Exercises 1.1

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MA111: Prepared by Dr.Archara Pacheenburawana 1
1.
Exercises 1.1
y
y = f(x)
1
x
For the function whose graph is given, state the value of the given quantity, if it does
not exist, explain why.
2.
(a) lim
x→0 −f(x)
(c) lim
x→0
f(x)
(e) lim
x→2 +f(x)
(g) lim
x→−3 f(x)
(b) lim
x→0 +f(x)
(d) lim
x→2 −f(x)
(f) lim f(x)
x→2
4y
3
2
1
0
-4 -3 -2 -1
-1
0 1 2 3 4 5
-2
y = f(x)
x
-3
-4
For the function whose graph is given, state the value of the given quantity, if it does
not exist, explain why.
(a) lim
x→1
f(x)
(c) lim
x→3 +f(x)
(e) f(3)
(g) lim
x→−2 +f(x)
(i) f(−2)
(b) lim
x→3 −f(x)
(d) lim f(x)
x→3
(f) lim
x→−2 −f(x)
(h) lim f(x)
x→−2

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MA111: Prepared by Dr.Archara Pacheenburawana 2
3.
4
3
2
y
y = f(x)
1
0
-4 -3 -2 -1 0 1 2 3 4
-1
x
For the function whose graph is given, state the value of the given quantity, if it does
not exist, explain why.
(a) lim
x→−3 f(x)
(c) f(−1)
(e) f(1)
(g) lim
x→1 +f(x)
(b) f(−3)
(d) lim
x→−1 f(x)
(f) lim
x→1 −f(x)
(h) lim f(x)
x→1
Answer to Exercises 1.1
1. (a) −2 (b) 2 (c) Does not exist (d) −1 (e) 3 (f) Does not exist (g) 2
2. (a) 3 (b) 2 (c) −2 (d) Does not exist (e) 1 (f) −1 (g) −1 (h) −1 (i) −3
3. (a) 2 (b) 1 (c) 2 (d) 5 2
(e) 2 (f) 2 (g) 1 (h) Does not exist
1. Given that
Exercise 1.2
limf(x) = −3, limg(x) = 0, lim
x→a x→a
h(x) = 8
x→a
find the limits that exist. If the limit does not exist, explain why.
[ ] [ ] 2
(a) lim f(x)+h(x) (b) lim f(x)
x→a x→a
(c) lim
x→a
3 √ h(x)
(e) lim
x→a
f(x)
h(x)
(g) lim
x→a
f(x)
g(x)
1
(d) lim
x→a f(x)
(f) lim
x→a
g(x)
f(x)
2f(x)
(h) lim
x→a h(x)−f(x)

MA111: Prepared by Dr.Archara Pacheenburawana 3
2. Evaluate the following limits.
(a) lim
x→0
(x 2 −3x+1)
(c) lim
x→2
x−5
x 2 +4
√
(e) lim x2 +2x+4
x→1
x 2 −x−12
(g) lim
x→−3 x+3
x 2 +x−2
(i) lim
x→1 x 2 −3x+2
(1+h) 4 −1
(k) lim
h→0 h
t−1
(m) lim √
t→1 t−1
t 2 +t−6
(o) lim
t→2 t 2 −4
(q) lim
x→2
x 4 −16
x−2
(s) lim
x→1
[ 1
x−1 − 2
x 2 −1
(3+h) −1 −3 −1
(u) lim
h→0 h
√ x−x
(w) lim
x→1 1− √ x
(y) lim
x→0
√ 3+x−
√
3
x
3. Evaluate lim
x→2
f(x) where f(x) =
]
(b) lim(x 3 +2)(x 2 −5x)
x→3
(d) lim
x→1
( x 4 +x 2 −6
x 4 +2x+3
(f) lim
x→4 − √
16−x
2
(h) lim
x→−2
(j) lim
x→1
x 3 −1
x 2 −1
x+2
x 2 −x−6
(2+h) 3 −8
(l) lim
h→0 h
9−t
(n) lim
t→9 3− √ t
√ √
2−t− 2
(p) lim
t→0 t
) 2
x 2 −81
(r) lim √
x→9 x−3
[
1
(t) lim
t→0 t √ 1+t − 1 ]
t
x
(v) lim
− 1 2
x−2
x
(x) lim √
x→0 1+3x−1
x→2
1
xe −2x+1
(z) lim
x→0 x 2 +1
{
3x 2 −2x+1 if x < 2
x 3 +1 if x ≥ 2
{ 2x if x < 2
4. Evaluate limf(x) where f(x) =
x→2 x 2 if x ≥ 2
{ x
5. Evaluate limf(x) where f(x) =
2 +1 if x < −1
x→0 3x+1 if x ≥ −1
6. Evaluate lim
x→−1 f(x) where f(x) = ⎧
⎨
⎩
2x+1 if x < −1
3 if −1 ≤ x < 1
2x+1 if x ≥ 1
7. Find the limit, if it exists. If the limit does not exist, explain why.

