 2z
= (x, y, z) | 0 ≤ z ≤ 2, − √ 4 − 2z ≤ x ≤ √
4 − 2z, x 2 ≤ y ≤ 4 − 2z
Then
E f(x, y, z) dV = 2
4
2−y/2
f(x, y, z) dz dy dx = 4
√ y
2−y/2
−2 x 2 0 0 − √ f(x, y, z) dz dx dy
y 0
= 4
0
= 2
−2
2−y/2
0
2 − x
2 /2
0
√ y
− √ f(x, y, z) dx dz dy = 2
y 0
4−2z
4−2z
f(x, y, z) dy dz dx = 2
√ 4−2z
x 2 0 − √ 4−2z
0
√ y
− √ f(x, y, z) dx dy dz
y
4−2z
f(x, y, z) dy dx dz
x 2

F.
TX.10
SECTION 16.6 TRIPLE INTEGRALS ET SECTION 15.6 ¤ 245
33.
The diagrams show the projections
of E on the xy-, yz-, and xz-planes.
Therefore
1
0
1√x
1 − y
f(x, y, z) dz dy dx = 1
0 0
= 1
0
= 1
0
y
2
0
1−y
y
2
0
(1−z)
2
0
1−y
f(x, y, z) dz dx dy = 1
0 0
0
f(x, y, z) dx dz dy = 1
0
1−z
√ x
f(x, y, z) dy dx dz
1−z
y
2
0
√ 1− x
1−z
√
0 x
0
f(x, y, z) dx dy dz
f(x, y, z) dy dz dx
35.
1
1
0 y
y f(x, y, z) dz dx dy = f(x, y, z) dV where E = {(x, y, z) | 0 ≤ z ≤ y, y ≤ x ≤ 1, 0 ≤ y ≤ 1}.
0 E
If D 1 , D 2 ,andD 3 are the projections of E on the xy-, yz-andxz-planes then
Thus we also have
Then
D 1 = {(x, y) | 0 ≤ y ≤ 1, y ≤ x ≤ 1} = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ x},
D 2 = {(y, z) | 0 ≤ y ≤ 1, 0 ≤ z ≤ y} = {(y, z) | 0 ≤ z ≤ 1, z ≤ y ≤ 1},and
D 3 = {(x, z) | 0 ≤ x ≤ 1, 0 ≤ z ≤ x} = {(x, z) | 0 ≤ z ≤ 1, z ≤ x ≤ 1}.
E = {(x, y, z) | 0 ≤ x ≤ 1, 0 ≤ y ≤ x, 0 ≤ z ≤ y} = {(x, y, z) | 0 ≤ y ≤ 1, 0 ≤ z ≤ y, y ≤ x ≤ 1}
= {(x, y, z) | 0 ≤ z ≤ 1, z ≤ y ≤ 1, y ≤ x ≤ 1} = {(x, y, z) | 0 ≤ x ≤ 1, 0 ≤ z ≤ x, z ≤ y ≤ x}
= {(x, y, z) | 0 ≤ z ≤ 1, z ≤ x ≤ 1, z ≤ y ≤ x} .
1
0
1
y f(x, y, z) dz dx dy = 1 x
y 0 0 0
= 1
0
= 1
0
1
z
1
z
y f(x, y, z) dz dy dx = 1 y
1
f(x, y, z) dx dz dy
0 0 0 y
1 f(x, y, z) dx dy dz = 1 x
x
f(x, y, z) dy dz dx
y 0 0 z
x
z
f(x, y, z) dy dx dz

F.
246 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
37. m = E ρ(x, y, z) dV = 1
0
= 1
0 2y +2xy + y
2 y= √ x
dx = 1
y=0 0
√ x
1+x+y
2 dz dy dx = 1
√ x
0 0 0
2 √ x +2x 3/2 + x
0
2(1 + x + y) dy dx
dx =
1
4
3 x3/2 + 4 5 x5/2 + 1 2 x2 = 79
30
0
M yz = xρ(x, y, z) dV = 1
√ x
1+x+y
2xdzdydx= 1
√ x
2x(1 + x + y) dy dx
E 0 0 0 0 0
= 1
0 2xy +2x 2 y + xy 2 y= √ x
dx =
1
1
y=0 0 (2x3/2 +2x 5/2 + x 2 4
) dx =
5 x5/2 + 4 7 x7/2 + 1 3 x3 = 179
105
0
M xz = E yρ(x, y, z) dV = 1
0
= 1
0
y 2 + xy 2 + 2 3 y3 y= √ x
y=0
M xy = E zρ(x, y, z) dV = 1
0
√ x
1+x+y
2ydzdydx= 1
√ x
0 0 0
dx = 1
0
x + x 2 + 2 3 x3/2
dx =
√ x
1+x+y
2zdzdydx= 1
0 0 0
2y(1 + x + y) dy dx
0
1
1
2 x2 + 1 3 x3 + 4 15 x5/2 = 11
10
0
√ x
0
z
2 z=1+x+y
= 1
√ x
(1 + 2x +2y +2xy + x 2 + y 2 ) dy dx = 1
0 0 0 y +2xy + y 2 + xy 2 + x 2 y + 1 y3 y= √ x
3
=
1 √x
0 +
7
3 x3/2 + x + x 2 + x 5/2 dx =
z=0
1
2
3 x3/2 + 14
15 x5/2 + 1 2 x2 + 1 3 x3 + 2 7 x7/2 = 571
210
0
Thus the mass is 79 and the center of mass is (x, y, z) = Myz
30
m , Mxz
m , Mxy 358
=
m 553 , 33
79 , 571
.
553
dy dx = 1
√ x
(1 + x + y) 2 dy dx
0 0
y=0
dx
39. m = a a
0 0
= a
0
M yz = a a
a
0 0 0
= a
0
a
0 (x2 + y 2 + z 2 ) dx dy dz = a a
1
0 0 3 x3 + xy 2 + xz 2 x=a
dy dz = a a
1
x=0 0 0 3 a3 + ay 2 + az 2 dy dz
1
3 a3 y + 1 3 ay3 + ayz 2 y=a
dz = a
2
y=0 0 3 a4 + a 2 z 2 dz = 2
3 a4 z + 1 3 a2 z 3 a
= 2 0 3 a5 + 1 3 a5 = a 5
x 3 + x(y 2 + z 2 ) dx dy dz = a
a
1
0 0 4 a4 + 1 2 a2 (y 2 + z 2 ) dy dz
1
4 a5 + 1 6 a5 + 1 2 a3 z 2 dz = 1 4 a6 + 1 3 a6 = 7 12 a6 = M xz = M xy by symmetry of E and ρ(x, y, z)
Hence (x, y, z) = 7
12 a, 7
12 a, 7 12 a .
41. I x = L L
L
0 0 0
k(y2 + z 2 ) dz dy dx = k L L
0 0
By symmetry, I x = I y = I z = 2 3 kL5 .
Ly 2 + 1 3 L3 dy dx = k L
0
2
3 L4 dx = 2 3 kL5 .
43. I z = E (x2 + y 2 ) ρ(x, y, z) dV =
h
0 k(x2 + y 2 ) dz dA =
k(x 2 + y 2 )hdA
x 2 +y 2 ≤a 2 x 2 +y 2 ≤a 2
= kh 2π a
0 0 (r2 ) rdrdθ= kh 2π
dθ a
0 0 r3 dr = kh(2π) 1
a
4 0 4 a4 = 1 2 πkha4
45. (a) m = 3
−3
√ √
9−x2
−
5−y
9−x 2 1
x2 + y 2 dz dy dx
Myz
(b) (x, y, z) =
m , Mxz
m , Mxy
where
m
M yz = 3
−3
M xy = 3
−3
(c) I z = 3
−3
√ √
9−x2
−
√ √
9−x2
−
√
√
9−x2
−
9−x 2 5−y
9−x 2 5−y
9−x 2 5−y
1
x x 2 + y 2 dz dy dx, M xz = 3
1
z x 2 + y 2 dz dy dx.
1
(x 2 + y 2 ) x 2 + y 2 dz dy dx = 3
−3
−3
√ 9−x
√
2
−
√ 9−x
√
2
−
9−x 2 5−y
9−x 2 5−y
1
y x 2 + y 2 dz dy dx, and
1
(x 2 + y 2 ) 3/2 dz dy dx

F.
TX.10
SECTION 16.6 TRIPLE INTEGRALS ET SECTION 15.6 ¤ 247
47. (a) m = 1
0
√ 1−x 2
0
y
0
(b) (x, y, z) = m −1 1 √ 1−x 2
0 0
(c) I z =
1
0
3π
(1 + x + y + z) dz dy dx = + 11
32 24
m −1 1
0
y
0
√ 1−x 2
0
x(1 + x + y + z) dz dy dx,
y
0
y(1 + x + y + z) dz dy dx,
m −1 1 √
1−x 2 y z(1 + x + y + z) dz dy dx 0 0 0
28 30π +128 45π +208
= , ,
9π +44 45π +220 135π + 660
√ 1−x 2
0
y
0
(x 2 + y 2 )(1 + x + y + z) dz dy dx =
68 + 15π
240
49. (a) f(x, y, z) is a joint density function, so we know R 3 f(x, y, z) dV =1.Herewehave
R 3 f(x, y, z) dV = ∞
−∞
∞
−∞
Then we must have 8C =1 ⇒ C = 1 8 .
(b) P (X ≤ 1,Y ≤ 1,Z ≤ 1) = 1
−∞
1
−∞
= 1 8
∞
−∞ f(x, y, z) dz dy dx = 2
= C 2
0 xdx 2
0 ydy 2
0 zdz= C 1
2 x2 2
0
1 f(x, y, z) dz dy dx = 1
1
1
−∞ 0 0 0
1
0 xdx 1
0 ydy 1
0 zdz= 1 8
1
2 x2 1
0
1
2 y2 1
0
2
2
0 0 0
1
2 y2 2
0
1
xyz dz dy dx
8
Cxyz dz dy dx
1 z2 2
=8C
2 0
1
2 z2 1
0 = 1 8
1
2
3
= 1 64
(c) P (X + Y + Z ≤ 1) = P ((X, Y, Z) ∈ E) where E is the solid region in the first octant bounded by the coordinate planes
and the plane x + y + z =1. The plane x + y + z =1meets the xy-plane in the line x + y =1,sowehave
1−x
1−x−y
P (X + Y + Z ≤ 1) = f(x, y, z) dV = 1
1
E 0 0 0
= 1 1
1−x
8
xy 1
z2 z=1−x−y
0 0 2
dy dx = 1 z=0 16
51. V (E) =L 3 ⇒ f ave = 1 L 3 L
0
= 1 1
1−x
16 0
= 1 1
16 0
8 xyzdzdydx
1
0
1−x
0
xy(1 − x − y) 2 dy dx
[(x 3 − 2x 2 + x)y +(2x 2 − 2x)y 2 + xy 3 ] dy dx
0
(x 3 − 2x 2 + x) 1 2 y2 +(2x 2 − 2x) 1 3 y3 + x 1
y4 y=1−x
dx
4 y=0
= 1 1 (x −
192 0 4x2 +6x 3 − 4x 4 + x 5 ) dx = 1 1
192 30 =
1
5760
L
L
0
0
xyz dx dy dz = 1 L
xdx
L 3
0
L
0
ydy
L
0
zdz
= 1 L 3 x
2
2
L
0
y
2
2
L
0
z
2
2
L
0
= 1 L 3 L 2
2
L 2
2
L 2
2 = L3
8
53. The triple integral will attain its maximum when the integrand 1 − x 2 − 2y 2 − 3z 2 is positive in the region E and negative
everywhere else. For if E contains some region F where the integrand is negative, the integral could be increased by excluding
F from E,andifE fails to contain some part G of the region where the integrand is positive, the integral could be increased
by including G in E. Sowerequirethatx 2 +2y 2 +3z 2 ≤ 1. This describes the region bounded by the ellipsoid
x 2 +2y 2 +3z 2 =1.

F.
248 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
16.7 Triple Integrals in Cylindrical Coordinates ET 15.7
1. (a)
(b)
x =2cos π 4 = √ 2, y =2sin π 4 = √ 2, z =1,
so the point is √ 2, √ 2, 1 in rectangular coordinates.
x =4cos − π 3
=2, y =4sin
−
π
3
= −2
√
3,
and z =5,sothepointis 2, −2 √ 3, 5 in rectangular
coordinates.
3. (a) r 2 = x 2 + y 2 =1 2 +(−1) 2 =2so r = √ 2; tan θ = y x = −1
1
= −1 and the point (1, −1) is in the fourth quadrant of
the xy-plane, so θ = 7π 4 +2nπ; z =4. Thus, one set of cylindrical coordinates is √ 2, 7π 4 , 4 .
(b) r 2 =(−1) 2 + − √ 3 2
=4so r =2; tan θ =
− √ 3
−1
= √ 3 and the point −1, − √ 3 is in the third quadrant of the
xy-plane, so θ = 4π 3 +2nπ; z =2. Thus, one set of cylindrical coordinates is 2, 4π 3 , 2 .
5. Since θ = π but r and z may vary, the surface is a vertical half-plane including the z-axis and intersecting the xy-plane in the
4
half-line y = x, x ≥ 0.
7. z =4− r 2 =4− (x 2 + y 2 ) or 4 − x 2 − y 2 , so the surface is a circular paraboloid with vertex (0, 0, 4), axis the z-axis, and
opening downward.
9. (a) x 2 + y 2 = r 2 , so the equation becomes z = r 2 .
(b) Substituting x 2 + y 2 = r 2 and y = r sin θ, the equation x 2 + y 2 =2y becomes r 2 =2r sin θ or r =2sinθ.
11. 0 ≤ r ≤ 2 and 0 ≤ z ≤ 1 describe a solid circular cylinder with
radius 2,axisthez-axis, and height 1,but−π/2 ≤ θ ≤ π/2 restricts
the solid to the first and fourth quadrants of the xy-plane, so we have
a half-cylinder.
13. We can position the cylindrical shell vertically so that its axis coincides with the z-axisanditsbaseliesinthexy-plane. If we
use centimeters as the unit of measurement, then cylindrical coordinates conveniently describe the shell as 6 ≤ r ≤ 7,
0 ≤ θ ≤ 2π, 0 ≤ z ≤ 20.

F.
SECTION 16.7 TX.10 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES ET SECTION 15.7 ¤ 249
15. The region of integration is given in cylindrical coordinates by
E = {(r, θ, z) | 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 4, r ≤ z ≤ 4}. This represents the solid
region bounded below by the cone z = r and above by the horizontal plane z =4.
4
2π
4 rdzdθdr = 4
2π
z=4
0 0 r 0 0 rz dθ dr = 4
2π
r(4 − r) dθ dr
z=r 0 0
= 4
0 (4r − r2 ) dr 2π
0
dθ = 2r 2 − 1 3 r3 4
0
= 32 − 64
3
(2π) =
64π
3
17. In cylindrical coordinates, E is given by {(r, θ, z) | 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 4, −5 ≤ z ≤ 4}. So
x2 + y
E
2 dV = 2π
4
4
√
r2 rdzdrdθ= 2π
dθ 4
0 0 −5 0 0 r2 dr 4
= θ 2π 1 r3 4 4
0 3 0 z =(2π)
64
−5 3 (9) = 384π
−5 dz
θ
2π
19. In cylindrical coordinates E is bounded by the paraboloid z =1+r 2 , the cylinder r 2 =5or r = √ 5,andthexy-plane,
so E is given by (r, θ, z) | 0 ≤ θ ≤ 2π, 0 ≤ r ≤ √ 5, 0 ≤ z ≤ 1+r 2 .Thus
E ez dV = 2π
0
√ 5
1+r
2
e z rdzdrdθ= 2π
√ 5
r e z z=1+r 2
dr dθ = 2π
√ 5
0 0 0 0 z=0 0
= 2π
dθ √ 5
0 0
re 1+r2 − r
dr =2π
√
1
2 e1+r2 − 1 5
2 r2 0
= π(e 6 − e − 5)
0
r(e 1+r2 − 1) dr dθ
21. In cylindrical coordinates, E is bounded by the cylinder r =1, the plane z =0, and the cone z =2r. So
E = {(r, θ, z) | 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 1, 0 ≤ z ≤ 2r} and
E x2 dV = 2π 1
2r
r 2 cos 2 θ r dz dr dθ = 2π 1
0 0 0 0 0 r 3 cos 2 θz z=2r
dr dθ = 2π 1
z=0 0 0 2r4 cos 2 θdrdθ
= 2π
2
0 5 r5 cos 2 θ r=1
dθ = 2 2π
r=0 5
cos 2 θdθ= 2 2π
1+cos2θ
dθ = 1 θ + 1 2π
0
5 2 5 2 sin 2θ = 2π 5
23. (a) The paraboloids intersect when x 2 + y 2 =36− 3x 2 − 3y 2 ⇒ x 2 + y 2 =9, so the region of integration
is D = (x, y) | x 2 + y 2 ≤ 9 . Then, in cylindrical coordinates,
E = (r, θ, z) | r 2 ≤ z ≤ 36 − 3r 2 , 0 ≤ r ≤ 3, 0 ≤ θ ≤ 2π and
V = 2π 3
36 − 3r
2
0 0 r 2
rdzdrdθ= 2π 3
0 0
0
36r − 4r
3
dr dθ = 2π
0 18r 2 − r 4 r=3
dθ = 2π
81 dθ =162π.
r=0 0
(b) For constant density K, m = KV =162πK from part (a). Since the region is homogeneous and symmetric,
M yz = M xz =0and
M xy = 2π
3
0 0
= K 2
2π
0
36−3r
2
r 2
(zK) rdzdrdθ= K 2π
3 r 1
z2 z=36−3r 2
0 0 2
dr dθ
z=r 2
3
0 r((36 − 3r2 ) 2 − r 4 ) dr dθ = K 2
= K (2π) 8
2 6 r6 − 216
4 r4 + 1296 r2 3
= πK(2430) = 2430πK
2 0
Myz
Thus (x, y, z) =
m , M xz
m , M
xy
m
2π
0
dθ 3
0 (8r5 − 216r 3 +1296r) dr
=
0, 0, 2430πK
162πK =(0, 0, 15).
0
0

F.
250 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
25. The paraboloid z =4x 2 +4y 2 intersects the plane z = a when a =4x 2 +4y 2 or x 2 + y 2 = 1 4
a. So, in cylindrical
coordinates, E = (r, θ, z) | 0 ≤ r ≤ 1 2
√ a, 0 ≤ θ ≤ 2π, 4r 2 ≤ z ≤ a .Thus
m =
2π
√ a/2 a
= K
0 0
2π
0
4r 2 Kr dz dr dθ = K
1
2 ar2 − r 4 r= √ a/2
r=0
Since the region is homogeneous and symmetric, M yz = M xz =0and
M xy =
Hence (x, y, z) = 0, 0, 2 3 a .
2π
√ a/2 a
= K
0 0
2π
0
4r 2 Krzdzdrdθ= K
1
4 a2 r 2 − 4 r6 r= √ a/2
dθ = K
3
r=0
2π
√ a/2
0
0
(ar − 4r 3 ) dr dθ
2π
1
dθ = K
16 a2 dθ = 1 8 a2 πK
0
2π
√ a/2
0 0
2π
0
1
2 a2 r − 8r 5 dr dθ
1
24 a3 dθ = 1 12 a3 πK
27. The region of integration is the region above the cone z = x 2 + y 2 ,orz = r, and below the plane z =2.Also,wehave
−2 ≤ y ≤ 2 with − 4 − y 2 ≤ x ≤ 4 − y 2 which describes a circle of radius 2 in the xy-plane centered at (0, 0). Thus,
2
−2
√ 4−y 2 2
√
xz dz dx dy =
− 4−y
√x 2 2 +y 2
2π
2
2
0
0
r
(r cos θ) z r dz dr dθ =
= 2π 2
0 0 r2 (cos θ) 1
z2 z=2
dr dθ = 1
2 z=r 2
4r 2 − r 4 dr = 1 2
= 1 2
2π
cos θdθ 2
0 0
2π
2
2
0
2π
0
0
[sin θ]2π
0
r
r 2 (cos θ) zdzdrdθ
2
0 r2 (cos θ) 4 − r 2 dr dθ
4
3 r3 − 1 5 r5 2
0 =0
29. (a) The mountain comprises a solid conical region C. The work done in lifting a small volume of material ∆V with density
g(P ) to a height h(P ) above sea level is h(P )g(P ) ∆V . Summing over the whole mountain we get
W = h(P )g(P ) dV .
C
(b) Here C is a solid right circular cone with radius R =62,000 ft, height H =12,400 ft,
and density g(P )=200lb/ft 3 at all points P in C. We use cylindrical coordinates:
W = 2π H
0 0
=400π
R(1−z/H)
z · 200rdrdzdθ=2π H
200z 1
r2 r=R(1−z/H)
dz
0 0 2 r=0
H
0
z R2
2
1 − z H
2
dz = 200πR
2
H
z
=200πR 2 2
H
2 − 2z3
3H +
z4
H
= 200πR 2 2
4H 2 0
2 − 2H2
3
= 50 3 πR2 H 2 = 50
3 π(62,000)2 (12,400) 2 ≈ 3.1 × 10 19 ft-lb
0
z − 2z2
H + z3
H 2
dz
+ H2
4
r
R = H − z
H
=1− z H

F.
SECTION 16.8TX.10
TRIPLE INTEGRALS IN SPHERICAL COORDINATES ET SECTION 15.8 ¤ 251
16.8 Triple Integrals in Spherical Coordinates ET 15.8
1. (a)
(b)
x = ρ sin φ cos θ =(1)sin0cos0=0,
y = ρ sin φ sin θ =(1)sin0sin0=0,and
z = ρ cos φ =(1)cos0=1so the point is
(0, 0, 1) in rectangular coordinates.
x =2sin π cos π = √ 2
, y 4 3 2 =2sinπ sin π = √ 6
,
4 3 2
z =2cos π = √ √2
2 so the point is , √ 6
, √
2 in
4 2 2
rectangular coordinates.
3. (a) ρ = x 2 + y 2 + z 2 = √ 1+3+12=4, cos φ = z ρ = 2 √ 3
4
cos θ =
x
ρ sin φ = 1
4sin(π/6) = 1 2
⇒ θ = π 3
=
√
3
2
⇒
φ = π 6 ,and
[since y>0]. Thus spherical coordinates are
4, π 3 , π
.
6
(b) ρ = √ 0+1+1= √ 2, cos φ = −1 √
2
⇒ φ = 3π 4 ,andcos θ = 0
√
2sin(3π/4)
=0 ⇒ θ = 3π 2
[since y

F.
252 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
13. ρ ≤ 1 represents the solid sphere of radius 1 centered at the origin.
3π
4 ≤ φ ≤ π restricts the solid to that portion on or below the cone φ = 3π 4 .
15. z ≥ x 2 + y 2 because the solid lies above the cone. Squaring both sides of this inequality gives z 2 ≥ x 2 + y 2 ⇒
2z 2 ≥ x 2 + y 2 + z 2 = ρ 2 ⇒ z 2 = ρ 2 cos 2 φ ≥ 1 2 ρ2 ⇒ cos 2 φ ≥ 1 . The cone opens upward so that the inequality is
2
cos φ ≥ 1 √
2
, or equivalently 0 ≤ φ ≤ π 4 . In spherical coordinates the sphere z = x2 + y 2 + z 2 is ρ cos φ = ρ 2
⇒
ρ =cosφ. 0 ≤ ρ ≤ cos φ because the solid lies below the sphere. The solid can therefore be described as the region in
spherical coordinates satisfying 0 ≤ ρ ≤ cos φ, 0 ≤ φ ≤ π 4 .
17. The region of integration is given in spherical coordinates by
E = {(ρ, θ, φ) | 0 ≤ ρ ≤ 3, 0 ≤ θ ≤ π/2, 0 ≤ φ ≤ π/6}. This represents the solid
region in the first octant bounded above by the sphere ρ =3and below by the cone
φ = π/6.
π/6
π/2
3
0 0 0 ρ2 sin φdρdθdφ = π/6
sin φdφ π/2
dθ 3
0 0 0 ρ2 dρ
= − cos φ π/6 π/2
0 θ 1 ρ3 3
0 3 0
19. The solid E is most conveniently described if we use cylindrical coordinates:
E = (r, θ, z) | 0 ≤ θ ≤ π , 0 ≤ r ≤ 3, 0 ≤ z ≤ 2 .Then
2
E f(x, y, z) dV = π/2 3
2
f(r cos θ, r sin θ, z) rdzdrdθ.
0 0 0
=
1 −
21. In spherical coordinates, B is represented by {(ρ, θ, φ) | 0 ≤ ρ ≤ 5, 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π }. Thus
B (x2 + y 2 + z 2 ) 2 dV = π
0
√
3
2
π
2
(9) = 9π 4
2π
5
0 0 (ρ2 ) 2 ρ 2 sin φdρdθdφ= π
sin φdφ 2π
dθ 5
0 0
2π
θ
= − cos φ π
0
= 312,500
7
π ≈ 140,249.7
0
1 ρ7 5
= (2)(2π) 78,125
7 0 7
23. In spherical coordinates, E is represented by (ρ, θ, φ) 1 ≤ ρ ≤ 2, 0 ≤ θ ≤
π
, 0 ≤ φ ≤ π
2 2 . Thus
E zdV = π/2
0
π/2
2 (ρ cos φ) 0 1 ρ2 sin φdρdθdφ= π/2
cos φ sin φdφ π/2
dθ 2
0 0 1 ρ3 dρ
= 1
2 sin2 φ π/2
0
θ
π/2
0
1 ρ4 2
=
1 π
15
4 1 2 2 4 =
15π
16
0 ρ6 dρ
2 −
√
3

F.
25.
E x2 dV = π
π
0 0
SECTION 16.8TX.10
TRIPLE INTEGRALS IN SPHERICAL COORDINATES ET SECTION 15.8 ¤ 253
4
3 (ρ sin φ cos θ)2 ρ 2 sin φdρdφdθ = π
0 cos2 θdθ π
0 sin3 φdφ 4
= 1
2 θ + 1 4 sin 2θ π
0
−
1
(2 + 3 sin2 φ)cosφ π 1 ρ5 4
=
π 2 + 2
0 5 3 2 3 3
3 ρ4 dρ
1
5 (45 − 3 5 )= 1562
15 π
27. The solid region is given by E = (ρ, θ, φ) | 0 ≤ ρ ≤ a, 0 ≤ θ ≤ 2π, π 6 ≤ φ ≤ π 3
and its volume is
V = E dV = π/3
π/6
=[− cos φ] π/3
1
π/6 [θ]2π 0
ρ3 a
= 3 0
2π
a
0 0 ρ2 sin φdρdθdφ= π/3
sin φdφ 2π
dθ a
π/6 0
− 1 2 + √ 3
2
(2π) 1
3 a3 = √ 3−1
3
πa 3
0 ρ2 dρ
29. (a) Since ρ =4cosφ implies ρ 2 =4ρ cos φ, the equation is that of a sphere of radius 2 with center at (0, 0, 2). Thus
V = 2π
0
π/3
4cosφ
ρ 2 sin φdρdφdθ = 2π
0 0 0
π/3
1 ρ3 ρ=4 cos φ
0 3 ρ=0
= 2π
0 −
16
3 cos4 φ φ=π/3
dθ = 2π
− 16 1
− 1 dθ =5θ
φ=0
0 3 16
(b) By the symmetry of the problem M yz = M xz =0.Then
M xy = 2π
0
Hence (x, y, z) =(0, 0, 2.1).
2π
0
sin φdφdθ = 2π
0
=10π
π/3
64
0 3
π/3
4cosφ
ρ 3 cos φ sin φdρdφdθ = 2π π/3
cos φ sin φ 64 cos 4 φ dφ dθ
0 0 0 0
= 2π
64 − 1 0 6 cos6 φ φ=π/3
dθ = 2π 21
dθ =21π
φ=0
0 2
31. By the symmetry of the region, M xy =0and M yz =0. Assuming constant density K,
and
m = KV = K π
π
E 0 0
= Kπ − cos φ π 1 ρ3 4
=2Kπ · 37 = 74
0 3 3 3
M xz = yKdV = K π
π
E 0 0
= K − cos θ π
0
Myz
Thus the centroid is (x, y, z) =
m , M xz
m , M xy
m
4
3
ρ2 sin φdρdφdθ = K π
0 dθ π
0 sin φdφ 4
3 ρ2 dρ
cos3 φ sin φdφdθ
πK 3
4 (ρ sin φ sin θ) 3 ρ2 sin φdρdφdθ = K π
sin θdθ π
0 0 sin2 φdφ 4
1 φ − 1 sin 2φ π 1 ρ4 4
= K(2)
π 1 175
(256 − 81) = πK
2 4 0 4 3 2 4 4
= 0, 175πK/4
74πK/3 , 0 = 0, 525 , 0 .
296
3 ρ3 dρ
33. (a) The density function is ρ(x, y, z) =K, a constant, and by the symmetry of the problem M xz = M yz =0.Then
M xy = 2π
0
π/2
a
0 0 Kρ3 sin φ cos φdρdφdθ = 1 2 πKa4 π/2
sin φ cos φdφ= 1 0 8 πKa4 .ButthemassisK(volume of
the hemisphere) = 2 3 πKa3 , so the centroid is 0, 0, 3 8 a .
(b) Place the center of the base at (0, 0, 0); the density function is ρ(x, y, z) =K. By symmetry, the moments of inertia about
any two such diameters will be equal, so we just need to find I x:
I x = 2π
0
= K 2π
0
π/2
a
0 0 (Kρ2 sin φ) ρ 2 (sin 2 φ sin 2 θ +cos 2 φ) dρ dφ dθ
π/2
0
(sin 3 φ sin 2 θ +sinφ cos 2 φ) 1
a5 dφ dθ
5
= 1 5 Ka5 2π
0 sin 2 θ − cos φ + 1 3 cos3 φ + − 1 3 cos3 φ φ=π/2
dθ = 1 φ=0 5 Ka5 2π
0
= 1
5 Ka5 2 1 θ − 1 sin 2θ + 1 θ 2π
= 1 3 2 4 3 0 5 Ka5 2
(π − 0) + 1 (2π − 0) = 4
3 3 15 Ka5 π
2
3 sin2 θ + 1 3 dθ

