AP Calculus AB – Semester Exam Review
No Calculator Portion
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION
Directions: DO NOT WRITE ON THIS TEST. After examining the form of the choices, decide which is the best of the
choices given and fill in the corresponding oval on the answer sheet. No partial credit will be given. Do not spend too
much time on any one problem. Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f(x) is a real number.
1. Let f(x) be the function whose graph is shown to the right:
A. lim f  x   
x 2
B. lim f  x   
C. lim f  x   3
x 2
x 2
D. lim f  x   3
E. none of the above
D. it varies
E. does not exist
x 2
2. The graph of f(x) is shown in the figure to the right.
What is lim f  x  ?
x 2
A. –1
B. 1
3. Which statement is true about the graph of y 
C. 2
4x 2  3x  2
?
6  x  2x 2
a. the line x = 2 is a vertical asymptote
b. the line y = 2 is a horizontal asymptote.
3
c. the line x  is a vertical asymptote.
2
d. the graph has two horizontal asymptotes
e. y  4x 2 is a parabolic asymptote.
3x 2  21x  30
?
x 2  3x  10
C. x = -5
D. x = 5
E. x = 2
C. III only
E. I and III only
4. Which of the following is a vertical asymptote for f  x  
A. x = -3
B. x = 3
5. The graph of the function f(x) is given to the right.
Which of the statements must be true?
I. lim f  x   1
x 2
II. f  2  1
III. f(x) is continuous at x = 2.
A. I only
B. II only
D. I and II only
6. Let f(x) be a function such that f(2) = 3. Which of the following statements must also be true?
I. lim f  x   3
II. f(x) is continuous at a x = 2
III. f(x) is differentiable at x = 2.
x 2
A. I only
B. II only
C. III only
D. all of the above
E. none of the above
7. Let f(x) be a continuous function on the interval –2 < x < 2. Use the table of values below to determine which of the
following statements must be true.
x
-2
-1
0
1
2
f(x)
-4
1
6
3
-5
I. f(x) takes on the value of 5
II. A zero of f(x) is between –2 and –1
III. A zero of f(x) is 6
A. I only
B. II only
C. III only
D. I and II only
E. I, II and III
8. Let f(x) be a continuous function on the interval –2 < x < 2. Use the table of values below to determine which of the
following statements must be true.
x
-2
-1
0
1
2
f(x)
-4
1
6
3
-5
I. For some value of x on the interval [-2, 2], f ‘ (x) = 0
II. f(x) has an absolute maximum value on the interval [-2, 2]
III. f(x) takes on the value of 5 in the interval [-2, 2]
A. I only
9. lim
x 
x 2
42 x
B. II only
C. III only
D. I and II only
C. 0
D.
E. I, II and III

