﻿ ﻿Miscellaneous Multiple-Choice Practice Questions - Calculus AB and Calculus BC

## Calculus AB and Calculus BC

CHAPTER 11 Miscellaneous Multiple-Choice Practice Questions

These questions provide further practice for Parts A and B of Section I of the examination.

1. Which of the following functions is continuous at x = 0?

(A)

(B) f (x) = [x] (greatest-integer function)

(C)

(D)

(E)

2. Which of the following statements about the graph of is not true?

(A) The graph is symmetric to the y-axis.

(B) The graph has two vertical asymptotes.

(C) There is no y-intercept.

(D) The graph has one horizontal asymptote.

(E) There is no x-intercept.

3.

(A) −1

(B) 0

(C) 1

(D) 2

(E) none of these

4. The x-coordinate of the point on the curve y = x2 − 2x + 3 at which the tangent is perpendicular to the line x + 3y + 3 = 0 is

(A)

(B)

(C)

(D)

(E) none of these

5.

(A) −3

(B) −1

(C) 1

(D) 3

(E) nonexistent

6. For polynomial function p, p (2) = −6, p ″(4) = 0, and p ″(5) = 3. Then p must:

(A) have an inflection point at x = 4

(B) have a minimum at x = 4

(C) have a root at x = 4

(D) be increasing on [2,5]

(E) none of these

7.

(A) 6

(B) 8

(C) 10

(D) 11

(E) 12

8.

(A)

(B)

(C) 1

(D) 3

(E) nonexistent

9. The maximum value of the function f (x) = x4 − 4x3 + 6 on [1, 4] is

(A) 1

(B) 0

(C) 3

(D) 6

(E) none of these

10. Let if x ≠ 5, and let f be continuous at x = 5. Then c =

(A)

(B) 0

(C)

(D) 1

(E) 6

11.

(A) −1

(B)

(C) 0

(D)

(E) 1

12. If sin x = ln y and 0 < x < π, then, in terms of x, equals

(A) esin x cos x

(B) e−sin x cos x

(C)

(D) ecos x

(E) esin x

13. If f (x) = x cos x, then equals

(A)

(B) 0

(C) −1

(D)

(E) 1

14. The equation of the tangent to the curve y = ex ln x, where x = 1, is

(A) y = ex

(B) y = ex + 1

(C) y = e(x − 1)

(D) y = ex + 1

(E) y = x − 1

15. If the displacement from the origin of a particle moving along the x-axis is given by s = 3 + (t − 2)4, then the number of times the particle reverses direction is

(A) 0

(B) 1

(C) 2

(D) 3

(E) none of these

16. equals

(A) 1 − e

(B)

(C) e − 1

(D)

(E) e + 1

17. If equals

(A) 7

(B)

(C)

(D) 9

(E)

18. If the position of a particle on a line at time t is given by s = t3 + 3t, then the speed of the particle is decreasing when

(A) − 1 < t < 1

(B) − 1 < t < 0

(C) t < 0

(D) t > 0

(E) |t| > 1

19. A rectangle with one side on the x-axis is inscribed in the triangle formed by the lines y = x, y = 0, and 2x + y = 12. The area of the largest such rectangle is

(A) 6

(B) 3

(C)

(D) 5

(E) 7

CHALLENGE

20. The x-value of the first-quadrant point that is on the curve of x2y2 = 1 and closest to the point (3, 0) is

(A) 1

(B)

(C) 2

(D) 3

(E) none of these

21. If y = ln(4x + 1), then is

(A)

(B)

(C)

(D)

(E)

22. The region bounded by the parabolas y = x2 and y = 6xx2 is rotated about the x-axis so that a vertical line segment cut off by the curves generates a ring. The value of x for which the ring of largest area is obtained is

(A) 4

(B) 3

(C)

(D) 2

(E)

23. equals

(A) ln (ln x) + C

(B)

(C)

(D) ln x + C

(E) none of these

24. The volume obtained by rotating the region bounded by x = y2 and x = 2 − y2 about the y-axis is equal to

(A)

(B)

(C)

(D)

(E)

25. The general solution of the differential equation is a family of

(A) straight lines

(B) circles

(C) hyperbolas

(D) parabolas

(E) ellipses

26. Estimate dx using the Left Rectangular Rule and two subintervals of equal width.

(A)

(B)

(C)

(D)

(E)

27.

