AP Calculus AB Midterm Review

Name: Date:

1. The gure below shows the graph of f . Use the (x + h)4 x4

gure to answer the following question(s). 5. lim =
h!0 h

A. 4x B. 4x3 C. 3x3 D. x3

sin( + h) sin
6. lim =
!0 h

A. cos B. sin
At which of the following x-values is f not
continuous? Choose the BEST answer. C. cos D. 2 cos
I. 6
1 1
II. 0
7. lim x+h x
=
h!0 h
III. 3

IV. 5 1 1 1
A. B. x2 C. D.
x2 x2 x
V. 7

VI. 10 1
8. lim =
x!7 (x 7)2
A. I, II, and IV B. I, IV, and VI
A. 1 B. 0
C. II, III, V D. I and IV
C. 7 D. no limit

3(x + h)2 3x2

2. lim = 1
h!0 h x3 4
9. lim =
x!64 x 64
A. 6xh B. 6 C. 6x D. 3
1 1 1
A. 4 B. 48 C. 8 D. 16
p p
x+h x
3. lim =
h x jx 3j
10. lim =
h!0
x!3 x 3
p p 1
A. 2 x B. 2 x C. p D. 2x
2 x A. 3 B. 3
1
C. 3 D. no limit
x 9
4. lim p =
h!9 x 3
n
7 for x = 4,
11. If f (x) = then
2x + 7 for x 6= 4
A. 0 B. 6 lim f (x) = .
x!4
C. 6x D. no limit

page 1
 sin x cos x 5x4 + 3x3 + 2x2 + 1
12. lim is 20. lim is
x!0 x x!1 4x4 + 5

A. 1 B. 0 C. 1 A. 5
B. 11
C. 1 D. 5
4 9

D. unde ned
2 ln 3x
21. lim is
x!1 x
sin 4x
13. lim is
x!0 2x
A. 1 B. 0 C. e
3 D. e
1
A. 2 B. 4 C. 2 D. 1
2 2x
22. lim is
x! 15 5x
x2
14. lim =
x!0 tan2 3x 1 2
A. 1 B. 2 C. 5 D. 5

1
A. 9 B. does not exist
4x
1 23. If f (x) = p , nd all horizontal asymptotes.
C. D. 6 x2 + 9

3 A. y= 1 B. y = 4 only
15. lim is
x!0 x
C. y= 4 D. y = 3

A. 1 B. 2 C. 1 D. 1
6x
24. Find all horizontal asymptotes of f (x) = p .
5 4x2 10
16. lim+ is
x!0 x2
A. y= 3 B. y = 6 only
A. 1 B. 0 C. e D. 1
C. y=0 D. y = 6

17. lim ln (3x 7) is

x!1 25. Consider f as de ned below:
(
x2 25
A. 1 B. 1 C. 1+ D. 1 f (x) = for x 6= 5,
x 5
7 for x = 5
3 Which of the following are true about f ?
18. lim p is
h!1 h 4
I. lim f (x) exists
x!5
1
A. 1 B. 0 C. 3 D. 1 II. f (5) exists

III. f (x) is continuous at x = 5

2x + 1
19. lim is
x!1 x
A. None B. II only

A. 1 B. 0 C. 3 D. 2 C. I and II only D. I, II, and III

page 2 AP Calculus AB Midterm Review

26. Let f be de ned as follows: 29. The function f is shown. Which of the following
8 are true for f on the open interval (b; d)?
< x2 c2
f (x) = for x 6= c, I. The domain of the derivative of f is the
: 2x c
c for x = c open interval (b; d).

Which of the following are true about f ? II. f is continuous on the open interval
(b; d).
I. lim f (x) exists
III. The derivative of f is positive on the
x!c

II. f (c) exists open interval (b; d).

III. f (x) is continuous at x = c

A. None B. I only

C. I and II only D. I, II, and III

27. f is continuous on [2; 4] and has the values shown.