A. –1
B.
1
2
1
4
E. does not exist
 x 2  3x  2
, when x  1

10. Let f  x    x 2  1
. If f(x) is continuous at x = -1, what must be the value of k?
k, when x  1

A. –2
B. –1
C.
1
2
D. 0
E. 1
11. Let f(x) be a continuous function on the interval –2 < x < 2. Use the table of values below to determine which of the
following statements must be true.
x
-2
-1
0
1
2
f(x)
-4
1
2
3
-5
A. f(x) is differentiable on the interval [-2, 3]
B. The absolute minimum value of f(x) is 5
C. f(x) has an absolute maximum value on the interval [-3, 3]
D. A zero of f(x) is 2
E. There are at least 2 zeroes of f(x).
12. Let f(x) be a continuous function on the interval –2 < x < 2. Use the table of values below to determine which of the
following statements must be true.
x
-2
-1
0
1
2
f(x)
-4
1
6
3
0
I. f ‘ (1) < 0
II. f ‘ (1) = -3
III. f ‘ (1) = 3
A. I only
B. II only
C. III only
D. I and II only
E. none of the above
C. ex 1
D. e2x 2
E. e2x 2
13. If g'  x   2g x  and g  1  1, then g  x  
A. e2x
B. e x
dy
at x = -2?
dx
-1
0
-5
0
14. Using the following table, which of the following is NOT an approximation of
x
f(x)
A. -4
B. -3
-6
25
-5
14
C. -2
-4
6
-3
1
-2
-3
D. 3/2
E. 2
For questions 15-16, let f(x) and g(x) be two continuous and differentiable functions that have the values shown in the
table below:
x
f(x)
f ‘ (x)
g(x)
g’(x)
-1
3
4
3
-2
0
7
-2
2
1
1
-2
1
-1
-4
2
6
0
3
3
3
5
1
5
2
15. Let h  x   2f  x   g  x  . What is the value of h'  2  ?
A. 0
B. 2
C. 3
D. 6
E. 16
C. 6
D. 18
E. 20
16. Let Q  x   f  g  x   . What is Q'  2 ?
A. 0
B. 3
17. The function defined by f  x   12  3x  x 3
A.
B.
C.
D.
E.
increases for x < -1, then decreases for x > -1
decreases for all x < 0, then increases for x > 0.
decreases for x < -1, then increases for x > -1
increases for x < -1, then decreases for –1 < x < 1, then increases for x > 1.
decreases for x < -1, then increases for –1 < x < 1, then decrease for x > 1
18. The graph of f ‘ (x) is shown in the figure to the right.
Which of the following must be true about the graph
of f at x = 3?
A. f is decreasing and concave down
B. f is decreasing and concave up
C. f is increasing and concave down
D. f is increasing and concave up
E. f is positive
19. The graph of f ' is shown in the figure to the right.
Based on the graph, which of the following
statements is true?
A. f is discontinuous at x = 3
B. f is increasing for 3 < x < 7
C. f has a local maximum at x = 5
D. f has a local minimum at x = 5
E. f is constant for 0 < x < 3
20. Suppose that g is a function with the following two properties: g  x   g  x  for all x, and g ‘ (a) exists. Which of the
following must necessarily be equal to g'  a  ?
A. g'  a 
B. g'  a 
C.
1
g'  a 
D.
1
g'  a 
E. none of these
21. The function f, whose graph consists of two line segments,
is shown to the right. Which of the following are true for
f on the open interval (a, b)?
I. The domain of the derivative of f is the open interval
(a, b).
II. f is continuous on the open interval (a, b).
III. The derivative of f is positive on the open interval
(a, b).
A. I only
B. II only
C. III only
D. II and III only
E. I, II and III



cos   h   cos
2
2

22. What is lim
?
h0
h
 2
A. –1
B.
2
C. 0
D. 1
E. does not exist
D. D
E. E
D. –3/14
E. 0
23. At which of the five points on the graph in the figure
dy
d2 y
and 2 both negative?
at the right are
dx
dx
A. A
B. B
C. C
24. The slope of the tangent to the curve y3 x  y2x 2  6 at (2, 1) is
A. –3/2
B. –1
C. –5/14
25. Which of the following statements about the function given by f  x   x 4  2x 3 is true?
A.
B.
C.
D.
E.
The function has no relative extremum.
The graph of the function has one point of inflection and the function has two relative extrema.
The graph of the function has two points of inflection and the function has one relative extremum.
The graph of the function has two points of inflection and the function has two relative extrema.
The graph of the function has two points of inflection and the function has three relative extrema.
26. If f  x   sin2 3  x  , then f ' 0  
A. –2 cos 3
27.