(A) −2

(B)

(C) 0

(D)

(E)

28.

(A) 0

(B)

(C)

(D)

(E)

BC ONLY

29.

(A) 0

(B)

(C) 1

(D) 2

(E)

30. The number of values of k for which f (x) = ex and g(x) = k sin x have a common point of tangency is

(A) 0

(B) 1

(C) 2

(D) large but finite

(E) infinite

CHALLENGE

31. The curve 2x2 y + y2 = 2x + 13 passes through (3, 1). Use the line tangent to the curve there to find the approximate value of y at x = 2.8.

(A) 0.5

(B) 0.9

(C) 0.95

(D) 1.1

(E) 1.4

32.

(A)

(B)

(C)

(D)

(E)

33. The region bounded by y = tan x, y = 0, and is rotated about the x-axis. The volume generated equals

(A)

(B)

(C)

(D)

(E) none of these

34. for the constant a > 0, equals

(A) 1

(B) a

(C) ln a

(D) log10 a

(E) a ln a

35. Solutions of the differential equation whose slope field is shown here are most likely to be

(B) cubic

(C) sinusoidal

(D) exponential

(E) logarithmic

36.

(A) 0

(B) 1

(C)

(D)

(E)

37. The graph of g, shown below, consists of the arcs of two quarter-circles and two straight-line segments. The value of is

(A) π + 2

(B)

(C)

(D)

(E)

38. Which of these could be a particular solution of the differential equation whose slope field is shown here?

(A)

(B) y = ln x

(C) y = ex

(D) y = ex

(E) y = ex2

39. What is the domain of the particular solution for containing the point where x = −1?

(A) x < 0

(B) x > −2

(C) − 2 < x < 2

(D) x ≠ ±2

(E) none of these; no solution exists for x = −1

40. The slope field shown here is for the differential equation

(A)

(B) y ′ = ln x

(C) y ′ = ex

(D) y ′ = y

(E) y ′ = −y2

41. If we substitute x = tan θ, which of the following is equivalent to

(A)

(B)

(C)

(D)

(E)

42. If x = 2 sin u and y = cos 2u, then a single equation in x and y is

(A) x2 + y2 = 1

(B) x2 + 4y2 = 4

(C) x2 + 2y = 2

(D) x2 + y2 = 4

(E) x2 − 2y = 2

BC ONLY

43. The area bounded by the lemniscate with polar equation r2 = 2 cos 2θ is equal to

(A) 4

(B) 1

(C)

(D) 2

(E) none of these

44.

(A) 0

(B)

(C) π

(D)

(E) none of these

45. The first four terms of the Maclaurin series (the Taylor series about x = 0) for are

(A) 1 + 2x + 4x2 + 8x3

(B) 1 − 2x + 4x2 − 8x3

(C) − 1 − 2x − 4x2 − 8x3

(D) 1 − x + x2x3

(E) 1 + x + x2 + x3

46.

(A)

(B)

(C)x2 e−x + 2xe−x + C

(D)x2 e−x − 2xex − 2ex + C

(E)x2 ex + 2xex − 2ex + C

47. is equal to

(A)

(B)

(C)

(D)

(E)

BC ONLY

48. A curve is given parametrically by the equations x = t, y = 1 − cos t. The area bounded by the curve and the x-axis on the interval 0 t 2π is equal to

(A) 2(π + 1)

(B) π

(C)

(D) π + 1

(E)

49. If x = a cot θ and y = a sin2 θ, then when is equal to

(A)

(B) −1

(C) 2

(D)

(E)

50. Which of the following improper integrals diverges?

(A)

(B)

(C)

(D)

(E)

51.