MA111: Prepared by Dr.Archara Pacheenburawana 4
(a) lim
x→−4 |x+4|
2x 2 −3x
(c) lim
x→1.5 |2x−3|
|x+4|
(b) lim
x→−4 − x+4
( 1
(d) lim
x→0 + x − 1 )
|x|
8. Use the Squeeze Theorem to find the following limits.
(a) lim
x→0
x 2 sin(1/x)
(b) lim
x→0
x 2 cos(20πx)
(c) lim
x→0
√
x3 +x 2 sin π x
Answer to Exercise 1.2
1. (a) 5 (b) 9 (c) 2 (d) − 1 3
(e) − 3 8
(f) 0 (g) Does not exist (h) − 6
11
2. (a) 1 (b) −174 (c) − 3 8
(d) 4 9
(e) √ 7 (f) 0 (g) −7 (h) − 1 5
(i) −3
(j) 3 (k) 4 (l) 12 (m) 2 (n) 6 (o) 5 (p) − √ 2
(q) 32 (r) 108 (s) 1 2 4 4 2
(t) − 1 (u) − 1 (v) − 1 (w) 1 (x) 2 (y) (z) 0
2 9 4 3
3. 9 4. 4 5. 1 6. Does not exist 7. (a) 0 (b) −1 (c) Does not exist (d) 0
8. (a) 0 (b) 0 (c) 0
1−22 Evaluate the limit, if it exists.
1
2 √ 3
Exercise 1.3
1. lim
x→∞
x+4
x 2 −2x+5
−x
3. lim √
x→−∞ 4+x
2
x 3 −2x+4
5. lim
x→∞ 3x 2 +3x−5
3x 3 −x+5
7. lim
x→∞ 4x 3 +4x 2 −1
9. lim
x→∞
(√
x2 +3−x
11. lim
x→∞
1− √ x
1+ √ x
)
13. lim
x→∞
(√
9x2 +x−3x
)
(1−x)(2+x)
2. lim
x→−∞ (1+2x)(2−3x)
x 3 −2x+1
4. lim
x→−∞ 3x 3 +4x−1
x 2 −sinx
6. lim
x→∞ x 2 +4x−1
x 4 −x 2 +1
8. lim
x→∞ x
√ 5 +x 3 −x
x2 +4x
10. lim
x→∞ 4x+1
12. lim
x→∞
(√
x2 +1− √ x 2 −1
14. lim
x→∞
√ x
)

MA111: Prepared by Dr.Archara Pacheenburawana 5
15. lim
x→∞
(
x−
√ x
)
17. lim
x→∞
x 7 −1
x 6 +1
19. lim
x→∞
sin2x
21. lim
x→∞
ln(2x)
16. lim
x→−∞ (x3 −5x 2 )
18. lim e 2x
x→∞
20. lim
x→∞
e −3x cos2x
22. lim
x→0 +(xln2x)
23−28 Find the horizontal asymptotes of each curve, if it exists.
23. f(x) = x
x+4
25. f(x) = x
4−x 2
27. f(x) =
x 3
x 2 +3x−10
24. f(x) =
x
√
4+x
2
26. f(x) = x3
4−x 2
28. f(x) =
x
4√
x4 +1
Answer to Exercise 1.3
1. 0 2. 1 6
3. 1 4. 1 3
5. ∞ 6. 1 7. 3 4
8. 0 9. 0 10. 1 4
11. −1 12. 0
13.
1
6
14. ∞ 15. ∞ 16. −∞ 17. ∞ 18. ∞ 19. Does not exist 20. 0
21. ∞ 22. 0 23. y = 1 24. y = −1,y = 1 25. y = 0
26. No horizontal asymptote 27. No horizontal asymptote 28. y = −1,y = 1
1−10 Evaluate the limits.
Exercise 1.4
sin5x
1. lim
x→0 3x
cosθ−1
3. lim
θ→0 sinθ
tanx
5. lim
x→0
7. lim
x→π/4
4x
sinx−cosx
cos2x
9. lim
x→0
x+sinx
tanx
sin8t
2. lim
t→0 sin9t
sin 2 x
4. lim
x→0 x
cot2x
6. lim
x→0 cscx
2xcot 2 x
8. lim
x→0 cscx
10. lim
x→0
sin(cosx)
secx
Answer to Exercise 1.4
1.
5
3
2. 8 9
3. 0 4. 0 5. 1 4
6. 1 2
7. − 1 √
2
8. 2 9. 2 10. sin1

MA111: Prepared by Dr.Archara Pacheenburawana 6
Exercise 1.5
1. Use the given graph to identify all discontinuility of the functions.
(a)
y
x
(b)
y
x
(c)
y
x
2. Use the definition of continuity and the properties of limits to show that the function
is continuous at the given number.
(a) f(x) = x 2 + √ 7−x, x = 4
(b) g(x) = (x+2x 3 ) 4 , x = −1
(c) h(x) = x+1
2x 2 +1 , x = 4
3. Use the definition of continuity and the properties of limits to show that the function
is continuous on the given interval.
(a) f(x) = 1
x+1 , (−1,∞)

MA111: Prepared by Dr.Archara Pacheenburawana 7
(b) f(x) = x−1
x 2 −4 , (−2,2)
(c) g(t) = √ 9−4t 2 , [− 3 2 , 3 2 ]
(d) h(z) = √ (z −1)(3−z), [1,3]
4. Explain why the function is discontinuous at the given number.
(a) f(x) = x
x−1 ; x = 1
(b) f(x) = sin 1 x ; x = 0
(c) f(x) = e 1/x ; x = 0
(d) f(x) = ln|x−2|; x = 2
{ 1
(e) f(x) = x−1 if x ≠ 1
2 if x = 1
; x = 1
(f) f(x) = x2 −1
x+1 ; x = −1
⎧
⎨ x 2 −2x−8
if x ≠ 4
(g) f(x) =
⎩
x−4 ; x = 4
3 if x = 4
{ 1−x if x ≤ 2
(h) f(x) =
x 2 −2x if x > 2 ; x = 2
⎧
⎨ x 2 if x < 2
(i) f(x) = 3 if x = 2 ; x = 2
⎩
3x−2 if x > 2
⎧
⎨
(j) f(x) =
⎩
x 2 if x < 0
−x if 0 ≤ x ≤ 1
x if x > 1
; x = 1
5. Determine the intervals on which f(x) is continuous.
(a) f(x) = 2x+ 3√ x (b) f(x) = 1
x+3
(c) f(x) = 1
x 2 +1
(e) f(x) = x2 +4
x−2
(g) f(x) = 3
x 2 −x
(i) f(x) = sinx
x 2
(d) f(x) = x−5
|x−5|
√
x+1
(f) f(x) = 3 x−1
(h) f(x) = x
4−x 2