F.
254 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
35. In spherical coordinates z = x 2 + y 2 becomes cos φ =sinφ or φ = π 4 .Then
V = 2π
0
M xy = 2π
0
π/4
1
0 0 ρ2 sin φdρdφdθ = 2π
dθ π/4
sin φdφ 1
0 0
π/4
0
Hence (x, y, z) =
1
0 ρ3 sin φ cos φdρdφdθ =2π − 1 cos 2φ π/4
4 0
0, 0,
3
8 2 − √ 2 .
0 ρ2 dρ = 1 π 2 − √ 2 ,
3
1
4 =
π
and by symmetry Myz = Mxz =0.
8
37. In cylindrical coordinates the paraboloid is given by z = r 2 and the plane by z =2r sin θ and they intersect in the circle
r =2sinθ. Then zdV = π 2sinθ
2r sin θ
rz dz dr dθ = 5π E 0 0 r 2 6
[using a CAS].
39. The region E of integration is the region above the cone z = x 2 + y 2 and below the sphere x 2 + y 2 + z 2 =2in the first
octant. Because E is in the first octant we have 0 ≤ θ ≤ π 2 . The cone has equation φ = π 4 (as in Example 4), so 0 ≤ φ ≤ π 4 ,
and 0 ≤ ρ ≤ √ 2. So the integral becomes
π/4
0
π/2
0
√ 2
0
(ρ sin φ cos θ)(ρ sin φ sin θ) ρ 2 sin φdρdθdφ
= π/4
sin 3 φdφ π/2
sin θ cos θdθ √ 2
ρ 4 dρ =
0 0 0
= 1
3 cos3 φ − cos φ π/4
· 1 · 1
0 2 5
√
2
5
=
√2
41. In cylindrical coordinates, the equation of the cylinder is r =3, 0 ≤ z ≤ 10.
The hemisphere is the upper part of the sphere radius 3, center (0, 0, 10), equation
π/4
1 − cos 2 φ
sin φdφ
1
2 sin2 θ π/2
0
−
√
2
− 1
− 1 · 2√ 2
= 4√ 2−5
12 2 3 5 15
r 2 +(z − 10) 2 =3 2 , z ≥ 10.InMaple,wecanusethecoords=cylindrical option
in a regular plot3d command. In Mathematica, we can use ParametricPlot3D.
0
1
5 ρ5√ 2
0
43. If E is the solid enclosed by the surface ρ =1+ 1 sin 6θ sin 5φ, it can be described in spherical coordinates as
5
E = (ρ, θ, φ) | 0 ≤ ρ ≤ 1+ 1 5 sin 6θ sin 5φ, 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π . Its volume is given by
V (E) = E dV = π
0
2π
1+(sin6θ sin 5φ)/5
ρ 2 sin φdρdθdφ= 136π [using a CAS].
0 0 99
45. (a) From the diagram, z = r cot φ 0 to z = √ a 2 − r 2 , r =0
to r = a sin φ 0 (or use a 2 − r 2 = r 2 cot 2 φ 0 ). Thus
V = 2π
0
a sin φ0
0
√ a 2 −r 2
r cot φ 0
rdzdrdθ
=2π a sin φ 0
√
0 r a2 − r 2 − r 2 cot φ 0 dr
a sin φ0
−(a 2 − r 2 ) 3/2 − r 3 cot φ 0
= 2π 3
= 2π 3
− a 2 − a 2 sin 2 φ 0
3/2
− a 3 sin 3 φ 0 cot φ 0 + a 3
= 2 3 πa3 1 − cos 3 φ 0 +sin 2 φ 0 cos φ 0
=
2
3 πa3 (1 − cos φ 0 )
0

F.
SECTION 16.9TX.10
CHANGE OF VARIABLES IN MULTIPLE INTEGRALS ET SECTION 15.9 ¤ 255
(b) The wedge in question is the shaded area rotated from θ = θ 1 to θ = θ 2.
Letting
V ij = volume of the region bounded by the sphere of radius ρ i
and the cone with angle φ j (θ = θ 1 to θ 2 )
and letting V be the volume of the wedge, we have
V =(V 22 − V 21 ) − (V 12 − V 11 )
= 1 (θ 3 2 − θ 1 ) ρ 3 2(1 − cos φ 2 ) − ρ 3 2(1 − cos φ 1 ) − ρ 3 1(1 − cos φ 2 )+ρ 3 1(1 − cos φ 1 )
= 1 3 (θ2 − θ1) ρ 3 2 − ρ1 3 (1 − cos φ2 ) − ρ 3 2 − ρ1 3 (1 − cos φ1 ) = 1 3 (θ2 − θ1)
ρ 3 2 − ρ 3 1 (cos φ1 − cos φ 2 )
θ2
θ 1
ρ2 sin φ 2
ρ 1 sin φ 1
r cot φ1
r cot φ 2
Or: Show that V =
rdzdrdθ.
(c) By the Mean Value Theorem with f(ρ) =ρ 3 there exists some ˜ρ with ρ 1 ≤ ˜ρ ≤ ρ 2 such that
f(ρ 2 ) − f(ρ 1 )=f 0 (˜ρ)(ρ 2 − ρ 1 ) or ρ 3 1 − ρ 3 2 =3˜ρ 2 ∆ρ. Similarly there exists φ with φ 1 ≤ ˜φ ≤ φ 2
such that cos φ 2 − cos φ 1 = − sin ˜φ
∆φ. Substituting into the result from (b) gives
∆V =(˜ρ 2 ∆ρ)(θ 2 − θ 1 )(sin ˜φ) ∆φ =˜ρ 2 sin ˜φ ∆ρ ∆φ ∆θ.
16.9 Change of Variables in Multiple Integrals ET 15.9
1. x =5u − v, y = u +3v.
∂(x, y)
The Jacobian is
∂(u, v) = ∂x/∂u ∂x/∂v
∂y/∂u ∂y/∂v = 5 −1
=5(3)− (−1)(1) = 16.
1 3 3. x = e −r sin θ, y = e r cos θ.
∂(x, y)
∂(r, θ) = ∂x/∂r ∂x/∂θ
∂y/∂r ∂y/∂θ = −e −r sin θ
e r cos θ
e −r cos θ
−e r sin θ = e−r e r sin 2 θ − e −r e r cos 2 θ =sin 2 θ − cos 2 θ or − cos 2θ
5. x = u/v, y = v/w, z = w/u.
∂x/∂u ∂x/∂v ∂x/∂w
∂(x, y, z)
1/v −u/v 2 0
∂(u, v, w) = ∂y/∂u ∂y/∂v ∂y/∂w
=
0 1/w −v/w 2
∂z/∂u ∂z/∂v ∂z/∂w
−w/u 2 0 1/u
= 1 1/w −v/w 2
v 0 1/u − − u 0 −v/w 2
v 2 −w/u 2 1/u +0 0 1/w
−w/u 2 0
= 1 1
v uw − 0 + u
0 − v
+0= 1
v 2 u 2 w uvw − 1
uvw =0

F.
256 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
7. The transformation maps the boundary of S to the boundary of the image R,sowefirstlookatsideS 1 in the uv-plane. S 1 is
described by v =0 [0 ≤ u ≤ 3], so x =2u +3v =2u and y = u − v = u. Eliminating u,wehavex =2y, 0 ≤ x ≤ 6. S 2
is the line segment u =3, 0 ≤ v ≤ 2,sox =6+3v and y =3− v. Thenv =3− y ⇒ x =6+3(3− y) =15− 3y,
6 ≤ x ≤ 12. S 3 is the line segment v =2, 0 ≤ u ≤ 3,sox =2u +6and y = u − 2,givingu = y +2 ⇒ x =2y +10,
6 ≤ x ≤ 12. Finally, S 4 is the segment u =0, 0 ≤ v ≤ 2,sox =3v and y = −v ⇒ x = −3y, 0 ≤ x ≤ 6. Theimageof
set S is the region R shown in the xy-plane, a parallelogram bounded by these four segments.
9. S 1 is the line segment u = v, 0 ≤ u ≤ 1,soy = v = u and x = u 2 = y 2 .Since0 ≤ u ≤ 1, the image is the portion of the
parabola x = y 2 , 0 ≤ y ≤ 1. S 2 is the segment v =1, 0 ≤ u ≤ 1,thusy = v =1and x = u 2 ,so0 ≤ x ≤ 1. Theimageis
thelinesegmenty =1, 0 ≤ x ≤ 1. S 3 is the segment u =0, 0 ≤ v ≤ 1,sox = u 2 =0and y = v ⇒ 0 ≤ y ≤ 1. The
image is the segment x =0, 0 ≤ y ≤ 1. Thus, the image of S is the region R in the first quadrant bounded by the parabola
x = y 2 ,they-axis, and the line y =1.
11.
∂(x, y)
∂(u, v) = 2 1
=3and x − 3y =(2u + v) − 3(u +2v) =−u − 5v. Tofind the region S in the uv-plane that
1 2 corresponds to R we first find the corresponding boundary under the given transformation. The line through (0, 0) and (2, 1) is
y = 1 x which is the image of u +2v = 1 2 2
(2u + v) ⇒ v =0; the line through (2, 1) and (1, 2) is x + y =3which is the
image of (2u + v)+(u +2v) =3 ⇒ u + v =1; the line through (0, 0) and (1, 2) is y =2x which is the image of
u +2v =2(2u + v) ⇒ u =0. Thus S is the triangle 0 ≤ v ≤ 1 − u, 0 ≤ u ≤ 1 in the uv-plane and
13.
R (x − 3y) dA = 1
1−u
(−u − 5v) |3| dv du = −3 1
0 0 0 uv +
5
v2 v=1−u
2
= −3 1
0
v=0
du
u − u 2 + 5 2 (1 − u)2 du = −3 1
2 u2 − 1 3 u3 − 5 6 (1 − u)3 1
0 = −3 1
2 − 1 3 + 5 6
= −3
∂(x, y)
∂(u, v) = 2 0
0 3 =6, x2 =4u 2 and the planar ellipse 9x 2 +4y 2 ≤ 36 is the image of the disk u 2 + v 2 ≤ 1. Thus
R x2 dA =
u 2 +v 2 ≤1
(4u 2 )(6) du dv = 2π
=24 1
2 x + 1 4 sin 2x 2π
0
0
1
0 (24r2 cos 2 θ) rdrdθ=24 2π
0
cos 2 θdθ 1
0 r3 dr
1
4 r4 1
0 =24(π) 1
4
=6π

F.
15.
SECTION 16.9TX.10
CHANGE OF VARIABLES IN MULTIPLE INTEGRALS ET SECTION 15.9 ¤ 257
∂(x, y)
∂(u, v) = 1/v −u/v 2
0 1 = 1 v , xy = u, y = x is the image of the parabola v2 = u, y =3x is the image of the parabola
v 2 =3u, and the hyperbolas xy =1, xy =3are the images of the lines u =1and u =3respectively. Thus
R
xy dA =
3
√ 3u
1
√ u
u 1
v
dv du =
3
1
u ln √ 3u − ln √
u du = 3
u ln √ 3 du =4ln √ 3=2ln3.
1
17. (a)
a 0 0
∂(x, y, z)
∂(u, v, w) = 0 b 0
= abc and since u = x a , v = y b , w = z the solid enclosed by the ellipsoid is the image of the
c
0 0 c
ball u 2 + v 2 + w 2 ≤ 1. So
E dV =
u 2 +v 2 +w 2 ≤ 1
(b) If we approximate the surface of the earth by the ellipsoid
abc du dv dw =(abc)(volume of the ball) = 4 3 πabc
x 2
6378 + y2
2 6378 + z2
=1, then we can estimate
2 63562 the volume of the earth by finding the volume of the solid E enclosed by the ellipsoid. From part (a), this is
E dV = 4 3 π(6378)(6378)(6356) ≈ 1.083 × 1012 km 3 .
19. Letting u = x − 2y and v =3x − y,wehavex = 1 (2v − u) and y = 1 y)
(v − 3u). Then∂(x,
5 5
∂(u, v) = −1/5 2/5
−3/5 1/5 = 1 5
and R is the image of the rectangle enclosed by the lines u =0, u =4, v =1,andv =8. Thus
R
x − 2y
4
3x − y dA =
0
8
1
u
v
1 5 dv du = 1 5
4
0
udu
8
1
1
v dv = 1 1 u2 4
5 2 0
8
ln |v| = 8 ln 8.
1 5
21. Letting u = y − x, v = y + x,wehavey = 1 (u + v), x = 1 y)
(v − u). Then∂(x,
2 2
∂(u, v) = −1/2 1/2
1/2 1/2 = −1 and R is the
2
image of the trapezoidal region with vertices (−1, 1), (−2, 2), (2, 2),and(1, 1). Thus
cos y − x 2
v
y + x dA = cos u v − 1 2 du dv = 1 2
v sin u u = v
dv = 1 2 v u = −v 2
R
1
−v
1
2
1
2v sin(1) dv = 3 2 sin 1
23. Let u = x + y and v = −x + y. Thenu + v =2y ⇒ y = 1 (u + v) and u − v =2x ⇒ x = 1 (u − v).
2 2
∂(x, y)
∂(u, v) = 1/2 −1/2
1/2 1/2 = 1 .Now|u| = |x + y| ≤ |x| + |y| ≤ 1 ⇒ −1 ≤ u ≤ 1,and
2
|v| = |−x + y| ≤ |x| + |y| ≤ 1 ⇒ −1 ≤ v ≤ 1. R is the image of the square
region with vertices (1, 1), (1, −1), (−1, −1),and(−1, 1).
So
R ex+y dA = 1 1
1
2 −1 −1 eu du dv = 1 2 e
u 1 1
−1 v = e − −1 e−1 .

F.
258 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
16 Review ET 15
1. (a) A double Riemann sum of f is m
i =1j =1
n
f
x ∗ ij,yij ∗ ∆A,where∆A is the area of each subrectangle and x
∗
ij ,yij ∗ is a
sample point in each subrectangle. If f(x, y) ≥ 0, this sum represents an approximation to the volume of the solid that lies
above the rectangle R and below the graph of f.
(b) f(x, y) dA = lim
R
m
m,n→∞ i =1j =1
n
f x ∗ ij,yij ∗ ∆A
(c) If f(x, y) ≥ 0, f(x, y) dA represents the volume of the solid that lies above the rectangle R and below the surface
R
z = f(x, y). Iff takes on both positive and negative values, f(x, y) dA is the difference of the volume above R but
R
below the surface z = f(x, y) and the volume below R but above the surface z = f(x, y).
(d) We usually evaluate f(x, y) dA as an iterated integral according to Fubini’s Theorem (see Theorem 16.2.4
R
[ET 15.2.4]).
(e) The Midpoint Rule for Double Integrals says that we approximate the double integral f(x, y) dA by the double
R
Riemann sum m
i =1j =1
n
f
x i , y j ∆A where the sample points xi , y j are the centers of the subrectangles.
(f ) f ave = 1
f(x, y) dA where A (R) is the area of R.
A (R) R
2. (a) See (1) and (2) and the accompanying discussion in Section 16.3 [ET 15.3].
(b) See (3) and the accompanying discussion in Section 16.3 [ET 15.3].
(c) See (5) and the preceding discussion in Section 16.3 [ET 15.3].
(d) See (6)–(11) in Section 16.3 [ET 15.3].
3. We may want to change from rectangular to polar coordinates in a double integral if the region R of integration is more easily
described in polar coordinates. To accomplish this, we use R f(x, y) dA = β
α
given by 0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β.
4. (a) m = ρ(x, y) dA
D
(b) M x = yρ(x, y) dA, M D y = xρ(x, y) dA
D
b
a
f(r cos θ, r sin θ) rdrdθwhere R is
(c) The center of mass is (x, y) where x = M y
m and y = M x
m .
(d) I x = D y2 ρ(x, y) dA, I y = D x2 ρ(x, y) dA, I 0 = D (x2 + y 2 )ρ(x, y) dA
5. (a) P (a ≤ X ≤ b, c ≤ Y ≤ d) = b
a
(b) f(x, y) ≥ 0 and R 2 f(x, y) dA =1.
d
f(x, y) dy dx
c
(c) The expected value of X is μ 1 = R 2 xf(x, y) dA; the expected value of Y is μ 2 = R 2 yf(x, y) dA.

F.
TX.10
CHAPTER 16 REVIEW ET CHAPTER 15 ¤ 259
6. (a) f(x, y, z) dV = lim
B l,m,n→∞
l
m
n
i =1j =1k =1
f x ∗ ijk,y ∗ ijk,z ∗ ijk
∆V
(b) We usually evaluate f(x, y, z) dV as an iterated integral according to Fubini’s Theorem for Triple Integrals
B
(see Theorem 16.6.4 [ET 15.6.4]).
(c) See the paragraph following Example 16.6.1 [ET 15.6.1].
(d) See (5) and (6) and the accompanying discussion in Section 16.6 [ET 15.6].
(e) See (10) and the accompanying discussion in Section 16.6 [ET 15.6].
(f ) See (11) and the preceding discussion in Section 16.6 [ET 15.6].
7. (a) m = ρ(x, y, z) dV
E
(b) M yz = xρ(x, y, z) dV , M E xz = yρ(x, y, z) dV , M E xy = zρ(x, y, z) dV .
E
(c) The center of mass is (x, y, z) where x = Myz
m , y = Mxz Mxy
,andz =
m m .
(d) I x = E (y2 + z 2 )ρ(x, y, z) dV , I y = E (x2 + z 2 )ρ(x, y, z) dV , I z = E (x2 + y 2 )ρ(x, y, z) dV .
8. (a) See Formula 16.7.4 [ET 15.7.4] and the accompanying discussion.
(b) See Formula 16.8.3 [ET 15.8.3] and the accompanying discussion.
(c) We may want to change from rectangular to cylindrical or spherical coordinates in a triple integral if the region E of
integration is more easily described in cylindrical or spherical coordinates or if the triple integral is easier to evaluate using
cylindrical or spherical coordinates.
∂ (x, y)
9. (a)
∂ (u, v) = ∂x/∂u
∂y/∂u
∂x/∂v
∂y/∂v = ∂x ∂y
∂u ∂v − ∂x ∂y
∂v ∂u
(b) See (9) and the accompanying discussion in Section 16.9 [ET 15.9].
(c) See (13) and the accompanying discussion in Section 16.9 [ET 15.9].
1. This is true by Fubini’s Theorem.
3. True by Equation 16.2.5 [ET 15.2.5].
5. True:
D 4 − x2 − y 2 dA = the volume under the surface x 2 + y 2 + z 2 =4and above the xy-plane
= 1 2 the volume of the sphere x 2 + y 2 + z 2 =4 = 1 · 4
2 3 π(2)3 = 16 π 3
7. The volume enclosed by the cone z = x 2 + y 2 and the plane z =2is, in cylindrical coordinates,
V = 2π 2
2
rdzdrdθ6= 2π 2
2
dz dr dθ,sotheassertionisfalse.
0 0 r 0 0 r

F.
260 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
1. Asshowninthecontourmap,wedivideR into 9 equally sized subsquares, each with area ∆A =1. Then we approximate
f(x, y) dA by a Riemann sum with m = n =3and the sample points the upper right corners of each square, so
3.
5.
R
R f(x, y) dA ≈ 3
i =1j =1
3
f(x i,y j) ∆A
= ∆A [f(1, 1) + f(1, 2) + f(1, 3) + f(2, 1) + f(2, 2) + f(2, 3) + f(3, 1) + f(3, 2) + f(3, 3)]
Using the contour lines to estimate the function values, we have
f(x, y) dA ≈ 1[2.7+4.7+8.0+4.7+6.7+10.0+6.7+8.6+11.9] ≈ 64.0
2
1
1
0
R
2 (y 0 +2xey ) dx dy = 2
1 xy + x 2 e y x=2
dy = 2
(2y x=0 1 +4ey ) dy = y 2 +4e y 2
1
=4+4e 2 − 1 − 4e =4e 2 − 4e +3
x
0 cos(x2 ) dy dx = 1
0 cos(x 2 )y y=x
dx = 1
x y=0 0 cos(x2 ) dx = 1 2 sin(x2 ) 1
= 1 sin 1
0 2
7.
π
0
1
0
√ 1−y 2
0
y sin xdzdydx = π
0
= π
0
1
z=
√1−y
0 (y sin x)z 2
dy dx = π
1 y 1 − y
z=0
0 0 2 sin xdydx
y=1
− 1 (1 − 3 y2 ) 3/2 sin x dx = π 1
sin xdx= − 1 cos x π
= 2
0 3 3 0 3
y=0
9. The region R is more easily described by polar coordinates: R = {(r, θ) | 2 ≤ r ≤ 4, 0 ≤ θ ≤ π}. Thus
R f(x, y) dA = π 4
f(r cos θ, r sin θ) rdrdθ.
0 2
11. The region whose area is given by π/2
sin 2θ
rdrdθis
0 0
(r, θ) | 0 ≤ θ ≤
π
, 0 ≤ r ≤ sin 2θ , which is the region contained in the
2
loop in the first quadrant of the four-leaved rose r =sin2θ.
13.
1
0
1
x cos(y2 ) dy dx = 1
y
0 0 cos(y2 ) dx dy
= 1
0 cos(y2 ) x x=y
x=0 dy = 1
0 y cos(y2 ) dy
= 1
2 sin(y2 ) 1
0 = 1 2 sin 1
15.
R yexy dA = 3
0
2
0 yexy dx dy = 3
0 e
xy x=2
dy = 3
x=0 0 (e2y − 1) dy = 1
2 e2y − y 3
= 1 0 2 e6 − 3 − 1 = 1 2 2 e6 − 7 2

F.
TX.10
CHAPTER 16 REVIEW ET CHAPTER 15 ¤ 261
17.
D
y
1
1+x dA = 2
= 1 2
√ x
0 0
1
0
y
1
1+x dy dx = 1 1 2 1+x y2 y= √ x
dx
2 2 y=0
x
1+x 2 dx = 1
4 ln(1 + x2 ) 1
0 = 1 4 ln 2
0
19.
D ydA= 2
8−y
2
ydxdy
0 y 2
= 2
0 y x x=8−y 2
x=y 2 dy = 2
0 y(8 − y2 − y 2 ) dy
= 2
0 (8y − 2y3 ) dy = 4y 2 − 1 2 y4 2
0 =8
21.
D
x 2 + y 2 3/2
dA =
π/3
3
0 0
(r 2 ) 3/2 rdrdθ
23.
25.
27.
E xy dV = 3
x
x+y
xy dz dy dx = 3
x xy z z=x+y
0 0 0 0 0 z=0
= 3
0
= 5 6
E y2 z 2 dV = 1
−1
x
0 (x2 y + xy 2 ) dy dx = 3
0
3
0 x4 dx = 1
6 x5 3
0 = 81
2 =40.5
= 2π
0
= 2π
0
E yz dV = 2
−2
29. V = 2
0
= 16
5
√ √
1−y2
−
1−y 2 1 − y
2 − z
2
=
= π 3
π/3
1
2 x2 y 2 + 1 xy3 y=x
dx = 3
3 y=0 0
0
y 2 z 2 dx dz dy = 1
−1
1
0 (r2 cos 2 θ)(r 2 sin 2 θ)(1 − r 2 ) rdrdθ= 2π
0
0
dθ
3
0
3 5
5 = 81π
5
r 4 dr = θ 3
π/3 1
0
5 r5 0
dy dx = 3
x
xy(x + y) dy dx
0 0
1
2 x4 + 1 x4 dx
3
√ 1−y 2
√1−y 2 y2 z 2 (1 − y 2 − z 2 ) dz dy
−
1
0
1
4 sin2 2θ(r 5 − r 7 ) dr dθ
1
(1 − cos 4θ) 1
8 6 r6 − 1 r8 r=1
dθ = 1
8 r=0 192 θ −
1
sin 4θ 2π
4
= 2π = π 0 192 96
√ 4−x 2
0
π
0 sin3 θdθ= 16
5
4
1 (x2 +4y 2 ) dy dx = 2
y yz dz dy dx = 2
√ 4−x 2 1
0 −2 0 2 y3 dy dx = π 2
0 0
0
− cos θ +
1
3 cos3 θ π
= 64
0 15
x 2 y + 4 3 y3 y=4
y=1 dx = 2
0 (3x2 +84)dx = 176
1
2 r3 (sin 3 θ) rdrdθ
31.
V = 2 y
(2−y)/2
dz dx dy = 2 y
0 0 0 0 0 1 −
1
y dx dy
2
= 2
0 y −
1
y2 dy = 2 2 3

F.
262 ¤ CHAPTER 16 MULTIPLE INTEGRALS ET CHAPTER 15
TX.10
33. Using the wedge above the plane z =0and below the plane z = mx and noting that we have the same volume for m0 (so use m>0), we have
V =2 a/3 √ a 2 −9y 2
mx dx dy =2 a/3 1
0 0 0 2 m(a2 − 9y 2 ) dy = m a 2 y − 3y 3 a/3
= m 1
0
3 a3 − 1 a3 = 2 9 9 ma3 .
35. (a) m = 1
1−y
2
ydxdy= 1
(y − 0 0 0 y3 ) dy = 1 − 1 = 1 2 4 4
(b) M y = 1 1 − y
2
xy dx dy = 1 1
y(1 − 0 0 0 2 y2 ) 2 dy = − 1 (1 − 12 y2 ) 3 1
= 1 , 0 12
M x = 1 1 − y
2
y 2 dx dy = 1
0 0 0 (y2 − y 4 ) dy = 2 .Hence(x, y) = 1
, 8
15 3 15 .
(c) I x = 1 1−y
2
y 3 dx dy = 1
0 0 0 (y3 − y 5 ) dy = 1 , 12
I y = 1 1−y
2
yx 2 dx dy = 1 1
y(1 − 0 0 0 3 y2 ) 3 dy = − 1 (1 − 24 y2 ) 4 1
= 1 , 0 24
I 0 = I x + I y = 1 8 , y 2 = 1/12
1/4 = 1 3
⇒ y = 1 √
3
,andx 2 = 1/24
1/4 = 1 6
⇒ x = 1 √
6
.
37. The equation of the cone with the suggested orientation is (h − z) = h a
x2 + y 2 , 0 ≤ z ≤ h. ThenV = 1 3 πa2 h is the
volume of one frustum of a cone; by symmetry M yz = M xz =0;and
√
h−(h/a) x 2 +y 2 2π
a
M xy =
zdzdA=
0
0 0 0
x 2 +y 2 ≤a 2
= πh2
a 2
a
Hence the centroid is (x, y, z) = 0, 0, 1 4 h .
0
(h/a)(a−r)
rz dz dr dθ = π
(a 2 r − 2ar 2 + r 3 ) dr = πh2 a
4
a 2 2 − 2a4
3 + a4
= πh2 a 2
4 12
a
0
r h2
a 2 (a − r)2 dr
39. 3
√ 9−x 2
(x
0 −
√9−x 3 + xy 2 ) dy dx =
2
41. From the graph, it appears that 1 − x 2 = e x at x ≈−0.71 and at
x =0,with1 − x 2 >e x on (−0.71, 0). So the desired integral is
D y2 dA ≈ 0
1−x
2
y 2 dy dx
−0.71 e x
0 [(1 − −0.71 x2 ) 3 − e 3x ] dx
= 1 3
= 1 3
x − x 3 + 3 5 x5 − 1 7 x7 − 1 3 e3x 0
−0.71 ≈ 0.0512
3
0
√ 9−x 2
x(x
−
√9−x 2 + y 2 ) dy dx
2
= π/2
−π/2
3
0 (r cos θ)(r2 ) rdrdθ
= π/2
−π/2 cos θdθ 3
0 r4 dr
= sin θ π/2 1 r5 3
−π/2 5
0 =2· 1
5
(243) =
486
5
=97.2

