Calculus AB and Calculus BC
CHAPTER 3 Differentiation
Practice Exercises
Part A. Directions: Answer these questions without using your calculator.
In each of Questions 1–20 a function is given. Choose the alternative that is the derivative, of the function.
1. y = x^{5} tan x
(A) 5x^{4} tan x
(B) x^{5} sec^{2} x
(C) 5x^{4} sec^{2} x
(D) 5x^{4} + sec^{2} x
(E) 5x^{4} tan x + x^{5} sec^{2} x
2.
(A)
(B)
(C)
(D)
(E)
3.
(A)
(B)
(C)
(D)
(E)
4.
(A)
(B) −30(5x + 1)^{−4}
(C)
(D)
(E)
5. y = 3x^{2/3} − 4x^{1/2} − 2
(A) 2x^{1/3} − 2x^{−1/2}
(B) 3x^{−1/3} − 2x^{−1/2}
(C)
(D)
(E) 2x^{−1/3} − 2x^{−1/2}
6.
(A)
(B) x^{−1/2} + x^{−3/2}
(C)
(D)
(E)
7.
(A)
(B) 4y(x + 1)
(C)
(D)
(E) none of these
8.
(A)
(B)
(C)
(D)
(E)
9.
(A)
(B)
(C)
(D) 0
(E)
10. y = tan^{−1}
(A)
(B)
(C)
(D)
(E)
11. y = ln (sec x + tan x)
(A) sec x
(B)
(C)
(D)
(E)
12.
(A) 0
(B) 1
(C)
(D)
(E)
13.
(A)
(B)
(C)
(D)
(E)
14.
(A)
(B)
(C)
(D)
(E) cos (ln x)
15.
(A) −csc 2x cot 2x
(B)
(C) −4 csc 2x cot 2x
(D)
(E) −csc^{2} 2x
16. y = e^{−}^{x} cos 2x
(A) −e^{−}^{x} (cos 2x + 2 sin 2x)
(B) e^{−}^{x} (sin 2x − cos 2x)
(C) 2e^{−}^{x} sin 2x
(D) −e^{−}^{x} (cos 2x + sin 2x)
(E) −e^{−}^{x} sin 2x
17. y = sec^{2} (x)
(A) 2 sec x
(B) 2 sec x tan x
(C) 2 sec^{2} x tan x
(D) sec^{2} x tan^{2} x
(E) tan x
18. y = x ln^{3} x
(A)
(B) 3 ln^{2} x
(C) 3x ln^{2} x + ln^{3} x
(D) 3(ln x + 1)
(E) none of these
19.
(A)
(B)
(C)
(D)
(E)
20. y = sin^{−1}
(A)
(B)
(C)
(D)
(E)
In each of Questions 21–24, y is a differentiable function of x. Choose the alternative that is the derivative
21. x^{3} − y^{3} = 1
(A) x
(B) 3x^{2}
(C)
(D)
(E)
22. x + cos(x + y) = 0
(A) csc(x + y) − 1
(B) csc(x + y)
(C)
(D)
(E)
23. sin x − cos y − 2 = 0
(A) −cot x
(B) −cot y
(C)
(D) −csc y cos x
(E)
24. 3x^{2} − 2xy + 5y^{2} = 1
(A)
(B)
(C) 3x + 5y
(D)
(E) none of these
25. If x = t^{2} + 1 and y = 2t^{3}, then
(A) 3t
(B) 6t^{2}
(C)
(D)
(E)
BC ONLY
26. If f (x) = x^{4} − 4x^{3} + 4x^{2} − 1, then the set of values of x for which the derivative equals zero is
(A) {1,2}
(B) {0,−1,−2}
(C) {−1, + 2}
(D) {0}
(E) {0,1,2}
27. If f (x) = then f ″(4) is equal to
(A) −32
(B) −16
(C) −4
(D) −2
(E)
28. If f (x) = ln x^{3} then f ″(3) is
(A)
(B) −1
(C) −3
(D) 1
(E) none of these
29. If a point moves on the curve x^{2} + y^{2} = 25, then, at (0,5), is
(A) 0
(B)
(C) −5
(D)
(E) nonexistent
30. If x = t^{2} − 1 and y = t^{4} − 2t^{3}, then, when t = 1, is
(A) 1
(B) −1
(C) 0
(D) 3
(E)
BC ONLY
31. If f (x) = 5^{x} and 5^{1.002} 5.016, which is closest to f ′(1)?
(A) 0.016
(B) 1.0
(C) 5.0
(D) 8.0
(E) 32.0
32. If y = e^{x} (x − 1), then y ″(0) equals
(A) −2
(B) −1
(C) 0
(D) 1
(E) none of these
33. If x = e^{θ} cos θ and y = e^{θ} sin θ, then, when is
(A) 1
(B) 0
(C) e^{π/2}
(D) nonexistent
(E) −1
BC ONLY
34. If x = cos t and y = cos 2t, then is
(A) 4 cos t
(B) 4
(C)
(D) −4
(E) −4 cot t
BC ONLY
35.
