5128_Ch03_pp098-184
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124 Chapter 3 Derivatives<br />
8. If f x 7 for all real numbers x, find<br />
(a) f 10. 7 (b) f 0. 7<br />
(c) f x h. 7 (d) lim f x f a<br />
. 0<br />
x→a x a<br />
9. Find the derivatives of these functions with respect to x.<br />
(a) f x p 0 (b) f x p 2 0 (c) f x p 15 0<br />
10. Find the derivatives of these functions with respect to x using<br />
the definition of the derivative.<br />
(a) f x p<br />
x <br />
f(x) <br />
1 <br />
(b) f x <br />
p<br />
x <br />
f(x) x2<br />
Section 3.3 Exercises<br />
In Exercises 1–6, find dydx.<br />
1. y x 2 3 dy/dx 2x 2. y x x dy/dx x<br />
3<br />
2 1<br />
3. y 2x 1 dy/dx 2 4. y x 2 x 1 dy/dx 2x 1<br />
3<br />
2<br />
5. y x x x 6. y 1 x x<br />
3 2<br />
2 x 3<br />
dy/dx x 2 x 1 dy/dx 1 2x 3x 2<br />
In Exercises 7–12, find the horizontal tangents of the curve.<br />
7. y x 3 2x 2 x 1 8. y x 3 4x 2 x 2<br />
At x 1/3, 1<br />
9. y x 4 4x 2 1 10. y 4x 3 6x 2 1 At x 0, 1<br />
11. y 5x 3 3x 5 At x 0, 2<br />
At x 1, 0, 1 12. y x 4 7x 3 2x 2 15<br />
13. Let y x 1x 2 1. Find dydx (a) by applying the<br />
Product Rule, and (b) by multiplying the factors first and then<br />
differentiating. (a) 3x 2 2x 1 (b) 3x 2 2x 1<br />
14. Let y x 2 3x. Find dydx (a) by using the Quotient<br />
Rule, and (b) by first dividing the terms in the numerator by<br />
the denominator and then differentiating.<br />
In Exercises 15–22, find dydx. Support your answer graphically.<br />
15. (x 3 x 1)(x 4 x 2 1) 16. (x 2 1)(x 3 1) 5x 4 3x 2 2x<br />
17. y 2 x 5 19<br />
18. y x2 5x 1 5 2<br />
<br />
3x<br />
2 (3x 2) 2<br />
x2<br />
x 2 x 3<br />
x 1x 2 x 1 3<br />
19. y <br />
x 4 20. y 1 x1 x 2 1<br />
x<br />
3<br />
x2 2x<br />
(1 <br />
1<br />
x 2 ) 2<br />
x<br />
21. y 2 x4<br />
2x<br />
1 x 3 22. y x 1x<br />
2<br />
12 6x <br />
2<br />
( 1 x3) 2<br />
x<br />
1x<br />
2<br />
(x<br />
2<br />
3x 2) 2<br />
23. Suppose u and v are functions of x that are differentiable<br />
at x 0, and that u0 5, u0 3, v0 1, v0 2.<br />
Find the values of the following derivatives at x 0.<br />
d<br />
d<br />
(a) uv 13 (b) d x<br />
d<br />
( x<br />
u v ) 7<br />
(c) d<br />
d<br />
x<br />
( v 7<br />
<br />
u )<br />
<br />
2 5<br />
3<br />
d<br />
(d) 7v 2u 20<br />
d x<br />
24. Suppose u and v are functions of x that are differentiable<br />
at x 2 and that u2 3, u2 4, v2 1, and<br />
v2 2. Find the values of the following derivatives<br />
at x 2.<br />
d<br />
d<br />
(a) uv 2 (b) d x<br />
d<br />
( x<br />
u v ) 10<br />
(c) d<br />
d<br />
x<br />
( v <br />
u )<br />
1 0<br />
<br />
9<br />
14. (a) x(2x) (<br />
x<br />
x 2 3)<br />
2<br />
x2 3<br />
3<br />
x (b) 1 <br />
2<br />
x 2<br />
d<br />
(d) 3u 2v 2uv<br />
d x<br />
12<br />
1 2<br />
36. y x 2, y x 3, y 6 , y<br />
x 4 (iv) 2 4<br />
x5<br />
25. Which of the following numbers is the slope of the line tangent<br />
to the curve y x 2 5x at x 3? (iii)<br />
i. 24 ii. 52 iii. 11 iv. 8<br />
26. Which of the following numbers is the slope of the line<br />
3x 2y 12 0? (iii)<br />
i. 6 ii. 3 iii. 32 iv. 23<br />
In Exercises 27 and 28, find an equation for the line tangent to the<br />
curve at the given point.<br />
27. y x3 y <br />
1<br />
1<br />
2<br />
, x<br />
x 1 2 x 1 2 y 2x 5<br />
28. y <br />
x4 2<br />
x , x 1<br />
2<br />
In Exercises 29–32, find dydx.<br />
29. y 4x 2 8x 1 8x 3 – 8<br />
30. y x 4<br />
x3 x2 x 1 3<br />
4 3 2<br />
x 5 x 4 – x 3 x 2<br />
31. y x 1 1<br />
1 1 1<br />
32. y 2x <br />
x<br />
1<br />
x x 2x<br />
3/2<br />
x(x 1)<br />
2<br />
In Exercises 33–36, find the first four derivatives of the function.<br />
y4x 3 3x 2 4x 1, y12x 2 6x 4, y 24x 6, y (iv) 24<br />
33. y x 4 x 3 2x 2 x 5 34. y x 2 x 3<br />
y 2x 1, y2, y 0, y (iv) 0<br />
35. y x 1 x 2 36. y x 1<br />
yx 2 2x, <br />
y2x 3 2, y6x 4 , y (iv) 24x 5 x<br />
In Exercises 37–42, support your answer graphically.<br />
37. Find an equation of the line perpendicular to the tangent to the<br />
curve y x 3 3x 1 at the point 2, 3. y 1 9 x 2 9<br />
<br />
9<br />
38. Find the tangents to the curve y x 3 x at the points where<br />
the slope is 4. What is the smallest slope of the curve? At what<br />
value of x does the curve have this slope? See page 126.<br />
39. Find the points on the curve y 2x 3 3x 2 12x 20 where<br />
the tangent is parallel to the x-axis. (1, 27) and (2, 0)<br />
40. Find the x- and y-intercepts of the line that is tangent to the<br />
curve y x 3 at the point 2, 8. x-intercept 4/3,<br />
y-intercept 16<br />
41. Find the tangents to Newton’s serpentine,<br />
4x<br />
At (0, 0): y 4x<br />
y x<br />
2<br />
,<br />
1 At (1, 2): y 2<br />
at the origin and the point 1, 2.<br />
42. Find the tangent to the witch of Agnesi,<br />
8<br />
y ,<br />
4 x 2 y 1 2 x 2<br />
at the point 2, 1.<br />
4 13 4 13<br />
21 377 21 377 <br />
8. At x 0.131, 2.535 12. At x 0, 0.198, 5.052<br />
3 3<br />
8<br />
8<br />
15. 7x 6 10x 4 4x 3 6x 2 2x 1