F.
TX.10
CHAPTER 16 REVIEW ET CHAPTER 15 ¤ 263
43. (a) f(x, y) is a joint density function, so we know that R 2 f(x, y) dA =1.Sincef(x, y) =0outside the rectangle
[0, 3] × [0, 2], we can say
R 2 f(x, y) dA = ∞
−∞
Then 15C =1 ⇒ C = 1 15 .
(b) P (X ≤ 2,Y ≥ 1) = 2
−∞
= 1 2
15 0
= C 3
0
∞ f(x, y) dy dx = 3 2
C(x + y) dy dx
−∞ 0 0
xy +
1
2 y2 y=2
y=0 dx = C 3
0 (2x +2)dx = C x 2 +2x 3
0 =15C
∞
f(x, y) dy dx = 2 2
1 0 1
x +
3
2 dx =
1 1
15 2 x2 + 3 x 2
= 1
2 0 3
1
(x, y) dy dx = 1 2
15 15 0 xy +
1
y2 y=2
dx
2 y=1
(c) P (X + Y ≤ 1) = P ((X, Y ) ∈ D) where D is the triangular region shown in
the figure. Thus
1−x
P (X + Y ≤ 1) = f(x, y) dA = 1
D 0 0
= 1 1
15 0 xy +
1
y2 y=1−x
2
= 1 1
15 0
= 1
30
y=0
dx
1
(x + y) dy dx
15
x(1 − x)+
1
(1 − x)2 dx
2
1 (1 −
0 x2 ) dx = 1
30 x −
1
x3 1
= 1
3 0 45
45.
1
1
1−y
f(x, y, z) dz dy dx = 1
−1 x 2 0 0
1−z
0
√ y
− √ f(x, y, z) dx dy dz
y
47. Since u = x − y and v = x + y, x = 1 (u + v) and y = 1 2 2
(v − u).
∂(x, y)
Thus
∂(u, v) = 1/2 1/2
−1/2 1/2 = 1
2
R
and x − y
4
0
x + y dA = u 1 4
dv
du dv = − = − ln 2.
2 −2 v 2
2 v
49. Let u = y − x and v = y + x so x = y − u =(v − x) − u ⇒ x = 1 (v − u) and y = v − 1 (v − u) = 1 2 2 2
(v + u).
∂(x, y)
∂(u, v) = ∂x ∂y
∂u ∂v − ∂x
∂y
∂v ∂u =
− 1 1
2 2 −
1 1
= − 1 = 1 . R is the image under this transformation of the square
2 2 2 2
with vertices (u, v) =(0, 0), (−2, 0), (0, 2),and(−2, 2). So
R
xy dA =
2
0
0
−2
v 2 − u 2
4
1
du dv = 1 2
2
8 0 v 2 u − 1 u3 u=0
dv = 1 2
3 u=−2 8 0
2v 2 − 8 3 dv =
1 2
8 3 v3 − 8 v 2
=0
3 0
This result could have been anticipated by symmetry, since the integrand is an odd function of y and R is symmetric about
the x-axis.
51. For each r such that D r lies within the domain, A(D r )=πr 2 , and by the Mean Value Theorem for Double Integrals there
exists (x r ,y r ) in D r such that f (x r ,y r )= 1
f(x, y) dA. But lim
πr (x r,y 2 r )=(a, b),
D r
r→0 +
1
so lim
f (x, y) dA = lim
r→0 + πr f(x r,y 2 r )=f(a, b) by the continuity of f.
D r
r→0 +

F.
TX.10

F.
TX.10
PROBLEMS PLUS
1. Let R = 5
i =1 R i,where
3. f ave = 1
b − a
b
a
f(x) dx = 1
1 − 0
R i = {(x, y) | x + y ≥ i +2,x+ y

F.
266 ¤ PROBLEMS PLUS
TX.10
a 7 = 1 < 0.003. By the Alternating Series Estimation Theorem from Section 12.5 [ ET 11.5], we have
73 |s − s 6 | ≤ a 7 < 0.003. This error of 0.003 will not affect the second decimal place, so we have s ≈ 0.90.
9. (a) x = r cos θ, y = r sin θ, z = z. Then ∂u
∂r = ∂u ∂x
∂x ∂r + ∂u ∂y
∂y ∂r + ∂u ∂z
∂z ∂r = ∂u
∂x
+sinθ
∂ 2 u ∂ 2
∂r =cosθ u ∂x
2 ∂x 2 ∂r +
∂2 u ∂y
∂y ∂x ∂r +
∂2 u ∂z
∂z ∂x ∂r
= ∂2 u
∂x 2 cos2 θ + ∂2 u
∂y 2 sin2 θ +2 ∂2 u
cos θ sin θ
∂y ∂x
Similarly ∂u
∂θ = −∂u ∂u
r sin θ + r cos θ and
∂x ∂y
∂ 2 u
∂θ 2
∂ 2 u
∂y 2 ∂y
∂r +
∂u
cos θ + sin θ and
∂y
∂2 u ∂x
∂x∂y ∂r + ∂2 u
∂z ∂y
=
∂2 u
∂x 2 r2 sin 2 θ + ∂2 u
∂y 2 r2 cos 2 θ − 2 ∂2 u
∂y ∂x r2 sin θ cos θ − ∂u ∂u
r cos θ − r sin θ. So
∂x ∂y
∂ 2 u
∂r + 1 ∂u
2 r ∂r + 1 ∂ 2 u
r 2 ∂θ 2 + ∂2 u
∂z = ∂2 u
2 ∂x 2 cos2 θ + ∂2 u
∂y 2 sin2 θ +2 ∂2 u
∂y ∂x
∂u cos θ
cos θ sin θ +
∂x r
+ ∂2 u
∂x 2 sin2 θ + ∂2 u
∂y 2 cos2 θ − 2 ∂2 u
sin θ cos θ
∂y ∂x
= ∂2 u
∂x + ∂2 u
2 ∂y + ∂2 u
2 ∂z 2
(b) x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ. Then
∂u
∂ρ = ∂u ∂x
∂x ∂ρ + ∂u ∂y
∂y ∂ρ + ∂u ∂z
∂z ∂ρ = ∂u
∂x
∂ 2 u
∂ρ 2 =sinφ cos θ ∂ 2 u
∂x 2 ∂x
∂ρ +
+sinφ sin θ
− ∂u cos θ
∂x r
∂u
∂u
sin φ cos θ + sin φ sin θ +
∂y ∂z
∂y
∂y ∂x ∂ρ + ∂z
∂z ∂x ∂ρ
∂2 u
∂ 2 u
∂y 2 ∂y
+cosφ
∂ρ +
∂ 2 u ∂z
∂z 2 ∂ρ +
∂2 u
∂2 u ∂x
∂x∂y ∂ρ + ∂2 u
∂z ∂y
− ∂u sin θ
+ ∂2 u
∂y r ∂z 2
cos φ,and
∂z
∂ρ
∂2 u ∂x
∂x∂z ∂ρ + ∂2 u
∂y ∂z
∂y
∂ρ
=2 ∂2 u
∂y ∂x sin2 φ sin θ cos θ +2 ∂2 u
∂z ∂x sin φ cos φ cos θ +2 ∂2 u
sin φ cos φ sin θ
∂y ∂z
+ ∂2 u
∂x 2 sin2 φ cos 2 θ + ∂2 u
∂y 2 sin2 φ sin 2 θ + ∂2 u
∂z 2 cos2 φ
Similarly ∂u
∂φ = ∂u
∂u
∂u
ρ cos φ cos θ + ρ cos φ sin θ − ρ sin φ,and
∂x ∂y ∂z
∂ 2 u
∂φ 2 =2 ∂2 u
∂y ∂x ρ2 cos 2 φ sin θ cos θ − 2 ∂2 u
∂x∂z ρ2 sin φ cos φ cos θ
∂z
∂r
− 2 ∂2 u
∂y ∂z ρ2 sin φ cos φ sin θ + ∂2 u
∂x 2 ρ2 cos 2 φ cos 2 θ + ∂2 u
∂y 2 ρ2 cos 2 φ sin 2 θ
+ ∂2 u
∂z 2 ρ2 sin 2 φ − ∂u
∂u
∂u
ρ sin φ cos θ − ρ sin φ sin θ −
∂x ∂y ∂z ρ cos φ
+ ∂u sin θ
∂y r

F.
TX.10
PROBLEMS PLUS ¤ 267
11.
x
y
0 0
And ∂u
∂θ = −∂u
∂u
ρ sin φ sin θ + ρ sin φ cos θ,while
∂x ∂y
Therefore
∂ 2 u
∂θ 2 = −2 ∂2 u
∂y ∂x ρ2 sin 2 φ cos θ sin θ + ∂2 u
∂x 2 ρ2 sin 2 φ sin 2 θ
∂ 2 u
∂ρ + 2 ∂u
2 ρ ∂ρ + cot φ
ρ 2
+ ∂2 u
∂y 2 ρ2 sin 2 φ cos 2 θ − ∂u
∂u
ρ sin φ cos θ − ρ sin φ sin θ
∂x ∂y
∂u
∂φ + 1 ∂ 2 u
ρ 2 ∂φ 2 + 1
ρ 2 sin 2 φ
∂ 2 u
∂θ 2
= ∂2 u
∂x 2
(sin 2 φ cos 2 θ)+(cos 2 φ cos 2 θ)+sin 2 θ
+ ∂2 u (sin 2 φ sin 2 θ)+(cos 2 φ sin 2 θ)+cos 2 θ + ∂2 u cos 2 φ +sin 2 φ
∂y 2 ∂z 2
+ ∂u
2sin 2 φ cos θ +cos 2 φ cos θ − sin 2 φ cos θ − cos θ
∂x
ρ sin φ
+ ∂u
2sin 2 φ sin θ +cos 2 φ sin θ − sin 2 φ sin θ − sin θ
∂y
ρ sin φ
But 2sin 2 φ cos θ +cos 2 φ cos θ − sin 2 φ cos θ − cos θ =(sin 2 φ +cos 2 φ − 1) cos θ =0andsimilarlythecoefficient of
∂u/∂y is 0. Alsosin 2 φ cos 2 θ +cos 2 φ cos 2 θ +sin 2 θ =cos 2 θ (sin 2 φ +cos 2 φ)+sin 2 θ =1,andsimilarlythe
coefficient of ∂ 2 u/∂y 2 is 1. So Laplace’s Equation in spherical coordinates is as stated.
z f (t) dt dz dy = f (t) dV ,where
0 E
E = {(t, z, y) | 0 ≤ t ≤ z, 0 ≤ z ≤ y, 0 ≤ y ≤ x}.
If we let D be the projection of E on the yt-plane then
D = {(y, t) | 0 ≤ t ≤ x, t ≤ y ≤ x}. And we see from the diagram
that E = {(t, z, y) | t ≤ z ≤ y, t ≤ y ≤ x, 0 ≤ t ≤ x}. So
x
y
0 0
z
0 f(t) dt dz dy = x
0
= x
0
x
t
y
t
f(t) dz dy dt = x
x
0
1
2 y2 − ty f(t) y = x
y = t
(y − t) f(t) dy dt
t
1
2 x2 − tx − 1 2 t2 + t 2 f(t) dt
dt = x
0
= x
1
0 2 x2 − tx + 1 t2 f(t) dt = x
1
2 0 2 x2 − 2tx + t 2 f(t) dt
x (x − 0 t)2 f(t) dt
= 1 2

F.
TX.10

F.
TX.10
17 VECTOR CALCULUS ET 16
17.1 Vector Fields ET 16.1
1. F(x, y) = 1 (i + j)
2
All vectors in this field are identical, with length 1 √
2
and
direction parallel to the line y = x.
3. F(x, y) =y i + 1 2 j
The length of the vector y i + 1 2 j is y 2 + 1 4 . Vectors
are tangent to parabolas opening about the x-axis.
5. F(x, y) = y i + x j
x2 + y 2
The length of the vector
y i + x j
is 1.
x2 + y2 7. F(x, y, z) =k
All vectors in this field are parallel to the z-axis and have
length 1.
9. F(x, y, z) =x k
At each point (x, y, z), F(x, y, z) is a vector of length |x|.
For x>0, all point in the direction of the positive z-axis,
while for x

F.
270 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
11. F(x, y) =hy, xi corresponds to graph II. In the first quadrant all the vectors have positive x-andy-components, in the second
quadrant all vectors have positive x-components and negative y-components, in the third quadrant all vectors have negative x-
and y-components, and in the fourth quadrant all vectors have negative x-components and positive y-components. In addition,
the vectors get shorter as we approach the origin.
13. F(x, y) =hx − 2,x+1i corresponds to graph I since the vectors are independent of y (vectors along vertical lines are
identical) and, as we move to the right, both the x-andthey-components get larger.
15. F(x, y, z) =i +2j +3k corresponds to graph IV, since all vectors have identical length and direction.
17. F(x, y, z) =x i + y j +3k corresponds to graph III; the projection of each vector onto the xy-plane is x i + y j, which points
away from the origin, and the vectors point generally upward because their z-components are all 3.
19.
The vector field seems to have very short vectors near the line y =2x.
For F(x, y) =h0, 0i we must have y 2 − 2xy =0and 3xy − 6x 2 =0.
The first equation holds if y =0or y =2x, and the second holds if
x =0or y =2x. So both equations hold [and thus F(x, y) =0]along
the line y =2x.
21. f(x, y) =xe xy ⇒
∇f(x, y) =f x (x, y) i + f y (x, y) j =(xe xy · y + e xy ) i +(xe xy · x) j =(xy +1)e xy i + x 2 e xy j
23. ∇f(x, y, z) =f x (x, y, z) i + f y (x, y, z) j + f z (x, y, z) k =
x
x2 + y 2 + z 2 i +
y
x2 + y 2 + z 2 j +
z
x2 + y 2 + z 2 k
25. f(x, y) =x 2 − y ⇒ ∇f(x, y) =2x i − j.
27. We graph ∇f along with a contour map of f.
The length of ∇f(x, y) is √ 4x 2 +1.Whenx 6= 0,the
vectors point away from the y-axis in a slightly downward
direction with length that increases as the distance from
the y-axis increases.
The graph shows that the gradient vectors are
perpendicular to the level curves. Also, the gradient
vectors point in the direction in which f is increasing and
are longer where the level curves are closer together.

F.
TX.10
SECTION 17.2 LINE INTEGRALS ET SECTION 16.2 ¤ 271
29. f(x, y) =x 2 + y 2 ⇒ ∇f(x, y) =2x i +2y j. Thus, each vector ∇f(x, y) has the same direction and twice the length of
the position vector of the point (x, y), so the vectors all point directly away from the origin and their lengths increase as we
move away from the origin. Hence, ∇f is graph II.
31. f(x, y) =(x + y) 2 ⇒ ∇f(x, y) =2(x + y) i +2(x + y) j. Thex-andy-components of each vector are equal, so all
vectors are parallel to the line y = x. The vectors are 0 along the line y = −x and their length increases as the distance from
this line increases. Thus, ∇f is graph II.
33. At t =3the particle is at (2, 1) so its velocity is V(2, 1) = h4, 3i. After 0.01 units of time, the particle’s change in
location should be approximately 0.01 V(2, 1) = 0.01 h4, 3i = h0.04, 0.03i, so the particle should be approximately at the
point (2.04, 1.03).
35. (a) We sketch the vector field F(x, y) =x i − y j along with
several approximate flow lines.The flow lines appear to be
hyperbolas with shape similar to the graph of y = ±1/x,
so we might guess that the flow lines have equations
y = C/x.
(b) If x = x(t) and y = y(t) are parametric equations of a flow line, then the velocity vector of the flow line at the
point (x, y) is x 0 (t) i + y 0 (t) j. Since the velocity vectors coincide with the vectors in the vector field, we have
x 0 (t) i + y 0 (t) j = x i − y j ⇒ dx/dt = x, dy/dt = −y. To solve these differential equations, we know
dx/dt = x ⇒ dx/x = dt ⇒ ln |x| = t + C ⇒ x = ±e t + C = Ae t for some constant A,and
dy/dt = −y ⇒ dy/y = −dt ⇒ ln |y| = −t + K ⇒ y = ±e −t + K = Be −t for some constant B. Therefore
xy = Ae t Be −t = AB = constant. If the flow line passes through (1, 1) then (1) (1) = constant =1 ⇒ xy =1 ⇒
y =1/x, x>0.
17.2 Line Integrals ET 16.2
1. x = t 3 and y = t, 0 ≤ t ≤ 2,sobyFormula3
C y3 ds =
2 dx
2
0 t3 dt + dy
2 2
dt dt =
0 t3 (3t 2 ) 2 +(1) 2 dt = 2
√
0 t3 9t 4 +1dt
= 1 · 2
36 3 9t 4 +1 3/2
2
√
= 1
54 (1453/2 − 1) or 1
54 145 145 − 1
0
3. Parametric equations for C are x =4cost, y =4sint, − π ≤ t ≤ π .Then
2 2
C xy4 ds = π/2
(4 cos t)(4 sin t)4 (−4sint)
−π/2 2 +(4cost) 2 dt = π/2
−π/2 45 cos t sin 4 t 16(sin 2 t +cos 2 t) dt
=4 5 π/2
−π/2 (sin4 t cos t)(4) dt =(4) 6 1
5 sin5 t π/2
= 2 · 46 =1638.4
−π/2 5

F.
Then
C xy dx +(x − y) dy = C 1
xy dx +(x − y) dy + C 2
xy dx +(x − y) dy
272 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
5. If we choose x as the parameter, parametric equations for C are x = x, y = √ x for 1 ≤ x ≤ 4 and
x 2 y 3 − √
x dy =
4
x 2 · ( √ x ) 3 − √
1
x
C
1
2 √ x dx = 1 4
2 1 x 3 − 1 dx
= 1 2
1
4 x4 − x 4
= 1
1 2 64 − 4 −
1
+1 = 243
4 8
7. C = C 1 + C 2
On C 1: x = x, y =0 ⇒ dy =0dx, 0 ≤ x ≤ 2.
On C 2: x = x, y =2x − 4 ⇒ dy =2dx, 2 ≤ x ≤ 3.
9. x =2sint, y = t, z = −2cost, 0 ≤ t ≤ π.ThenbyFormula9,
C xyz ds =
π
(2 sin t)(t)(−2cost) dx
2
0 dt + dy
2
dt + dz
2
dt dt
= 2
0 (0 + 0) dx + 3
2 [(2x2 − 4x)+(−x +4)(2)]dx
= 3
2 (2x2 − 6x +8)dx = 17 3
= π
−4t sin t cos t (2 cos t)
0 2 +(1) 2 +(2sint) 2 dt = π
−2t sin 2t 4(cos
0 2 t +sin 2 t)+1dt
= −2 √ 5 π
0 t sin 2tdt= −2 √ 5 − 1 2 t cos 2t + 1 4 sin 2t π
0
= −2 √ 5 − π 2 − 0 = √ 5 π
11. Parametric equations for C are x = t, y =2t, z =3t, 0 ≤ t ≤ 1. Then
C xeyz ds = 1
te(2t)(3t)√ 1
0 2 +2 2 +3 2 dt = √ 14 1
0 te6t2 dt = √
14
13.
C x2 y √ zdz= 1
0 (t3 ) 2 (t) √ t 2 · 2tdt= 1
0 2t9 dt = 1 5 t10 1
0 = 1 5
1
e6t2 1
12
0
integrate by parts with
u = t, dv =sin2tdt
= √ 14
12 (e6 − 1).
15. On C 1 : x =1+t ⇒ dx = dt, y =3t ⇒
dy =3dt, z =1 ⇒ dz =0dt, 0 ≤ t ≤ 1.
On C 2 : x =2 ⇒ dx =0dt, y =3+2t ⇒
dy =2dt, z =1+t ⇒ dz = dt, 0 ≤ t ≤ 1.
Then
(x + yz) dx +2xdy+ xyz dz
C
= C 1
(x + yz) dx +2xdy+ xyz dz + C 2
(x + yz) dx +2xdy+ xyz dz
= 1
(1 + t +(3t)(1)) dt +2(1+t) · 3 dt +(1+t)(3t)(1) · 0 dt
0
+ 1
(2 + (3 + 2t)(1 + t)) · 0 dt +2(2)· 2 dt +(2)(3+2t)(1 + t) dt
0
= 1
0 (10t +7)dt + 1
0 (4t2 +10t + 14) dt = 5t 2 +7t 1
0 + 4
3 t3 +5t 2 +14t 1
0 =12+61 3 = 97 3

F.
TX.10
SECTION 17.2 LINE INTEGRALS ET SECTION 16.2 ¤ 273
17. (a) Along the line x = −3, the vectors of F have positive y-components, so since the path goes upward, the integrand F · T is
always positive. Therefore C 1
F · dr = C 1
F · T ds is positive.
(b) All of the (nonzero) field vectors along the circle with radius 3 are pointed in the clockwise direction, that is, opposite the
direction to the path. So F · T is negative, and therefore C 2
F · dr = C 2
F · T ds is negative.
19. r(t) =11t 4 i + t 3 j,soF(r(t)) = (11t 4 )(t 3 ) i +3(t 3 ) 2 j =11t 7 i +3t 6 j and r 0 (t) =44t 3 i +3t 2 j.Then
C F · dr = 1
F(r(t)) · 0 r0 (t) dt = 1
0 (11t7 · 44t 3 +3t 6 · 3t 2 ) dt = 1
0 (484t10 +9t 8 ) dt = 44t 11 + t 9 1
=45. 0
21.
C F · dr = 1
0 sin t 3 , cos(−t 2 ),t 4 · 3t 2 , −2t, 1 dt
= 1
0 (3t2 sin t 3 − 2t cos t 2 + t 4 ) dt = − cos t 3 − sin t 2 + 1 t5 1
= 6 − cos 1 − sin 1
5 0 5
23. F(r(t)) = (e t ) e −t2
i +sin e −t2
j = e t−t2 i +sin e −t2 j, r 0 (t) =e t i − 2te −t2 j.Then
C
F · dr =
=
2
1
2
25. x = t 2 , y = t 3 , z = t 4 so by Formula 9,
1
F(r(t)) · r 0 (t) dt =
e 2t−t2 − 2te −t2 sin
2
1
e t−t2 e t +sin e −t2
· −2te −t2 dt
e −t2 dt ≈ 1.9633
C x sin(y + z) ds = 5
0 (t2 )sin(t 3 + t 4 ) (2t) 2 +(3t 2 ) 2 +(4t 3 ) 2 dt
= 5
0 t2 sin(t 3 + t 4 ) √ 4t 2 +9t 4 +16t 6 dt ≈ 15.0074
27. We graph F(x, y) =(x − y) i + xy j and the curve C. We see that most of the vectors starting on C point in roughly the same
direction as C, so for these portions of C the tangential component F · T is positive. Although some vectors in the third
quadrant which start on C point in roughly the opposite direction, and hence give negative tangential components, it seems
reasonable that the effect of these portions of C is outweighed by the positive tangential components. Thus, we would expect
F · dr = F · T ds to be positive.
C C
To verify, we evaluate C F · dr. ThecurveC can be represented by r(t) =2cost i +2sint j, 0 ≤ t ≤ 3π 2 ,
so F(r(t)) = (2 cos t − 2sint) i +4cost sin t j and r 0 (t) =−2sint i +2cost j. Then
C F · dr = 3π/2
0
F(r(t)) · r 0 (t) dt
= 3π/2
0
[−2sint(2 cos t − 2sint)+2cost(4 cos t sin t)] dt
=4 3π/2
(sin 2 t − sin t cos t +2sint cos 2 t) dt
0
=3π + 2 [using a CAS]
3

F.
274 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
29. (a) F · dr =
1
e t2−1 ,t 5 C 0
TX.10
· 2t,
3t 2 dt =
1
1
2te t2−1 +3t 7 dt = e t2−1 + 3 0
8 t8 = 11 − 1/e
8
0
(b) r(0) = 0, F(r(0)) = e −1 , 0 ;
1
r √
1
2
= , 1
2 2 √ 1
, F r √
2
2
=
r(1) = h1, 1i, F(r(1)) = h1, 1i.
e −1/2 ,
1
4 √ ;
2
In order to generate the graph with Maple, we use the PLOT command
(not to be confused with the plot command) to define each of the vectors.
For example,
v1:=PLOT(CURVES([[0,0],[evalf(1/exp(1)),0]]));
generates the vector from the vector field at the point (0, 0) (but without an arrowhead) and gives it the name v1. Toshow
everything on the same screen, we use the display command. In Mathematica, we use ListPlot (with the
PlotJoined - > True option) to generate the vectors, and then Show to show everything on the same screen.
31. x = e −t cos 4t, y = e −t sin 4t, z = e −t , 0 ≤ t ≤ 2π .
Then dx
dt = e−t (− sin 4t)(4) − e −t cos 4t = −e −t (4 sin 4t +cos4t),
dy
dt = e−t (cos 4t)(4) − e −t sin 4t = −e −t (−4cos4t +sin4t),and dz
dt = −e−t ,so
dx 2
+
dt
2 dy
+
dt
2 dz
= (−e
dt
−t ) 2 [(4 sin 4t +cos4t) 2 +(−4cos4t +sin4t) 2 +1]
= e −t 16(sin 2 4t +cos 2 4t)+sin 2 4t +cos 2 4t +1=3 √ 2 e −t
Therefore
C x3 y 2 zds= 2π
0 (e−t cos 4t) 3 (e −t sin 4t) 2 (e −t )(3 √ 2 e −t ) dt
= 2π
3 √ √
2 e −7t cos 3 4t sin 2 4tdt= 172,704
0 5,632,705 2(1− e −14π )
33. We use the parametrization x =2cost, y =2sint, − π ≤ t ≤ π .Then
2 2
ds = dx
2
dt + dy
2dt
dt = (−2sint)2 +(2cost) 2 dt =2dt,som = kds=2k π/2
dt =2k(π),
C −π/2
x = 1
1 π/2
1
π/2
xk ds = (2 cos t)2 dt =
2πk
C 2π −π/2 2π 4sint = 4 , y = 1
1 π/2
yk ds = (2 sin t)2 dt =0.
−π/2 π 2πk
C 2π −π/2
Hence (x, y) = 4
, 0 .
π
35. (a) x = 1
xρ(x, y, z) ds , y = 1
m C m
C
yρ(x, y, z) ds, z = 1 m
(b) m = kds = k
2π
C 0 4sin 2 t +4cos 2 t +9dt = k √ 13 2π
dt =2πk √ 13,
0
1 2π
x =
2πk √ 2k √
1 2π
13 sin tdt =0, y =
13
2πk √ 2k √ 13 cos tdt =0,
13
z =
1
2πk √ 13
0
2π
0
k √
13 (3t) dt = 3
2π
2 =3π. Hence (x, y, z) =(0, 0, 3π).
2π
0
C
zρ(x, y, z) ds where m = ρ(x, y, z) ds.
C

F.
TX.10
SECTION 17.2 LINE INTEGRALS ET SECTION 16.2 ¤ 275
37. From Example 3, ρ(x, y) =k(1 − y), x =cost, y =sint,andds = dt, 0 ≤ t ≤ π ⇒
I x = C y2 ρ(x, y) ds = π
0 sin2 t [k(1 − sin t)] dt = k π
0 (sin2 t − sin 3 t) dt
= 1 k π
(1 − cos 2t) dt − k π
(1 − 2 0 0 cos2 t)sintdt
= k
π
+ −1
2
1
(1 − u 2 ) du
= k π
−
4
2 3
Let u =cost, du = − sin tdt
in the second integral
I y = C x2 ρ(x, y) ds = k π
0 cos2 t (1 − sin t) dt = k π (1 + cos 2t) dt − k π
2 0 0 cos2 t sin tdt
= k π
2 − 2 3
, using the same substitution as above.
39. W = C F · dr = 2π
0
ht − sin t, 3 − cos ti·h1 − cos t, sin ti dt
= 2π
(t − t cos t − sin t +sint cos t +3sint − sin t cos t) dt
0
= 2π
0 (t − t cos t +2sint) dt = 1
2 t2 − (t sin t +cost) − 2cost 2π
0
=2π 2
integrate by parts
in the second term
41. r(t) =h1+2t, 4t, 2ti, 0 ≤ t ≤ 1,
W = F · d r = 1
h6t, 1+4t, 1+6ti·h2, 4, 2i dt = 1
(12t +4(1+4t)+2(1+6t)) dt
C 0 0
= 1
0 (40t +6)dt = 20t 2 +6t 1
0 =26
43. Let F =185k. To parametrize the staircase, let x =20cost, y =20sint, z = 90 t = 15
6π π
t, 0 ≤ t ≤ 6π ⇒
W = F · dr = 6π
h0, 0, 185i·−20
sin t, 20 cos t, 15
C 0 π dt =(185)
15 6π
dt = (185)(90) ≈ 1.67 × 10 4 ft-lb
π 0
45. (a) r(t) =hcos t, sin ti, 0 ≤ t ≤ 2π,andletF = ha, bi. Then
W = F · d r = 2π
ha, bi·h− sin t, cos ti dt = 2π
(−a sin t + b cos t) dt = a cos t + b sin t 2π
C 0 0 0
= a +0− a +0=0
(b) Yes. F (x, y) =k x = hkx, kyi and
W = C F · d r = 2π
0
hk cos t, k sin ti·h− sin t, cos ti dt = 2π
0 (−k sin t cos t + k sin t cos t) dt = 2π
0
0 dt =0.
47. The work done in moving the object is F · dr = F · T ds. We can approximate this integral by dividing C into
C C
7 segments of equal length ∆s =2and approximating F · T, that is, the tangential component of force, at a point (x ∗ i ,y ∗ i ) on
each segment. Since C is composed of straight line segments, F · T is the scalar projection of each force vector onto C.
If we choose (x ∗ i ,y ∗ i ) to be the point on the segment closest to the origin, then the work done is
C F · T ds ≈ 7 [F(x ∗ i ,yi ∗ ) · T(x ∗ i ,yi ∗ )] ∆s =[2+2+2+2+1+1+1](2)=22. Thus, we estimate the work done to
i =1
be approximately 22 J.

