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, Image of the function.

1. y = x5 tan x

(A) 5x4 tan x

(B) x5 sec2 x

(C) 5x4 sec2 x

(D) 5x4 + sec2 x

(E) 5x4 tan x + x5 sec2 x

2. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

3. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

4. Image

(A) Image

(B) −30(5x + 1)−4

(C) Image

(D) Image

(E) Image

5. y = 3x2/3 − 4x1/2 − 2

(A) 2x1/3 − 2x−1/2

(B) 3x−1/3 − 2x−1/2

(C) Image

(D) Image

(E) 2x−1/3 − 2x−1/2

6. Image

(A) Image

(B) x−1/2 + x−3/2

(C) Image

(D) Image

(E) Image

7. Image

(A) Image

(B) 4y(x + 1)

(C) Image

(D) Image

(E) none of these

8. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

9. Image

(A) Image

(B) Image

(C) Image

(D) 0

(E) Image

10. y = tan−1 Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

11. y = ln (sec x + tan x)

(A) sec x

(B) Image

(C) Image

(D) Image

(E) Image

12. Image

(A) 0

(B) 1

(C) Image

(D) Image

(E) Image

13. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

14. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) cos (ln x)

15. Image

(A) −csc 2x cot 2x

(B) Image

(C) −4 csc 2x cot 2x

(D) Image

(E) −csc2 2x

16. y = ex cos 2x

(A)ex (cos 2x + 2 sin 2x)

(B) ex (sin 2x − cos 2x)

(C) 2ex sin 2x

(D)ex (cos 2x + sin 2x)

(E)ex sin 2x

17. y = sec2 (x)

(A) 2 sec x

(B) 2 sec x tan x

(C) 2 sec2 x tan x

(D) sec2 x tan2 x

(E) tan x

18. y = x ln3 x

(A) Image

(B) 3 ln2 x

(C) 3x ln2 x + ln3 x

(D) 3(ln x + 1)

(E) none of these

19. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

20. y = sin−1 Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

In each of Questions 21–24, y is a differentiable function of x. Choose the alternative that is the derivative Image

21. x3y3 = 1

(A) x

(B) 3x2

(C) Image

(D) Image

(E) Image

22. x + cos(x + y) = 0

(A) csc(x + y) − 1

(B) csc(x + y)

(C) Image

(D) Image

(E) Image

23. sin x − cos y − 2 = 0

(A) −cot x

(B) −cot y

(C) Image

(D) −csc y cos x

(E) Image

24. 3x2 − 2xy + 5y2 = 1

(A) Image

(B) Image

(C) 3x + 5y

(D) Image

(E) none of these

25. If x = t2 + 1 and y = 2t3, then Image

(A) 3t

(B) 6t2

(C) Image

(D) Image

(E) Image

BC ONLY

26. If f (x) = x4 − 4x3 + 4x2 − 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) = Image then f (4) is equal to

(A) −32

(B) −16

(C) −4

(D) −2

(E) Image

28. If f (x) = ln x3 then f (3) is

(A) Image

(B) −1

(C) −3

(D) 1

(E) none of these

29. If a point moves on the curve x2 + y2 = 25, then, at (0,5), Image is

(A) 0

(B) Image

(C) −5

(D) Image

(E) nonexistent

30. If x = t2 − 1 and y = t4 − 2t3, then, when t = 1, Image is

(A) 1

(B) −1

(C) 0

(D) 3

(E) Image

BC ONLY

31. If f (x) = 5x and 51.002 Image 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 = ex (x − 1), then y (0) equals

(A) −2

(B) −1

(C) 0

(D) 1

(E) none of these

33. If x = eθ cos θ and y = eθ sin θ, then, when Image is

(A) 1

(B) 0

(C) eπ/2

(D) nonexistent

(E) −1

BC ONLY

34. If x = cos t and y = cos 2t, then Image is

(A) 4 cos t

(B) 4

(C) Image

(D) −4

(E) −4 cot t

BC ONLY

35. Image

(A) 0

(B) 1

(C) 6

(D)

(E) nonexistent

36. Image

(A) 0

(B) Image

(C) 1

(D) 192

(E)

37. Image

(A) 0

(B) Image

(C) 1

(D) e

(E) nonexistent

38. Image

(A) −1

(B) 0

(C) 1

(D)

(E) none of these

39. Image which of these statements are true?

I. Image 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. Image which of these statements are true?

I. Image 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) = x2/3 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) = 2x3 − 6x, at what point on the interval 0 ≤ xImage if any, is the tangent to the curve parallel to the secant line on that interval?