MA111: Prepared by Dr.Archara Pacheenburawana 8
6. Find the constant c that makes f(x) continuous on (−∞,∞).
{ x+c if x < 0
(a) f(x) =
4−x 2 if x ≥ 0
{ c
(b) f(x) =
2 −x 2 if x < 0
2(x−c) 2 if x ≥ 0
{
cx+1 if x ≤ 3
(c) f(x) =
cx 2 −1 if x > 3
{ x
(d) f(x) =
2 −c 2 if x < 4
cx+20 if x ≥ 4
7. Find the values a and b so that f is continuous on (−∞,∞).
⎧
⎨ x+1 if x < 1
f(x) = ax+b if 1 ≤ x < 2
⎩
3x+1 if x ≥ 2
8. Let
Show that
f(x) =
{ x 2 , x ≠ 0
4, x = 0
and g(x) = 2x.
lim f( g(x) ) ( )
≠ f lim g(x) .
x→0 x→0
9. Use the Intermediate Value Theorem to verify that f(x) has a zero in the given
interval.
(a) f(x) = x 2 −7; [2,3]
(b) f(x) = x 3 −4x−2; [−1,0]
(c) f(x) = cosx−x; [0,1]
10. Use the Intermediate Value Theorem to show that x 3 + 3x − 2 = 0 has a real root
between 0 and 1.
11. Use the Intermediate Value Theorem to show that (cost)t 3 + 6sin 5 t−3 = 0 has a
real root between 0 and 2π.
12. Show that the equation x 5 +4x 3 −7x+14 = 0 has at least one real root.
Answer to Exercise 1.5
1. (a) x = −2,2 (b) x = −2,1,4 (c) x = −2,2,4

MA111: Prepared by Dr.Archara Pacheenburawana 9
4. (a) f(1) is not defined (b) f(0) is not defined (c) f(0) is not defined
(d) f(2) is not defined (e) limf(x) is not defined (f) f(−1) is not defined
x→1
(g) limf(x) ≠ f(4) (h) limf(x) is not defined (i) limf(x) ≠ f(2)
x→4 x→2 x→2
(j) limf(x) is not defined
x→1
5. (a) R (b) (−∞,−3)∪(−3,∞) (c) R (d) (−∞,5)∪(5,∞) (e) (−∞,2)∪(2,∞)
(f) (−∞,1)∪(1,∞) (g) (−∞,0)∪(0,1)∪(1,∞) (h) (−∞,−2)∪(−2,2)∪(2,∞)
(i) (−∞,0)∪(0,∞)
6. (a) 4 (b) 0 (c) 1 3
(d) −2 7. a = 5, b = −3
1. A curve has equation y = f(x).
Exercise 2.1
(a) Write an equation for the slope of the secant line through the points P ( 3,f(3) )
and Q ( x,f(x) ) .
(b) Write an equation for the slope of the tangent line at P.
2. Suppose an object moves with position function s = f(t).
(a) Write an equation for the average velocity of the object in the time interval from
t = a to t = a+h.
(b) Write an equation for the instantaneous velocity at time t = a.
3. Find an equation of the tangent line to the curve at the given point.
(a) y = x 2 −2, (1,−1)
(b) y = x 2 −3x, (−2,10)
(c) y = 1−2x−3x 2 , (−2,−7) (c) y = 1 x 2, (−2, 1 4 )
(e) y = 2
x+1 , (1,1) (f) y = √ x+3, (−2,1)
4. (a) Find the slope of the tangent to the curve y = 2/(x + 3) at the point where
x = a.
(b) Find the slopes of the tangent lines at the points whose x-coordinates are
(i) −1, (ii) 0, and (iii) 1.
5. (a) Find the slope of the tangent to the curve y = x 3 − 4x + 1 at the point where
x = a.
(b) Find equations of the tangent lines at the point (1,−2) and (2,1).
6. The following functions represent the position of an object at time t seconds. Find
the average velocity between (i) t = 0 and t = 2 (ii) t = 1 and t = 2 (iii) t = 1.9
and t = 2 and (iv) t = 1.99 and t = 2 and (v) estimate the instantaneous velocity
at t = 2.

MA111: Prepared by Dr.Archara Pacheenburawana 10
(a) s = f(t) = 16t 2 +10
(b) s = f(t) = √ t 2 +8t
7. Use the position function f(t) to find the velocity at time t = a.
(a) f(t) = −16t 2 +5, a = 1
(b) f(t) = √ t+16, a = 0
8. The displacement (in meters) of the particle moving in a straight line is given by the
equation of motion s = f(t) = 4t 3 +6t+2, where t is measured in seconds. Find the
velocity of the particle at times t = a, t = 1, t = 2, and t = 3.
1. (a) f(x)−f(3)
x−3
Answer to Exercise 2.1
(b) lim
x→3
f(x)−f(3)
x−3
2. (a) f(a+h)−f(a)
h
(b) lim
h→0
f(a+h)−f(a)
h
3. (a) y = 2x−3 (b) y = −7x−4 (c) y = 10x+13 (d) y = 1 4 x+ 3 4
(e) y = − 1 2 x+ 3 2
(f) y = 1 2 x+2
4. (a) −2/(a+3) 2 (b) (i) − 1 2
(ii) − 2 9
(iii) − 1 8
5. (a) 3a 2 −4 (b) y = −x−1, y = 8x−15
6. (a) (i) 32 (ii) 48 (iii) 62.4 (iv) 63.84 (v) 64
(b) (i) 2.236 (ii) 1.472 (iii) 1.351 (iv) 1.343 (v) 1.342
7. (a) −32 (b) 1 8
8. 12a 2 +6, 18 m/s, 54 m/s, 114 m/s
Exercise 2.2
1. If f(x) = 3x 2 −5x, find f ′ (2) and use it to find an equation of the tangent line to the
parabola y = 3x 2 −5x at the point (2,2).
2. If g(x) = x 3 −5x+1, find g ′ (1) and use it to find an equation of the tangent line to
the curve y = x 3 −5x+1 at the point (1,−3).
3. Compute f ′ (a) using Definition3.2.
(a) f(x) = 3x+1, a = 1 (b) f(x) = √ 3x+1, a = 1
(c) f(x) = x 2 +2x, a = 0 (d) f(x) = x 3 +4, a = −1
(e) f(x) = x
2x−1 , a = 1 (f) f(x) = 2
√ , a = −1 3−x
4. Compute the derivative function f ′ (x) using Definition3.3.
(a) f(x) = 3x 2 +1 (b) f(x) = 3
x+1
(c) f(x) = √ 3x+1
(d) f(x) = x 3 +2x−1