F.
276 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
17.3 The Fundamental Theorem for Line Integrals ET 16.3
1. C appears to be a smooth curve, and since ∇f is continuous, we know f is differentiable. Then Theorem 2 says that the value
of ∇f · dr is simply the difference of the values of f at the terminal and initial points of C. From the graph, this is
C
50 − 10 = 40.
3. ∂(2x − 3y)/∂y = −3 =∂(−3x +4y − 8)/∂x and the domain of F is R 2 which is open and simply-connected, so by
Theorem 6 F is conservative. Thus, there exists a function f such that ∇f = F, thatis,f x(x, y) =2x − 3y and
f y (x, y) =−3x +4y − 8. Butf x (x, y) =2x − 3y implies f(x, y) =x 2 − 3xy + g(y) and differentiating both sides of this
equation with respect to y gives f y (x, y) =−3x + g 0 (y). Thus−3x +4y − 8=−3x + g 0 (y) so g 0 (y) =4y − 8 and
g(y) =2y 2 − 8y + K where K is a constant. Hence f(x, y) =x 2 − 3xy +2y 2 − 8y + K is a potential function for F.
5. ∂(e x sin y)/∂y = e x cos y = ∂(e x cos y)/∂x and the domain of F is R 2 . Hence F is conservative so there exists a function f
such that ∇f = F. Thenf x(x, y) =e x sin y implies f(x, y) =e x sin y + g(y) and f y(x, y) =e x cos y + g 0 (y). But
f y (x, y) =e x cos y so g 0 (y) =0 ⇒ g(y) =K. Thenf(x, y) =e x sin y + K is a potential function for F.
7. ∂(ye x +siny)/∂y = e x +cosy = ∂(e x + x cos y)/∂x and the domain of F is R 2 . Hence F is conservative so there
exists a function f such that ∇f = F. Thenf x(x, y) =ye x +siny implies f(x, y) =ye x + x sin y + g(y) and
f y (x, y) =e x + x cos y + g 0 (y). Butf y (x, y) =e x + x cos y so g(y) =K and f(x, y) =ye x + x sin y + K is a potential
function for F.
9. ∂(ln y +2xy 3 )/∂y =1/y +6xy 2 = ∂(3x 2 y 2 + x/y)/∂x and the domain of F is {(x, y) | y>0} whichisopenandsimply
connected. Hence F is conservative so there exists a function f such that ∇f = F. Thenf x(x, y) =lny +2xy 3 implies
f(x, y) =x ln y + x 2 y 3 + g(y) and f y (x, y) =x/y +3x 2 y 2 + g 0 (y). Butf y (x, y) =3x 2 y 2 + x/y so g 0 (y) =0
⇒
g(y) =K and f(x, y) =x ln y + x 2 y 3 + K is a potential function for F.
11. (a) F has continuous first-order partial derivatives and ∂ ∂y
2xy =2x =
∂
∂x (x2 ) on R 2 , which is open and simply-connected.
Thus, F is conservative by Theorem 6. Then we know that the line integral of F is independent of path; in particular, the
value of F · dr depends only on the endpoints of C. Since all three curves have the same initial and terminal points,
C
F · dr willhavethesamevalueforeachcurve.
C
(b) We first find a potential function f,sothat∇f = F. Weknowf x (x, y) =2xy and f y (x, y) =x 2 .Integrating
f x(x, y) with respect to x, wehavef(x, y) =x 2 y + g(y). Differentiating both sides with respect to y gives
f y (x, y) =x 2 + g 0 (y), sowemusthavex 2 + g 0 (y) =x 2 ⇒ g 0 (y) =0 ⇒ g(y) =K, a constant.
Thus f(x, y) =x 2 y + K. All three curves start at (1, 2) and end at (3, 2), so by Theorem 2,
F · dr = f(3, 2) − f(1, 2) = 18 − 2=16for each curve.
C
13. (a) f x (x, y) =xy 2 implies f(x, y) = 1 2 x2 y 2 + g(y) and f y (x, y) =x 2 y + g 0 (y). Butf y (x, y) =x 2 y so g 0 (y) =0 ⇒
g(y) =K, a constant. We can take K =0,sof(x, y) = 1 2 x2 y 2 .

F.
SECTION 17.3 THE TX.10 FUNDAMENTAL THEOREM FOR LINE INTEGRALS ET SECTION 16.3 ¤ 277
(b) The initial point of C is r(0) = (0, 1) and the terminal point is r(1) = (2, 1), so
F · dr = f(2, 1) − f(0, 1) = 2 − 0=2.
C
15. (a) f x(x, y, z) =yz implies f(x, y, z) =xyz + g(y, z) and so f y(x, y, z) =xz + g y(y, z). Butf y(x, y, z) =xz so
g y(y, z) =0 ⇒ g(y, z) =h(z). Thusf(x, y, z) =xyz + h(z) and f z(x, y, z) =xy + h 0 (z). But
f z(x, y, z) =xy +2z,soh 0 (z) =2z ⇒ h(z) =z 2 + K. Hence f(x, y, z) =xyz + z 2 (taking K =0).
(b) F · dr = f(4, 6, 3) − f(1, 0, −2) = 81 − 4=77.
C
17. (a) f x(x, y, z) =y 2 cos z implies f(x, y, z) =xy 2 cos z + g(y, z) and so f y(x, y, z) =2xy cos z + g y(y, z). But
f y(x, y, z) =2xy cos z so g y(y, z) =0 ⇒ g(y, z) =h(z). Thus f(x, y, z) =xy 2 cos z + h(z) and
f z (x, y, z) =−xy 2 sin z + h 0 (z). Butf z (x, y, z) =−xy 2 sin z, soh 0 (z) =0 ⇒ h(z) =K. Hence
f(x, y, z) =xy 2 cos z (taking K =0).
(b) r(0) = h0, 0, 0i, r(π) = π 2 , 0,π so F · dr = C f(π2 , 0,π) − f(0, 0, 0) = 0 − 0=0.
19. Here F(x, y) =tany i + x sec 2 y j. Thenf(x, y) =x tan y is a potential function for F,thatis,∇f = F so
F is conservative and thus its line integral is independent of path. Hence
tan ydx+ x C sec2 ydy= F · d r =f
2, π C 4 − f(1, 0) = 2 tan
π
− tan 0 = 2.
4
21. F(x, y) =2y 3/2 i +3x y j, W = C F · d r. Since∂(2y3/2 )/∂y =3 √ y = ∂(3x y )/∂x, there exists a function f such
that ∇f = F. Infact,f x(x, y) =2y 3/2 ⇒ f(x, y) =2xy 3/2 + g(y) ⇒ f y(x, y) =3xy 1/2 + g 0 (y). But
f y (x, y) =3x √ y so g 0 (y) =0or g(y) =K. We can take K =0 ⇒ f(x, y) =2xy 3/2 .Thus
W = F · d r = f(2, 4) − f(1, 1) = 2(2)(8) − 2(1) = 30.
C
23. We know that if the vector field (call it F) is conservative, then around any closed path C, F · dr =0.ButtakeC to be a
C
circle centered at the origin, oriented counterclockwise. All of the field vectors that start on C are roughly in the direction of
motion along C, so the integral around C will be positive. Therefore the field is not conservative.
25. From the graph, it appears that F is conservative, since around all closed
paths, the number and size of the field vectors pointing in directions similar
to that of the path seem to be roughly the same as the number and size of the
vectors pointing in the opposite direction. To check, we calculate
∂
∂
(sin y) =cosy = (1 + x cos y). ThusF is conservative, by
∂y ∂x
Theorem 6.
27. Since F is conservative, there exists a function f such that F = ∇f,thatis,P = f x, Q = f y ,andR = f z.SinceP ,
Q and R have continuous first order partial derivatives, Clairaut’s Theorem says that ∂P/∂y = f xy = f yx = ∂Q/∂x,
∂P/∂z = f xz = f zx = ∂R/∂x,and∂Q/∂z = f yz = f zy = ∂R/∂y.

F.
278 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
29. D = {(x, y) | x>0, y>0} = the first quadrant (excluding the axes).
(a) D is open because around every point in D we can put a disk that lies in D.
(b) D is connected because the straight line segment joining any two points in D lies in D.
(c) D is simply-connected because it’s connected and has no holes.
31. D = (x, y) | 1

F.
TX.10
(b) xy dx + C x2 y 3 dy =
∂
D ∂x (x2 y 3 ) − ∂ (xy) ∂y
dA = 1
2x
0 0 (2xy3 − x) dy dx
= 1
1
0 2 xy4 − xy y=2x
dx = 1
y=0 0 (8x5 − 2x 2 ) dx = 4 − 2 = 2 3 3 3
SECTION 17.4 GREEN’S THEOREM ET SECTION 16.4 ¤ 279
5. The region D enclosed by C is given by {(x, y) | 0 ≤ x ≤ 2,x≤ y ≤ 2x},so
C xy2 dx +2x 2 ydy =
∂
D ∂x (2x2 y) − ∂
∂y (xy2 ) dA
= 2
0
= 2
0
2x
x
(4xy − 2xy) dy dx
xy
2 y=2x
dx
y=x
= 2
0 3x3 dx = 3 4 x4 2
0 =12
7.
C
y + e √
x
dx +(2x +cosy 2 ) dy = D
∂
(2x ∂x +cosy2 ) − ∂
∂y
y + e √
x
dA
= 1 √ y
(2 − 1) dx dy = 1
0 y 2 0 (y1/2 − y 2 ) dy = 1 3
9.
C y3 dx − x 3 dy =
∂
D ∂x (−x3 ) − ∂
∂y (y3 ) dA = D (−3x2 − 3y 2 ) dA = 2π
2
0 0 (−3r2 ) rdrdθ
= −3 2π
0
dθ 2
0 r3 dr = −3(2π)(4) = −24π
11. F(x, y) = √ x + y 3 ,x 2 + √ y and the region D enclosed by C is given by {(x, y) | 0 ≤ x ≤ π, 0 ≤ y ≤ sin x}.
C is traversed clockwise, so −C gives the positive orientation.
F · d r = − √
C −C x + y
3
dx + x 2 + √ y dy = −
∂
x 2 + √
y − ∂
D ∂x
∂y x + y
3
dA
= − π
sin x
(2x − 3y 2 ) dy dx = − π
0 0 0 2xy − y
3 y=sin x
dx
= − π
0 (2x sin x − sin3 x) dx = − π
0 (2x sin x − (1 − cos2 x)sinx) dx
= − 2sinx − 2x cos x +cosx − 1 3 cos3 x π
= −
2π − 2+ 2 3 =
4
− 2π 3
y=0
0
[integrate by parts in the first term]
13. F(x, y) = e x + x 2 y, e y − xy 2 and the region D enclosed by C is the disk x 2 + y 2 ≤ 25.
C is traversed clockwise, so −C gives the positive orientation.
F · d r = − C −C (ex + x 2 y) dx +(e y − xy 2 ) dy = −
∂
D ∂x (ey − xy 2 ) − ∂
∂y (ex + x 2 y) dA
= − D (−y2 − x 2 ) dA = D (x2 + y 2 ) dA = 2π
5
0 0 (r2 ) rdrdθ
= 2π
dθ 5
0 0 r3 dr =2π 1
r4 5
= 625 π
4 0 2

F.
280 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
15. Here C = C 1 + C 2 where
C 1 can be parametrized as x = t, y =1, −1 ≤ t ≤ 1,and
C 2 is given by x = −t, y =2− t 2 , −1 ≤ t ≤ 1.
Thenthelineintegralis
C 1 +C 2
y 2 e x dx + x 2 e y dy = 1
−1 [1 · et + t 2 e · 0] dt
+ 1
−1 [(2 − t2 ) 2 e −t (−1) + (−t) 2 e 2−t2 (−2t)] dt
= 1
−1 [et − (2 − t 2 ) 2 e −t − 2t 3 e 2−t2 ] dt = −8e +48e −1
according to a CAS. The double integral is
∂Q
∂x − ∂P 1
2−x
2
dA = (2xe y − 2ye x ) dy dx = −8e +48e −1 , verifying Green’s Theorem in this case.
∂y
D
−1
1
17. By Green’s Theorem, W = C F · dr = C x(x + y) dx + xy2 dy = D (y2 − x) dy dx where C is the path described in the
question and D is the triangle bounded by C. So
W = 1 1−x
(y 2 − x) dy dx = 1
1
0 0 0 3 y3 − xy y =1−x
dx = 1
1 (1 − y =0 0 3 x)3 − x(1 − x) dx
= − 1 (1 − 12 x)4 − 1 2 x2 + 1 x3 1
=
3
− 1 + 1
0 2 3 − −
1
12 = −
1
12
19. Let C 1 be the arch of the cycloid from (0, 0) to (2π, 0), which corresponds to 0 ≤ t ≤ 2π,andletC 2 be the segment from
(2π, 0) to (0, 0),soC 2 is given by x =2π − t, y =0, 0 ≤ t ≤ 2π. ThenC = C 1 ∪ C 2 is traversed clockwise, so −C is
oriented positively. Thus −C encloses the area under one arch of the cycloid and from (5) we have
A = − −C ydx= C 1
ydx+ C 2
ydx= 2π
0 (1 − cos t)(1 − cos t) dt + 2π
0
0(−dt)
= 2π
(1 − 2cost 0 +cos2 t) dt +0= t − 2sint + 1 t + 1 sin 2t 2π
=3π
2 4 0
21. (a) Using Equation 17.2.8 [ ET 16.2.8], we write parametric equations of the line segment as x =(1− t)x 1 + tx 2 ,
y =(1− t)y 1 + ty 2 , 0 ≤ t ≤ 1. Thendx =(x 2 − x 1 ) dt and dy =(y 2 − y 1 ) dt,so
C xdy− ydx= 1
0 [(1 − t)x 1 + tx 2 ](y 2 − y 1 ) dt +[(1− t)y 1 + ty 2 ](x 2 − x 1 ) dt
= 1
(x1(y2 − y1) − y1(x2 − x1)+t[(y2 − y1)(x2 − x1) − (x2 − x1)(y2 − y1)]) dt
0
= 1
(x 0 1y 2 − x 2 y 1 ) dt = x 1 y 2 − x 2 y 1
(b) We apply Green’s Theorem to the path C = C 1 ∪ C 2 ∪ ···∪ C n ,whereC i is the line segment that joins (x i ,y i ) to
(x i+1 ,y i+1 ) for i =1, 2, ..., n − 1, andC n is the line segment that joins (x n ,y n ) to (x 1 ,y 1 ).From(5),
1
xdy− ydx= dA,whereD is the polygon bounded by C. Therefore
2 C D
area of polygon = A(D) = dA =
1 xdy− ydx
D 2 C
= 1 2 C 1
xdy− ydx+ C 2
xdy− ydx+ ···+ C n−1
xdy− ydx+
C n
xdy− ydx
To evaluate these integrals we use the formula from (a) to get
A(D) = 1 2 [(x 1y 2 − x 2 y 1 )+(x 2 y 3 − x 3 y 2 )+···+(x n−1 y n − x n y n−1 )+(x n y 1 − x 1 y n )].

F.
TX.10
SECTION 17.4 GREEN’S THEOREM ET SECTION 16.4 ¤ 281
(c) A = 1 2
[(0 · 1 − 2 · 0) + (2 · 3 − 1 · 1) + (1 · 2 − 0 · 3) + (0 · 1 − (−1) · 2) + (−1 · 0 − 0 · 1)]
= 1 2 (0+5+2+2)= 9 2
23. We orient the quarter-circular region as shown in the figure.
A = 1 4 πa2 so x = 1
x 2 dy and y = − 1
πa 2 /2
πa 2 /2
C
C
y 2 dx.
Here C = C 1 + C 2 + C 3 where C 1 : x = t, y =0, 0 ≤ t ≤ a;
C 2 : x = a cos t, y = a sin t, 0 ≤ t ≤ π 2 ;and
C 3 : x =0, y = a − t, 0 ≤ t ≤ a. Then
C x2 dy = C 1
x 2 dy + C 2
x 2 dy + C 3
x 2 dy = a
0 0 dt + π/2
0
(a cos t) 2 (a cos t) dt + a
0 0 dt
so x = 1
πa 2 /2
= π/2
a 3 cos 3 tdt= a 3 π/2
(1 − sin 2 t)costdt= a 3 sin t − 1 0 0 3 sin3 t π/2
= 2 0 3 a3
x 2 dy = 4a
C 3π .
C y2 dx = C 1
y 2 dx + C 2
y 2 dx + C 3
y 2 dx = a
0 0 dt + π/2
0
(a sin t) 2 (−a sin t) dt + a
0 0 dt
so y = − 1
πa 2 /2
= π/2
(−a 3 sin 3 t) dt = −a 3 π/2
(1 − cos 2 t)sintdt= −a 3 1
0 0 3 cos3 t − cos t π/2
= − 2 0 3 a3 ,
y 2 dx = 4a
4a
C 3π . Thus (x, y) = 3π , 4a
3π
25. By Green’s Theorem, − 1 3 ρ C y3 dx = − 1 3 ρ D (−3y2 ) dA = D y2 ρdA = I x and
1
ρ 3 C x3 dy = 1 ρ 3 D (3x2 ) dA = D x2 ρdA= I y .
27. Since C is a simple closed path which doesn’t pass through or enclose the origin, there exists an open region that doesn’t
.
contain the origin but does contain D. ThusP = −y/(x 2 + y 2 ) and Q = x/(x 2 + y 2 ) have continuous partial derivatives on
this open region containing D and we can apply Green’s Theorem. But by Exercise 17.3.33(a) [ ET 16.3.33(a)],
∂P/∂y = ∂Q/∂x,so F · dr = 0 dA =0.
C D
29. Using the firstpartof(5),wehavethat dx dy = A(R) = ∂h ∂h
xdy.Butx = g(u, v),anddy = du +
R ∂R
∂u ∂v dv,
and we orient ∂S by taking the positive direction to be that which corresponds, under the mapping, to the positive direction
along ∂R,so
∂R
∂h
xdy= g(u, v)
∂S ∂u
= ± S
= ± S
= ± S
∂
∂u
∂h
du +
∂v dv = g(u, v) ∂h
∂S ∂u
g(u, v)
∂h
g(u, v)
∂h
∂v
−
∂
∂v
∂g ∂h
+ g(u, v) ∂2 h
− ∂g ∂h
− g(u, v) ∂2 h
∂u ∂v ∂u ∂v ∂v ∂u ∂v ∂u
∂x ∂y
− ∂x ∂y
∂u ∂v ∂v ∂u
du + g(u, v)
∂h
∂v dv
∂u dA [using Green’s Theorem in the uv-plane]
dA [using the Chain Rule]
dA [by the equality of mixed partials] = ±
S
∂(x,y)
du dv
∂(u,v)
The sign is chosen to be positive if the orientation that we gave to ∂S corresponds to the usual positive orientation, and it is
negative otherwise. In either case, since A(R) is positive, the sign chosen must be the same as the sign of
Therefore A(R) = dx dy =
R
S
∂(x, y)
∂(u, v) du dv.
∂ (x, y)
∂(u, v) .

F.
282 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
17.5 Curl and Divergence ET 16.5
i j k
1. (a) curl F = ∇ × F =
∂/∂x ∂/∂y ∂/∂z
=(−x 2 − 0) i − (−2xy − xy) j +(0− xz) k
xyz 0 −x 2 y
= −x 2 i +3xy j − xz k
(b) div F = ∇ · F = ∂
∂x (xyz)+ ∂ ∂y (0) + ∂ ∂z (−x2 y)=yz +0+0=yz
i j k
3. (a) curl F = ∇ × F =
∂/∂x ∂/∂y ∂/∂z
1 x + yz xy − √ =(x − y) i − (y − 0) j +(1− 0) k
z
=(x − y) i − y j + k
(b) div F = ∇ · F = ∂
∂x (1) + ∂ ∂y (x + yz)+ ∂
xy − √
z = z − 1
∂z
2 √ z
i j k
∂/∂x ∂/∂y ∂/∂z
5. (a) curl F = ∇ × F =
(b) div F = ∇ · F = ∂
∂x
x
x2 + y 2 + z 2
y
x2 + y 2 + z 2
z
x2 + y 2 + z 2
1
=
[(−yz + yz) i − (−xz + xz) j +(−xy + xy) k] =0
(x 2 + y 2 + z 2 )
3/2
x
+ ∂ y
+ ∂
x2 + y 2 + z 2 ∂y x2 + y 2 + z 2 ∂z
z
x2 + y 2 + z 2
= x2 + y 2 + z 2 − x 2
(x 2 + y 2 + z 2 ) 3/2 + x2 + y 2 + z 2 − y 2
(x 2 + y 2 + z 2 ) 3/2 + x2 + y 2 + z 2 − z 2
(x 2 + y 2 + z 2 ) 3/2 = 2x2 +2y 2 +2z 2
(x 2 + y 2 + z 2 ) 3/2 = 2
x2 + y 2 + z 2
i j k
7. (a) curl F = ∇ × F =
∂/∂x ∂/∂y ∂/∂z
ln x ln(xy) ln(xyz)
xz
yz
y 1
=
xyz − 0 i −
xyz − 0 j +
xy − 0 k =
y , − 1 x , 1
x
(b) div F = ∇ · F = ∂
∂x (ln x)+ ∂ ∂y (ln(xy)) + ∂ ∂z (ln(xyz)) = 1 x + x xy + xy
xyz = 1 x + 1 y + 1 z
9. If the vector field is F = P i + Q j + R k, then we know R =0. In addition, the x-component of each vector of F is 0,so
P =0, hence ∂P
∂x = ∂P
∂y = ∂P
∂z = ∂R
∂x = ∂R
∂y = ∂R
∂Q
=0. Q decreases as y increases, so < 0,butQ doesn’t change
∂z ∂y
in the x-orz-directions, so ∂Q
∂x = ∂Q
∂z =0.
(a) div F = ∂P
∂x + ∂Q
∂y + ∂R
(b) curl F =
∂R
∂y − ∂Q
∂z
∂z =0+∂Q ∂y +0< 0
∂P
i +
∂z − ∂R
j +
∂x
∂Q
∂x − ∂P
∂y
k =(0− 0) i +(0− 0) j +(0− 0) k = 0

F.
TX.10
SECTION 17.5 CURL AND DIVERGENCE ET SECTION 16.5 ¤ 283
11. If the vector field is F = P i + Q j + R k, then we know R =0. In addition, the y-component of each vector of F is 0,so
Q =0,hence ∂Q
∂x = ∂Q
∂y = ∂Q
∂z = ∂R
∂x = ∂R
∂y = ∂R
∂P
=0. P increases as y increases, so
∂z ∂y
the x-orz-directions, so ∂P
∂x = ∂P
∂z =0.
(a) div F = ∂P
∂x + ∂Q
∂y + ∂R
∂R
(b) curl F =
∂y − ∂Q
∂z
Since ∂P
∂y
∂z =0+0+0=0
∂P
i +
∂z − ∂R
∂x
∂Q
j +
∂x − ∂P
k =(0− 0) i +(0− 0) j +
∂y
> 0, −∂P k is a vector pointing in the negative z-direction.
∂y
> 0,butP doesn’t change in
0 − ∂P
∂y
k = − ∂P
∂y k
i j k 13. curl F = ∇ × F =
∂/∂x ∂/∂y ∂/∂z =(6xyz 2 − 6xyz 2 ) i − (3y 2 z 2 − 3y 2 z 2 ) j +(2yz 3 − 2yz 3 ) k = 0
y 2 z 3 2xyz 3 3xy 2 z 2
and F is definedonallofR 3 with component functions which have continuous partial derivatives, so by Theorem 4,
F is conservative. Thus, there exists a function f such that F = ∇f. Thenf x(x, y, z) =y 2 z 3 implies
f(x, y, z) =xy 2 z 3 + g(y, z) and f y(x, y, z) =2xyz 3 + g y(y, z). Butf y(x, y, z) =2xyz 3 ,sog(y, z) =h(z) and
f(x, y, z) =xy 2 z 3 + h(z). Thus f z (x, y, z) =3xy 2 z 2 + h 0 (z) but f z (x, y, z) =3xy 2 z 2 so h(z) =K, a constant.
Hence a potential function for F is f(x, y, z) =xy 2 z 3 + K.
i j k 15. curl F = ∇ × F =
∂/∂x ∂/∂y ∂/∂z =(2y − 2y) i − (0 − 0) j +(2x − 2x) k = 0, F is definedonallofR 3 ,
2xy x 2 +2yz y 2
and the partial derivatives of the component functions are continuous, so F is conservative. Thus there exists a function f
such that ∇f = F. Thenf x(x, y, z) =2xy implies f(x, y, z) =x 2 y + g(y, z) and f y(x, y, z) =x 2 + g y(y, z). But
f y(x, y, z) =x 2 +2yz,sog(y, z) =y 2 z + h(z) and f(x, y, z) =x 2 y + y 2 z + h(z). Thusf z(x, y, z) =y 2 + h 0 (z) but
f z(x, y, z) =y 2 so h(z) =K and f(x, y, z) =x 2 y + y 2 z + K.
i j k
17. curl F = ∇ × F =
∂/∂x ∂/∂y ∂/∂z
=(0− 0) i − (0 − 0) j +(−e −x − e −x ) k = −2e −x k 6=0,
ye −x e −x 2z
so F is not conservative.
19. No. Assume there is such a G. Thendiv(curl G) = ∂
∂x (x sin y)+ ∂ ∂y (cos y)+ ∂ (z − xy) =siny − sin y +16= 0,
∂z
which contradicts Theorem 11.
i j k
21. curl F =
∂/∂x ∂/∂y ∂/∂z
=(0− 0) i +(0− 0) j +(0− 0) k = 0. Hence F = f(x) i + g(y) j + h(z) k
f(x) g(y) h(z)
is irrotational.

F.
284 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
For Exercises 23–29, let F(x, y, z) =P 1 i + Q 1 j + R 1 k and G(x, y, z) =P 2 i + Q 2 j + R 2 k.
23. div(F + G) =divhP 1 + P 2 ,Q 1 + Q 2 ,R 1 + R 2 i =
∂(P1 + P2)
∂x
+
∂(Q1 + Q2)
∂y
= ∂P 1
∂x + ∂P 2
∂x + ∂Q 1
∂y + ∂Q 2
∂y + ∂R 1
∂z + ∂R
2
∂z
= ∂P1
∂x + ∂Q 1
∂y + ∂R 1
∂z
=divhP 1,Q 1,R 1i +divhP 2,Q 2,R 2i =divF +divG
∂(R1 + R2)
+
∂z
+
25. div(fF) =div(f hP 1,Q 1,R 1i) =divhfP 1,fQ 1,fR 1i = ∂(fP 1)
+ ∂(fQ 1)
+ ∂(fR 1)
∂x ∂y ∂z
= f ∂P
1
∂x + ∂f
P1 + f ∂Q
1
∂x ∂y + ∂f
Q1 + f ∂R
1
∂y ∂z
+ ∂f
R1
∂z
∂P1
= f
∂x + ∂Q 1
∂y + ∂R
1
∂f
+ hP 1,Q 1,R 1i·
∂z
∂x , ∂f
∂y , ∂f
= f div F + F · ∇f
∂z
∂P2
∂x + ∂Q 2
∂y + ∂R 2
∂z
∂/∂x ∂/∂y ∂/∂z 27. div(F × G)=∇ · (F × G) =
P 1 Q 1 R 1 = ∂
Q 1 R 1 − ∂ P 1 R 1 + ∂ P 1 Q 1
∂x Q
2 R 2
∂y P 2 R 2
∂z P 2 Q 2
P 2 Q 2 R 2
∂R 2
= Q 1
∂x + R ∂Q 1
2
∂x − Q ∂R 1
2
∂x − R ∂Q 2 ∂R 2
1 − P 1
∂x ∂y + R ∂P 1
2
∂y − P ∂R 1
2
∂y − R ∂P 2
1
∂y
∂Q 2
+ P 1
∂z + Q ∂P 1
2
∂z − P ∂Q 1
2
∂z − Q 1
∂P 2
∂z
∂R1
= P 2
∂y − ∂Q
1 ∂P1
+ Q 2
∂z ∂z − ∂R
1 ∂Q1
+ R 2
∂x ∂x − ∂P
1
∂y
∂R2
− P 1
∂y − ∂Q
2 ∂P2
+ Q 1
∂z ∂z − ∂R
2 ∂Q2
+ R 1
∂x ∂x − ∂P
2
∂y
= G · curl F − F · curl G
i j k
29. curl(curl F) =∇ × (∇ × F) =
∂/∂x ∂/∂y ∂/∂z
∂R 1 /∂y − ∂Q 1 /∂z ∂P 1 /∂z − ∂R 1 /∂x ∂Q 1 /∂x − ∂P 1 /∂y
∂ 2 Q 1
=
∂y∂x − ∂2 P 1
∂y − ∂2 P 1
2 ∂z + ∂2 R 1 ∂ 2 R 1
i +
2 ∂z∂x ∂z∂y − ∂2 Q 1
∂z − ∂2 Q 1
2 ∂x + ∂2 P 1
j
2 ∂x∂y
∂ 2 P 1
+
∂x∂z − ∂2 R 1
∂x − ∂2 R 1
2 ∂y + ∂2 Q 1
k
2 ∂y∂z
Now let’s consider grad(div F) −∇ 2 F and compare with the above.
(Note that ∇ 2 F is defined on page 1102 [ ET 1066].)