B. –2 sin 3 cos 3
C. 6 cos 3
D. 2 sin 3 cos 3
B. 3
C. 3x
D.
E. 6 sin 3 cos 3

d
ln e3 x 
dx
A. 1
28. lim
x 
1
e3x
E.
3
e3x
9x 2  2

4x  3
A.
3
2
B.
3
4
C.
2
3
D. 1
E. does not exist
Questions 29-30 refer to the following graph and information.
A bug is crawling along a straight wire. The velocity, v(t), of the bug at time t, 0 < t < 11, is given in the graph above.
29. According to the graph, at what time t does the bug change direction?
A. 2
B. 5
C. 6
D. 8
E. 10
30. According to the graph, at what time t is the speed of the bug greatest?
A. 2
B. 5
C. 6
D. 8
E. 10
x2  4
?
x  2  x  4x 2
31. What is lim
1
1
C.
D. 1
4
2
32. If f '  x   ln  x  2, then the graph of y = f(x) is decreasing if and only if
A. –2
B.
A. 2 < x < 3
B. 0 < x
C. 0 < x < 1
D. x > 1
E. does not exist
E. x > 2
33. If r is positive and increasing, for what value of r is the rate of increase of r 3 twelve times that of r?
A. 3 4
B. 2
C. 3 12
D. 2 3
E. 6
34. lim
sin    h  sin 
h0
h
=?
A. 1
B. 0
C. –1
D. 
E. 
35. The slope of the line tangent to the curve y3  x 2y2  3x3  9 at (1, 2) is:
A.
1
16
B.
7
16
C.
25
16
D.
5
16
E.
7
16
36. The position of a particle moving along a straight line at any time t is given by s  t   2t3  4t 2  2t  1 . What is the
acceleration of the particle when t = 2?
A. 32
B. 16
C. 4
D. 8
E. 0
37. Let f  x   lnx  e x . Which of the following is TRUE at x = 1
A. f is increasing
B. f is decreasing
C. f is discontinuous D. f has a relative min E. f has a relative max
38. A leaf falls from a tree into a swirling wind. The graph below shows the vertical distance (feet) above the ground
plotted against time (seconds). According to the graph, in what time interval is the speed of the leaf the greatest?
A. 1 < t < 3
B. 3 < t < 5
C. 5 < t < 7
D. 7 < t < 9
E. none of these
39. Water is flowing into a spherical tank with six foot radius at the constant rate of 30 cu ft per hour. When the water is
h2
h feet deep, the volume of the water in the tank is given by V 
18  h . What is the rate at which the depth of
3
the water in the tank is increasing at the moment when the water is 2 feet deep, in feet per hour?
A. 0.5
B. 1.0
C. 1.5
D. 2.0
E. 2.5
40. The volume V (cubic inches) of unmelted ice remaining from a melting ice cube after t seconds is
V  2000  40t  0.2t 2 . How fast is the volume changing when t = 40 seconds?
in3
in3
in3
in3
in3
A. –26
B. 24
C. 120
D. 0
E. –24
sec
sec
sec
sec
sec
41. The graph of the function f  x   2x5 / 3  5x2/ 3 is increasing on which of the following intervals?
A. I only
I. 1 < x
B. II only
II. 0 < x < 1
C. III only
III. x < 0
D. I and II only
E. I and III only

42.
sin 2x
dx =? ?
sin x


A. 4
B. 2
C.

2
D. 2
E. 0
43. Which of the following statements is/are true?
I. If f is continuous everywhere, then f is differentiable everywhere.
II. If f is differentiable everywhere, the f is continuous everywhere.
III. If f is continuous and f  x   2 for every x in [3, 7], then
7
 f  x  dx  8
3
A. I only
B. II only
C. III only
dy
at the point (a, b).
dx
b  2a
a  2b
B.
C.
2b  a
2a  b
D. I and III only
E. II and III only
44. Suppose x 2  xy  y2  3 . Find
A.
a  2b
2a  b
D.
b  2a
2b  a
E.
b  2a
2b  a
45. For which of the following intervals is the graph of y  3x5  10x 4  10x3  60x2  12 concave down?
A. (-3, -1)
B. (-2, -1)
C.
1,  
D. (-1, 1)
E.
 ,  1
A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAMINATION
Directions: DO NOT WRITE ON THIS TEST. After examining the form of the choices, decide which is the best of the
choices given and fill in the corresponding oval on the answer sheet. No partial credit will be given. Do not spend too
much time on any one problem. Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f(x) is a real number.
In this test:
(1) The exact numerical value of the correct answer does not always appear among the choices given. When this
happens, select from among the choices the number that best approximates the exact numerical value.
(2) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a
real number.
46. If a  0, lim
x a
A.
1
a2
x 2  a2
?
x 4  a4
B.
1
2a 2
C.
1
6a 2
D. 0
E. nonexistent
47. If g is a differentiable function such that g(x) < 0 for all real numbers x and if f '  x    x 2  4  g  x  , which of the
following is true?
A. f has a relative maximum at x = -2 and a relative minimum at x = 2.
B. f has a relative minimum at x = -2 and a relative maximum at x = 2
C. f has relative minima at x = -2 and at x = 2
D. f has relative maxima at x = -2 and at x = 2.
E. It cannot be determined if f has any relative extrema.
48. If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at a rate of 3
inches per minute, which of the following must be true about the area A of the triangle?
A. A is always increasing
B. A is always decreasing
C. A is decreasing only when b < h.
D. A is decreasing only when b > h.
E. A remains constant.
49.
The graphs of the derivatives of the functions f, g and h are shown above. Which of the functions f, g, or h have a
relative maximum on the open interval (a, b)?
A. f only
B. g only
C. h only
D. f and g only
E. f, g, and h
50. The first derivative of the function f is given by f "  x  
interval (0, 10)?
A. One
B. Three
cos2 x 1
 . How many critical values does f have on the open
x
5
C. Four
D. Five
E. Seven
51. Let f be the function given by f  x   x . Which of the following statements about f are true?
I. f is continuous at x = 0
II. f is differentiable at x = 0.
III. f has an absolute minimum at x = 0
A. I only
B. II only
C. III only
D. I and III only
E. II and III only
52. The area of the region in the first quadrant between the graph of y  x 4  x 2 and the x-axis is:
A. 0.943
B. 2.667
C. 2.828
D. 3.464
E. 5.333
53. Let f be a function that is differentiable on the open interval (1, 10). If f(2) = -5, f(5) = 5, and f(9) = -5, which of the
following must be true?
I. f has at least 2 zeros
II. The graph of f has at least one horizontal tangent.
III. For some c, 2 < c < 5, f( c) = 3
A. None
B. I only
C. I and II only
54. If y = 2x – 8, what is the minimum value of the product xy?
A. –16
B. –8
C. –4
D. I and III only
E. I, II, and III
D. 0
E. 2
55. The function f is continuous on the closed interval [2, 8] and has values that are given in the table below. Using the
8
subintervals [2, 5], [5, 7], [7, 8], what is the trapezoidal approximation of
 f  x  dx ?
2
x
F(x)
A. 110
B. 130
2
10
5
30
7
40
C. 160
8
20
D. 190
E. 210
56. Which of the following is an equation of the line tangent to the graph of f  x   x 4  2x 2 at the point where f '  x   1 ?
A. y = 8x – 5
B. y = x + 7
C. y = x + 0.763
D. y = x – 0.122
E. y = x – 2.146
57. What is the area, in square units, of the largest rectangle that can be inscribed under the graph of