MA111: Prepared by Dr.Archara Pacheenburawana 11
5. Determine whether or not f ′ (0) exists.
{ 2x+1 x < 0
(a) f(x) =
3x+1 x ≥ 0
{
xsin 1 x ≠ 0
(b) f(x) = x
0 x = 0
Answer to Exercise 2.2
1. 7, y = 7x−12 2. −2, y = −2x−1 3. (a) 3 (b) 3 4
(c) 2 (d) 3 (e) −1 (f) 1 8
4. (a) 6x (b)
−3
(x+1) 2
(c)
3
2 √ 3x+1
5. (a) Does not exist (b) Does not exist
(d) 3x 2 +2
Exercise 2.3
1. Differentiate the function.
(a) f(x) = x 3 −2x+1
(b) f(x) = 3x 2 −4
(c) f(x) = 4
(d) f(x) = 3x 3 −2 √ x
(e) f(x) = 3 −8x+1
x
10
(f) f(x) = √ −2x x
(g) f(x) = 2x 3/2 −3x −1/3 (h) f(x) = 2 3√ x+3
(i) f(x) = x(3x 2 − √ x) (j) f(x) = 3x2 −3x+1
2x
(k) f(x) = x+ 5√ x 2 (l) f(x) = x √ x+ 1
x 2√ x
2. Find an equation of the tangent line to the curve at the given point.
(a) y = x+ 4 x , (2,4) (b) y = x+√ x, (1,2)
3. Find the points on the curve y = x 3 −x 2 −x+1 where the tangent is horizontal.
4. Show that the curve y = 6x 3 +5x−3 has no tangent line with slope 4.
5. Let
Is f differentiable at x = 1
f(x) =
{
2−x if x ≤ 1
x 2 −2x+2 if x > 1
6. For what values of x is the function f(x) = |x 2 −9| differentiable

MA111: Prepared by Dr.Archara Pacheenburawana 12
7. For what values of a and b is the line 2x + y = b tangent to the parabola y = ax 2
when x = 2
8. Find a cubic function y = ax 3 +bx 2 +cx+d whose graph has horizontal tangents at
the points (−2,6) and (2,0).
9. Differentiate.
(a) f(x) = (x 2 +3)(x 3 −3x+1)
(b) f(x) = (3x+4)(x 3 −2x 2 +x)
(c) f(x) = ( √ (
x+3x) 5x 2 − 3 )
x
(d) f(x) = 3x−2
5x+1
(f) f(x) = 3x−6√ x
5x 2 −2
(h) f(x) = x2 +3x−2
√ x
(j) f(x) = (x 2 +1) x3 +3x 2
x 2 +2
(e) f(x) = x−2
x 2 +x+1
(g) f(x) = (x+1)(x−2)
x 2 −5x+1
(i) f(x) = x( 3√ x+3)
(k) f(x) = x
x+ c x
10. Find an equation of the tangent line to the curve at the given point.
(a) y = 2x
1
, (1,1) (b) y =
x+1 1+x , (−1, 1)
2 2
11. Write put the product rule for the function f(x)g(x)h(x).
12. Find the derivative of each function using the general product rule developed in exercise
12.
(a) f(x) = x 2/3 (x 2 −2)(x 3 −x+1)
(b) f(x) = (x+1)(x 3 +4x)(x 5 −3x 2 +1)
13. If f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) =
f(x)/g(x).
(a) Find u ′ (1). (b) Find v ′ (1).
y
1
f
g
1
x

MA111: Prepared by Dr.Archara Pacheenburawana 13
14. Suppose that f(5) = 1, f ′ (5) = 6, g(5) = −3, and g ′ (5) = 2. Find the values of (a)
(fg) ′ (5), (b) (f/g) ′ (5), and (c) (g/f) ′ (5).
Answer to Exercise 2.3
1. (a) 3x 2 −2 (b) 6x (c) 0 (d) 9x 2 − 1 √ x
(e) − 3
x 2 −8 (f) −5x −3/2 −2
(g) 3x 1/2 +x −4/3 (h) 2 3 x−2/3 (i) 9x 2 − 3 2 x1/2 (j) 3 2 − 1 2 x−2 (k) 1+2/(5 5√ x 3 )
(l) 3 2√ x−5/(2x
3 √ x)
2. (a) y = 4 (b) y = 3x+ 1 3. (1,0),(− 1, 32 ) 5. No
2 2 3 27
6. Not differentiable at x = −3 or 3 7. a = − 1 3
,b = 2 8. y =
2 16 x3 − 9 x+3 4
9. (a) 2x(x 3 −3x+1)+(x 2 +3)(3x 2 −3) (b) 3(x 3 −2x 2 +x)+(3x+4)(3x 2 −4x+1)
( )( 1
(c)
2 x−1/2 +3 5x 2 − 3 )
+( √ x+3x)(10x+3x −2 13
) (d)
x
(5x+1) 2
(e) −x2 +4x+3
(f) (3−3x−1/2 )(5x 2 −2)−(3x−6 √ x)(10x)
(x 2 +x+1) 2 (5x−2) 2
(h) 3 2 x1/2 + 3 2 x−1/2 +x −3/2 (i) 4 3 x1/3 +3
(j) 2x x3 +3x 2
x 2 +2 +(x2 −1) (3x2 +6x)(x 2 +2)−(x 3 +3x 2 )(2x)
(x 2 +2) 2 (k)
(g) −4x2 +6x−11
(x 2 −5x+1) 2
2cx
(x 2 +c) 2
10. (a) y = 1 2 x+ 1 2
(b) y = 1 2 x+1 11. f′ (x)g(x)h(x)+f(x)g ′ (x)h(x)+f(x)g(x)h ′ (x)
12. (a) 2 3 x−1/3 (x 2 −2)(x 3 −x+1)+2x 4/3 (x 3 −x+1)+x 2/3 (x 2 −2)(3x 2 −1)
(b) (x 3 +4x)(x 5 −3x 2 +1)+(x+1)(3x 2 +4)(x 5 −3x 2 +1)+(x+1)(x 3 +4x)(5x 4 −6x)
13. (a) 0 (b) − 2 3
14. (a) −6 (b) − 20 9
(c) 20
1. Find the derivative of each function.
(a) f(x) = 4sinx−x
Exercise 2.4
(c) f(x) = xcosx
(e) f(x) = sinx
x
(g) f(x) = cosx−1 (h) f(x) =
x 2
(i) f(x) = cscxcotx
(k) f(x) = 2sinxcosx
(m) f(x) = 4sin 2 x+4cos 2 x
(b) f(x) = tanx−cscx
(d) f(x) = 4 √ x−2sinx
(f) f(x) = tanx
x
x
sinx+cosx
(j) f(x) = sinxsecx
(l) f(x) = 4x 2 tanx

