AP Calculus AB
Name ________________________________________________
Semester Exam Review
1.
1
Let 𝑓𝑓 −5 = 0, 𝑓𝑓 ′ −5 = −10, 𝑔𝑔 −5 = 1, and 𝑔𝑔′ −5 = − 5
𝑓𝑓 𝑥𝑥
(A)
(C)
2.
(A)
(C)
3.
(A)
(C)
4.
(A)
(C)
5.
(A)
(C)
Find ℎ′ (−5) if ℎ 𝑥𝑥 = 𝑔𝑔 𝑥𝑥
10
−10
(B)
(D)
−2
50
Find an equation of the tangent line to the graph of 𝑓𝑓 𝜃𝜃 = tan 𝜃𝜃 at the point
(B)
4𝑥𝑥 − 4𝑦𝑦 = 𝜋𝜋 − 4
(D)
Find the limit.
𝑥𝑥 2 + 11𝑥𝑥 + 28
lim
𝑥𝑥→−4
𝑥𝑥 + 4
(B)
4 2 𝑥𝑥 − 4𝑦𝑦 = 𝜋𝜋 − 4
0
3
𝑥𝑥 + ℎ 3 − 𝑥𝑥 3
equals
lim
ℎ→0
ℎ
3𝑥𝑥 2
0
𝑥𝑥 3 − 8
lim
equals
𝑥𝑥→2 𝑥𝑥 − 2
0
0
16
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(D)
(B)
(D)
(B)
(D)
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4𝑥𝑥 − 2𝑦𝑦 = 𝜋𝜋 − 2
𝑦𝑦 = 𝑥𝑥
0
Does not exist
𝑥𝑥 3
3𝑥𝑥 2 ℎ
4
12
𝜋𝜋
,1
4
6.
lim +
𝑥𝑥→−7
(A)
(C)
7.
(A)
(C)
8.
(A)
(C)
9.
(A)
(C)
10.
(A)
(C)
−1
𝑥𝑥 + 7
equals
𝑥𝑥 + 7
(B)
(D)
0
Find the limit.
𝑥𝑥 + 9 − 3
lim
𝑥𝑥→0
𝑥𝑥
(B)
∞
(D)
1
3𝑥𝑥 + sin 𝑥𝑥
𝑥𝑥→0
𝑥𝑥
1
∞
0
1
6
lim
(B)
4
(D)
1
3𝑥𝑥 2 + 10𝑥𝑥 − 8
equals
𝑥𝑥→−4 𝑥𝑥 2 + 𝑥𝑥 − 12
3
0
lim
−
2
14
3
(B)
(D)
14
3
∞
𝑥𝑥−3
At which value(s) of 𝑥𝑥 is 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2−2𝑥𝑥 −8 discontinuous?
3
−4, −2, 3, 4
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(B)
(D)
−2, 3,4
−2, 4
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11.
(A)
(C)
12.
(A)
(B)
(C)
(D)
13.
Determine the value of 𝑐𝑐 so that 𝑓𝑓 𝑥𝑥 is continuous on the entire real number line when
𝑥𝑥 − 2, ; 𝑥𝑥 ≤ 5
𝑓𝑓 𝑥𝑥 = �
𝑐𝑐𝑐𝑐 − 3 ; 𝑥𝑥 > 5
(B)
0
5
6
(D)
(C)
14.
(A)
(C)
1
Which of the following statements is not true of 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 − 49 ?
𝑓𝑓 is continuous on the interval (−∞, −7]
𝑓𝑓 is continuous on the interval [−7, 7]
𝑓𝑓 is continuous at 𝑥𝑥 = 14
𝑓𝑓 is continuous on the interval [7, ∞)
Find the limit: lim
𝑥𝑥→1
(A)
6
5
5
𝑥𝑥 − 1 2
0
1
(B)
(D)
6𝑥𝑥 2 − 13𝑥𝑥 − 5
equals
𝑥𝑥→−∞ 4𝑥𝑥 2 + 19𝑥𝑥 − 63
−∞
∞
lim
0
3
2
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(B)
(D)
2
3
5
63
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15.
(A)
(C)
16.
