5128_Ch03_pp098-184

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