(A)

(B)

(C)

(D)

(E)

52.

(A) −∞

(B) 0

(C) 1

(D)

(E) nonexistent

53. A particle moves along the parabola x = 3yy2 so that at all time t. The speed of the particle when it is at position (2, 1) is equal to

(A) 0

(B) 3

(C)

(D)

(E) none of these

54.

(A) −∞

(B) −1

(C) 0

(D) 1

(E)

55. When rewritten as partial fractions, includes which of the following?

I.

II.

III.

(A) none

(B) I only

(C) II only

(D) III only

(E) I and III

56. Using two terms of an appropriate Maclaurin series, estimate

(A)

(B)

(C)

(D)

(E) undefined; the integral is improper

BC ONLY

57. The slope of the spiral r = θ at

(A)

(B) −1

(C) 1

(D)

(E) undefined

Part B. Directions: Some of these questions require the use of a graphing calculator.

58. The graph of function h is shown here. Which of these statements is (are) true?

I. The first derivative is never negative.

II. The second derivative is constant.

III. The first and second derivatives equal 0 at the same point.

(A) I only

(B) III only

(C) I and II

(D) I and III

(E) all three

59. Graphs of functions f (x), g(x), and h(x) are shown below.

Consider the following statements:

I. g(x) = f ′(x)

II. f (x) = g ′(x)

III. h(x) = g (x)

Which of these statements is (are) true?

(A) I only

(B) II only

(C) II and III only

(D) all three

(E) none of these

60.

(A)

(B)

(C)

(D)

(E) 0

61. If

(A) −6

(B) −5

(C) 5

(D) 6

(E) 7

62. At what point in the interval [1, 1.5] is the rate of change of f (x) = sin x equal to its average rate of change on the interval?

(A) 0.995

(B) 1.058

(C) 1.239

(D) 1.253

(E) 1.399

63. Suppose f ′(x) = x2 (x − 1). Then f (x) = x (3x − 2). Over which interval(s) is the graph of f both increasing and concave up?

I. x < 0

II.

III.

IV. x > 1

(A) I only

(B) II only

(C) II and IV

(D) I and III

(E) IV only

64. Which of the following statements is true about the graph of f (x) in Question 62?

(A) The graph has no relative extrema.

(B) The graph has one relative extremum and one inflection point.

(C) The graph has two relative extrema and one inflection point.

(D) The graph has two relative extrema and two inflection points.

(E) None of the preceding statements is true.

65. The nth derivative of ln (x + 1) at x = 2 equals

(A)

(B)

(C)

(D)

(E)

66. If f (x) is continuous at the point where x = a, which of the following statements may be false?

(A)

(B)

(C) f ′(a) exists.

(D) f (a) is defined.

(E)

67. Suppose where k is a constant. Then equals

(A) 3

(B) 4 − k

(C) 4

(D) 4 + k

(E) none of these

68. The volume, in cubic feet, of an “inner tube” with inner diameter 4 ft and outer diameter 8 ft is

(A)2

(B) 12π2

(C)2

(D) 24π2

(E)2

CHALLENGE

69. If f (u) = tan−1 u2 and g(u) = eu, then the derivative of f (g (u)) is

(A)

(B)

(C)

(D)

(E)

70. If sin (xy) = y, then equals

(A) sec (xy)

(B) y cos (xy) − 1

(C)

(D)

(E) cos (xy)

71. Let x > 0. Suppose

(A) f (x4)

(B) f (x2)

(C) 2xg(x2)

(D)

(E) 2g(x2) + 4x2 f (x)

72. The region bounded by y = ex, y = 1, and x = 2 is rotated about the x-axis. The volume of the solid generated is given by the integral

(A)

(B)

(C)

(D)

(E)

73. Suppose the function f is continuous on 1 x 2, that f ′(x) exists on 1 < x < 2, that f (1) = 3, and that f (2) = 0. Which of the following statements is not necessarily true?