F.
TX.10 SECTION 17.5 CURL AND DIVERGENCE ET SECTION 16.5 ¤ 285
∂
grad(div F) −∇ 2 2 P 1
F =
∂x + ∂2 Q 1
2 ∂x∂y + ∂2 R 1 ∂ 2 P 1
i +
∂x∂z ∂y∂x + ∂2 Q 1
∂y + ∂2 R 1 ∂ 2 P 1
j +
2 ∂y∂z ∂z∂x + ∂2 Q 1
∂z∂y + ∂2 R 1
k
∂z 2
∂ 2 P 1
−
∂x + ∂2 P 1
2 ∂y + ∂2 P 1 ∂ 2 Q 1
i +
2 ∂z 2 ∂x + ∂2 Q 1
2 ∂y + ∂2 Q 1
j
2 ∂z 2
∂ 2 R 1
+
∂x + ∂2 R 1
2 ∂y + ∂2 R 1
k
2 ∂z 2
∂ 2 Q 1
=
∂x∂y + ∂2 R 1
∂x∂z − ∂2 P 1
∂y − ∂2 P 1 ∂ 2 P 1
i +
2 ∂z 2 ∂y∂x + ∂2 R 1
∂y∂z − ∂2 Q 1
∂x − ∂2 Q 1
j
2 ∂z 2
∂ 2 P 1
+
∂z∂x + ∂2 Q 1
∂z∂y − ∂2 R 1
∂x − ∂2 R 2
k
2 ∂y 2
Then applying Clairaut’s Theorem to reverse the order of differentiation in the second partial derivatives as needed and
comparing, we have curl curl F =graddivF −∇ 2 F as desired.
31. (a) ∇r = ∇ x 2 + y 2 + z 2 =
(b) ∇ × r =
∂
∂x
1
(c) ∇ = ∇
r
i j k
∂
∂y
∂
∂z
x y z
1
x2 + y 2 + z 2
x
x2 + y 2 + z 2 i +
1
−
2 x
=
2 + y 2 + z (2x)
2 i −
x 2 + y 2 + z 2
= − x i + y j + z k
(x 2 + y 2 + z 2 ) 3/2 = − r
r 3
y
x2 + y 2 + z 2 j +
z
x2 + y 2 + z k = x i + y j + z k
2 x2 + y 2 + z = r 2 r
∂
=
∂y (z) − ∂ ∂
∂z (y) i +
∂z (x) − ∂ ∂
∂x (z) j +
∂x (y) − ∂
∂y (x) k = 0
1
2 x 2 + y 2 + z 2 (2y)
x 2 + y 2 + z 2
(d) ∇ ln r = ∇ ln(x 2 + y 2 + z 2 ) 1/2 = 1 2 ∇ ln(x2 + y 2 + z 2 )
=
x
x 2 + y 2 + z 2 i +
y
x 2 + y 2 + z 2 j +
j −
1
2 x 2 + y 2 + z 2 (2z)
x 2 + y 2 + z 2
z
x 2 + y 2 + z k = x i + y j + z k
2 x 2 + y 2 + z = r
2 r 2
33. By (13), C f(∇g) · n ds = D div(f∇g) dA = D [f div(∇g)+∇g · ∇f] dA by Exercise 25. But div(∇g) =∇2 g.
Hence D f∇2 gdA= f(∇g) · n ds − ∇g · ∇f dA.
C D
k
35. Let f(x, y) =1.Then∇f = 0 and Green’s first identity (see Exercise 33) says
D ∇2 gdA= (∇g) · n ds − 0 · ∇gdA ⇒ C D D ∇2 gdA= ∇g · n ds. Butg is harmonic on D,so
C
∇ 2 g =0 ⇒ ∇g · n ds =0and D C C ngds= (∇g · n) ds =0.
C
37. (a) We know that ω = v/d, and from the diagram sin θ = d/r ⇒ v = dω =(sinθ)rω = |w × r|. Butv is perpendicular
to both w and r,sothatv = w × r.

F.
286 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
i j k
(b) From (a), v = w × r =
0 0 ω
=(0· z − ωy) i +(ωx − 0 · z) j +(0· y − x · 0) k = −ωy i + ωx j
x y z
i j k
(c) curl v = ∇ × v =
∂/∂x ∂/∂y ∂/∂z
−ωy ωx 0
∂
=
∂y (0) − ∂
∂z (ωx) i +
=[ω − (−ω)] k =2ω k =2w
TX.10
∂
∂z (−ωy) − ∂
∂x (0)
j +
∂
∂x (ωx) − ∂ ∂y (−ωy)
k
39. For any continuous function f on R 3 ,defineavectorfield G(x, y, z) =hg(x, y, z), 0, 0i where g(x, y, z) = x
f (t, y, z) dt.
0
Then div G = ∂
∂x (g(x, y, z)) + ∂ ∂y (0) + ∂ ∂z (0) = ∂ x
f(t, y, z) dt = f(x, y, z) by the Fundamental Theorem of
∂x
0
Calculus. Thus every continuous function f on R 3 is the divergence of some vector field.
17.6 Parametric Surfaces and Their Areas ET 16.6
1. P (7, 10, 4) lies on the parametric surface r(u, v) =h2u +3v, 1+5u − v, 2+u + vi if and only if there are values for u
and v where 2u +3v =7, 1+5u − v =10,and2+u + v =4.Butsolvingthefirst two equations simultaneously gives
u =2, v =1and these values do not satisfy the third equation, so P does not lie on the surface.
Q(5, 22, 5) lies on the surface if 2u +3v =5, 1+5u − v =22,and2+u + v =5for some values of u and v. Solving the
first two equations simultaneously gives u =4, v = −1 and these values satisfy the third equation, so Q lies on the surface.
3. r(u, v) =(u + v) i +(3− v) j +(1+4u +5v) k = h0, 3, 1i + u h1, 0, 4i + v h1, −1, 5i. From Example 3, we recognize
this as a vector equation of a plane through the point (0, 3, 1) and containing vectors a = h1, 0, 4i and b = h1, −1, 5i. Ifwe
i j k
wish to find a more conventional equation for the plane, a normal vector to the plane is a × b =
1 0 4
=4i − j − k
1 −1 5
andanequationoftheplaneis4(x − 0) − (y − 3) − (z − 1) = 0 or 4x − y − z = −4.
5. r(s, t) = s, t, t 2 − s 2 , so the corresponding parametric equations for the surface are x = s, y = t, z = t 2 − s 2 .Forany
point (x, y, z) onthesurface,wehavez = y 2 − x 2 . With no restrictions on the parameters, the surface is z = y 2 − x 2 ,which
we recognize as a hyperbolic paraboloid.

F.
SECTIONTX.10
17.6 PARAMETRIC SURFACES AND THEIR AREAS ET SECTION 16.6 ¤ 287
7. r(u, v) = u 2 +1,v 3 +1,u+ v , −1 ≤ u ≤ 1, −1 ≤ v ≤ 1.
The surface has parametric equations x = u 2 +1, y = v 3 +1, z = u + v,
−1 ≤ u ≤ 1, −1 ≤ v ≤ 1. In Maple, the surface can be graphed by entering
plot3d([uˆ2+1,vˆ3+1,u+v],u=-1..1,v=-1..1);. In
Mathematica we use the ParametricPlot3D command. If we keep u
constant at u 0, x = u 2 0 +1, a constant, so the corresponding grid curves must
be the curves parallel to the yz-plane. If v is constant, we have y = v0 3 +1,
a constant, so these grid curves are the curves parallel to the xz-plane.
9. r(u, v) = u cos v, u sin v, u 5 .
The surface has parametric equations x = u cos v, y = u sin v,
z = u 5 , −1 ≤ u ≤ 1, 0 ≤ v ≤ 2π. Note that if u = u 0 is constant
then z = u 5 0 is constant and x = u 0 cos v, y = u 0 sin v describe a
circle in x, y of radius |u 0 |, so the corresponding grid curves are
circles parallel to the xy-plane. If v = v 0 , a constant, the parametric
equations become x = u cos v 0, y = u sin v 0, z = u 5 .Then
y =(tanv 0 )x, so these are the grid curves we see that lie in vertical
planes y = kx through the z-axis.
11. x =sinv, y =cosu sin 4v, z =sin2u sin 4v, 0 ≤ u ≤ 2π, − π ≤ v ≤ π .
2 2
Note that if v = v 0 is constant, then x =sinv 0 is constant, so the
corresponding grid curves must be parallel to the yz-plane. These
are the vertically oriented grid curves we see, each shaped like a
“figure-eight.” When u = u 0 is held constant, the parametric
equations become x =sinv, y =cosu 0 sin 4v,
z =sin2u 0 sin 4v. Sincez is a constant multiple of y,the
corresponding grid curves are the curves contained in planes
z = ky that pass through the x-axis.
13. r(u, v) =u cos v i + u sin v j + v k. The parametric equations for the surface are x = u cos v, y = u sin v, z = v. We look at
the grid curves first; if we fix v,thenx and y parametrizeastraightlineintheplanez = v which intersects the z-axis. If u is
held constant, the projection onto the xy-plane is circular; with z = v, each grid curve is a helix. The surface is a spiraling
ramp, graph I.
15. r(u, v) =sinv i +cosu sin 2v j +sinu sin 2v k. Parametric equations for the surface are x =sinv, y =cosu sin 2v,
z =sinu sin 2v. Ifv = v 0 is fixed, then x =sinv 0 is constant, and y =(sin2v 0 )cosu and z =(sin2v 0 )sinu describe a
circle of radius |sin 2v 0 |, so each corresponding grid curve is a circle contained in the vertical plane x =sinv 0 parallel to the

F.
288 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
yz-plane. The only possible surface is graph II. The grid curves we see running lengthwise along the surface correspond to
holding u constant, in which case y =(cosu 0)sin2v, z =(sinu 0)sin2v ⇒ z = (tan u 0)y,soeachgridcurveliesina
plane z = ky that includes the x-axis.
17. x =cos 3 u cos 3 v, y =sin 3 u cos 3 v, z =sin 3 v.Ifv = v 0 is held constant then z =sin 3 v 0 is constant, so the
corresponding grid curve lies in a horizontal plane. Several of the graphs exhibit horizontal grid curves, but the curves for this
surface are neither circles nor straight lines, so graph III is the only possibility. (In fact, the horizontal grid curves here are
members of the family x = a cos 3 u, y = a sin 3 u and are called astroids.) The vertical grid curves we see on the surface
correspond to u = u 0 held constant, as then we have x =cos 3 u 0 cos 3 v, y =sin 3 u 0 cos 3 v so the corresponding grid curve
lies in the vertical plane y =(tan 3 u 0 )x through the z-axis.
19. From Example 3, parametric equations for the plane through the point (1, 2, −3) that contains the vectors a = h1, 1, −1i and
b = h1, −1, 1i are x =1+u(1) + v(1) = 1 + u + v, y =2+u(1) + v(−1) = 2 + u − v,
z = −3+u(−1) + v(1) = −3 − u + v.
21. Solving the equation for y gives y 2 =1− x 2 + z 2 ⇒ y = √ 1 − x 2 + z 2 . (We choose the positive root since we want the
part of the hyperboloid that corresponds to y ≥ 0.) If we let x and z be the parameters, parametric equations are x = x, z = z,
y = √ 1 − x 2 + z 2 .
23. Since the cone intersects the sphere in the circle x 2 + y 2 =2, z = √ 2 and we want the portion of the sphere above this, we
can parametrize the surface as x = x, y = y, z = 4 − x 2 − y 2 where x 2 + y 2 ≤ 2.
Alternate solution: Using spherical coordinates, x =2sinφ cos θ, y =2sinφ sin θ, z =2cosφ where 0 ≤ φ ≤ π and 4
0 ≤ θ ≤ 2π.
25. Parametric equations are x = x, y =4cosθ, z =4sinθ, 0 ≤ x ≤ 5, 0 ≤ θ ≤ 2π.
27. The surface appears to be a portion of a circular cylinder of radius 3 with axis the x-axis. An equation of the cylinder is
y 2 + z 2 =9, and we can impose the restrictions 0 ≤ x ≤ 5, y ≤ 0 to obtain the portion shown. To graph the surface on a
CAS, we can use parametric equations x = u, y =3cosv, z =3sinv with the parameter domain 0 ≤ u ≤ 5, π ≤ v ≤ 3π .
2 2
Alternatively, we can regard x and z as parameters. Then parametric equations are x = x, z = z, y = − √ 9 − z 2 ,where
0 ≤ x ≤ 5 and −3 ≤ z ≤ 3.
29. Using Equations 3, we have the parametrization x = x, y = e −x cos θ, z = e −x sin θ, 0 ≤ x ≤ 3, 0 ≤ θ ≤ 2π.

F.
SECTIONTX.10
17.6 PARAMETRIC SURFACES AND THEIR AREAS ET SECTION 16.6 ¤ 289
31. (a) Replacing cos u by sin u and sin u by cos u gives parametric equations
x =(2+sinv)sinu, y =(2+sinv)cosu, z = u +cosv. From the graph, it
appears that the direction of the spiral is reversed. We can verify this observation by
noting that the projection of the spiral grid curves onto the xy-plane, given by
x =(2+sinv)sinu, y =(2+sinv)cosu, z =0,drawsacircleintheclockwise
direction for each value of v. The original equations, on the other hand, give circular
projections drawn in the counterclockwise direction. The equation for z is identical in
both surfaces, so as z increases, these grid curves spiral up in opposite directions for
the two surfaces.
(b) Replacing cos u by cos 2u and sin u by sin 2u gives parametric equations
x =(2+sinv)cos2u, y =(2+sinv)sin2u, z = u +cosv. From the graph, it
appears that the number of coils in the surface doubles within the same parametric
domain. We can verify this observation by noting that the projection of the spiral grid
curves onto the xy-plane, given by x =(2+sinv)cos2u, y =(2+sinv)sin2u,
z =0(where v is constant), complete circular revolutions for 0 ≤ u ≤ π while the
original surface requires 0 ≤ u ≤ 2π for a complete revolution. Thus, the new
surface winds around twice as fast as the original surface, and since the equation for z
is identical in both surfaces, we observe twice as many circular coils in the same
z-interval.
33. r(u, v) =(u + v) i +3u 2 j +(u − v) k.
r u = i +6u j + k and r v = i − k,sor u × r v = −6u i +2j − 6u k.
Since the point (2, 3, 0) corresponds to u =1, v =1, a normal vector
to the surface at (2, 3, 0) is −6 i +2j − 6 k, and an equation of the
tangent plane is −6x +2y − 6z = −6 or 3x − y +3z =3.
35. r(u, v) =u 2 i +2u sin v j + u cos v k ⇒ r(1, 0) = (1, 0, 1).
r u =2u i +2sinv j +cosv k and r v =2u cos v j − u sin v k,
so a normal vector to the surface at the point (1, 0, 1) is
r u (1, 0) × r v (1, 0) = (2 i + k) × (2 j) =−2 i +4k.
Thus an equation of the tangent plane at (1, 0, 1) is
−2(x − 1) + 0(y − 0) + 4(z − 1) = 0 or −x +2z =1.

F.
290 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
37. The surface S is given by z = f(x, y) =6− 3x − 2y which intersects the xy-plane in the line 3x +2y =6,soD is the
triangular region given by (x, y) 0 ≤ x ≤ 2, 0 ≤ y ≤ 3 −
3
x 2
. By Formula 9, the surface area of S is
2 2 ∂z ∂z
A(S)= 1+ + dA
D ∂x ∂y
=
1+(−3)2 +(−2)
D
2 dA = √ 14 dA = √ 14 A(D) = √ 14 1 · 2 · 3 =3 √ 14.
D 2
39. z = f(x, y) = 2 3 (x3/2 + y 3/2 ) and D = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 }. Thenf x = x 1/2 , f y = y 1/2 and
A(S)=
1+( √ x ) 2 + √ 2
D
y dA = 1
1 √
0 0 1+x + ydydx
=
y=1
1 2
(x + y
0 3 +1)3/2 dx = 2 1
(x +2) 3/2 3
− (x +1) 3/2 dx
0
y=0
1
= 2 2
(x 3 5 +2)5/2 − 2 (x 5 +1)5/2 = 4 15 (35/2 − 2 5/2 − 2 5/2 +1)= 4
15 (35/2 − 2 7/2 +1)
0
41. z = f(x, y) =xy with 0 ≤ x 2 + y 2 ≤ 1,sof x = y, f y = x ⇒
A(S)= D
= 2π
0
1+y2 + x 2 dA = 2π
0
1
3
2
√
2 − 1
dθ =
2π
3
1
0
√
r2 +1rdrdθ= 2π
0
2
√
2 − 1
r=1
1
3 (r2 +1) 3/2 dθ
r=0
43. z = f(x, y) =y 2 − x 2 with 1 ≤ x 2 + y 2 ≤ 4. Then
A(S)=
1+4x2 +4y
D
2 dA = 2π 2
√
1+4r2 rdrdθ= 2π
dθ 2
r √ 1+4r
0 1 0 1 2 dr
= θ
2π
2
1
(1 + √ √
0 12 4r2 ) 3/2 = π 6 17 17 − 5 5
1
45. A parametric representation of the surface is x = x, y =4x + z 2 , z = z with 0 ≤ x ≤ 1, 0 ≤ z ≤ 1.
Hence r x × r z =(i +4j) × (2z j + k) =4i − j +2z k.
Note: In general, if y = f(x, z) then r x × r z = ∂f
∂x i − j + ∂f
2 2 ∂f ∂f
∂z
D
k and A (S) = 1+ + dA. Then
∂x ∂z
A(S)= 1 1
√
0 0 17 + 4z2 dx dz = 1
√
0 17 + 4z2 dz
√
= 1 2 z 17 + 4z2 + 17 ln √
2 2z + 4z2 +17 1
= √ √ √
21
0 2
+ 17 4 ln 2+ 21 − ln 17
47. r u = h2u, v, 0i, r v = h0,u,vi,andr u × r v = v 2 , −2uv, 2u 2 .Then
A(S)= D |ru × rv| dA = 1 2
0 0
= 1
0
2
0 (v2 +2u 2 ) dv du = 1
0
√
v4 +4u 2 v 2 +4u 4 dv du = 1 2
0 0
(v2 +2u 2 ) 2 dv du
1
3 v3 +2u 2 v v=2
du = 1
8 +4u2 du = 8
u + 4 u3 1
=4
v=0 0 3 3 3 0
49. z = f(x, y) =e −x2 −y 2 with x 2 + y 2 ≤ 4.
A(S) =
1+ −2xe
D
−x2 −y 22 + −2ye −x2 −y 22 dA =
1+4(x2 + y
D 2 )e −2(x2 +y 2 )
dA
= 2π
2
1+4r2 e
0 0 −2r2 rdrdθ= 2π
dθ 2
r 1+4r
0 0 2 e −2r2 dr =2π 2
r 1+4r
0 2 e −2r2 dr ≈ 13.9783

F.
51. (a) A(S) = 1+
D
2 ∂z
+
∂x
∂z
∂y
Using the Midpoint Rule with f(x, y) =
A(S) ≈
3
i =1j =1
(b) Using a CAS we have A(S) =
to the first decimal place.
53. z =1+2x +3y +4y 2 ,so
A(S) = 1+
D
Using a CAS, we have
4
1
1
0
or 45
8
SECTIONTX.10
17.6 PARAMETRIC SURFACES AND THEIR AREAS ET SECTION 16.6 ¤ 291
2
dA =
6
0
4
0
1+ 4x2 +4y 2
dy dx.
(1 + x 2 + y 2 )
4
1+ 4x2 +4y 2
, m =3, n =2we have
(1 + x 2 + y 2 )
4
2
f
x i , y j ∆A =4[f(1, 1) + f(1, 3) + f(3, 1) + f(3, 3) + f(5, 1) + f(5, 3)] ≈ 24.2055
2 ∂z
+
∂x
6
4
0
∂z
∂y
14 + 48y +64y2 dy dx = 45
8
√
14 +
15
ln 11 √ 5+3 √ 70
16 3 √ 5+ √ . 70
0
2
dA =
4
1+ 4x2 +4y 2
dy dx ≈ 24.2476. This agrees with the estimate in part (a)
(1 + x 2 + y 2 )
4
1
1
0
1+4+(3+8y)2 dy dx =
4
1
1
√
14 +
15
16 ln 11 √ 5+3 √ 14 √ 5 − 15
16 ln 3 √ 5+ √ 14 √ 5
0
14 + 48y +64y2 dy dx.
55. (a) x = a sin u cos v, y = b sin u sin v, z = c cos u ⇒
x 2
a + y2
2 b + z2
2 c =(sinu cos 2 v)2 +(sinu sin v) 2 +(cosu) 2
=sin 2 u +cos 2 u =1
and since the ranges of u and v are sufficient to generate the entire graph,
the parametric equations represent an ellipsoid.
(b)
(c) From the parametric equations (with a =1, b =2,andc =3), we calculate
r u =cosu cos v i +2cosu sin v j − 3sinu k and r v = − sin u sin v i +2sinu cos v j. So
r u × r v =6sin 2 u cos v i +3sin 2 u sin v j +2sinu cos u k, and the surface area is given by
A(S) = 2π
0
π
0
|ru × rv| du dv = 2π
0
π
0
36 sin 4 u cos 2 v +9sin 4 u sin 2 v +4cos 2 u sin 2 ududv
57. To find the region D: z = x 2 + y 2 implies z + z 2 =4z or z 2 − 3z =0.Thusz =0or z =3are the planes where the
surfaces intersect. But x 2 + y 2 + z 2 =4z implies x 2 + y 2 +(z − 2) 2 =4,soz =3intersects the upper hemisphere.
Thus (z − 2) 2 =4− x 2 − y 2 or z =2+ 4 − x 2 − y 2 . Therefore D is the region inside the circle x 2 + y 2 +(3− 2) 2 =4,
that is, D = (x, y) | x 2 + y 2 ≤ 3 .
A(S) = 1+[(−x)(4 − x 2 − y 2 ) −1/2 ] 2 +[(−y)(4 − x 2 − y 2 ) −1/2 ] 2 dA
D
2π
√
3
2π
√
=
1+ r2
3
4 − r rdrdθ= 2rdr
2π
√
−2(4 dθ = − r 2 ) 1/2 r= √ 3
2 4 − r
2 r=0
0
0
= 2π
0 (−2+4)dθ =2θ 2π
0
=4π
0
0
0
dθ

F.
292 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
59. Let A(S 1) be the surface area of that portion of the surface which lies above the plane z =0.ThenA(S) =2A(S 1).
Following Example 10, a parametric representation of S 1 is x = a sin φ cos θ, y = a sin φ sin θ,
z = a cos φ and |r φ × r θ | = a 2 sin φ. ForD, 0 ≤ φ ≤ π 2 and for each fixed φ, x − 1 2 a2 + y 2 ≤ 1
2 a2 or
a sin φ cos θ −
1
2 a2 + a 2 sin 2 φ sin 2 θ ≤ (a/2) 2 implies a 2 sin 2 φ − a 2 sin φ cos θ ≤ 0 or
sin φ (sin φ − cos θ) ≤ 0. But0 ≤ φ ≤ π 2 ,socos θ ≥ sin φ or sin π
2 + θ ≥ sin φ or φ − π 2 ≤ θ ≤ π 2 − φ.
Hence D = (φ, θ) | 0 ≤ φ ≤ π 2 , φ − π 2 ≤ θ ≤ π 2 − φ .Then
Thus A(S) =2a 2 (π − 2).
A(S 1 )= π/2 (π/2) − φ
0 φ − (π/2) a2 sin φdθdφ= a 2 π/2
(π − 2φ)sinφdφ
0
= a 2 [(−π cos φ) − 2(−φ cos φ +sinφ)] π/2
0
= a 2 (π − 2)
Alternate solution: Working on S 1 we could parametrize the portion of the sphere by x = x, y = y, z = a 2 − x 2 − y 2 .
x 2
Then |r x × r y| = 1+
a 2 − x 2 − y + y 2
2 a 2 − x 2 − y = a
2 a2 − x 2 − y and
2
A(S 1 )=
0 ≤ (x − (a/2)) 2 + y 2 ≤ (a/2) 2 a
= π/2
−π/2
Thus A(S) =4a 2 π
2 − 1 =2a 2 (π − 2).
Notes:
−a(a2 − r 2 ) 1/2 r = a cos θ
a2 − x 2 − y 2 dA = π/2
r =0
−π/2
a cos θ
0
a
√
a2 − r 2 rdrdθ
dθ = π/2
−π/2 a2 [1 − (1 − cos 2 θ) 1/2 ] dθ
= π/2
−π/2 a2 (1 − |sin θ|) dθ =2a 2 π/2
0
(1 − sin θ) dθ =2a 2 π
2 − 1
(1) Perhaps working in spherical coordinates is the most obvious approach here. However, you must be careful
in setting up D.
(2) In the alternate solution, you can avoid having to use |sin θ| by working in the first octant and then
multiplying by 4. However, if you set up S 1 as above and arrived at A(S 1)=a 2 π, you now see your error.
17.7 Surface Integrals ET 16.7
1. The faces of the box in the planes x =0and x =2have surface area 24 and centers (0, 2, 3), (2, 2, 3). The faces in y =0and
y =4have surface area 12 and centers (1, 0, 3), (1, 4, 3), and the faces in z =0and z =6have area 8 and centers (1, 2, 0),
(1, 2, 6). For each face we take the point Pij ∗ to be the center of the face and f(x, y, z) =e −0.1(x+y+z) ,sobyDefinition 1,
f(x, y, z) dS ≈ [f(0, 2, 3)](24) + [f(2, 2, 3)](24) + [f(1, 0, 3)](12)
S
+[f(1, 4, 3)](12) + [f(1, 2, 0)](8) + [f(1, 2, 6)](8)
=24(e −0.5 + e −0.7 ) + 12(e −0.4 + e −0.8 )+8(e −0.3 + e −0.9 ) ≈ 49.09

F.
TX.10
SECTION 17.7 SURFACE INTEGRALS ET SECTION 16.7 ¤ 293
3. We can use the xz-andyz-planes to divide H into four patches of equal size, each with surface area equal to 1 8
the surface
area of a sphere with radius √ 50,so∆S = 1 8 (4)π√ 50 2
=25π. Then(±3, ±4, 5) are sample points in the four patches,
and using a Riemann sum as in Definition 1, we have
f(x, y, z) dS ≈ f(3, 4, 5) ∆S + f(3, −4, 5) ∆S + f(−3, 4, 5) ∆S + f(−3, −4, 5) ∆S
5. z =1+2x +3y so ∂z ∂z
=2and
∂x
H
S x2 yz dS =
D
= √ 14 3
0
=(7+8+9+12)(25π) =900π ≈ 2827
∂y =3.ThenbyFormula4,
2 ∂z ∂z
+
∂x ∂y
x 2 yz
2
+1dA = 3
2
0 0 x2 y(1 + 2x +3y) √ 4+9+1dy dx
2
0 (x2 y +2x 3 y +3x 2 y 2 ) dy dx = √ 14 3
1
0 2 x2 y 2 + x 3 y 2 + x 2 y 3 y=2
dx y=0
= √ 14 3
0 (10x2 +4x 3 ) dx = √ 14 10
3 x3 + x 4 3
0 = 171 √ 14
7. S is the part of the plane z =1− x − y over the region D = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x}. Thus
S yz dS = y(1 − x − y) (−1)
D 2 +(−1) 2 +1dA = √ 3 1
1−x
0 0 y − xy − y
2
dy dx
= √ 3 1
1
0 2 y2 − 1 2 xy2 − 1 y3 y=1−x
3
dx = √ 3 1 1
(1 − y=0 0 6 x)3 dx = − √ 1
3
(1 − 24 x)4 = √ 3
24
0
9. r(u, v) =u 2 i + u sin v j + u cos v k, 0 ≤ u ≤ 1, 0 ≤ v ≤ π/2, so
r u × r v =(2u i +sinv j +cosv k) × (u cos v j − u sin v k) =−u i +2u 2 sin v j +2u 2 cos v k and
|r u × r v| = u 2 +4u 4 sin 2 v +4u 4 cos 2 v = u 2 +4u 4 (sin 2 v +cos 2 v)=u √ 1+4u 2 (since u ≥ 0). Then by
Formula 2,
S yz dS = (u sin v)(u cos v) D |ru × rv| dA = π/2
1 (u sin v)(u cos v) · u √ 1+4u
0 0 2 du dv
= 1
√
0 u3 1+4u 2 du π/2
sin v cos vdv
0 let t =1+4u
2
⇒ u 2 = 1 (t − 1) and 1 dt = udu
4 8
= 5 1 · 1 (t − 1)√ tdt π/2
sin v cos vdv= 1 5
t 3/2 − √
t dt π/2
sin v cos vdv
1 8 4 0 32 1
0
= 1 32
5
2
5 t5/2 − 2 1
3 t3/2 2 sin2 v π/2
= 1
1
0 32
2
5 (5)5/2 − 2 3 (5)3/2 − 2 + 2 5 3
√
· 1
5
(1 − 0) =
2 48 5+
1
240
11. S is the portion of the cone z 2 = x 2 + y 2 for 1 ≤ z ≤ 3, or equivalently, S is the part of the surface z = x 2 + y 2 over the
region D = (x, y) | 1 ≤ x 2 + y 2 ≤ 9 .Thus
2
x 2 z 2 dS = x 2 (x 2 + y 2 )
x
+
S
D
x2 + y 2
= x 2 (x 2 + y 2 )
D
x 2 + y 2
x 2 + y 2 +1dA = D
y
x2 + y 2 2
+1dA
√
2 x 2 (x 2 + y 2 ) dA = √ 2
2π
0
3
1
(r cos θ) 2 (r 2 ) rdrdθ
= √ 2 2π
cos 2 θdθ 3
0 1 r5 dr = √ 2 1
θ + 1 sin 2θ 2π 1 r6 3
= √
2 4 0 6
2(π) · 1
1 6 (36 − 1) = 364 √ 2
π
3