MA111: Prepared by Dr.Archara Pacheenburawana 14
2. Prove, using the definition of derivative, that if f(x) = cosx, then f ′ (x) = −sinx.
3. Find an equation of the tangent line to the given curve at the specified point.
(a) y = sinx, ( π 2 ,1) (b) y = cosx, (π 2 ,0)
(c) y = tanx, ( π ,1) (d) y = x+cosx, (0,1)
4
(e) y = xcosx, (π,−π) (f) y = 3tanx−2cscx at x = π 3
4. For what values of x does the graph of f(x) = x+2sinx have a horizontal tangent.
5. Using the identities sin2x = 2sinxcosx and cos2x = cos 2 x − sin 2 x, prove that if
f(x) = sin2x, then f ′ (x) = 2cos2x.
6. Using the identities sin2x = 2sinxcosx and cos2x = cos 2 x − sin 2 x, prove that if
f(x) = cos2x, then f ′ (x) = −2sin2x.
Answer to Exercise 2.4
1. (a) 4cosx−1 (b) sec 2 x+cscxcotx (c) cosx−xsinx (d) 2x −1/2 −2cosx
(e) xcosx−sinx
x 2
(f) xsec2 x−tanx
x
(g) −x2 sinx−(cosx−1)2x
x 4
(h) sinx+cosx+xsinx−xcosx (i) −cscxcot 2 x−csc 3 x (j) sec 2 x
1+sin2x
(k) 2cos 2 x−2sin 2 x (l) 8xtanx+4x 2 sec 2 x (m) 0
3. (a) y = 1 (b) y = −x+ π (c) y = 2x+1− π (d) y = x+1 (e) y = −x
2 2
( √
(f) y = 40 3 x−
π
3)
+3 3− √3 4
4. (2n+1)π ±π/3, n an integer
1. Find the derivative of each function.
Exercise 2.5
(a) f(x) = (x 3 +4x) 7 (b) f(x) = (x 3 +x−1) 3
(c) f(x) = √ x 2 −7x (d) f(x) = √ x 2 +4
(
(e) f(x) = x−
x) 1 3/2
(f) f(x) = sin(2x 2 +3)
(g) f(x) = cos(a 3 +x 3 )
(h) f(x) = tan 2 x
(i) f(x) = (3x−2) 10 (5x 2 −x+1) 12 (i) f(x) = (2x−5) 4 (8x 2 −5)
( ) −3
3 x−6
1
(k) f(x) =
(l) f(x) =
x+7
5√ 2x−1

MA111: Prepared by Dr.Archara Pacheenburawana 15
(m) f(x) = tan(cosx)
(o) f(x) = sec 3 4x
(
(q) f(x) = sin tan √ )
sinx
(n) f(x) = x 2 sin4x
(p) f(x) = √ x+ √ x
2. Find an equation of the tangent line to y = f(x) at x = a.
(a) y = √ 8
x 2 +16, a = 3 (b) y = √ , a = 4 4+3x
(c) y = sin(sinx), a = π (d) y =
2
1+e −x , a = 0
3. Findall pointsonthegraphofthefunctionf(x) = 2sinx+sin 2 xatwhich thetangent
line is horizontal.
4. Suppose that F(x) = f ( g(x) ) and g(3) = 6, g ′ (3) = 4, f ′ (3) = 2, and f ′ (6) = 7. Find
F ′ (3).
5. If f and g are the functions whose graphs are shown, let u(x) = f ( g(x) ) , v(x) =
g ( f(x) ) , and w(x) = g ( g(x) ) . Find each derivative, if it exists. If it dose not exist,
explain why.
(a) u ′ (1) (b) v ′ (1) (c) w ′ (1)
y
f
1
g
1
x
6. A table of values for f, g, f ′ , and g ′ is given.
(a) If h(x) = f ( g(x) ) , find h ′ (1).
(b) If H(x) = g ( f(x) ) , find H ′ (1).
x f(x) g(x) f ′ (x) g ′ (x)
1 3 2 4 6
2 1 8 5 7
3 7 2 7 9