(A) The Mean-Value Theorem applies to f on 1 x 2.

(B) exists.

(C) There exists a number c in the closed interval [1,2] such that f ′(c) = 0.

(D) If k is any number between 0 and 3, there is a number c between 1 and 2 such that f (c) = k.

(E) If c is any number such that 1 < c < 2, then exists.

74. The region S in the figure is bounded by y = sec x, the y-axis, and y = 4. What is the volume of the solid formed when S is rotated about the y-axis?

(A) 0.791

(B) 2.279

(C) 5.692

(D) 11.385

(E) 17.217

75. If 40 g of a radioactive substance decomposes to 20 g in 2 yr, then, to the nearest gram, the amount left after 3 yr is

(A) 10

(B) 12

(C) 14

(D) 16

(E) 17

76. An object in motion along a line has acceleration and is at rest when t = 1. Its average velocity from t = 0 to t = 2 is

(A) 0.362

(B) 0.274

(C) 3.504

(D) 7.008

(E) 8.497

77. Find the area bounded by y = tan x and x + y = 2, and above the x-axis on the interval [0, 2],

(A) 0.919

(B) 0.923

(C) 1.013

(D) 1.077

(E) 1.494

78. An ellipse has major axis 20 and minor axis 10. Rounded off to the nearest integer, the maximum area of an inscribed rectangle is

(A) 50

(B) 79

(C) 80

(D) 82

(E) 100

79. The average value of y = x ln x on the interval 1 x e is

(A) 0.772

(B) 1.221

(C) 1.359

(D) 1.790

(E) 2.097

80. Let for 0 x 2π. On which interval is f increasing?

(A) 0 < x < π

(B) 0.654 < x < 5.629

(C) 0.654 < x < 2π

(D) π < x < 2π

(E) none of these

81. The table shows the speed of an object (in ft/sec) during a 3-sec period. Estimate its acceleration (in ft/sec2) at t = 1.5 sec.

 time, sec 0 1 2 3 speed, ft/sec 30 22 12 0

(A) −17

(B) −13

(C) −10

(D) −5

(E) 17

82. A maple-syrup storage tank 16 ft high hangs on a wall. The back is in the shape of the parabola y = x2 and all cross sections parallel to the floor are squares. If syrup is pouring in at the rate of 12 ft3 /hr, how fast (in ft/hr) is the syrup level rising when it is 9 ft deep?

(A)

(B)

(C)

(D) 36

(E) 162

83. In a protected area (no predators, no hunters), the deer population increases at a rate of where P(t) represents the population of deer at t yr. If 300 deer were originally placed in the area and a census showed the population had grown to 500 in 5 yr, how many deer will there be after 10 yr?

(A) 608

(B) 643

(C) 700

(D) 833

(E) 892

84. Shown is the graph of

Let The local linearization of H at x = 1 is H(x) equals

(A) 2x

(B) −2x − 4

(C) 2x + π − 2

(D) −2x + π + 2

(E) 2x + ln 16 + 2

85. A smokestack 100 ft tall is used to treat industrial emissions. The diameters, measured at 25-ft intervals, are shown in the table. Using the midpoint rule, estimate the volume of the smokestack to the nearest 100 ft3.

(A) 8100

(B) 9500

(C) 9800

(D) 12,500

(E) 39,300

For Questions 86–90 the table shows the values of differentiable functions f and g.