(A) 0
(B) 1
(C) 6
(D) ∞
(E) nonexistent
36.
(A) 0
(B)
(C) 1
(D) 192
(E) ∞
37.
(A) 0
(B)
(C) 1
(D) e
(E) nonexistent
38.
(A) −1
(B) 0
(C) 1
(D) ∞
(E) none of these
39. which of these statements are true?
I. exists.
II. f is continuous at x = 1.
III. f is differentiable at x = 1.
(A) none
(B) I only
(C) I and II only
(D) I and III only
(E) I, II, and III
40. which of these statements are true?
I. exists.
II. g is continuous at x = 3.
III. g is differentiable at x = 3.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III
41. The function f (x) = x^{2/3} on [−8, 8] does not satisfy the conditions of the Mean Value Theorem because
(A) f (0) is not defined
(B) f (x) is not continuous on [−8, 8]
(C) f ′(−1) does not exist
(D) f (x) is not defined for x < 0
(E) f ′(0) does not exist
42. If f (x) = 2x^{3} − 6x, at what point on the interval 0 ≤ x ≤ if any, is the tangent to the curve parallel to the secant line on that interval?
(A) 1
(B) −1
(C)
(D) 0
(E) nowhere
43. If h is the inverse function of f and if then h ′(3) =
(A) −9
(B)
(C)
(D) 3
(E) 9
44. equals
(A) 0
(B) 1
(C)
(D) ∞
(E) none of these
BC ONLY
45. If sin(xy) = x, then
(A) sec(xy)
(B)
(C)
(D)
(E) sec(xy) − 1
46.
(A) 1
(B) 2
(C)
(D) 0
(E) ∞
47.
(A) 1
(B)
(C)
(D) 0
(E) nonexistent
48.
(A) nonexistent
(B) 1
(C) 2
(D) ∞
(E) none of these
49.
(A)
(B) 0
(C) 1
(D) π
(E) ∞
50.
(A) is 1
(B) is 0
(C) is ∞
(D) oscillates between −1 and 1
(E) is none of these
51. The graph in the xy-plane represented by x = 3 + 2 sin t and y = 2 cos t − 1, for −π ≤ t ≤ π, is
(A) a semicircle
(B) a circle
(C) an ellipse
(D) half of an ellipse
(E) a hyperbola
BC ONLY
52. equals
(A) 0
(B)
(C) 1
(D) 2
(E) none of these
In each of Questions 53–56 a pair of equations that represent a curve parametrically is given. Choose the alternative that is the derivative
53. x = t − sin t and y = 1 − cos t
(A)
(B)
(C)
(D)
(E)
BC ONLY
54. x = cos^{3} θ and y = sin^{3} θ
(A) tan^{3} θ
(B) −cot θ
(C) cot θ
(D) −tan θ
(E) −tan^{2} θ
BC ONLY
55. x = 1 − e^{−}^{t} and y = t + e^{−}^{t}
(A)
(B) e^{−}^{t} − 1
(C) e^{t} + 1
(D) e^{t} − e^{−2}^{t}
(E) e^{t} − 1
56. and y = 1 − ln(1 − t) (t < 1)
(A)
(B) t − 1
(C)
(D)
(E) 1 + ln x
Part B. Directions: Some of the following questions require the use of a graphing calculator.
In Questions 57–64, differentiable functions f and g have the values shown in the table.
x |
f |
f ′ |
g |
g ′ |
0 |
2 |
1 |
5 |
−4 |
1 |
3 |
2 |
3 |
−3 |
2 |
5 |
3 |
1 |
−2 |
3 |
10 |
4 |
0 |
−1 |
57. If A = f + 2g, then A ′(3) =
(A) −2
(B) 2
(C) 7
(D) 8
(E) 10
58. If B = f · g, then B ′(2) =
(A) −20
(B) −7
(C) −6
(D) −1
(E) 13
59. If then D ′(1) =
(A)
(B)
(C)
(D)
(E)
60. If H(x) = then H ′(3) =
(A)
(B)
(C) 2
(D)
(E)
61. If K(x) = then K ′(0) =
(A)
(B)
(C)
(D)
(E)
62. If M(x) = f (g(x)), then M ′(1) =
(A) −12
(B) −6
(C) 4
(D) 6
(E) 12
63. If P(x) = f (x^{3}), then P ′(1) =
(A) 2
(B) 6
(C) 8
(D) 12
(E) 54
64. If S(x) = f ^{−1}(x), then S ′(3) =
(A) −2
(B)
(C)
(D)
(E) 2
65. The graph of g ′ is shown here. Which of the following statements is (are) true of g at x = a ?
I. g is continuous.
II. g is differentiable.
III. g is increasing.
(A) I only
(B) III only
(C) I and III only
(D) II and III only
(E) I, II, and III
66. A function f has the derivative shown. Which of the following statements must be false?
(A) f is continuous at x = a.
