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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

230 Chapter 2 • The Derivative and Its Properties

In Problems 19–60, find the derivative of each function. Treat a and b, In Problems 67–70, for each function:
if present, as constants. (a) Find an equation of the tangent line to the graph of the function
at the indicated point.
3
5
19. f (x) = x 20. f (x) = ax
(b) Find an equation of the normal line to the function at the
x 4 indicated point.
21. f (x) = 22. f (x) = −6x 2
4 (c) Graph the function, the tangent line, and the normal line on the
same screen.
2
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2
2
3
23. f (x) = 3x − 4x 24. f (x) = 2x + x − 6x + 8 67. f (x) = 2x − 3x + 7 at (−1, 12)
2
3
2
5(x + 6) x + 1 5
2
25. F(x) = 7(x − 4) 26. F(x) = 68. y = at 2,
7 2x − 1 3
x
2
2
2
3
27. f (x) = 5(x − 3x)(x − 6) 28. f (x) = (2x + x)(x − 5) 69. f (x) = x − e at (0, −1)
70. s(t) = 1 + 2 sin t at (π, 1)
4
6x − 9x 2 2x + 2
29. f (x) = 30. f (x) = 71. Motion on a Line As an object in moves on a line, its signed
3x 3 5x − 3
distance s (in meters) from the origin at time t (in seconds) is
7x −12 given by the position function
31. f (x) = 32. f (x) = 2x
x − 5
2
s = f (t) = t − 6t
3 4
2
33. f (x) = 2x − 5x − 2 34. f (x) = 2 + +
x x 2 (a) Find the average velocity of the object from 0 to 5 s.
a b
3
35. f (x) = − 36. f (x) = (x − 1) 2 (b) Find the velocity at t = 0, at t = 5, and at any time t.
x x 3
3 x 2 (c) Find the acceleration at any time t.
37. f (x) = 2 2 38. f (x) =
(x − 3x) x + 1 72. Motion on a Line As an object moves on a line, its signed
t 3 −2 −1 distance s from the origin at time t is given by the position
2
39. s(t) = 40. f (x) = 3x + 2x + 1 function s(t) = t − t , where s is in centimeters and t is in
t − 2
seconds.
1 v − 1
41. F(z) = 42. f (v) = (a) Find the average velocity of the object from 1 to 3 s.
2
z + 1 v + 1
2
1 x 2 (b) Find the velocity of the object at t = 1 s and t = 3 s.
43. g(z) = 44. f (x) = 3e + x
1 − z + z 2
(c) What is its acceleration at t = 1 and t = 3?
x
2
45. s(t) = 1 − e t 46. f (x) = ae (2x + 7x)
73. Business The price p in dollars per pound when x pounds
x
47. f (x) = (1 + x) ln x 48. f (x) = 2x ln x + e tan x of a commodity are demanded is modeled by the function
2
49. f (x) = x sin x 50. s(t) = cos t
10,000
1 p(x) = − 5
51. G(u) = tan u + sec u 52. g(v) = sin v − cos v 5x + 100
3
x
x
53. f (x) = e sin x 54. f (x) = e csc x when between 0 and 90 lb are demanded (purchased).
x
55. f (x) = 2 sin x cos x 56. f (x) = (e + b) cos x (a) Find the rate of change of price with respect to demand.
sin x 1 − cot x (b) What is the revenue function R? (Recall, revenue R equals
57. f (x) = 58. f (x) =
csc x 1 + cot x price times amount purchased.)
cos θ
′
59. f (θ) = 60. f (θ) = 4θ cot θ tan θ (c) What is the marginal revenue R at x = 10 and at x = 40 lb?
2e θ
x − 1
′
In Problems 61–66, find the first derivative and the second derivative 74. If f (x) = for all x 6= −1, find f (1).
x + 1
of each function.
75. If f (x) = 2 + |x − 3| for all x, determine whether the
61. f (x) = (5x + 3) 2 62. f (x) = xe x derivative f exists at x = 3. Justify your reason.
′
u x
63. g(u) = 64. F(x) = e (sin x + 2 cos x)
2u + 1
cos u sin x
65. f (u) = 66. F(x) =
e u x
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