F.
294 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
13. Using x and z as parameters, we have r(x, z) =x i +(x 2 + z 2 ) j + z k, x 2 + z 2 ≤ 4. Then
r x × r z =(i +2x j) × (2z j + k) =2x i − j +2z k and |r x × r z | = √ 4x 2 +1+4z 2 = 1+4(x 2 + z 2 ).Thus
ydS =
S
x 2 +z 2 ≤4
(x 2 + z 2 ) 1+4(x 2 + z 2 ) dA = 2π
=2π 2
0 r2 √ 1+4r 2 rdr
1
0
let u =1+4r
2
=2π 17 1
(u − 1)√ u · 1 du = 1 π 17
1 4 8 16 1 (u3/2 − u 1/2 ) du
17
= 1 π 2
16 5 u5/2 − 2 3 u3/2 = 1 π 2
16 5 (17)5/2 − 2 3 (17)3/2 − 2 + 2 5 3
2
0 r2 √ 1+4r 2 rdrdθ= 2π
0
dθ 2
0 r2 √ 1+4r 2 rdr
⇒
r 2 = 1 (u − 1) and 1 du = rdr
4 8
= π √
391 17 + 1
60
15. Using spherical coordinates and Example 17.6.10 [ ET 16.6.10] we have r(φ, θ) =2sinφ cos θ i +2sinφ sin θ j +2cosφ k
and |r φ × r θ | =4sinφ. Then S (x2 z + y 2 z) dS = 2π π/2
(4 sin 2 φ)(2 cos φ)(4 sin φ) dφ dθ =16π sin 4 φ π/2
=16π.
0 0 0
17. S is given by r(u, v) =u i +cosv j +sinv k, 0 ≤ u ≤ 3, 0 ≤ v ≤ π/2. Then
r u × r v = i × (− sin v j +cosv k) =− cos v j − sin v k and |r u × r v | = cos 2 v +sin 2 v =1,so
S (z + x2 y) dS = π/2
3 (sin v + 0 0 u2 cos v)(1) du dv = π/2
(3 sin v +9cosv) dv
0
=[−3cosv +9sinv] π/2
0
=0+9+3− 0=12
19. F(x, y, z) =xy i + yz j + zxk, z = g(x, y) =4− x 2 − y 2 ,andD is the square [0, 1] × [0, 1], so by Equation 10
S F · dS = [−xy(−2x) − yz(−2y)+zx] dA = 1
1
D 0 0 [2x2 y +2y 2 (4 − x 2 − y 2 )+x(4 − x 2 − y 2 )] dy dx
= 1
1
0 3 x2 + 11 x −
3 x3 + 34
15 dx =
713
180
21. F(x, y, z) =xze y i − xze y j + z k, z = g(x, y) =1− x − y,andD = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x}. SinceS has
downward orientation, we have
S F · dS = − D [−xzey (−1) − (−xze y )(−1) + z] dA = − 1
1−x
(1 − x − y) dy dx
0 0
= − 1
1
0 2 x2 − x + 1 2 dx = −
1
6
23. F(x, y, z) =x i − z j + y k, z = g(x, y) = 4 − x 2 − y 2 and D is the quarter disk
(x, y)
0 ≤ x ≤ 2, 0 ≤ y ≤
√
4 − x
2 . S has downward orientation, so by Formula 10,
S F · dS = − D
= −
D
−x · 1 (4 − 2 x2 − y 2 ) −1/2 (−2x) − (−z) · 1 (4 − 2 x2 − y 2 ) −1/2 (−2y)+y
x 2
4 − x2 − y 2 − 4 − x 2 − y 2 ·
y
4 − x2 − y 2 + y
dA
= − D x2 (4 − (x 2 + y 2 )) −1/2 dA = − π/2
2 (r cos 0 0 θ)2 (4 − r 2 ) −1/2 rdrdθ
dA
= − π/2
cos 2 θdθ 2
0 0 r3 (4 − r 2 ) −1/2 dr
= − π/2
0
let u =4− r
2
1
2 + 1 2 cos 2θ dθ 0
4 − 1 2 (4 − u)(u)−1/2 du
⇒
r 2 =4− u and − 1 du = rdr
2
= − 1
θ + 1 sin 2θ π/2
2 4 0 −
1
8 √ 0
u − 2 2 3 u3/2 = − π 4
4
−
1
2
−16 +
16
3 = −
4
π 3

F.
TX.10
SECTION 17.7 SURFACE INTEGRALS ET SECTION 16.7 ¤ 295
25. Let S 1 be the paraboloid y = x 2 + z 2 , 0 ≤ y ≤ 1 and S 2 the disk x 2 + z 2 ≤ 1, y =1.SinceS is a closed
surface, we use the outward orientation.
On S 1 : F(r(x, z)) = (x 2 + z 2 ) j − z k and r x × r z =2x i − j +2z k (since the j-component must be negative on S 1 ). Then
S 1
F · dS = [−(x 2 + z 2 ) − 2z 2 ] dA = − 2π 1
0 (r2 +2r 2 cos 2 θ) rdrdθ
x 2 + z 2 ≤ 1
= − 2π 1
(1 + 2 0 4 cos2 θ) dθ = − π
+ π
2 2 = −π
On S 2 : F(r(x, z)) = j − z k and r z × r x = j. Then S 2
F · dS =
(1) dA = π.
Hence F · dS = −π + π =0.
S
0
x 2 + z 2 ≤ 1
27. Here S consists of the six faces of the cube as labeled in the figure. On S 1:
F = i +2y j +3z k, r y × r z = i and S 1
F · dS = 1
1
dy dz =4;
−1 −1
S 2 : F = x i +2j +3z k, r z × r x = j and S 2
F · dS = 1
1
2 dx dz =8;
−1 −1
S 3: F = x i +2y j +3k, r x × r y = k and S 3
F · dS = 1
1
3 dx dy =12;
−1 −1
S 4: F = −i +2y j +3z k, r z × r y = −i and S 4
F · dS =4;
S 5 : F = x i − 2 j +3z k, r x × r z = −j and S 5
F · dS =8;
S 6: F = x i +2y j − 3 k, r y × r x = −k and S 6
F · dS = 1
1
3 dx dy =12.
−1 −1
Hence F · dS = 6
S i=1 S i
F · dS =48.
29. Here S consists of four surfaces: S 1 , the top surface (a portion of the circular cylinder y 2 + z 2 =1); S 2 , the bottom surface
(a portion of the xy-plane); S 3, the front half-disk in the plane x =2,andS 4, the back half-disk in the plane x =0.
On S 1: The surface is z = 1 − y 2 for 0 ≤ x ≤ 2, −1 ≤ y ≤ 1 with upward orientation, so
2
1
F · dS =
−x 2 (0) − y 2 y
2
1
− + z 2 y 3
dy dx = +1−
S 1 0 −1
1 − y
2
0 −1 1 − y
2 y2 dy dx
= 2
− y=1
1 − y
0
2 + 1 (1 − 3 y2 ) 3/2 + y − 1 3 y3 dx = 2 4
dx = 8
0 3 3
On S 2 : The surface is z =0with downward orientation, so
S 2
F · dS = 2
0
y=−1
1
−1
−z
2 dy dx = 2
0
1
(0) dy dx =0
−1
On S 3 :Thesurfaceisx =2for −1 ≤ y ≤ 1, 0 ≤ z ≤ 1 − y 2 , oriented in the positive x-direction. Regarding y and z as
parameters, we have r y × r z = i and
S 3
F · dS = 1
√ 1−y 2
x 2 dz dy = 1
√ 1−y 2
4 dz dy =4A (S
−1 0 −1 0 3)=2π
On S 4 :Thesurfaceisx =0for −1 ≤ y ≤ 1, 0 ≤ z ≤ 1 − y 2 , oriented in the negative x-direction. Regarding y and z as
parameters, we use − (r y × r z )=−i and
Thus S F · dS = 8 3 +0+2π +0=2π + 8 3 .
S 4
F · dS = 1
√ 1−y 2
x 2 dz dy = 1
√ 1−y 2
(0) dz dy =0
−1 0 −1 0

F.
296 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
31. z = xy ⇒ ∂z/∂x = y, ∂z/∂y = x, so by Formula 4, a CAS gives
S xyz dS = 1 1 xy(xy) y
0 0 2 + x 2 +1dx dy ≈ 0.1642.
33. We use Formula 4 with z =3− 2x 2 − y 2 ⇒ ∂z/∂x = −4x, ∂z/∂y = −2y. The boundaries of the region
3 − 2x 2 − y 2 3
≥ 0 are − ≤ x ≤ 3
and −√ 3 − 2x
2 2 2 ≤ y ≤ √ 3 − 2x 2 ,soweuseaCAS(withprecisionreducedto
seven or fewer digits; otherwise the calculation may take a long time) to calculate
S
√
x 2 y 2 z 2 3/2
√ 3 − 2x 2
dS = √ x
− 3/2 −
√3 2 y 2 (3 − 2x 2 − y 2 ) 2 16x 2 +4y 2 +1dy dx ≈ 3.4895
− 2x 2
35. If S is given by y = h(x, z), thenS is also the level surface f(x, y, z) =y − h(x, z) =0.
∇f(x, y, z)
n =
|∇f(x, y, z)| = −h x i + j − h z k
√ ,and−n is the unit normal that points to the left. Now we proceed as in the
h
2 x +1+h 2 z
derivation of (10), using Formula 4 to evaluate
S
∂h
F · dS = F · n dS = (P i + Q j + R k) ∂x i − j + ∂h
∂z k ∂h
∂h
S
D
2 2 ∂h
∂x
+1+
∂x
∂z
2
+1+
where D is the projection of S onto the xz-plane. Therefore F · dS = P ∂h
S
D ∂x − Q + R ∂h
dA.
∂z
37. m = S KdS = K · 4π 1
2 a2 =2πa 2 K;bysymmetryM xz = M yz =0,and
M xy = zK dS = K 2π π/2
(a cos φ)(a 2 sin φ) dφ dθ =2πKa 3 − 1 cos 2φ π/2
S 0 0 4
= πKa 3 .
0
Hence (x, y, z) = 0, 0, 1 2 a .
39. (a) I z = S (x2 + y 2 )ρ(x, y, z) dS
(b) I z = S (x2 + y 2 )
10 −
x 2 + y 2 dS =
= 2π
0
4
1
√
2(10r 3 − r 4 ) dr dθ =2 √ 2 π 4329
10
1 ≤ x 2 + y 2 ≤ 16
=
4329
5
(x 2 + y 2 ) 10 − √2
x 2 + y 2 dA
√
2 π
2 ∂h
dA
∂z
41. Therateofflow through the cylinder is the flux ρv · n dS = ρv · dS. We use the parametric representation
S S
r(u, v) =2cosu i +2sinu j + v k for S,where0 ≤ u ≤ 2π, 0 ≤ v ≤ 1,sor u = −2sinu i +2cosu j, r v = k,andthe
outward orientation is given by r u × r v =2cosu i +2sinu j. Then
S ρv · dS = ρ 2π
1
0 0 v i +4sin 2 u j +4cos 2 u k · (2 cos u i +2sinu j) dv du
= ρ 2π
0
1
0
= ρ sin u +8 − 1 3
2v cos u +8sin 3 u dv du = ρ 2π
0 cos u +8sin 3 u du
(2 + sin 2 u)cosu 2π
0
=0kg/s

F.
TX.10
SECTION 17.8 STOKES’ THEOREM ET SECTION 16.8 ¤ 297
43. S consists of the hemisphere S 1 given by z = a 2 − x 2 − y 2 and the disk S 2 given by 0 ≤ x 2 + y 2 ≤ a 2 , z =0.
On S 1 : E = a sin φ cos θ i + a sin φ sin θ j +2a cos φ k,
T φ × T θ = a 2 sin 2 φ cos θ i + a 2 sin 2 φ sin θ j + a 2 sin φ cos φ k. Thus
S 1
E · dS = 2π π/2
(a 3 sin 3 φ +2a 3 sin φ cos 2 φ) dφ dθ
0 0
= 2π π/2
(a 3 sin φ + a 3 sin φ cos 2 φ) dφ dθ =(2π)a 3
1+ 1 0 0 3 =
8
3 πa3
On S 2: E = x i + y j,andr y × r x = −k so S 2
E · dS =0. Hence the total charge is q = ε 0
S E · dS = 8 3 πa3 ε 0.
45. K∇u =6.5(4y j +4z k). S is given by r(x, θ) =x i + √ 6cosθ j + √ 6sinθ k and since we want the inward heat flow, we
use r x × r θ = − √ 6cosθ j − √ 6sinθ k. Then the rate of heat flow inward is given by
S (−K ∇u) · dS = 2π 4
−(6.5)(−24) dx dθ =(2π)(156)(4) = 1248π.
0 0
47. Let S be a sphere of radius a centered at the origin. Then |r| = a and F(r) =cr/ |r| 3 = c/a 3 (x i + y j + z k). A
parametric representation for S is r(φ, θ) =a sin φ cos θ i + a sin φ sin θ j + a cos φ k, 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π. Then
r φ = a cos φ cos θ i + a cos φ sin θ j − a sin φ k, r θ = −a sin φ sin θ i + a sin φ cos θ j, and the outward orientation is given
by r φ × r θ = a 2 sin 2 φ cos θ i + a 2 sin 2 φ sin θ j + a 2 sin φ cos φ k. Theflux of F across S is
S F · dS = π
2π c
(a sin φ cos θ i + a sin φ sin θ j + a cos φ k)
0 0
a3 = c
a 3 π
0
Thus the flux does not depend on the radius a.
· a 2 sin 2 φ cos θ i + a 2 sin 2 φ sin θ j + a 2 sin φ cos φ k dθ dφ
2π
a 3 sin 3 φ +sinφ cos 2 φ dθ dφ = c π
2π
sin φdθdφ=4πc
0 0 0
17.8 Stokes' Theorem ET 16.8
1. Both H and P are oriented piecewise-smooth surfaces that are bounded by the simple, closed, smooth curve x 2 + y 2 =4,
z =0(whichwecantaketobeorientedpositivelyforbothsurfaces).ThenH and P satisfy the hypotheses of Stokes’
Theorem, so by (3) we know curl F · dS = F · dr = curl F · dS (where C is the boundary curve).
H C P
3. The paraboloid z = x 2 + y 2 intersects the cylinder x 2 + y 2 =4in the circle x 2 + y 2 =4, z =4. This boundary curve C
should be oriented in the counterclockwise direction when viewed from above, so a vector equation of C is
r(t) =2cost i +2sint j +4k, 0 ≤ t ≤ 2π. Thenr 0 (t) =−2sint i +2cost j,
F(r(t)) = (4 cos 2 t)(16) i +(4sin 2 t)(16) j +(2cost)(2 sin t)(4) k =64cos 2 t i +64sin 2 t j +16sint cos t k,
andbyStokes’Theorem,
S curl F · dS = C F · dr = 2π
0
F(r(t)) · r 0 (t) dt = 2π
0 (−128 cos2 t sin t +128sin 2 t cos t +0)dt
=128 1
3 cos3 t + 1 3 sin3 t 2π
=0
0

F.
298 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
5. C is the square in the plane z = −1. By(3), S 1
curl F · dS = C F · dr = S 2
curl F · dS where S 1 is the original cube
without the bottom and S 2 is the bottom face of the cube. curl F = x 2 z i +(xy − 2xyz) j +(y − xz) k. ForS 2 , we choose
n = k so that C has the same orientation for both surfaces. Then curl F · n = y − xz = x + y on S 2 ,wherez = −1. Thus
S 2
curl F · dS = 1
−1
1
−1 (x + y) dx dy =0so S 1
curl F · dS =0.
7. curl F = −2z i − 2x j − 2y k and we take the surface S to be the planar region enclosed by C,soS is the portion of the plane
x + y + z =1over D = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x}. SinceC is oriented counterclockwise, we orient S upward.
Using Equation 17.7.10 [ ET 16.7.10], we have z = g(x, y) =1− x − y, P = −2z, Q = −2x, R = −2y,and
F · dr = curl F · dS = [−(−2z)(−1) − (−2x)(−1) + (−2y)] dA
C S D
= 1
0
1−x
0
(−2) dy dx = −2 1
0
(1 − x) dx = −1
9. curl F =(xe xy − 2x) i − (ye xy − y) j +(2z − z) k and we take S to be the disk x 2 + y 2 ≤ 16, z =5.SinceC is oriented
counterclockwise (from above), we orient S upward. Then n = k and curl F · n =2z − z on S,wherez =5. Thus
F · dr =
S curl F · n dS= S (2z − z) dS = S (10 − 5) dS =5(areaofS) =5(π · 42 )=80π
11. (a) The curve of intersection is an ellipse in the plane x + y + z =1with unit normal n = 1 √
3
(i + j + k),
curl F = x 2 j + y 2 k,andcurl F · n = 1 √
3
(x 2 + y 2 ).Then
C F · dr = S
1
√
3
x 2 + y 2 dS = x 2 + y 2 ≤ 9
x 2 + y 2 dx dy = 2π
0
3
0 r3 dr dθ =2π
81
4 =
81π
2
(b)
(c) One possible parametrization is x =3cost, y =3sint,
z =1− 3cost − 3sint, 0 ≤ t ≤ 2π.
13. The boundary curve C is the circle x 2 + y 2 =1, z =1oriented in the counterclockwise direction as viewed from above.
We can parametrize C by r(t) =cost i +sint j + k, 0 ≤ t ≤ 2π, andthenr 0 (t) =− sin t i +cost j. Thus
F(r(t)) = sin 2 t i +cost j + k, F(r(t)) · r 0 (t) =cos 2 t − sin 3 t,and
C F · dr = 2π
0 (cos2 t − sin 3 t) dt = 2π 1
(1 + cos 2t) dt − 2π
(1 − 0 2 0 cos2 t)sintdt
= 1 2 t +
1
sin 2t 2π
−
2
− cos t + 1 0 3 cos3 t 2π
= π
0
Now curl F =(1− 2y) k, and the projection D of S on the xy-plane is the disk x 2 + y 2 ≤ 1, so by Equation 17.7.10
[ ET 16.7.10] with z = g(x, y) =x 2 + y 2 we have
S curl F · dS = D (1 − 2y) dA = 2π
0
1
0 (1 − 2r sin θ) rdrdθ= 2π
0
1
2 − 2 3 sin θ dθ = π.
15. The boundary curve C is the circle x 2 + z 2 =1, y =0oriented in the counterclockwise direction as viewed from the positive
y-axis. Then C can be described by r(t) =cost i − sin t k, 0 ≤ t ≤ 2π,andr 0 (t) =− sin t i − cos t k. Thus

F.
F(r(t)) = − sin t j +cost k, F(r(t)) · r 0 (t) =− cos 2 t,and C
TX.10 SECTION 17.9 THE DIVERGENCE THEOREM ET SECTION 16.9 ¤ 299
Now curl F = −i − j − k, andS can be parametrized (see Example 17.6.10 [ ET 16.6.10]) by
r(φ, θ) =sinφ cos θ i +sinφ sin θ j +cosφ k, 0 ≤ θ ≤ π, 0 ≤ φ ≤ π. Then
r φ × r θ =sin 2 φ cos θ i +sin 2 φ sin θ j +sinφ cos φ k and
curl F · dS = curl F · (r
S φ × r θ ) dA = π
0
x 2 +z 2 ≤1
F · dr = 2π
0
− cos 2 tdt= − 1 2 t − 1 4 sin 2t 2π
0
= −π.
π
0 (− sin2 φ cos θ − sin 2 φ sin θ − sin φ cos φ) dθ dφ
= π
0 (−2sin2 φ − π sin φ cos φ) dφ = 1
2 sin 2φ − φ − π 2 sin2 φ π
0 = −π
17. It is easier to use Stokes’ Theorem than to compute the work directly. Let S be the planar region enclosed by the path of the
particle, so S is the portion of the plane z = 1 y for 0 ≤ x ≤ 1, 0 ≤ y ≤ 2,withupwardorientation.
2
curl F =8y i +2z j +2y k and
F · dr = curl F · dS = C S D
= 1
0
2
0
3
ydydx= 1
2 0
−8y (0) − 2z
1
1
2 +2y dA =
0
3
4 y2 y=2
y=0 dx = 1
0
3 dx =3
2
0
2y −
1
2 y dy dx
19. Assume S is centered at the origin with radius a and let H 1 and H 2 be the upper and lower hemispheres, respectively, of S.
Then S curl F · dS = H 1
curl F · dS + H 2
curl F · dS = C 1
F · dr + C 2
F · dr by Stokes’ Theorem. But C 1 is the
circle x 2 + y 2 = a 2 oriented in the counterclockwise direction while C 2 is the same circle oriented in the clockwise direction.
Hence C 2
F · dr = − C 1
F · dr so curl F · dS =0as desired.
S
17.9 The Divergence Theorem ET 16.9
1. div F =3+x +2x =3+3x,so
div F dV = 1 1
1 (3x +3)dx dy dz = 9 (notice the triple integral is
E 0 0 0 2
three times the volume of the cube plus three times x).
To compute S F · dS,on
S 1: n = i, F =3i + y j +2z k,and S 1
F · dS = S 1
3 dS =3;
S 2 : F =3x i + x j +2xz k, n = j and S 2
F · dS = S 2
xdS = 1 2 ;
S 3: F =3x i + xy j +2x k, n = k and S 3
F · dS = S 3
2xdS =1;
S 4 : F = 0, S 4
F · dS =0; S 5 : F =3x i +2x k, n = −j and S 5
F · dS = S 5
0 dS =0;
S 6 : F =3x i + xy j, n = −k and S 6
F · dS = S 6
0 dS =0.Thus S F · dS = 9 2 .
3. div F = x + y + z,so
E div F dV = 2π
0
= 2π
0
1
1 (r cos θ + r sin θ + z) rdzdrdθ= 2π
1
0 0 0 0 r 2 cos θ + r 2 sin θ + 1 r 2
dr dθ
1 cos θ + 1 sin θ + 1
3 3 4 dθ =
1
(2π) = π 4 2
Let S 1 be the top of the cylinder, S 2 the bottom, and S 3 the vertical edge. On S 1, z =1, n = k,andF = xy i + y j + x k,so

F.
300 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
S 1
F · dS = S 1
F · n dS = S 1
xdS = 2π
0
On S 2 , z =0, n = −k,andF = xy i so S 2
F · dS = S 2
0 dS =0.
1 (r cos θ) rdrdθ= sin θ 2π 1 r3 1
=0.
0 0 3 0
S 3 is given by r(θ, z) =cosθ i +sinθ j + z k, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 1. Thenr θ × r z =cosθ i +sinθ j and
S 3
F · dS = F · (r D θ × r z ) dA = 2π
1
0 0 (cos2 θ sin θ + z sin 2 θ) dz dθ
= 2π
0
Thus S F · dS =0+0+π 2 = π 2 .
cos 2 θ sin θ + 1 2 sin2 θ dθ =
− 1 3 cos3 θ + 1 4 θ −
1
sin 2θ 2π
= π 2 0 2
5. div F = ∂
∂x (ex sin y)+ ∂
∂y (ex cos y)+ ∂ ∂z (yz2 )=e x sin y − e x sin y +2yz =2yz, so by the Divergence Theorem,
S F · dS = E div F dV = 1
0
1
0
2
0 2yz dz dy dx =2 1
0 dx 1
0 ydy 1
0 zdz=2 x 1
0
1 y2 1 1 z2 2
=2.
2 0 2 0
7. div F =3y 2 +0+3z 2 , so using cylindrical coordinates with y = r cos θ, z = r sin θ, x = x we have
S F · dS = E (3y2 +3z 2 ) dV = 2π
1
2
0 0 −1 (3r2 cos 2 θ +3r 2 sin 2 θ) rdxdrdθ
=3 2π
dθ 1
0 0 r3 dr 2
dx =3(2π)
1
−1 4 (3) =
9π
2
9. div F = y sin z +0− y sin z =0, so by the Divergence Theorem, F · dS = 0 dV =0.
S E
11. div F = y 2 +0+x 2 = x 2 + y 2 so
S F · dS = E (x2 + y 2 ) dV = 2π
0
2
0
4
r 2 · rdzdrdθ= 2π
2
r 2 0 0
= 2π
0
dθ 2
0 (4r3 − r 5 ) dr =2π r 4 − 1 6 r6 2
0 = 32 3 π
r3 (4 − r 2 ) dr dθ
13. div F =12x 2 z +12y 2 z +12z 3 so
S F · dS = E 12z(x2 + y 2 + z 2 ) dV = 2π
π
R 12(ρ cos 0 0 0
φ)(ρ2 )ρ 2 sin φdρdφdθ
=12 2π
dθ π
sin φ cos φdφ R
0 0 0 ρ5 dρ =12(2π) 1
2 sin2 φ π 1 ρ6 R
=0
0 6 0
15. F · dS = √
S E 3 − x2 dV = 1
1
−1 −1
2 − x
4 − y
4
0
√
3 − x2 dz dy dx = 341
60
√ √3
2+
81
20 sin−1 3
17. For S 1 we have n = −k,soF · n = F · (−k) =−x 2 z − y 2 = −y 2 (since z =0on S 1 ). So if D is the unit disk, we get
S 1
F · dS = S 1
F · n dS = D (−y2 ) dA = − 2π 1
0 0 r2 (sin 2 θ) rdrdθ = − 1 π.NowsinceS 4 2 is closed, we can use
the Divergence Theorem. Since div F = ∂
∂x (z2 x)+ ∂ 1
∂y 3 y3 +tanz + ∂ ∂z (x2 z + y 2 )=z 2 + y 2 + x 2 , we use spherical
coordinates to get S 2
F · dS = E div F dV = 2π
0
S F · dS = S 2
F · dS − S 1
F · dS = 2 5 π − − 1 4 π = 13
20 π.
π/2
1
0 0 ρ2 · ρ 2 sin φdρdφdθ = 2 π.Finally
5
19. The vectors that end near P 1 are longer than the vectors that start near P 1 ,sothenetflow is inward near P 1 and div F(P 1 ) is
negative. The vectors that end near P 2 are shorter than the vectors that start near P 2, so the net flow is outward near P 2 and
div F(P 2 ) is positive.

F.
TX.10
CHAPTER 17 REVIEW ET CHAPTER 16 ¤ 301
21. From the graph it appears that for points above the x-axis, vectors starting near a
particular point are longer than vectors ending there, so divergence is positive.
The opposite is true at points below the x-axis, where divergence is negative.
F (x, y) = xy, x + y 2 ⇒ div F = ∂ (xy)+ ∂
∂x ∂y x + y
2
= y +2y =3y.
Thus div F > 0 for y>0,anddiv F < 0 for y

F.
302 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
3. (a) See Definition 17.2.2 [ ET 16.2.2].
(b) We normally evaluate the line integral using Formula 17.2.3 [ ET 16.2.3].
(c) The mass is m = C ρ (x, y) ds,andthecenterofmassis(x, y) where x = 1 m
C xρ (x, y) ds, y = 1 m
(d) See (5) and (6) in Section 17.2 [ ET 16.2] for plane curves; we have similar definitions when C is a space curve
(see the equation preceding (10) in Section 17.2 [ ET 16.2]).
(e) For plane curves, see Equations 17.2.7 [ ET 16.2.7]. We have similar results for space curves
(see the equation preceding (10) in Section 17.2 [ ET 16.2]).
4. (a) See Definition 17.2.13 [ ET 16.2.13].
(b) If F is a force field, F · dr represents the work done by F in moving a particle along the curve C.
C
(c) F · dr = Pdx+ Qdy+ Rdz
C C
5. See Theorem 17.3.2 [ ET 16.3.2].
C
yρ (x, y) ds.
6. (a) F · dr is independent of path if the line integral has the same value for any two curves that have the same initial and
C
terminal points.
(b) See Theorem 17.3.4 [ ET 16.3.4].
7. See the statement of Green’s Theorem on page 1091 [ ET 1055].
8. See Equations 17.4.5 [ ET 16.4.5].
∂R
9. (a) curl F =
∂y − ∂Q ∂P
i +
∂z ∂z − ∂R ∂Q
j +
∂x ∂x − ∂P
k = ∇ × F
∂y
(b) div F = ∂P
∂x + ∂Q
∂y + ∂R
∂z = ∇ · F
(c) For curl F, see the discussion accompanying Figure 1 on page 1100 [ ET 1064] as well as Figure 6 and the accompanying
discussion on page 1132 [ ET 1096]. For div F, see the discussion following Example 5 on page 1102 [ ET 1066] as well
as the discussion preceding (8) on page 1139 [ ET 1103].
10. See Theorem 17.3.6 [ ET 16.3.6]; see Theorem 17.5.4 [ ET 16.5.4].
11. (a) See (1) and (2) and the accompanying discussion in Section 17.6 [ ET 16.6] ; See Figure 4 and the accompanying
discussion on page 1107 [ ET 1071] .
(b) See Definition 17.6.6 [ ET 16.6.6 ].
(c) See Equation 17.6.9 [ ET 16.6.9].
12. (a)See(1)inSection17.7[ET16.7].
(b) We normally evaluate the surface integral using Formula 17.7.2 [ ET 16.7.2].
(c) See Formula 17.7.4 [ ET 16.7.4].
TX.10
(d) The mass is m = ρ(x, y, z) dS and the center of mass is (x, y, z) where x = 1 S m
y = 1 yρ(x, y, z) dS, z =
1
m S m
zρ(x, y, z) dS.
S
S
xρ(x, y, z) dS,

F.
TX.10
CHAPTER 17 REVIEW ET CHAPTER 16 ¤ 303
13. (a) See Figures 6 and 7 and the accompanying discussion in Section 17.7 [ ET 16.7]. A Möbius strip is a nonorientable
surface; see Figures 4 and 5 and the accompanying discussion on page 1121 [ ET 1085].
(b) See Definition 17.7.8 [ ET 16.7.8].
(c) See Formula 17.7.9 [ ET 16.7.9].
(d) See Formula 17.7.10 [ ET 16.7.10].
14. See the statement of Stokes’ Theorem on page 1129 [ ET 1093.].
15. See the statement of the Divergence Theorem on page 1135 [ ET 1099].
16. In each theorem, we have an integral of a “derivative” over a region on the left side, while the right side involves the values of
the original function only on the boundary of the region.
1. False; div F is a scalar field.
3. True, by Theorem 17.5.3 [ ET 16.5.3] and the fact that div 0 =0.
5. False. See Exercise 17.3.33 [ ET 16.3.33]. (But the assertion is true if D is simply-connected; see Theorem 17.3.6
[ ET 16.3.6].)
7. True. Apply the Divergence Theorem and use the fact that div F =0.
1. (a) Vectors starting on C point in roughly the direction opposite to C, so the tangential component F · T is negative.
Thus F · dr = F · T ds is negative.
C C
(b) The vectors that end near P are shorter than the vectors that start near P ,sothenetflow is outward near P and
div F (P ) is positive.
3.
C yz cos xds = π
(3 cos t)(3sint)cost (1)
0 2 +(−3sint) 2 +(3cost) 2 dt = π
(9 0 cos2 t sin t) √ 10 dt
=9 √ 10 − 1 3 cos3 t π
= −3 √ 10 (−2) = 6 √ 10
0
5.
C y3 dx + x 2 dy = 1
−1 y 3 (−2y)+(1− y 2 ) 2 dy = 1
−1 (−y4 − 2y 2 +1)dy
= − 1 5 y5 − 2 3 y3 + y 1
−1 = − 1 5 − 2 3 +1− 1 5 − 2 3 +1= 4 15
7. C: x =1+2t ⇒ dx =2dt, y =4t ⇒ dy =4dt, z = −1+3t ⇒ dz =3dt, 0 ≤ t ≤ 1.
C xy dx + y2 dy + yz dz = 1
[(1 + 2t)(4t)(2) + 0 (4t)2 (4) + (4t)(−1+3t)(3)] dt
9. F(r(t)) = e −t i + t 2 (−t) j +(t 2 + t 3 ) k, r 0 (t) =2t i +3t 2 j − k and
= 1
0 (116t2 − 4t) dt = 116
3 t3 − 2t 2 1
= 116
110
− 2=
0 3 3
C F · dr = 1
0 (2te−t − 3t 5 − (t 2 + t 3 )) dt = −2te −t − 2e −t − 1 2 t6 − 1 3 t3 − 1 4 t4 1
0 = 11
12 − 4 e .