The equation f (x) = 3 must have at least 2 solutions

on [2; 4] for k = .
A. I only B. II only

A. 3 B. 4 x 2 3 4 C. I and II only D. I, II, and III

C. 2 D. 6 f (x) 5 k 9
x+c for x < 3,
30. Consider f (x) =
cx2 + 5 for x 3
28. Let f be de ned as follows:
For what value of the constant c is f continuous
f (x) = x2 + 5 for x > 5, for all real numbers?
3ax for x 5

For what value of a is the function continuous? 31. By applying the Intermediate Value Theorem
choose the interval over which x4 = 7x3 5 will
have a solution.
A. 2 B. 5 C. 15 D. 30

A. [ 2; 1] B. [ 1; 0]

C. [0; 1] D. [2; 3]

32. The Intermediate Value Theorem states that IF

I. f is continuous throughout [a; b] AND

II. f (a) m f (b) if f (a) f (b) OR

f (b) m f (a) if f (b) f (a)

THEN there is at least one number c in [a; b]

such that f (c) = m.

If f (x) = x2 + 7x on [ 3; 0] and m = 3, then nd

the value of c.

page 3 AP Calculus AB Midterm Review

33. At which of the ve points shown on the graph 34.
dy d2 y
are and 2 both positive?
dx dx

Given the graph of f above, which of the following

is the graph of the derivative, f 0 ?

A. A B. C C. D D. E
A.

B.

C.

D.

page 4 AP Calculus AB Midterm Review

35. 37. Let f be de ned by f (x) = (x2 1)3 for all real
numbers x. For what values of x is the function
increasing?

A. (1; 1) B. (0; 1)

C. ( 1; 1) D. ( 1; 0]

38. Determine the interval where f (x) = x4 4x3 is

The graph of f 0 is shown. If f (2) = 17, then increasing.
f (0) = .
1 1
A. 0<x< p B. x<
A. 1 B. 1 C. 2 D. 10 3 3
r
3 1
C. x< p D. x <
36. Which of the following tables goes best with the 3 3
graph of f ?

39. Given that f is increasing, g is increasing, and that

the range of g is included in f .

What can be said about f g?

A. it increases then decreases

B. it is decreasing

C. it is increasing

D. not enough information

A. x f 0 (x) 40. Given that f is continuous and the following

a 0 information:
b 0 Intervals x<4 4<x<9 9<x
c 4 Sign of f 0 + +

B. x f 0 (x) a) For what x-coordinate(s) is there a local

a 0 minimum?

b 0 b) For what x-coordinate(s) is there a local

c 2 maximum?

C. x f 0 (x) 41. Given the function f (x) = x(x2 8) 5 satis es the

a does not exist hypothesis of the Mean Value Theorem on the
b 0 interval [1; 4], nd a number C in the interval
(1; 4) which satis es this theorem.
c 6.2
p p
D. x f 0 (x) A. 5 B. 7 C. 7 D. 5
a does not exist
b does not exist
c 1

page 5 AP Calculus AB Midterm Review

42. Which of the following functions does not satisfy 49. The graph y = ax3 6ax2 +
ax
1, a 2 I, has a
the conditions of the Mean Value Theorem on 2
point of in ection at x =
[3; 8]?

p 1
A. f (x) = jx + 5j B. f (x) = x+5 A. 1 B. 2 C. a2 D.
a
C. f (x) = jx 5j D. f (x) = x2 5x
50. Find the x-coordiante(s) of the points of in ection
2x
43. Mael borrowed his dad's car. He drove 22 km on the curve f (x) = 2 .
x +3
across town to pick up his friend. This only took
him 15 minutes. Assuming the speed limit is
60 km/h, explain why Mael should get a speeding A. 1 only B. 0 only
ticket. p
C. 3, 3, 0 D. 0 and 3
p3
x
44. Given f 0 (x) = and f (8) = 10, then a local 51. Find all intervals on which the function
4 x2
linearization estimate of f (8:1) is . y = 8x3 2x4 is concave upward.

A. 11.597 B. 11.024 A. ( 1; 0) and (2; 1)

C. 10.003 D. 10.685 B. ( 1; 24) and (48; 1)

C. ( 1; 2) and (8; 1)
45. What is the average rate of change over 2 t 4?
D. (0; 2)
t 2 3 4 5 6
f (t) 1.8 3.4 4.6 6.4 8.4 52. Let f (x) = x3 x2 + 3. Determine the critical
numbers.
A. 2.8 B. 1.4 C. 1.4 D. 0.714
2 2 2
A. 0, 3 B. 0, 3 C. 3, 3 D. 1, 3
46. Find the average velocity of an object for the
interval 2 t 4, if its position is given by
s = t2 + 5t 30. 53. Given that f (x) = x2 + 12x 28 has a relative
maximum at x = 6, choose the correct statement.