y  2cos x for
x ?
2
2
A. 2.20
B. 2.24
C. 2.28
D. 2.32
E. 2.36
58. The graph of the function y  x3  6x 2  7x  2cos x changes concavity at x =
A. –1.58
B. –1.63
C. –1.67
D. –1.89
E. –2.33
59. What is the area of the region in the first quadrant enclosed by the graphs of y = cos x, y = x, and the y-axis?
A. 0.127
B. 0.385
C. 0.400
D. 0.600
E. 0.947
60. A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and
watches an eastbound train traveling at 60 meters per second. At how many meters per second is the train moving
away from the observer 4 seconds after it passes through the intersection?
A. 57.60
B. 57.88
C. 59.20
D. 60.00
E. 67.40
61. At time t > 0, the acceleration of a particle moving on the x-axis is a(t) = t + sin t. At t = 0, the velocity of the particle is
–2. For what value of t will the velocity of the particle be zero?
A. 1.02
B. 1.48
C. 1.85
D. 2.81
E. 3.14
62. Let f be a function such that lim
f  2  h  f  2 
h0
h
 5 . Which of the following must be true?
I. f is continuous at x = 2
II. f is differentiable at x = 2
III. The derivative of f is continuous at x = 2
A. I only
B. II only
C. I and II only
D. I and III only
E. II and III only
Problems 63-64 refer to the table below:
x
-2
-1
2
3
f(x)
3
4
5
2
f’(x)
5
3
-1
4
g(x)
-1
2
-2
-1
g’(x)
2
-2
3
-2
63. Let h  x   f  g  x   . Find h'  2  .
A. –3
B. 4
C. 5
D. 12
E. 15
B. –8
C. –6
D. 3
E. 5
B. 2tanx csc 2 x
C.
2 sin x
cos 2 x
D.
2 sin x
cos3 x
E. tanx sec x
C.
 xy
xy
D.
2x
xy
E.
64. Let h  x   f  g  x   . Find h'  3  .
A. –10
65. Find
dy
if y  tan2 x
dx
A. 2sec 2 x
66. If 2xy  y2  5, then
A.
y
xy
dy

dx
B.
x
y
x
1 x
67. A balloon rises at a rate of 4 meters per second from a point on the ground 40 meters from an observer. What is the
rate of change of the angle of elevation (in rad/sec) when the balloon is 40 meters above the ground?
A. 0.05
B. 0.10
C. 0.25
68. Suppose a particle moves along the curve xy  20 . If x = 4 and
A.
92
5
B. –10
C.
5
2
D. 0.785
E. 2.0
dy
 2 , what is the value of
dt
8
D.
E.
5
dx
?
dt
5
2
69. Given the graph of f(x) to the right, which of the following
statements is true?
A. f '  2  f ' 1
B. f " 1  f "  1
f  2  f 0 
2
20
D. f '  1  f "  2
C.
E.
f  2
4
1
70. Let f  x   x 2 
1
x
. What is the x-coordinate of the point of inflection?
2
 1 5
A.  
4
5
2
 3 5
B.  
8
5
C. 1
 3 2
D.  
8
 3 2
E.  
2
C. 0
D. 7
E. does not exist
71. The graph of g ‘ (x) is shown in the figure.
Which of the following has the largest value?
A. g(0)
B. g(1)
C. g(2)
D. g(3)
E. g(4)
72. lim
h0
49  h  7