(A)
(C)
17.
(A)
(C)
18.
(A)
(C)
19.
(A)
(C)
𝑥𝑥−2
Find all vertical asymptotes of the graph of 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2−3𝑥𝑥 −10
(B)
𝑥𝑥 = −2, 𝑥𝑥 = 5
(D)
𝑦𝑦 = 0
𝑦𝑦 = 1
𝑥𝑥 = 2, 𝑦𝑦 = 5
If 𝑓𝑓 2 = 3 and 𝑓𝑓 ′ 2 = −1, find an equation of the tangent line when 𝑥𝑥 = 2.
(B)
𝑦𝑦 = 2(𝑥𝑥 + 1) + 3
𝑦𝑦 = −1 𝑥𝑥 − 2 + 3
𝑦𝑦 = 3 𝑥𝑥 + 1 + 2
(D)
𝑦𝑦 = 2 𝑥𝑥 − 2 − 1
(B)
12𝑥𝑥 3 − 18𝑥𝑥 2 + 3
Find 𝑓𝑓 ′ 𝑥𝑥 : 𝑓𝑓 𝑥𝑥 = 3𝑥𝑥 4 − 6𝑥𝑥 3 + 3𝑥𝑥 − 2
3𝑥𝑥 3 − 6𝑥𝑥 2 + 3
3𝑥𝑥 4 − 6𝑥𝑥 3 + 3𝑥𝑥
Find 𝑓𝑓 ′ 𝑥𝑥 : 𝑓𝑓 𝑥𝑥 =
(D)
12𝑥𝑥 3 − 18𝑥𝑥 2 + 3𝑥𝑥 − 2
𝑥𝑥 2 − 4𝑥𝑥
𝑥𝑥
3𝑥𝑥 − 4
2𝑥𝑥 1/2
2𝑥𝑥 − 4
1
2 𝑥𝑥
(B)
2𝑥𝑥 − 4
𝑥𝑥
(D)
𝑥𝑥 3/2 − 4𝑥𝑥1/2
(B)
−5 sin 𝑥𝑥 − 3 cos 𝑥𝑥 + 4
𝑑𝑑𝑑𝑑
Find 𝑑𝑑𝑑𝑑 : 𝑦𝑦 = 5 cos 𝑥𝑥 − 3 sin 𝑥𝑥 + 4𝑥𝑥
5 sin 𝑥𝑥 − 3 cos 𝑥𝑥 + 4
−5 sin 𝑥𝑥 − 3 cos 𝑥𝑥
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(D)
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−5 cos 𝑥𝑥 + 3 sin 𝑥𝑥
20.
(A)
(C)
21.
(A)
(C)
22.
(A)
(C)
23.
(A)
Find an equation for the tangent line to the graph of 𝑓𝑓 𝑥𝑥 = 2𝑥𝑥 2 − 2𝑥𝑥 + 3 at the point
where 𝑥𝑥 = 1.
(B)
𝑦𝑦 = 2𝑥𝑥 − 2
𝑦𝑦 = 4𝑥𝑥 2 − 6𝑥𝑥 + 2
(D)
(B)
252 ft/sec
(D)
264 ft/sec
24 ft/sec
240 ft/sec
The velocity function for an object is given by 𝑠𝑠 ′ 𝑡𝑡 = −15𝑡𝑡 2 + 8, where 𝑠𝑠 is measured in feet and
𝑡𝑡 is measured in seconds. What is the instantaneous velocity when 𝑡𝑡 = 3?
(B)
−15 ft/sec
(D)
−37 ft/sec
−135 ft/sec
−127 ft/sec
An object is thrown straight down from a 225-foot-tall building with an initial velocity of
46 feet per second. The position function for the movement described is
𝑠𝑠 𝑡𝑡 = −16𝑡𝑡 2 − 46𝑡𝑡 + 225
𝑠𝑠 𝑡𝑡 = 16𝑡𝑡 2 − 46𝑡𝑡 + 225
24.