A. 5 B. 11 C. 11 D. 5.5
A. f 0 is negative on the interval ( 1; 6)

B. f 0 is negative on the interval (6; 1)

47. Given the position function s = t2 10t + 7, what
is the instantaneous rate of change at t = 4? C. f 0 is positive on the interval (6; 1)

D. f 0 is negative for all real values

A. 2t 10 B. 2t

C. 8 D. 2
54. Given a function de ned by f (x) = 3x5 5x3 + 12,
for what value(s) of x is there a relative maximum?
48. Find all points of in ection: f (x) = x3 12x
A. 1 only B. 1 only
A. (0; 0) B. (2; 0), ( 2; 0) C. 0 and 1 D. 1 and 1
C. (2; 16), ( 2; 16) D. (0; 0), (2; 16)

page 6 AP Calculus AB Midterm Review

 x ex e x
55. Given a function is de ned by f (x) = . Find 61. A function f is de ned by f (x) = . Find
+1 x2 ex + e x
all relative maximum and/or relative minimum f 0 (0).
points.

A. 1 B. 2e C. 4 D. 2e + 1
A. (1; 21 ) min only
1
B. ( 1; 2) max only 62. If f (x) = x ln x2 , then f 0 (e) =

C. (1; 12 ) max; (1; 1

2) min p
A. e B. 2 C. 4 D. e
D. (1; 12 ) max; ( 1; 1
2) min

63. Assume f (7) = 0, f 0 (7) = 14, g(7) = 1, and

56. If f (x) = sin4 x, then f 0 ( 3 ) = f (x)
g0 (7) = 17 . Find h0 (7) given h(x) = .
g(x)
p p p
27 3 3 3
A. 9 3 B. C. D.
4 4 4 A. 14 B. 2 C. 14 D. 98

57. If f (x) = sin x cos x, then f 0 ( 6 ) =

64. Given f (2) = 3; f 0 (2) = 4; g(2) = 2, and g0 (2) = 4.
Find h0 (2) if h(x) = f (x) g(x).
p 2
A. 3 1 B. 3 1
2
A. 16 B. 20 C. 20 D. 0
1
C. 2 D. 3

65. The graph f (x) has horizontal tangents when x =

dy x2 2
58. Find the derivative, , of f (x) = 2 .
dx x +2

8x
A. 1 B.
(x2 + 2)2

8x 8x2 8x
C. D.
(x2 + 2)2 (x2 + 2)2

r A. 3, 0, 3 B. 4, 2, 2, 4
1 x dy
59. If y = ln , then = C. 4, 2, 4 D. 2, 4
1+x dx

1 1 66. Find an equation for the tangent line to the graph

A. B. p
1 x2 1 + x2 of f (x) = x + 1 at the point where x = 3.
1 1
C. D.
1 + x2 1 x2 A. x 4y = 5 B. 4x y=5

C. x 4y = 5 D. 4x y= 8
x3 (x 2)
60. If f (x) = , then f 0 (5) is
x+1
3x2
67. The graph of f (x) = has a horizontal
75 525 575 575 16 x2
A. 36 B. 36 C. 36 D. 12 tangent at y =

A. 4 B. 3 C. 3 D. 0

page 7 AP Calculus AB Midterm Review

 p
68. Given a function is de ned by f (x) = x + 4, for 74. Find the derivative: s(t) = cos( 3t )
what value(s) of x does the function have one or
more vertical tangents?
A. sin( 3t ) B. 3 sin( 3t )
1 1
A. 4 only B. 4 only C. 3 sin( 3t ) D. 3 sin( 3t )

C. 0 and 4 D. 0 and 4
p
75. Di erentiate: s(t) = sec t

69. The x coordinate(s) of the point(s) on the graph of p p

f (x) = x2=3 (x2 4) where vertical tangent(s) exist p sec t tan t
A. tan2 t B. p
are 2 t
p
1 1 csc t
A. 1 B. 2, 2 C. 4 D. 0 C. sec p tan p D. p
2 t 2 t t

