Page 63 - 2024-calc4e-SE proofs-4e.indd

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

Section 2.4 • Differentiating the Product and the Quotient of Two Functions; Higher-Order Derivatives 213

27. s(t) = t −3 28. G(u) = u −4 In Problems 69–72:
4 3 (a) Find the slope of the tangent line for each function f at the given
29. f (x) = − 30. f (x) =
e x 4e x point.
10 3 2 3 (b) Find an equation of the tangent line to the graph of each
PAGE
208 31. f (x) = 4 + 2 32. f (x) = 5 − 3
x x x x function f at the given point.
1 5 (c) Find the points, if any, where the graph of the function has a
3 5
33. f (x) = 3x − 34. f (x) = x − horizontal tangent line.
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3x 2 x 5
(d) Graph each function, the tangent line found in (b), and any
1 1 1 1 1 1
35. s(t) = − + 36. s(t) = + + tangent lines found in (c) on the same set of axes.
t t 2 t 3 t t 2 t 3
e x x 2 x 2 1 x
37. f (x) = 2 38. f (x) = 69. f (x) = at −1, − 70. f (x) = at (0, 0)
x e x x − 1 2 x + 1
2
x + 1 xe x
39. f (x) = 40. f (x) = 3 2
2
xe x x − x x 1 x + 1 5
71. f (x) = at 1, 72. f (x) = at 2,
In Problems 41–54, find f and f for each function. x + 1 2 x 2
′′
′
PAGE 2 2
209 41. f (x) = 5x + x − 8 42. f (x) = −4x − 7x In Problems 73–80:
43. f (x) = 3 ln x − 3 44. f (x) = x − 4 ln x
(a) Find the points, if any, at which the graph of each function f has a
PAGE x 4 x horizontal tangent line.
210 45. f (x) = (x + 5)e 46. f (x) = 3x e
(b) Find an equation for each horizontal tangent line.
3
2
47. f (x) = (2x + 1)(x + 5) 48. f (x) = (3x − 5)(x − 2)
′
(c) Solve the inequality f (x) > 0.
1 1
′
49. f (x) = x + 50. f (x) = x − (d) Solve the inequality f (x) < 0.
x x
2
t − 1 u + 1 (e) Graph f and any horizontal lines found in (b) on the same set of
51. f (t) = 52. f (u) = axes.
t u
x
e + x e x (f) Describe the graph of f for the results obtained in (c) and (d).
53. f (x) = 54. f (x) =
x x
2
2
1 2x − 5 73. f (x) = (x + 1)(x − x − 11) 74. f (x) = (3x − 2)(2x + 1)
′ ′′
55. Find y and y for (a) y = and (b) y = .
x x
2
x 2 x + 1
2
dy d y 5 2 − 3x 75. f (x) = 76. f (x) =
56. Find and for (a) y = and (b) y = . x + 1 x
dx dx 2 x 2 x
2 x
77. f (x) = xe x 78. f (x) = x e
Motion on a Line In Problems 57–60, find the velocity v = v(t) and
2
acceleration a = a(t) of an object moving on a line whose signed x − 3 e x
79. f (x) = 80. f (x) =
2
distance s from the origin at time t is modeled by the position e x x + 1
function s = s(t).
In Problems 81 and 82, use the graphs to determine each derivative.
2
2
57. s(t) = 16t + 20t 58. s(t) = 16t + 10t + 1
g(x)
2
2
59. s(t) = 4.9t + 4t + 4 60. s(t) = 4.9t + 5t 81. Let u(x) = f (x) · g(x) and v(x) = f (x) .
y
In Problems 61–68, find the indicated derivative. (4, 5)
4 y f (x)
3
2
61. f (4) (x) if f (x) = x − 6x + 8x − 5 (6, 4)
(1, 2)
3
2
62. f (5) (x) if f (x) = 6x + x − 4 (1, 2) (1, 1)
d 8 1 1 d 6
7
6
5
5
8
63. t − t + t − t 3 64. (t + 5t − 2t + 4) 4 2 2 4 6 x
dt 8 8 7 dt 6 (5, 2)
2
y g(x)
d 7 d 10 (4, 3)
u
u
2
65. (e + u ) 66. (2e )
du 7 du 10
(a) u (0) (b) u (4)
′
′
d 5 d 8
x
x
′
′
67. (−e ) 68. (12x − e ) (c) v (−2) (d) v (6)
dx 5 dx 8
d 1 d 1
(e) at x = −2 (f) at x = 4
dx f (x) dx g(x)
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