86. If then P ′(3) =

(A) −2

(B)

(C)

(D)

(E) 2

87. If H(x) = f (g (x)), then H (3) =

(A) 1

(B) 2

(C) 3

(D) 6

(E) 9

88. If M(x) = f (x) · g (x), then M (3) =

(A) 2

(B) 6

(C) 8

(D) 14

(E) 16

89. If K(x) = g−1 (x), then K (3) =

(A)

(B)

(C)

(D)

(E) 2

90. If R (x) = then R (3) =

(A)

(B)

(C)

(D)

(E) 2

91. Water is poured into a spherical tank at a constant rate. If W(t) is the rate of increase of the depth of the water, then W is

(A) constant

(B) linear and increasing

(C) linear and decreasing

(D) concave up

(E) concave down

92. The graph of f ′ is shown below. If f (7) = 3 then f (1) =

(A) −10

(B) −4

(C) −3

(D) 10

(E) 16

93. At an outdoor concert, the crowd stands in front of the stage filling a semicircular disk of radius 100 yd. The approximate density of the crowd x yd from the stage is given by

people per square yard. About how many people are at the concert?

(A) 200

(B) 19,500

(C) 21,000

(D) 165,000

(E) 591,000

94. The Centers for Disease Control announced that, although more AIDS cases were reported this year, the rate of increase is slowing down. If we graph the number of AIDS cases as a function of time, the curve is currently

(A) increasing and linear

(B) increasing and concave down

(C) increasing and concave up

(D) decreasing and concave down

(E) decreasing and concave up

The graph below is for Questions 95–97. It shows the velocity, in feet per second, for 0 < t < 8, of an object moving along a straight line.

95. The object’s average speed (in ft/sec) for this 8-sec interval was

(A) 0

(B)

(C) 1

(D)

(E) 8

96. When did the object return to the position it occupied at t = 2?

(A) t = 4

(B) t = 5

(C) t = 6

(D) t = 8

(E) never

97. The object’s average acceleration (in ft/sec2) for this 8-sec interval was

(A) −2

(B)

(C) 0

(D)

(E) 1

98. If a block of ice melts at the rate of cm3 /min, how much ice melts during the first 3 min?

(A) 8 cm3

(B) 16 cm3

(C) 21 cm3

(D) 40 cm3

(E) 79 cm3

99. A particle moves counterclockwise on the circle x2 + y2 = 25 with a constant speed of 2 ft/sec. Its velocity vector, v, when the particle is at (3, 4), equals

(A)

(B)

(C)

(D)

(E)

BC ONLY

100. Let R = a cos kti + a sin ktj be the (position) vector xi + yj from the origin to a moving point P(x, y) at time t, where a and k are positive constants. The acceleration vector, a, equals

(A) −k2 R

(B) a2 k2 R

(C)aR

(D)

(E)R

101. The length of the curve y = 2x between (0, 1) and (2, 4) is

(A) 3.141

(B) 3.664

(C) 4.823

(D) 5.000

(E) 7.199

102. The position of a moving object is given by P(t) = (3t, et). Its acceleration is

(A) undefined

(B) constant in both magnitude and direction

(C) constant in magnitude only

(D) constant in direction only

(E) constant in neither magnitude nor direction

BC ONLY

103. Suppose we plot a particular solution of from initial point (0, 1) using Euler’s method. After one step of size Δx = 0.1, how big is the error?

(A) 0.09

(B) 1.09

(C) 1.49

(D) 1.90

(E) 2.65

104. We use the first three terms to estimate Which of the following statements is (are) true?

I. The estimate is 0.7.

II. The estimate is too low.

III. The estimate is off by less than 0.1.

(A) I only

(B) III only

(C) I and II

(D) I and III

(E) all three

105. Which of these diverges?

(A)

(B)

(C)

(D)

(E)

106. Find the radius of convergence of

(A) 0

(B)

(C) 1

(D) e

(E)

107. When we use to estimate the Lagrange remainder is no greater than

(A) 0.021

(B) 0.034

(C) 0.042

(D) 0.067

(E) 0.742

108. An object in motion along a curve has position P(t) = (tan t, cos 2t) for 0 t 1. How far does it travel?

(A) 0.96

(B) 1.73

(C) 2.10

(D) 2.14

(E) 3.98

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