(A) 1

(B) −1

(C) Image

(D) 0

(E) nowhere

43. If h is the inverse function of f and if Image then h (3) =

(A) −9

(B) Image

(C) Image

(D) 3

(E) 9

44. Image equals

(A) 0

(B) 1

(C) Image

(D)

(E) none of these

BC ONLY

45. If sin(xy) = x, then Image

(A) sec(xy)

(B) Image

(C) Image

(D) Image

(E) sec(xy) − 1

46. Image

(A) 1

(B) 2

(C) Image

(D) 0

(E)

47. Image

(A) 1

(B) Image

(C) Image

(D) 0

(E) nonexistent

48. Image

(A) nonexistent

(B) 1

(C) 2

(D)

(E) none of these

49. Image

(A) Image

(B) 0

(C) 1

(D) π

(E)

50. Image

(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. Image equals

(A) 0

(B) Image

(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 Image

53. x = t − sin t and y = 1 − cos t

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

BC ONLY

54. x = cos3 θ and y = sin3 θ

(A) tan3 θ

(B) −cot θ

(C) cot θ

(D) −tan θ

(E) −tan2 θ

BC ONLY

55. x = 1 − et and y = t + et

(A) Image

(B) et − 1

(C) et + 1

(D) ete−2t

(E) et − 1

56. Image and y = 1 − ln(1 − t) (t < 1)

(A) Image

(B) t − 1

(C) Image

(D) Image

(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 Image then D (1) =

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

60. If H(x) = Image then H (3) =

(A) Image

(B) Image

(C) 2

(D) Image

(E) Image

61. If K(x) = Image then K (0) =

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

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 (x3), 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) Image

(C) Image

(D) Image

(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.

Image

(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?

Image

(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 =

Image

(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

Image

(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.

Image

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) Image

(B) Image

(C) 1

(D) 2

(E) Image

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 = x2 − 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) Image

(B) Image

(C) Image

(D) Image

(E) Image

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) = x2 − 9, then (f ° g) (3) equals

(A) 0

(B) 1

(C) 6

(D) 9

(E) none of these

80. If Image 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 Image and x = −1

(E) all reals except x = 1

81. If Image then the derivative of y2 with respect to x2 is

(A) 1

(B) Image

(C) Image

(D) Image

(E) Image

BC ONLY

82. If y = x2 + x, then the derivative of y with respect to Image is

(A) (2x + 1)(x − 1)2

(B) Image

(C) 2x + 1

(D) Image

(E) none of these

BC ONLY

83. If Image and g(x) = Image then the derivative of f (g(x)) is

(A) Image

(B) −(x + 1)−2

(C) Image

(D) Image

(E) Image

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) = 2x3 − 3x. If h(x) is the inverse function of f, then h (−1) =

(A) −1

(B) Image

(C) Image

(D) 1

(E) 3

86. Suppose f (1) = 2, f (1) = 3, and f (2) = 4. Then (f −1) (2)

(A) equals Image

(B) equals Image

(C) equals Image

(D) equals Image

(E) cannot be determined

87. If f (x) = x3 − 3x2 + 8x + 5 and g(x) = f −1(x), then g (5) =

(A) 8

(B) Image

(C) 1

(D) Image

(E) 53

88. Suppose Image 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.

Image

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 Image

(A) −3

(B) −1.3

(C) 0

(D) 0.7

(E) 2.7

Use the following definition of the symmetric difference quotient for f (x0) for Questions 91–93: For small values of h,

Image

91. For f (x) = 5x, 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) = 4x and h = 0.08?

(A) 1

(B) 2

(C) 3

(D) 4

(E) more than 4

93. To how many places is f (x0) accurate when it is used to approximate f (0) for f (x) = 4x 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 Image and h(x) = sin x, then Image 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) = 3xx3. 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.

Image

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) Image

(B) −1

(C) 1

(D) Image

(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?

Image

(A) none

(B) 1

(C) 2

(D) 3

(E) 4