(B) f (a) = 0.
(C) f has a vertical asymptote at x = a.
(D) f has a jump discontinuity at x = a.
(E) f has a removable discontinuity at x = a.
67. The function f whose graph is shown has f ′ = 0 at x =
(A) 2 only
(B) 2 and 5
(C) 4 and 7
(D) 2, 4, and 7
(E) 2, 4, 5, and 7
68. A differentiable function f has the values shown. Estimate f ′(1.5).
x |
1.0 |
1.2 |
1.4 |
1.6 |
f (x) |
8 |
10 |
14 |
22 |
(A) 8
(B) 12
(C) 18
(D) 40
(E) 80
69. Water is poured into a conical reservoir at a constant rate. If h(t) is the rate of change of the depth of the water, then h is
(A) constant
(B) linear and increasing
(C) linear and decreasing
(D) nonlinear and increasing
(E) nonlinear and decreasing
Use the figure to answer Questions 70–72. The graph of f consists of two line segments and a semicircle.
70. f ′(x) = 0 for x =
(A) 1 only
(B) 2 only
(C) 4 only
(D) 1 and 4
(E) 2 and 6
71. f ′(x) does not exist for x =
(A) 1 only
(B) 2 only
(C) 1 and 2
(D) 2 and 6
(E) 1, 2, and 6
72. f ′(5) =
(A)
(B)
(C) 1
(D) 2
(E)
73. At how many points on the interval [−5,5] is a tangent to y = x + cos x parallel to the secant line?
(A) none
(B) 1
(C) 2
(D) 3
(E) more than 3
74. From the values of f shown, estimate f ′(2).
x |
1.92 |
1.94 |
1.96 |
1.98 |
2.00 |
f (x) |
6.00 |
5.00 |
4.40 |
4.10 |
4.00 |
(A) −0.10
(B) −0.20
(C) −5
(D) −10
(E) −25
75. Using the values shown in the table for Question 74, estimate (f ^{−1}) ′(4).
(A) −0.2
(B) −0.1
(C) −5
(D) −10
(E) −25
76. The “left half” of the parabola defined by y = x^{2} − 8x + 10 for x ≤ 4 is a one-to-one function; therefore its inverse is also a function. Call that inverse g. Find g ′(3).
(A)
(B)
(C)
(D)
(E)
77. The table below shows some points on a function f that is both continuous and differentiable on the closed interval [2,10].
x |
2 |
4 |
6 |
8 |
10 |
f (x) |
30 |
25 |
20 |
25 |
30 |
Which must be true?
(A) f (x) > 0 for 2 < x < 10
(B) f ′(6) = 0
(C) f ′(8) > 0
(D) The maximum value of f on the interval [2,10] is 30.
(E) For some value of x on the interval [2,10] f ′(x) = 0.
78. If f is differentiable and difference quotients overestimate the slope of f at x = a for all h > 0, which must be true?