F.
304 ¤ CHAPTER 17 VECTOR CALCULUS ET CHAPTER 16
TX.10
11.
∂
[(1 + ∂y xy)exy ]=2xe xy + x 2 ye xy =
∂x ∂ e y + x 2 e xy and the domain of F is R 2 ,soF is conservative. Thus there
exists a function f such that F = ∇f. Thenf y (x, y) =e y + x 2 e xy implies f(x, y) =e y + xe xy + g(x) and then
f x(x, y) =xye xy + e xy + g 0 (x) =(1+xy)e xy + g 0 (x). Butf x(x, y) =(1+xy)e xy ,sog 0 (x) =0 ⇒ g(x) =K.
Thus f(x, y) =e y + xe xy + K is a potential function for F.
13. Since ∂
∂y (4x3 y 2 − 2xy 3 )=8x 3 y − 6xy 2 = ∂
∂x (2x4 y − 3x 2 y 2 +4y 3 ) and the domain of F is R 2 , F is conservative.
Furthermore f(x, y) =x 4 y 2 − x 2 y 3 + y 4 is a potential function for F. t =0corresponds to the point (0, 1) and t =1
corresponds to (1, 1),so F · dr = f(1, 1) − f(0, 1) = 1 − 1=0.
C
15. C 1 : r(t) =t i + t 2 j, −1 ≤ t ≤ 1;
C 2 : r(t) =−t i + j, −1 ≤ t ≤ 1.
Then
C xy2 dx − x 2 ydy= 1
−1 (t5 − 2t 5 ) dt + 1
−1 tdt
Using Green’s Theorem, we have
C
= − 1 6 t6 1
−1 + 1
2 t2 1
−1 =0
∂
xy 2 dx − x 2 ydy=
D ∂x (−x2 y) − ∂
1
1
∂y (xy2 ) dA = (−2xy − 2xy) dA =
D
−1
17.
C x2 ydx− xy 2 dy =
= 1
−1
x 2 + y 2 ≤ 4
−4xy dy dx
x 2
−2xy
2 y=1
dx = 1
y=x 2 −1 (2x5 − 2x) dx = 1
3 x6 − x 2 1
=0 −1
∂
∂x (−xy2 ) − ∂
∂y (x2 y) dA = (−y 2 − x 2 ) dA = − 2π 2
0 r3 dr dθ = −8π
x 2 + y 2 ≤ 4
19. If we assume there is such a vector field G,thendiv(curl G) =2+3z − 2xz. Butdiv(curl F) =0for all vector fields F.
Thus such a G cannot exist.
21. For any piecewise-smooth simple closed plane curve C bounding a region D, we can apply Green’s Theorem to
F(x, y) =f(x) i + g(y) j to get f(x) dx + g(y) dy =
∂
g(y) − ∂ f(x) dA = 0 dA =0.
C D ∂x ∂y D
23. ∇ 2 f =0means that ∂2 f
∂x 2 + ∂2 f
∂y 2
=0.NowifF = f y i − f x j and C is any closed path in D,thenapplyingGreen’s
Theorem, we get
F · dr = f C C y dx − f x dy =
∂
(−f D ∂x x) − ∂ (f ∂y y) dA = − (f D xx + f yy ) dA = − 0 dA =0
D
Therefore the line integral is independent of path, by Theorem 17.3.3 [ ET 16.3.3].
25. z = f (x, y) =x 2 +2y with 0 ≤ x ≤ 1, 0 ≤ y ≤ 2x. Thus
A(S) = D
√
1+4x2 +4dA = 1
0
2x
0
√
5+4x2 dy dx = 1
0 2x √ 5+4x 2 dx = 1 6 (5 + 4x2 ) 3/2 1
0
= 1 6
27 − 5
√
5
.
0

F.
TX.10
CHAPTER 17 REVIEW ET CHAPTER 16 ¤ 305
27. z = f(x, y) =x 2 + y 2 with 0 ≤ x 2 + y 2 ≤ 4 so r x × r y = −2x i − 2y j + k (using upward orientation). Then
zdS = (x
S 2 + y 2 ) 4x 2 +4y 2 +1dA = 2π 2 r3√ 1+4r
0 0 2 dr dθ = 1 π 391 √ 17 + 1
60
x 2 + y 2 ≤ 4
(Substitute u =1+4r 2 and use tables.)
29. Since the sphere bounds a simple solid region, the Divergence Theorem applies and
S F · dS = E (z − 2) dV = E zdV − 2 E dV = mz − 2 4
3 π23 = − 64 3 π.
Alternate solution: F(r(φ, θ)) = 4 sin φ cos θ cos φ i − 4sinφ sin θ j +6sinφ cos θ k,
r φ × r θ =4sin 2 φ cos θ i +4sin 2 φ sin θ j +4sinφ cos φ k, and
F · (r φ × r θ )=16sin 3 φ cos 2 θ cos φ − 16 sin 3 φ sin 2 θ +24sin 2 φ cos φ cos θ. Then
S F · dS = 2π π (16 0 0 sin3 φ cos φ cos 2 θ − 16 sin 3 φ sin 2 θ +24sin 2 φ cos φ cos θ) dφ dθ
= 2π
0
4
(−16 3 sin2 θ) dθ = − 64 π 3
31. Since curl F = 0, (curl F) · dS =0. We parametrize C: r(t) =cost i +sint j, 0 ≤ t ≤ 2π and
S
F · dr = 2π
(− C 0 cos2 t sin t +sin 2 t cos t) dt = 1 3 cos3 t + 1 3 sin3 t 2π
=0.
0
33. The surface is given by x + y + z =1or z =1− x − y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 − x and r x × r y = i + j + k. Then
C F · dr = S curl F · dS = D (−y i − z j − x k) · (i + j + k) dA = D (−1) dA = −(area of D) =− 1 2 .
35. div F dV =
E
x 2 + y 2 + z 2 ≤ 1
3 dV =3(volume of sphere) =4π. Then
F(r(φ, θ)) · (r φ × r θ )=sin 3 φ cos 2 θ +sin 3 φ sin 2 θ +sinφ cos 2 φ =sinφ and
S F · dS = 2π
0
π
0
sin φdφdθ =(2π)(2) = 4π.
37. Because curl F = 0, F is conservative, and if f(x, y, z) =x 3 yz − 3xy + z 2 ,then∇f = F. Hence
F · dr = ∇f · dr = f(0, 3, 0) − f(0, 0, 2) = 0 − 4=−4.
C C
39. By the Divergence Theorem, F · n dS = div F dV =3(volumeofE) =3(8− 1) = 21.
S E
41. Let F = a × r = ha 1 ,a 2 ,a 3 i×hx, y, zi = ha 2 z − a 3 y, a 3 x − a 1 z, a 1 y − a 2 xi. Then curl F = h2a 1 , 2a 2 , 2a 3 i =2a,
and 2a · dS = curl F · dS = F · dr = (a × r) · dr by Stokes’ Theorem.
S S C C

F.
TX.10

F.
TX.10
PROBLEMS PLUS
1. Let S 1 be the portion of Ω(S) between S(a) and S,andlet∂S 1 be its boundary. Also let S L be the lateral surface of S 1 [that
r · n
r
is, the surface of S 1 except S and S(a)]. Applying the Divergence Theorem we have
dS = ∇ ·
∂S 1
r 3 S 1
r dV . 3
But
⇒
∇ ·
∂S 1
r · n
r 3 dS =
r ∂
r = 3 ∂x , ∂ ∂y , ∂
x
·
∂z (x 2 + y 2 + z 2 ) , y
3/2 (x 2 + y 2 + z 2 ) , z
3/2 (x 2 + y 2 + z 2 ) 3/2
= (x2 + y 2 + z 2 − 3x 2 )+(x 2 + y 2 + z 2 − 3y 2 )+(x 2 + y 2 + z 2 − 3z 2 )
(x 2 + y 2 + z 2 ) 5/2 =0
S 1
0 dV =0. On the other hand, notice that for the surfaces of ∂S 1 other than S(a) and S,
r · n =0 ⇒
r · n
0=
dS =
∂S 1
r 3
r · n
r 3
S
S
r · n
r · n
r · n r · n
r · n
dS +
dS +
dS = dS +
dS ⇒
r 3 S(a) r 3 S L
r 3 S r 3 S(a) r 3
r · n
dS = −
dS. NoticethatonS(a), r = a ⇒ n = − r
S(a) r 3 r = − r a and r · r = r2 = a 2 ,so
r · n
r · r
that −
dS =
S(a) r 3 S(a) a dS = a
S(a)
2
4 a dS = 1 area of S (a)
dS = = |Ω(S)|.
4 a
S(a)
2 a 2
r · n
Therefore |Ω(S)| = dS.
S r 3
3. The given line integral 1 (bz − cy) dx +(cx − az) dy +(ay − bx) dz canbeexpressedas F · dr if we define the vector
2 C C
field F by F(x, y, z) =P i + Q j + R k = 1 (bz − cy) i + 1 (cx − az) j + 1 (ay − bx) k. Thendefine S to be the planar
2 2 2
interior of C,soS is an oriented, smooth surface. Stokes’ Theorem says F · dr = curl F · dS = curl F · n dS.
C S S
Now
curl F =
∂R
∂y − ∂Q ∂P
i +
∂z ∂z − ∂R ∂Q
j +
∂x ∂x − ∂P
k
∂y
= 1
2 a + 1 2 a i + 1
2 b + 1 2 b j + 1
2 c + 1 2 c k = a i + b j + c k = n
so curl F · n = n · n = |n| 2 =1, hence curl F · n dS = dS whichissimplythesurfaceareaofS. Thus,
S S
C F · dr = 1 2
C (bz − cy) dx +(cx − az) dy +(ay − bx) dz is the plane area enclosed by C. 307

F.
308 ¤ PROBLEMS PLUS
TX.10
∂
5. (F · ∇) G = P 1
∂x + Q ∂
1
∂y + R ∂
1 (P 2 i + Q 2 j+R 2 k)
∂z
∂P 2
= P 1
∂x + ∂P 2
Q1
∂y + ∂P 2 ∂Q 2
R1 i + P 1
∂z
∂x + ∂Q 2
Q1
∂y + ∂Q 2
R1 j
∂z
=(F · ∇P 2 ) i +(F · ∇Q 2 ) j +(F · ∇R 2 ) k.
Similarly, (G · ∇) F =(G · ∇P 1 ) i +(G · ∇Q 1 ) j +(G · ∇R 1 ) k. Then
i
F × curl G =
P 1
∂R 2 /∂y − ∂Q 2 /∂z
and
=
G × curl F =
Then
(F · ∇) G + F × curl G =
and
(G · ∇) F + G × curl F =
j
Q 1
∂P 2 /∂z − ∂R 2 /∂x
∂Q 2
Q 1
∂x − Q ∂P 2
1
∂y − R ∂P 2
1
∂z + R 1
∂R 2
∂x
i +
∂R 2
+ P 1
∂x + Q ∂R 2
1
∂y + R ∂R 2
1 k
∂z
k
R 1
∂Q 2 /∂x − ∂P 2 /∂y
∂R 2
R 1
∂y − R ∂Q 2
1
∂Q 1
Q 2
∂x − ∂P 1
Q2
∂y − ∂P 1
R2
∂z + ∂R 1 ∂R 1
R2 i + R 2
∂x
∂P 2
P 1
∂x + ∂Q 2
Q1
∂x + ∂R 2 ∂P 2
R1 i + P 1
∂x
∂P 1
P 2
∂x + Q ∂Q 1
2
∂x + R ∂R 1 ∂P 1
2 i + P 2
∂x
Hence
(F · ∇) G + F × curl G +(G · ∇) F + G × curl F
∂P 2
= P 1
∂x + P ∂P 1
2 +
∂x
+
∂P 2
P 1
∂y + P ∂P 1
2
∂y
+
Q 1
∂Q 2
∂x + Q 2
+
∂P 2
P 1
∂z + P ∂P 1
2
∂z
= ∇(P 1P 2 + Q 1Q 2 + R 1R 2)=∇(F · G).
+
∂P 2
j
∂y
∂z − P ∂Q 2
1
∂x + P 1
∂P 2
+ P 1
∂z − ∂R 2
P1
∂x − ∂R 2
Q1
∂y + ∂Q 2
Q1
∂z
∂y − ∂Q 1
R2
∂z − ∂Q 1
P2
∂x + ∂P 1
P2
∂y
∂P 1
P 2
∂z − P ∂R 1
2
∂x − Q ∂R 1
2
∂y + Q 2
j
∂y + ∂Q 2
Q1
∂y + ∂R 2
R1
∂y
∂P 2
+ P 1
∂z + Q ∂Q 2
1
∂z + R 1
∂y + Q 2
+
∂Q 1 ∂R 2
+ R 1
∂y
Q 1
∂Q 2
∂y + Q 2
+
∂Q 1
∂y
Q 1
∂Q 2
∂z + Q 2
∂Q 1
∂y + R 2
∂R 1
j
∂y
∂P 1
P 2
∂z + Q ∂Q 1
2
∂z + R 2
∂x + R 2
+
∂R 1
i
∂x
R 1
∂R 2
∂y + R 2
∂Q 1
∂z
+
∂R 1
j
∂y
R 1
∂R 2
∂z + R 2
j
k
∂Q 1
k.
∂z
∂R 2
k
∂z
∂R 1
k.
∂z
∂R 1
k
∂z

F.
TX.10
18 SECOND-ORDER DIFFERENTIAL EQUATIONS ET 17
18.1 Second-Order Linear Equations ET 17.1
1. The auxiliary equation is r 2 − r − 6=0 ⇒ (r − 3)(r +2)=0 ⇒ r =3, r = −2. Then by (8) the general solution is
y = c 1 e 3x + c 2 e −2x .
3. The auxiliary equation is r 2 +16=0 ⇒ r = ±4i. Then by (11) the general solution is
y = e 0x (c 1 cos 4x + c 2 sin 4x) =c 1 cos 4x + c 2 sin 4x.
5. The auxiliary equation is 9r 2 − 12r +4=0 ⇒ (3r − 2) 2 =0 ⇒ r = 2 . Then by (10), the general solution is
3
y = c 1e 2x/3 + c 2xe 2x/3 .
7. The auxiliary equation is 2r 2 − r = r(2r − 1) = 0 ⇒ r =0, r = 1 2 ,soy = c 1e 0x + c 2 e x/2 = c 1 + c 2 e x/2 .
9. The auxiliary equation is r 2 − 4r +13=0 ⇒ r = 4 ± √ −36
2
=2± 3i,soy = e 2x (c 1 cos 3x + c 2 sin 3x).
11. The auxiliary equation is 2r 2 +2r − 1=0 ⇒ r = −2 ± √ 12
4
= − 1 2 ± √
3
2 ,so
y = c 1e (−1/2+√ 3/2)t + c 2e (−1/2−√ 3/2)t .
13. The auxiliary equation is 100r 2 +200r + 101 = 0 ⇒ r = −200 ± √ −400
200
P = e −t c 1 cos 1
10 t + c 2 sin 1
10 t .
15. The auxiliary equation is 5r 2 − 2r − 3=(5r +3)(r − 1) = 0 ⇒ r = − 3 5 ,
r =1, so the general solution is y = c 1 e −3x/5 + c 2 e x . We graph the basic
solutions f(x) =e −3x/5 , g(x) =e x as well as y = e −3x/5 +2e x ,
y = e −3x/5 − e x ,andy = −2e −3x/5 − e x . Each solution consists of a single
continuous curve that approaches either 0 or ±∞ as x → ±∞.
= −1 ± 1 10 i,so
17. 2r 2 +5r +3=(2r +3)(r +1)=0,sor = − 3 2 , r = −1 and the general solution is y = c 1e −3x/2 + c 2 e −x .
Then y(0) = 3 ⇒ c 1 + c 2 =3and y 0 (0) = −4 ⇒ − 3 c1 − c2 = −4,soc1 =2and c2 =1. Thus the solution to the
2
initial-value problem is y =2e −3x/2 + e −x .
19. 4r 2 − 4r +1=(2r − 1) 2 =0 ⇒ r = 1 2 and the general solution is y = c 1e x/2 + c 2 xe x/2 .Theny(0) = 1 ⇒ c 1 =1
and y 0 (0) = −1.5 ⇒ 1 2 c1 + c2 = −1.5,soc2 = −2 and the solution to the initial-value problem is y = ex/2 − 2xe x/2 .
309

F.
310 ¤ CHAPTER 18 SECOND-ORDER DIFFERENTIAL EQUATIONS ETX.10
CHAPTER 17
21. r 2 +16=0 ⇒ r = ±4i and the general solution is y = e 0x (c 1 cos 4x + c 2 sin 4x) =c 1 cos 4x + c 2 sin 4x. Then
y
π
4 = −3 ⇒ −c1 = −3 ⇒ c 1 =3and y 0
π
4 =4 ⇒ −4c2 =4 ⇒ c 2 = −1, so the solution to the
initial-value problem is y =3cos4x − sin 4x.
23. r 2 +2r +2=0 ⇒ r = −1 ± i and the general solution is y = e −x (c 1 cos x + c 2 sin x). Then2=y(0) = c 1 and
1=y 0 (0) = c 2 − c 1 ⇒ c 2 =3and the solution to the initial-value problem is y = e −x (2 cos x +3sinx).
25. 4r 2 +1=0 ⇒ r = ± 1 2 i and the general solution is y = c1 cos 1
2 x + c 2 sin 1
2 x .Then3=y(0) = c 1 and
−4 =y(π) =c 2 , so the solution of the boundary-value problem is y =3cos 1
2 x − 4sin 1
2 x .
27. r 2 − 3r +2=(r − 2)(r − 1) = 0 ⇒ r =1, r =2and the general solution is y = c 1 e x + c 2 e 2x .Then
1=y(0) = c 1 + c 2 and 0=y(3) = c 1 e 3 + c 2 e 6 so c 2 =1/(1 − e 3 ) and c 1 = e 3 /(e 3 − 1). The solution of the
boundary-value problem is y = ex+3
e 3 − 1 +
e2x
1 − e . 3
29. r 2 − 6r +25=0 ⇒ r =3± 4i and the general solution is y = e 3x (c 1 cos 4x + c 2 sin 4x). But1=y(0) = c 1 and
2=y(π) =c 1 e 3π ⇒ c 1 =2/e 3π , so there is no solution.
31. r 2 +4r +13=0 ⇒ r = −2 ± 3i and the general solution is y = e −2x (c 1 cos 3x + c 2 sin 3x). But2=y(0) = c 1
and 1=y π
2
= e −π (−c 2 ), so the solution to the boundary-value problem is y = e −2x (2 cos 3x − e π sin 3x).
33. (a) Case 1 (λ =0): y 00 + λy =0 ⇒ y 00 =0which has an auxiliary equation r 2 =0 ⇒ r =0 ⇒ y = c 1 + c 2 x
where y(0) = 0 and y(L) =0.Thus,0=y(0) = c 1 and 0=y(L) =c 2L ⇒ c 1 = c 2 =0. Thus y =0.
Case 2 (λ

F.
SECTION TX.10 18.2 NONHOMOGENEOUS LINEAR EQUATIONS ET SECTION 17.2 ¤ 311
18.2 Nonhomogeneous Linear Equations ET 17.2
1. The auxiliary equation is r 2 +3r +2=(r +2)(r +1)=0, so the complementary solution is y c (x) =c 1 e −2x + c 2 e −x .
We try the particular solution y p (x) =Ax 2 + Bx + C,soy 0 p =2Ax + B and y 00
p =2A. Substituting into the differential
equation, we have (2A)+3(2Ax + B)+2(Ax 2 + Bx + C) =x 2 or 2Ax 2 +(6A +2B)x +(2A +3B +2C) =x 2 .
Comparing coefficients gives 2A =1, 6A +2B =0,and2A +3B +2C =0,soA = 1 2 , B = − 3 2 ,andC = 7 4 .Thusthe
general solution is y(x) =y c(x)+y p(x) =c 1e −2x + c 2e −x + 1 2 x2 − 3 2 x + 7 4 .
3. The auxiliary equation is r 2 − 2r = r(r − 2) = 0, so the complementary solution is y c (x) =c 1 + c 2 e 2x . Try the particular
solution y p (x) =A cos 4x + B sin 4x,soy 0 p = −4A sin 4x +4B cos 4x and y 00
p = −16A cos 4x − 16B sin 4x. Substitution
into the differential equation gives (−16A cos 4x − 16B sin 4x) − 2(−4A sin 4x +4B cos 4x) =sin4x
⇒
(−16A − 8B)cos4x +(8A − 16B)sin4x =sin4x. Then−16A − 8B =0and 8A − 16B =1 ⇒ A = 1 40 and
B = − 1 . Thus the general solution is y(x) =y 20 c(x)+y p (x) =c 1 + c 2 e 2x + 1 cos 4x − 1 sin 4x.
40 20
5. The auxiliary equation is r 2 − 4r +5=0with roots r =2± i, so the complementary solution is
y c (x) =e 2x (c 1 cos x + c 2 sin x). Tryy p (x) =Ae −x ,soy 0 p = −Ae −x and y 00
p = Ae −x . Substitution gives
Ae −x − 4(−Ae −x )+5(Ae −x )=e −x ⇒ 10Ae −x = e −x ⇒ A = 1 10
. Thus the general solution is
y(x) =e 2x (c 1 cos x + c 2 sin x)+ 1
10 e−x .
7. The auxiliary equation is r 2 +1=0with roots r = ±i, so the complementary solution is y c(x) =c 1 cos x + c 2 sin x.
For y 00 + y = e x try y p1 (x) =Ae x .Theny 0 p 1
= y 00
p 1
= Ae x and substitution gives Ae x + Ae x = e x ⇒ A = 1 2 ,
so y p1 (x) = 1 2 ex .Fory 00 + y = x 3 try y p2 (x) =Ax 3 + Bx 2 + Cx + D. Theny 0 p 2
=3Ax 2 +2Bx + C and
y 00
p 2
=6Ax +2B. Substituting, we have 6Ax +2B + Ax 3 + Bx 2 + Cx + D = x 3 ,soA =1, B =0,
6A + C =0 ⇒ C = −6, and2B + D =0 ⇒ D =0. Thus y p2 (x) =x 3 − 6x and the general solution is
y(x) =y c(x)+y p1 (x)+y p2 (x) =c 1 cos x + c 2 sin x + 1 2 ex + x 3 − 6x. But2=y(0) = c 1 + 1 2
⇒
c 1 = 3 and 2 0=y0 (0) = c 2 + 1 − 6 ⇒ c 2 2 = 11 . Thus the solution to the initial-value problem is
2
y(x) = 3 2
cos x +
11
2 sin x + 1 2 ex + x 3 − 6x.
9. The auxiliary equation is r 2 − r =0with roots r =0, r =1so the complementary solution is y c(x) =c 1 + c 2e x .
Try y p (x) =x(Ax + B)e x so that no term in y p is a solution of the complementary equation. Then
y 0 p =(Ax 2 +(2A + B)x + B)e x and y 00
p =(Ax 2 +(4A + B)x +(2A +2B))e x . Substitution into the differential equation
gives (Ax 2 +(4A + B)x +(2A +2B))e x − (Ax 2 +(2A + B)x + B)e x = xe x ⇒ (2Ax +(2A + B))e x = xe x ⇒
A = 1 2 , B = −1. Thus yp(x) = 1
2 x2 − x e x and the general solution is y(x) =c 1 + c 2e x + 1
2 x2 − x e x .But
2=y(0) = c 1 + c 2 and 1=y 0 (0) = c 2 − 1, soc 2 =2and c 1 =0. The solution to the initial-value problem is
y(x) =2e x + 1
2 x2 − x e x = e x 1
2 x2 − x +2 .