MA111: Prepared by Dr.Archara Pacheenburawana 16
Answer to Exercise 2.5
1. (a) 7(x 3 +4x) 6 (3x 2 +4) (b) 3(3x 2 +1)(x 3 +x−1) 2 (c)
2x−7
2 √ x 2 −7x
(d) x(x 2 +4) −1/2 (e) 3 2 (x−1/x)1/2 (1+1/x 2 ) (f) 4xcos(2x 2 +3)
(g) −3x 2 sin(a 3 +x 3 ) (h) 2tanxsec 2 x (i) 6(3x−2) 9 (5x 2 −x+1) 11 (85x 2 −51x+9)
(j) 8(2x−5) 3 (8x 2 −5) −4 (−4x 2 +30x−5) (k) 39(x−6)2
(x+7) 4
(l) − 2 5 (2x−1)−6/5
(m) −sinxsec 2 (cosx) (n) 2xsin4x+4x 2 cos4x (o) 12sec 3 4xtan4x
(p) [1+1/(2 √ x)]/(2 √ x+ √ x)
(q) cos ( tan √ sinx ) (sec 2√ sinx) [ 1/(2 √ sinx) ] (cosx)
2. (a) y = 3 16
x+ (b) y = − 3 11
x+ (c) y = −x+π (d) y = 1x+1
5 5 16 4 2
3. ( π
2 +2nπ,3) , ( 3π
2 +2nπ,−1) ,n an integer 4. 28
5. (a) 3 4
(b) Does not exist (c) −2 6. (a) 30 (b) 36
1. Find the derivative y ′ (x) implicitly.
Exercise 2.6
(a) x 2 +y 2 = 1 (b) x 3 +x 2 y +4y 2 = 6
(c) x 2 y +xy 2 = 3x (d) x 2 y 2 +3y = 4x
(e) √ xy −4y 2 = 12 (f) x+3 = 4x+y 2
y
(g) √ x+y −4x 2 y
= y (h)
x−y = x2 +1
(i) √ xy = 1+xy (j) 4cosxsiny = 1
(k) xy = cot(xy)
2. Find an equation of the tangent line to the curve at the given point.
(a) x 2 −4y 2 = 0, (2,1) (b) x 2 −4y 3 = 0, (2,1)
(c) x 2 y 2 = 4y, (2,1) (d) x 3 y 3 = 9y, (1,3)
(e) x2
16 − y2
9 = 1, (−5, 9 4 ) (f) y2 = x 3 (2−x), (1,1)
(g) 2(x 2 +y 2 ) 2 = 25(x 2 −y 2 ), (3,1) (h) y 2 = x 3 +3x 2 , (1,2)
(i) y(y 2 −1)(y −2) = x(x−1)(x−2), (0,1)

MA111: Prepared by Dr.Archara Pacheenburawana 17
3. Find an equation of the tangent line to the hyperbola
at the point (x 0 ,y 0 ).
x 2
a 2 − y2
b 2 = 1
4. Find all points on the curve x 2 y 2 +xy = 2 where the slope of the tangent line is −1.
5. If x [ f(x) ] 3
+xf(x) = 6 and f(3) = 1, then find f ′ (3).
1. (a) −x
y
(f)
y −4y 2
x+3+2y 3
(j) tanxtany
(b) −x(3x+2y)
x 2 +8y
Answer to Exercise 2.6
(c)
(g) 16x√ x+y −1
1−2 √ x+y
(k) −y
x
3−2xy −y2
x 2 +2xy
(d) 4−2xy2
3+2x 2 y
(h) 2x(x−y)2 +y
x
(e)
y
16y √ xy −x
(i) 2y√ xy −y
x−2x √ xy
2. (a) y = 1 2 x (b) y = 1 3 x+ 1 3
(c) y = −x+3 (d) y = − 9 2 x+ 15 2
(e) y = − 5 4 x−4
3.
(f) y = x (g) y = − 9 40
x+ (h) y = 9 x− 5 (i) y = −x+1
13 13 2 2
x 0 x
a − y 0y
= 1 4. (−1,−1),(1,1) 5. − 1 2 b 2 6
Exercise 2.7
Find the derivative of each function.
1. y = sin −1 (x 2 ) 2. y = tan −1 (e x )
3. y = cos −1 (x 3 ) 4. y = sec −1 (x 2 )
5. H(x) = (1+x 2 )tan −1 x 6. g(t) = sin −1 (4/t)
7. y = x 2 cot −1 (3x) 8. f(x) = e x −x 2 tan −1 x
9. y = tan −1 (cos2x) 10. y = xcos −1 (2x)
11. y = cos −1 (sinx) 12. y = tan −1 (secx)
Answer to Exercise 2.7
1.
2x
√
1−x
4
2.
e x
1+e 2x 3.
−3x 2
√
1−x
6
4.
2
x √ x 4 −1
7. 2xcot −1 (3x)− 3x2
1+9x 2 8. e x − x2
1+x 2 −2xtan−1 x 9.
10. cos −1 2x−
2x
√
1−4x
2
11.
−cosx
√
1−sin 2 x
= ±1 12.
secxtanx
1+sec 2 x
5. 1+2xtan −1 x 6.
−2sin2x
1+cos 2 2x
−4
√
t4 −16t 2

MA111: Prepared by Dr.Archara Pacheenburawana 18
Exercise 2.8
1. Find the derivative of each function.
(a) f(x) = 4e x −x
(b) f(x) = xe x
(c) f(x) = x−2 x (d) f(x) = 2e x+1
(e) f(x) = (1/3) x
(f) f(x) = 4 −x+1
(g) f(x) = e 2x
(i) f(x) = ex
x
2. Differentiate the function.
(h) f(x) = x 2 e −x
(a) f(x) = ln(2x) (b) f(x) = ln(x 3 )
(c) f(θ) = ln(cosθ) (d) f(x) = log 3 (x 2 −4)
(e) f(x) = ln √ x
( ) a−x
(g) f(x) = ln
a+x
(i) f(x) = lnx
1+x
(f) f(x) = √ xlnx
(h) f(x) = e x lnx
(j) f(x) = |x 3 −x 2 |
(k) f(x) = ln(e −x +xe −x ) (l) f(x) = x 2 ln(1−x 2 )
3. Find an equation of the tangent line to the curve at the given point.
(a) y = x 2 lnx, (1,0)
(b) y = ln(lnx), (e,0)
4. Use logarithmic differentiation to find the derivative of the function.
(a) y = (2x+1) 5 (x 4 −3) 6 (b) y = sin2 xtan 4 x
(x 2 +1) 2
(c) y = x x
(d) y = x sinx
(e) y = (lnx) x
(f) y = x ex
Answer to Exercise 2.8
1. (a) 4e x −1 (b) e x +xe x (c) 1+(ln2)2 x (d) 2e x+1 (e) ( ln 3)( 1 1
3
(f) −(ln4)4 −x+1 (g) 2e 2x (h) 2xe −x −x 2 e −x (i) xex −e x
2. (a) f ′ (x) = 1 x (b) f′ (x) = 3 x (c) f′ (θ) = −tanθ (d) f ′ (x) =
(e) f ′ (x) = 1
2x (f) f′ (x) = 2+lnx
2 √ x
x 2
(g) f ′ (x) = −2a
a 2 −x 2
) x
2x
(x 2 −4)ln3

