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.

Part A. Directions: Answer these questions without using your calculator.

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

(A) Image

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

(C) Image

(D) Image

(E) Image

2. Which of the following statements about the graph of Image 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. Image

(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) Image

(B) Image

(C) Image

(D) Image

(E) none of these

5. Image

(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. Image

(A) 6

(B) 8

(C) 10

(D) 11

(E) 12

8. Image

(A) Image

(B) Image

(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 Image if x ≠ 5, and let f be continuous at x = 5. Then c =

(A) Image

(B) 0

(C) Image

(D) 1

(E) 6

11. Image

(A) −1

(B) Image

(C) 0

(D) Image

(E) 1

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

(A) esin x cos x

(B) e−sin x cos x

(C) Image

(D) ecos x

(E) esin x

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

(A) Image

(B) 0

(C) −1

(D) Image

(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. Image equals

(A) 1 − e

(B) Image

(C) e − 1

(D) Image

(E) e + 1

17. If Image equals

(A) 7

(B) Image

(C) Image

(D) 9

(E) Image

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) Image

(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) Image

(C) 2

(D) 3

(E) none of these

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

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

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) Image

(D) 2

(E) Image

23. Image equals

(A) ln (ln x) + C

(B) Image

(C) Image

(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) Image

(B) Image

(C) Image

(D) Image

(E) Image

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

(A) straight lines

(B) circles

(C) hyperbolas

(D) parabolas

(E) ellipses

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

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

27. Image

(A) −2

(B) Image

(C) 0

(D) Image

(E) Image

28. Image

(A) 0

(B) Image

(C) Image

(D) Image

(E)

BC ONLY

29. Image

(A) 0

(B) Image

(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. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

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

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

34. Image 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

Image

(A) quadratic

(B) cubic

(C) sinusoidal

(D) exponential

(E) logarithmic

36. Image

(A) 0

(B) 1

(C) Image

(D) Image

(E) Image

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

Image

(A) π + 2

(B) Image

(C) Image

(D) Image

(E) Image

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

Image

(A) Image

(B) y = ln x

(C) y = ex

(D) y = ex

(E) y = ex2

39. What is the domain of the particular solution for Image 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

Image

(A) Image

(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 Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

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) Image

(D) 2

(E) none of these

44. Image

(A) 0

(B) Image

(C) π

(D)

(E) none of these

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

(A) Image

(B) Image

(C)x2 e−x + 2xe−x + C

(D)x2 e−x − 2xex − 2ex + C

(E)x2 ex + 2xex − 2ex + C

47. Image is equal to

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

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 Image t Image 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 Image when Image is equal to

(A) Image

(B) −1

(C) 2

(D) Image

(E) Image

50. Which of the following improper integrals diverges?

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

51. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

52. Image

(A) −∞

(B) 0

(C) 1

(D)

(E) nonexistent

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

(A) 0

(B) 3

(C) Image

(D) Image

(E) none of these

54. Image

(A) −∞

(B) −1

(C) 0

(D) 1

(E)

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

I. Image

II. Image

III. Image

(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 Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) undefined; the integral is improper

BC ONLY

57. The slope of the spiral r = θ at Image

(A) Image

(B) −1

(C) 1

(D) Image

(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

Image

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

Image

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. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) 0

61. If Image

(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. Image

III. Image

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) Image

(B) Image

(C) Image

(D) Image

(E) Image

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

(A) Image

(B) Image

(C) f ′(a) exists.

(D) f (a) is defined.

(E) Image

67. Suppose Image where k is a constant. Then Image 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) Image

(B) Image

(C) Image

(D) Image

(E) Image

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

(A) sec (xy)

(B) y cos (xy) − 1

(C) Image

(D) Image

(E) cos (xy)

71. Let x > 0. Suppose Image

(A) f (x4)

(B) f (x2)

(C) 2xg(x2)

(D) Image

(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) Image

(B) Image

(C) Image

(D) Image

(E) Image

73. Suppose the function f is continuous on 1 Image x Image 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 Image x Image 2.

(B) Image 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 Image 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?

Image

(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 Image 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 Image x Image e is

(A) 0.772

(B) 1.221

(C) 1.359

(D) 1.790

(E) 2.097

80. Let Image for 0 Image x Image 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?

Image

(A) Image

(B) Image

(C) Image

(D) 36

(E) 162

83. In a protected area (no predators, no hunters), the deer population increases at a rate of Image 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 Image

Image

Let Image 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.

Image

(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.

Image

86. If Image then P ′(3) =

(A) −2

(B) Image

(C) Image

(D) Image

(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) Image

(B) Image

(C) Image

(D) Image

(E) 2

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

(A) Image

(B) Image

(C) Image

(D) Image

(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) =

Image

(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

Image

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

Image

(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.

Image

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

(A) 0

(B) Image

(C) 1

(D) Image

(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) Image

(C) 0

(D) Image

(E) 1

98. If a block of ice melts at the rate of Image 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) Image

(B) Image

(C) Image

(D) Image

(E) Image

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) Image

(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 Image 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 Image 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) Image

(B) Image

(C) Image

(D) Image

(E) Image

106. Find the radius of convergence of Image

(A) 0

(B) Image

(C) 1

(D) e

(E)

107. When we use Image to estimate Image 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 Image t Image 1. How far does it travel?

(A) 0.96

(B) 1.73

(C) 2.10

(D) 2.14

(E) 3.98