h
1
A.
14
B.
1
7
 2
x  3
3x  27,

73. Let f  x    9  x 2 ,
3  x  2.4 . Which of the following statements must be true?
 4
 x  5,
x  2.4
3
I. f(x) is differentiable at x = -3
II. f(x) is differentiable at x = 2.4
III. f(x) is continuous at x = -3
IV. f(x) is continuous at x = 2.4
A. I and IV only
B. II and IV only
C. I, III, and IV only
D. II, III, and IV only E. all are true.
74. Let f(x) be a function that is differentiable at x = 3. Which of the following statements must also be true?
I. f(x) is continuous at x = 3.
II. f(x) is defined at x = 3.
III. f "  x  exists at x = 3.
A. I only
B. II only
C. III only
75. Given f(x) as shown in the figure to the right. At which
point is the function continuous but not differentiable?
A. a
B. b
C. c
D. d
E. e
D. I and II only
E. II and III only
76. Let f  x   3  x 2  4 . Which of the following statements is true?
A. limit as x approaches 2 from the right does not exist.
B. f is differentiable at x = -2.
C. f ' 0  0
D. f ' 1  2
E. f '  1  2
77. If f  x    x  2  tan2x, f ' 0  
2
A. –8
B. –4
C. 0
D. 4
E. 8
78. What is the slope of the line normal to the graph of 2x 2y  3x  18 at the point (2, 3)?
A.
21
8
B.
8
21
C.
3
8
D.
8
21
E.
8
3
79. Using local linearization at x = 0 for f  x   4  tan2x , what is the best estimate for f(0.04)?
A. 1.98
B. 2.01
C. 2.02
80. Let f be a function whose derivative is given by f '  x  
D. 2.04

E. 2.08

x
 sin e0.2x . How many relative maximum points does f(x)
15
have in the interval 0 < x < 12?
A. None
B. One
C. Two
D. Three
E. More than three
81. Let f(x) be a continuous function defined on [-3, 5]. The
graph of the second derivative of f(x) is shown below.
The graph of f(x) has a point of inflection:
A. at x = 0 only
B. at x = 2 only
C. at x = 0 and x = 2
D. at no values of x
E. not enough information is given to determine the answer
82. The graph of the derivative of f(x) is shown to the right. For
what value of x is f(x) concave down?
A. 2
B. 3
C. 4
D. 5
E. 6
83. Use the graph of the function to the right, f(x), to determine the
intervals for which the derivative of f(x) is increasing.
A. (1, 3)
B. (2, 4)
C. (3, 5)
D. (0, 3)
E. (0, 1)
84. Let f be a twice-differentiable function whose derivative, f '  x  , is decreasing for all x. Which of the following must be
true for all x?
I. f(x) < 0
II. f ‘ (x) < 0
III. f “ (x) < 0
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
85. The graph of v(t) is shown to the right for [0,6].
2

0t2
4  t ,
y

 t  4  2, 2  t  6
For how many values of t is a(t) undefined?
A. one
B. two
C. three
4x
at x = -2?
x 2
C. 10x + y = -16
D. 2x + 3y = -16
86. Which of the following is an equation of the line tangent to y 
A. 6x + y = -16
B. x + y = -2
D. four
E. none
2
E. 6x – y = -8
87. Let f(x) be a differentiable function defined for all real numbers. The table below gives the value of f(x) and its
derivative f ‘ (x) for several values of x.
x
-3
-2
-1
0
1
2
3
f(x)
8
5
0
1
0
5
8
f ‘ (x)
-6
-4
-2
0
2
4
6
Which of the following statements are true?
I. At x = 2, the function is increasing.
II. There is a relative minimum in the interval –1 < x < 1, but not necessarily at x = 0.
III. There is a relative maximum in the interval –1 < x < 1.
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
1
88.

0
xdx
is approximately:
4  x2
A. –0.268
B. 0.268
C. 0.536
D. –0.536
E. –3.732
89. The graph shown to the right represents y = f(x). Which
one of the following is NOT true?
A. f is continuous on (-2, 2)
B. lim f  x   f  0 
x 0
C. f is differentiable on (-2, 2)
D. lim f  x   f 0
x 0
E. f '  x   0 for x < 0
90. The acceleration of a particle moving on a line is a  t 1/ 2  3t1/ 2 . What velocity did the particle have from
t = 0 to t = 9?
A. 60
B. 63
C. -1
D. 1
E. 46.5