MA111: Prepared by Dr.Archara Pacheenburawana 19
(
(h) f ′ (x) = e x lnx+ 1 )
x
(i) f ′ (x) = 1+x−xlnx
x(1+x) 2
(k) f ′ (x) = −x
1+x (l) f′ (x) = 2xln(1−x 2 )− 2x3
1−x 2
(j) f ′ (x) = 3x−2
x(x−1)
( 10
3. (a) y = x−1 (b) x−ey = e 4. (a) y ′ = (2x+1) 5 (x 4 −3) 6 2x+1 + 24x3
x 4 −3
(
(b) y ′ = sin2 xtan 4 x
2cotx+ 4sec2 x
(x 2 +1) 2 tanx − 4x )
(c) y ′ = x x (lnx+1)
x 2 +1
[
(d) y ′ = x sinx cosxlnx+ sinx
x
(
(f) y ′ = e x x ex lnx+ 1 )
x
]
(e) y ′ = (lnx) x (
lnlnx+ 1
lnx
)
)
Differentiate the function.
1. f(x) = xcoshx+sinh(x 2 )
2. g(x) = 1−coshx
1+coshx
3. h(x) = coth √ 1+x 2 −tanh(e x )
4. y = e cosh3x +tanh −1√ x
5. f(x) = xsinh −1 (x/3)− √ 9+x 2
6. g(x) = coth −1√ x 2 +1
Exercise 2.9
Answer to Exercise 2.9
1. f ′ (x) = xsinhx+coshx+2xcosh(x 2 ) 2. g ′ (x) = −2sinhx
(1+coshx) 2
3. h ′ (x) = −x·csch 2√ 1+x 2
√
1+x
2
5. f ′ (x) = sinh −1 (x/3) 6. g ′ −1
(x) =
x √ x 2 +1
+e x sech 2 (e x ) 4. y ′ = 3e cosh3x sinhx+
1
2 √ x(1−x)

MA111: Prepared by Dr.Archara Pacheenburawana 20
Exercise 2.10
1. Find the first and second derivatives of the function.
(a) f(x) = x 4 +3x 2 −2 (b) f(x) = x 5 +6x 2 −7x
(c) f(x) = x 6 + √ x (d) f(x) = √ 2x+1
(e) f(x) = e 2x
(f) y = cos2θ
(g) h(x) = √ x 2 +1 (h) F(s) = (3s+5) 8
(i) y = x
1−x
(k) H(t) = tan3t
(j) y = (1−x 2 ) 3/4
(l) g(t) = t 3 e 5t
2. If f(x) = (2−3x) −1/2 , find f(0), f ′ (0), f ′′ (0), and f ′′′ (0).
3. If f(θ) = cotθ, find f ′′′ (π/6).
4. Find a formula for f (n) (x).
(a) f(x) = 1 x
(b) f(x) = e 2x (c) f(x) = 1
3x 3
5. Find a second-degree polynomial P such that P(2) = 5, P ′ (2) = 3, and P ′′ (2) = 2.
6. For what values of r does the function y = e rx satisfy the equation y ′′ +5y ′ −6y = 0
1. (a) f ′ (x) = 4x 3 +6x, f ′′ (x) = 12x 2 +6
2.
Answer to Exercise 2.10
(b) f ′ (x) = 5x 4 +12x−7, f ′′ (x) = 20x 3 +12
(c) f ′ (x) = 6x 5 + 1 2 x−1/2 , f ′′ (x) = 30x 4 − 1 4 x−3/2
(d) f ′ (x) = (2x+1) −1/2 , f ′′ (x) = −(2x+1) −3/2
(e) f ′ (x) = 2e 2x , f ′′ (x) = 4e 2x (f) y ′ = −2sin2θ, y ′′ = −4cos2θ
(g) h ′ x
(x) = √
x2 +1 , 1
f′′ (x) =
(x 2 +1) 3/2
(h) F ′ (s) = 24(3s+5) 7 , F ′′ (s) = 504(3s+5) 6 (i) y ′ =
(j) y ′ = − 3 2 x(1−x2 ) −1/4 , y ′′ = 3 4 (1−x2 ) −5/4 (x 2 −2)
(k) H ′ (t) = 3sec 2 3t, H ′′ (t) = 18sec 2 3ttan3t
(l) g ′ (t) = t 2 e 5t (5t+3), g ′′ (t) = te 5t (25t 2 +30t+6)
1
√
2
,
3
4 √ 2 , 27
16 √ 2 , 405
64 √ 2
5. P(x) = x 2 −x+3 6. r = 1,−6
3. −80 4. (a) (−1)n n!
x n+1
1
(1−x) 2, y′′ =
2
(1−x) 3
(b) 2 n e 2x (c) (−1)n (n+2)!
6x n+3