d3 y 2 76. Find the derivative of y = sin(x2

70. Find given y = 2 . 4 ).
dx3 x 3

A. cos 2x B. 2x cos(x2 4)
48(x2 3) 48x(x2 + 3)
A. B.
(x2 + 3)4 (x2 3)4 C. 2 cos(x2 4) D. 2x cos2 2x

24x(x2 3) 48x
C. D.
(x2 + 3)4 (x2 3)4 77. Di erentiate: y = sec2 x + tan2 x

71. Given y3 = x3 1, nd y00 . A. 0

B. sec2 x(sec2 x + tan2 x)

2x4
A. 2xy 2 (1 x3 y 3 ) B. 2xy2
y5 C. 4 sec2 x tan x

x3 2xy2 x2 y D. 2 sec x csc x + 2

C. 2xy2 1 D.
y3 y4

78. Find f 0 (x) given f (x) = sin3 (4x).

72. If f (x) = ln(sin x), then f 00 (x) =
A. 4 cos3 (4x) B. 3 sin2 4x cos(4x)
A. sin x2 + 1 B. (cot x)2 1
C. 12 sin2 4x cos(4x) D. 12 cos2 (4x)
C. cot x2 +1 D. tan x2 1

p 79. If x = y + 3y2 + 5y3 , then y0 =

3
73. Find the derivative of y = x2 + x.
3 1
A. B.
A. 1 2 2=3 (2x 1 + 6y + 15y2 6y + 15y2
3 (x + x) + 1)
3 2 x 1
B. 2 (x + x)2=3 (2x + 1) C. D.
1 + 6y + 15y2 1 + 6y + 15y2
C. 2=3 (2x
3 (x + 1) + 1)
x

1 2
D. 3 (x + x)2=3 (2x + 1)

page 8 AP Calculus AB Midterm Review

 dy 86. The graph shows the velocity of Elvis' hound dog
80. Find given x = sin (x + y).
dx that is moving along a straight line for t on [0; 7].

On the interval (0; 7), how many times is the

1 cos (x + y) acceleration of the object unde ned?
A. B. cos (x + y)
cos (x + y)
1
C. 1 D.
cos(x + y)

81. What is the slope of the tangent line to

x2 y + xy2 = 12 at the point (1; 3)?

15 2 1 1
A. 7 B. 3 C. 3 D. 3

82. The point A(0; 3) is nearest to what point on the

curve f (x) = x2 ?

p
5 5
A. (2; 4) B. ( ; ) A. 3 B. 4 C. 5 D. 7
2 4
C. (3; 9) D. ( 4; 16)
87. A mouse is running through a straight pipe. The
velocity, v(t), of the mouse is given at time t for
83. The displacement x(t) of a particle from the origin 0 t 7.
at time t, t 0, is x(t) = t3 + 3t(t 3) + 10. The
According to the graph, at what time t is the
minimum velocity is
mouse's speed the greatest?

A. 6 units/sec B. 8 units/sec
A. 1 B. 3 C. 4 D. 6
C. 12 units/sec D. 16 units/sec
88. The position of a particle at any time t is given by
84. A mother has 200 meters of fencing. She wants to s = t3 92 t2 12t + 4. What is the velocity after
build and enclose three adjacent rectangular ower 5 seconds?
gardens. What are the overall dimensions of the
entire garden that maximize the area? A. 84 B. 18 C. 6 D. 84

50
A. 25 m by 3 m B. 25 m by 50 m
89. A projectile starts at time t = 0 and moves along
C. 50 m by 50 m D. 20 m by 80 m the x-axis so that its position at any time t 0 is
x(t) = (2t2 5t + 3)(t 1)2 . What is the velocity
of the particle at time t = 3.
85. If the radius of the circle shown is 10, then what
is the area of the largest rectangle that can be
A. 13 B. 18 C. 20 D. 52
inscribed in the circle?

A. 25 units2 90. A prairie dog travels along its tunnel so that its
position at any time is s(t) = t3 12t2 + 36t 20
B. 200 units2 on the interval [0; 7]. How many times does the
p prairie dog change direction?
C. 25 2 units2
p
D. 25 2 units2 A. 1 B. 2 C. 4 D. 0

page 9 AP Calculus AB Midterm Review

91. The motion of an object is x(t) = t4 3t3 , t 0, 96. A man 2 m tall walks away from a lamppost
where t is in seconds and distance in meters. Find whose light is 5 m above the ground. If he walks
the local minimum velocity. at a speed of 1.4 m/s, at what rate is his shadow
growing when he is 10 m from the lamppost?
27 27
A. 4 m=s B. 4 m=s
14 5
A. 15 m=s B. 7 m=s
3
C. 2 m=s D. 6 m=s
15 2
C. 14 m=s D. 5 m=s

92. An amusement park patron is riding the Ferris

wheel. Their height above the hub of the 97. In triangle ABC, AB = 10 inches, AC = 15 inches.
wheel at any given time is given by the formula The angle between AB and BC is increasing at
h(t) = 20 sin 5t for t 0. The vertical acceleration 60 radians/sec. How fast is the length of the third
is side changing when the angle between the sides is
2
3 radians?