F.
312 ¤ CHAPTER 18 SECOND-ORDER DIFFERENTIAL EQUATIONS ETTX.10
CHAPTER 17
11. The auxiliary equation is r 2 +3r +2=(r +1)(r +2)=0,sor = −1, r = −2 and y c(x) =c 1e −x + c 2e −2x .
Try y p = A cos x + B sin x ⇒ y 0 p = −A sin x + B cos x, y 00
p = −A cos x − B sin x. Substituting into the differential
equation gives (−A cos x − B sin x)+3(−A sin x + B cos x)+2(A cos x + B sin x) =cosx or
(A +3B)cosx +(−3A + B)sinx =cosx. Then solving the equations
A +3B =1, −3A + B =0gives A = 1 , B = 3 and the general
10 10
solution is y(x) =c 1 e −x + c 2 e −2x + 1 cos x + 3 sin x. Thegraph
10 10
shows y p and several other solutions. Notice that all solutions are
asymptotic to y p as x →∞.Exceptfory p , all solutions approach either ∞
or −∞ as x →−∞.
13. Here y c (x) =c 1 cos 3x + c 2 sin 3x. Fory 00 +9y = e 2x try y p1 (x) =Ae 2x and for y 00 +9y = x 2 sin x
try y p2 (x) =(Bx 2 + Cx + D)cosx +(Ex 2 + Fx+ G)sinx. Thus a trial solution is
y p (x) =y p1 (x)+y p2 (x) =Ae 2x +(Bx 2 + Cx + D)cosx +(Ex 2 + Fx+ G)sinx.
15. Here y c (x) =c 1 + c 2 e −9x .Fory 00 +9y 0 =1try y p1 (x) =Ax (since y = A is a solution to the complementary equation)
and for y 00 +9y 0 = xe 9x try y p2 (x) =(Bx + C)e 9x .
17. Since y c (x) =e −x (c 1 cos 3x + c 2 sin 3x) we try y p (x) =x(Ax 2 + Bx + C)e −x cos 3x + x(Dx 2 + Ex + F )e −x sin 3x
(sothatnotermofy p is a solution of the complementary equation).
Note: Solving Equations (7) and (9) in The Method of Variation of Parameters gives
u 0 Gy 2
1 = −
a (y 1 y2 0 − y 2y1 0 ) and u 0 Gy 1
2 =
a (y 1 y2 0 − y 2y1 0 )
We will use these equations rather than resolving the system in each of the remaining exercises in this section.
19. (a) Here 4r 2 +1=0 ⇒ r = ± 1 2 i and y c(x) =c 1 cos 1
2 x + c 2 sin 1
2 x . We try a particular solution of the form
y p(x) =A cos x + B sin x ⇒ y 0 p = −A sin x + B cos x and y 00
p = −A cos x − B sin x. Then the equation
4y 00 + y =cosx becomes 4(−A cos x − B sin x)+(A cos x + B sin x) =cosx or
−3A cos x − 3B sin x =cosx ⇒ A = − 1 , B =0.Thus,yp(x) =− 1 cos x and the general solution is
3 3
y(x) =y c (x)+y p (x) =c 1 cos 1
x + c
2 2 sin 1
x − 1 cos x.
2 3
(b) From (a) we know that y c(x) =c 1 cos x 2 + c2 sin x 2 .Settingy1 =cosx 2 , y2 =sinx 2 ,wehave
y 1 y 0 2 − y 2 y 0 1 = 1 2 cos2 x 2 + 1 2 sin2 x 2 = 1 2 .Thusu0 1 = − cos x sin x 2
4 · 1
2
= − 1 cos
2 · x
2 2 sin
x
= − 1
2 2 2cos
2 x
− 1 sin x 2 2
and u 0 2 = cos x cos x 2
4 · 1
2
= 1 cos
2
2 · x
2 cos
x
= 1
2 2 1 − 2sin
2 x
2 cos
x
.Then 2
u 1(x) = 1
2 sin x 2 − cos2 x 2 sin x 2
dx = − cos
x
2 + 2 3 cos3 x 2 and

F.
u 2(x) = 1
cos x − 2 2 sin2 x cos x
2 2 dx =sin
x
− 2 2 3 sin3 x . Thus 2
y p (x)= − cos x + 2 2 3 cos3 x 2
= − cos
2 · x
2 +
2
3
SECTION TX.10 18.2 NONHOMOGENEOUS LINEAR EQUATIONS ET SECTION 17.2 ¤ 313
cos
x
+ sin x − 2 2 2 3 sin3 x 2
cos
2 x
2 +sin2 x 2
cos
2 x
2 − sin2 x 2
and the general solution is y(x) =y c(x)+y p(x) =c 1 cos x + 2 c2 sin x − 1 2 3
cos x.
sin
x
= − cos 2 x −
2 2 sin2 x 2 +
2
3 cos
4 x
−
2 sin4 x 2
= − cos x +
2
3 cos x = − 1 3 cos x
21. (a) r 2 − 2r +1=(r − 1) 2 =0 ⇒ r =1, so the complementary solution is y c (x) =c 1 e x + c 2 xe x . A particular solution
is of the form y p (x) =Ae 2x . Thus 4Ae 2x − 4Ae 2x + Ae 2x = e 2x ⇒ Ae 2x = e 2x ⇒ A =1 ⇒ y p (x) =e 2x .
So a general solution is y(x) =y c (x)+y p (x) =c 1 e x + c 2 xe x + e 2x .
(b) From (a), y c (x) =c 1 e x + c 2 xe x ,sosety 1 = e x , y 2 = xe x . Then, y 1 y 0 2 − y 2 y 0 1 = e 2x (1 + x) − xe 2x = e 2x and so
u 0 1 = −xe x ⇒ u 1 (x) =− xe x dx = −(x − 1)e x [by parts] and u 0 2 = e x ⇒ u 2 (x) = e x dx = e x . Hence
y p (x) =(1− x)e 2x + xe 2x = e 2x and the general solution is y(x) =y c (x)+y p (x) =c 1 e x + c 2 xe x + e 2x .
23. As in Example 5, y c(x) =c 1 sin x + c 2 cos x,sosety 1 =sinx, y 2 =cosx. Theny 1y 0 2 − y 2y 0 1 = − sin 2 x − cos 2 x = −1,
so u 0 1 = − sec2 x cos x
−1
and u 0 2 = sec2 x sin x
−1
=secx ⇒ u 1 (x) = sec xdx =ln(secx +tanx) for 0

F.
314 ¤ CHAPTER 18 SECOND-ORDER DIFFERENTIAL EQUATIONS ETX.10
CHAPTER 17
18.3 Applications of Second-Order Differential Equations ET 17.3
1. By Hooke’s Law k(0.25) = 25 so k =100is the spring constant and the differential equation is 5x 00 + 100x =0.
The auxiliary equation is 5r 2 + 100 = 0 with roots r = ±2 √ 5 i, so the general solution to the differential equation is
x(t) =c 1 cos 2 √ 5 t + c 2 sin 2 √ 5 t . We are given that x(0) = 0.35 ⇒ c 1 =0.35 and x 0 (0) = 0 ⇒
2 √ 5 c 2 =0 ⇒ c 2 =0, so the position of the mass after t seconds is x(t) =0.35 cos 2 √ 5 t .
3. k(0.5) = 6 or k =12is the spring constant, so the initial-value problem is 2x 00 +14x 0 +12x =0, x(0) = 1, x 0 (0) = 0.
The general solution is x(t) =c 1e −6t + c 2e −t .But1=x(0) = c 1 + c 2 and 0=x 0 (0) = −6c 1 − c 2. Thus the position is
given by x(t) =− 1 5 e−6t + 6 5 e−t .
5. For critical damping we need c 2 − 4mk =0or m = c 2 /(4k) =14 2 /(4 · 12) = 49
12 kg.
7. We are given m =1, k = 100, x(0) = −0.1 and x 0 (0) = 0. From (3), the differential equation is d2 x
dt 2
with auxiliary equation r 2 + cr +100=0.
If c =10,wehavetwocomplexrootsr = −5 ± 5 √ 3 i, so the motion is underdamped and the solution is
+ c dx + 100x =0
dt
x = e −5t c 1 cos 5 √ 3 t + c 2 sin 5 √ 3 t .Then−0.1 =x(0) = c 1 and 0=x 0 (0) = 5 √ 3 c 2 − 5c 1 ⇒ c 2 = − 1
10 √ , 3
so x = e −5t −0.1cos 5 √ 3 t − 1
10 √ sin 5 √ 3 t .
3
If c =15, we again have underdamping since the auxiliary equation has roots r = − 15
x = e
c −15t/2 5 1 cos
√
7
t 5
2
+ c 2 sin
√
7
t 2
Thus x = e
−0.1cos
−15t/2 5 √
7
t 2
− 3
10 √ sin 5 √ 7
t 7 2
.
± 5 √ 7
2 2
,so−0.1 =x (0) = c 1 and 0=x 0 (0) = 5 √ 7
2
c2 −
15
2
For c =20, we have equal roots r 1 = r 2 = −10, so the oscillation is critically damped and the solution is
i. The general solution is
c1 ⇒ c2 = −
3
10 √ 7 .
x =(c 1 + c 2 t)e −10t .Then−0.1 =x(0) = c 1 and 0=x 0 (0) = −10c 1 + c 2 ⇒ c 2 = −1,sox =(−0.1 − t)e −10t .
If c =25the auxiliary equation has roots r 1 = −5, r 2 = −20, so we have overdamping and the solution is
x = c 1 e −5t + c 2 e −20t .Then−0.1 =x(0) = c 1 + c 2 and 0=x 0 (0) = −5c 1 − 20c 2 ⇒ c 1 = − 2 15 and c 2 = 1
30 ,
so x = − 2 15 e−5t + 1 30 e−20t .
If c =30we have roots r = −15 ± 5 √ 5, so the motion is
overdamped and the solution is x = c 1 e (−15 + 5 √ 5 )t + c 2 e (−15 − 5 √ 5 )t .
Then −0.1 =x(0) = c 1 + c 2 and
0=x 0 (0) = −15 + 5 √ 5 c 1 + −15 − 5 √ 5 c 2 ⇒
c 1 = −5 − 3 √ 5
and c
100 2 = −5+3√ 5
x =
e (−15 + 5 √
5)t +
−5 − 3 √ 5
100
100
,so
−5+3 √ 5
100
e (−15 − 5 √ 5)t .

F.
SECTION 18.3 APPLICATIONS TX.10 OF SECOND-ORDER DIFFERENTIAL EQUATIONS ET SECTION 17.3 ¤ 315
9. The differential equation is mx 00 + kx = F 0 cos ω 0t and ω 0 6=ω = k/m. Here the auxiliary equation is mr 2 + k =0
with roots ± k/m i = ±ωi so x c (t) =c 1 cos ωt + c 2 sin ωt. Sinceω 0 6=ω,tryx p (t) =A cos ω 0 t + B sin ω 0 t.
Then we need (m) −ω 2 0 (A cos ω0 t + B sin ω 0 t)+k(A cos ω 0 t + B sin ω 0 t)=F 0 cos ω 0 t or A k − mω 2 0 = F0 and
B
k − mω 2 F 0
F 0
0 =0. Hence B =0and A = =
k − mω 2 0 m(ω 2 − ω 2 0) since ω2 = k . Thus the motion of the mass is given
m
by x(t) =c 1 cos ωt + c 2 sin ωt +
F 0
m(ω 2 − ω 2 0 ) cos ω 0t.
11. From Equation 6, x(t) =f(t)+g(t) where f(t) =c 1 cos ωt + c 2 sin ωt and g(t) =
F 0
m(ω 2 − ω 2 0 ) cos ω 0t. Thenf
is periodic, with period 2π ,andifω 6=ω ω 0, g is periodic with period 2π
ω 0
.If ω ω 0
is a rational number, then we can say
ω
ω 0
= a b
⇒ a = bω
ω 0
where a and b are non-zero integers. Then
x t + a · 2π ω
so x(t) is periodic.
= f
t + a · 2π
ω
+ g t + a · 2π
ω = f(t)+g
t + bω
ω 0
· 2π = f(t)+g t + b · 2π
ω
ω 0
= f(t)+g(t) =x(t)
13. Here the initial-value problem for the charge is Q 00 +20Q 0 +500Q =12, Q(0) = Q 0 (0) = 0. Then
Q c (t) =e −10t (c 1 cos 20t + c 2 sin 20t) and try Q p (t) =A ⇒ 500A =12or A = 3
125 .
The general solution is Q(t) =e −10t (c 1 cos 20t + c 2 sin 20t)+ 3
125 .But0=Q(0) = c 1 + 3
125 and
Q 0 (t) =I(t) =e −10t [(−10c 1 +20c 2) cos 20t +(−10c 2 − 20c 1)sin20t] but 0=Q 0 (0) = −10c 1 +20c 2. Thus the charge
is Q(t) =− 1
250 e−10t (6 cos 20t +3sin20t)+ 3
125 and the current is I(t) =e−10t 3
5
sin 20t.
15. As in Exercise 13, Q c (t) =e −10t (c 1 cos 20t + c 2 sin 20t) but E(t) =12sin10t so try
Q p(t) =A cos 10t + B sin 10t. Substituting into the differential equation gives
(−100A +200B +500A)cos10t +(−100B − 200A + 500B)sin10t =12sin10t
⇒
400A +200B =0and 400B − 200A =12. Thus A = − 3 , B = 3 and the general solution is
250 125
Q(t) =e −10t (c 1 cos 20t + c 2 sin 20t) − 3
3
cos 10t + sin 10t. But0=Q(0) = c 250 125 1 − 3 so c 250 1 = 3 . 250
Also Q 0 (t) = 3 25 sin 10t + 6 25 cos 10t + e−10t [(−10c 1 +20c 2) cos 20t +(−10c 2 − 20c 1)sin20t] and
0=Q 0 (0) = 6 − 10c 25 1 +20c 2 so c 2 = − 3 . Hence the charge is given by
500
Q(t) =e −10t 3
3
cos 20t − sin 20t − 3
3
cos 10t + sin 10t.
250 500 250 125
c1
17. x(t) =A cos(ωt + δ) ⇔ x(t) =A[cos ωt cos δ − sin ωt sin δ ] ⇔ x(t) =A
A cos ωt + c
2
A sin ωt where
cos δ = c 1/A and sin δ = −c 2/A ⇔ x(t) =c 1 cos ωt + c 2 sin ωt. [Notethatcos 2 δ +sin 2 δ =1 ⇒ c 2 1 + c 2 2 = A 2 .]

F.
316 ¤ CHAPTER 18 SECOND-ORDER DIFFERENTIAL EQUATIONS ETX.10
CHAPTER 17
18.4 Series Solutions ET 17.4
1. Let y(x) = ∞ c nx n .Theny 0
(x) = ∞ nc nx n−1 and the given equation, y 0 − y =0, becomes
n=0
n=1
∞
nc n x n−1 − ∞
c n x n =0. Replacing n by n +1in the first sum gives ∞
(n +1)c n+1 x n − ∞ c n x n =0,so
n=1
n=0
∞
[(n +1)c n+1 − c n ]x n =0. Equating coefficients gives (n +1)c n+1 − c n =0, so the recursion relation is
n=0
c n+1 =
cn
n +1 , n =0, 1, 2,....Thenc 1 = c 0 , c 2 = 1 2 c 1 = c0 2 , c 3 = 1 3 c 2 = 1 3 · 1
2 c 0 = c0
3! , c 4 = 1 4 c 3 = c0
4! ,and
in general, c n = c 0
n! . Thus, the solution is y(x) = ∞
3. Assuming y(x) = ∞
−x 2 y = − ∞
n =0
n =0
c n x n ,wehavey 0 (x) =
n =2
n=0
∞
n =1
c n x n = ∞
n=0
nc n x n−1 =
n=0
c 0
∞
n! xn = c 0
∞
n =0
n=0
x n
n! = c 0e x .
(n +1)c n+1 x n and
c nx n+2
= − ∞ c n−2x n . Hence, the equation y 0 = x 2
y becomes ∞ (n +1)c n+1x n
− ∞ c n−2x n =0
or c 1 +2c 2 x + ∞ [(n +1)c n+1 − c n−2 ] x n =0. Equating coefficients gives c 1 = c 2 =0and c n+1 = cn−2
n +1
n =2
for n =2, 3, ....Butc 1 =0,soc 4 =0and c 7 =0and in general c 3n+1 =0. Similarly c 2 =0so c 3n+2 =0. Finally
c 3 = c0 3 , c 6 = c3 6 =
c0
6 · 3 = c0
3 2 · 2! , c 9 = c6 9 = c0
9 · 6 · 3 = c0
3 3 · 3! , ...,andc 3n = c0
. Thus, the solution
3 n · n!
is y (x) =
∞
c n x n =
∞
c 3n x 3n =
∞ c 0
∞ x 3n
3 n · n! x3n = c 0
3 n n! = c ∞ x 3 /3 n
0
= c 0 e x3 /3 .
n!
n =0
n =0
n =0
5. Let y (x) = ∞ c nx n ⇒ y 0
(x) = ∞ nc nx n−1 and y 00
(x) = ∞ (n +2)(n +1)c n+2x n . The differential equation
n=0
n=1
becomes ∞ (n +2)(n +1)c n+2 x n
+ x ∞ nc n x n−1
+ ∞ c n x n
=0or ∞ [(n +2)(n +1)c n+2 + nc n + c n ]x n =0
since
n=0
∞
n=1
n=1
n=0
nc nx n
= ∞ nc nx
. n Equating coefficients gives (n +2)(n +1)c n+2 +(n +1)c n =0, thus the
n=0
recursion relation is c n+2 =
n =0
n=0
n=0
n =0
n =0
−(n +1)cn
(n +2)(n +1) = − c n
, n =0, 1, 2,.... Then the even
n +2
coefficients are given by c 2 = − c 0
2 , c4 = − c 2
4 = c 0
2 · 4 , c6 = −c 4
6 = − c 0
, and in general,
2 · 4 · 6
c 2n =(−1) n c 0
2 · 4 ·····2n = (−1)n c 0
. The odd coefficients are c
2 n 3 = − c 1
n!
3 , c5 = −c 3
5 = c 1
3 · 5 , c7 = −c 5
7 = − c 1
3 · 5 · 7 ,
and in general, c 2n+1 =(−1) n c 1
3 · 5 · 7 ·····(2n +1) = (−2)n n! c 1
. The solution is
(2n +1)!
y (x) =c 0
∞
n=0
(−1) n
2 n n! x2n + c 1
∞
n=0
(−2) n n!
(2n +1)! x2n+1 .
7. Let y (x) = ∞ c nx n ⇒ y 0
(x) = ∞ nc nx n−1
= ∞ (n +1)c n+1x n and y 00
(x) = ∞ (n +2)(n +1)c n+2x n .Then
n=0
n=1
n=0
(x−1)y 00
(x) = ∞ (n+2)(n+1)c n+2 x n+1
− ∞ (n+2)(n+1)c n+2 x n
= ∞ n(n+1)c n+1 x n
− ∞ (n+2)(n+1)c n+2 x n .
n=0
n=0
n=1
n=0
n=0
n=0
n =2

F.
TX.10
SECTION 18.4 SERIES SOLUTIONS ET SECTION 17.4 ¤ 317
Since ∞ n(n +1)c n+1 x n
= ∞ n(n +1)c n+1 x n , the differential equation becomes
n=1
n=0
∞
n(n +1)c n+1 x n
− ∞ (n +2)(n +1)c n+2 x n
+ ∞ (n +1)c n+1 x n =0 ⇒
n=0
n=0
n=0
∞
[n(n +1)c n+1 − (n +2)(n +1)c n+2 +(n +1)c n+1]x n =0 or
n=0
∞
[(n +1) 2 c n+1 − (n +2)(n +1)c n+2]x n =0.
Equating coefficients gives (n +1) 2 c n+1 − (n +2)(n +1)c n+2 =0for n =0, 1, 2, .... Then the recursion relation is
c n+2 =
(n +1) 2
n +1
cn+1 =
(n +2)(n +1) n +2 cn+1,sogivenc0 and c1,wehavec2 = 1 c1, 2 c3 = 2 3 c2 = 1 c1, 3 c4 = 3 4 c3 = 1 c1,and
4
in general c n = c 1
n , n =1, 2, 3, .... Thus the solution is y(x) =c ∞
0 + c 1
c 0 − c 1 ln(1 − x) for |x| < 1.
9. Let y(x) = ∞
y 00 (x) =
becomes
c n+2 =
y(0) = ∞
∞
n =0
∞
n =0
n =0
c n x n .Then−xy 0 (x) =−x ∞
n =1
nc n x n−1 = − ∞
n =1
n=0
(n +2)(n +1)c n+2x n , and the equation y 00 − xy 0 − y =0
n=1
x n
. Note that the solution can be expressed as
n
nc n x n = − ∞
n =0
[(n +2)(n +1)c n+2 − nc n − c n ]x n =0. Thus, the recursion relation is
nc n x n ,
nc n + c n
(n +2)(n +1) = c n(n +1)
(n +2)(n +1) = c n
for n =0, 1, 2, .... One of the given conditions is y(0) = 1. But
n +2
n=0
c n(0) n = c 0 +0+0+···= c 0,soc 0 =1. Hence, c 2 = c 0
2 = 1 2 , c4 = c 2
4 = 1
2 · 4 , c6 = c 4
6 = 1
2 · 4 · 6 , ...,
c 2n = 1
2 n n! . The other given condition is y0 (0) = 0. Buty 0 (0) = ∞
By the recursion relation, c 3 = c 1
3
problem is y(x) =
∞
n =0
11. Assuming that y(x) = ∞
y 00 (x)=
c n x n =
n =0
∞
n =2
n=1
nc n (0) n−1 = c 1 +0+0+··· = c 1 ,soc 1 =0.
=0, c5 =0, ..., c2n+1 =0for n =0, 1, 2, .... Thus, the solution to the initial-value
∞
n =0
c 2n x 2n =
∞
n =0
c nx n ,wehavexy = x ∞
n(n − 1)c n x n−2 =
x 2n
2 n n! = ∞
n =0
∞
n=−1
c nx n =
=2c 2 + ∞ (n +3)(n +2)c n+3 x n+1 ,
n =0
and the equation y 00 + x 2 y 0 + xy =0becomes 2c 2 + ∞
recursion relation is c n+3 =
n =0
(x 2 /2) n
= e x2 /2 .
n!
∞
n =0
c nx n+1 , x 2 y 0 = x 2
∞
n =1
nc nx n−1 =
(n +3)(n +2)c n+3 x n+1 [replace n with n +3]
n =0
∞
n =0
nc nx n+1 ,
[(n +3)(n +2)c n+3 + nc n + c n] x n+1 =0.Soc 2 =0and the
−nc n − c n (n +1)cn
= − , n =0, 1, 2, ....Butc0 = y(0) = 0 = c2 and by the
(n +3)(n +2) (n +3)(n +2)
recursion relation, c 3n = c 3n+2 =0for n =0, 1, 2, ....Also,c 1 = y 0 (0) = 1, soc 4 = − 2c 1
4 · 3 = − 2
4 · 3 ,
c 7 = − 5c4
2 · 5
2 2 5 2
7 · 6 =(−1)2 7 · 6 · 4 · 3 =(−1)2 , ..., c 3n+1 =(−1) n 2 2 5 2 ·····(3n − 1) 2
. Thus, the solution is
7!
(3n +1)!
y(x) =
∞ c n x n
= x +
(−1) ∞ n 2 2 5 2 ·····(3n − 1) 2 x 3n+1
.
(3n +1)!
n =0
n =1

F.
318 ¤ CHAPTER 18 SECOND-ORDER DIFFERENTIAL EQUATIONS ETX.10
CHAPTER 17
18 Review ET 17
1. (a) ay 00 + by 0 + cy =0where a, b,andc are constants.
(b) ar 2 + br + c =0
(c) If the auxiliary equation has two distinct real roots r 1 and r 2 , the solution is y = c 1 e r1x + c 2 e r2x . If the roots are real and
equal, the solution is y = c 1 e rx + c 2 xe rx where r is the common root. If the roots are complex, we can write r 1 = α + iβ
and r 2 = α − iβ, and the solution is y = e αx (c 1 cos βx + c 2 sin βx).
2. (a) An initial-value problem consists of finding a solution y of a second-order differential equation that also satisfies given
conditions y(x 0)=y 0 and y 0 (x 0)=y 1,wherey 0 and y 1 are constants.
(b) A boundary-value problem consists of finding a solution y of a second-order differential equation that also satisfies given
boundary conditions y(x 0)=y 0 and y(x 1)=y 1.
3. (a) ay 00 + by 0 + cy = G(x) where a, b,andc are constants and G is a continuous function.
(b) The complementary equation is the related homogeneous equation ay 00 + by 0 + cy =0.Ifwefind the general solution y c
of the complementary equation and y p is any particular solution of the original differential equation, then the general
solution of the original differential equation is y(x) =y p(x)+y c(x).
(c) See Examples 1–5 and the associated discussion in Section 18.2 [ ET 17.2].
(d) See the discussion on pages 1158–1160 [ ET 1122–1124].
4. Second-order linear differential equations can be used to describe the motion of a vibrating spring or to analyze an electric
circuit; see the discussion in Section 18.3 [ ET 17.3].
5. See Example 1 and the preceding discussion in Section 18.4 [ ET 17.4].
.
1. True. See Theorem 18.1.3 [ ET 17.1.3].
3. True. cosh x and sinh x are linearly independent solutions of this linear homogeneous equation.
1. The auxiliary equation is r 2 − 2r − 15 = 0 ⇒ (r − 5)(r +3)=0 ⇒ r =5, r = −3. Then the general solution
is y = c 1 e 5x + c 2 e −3x .
3. The auxiliary equation is r 2 +3=0 ⇒ r = ± √ 3 i. Then the general solution is y = c 1 cos √ 3 x + c 2 sin √ 3 x .
5. r 2 − 4r +5=0 ⇒ r =2± i,soy c (x) =e 2x (c 1 cos x + c 2 sin x). Tryy p (x) =Ae 2x ⇒ y 0 p =2Ae 2x
and y 00
p =4Ae 2x . Substitution into the differential equation gives 4Ae 2x − 8Ae 2x +5Ae 2x = e 2x ⇒ A =1and
the general solution is y(x) =e 2x (c 1 cos x + c 2 sin x)+e 2x .

F.
TX.10
CHAPTER 18 REVIEW ET CHAPTER 17 ¤ 319
7. r 2 − 2r +1=0 ⇒ r =1and y c(x) =c 1e x + c 2xe x .Tryy p(x) =(Ax + B)cosx +(Cx + D)sinx ⇒
y 0 p =(C − Ax − B)sinx +(A + Cx + D)cosx and y 00
p =(2C − B − Ax)cosx +(−2A − D − Cx)sinx. Substitution
gives (−2Cx +2C − 2A − 2D)cosx +(2Ax − 2A +2B − 2C)sinx = x cos x ⇒ A =0, B = C = D = − 1 2 .
The general solution is y(x) =c 1 e x + c 2 xe x − 1 cos x − 1 (x +1)sinx.
2 2
9. r 2 − r − 6=0 ⇒ r = −2, r =3and y c(x) =c 1e −2x + c 2e 3x .Fory 00 − y 0 − 6y =1,tryy p1 (x) =A. Then
y 0 p 1
(x) =y 00
p 1
(x) =0and substitution into the differential equation gives A = − 1 6 .Fory00 − y 0 − 6y = e −2x try
y p2 (x) =Bxe −2x
[since y = Be −2x satisfies the complementary equation]. Then y 0 p 2
=(B − 2Bx)e −2x and
yp 00
2
=(4Bx − 4B)e −2x , and substitution gives −5Be −2x = e −2x ⇒ B = − 1 5
. The general solution then is
y(x) =c 1 e −2x + c 2 e 3x + y p1 (x)+y p2 (x) =c 1 e −2x + c 2 e 3x − 1 6 − 1 5 xe−2x .
11. The auxiliary equation is r 2 +6r =0and the general solution is y(x) =c 1 + c 2 e −6x = k 1 + k 2 e −6(x−1) .But
3=y(1) = k 1 + k 2 and 12 = y 0 (1) = −6k 2 . Thus k 2 = −2, k 1 =5and the solution is y(x) =5− 2e −6(x−1) .
13. The auxiliary equation is r 2 − 5r +4=0and the general solution is y(x) =c 1e x + c 2e 4x .But0=y(0) = c 1 + c 2
and 1=y 0 (0) = c 1 +4c 2 , so the solution is y(x) = 1 3 (e4x − e x ).
15. Let y(x) = ∞ c n x n .Theny 00
(x) = ∞ n(n − 1)c n x n−2
= ∞ (n +2)(n +1)c n+2 x n and the differential equation
n=0
n=0
becomes ∞ [(n +2)(n +1)c n+2 +(n +1)c n ]x n =0. Thus the recursion relation is c n+2 = −c n /(n +2)
n=0
for n =0, 1, 2, ....Butc 0 = y(0) = 0,soc 2n =0for n =0, 1, 2, ....Alsoc 1 = y 0 (0) = 1,soc 3 = − 1 3 , c 5 = (−1)2
3 · 5 ,
c 7 = (−1)3
3 · 5 · 7 = (−1)3 2 3 3!
, ..., c 2n+1 = (−1)n 2 n n!
7!
(2n +1)!
is y(x) = ∞ c n x n
= ∞
n=0
n=0
(−1) n 2 n n!
(2n +1)!
x 2n+1 .
n=0
for n =0, 1, 2,.... Thus the solution to the initial-value problem
17. Here the initial-value problem is 2Q 00 +40Q 0 + 400Q =12, Q (0) = 0.01, Q 0 (0) = 0. Then
Q c(t) =e −10t (c 1 cos 10t + c 2 sin 10t) and we try Q p(t) =A. Thus the general solution is
Q(t) =e −10t (c 1 cos 10t + c 2 sin 10t)+ 3
100 .But0.01 = Q0 (0) = c 1 +0.03 and 0=Q 00 (0) = −10c 1 +10c 2 ,
so c 1 = −0.02 = c 2 . Hence the charge is given by Q(t) =−0.02e −10t (cos 10t +sin10t)+0.03.
19. (a) Since we are assuming that the earth is a solid sphere of uniform density, we can calculate the density ρ as follows:
ρ =
mass of earth
volume of earth =
M
4
3 πR3 .IfV r is the volume of the portion of the earth which lies within a distance r of the
center, then V r = 4 3 πr3 and M r = ρV r = Mr3
R .ThusF 3
r = − GM rm
= − GMm r.
r 2 R 3

F.
320 ¤ CHAPTER 18 SECOND-ORDER DIFFERENTIAL EQUATIONS ETX.10
CHAPTER 17
(b) The particle is acted upon by a varying gravitational force during its motion. By Newton’s Second Law of Motion,
m d2 y
dt 2
= Fy = −GMm
R 3
g = GM
R 2 . Therefore k2 = g R .
y,soy 00 (t) =−k 2 y (t) where k 2 = GM
R 3
. At the surface, −mg = FR = −GMm ,so
(c) The differential equation y 00 + k 2 y =0has auxiliary equation r 2 + k 2 =0. (This is the r of Section 18.1 [ ET 17.1],
not the r measuring distance from the earth’s center.) The roots of the auxiliary equation are ±ik,soby(11)in
Section 18.1 [ ET 17.1], the general solution of our differential equation for t is y(t) =c 1 cos kt + c 2 sin kt. Itfollowsthat
y 0 (t) =−c 1 k sin kt + c 2 k cos kt. Nowy (0) = R and y 0 (0) = 0,soc 1 = R and c 2 k =0.Thusy(t) =R cos kt and
y 0 (t) =−kR sin kt. This is simple harmonic motion (see Section 18.3 [ ET 17.3]) with amplitude R, frequency k, and
phase angle 0. The period is T =2π/k. R ≈ 3960 mi = 3960 · 5280 ft and g =32ft/s 2 ,so
k = g/R ≈ 1.24 × 10 −3 s −1 and T =2π/k ≈ 5079 s ≈ 85 min.
(d) y(t) =0 ⇔ cos kt =0 ⇔ kt = π 2 + πn for some integer n ⇒ y0 (t) =−kR sin π
2 + πn = ±kR. Thusthe
particle passes through the center of the earth with speed kR ≈ 4.899 mi/s ≈ 17,600 mi/h.
R 2