MA111: Prepared by Dr.Archara Pacheenburawana 21
Exercise 2.11
1. Find the linearization L(x) of the function at a.
(a) f(x) = x 3 , a = 1 (b) f(x) = lnx, a = 1
(c) f(x) = e −2x , a = 0
2. Find the differential of the function.
(d) f(x) = 3√ x, a = −8
(a) y = x 2 +x−3 (b) y = x 4 +5x
(c) y = (2x+3) −4 (d) y = xlnx
(e) y = (sinx+cosx) 3 (f) y = x+1
x−1
(g) y = (7x 2 +3x−1) −3/2 (h) y = 5√ (x−2) 2
3. Use a linear approximation to estimate the given number.
(a) √ 402 (b) √ 35.9
(c) √ 36.1 (d) 3√ 1.02+ 4√ 1.02
1
(e)
10.1
(f) sin59 ◦
1
(g) 3√
9
(h) 3√ 8.01− 3√ 1
8.01
Answer to Exercise 2.11
1. (a) L(x) = 3x−2 (b) L(x) = x−1 (c) L(x) = 1−2x (d) L(x) = 1
12 x− 4 3
2. (a) dy = (2x+1)dx (b) dy = (4x 3 +5)dx (c) dy = −8(2x+3) −5 dx
(d) dy = (1+lnx)dx (e) dy = 3(sinx+cosx) 2 (cosx−sinx)dx
(f) dy = −2/(x−1) 2 dx (g) dy = − 3 2 (7x2 +3x−1) −5/2 (14x+3)dx
(h) dy =
2
5 5√ (x−2) 3 dx
3. (a) 20.05 (b) 5.9917 (c) 6.0083 (d) 2.0117 (e) 0.099 (f) 0.857 (g) 0.479
(h) 1.501
Exercise 4.1
1. If A is the area of a circle with radius r and the circle expands as time passed, find
dA/dt in terms of dr/dt.
2. If V is the volume of a cube with edge length x and the cube expands as time passed,
find dV/dt in terms of dx/dt.

MA111: Prepared by Dr.Archara Pacheenburawana 22
3. Aplaneflying horizontallyatthealtitude1mi andaspeed of500mi/hpassed directly
over a radar station. Find the rate at which the distance from the plane to the station
is increasing when it is 2 mi away from the station.
4. A street light is mounted at the top of a 15-ft-tall pole. A men 6 ft tall walks away
from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his
shadow moving when he is 40 ft from the pole
5. A man starts walking north at 4 ft/s from a point P. Five minutes later a woman
starts walking south at 5 ft/s from a point 500 ft due east to P. At what rate are the
people moving apart 15 min after the women starts walking
6. The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the
triangle is increasing at a rate of 2 cm 2 /min. At what rate is the base of the triangle
changing when the altitude is 10 cm and the area is 100 cm 2
7. At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km/h and ship
B is sailing north at 25 km/h. How fast is the distance between the ships changing
at 4:00pm.
8. Two sides of a triangle are 4 m and 5 m in length and the angle between them is
increasing at a rate 0.06 rad/s. Find the rate at which the area of the triangle is
increasing when the angle between the sides of fixed length is π/3.
9. A plane flying with a constant speed of 300 km/h passes over a ground radar station
at an altitude of 1 km and climbs at an angle 30 ◦ . At what rate is the distance from
the plane to the radar station increasing a minute later
10. Water is leaking out of an inverted conical tank at a rate of 10,000 cm 3 /min at the
same time that water is being pumped into the tank at a constant rate. The tank has
height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate
20 cm/min when the height of the water is 2 m, find the rate at which water is being
pumped into the tank.
11. A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate
is the angle between the string and the horizontal decreasing when 200 ft of string
have been let out
12. Boyle’s Lawstatesthatwhenasample ofgasiscompressed ataconstant temperature,
the pressure P and volume V satisfy the equation PV = C, where C is a constant.
Suppose that at a certain instant the volume is 600 cm 3 , the pressure is 150 kPa,
and the pressure is increasing at a rate of 20 kPa/min. At what rate is the volume
decreasing at this instant
13. Suppose that the average yearly cost per item for producing x items of a business
product is C(x) = 10 + 100 . If the current production is x = 10 and production is
x
increasing at a rate of 2 items per year, find the rate of change of the average cost.

MA111: Prepared by Dr.Archara Pacheenburawana 23
1.
dA
dt
= 2πr
dr
dt
2. dV
dt = 3x2dx dt
Answer to Exercise 4.1
3. 250 √ 3 mi/h 4. 25
3 ft/s
5. 837/ √ 8674 ≈ 8.99 ft/s 6. −1.6 cm/min 7. 720
13 ≈ 55.4 km/h 8. 0.3 m2 /s
9. 1650/ √ 31 ≈ 296 km/h 10. ≈ 2.79×10 5 cm 3 /min 11. 2 √ 3 ft/s
12. 80 cm 3 /min 13. −2 dollars per year
Exercise 4.2
Find the following limit.
x+2
1. lim
x→−2 x 2 −4
3x 2 +2
3. lim
x→∞ x 2 −4
e x −1
5. lim
x→0 sinx
sinx−x
7. lim
x→0 x 3
x 3
9. lim
x→∞ e x
11. lim
x→1
sinπx
x−1
13. lim
x→∞
xe −x
15. lim
x→0 +xlnx
(√ )
17. lim x2 −1−x
x→∞
19. lim
(lnx+ 1 )
x→0 + x
21. lim
x→0 +xsinx
2. lim
x→−2
x+1
x 2 +4x+3
x+1
4. lim
x→−∞ x 2 +4x+3
sinx
6. lim
x→0 x 3
√ x−1
8. lim
x→1 x−1
e x −1
10. lim
x→0 x
lnx
12. lim
x→∞ x 2
xsinx
14. lim
x→0 cosx−1
lnx
16. lim
x→0 + cotx
18. lim
x→∞
(1+ 1 x
20. lim
x→0 +(1/x)x
) x
22. lim
x→0
(1−2x) 1/x
Answer to Exercise 4.2
1. − 1 4
2. 1 3. 3 4. 0 5. 1 6. ∞ 7. − 1 6
8. 1 2
9. 0 10. 1 11. −π
12. 0 13. 0 14. −2 15. 0 16. 0 17. 0 18. e 19. ∞ 20. 1 21. 1 22. e −2