A. 100 cos 5 t B. 500 3 sin 5 t p

3 4
C. 500 sin 5t D. 500 sin 5t A. p in=sec B. in=sec
4 19 3

C. p in=sec D. p in=sec
93. A particle starts at time t = 0 and moves along 19 2 19
the x-axis so that its position at any time t 0 is
x(t) = (3t2 5t 2)(t 2)2 . Find the value of t
when the acceleration is zero and the particle is 98. Sand is falling of a conveyor onto a conical pile at
moving. the rate of 15 feet3 per minute. The diameter of
the base of the cone is twice the altitude. At what
rate is the height of the pile changing when it is
5 1
A. 1 B. 0 C. 6 D. 3 10 feet high?

3 20
94. How fast is the area of a square increasing when A. 20 ft=min B. 3 ft=min
the side is 3 m in length and growing at a rate of
20 3
0.8 m/min? C. 3 ft=min D. 20 ft=min

A. 4:8 m2 =min B. 7:5 m2 =min 99. Sand is falling of a conveyor onto a conical pile at
the rate of 20 feet3 per minute. The diameter of
C. 3:25 m2 =min D. 5:5 m2 =min the base of the cone is four times the altitude. At
what rate is the height of the pile changing when
it is 8 feet high?
95. The radius of a circle is increasing at the rate of
5 inches per minute. At what rate is the area
5 64
increasing when the radius is 10 inches? A. 64 ft=min B. 5 ft=min
5 5
C. 16 ft=min D. 24 ft=min
A. 100 in2 =min B. 50 in2 =min

C. 5 in2 =min D. 50 in2 =min

page 10 AP Calculus AB Midterm Review

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AP Calculus AB Midterm Review 10/19/2017

1. 21.
Answer: C Answer: B
2. 22.
Answer: C Answer: D
3. 23.
Answer: C Answer: C
4. 24.
Answer: B Answer: A
5. 25.
Answer: B Answer: C
6. 26.
Answer: C Answer: C
7. 27.
Answer: C Answer: C
8. 28.
Answer: A Answer: A
9. 29.
Answer: B Answer: B
10. 30.
1
Answer: D Answer: 4

11. 31.
Answer: 15 Answer: B
12. 32. p
Answer: A 7 + 37
Answer:
13. 2
Answer: C 33.
14. Answer: C
Answer: A 34.
15. Answer: A
Answer: C 35.
16. Answer: A
Answer: A 36.
17. Answer: B
Answer: D 37.
18. Answer: B
Answer: B 38.
19. Answer: A
Answer: D 39.
20. Answer: C
Answer: A
 Teacher's Key Page 2

40. 62.
Answer: 9, 4 Answer: C

41. 63.
Answer: B Answer: C

42. 64.
Answer: C Answer: C

43. 65.
Answer: Answer: B

44. 66.
Answer: C Answer: A

45. 67.
Answer: B Answer: D

46. 68.
Answer: C Answer: B

47. 69.
Answer: D Answer: D
70.
48.
Answer: B
Answer: A
71.
49.
Answer: A
Answer: B
72.
50.
Answer: B
Answer: C
73.
51.
Answer: A
Answer: D
74.
52. Answer: D
Answer: A
75.
53. Answer: B
Answer: B
76.
54. Answer: B
Answer: B
77.
55. Answer: C
Answer: D
78.
56. Answer: C
Answer: D
79.
57. Answer: D
Answer: C
80.
58. Answer: A
Answer: B
81.
59. Answer: A
Answer: D
82.
60. Answer: B
Answer: D
83.
61. Answer: C
Answer: A
84.
Answer: B
 Teacher's Key Page 3

85.
Answer: B
86.
Answer: C
87.
Answer: D
88.
Answer: B
89.
Answer: D
90.
Answer: B
91.
Answer: A
92.
Answer: C
93.
Answer: C
94.
Answer: A
95.
Answer: A
96.
Answer: A
97.
Answer: A
98.
Answer: A
99.
Answer: A