Differentiate: 𝑦𝑦 = 𝑥𝑥 2
(C)
𝑦𝑦 = 2𝑥𝑥 + 1
The position function for an object is given by 𝑠𝑠 𝑡𝑡 = 6𝑡𝑡 2 + 240 𝑡𝑡, where 𝑠𝑠 is measured in feet
and 𝑡𝑡 is measured in seconds. Find the velocity of the object when 𝑡𝑡 = 2 seconds.
(C)
(A)
𝑦𝑦 = 4𝑥𝑥 2 − 6𝑥𝑥 + 5
(B)
(D)
𝑠𝑠 𝑡𝑡 = −9.8𝑡𝑡 2 + 46𝑡𝑡 + 225
𝑠𝑠 𝑡𝑡 = 9.8𝑡𝑡 2 − 46𝑡𝑡 + 225
5𝑥𝑥
+1
5
2𝑥𝑥
5
1 + 𝑥𝑥 2
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(B)
(D)
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5(1 − 𝑥𝑥 2 )
1 + 𝑥𝑥 2 2
5𝑥𝑥 2 − 5
1 + 𝑥𝑥 2 3
25.
(A)
(C)
26.
(A)
(C)
27.
(A)
(C)
28.
(A)
(C)
Differentiate: 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 3 + 3 cot 𝑥𝑥
3𝑥𝑥 2 + 3 cot 𝑥𝑥
3𝑥𝑥 2 − 3 csc 2 𝑥𝑥
(D)
3𝑥𝑥 2 + 3 csc 2 𝑥𝑥
3𝑥𝑥 − 3 csc 2 𝑥𝑥
Find 𝑓𝑓 ′ (𝑥𝑥) for 𝑓𝑓 𝑥𝑥 = 2𝑥𝑥 2 + 5 7
7 4𝑥𝑥 6
Find the derivative: 𝑓𝑓 𝑥𝑥 = sec
4𝑥𝑥 7
(B)
7 2𝑥𝑥 2 + 5 6
(D)
28𝑥𝑥 2𝑥𝑥 2 + 5 6
(B)
sec
𝑥𝑥
2
1
𝑥𝑥
𝑥𝑥
sec
tan
2
2
2
1
𝑥𝑥
𝑥𝑥
tan
− sec
2
2
2
(D)
−
𝑥𝑥
𝑥𝑥
tan
2
2
1
𝑥𝑥
csc 2
2
2
Differentiate: 𝑦𝑦 = sin2 𝑡𝑡 − cos2 𝑡𝑡
2 sin(2𝑡𝑡)
−4 sin 𝑡𝑡 cos 𝑡𝑡
29.
Find 𝑓𝑓 ′′ (𝑥𝑥) if 𝑓𝑓 𝑥𝑥 = sin 𝑥𝑥 2
(A)
2(cos 𝑥𝑥 2 − 2𝑥𝑥 2 sin 𝑥𝑥 2 )
(C)
(B)
2𝑥𝑥 cos 𝑥𝑥 2
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(B)
(D)
(B)
(D)
4 sin 𝑡𝑡 cos 𝑡𝑡
1
−4𝑥𝑥 sin 𝑥𝑥 2
2(cos 𝑥𝑥 2 − 𝑥𝑥 sin 𝑥𝑥 2 )
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30.
(A)
(C)
31.
(A)
(C)
32.
(A)
(C)
33.
(A)
(C)
34.
(A)
(C)
𝑑𝑑𝑑𝑑
Find 𝑑𝑑𝑑𝑑 for 2𝑥𝑥 2 + 𝑥𝑥𝑥𝑥 + 3𝑦𝑦 2 = 0
−4𝑥𝑥 − 𝑦𝑦
6𝑦𝑦
4𝑥𝑥 + 𝑦𝑦 + 6𝑦𝑦
(B)
(D)
− 4𝑥𝑥 + 𝑦𝑦
𝑥𝑥 + 6𝑦𝑦
4𝑥𝑥 + 6𝑦𝑦
−𝑥𝑥
𝑑𝑑𝑑𝑑
Find 𝑑𝑑𝑑𝑑, then evaluate the derivative at the point 0, 2 : 5x 2 − 3xy + y = 2.