(A) f ′(a) > 0
(B) f ′(a) < 0
(C) f ″(a) > 0
(D) f ″(a) < 0
(E) none of these
79. If f (u) = sin u and u = g(x) = x^{2} − 9, then (f ° g) ′(3) equals
(A) 0
(B) 1
(C) 6
(D) 9
(E) none of these
80. If then the set of x’s for which f ′(x) exists is
(A) all reals
(B) all reals except x = 1 and x = −1
(C) all reals except x = −1
(D) all reals except and x = −1
(E) all reals except x = 1
81. If then the derivative of y^{2} with respect to x^{2} is
(A) 1
(B)
(C)
(D)
(E)
BC ONLY
82. If y = x^{2} + x, then the derivative of y with respect to is
(A) (2x + 1)(x − 1)^{2}
(B)
(C) 2x + 1
(D)
(E) none of these
BC ONLY
83. If and g(x) = then the derivative of f (g(x)) is
(A)
(B) −(x + 1)^{−2}
(C)
(D)
(E)
84. If f (a) = f (b) = 0 and f (x) is continuous on [a, b], then
(A) f (x) must be identically zero
(B) f ′(x) may be different from zero for all x on [a, b]
(C) there exists at least one number c, a < c < b, such that f ′(c) = 0
(D) f ′(x) must exist for every x on (a, b)
(E) none of the preceding is true
85. Suppose y = f (x) = 2x^{3} − 3x. If h(x) is the inverse function of f, then h ′(−1) =
(A) −1
(B)
(C)
(D) 1
(E) 3
86. Suppose f (1) = 2, f ′(1) = 3, and f ′(2) = 4. Then (f ^{−1}) ′(2)
(A) equals
(B) equals
(C) equals
(D) equals
(E) cannot be determined
87. If f (x) = x^{3} − 3x^{2} + 8x + 5 and g(x) = f ^{−1}(x), then g ′(5) =
(A) 8
(B)
(C) 1
(D)
(E) 53
88. Suppose It follows necessarily that
(A) g is not defined at x = 0
(B) g is not continuous at x = 0
(C) the limit of g(x) as x approaches 0 equals 1
(D) g ′(0) = 1
(E) g ′(1) = 0
Use this graph of y = f (x) for Questions 89 and 90.
89. f ′(3) is most closely approximated by
(A) 0.3
(B) 0.8
(C) 1.5
(D) 1.8
(E) 2
90. The rate of change of f (x) is least at x
(A) −3
(B) −1.3
(C) 0
(D) 0.7
(E) 2.7
Use the following definition of the symmetric difference quotient for f ′ (x_{0}) for Questions 91–93: For small values of h,
91. For f (x) = 5^{x}, what is the estimate of f ′(2) obtained by using the symmetric difference quotient with h = 0.03?
(A) 25.029
(B) 40.236
(C) 40.252
(D) 41.223
(E) 80.503
92. To how many places is the symmetric difference quotient accurate when it is used to approximate f ′(0) for f (x) = 4^{x} and h = 0.08?
(A) 1
(B) 2
(C) 3
(D) 4
(E) more than 4
93. To how many places is f ′(x_{0}) accurate when it is used to approximate f ′(0) for f (x) = 4^{x} and h = 0.001?
(A) 1
(B) 2
(C) 3
(D) 4
(E) more than 4
94. The value of f ′(0) obtained using the symmetric difference quotient with f (x) = |x| and h = 0.001 is
(A) −1
(B) 0
(C) ±1
(D) 1
(E) indeterminate
95. If and h(x) = sin x, then equals
(A) g(sin x)
(B) cos x · g(x)
(C) g ′(x)
(D) cos x · g (sin x)
(E) sin x · g(sin x)
96. Let f (x) = 3^{x} − x^{3}. The tangent to the curve is parallel to the secant through (0,1) and (3,0) for x =
(A) 0.984 only
(B) 1.244 only
(C) 2.727 only
(D) 0.984 and 2.804 only
(E) 1.244 and 2.727 only
Questions 97–101 are based on the following graph of f (x), sketched on −6 ≤ x ≤ 7. Assume the horizontal and vertical grid lines are equally spaced at unit intervals.
97. On the interval 1 < x < 2, f (x) equals
(A) −x − 2
(B) −x − 3
(C) −x − 4
(D) −x + 2
(E) x − 2
98. Over which of the following intervals does f ′(x) equal zero?
I. (−6,−3)
II. (−3,−1)
III. (2,5)
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) II and III only
99. How many points of discontinuity does f ′(x) have on the interval −6 < x < 7?
(A) none
(B) 2
(C) 3
(D) 4
(E) 5
100. For −6 < x < −3, f ′(x) equals
(A)
(B) −1
(C) 1
(D)
(E) 2
101. Which of the following statements about the graph of f ′(x) is false?
(A) It consists of six horizontal segments.
(B) It has four jump discontinuities.
(C) f ′(x) is discontinuous at each x in the set {−3,−1,1,2,5}.
(D) f ′(x) ranges from −3 to 2.
(E) On the interval −1 < x < 1, f ′(x) = −3.
102. The table gives the values of a function f that is differentiable on the interval [0,1]:
x |
0.10 |
0.20 |
0.30 |
0.40 |
0.50 |
0.60 |
f (x) |
0.171 |
0.288 |
0.357 |
0.384 |
0.375 |
0.336 |
According to this table, the best approximation of f ′(0.10) is
(A) 0.12
(B) 1.08
(C) 1.17
(D) 1.77
(E) 2.88
103. At how many points on the interval [a, b] does the function graphed satisfy the Mean Value Theorem?
(A) none
(B) 1
(C) 2
(D) 3
(E) 4