−2
0
(B)
(D)
−
6
5
4
The volume of a cube is changing at the rate of 18 cubic centimeters per second. How fast is the
edge of the cube expanding when each edge is 3 centimeters?
2
cm/sec
3
3
9 cm/sec
(B)
(D)
3 cm/sec
3
18 cm/sec
A spherical balloon is inflated at the rate of 16 cubic feet per minute. How fast is the radius of the
balloon changing at the instant the radius is 2 feet?
4𝜋𝜋 ft/min
1
ft/min
𝜋𝜋
(B)
(D)
32𝜋𝜋
ft/min
3
2𝜋𝜋 ft/min
The absolute maximum of 𝑓𝑓 𝑥𝑥 = 5 − 6𝑥𝑥 2 − 2𝑥𝑥 3 on [−3, 1] equals
5
1
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(B)
(D)
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−3
3
35.
(A)
(C)
36.
(A)
(C)
37.
(A)
(C)
38.
(A)
(C)
39.
(A)
(C)
𝑓𝑓 𝑥𝑥 = 6𝑥𝑥 2 − 9𝑥𝑥 + 5 has a local minimum at
𝑥𝑥 =
𝑥𝑥 =
1
2
3
4
(B)
(D)
𝑥𝑥 = 1
𝑥𝑥 =
3
2
The set of all critical numbers of 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 2 − 𝑥𝑥 − 2 1/3 equals
{−1, 2}
{0, 2}
(B)
(D)
{0, −1}
1
−1, 2 , 2
The set of all numbers 𝑐𝑐 in (0, 4) satisfying the conclusion of the Mean Value Theorem for the
function 𝑓𝑓 𝑥𝑥 = 3𝑥𝑥 2 − 12𝑥𝑥 + 11 on the interval [0, 4] equals
{2}
{1, 3}
(B)
(D)
{1}
{1, 2}
The set of all numbers 𝑐𝑐 in (0, 4) satisfying the conclusion of Rolle’s Theorem for the function
𝑓𝑓 𝑥𝑥 = 3𝑥𝑥 2 − 12𝑥𝑥 + 11 on the interval [0, 4] equals
{1}
{3}
(B)
(D)
{2}
{1, 2}
Find all the points of inflection of the graph of the function 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 3 − 3𝑥𝑥 2 − 𝑥𝑥 + 7
(0,0) and (1, 4)
(1, 4)
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(B)
(D)
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(0,0)
(2, 1)
40.
Find all the interval(s) on which the graph of the function 𝑓𝑓 𝑥𝑥 = 𝑥𝑥 4 − 4𝑥𝑥 3 + 2 is concave
downward.
(A)
(C)
41.
(A)
(D)
(0, 2) and (2, ∞)
(−∞, 0) and (0, 2)
(0, 2)
Let 𝑓𝑓 be a continuous function whose derivative is given by 𝑓𝑓 ′ 𝑥𝑥
= �1
For what values of 𝑥𝑥 does the graph of 𝑓𝑓 have a point of inflection?
(B)
0 only
(C)
42.
(B)
(−∞, 0) and (2, ∞)
(D)
4 only
0 and 2 only
Let 𝑓𝑓 be a function that is differentiable on the open interval (𝑎𝑎, 𝑏𝑏). If 𝑓𝑓 has a relative minimum
at 𝑐𝑐, 𝑓𝑓 𝑐𝑐 and 𝑎𝑎 < 𝑐𝑐 < 𝑏𝑏, which of the following must be true?
II. 𝑓𝑓 ′′ 𝑐𝑐 must exist
III. If 𝑓𝑓 ′′ (𝑐𝑐) exists, the 𝑓𝑓 ′′ 𝑐𝑐 > 0
(A)
(C)
(B)
I only
(C)
(A)
𝑥𝑥 + 3 ; 𝑥𝑥 > 2
2 only
I. 𝑓𝑓 ′ 𝑐𝑐 = 0
43.
2
𝑥𝑥 2 ; 𝑥𝑥 ≤ 2
(D)
III only
4
�
2
𝑑𝑑
3𝑡𝑡 2 + 2𝑡𝑡 − 1
𝑑𝑑𝑑𝑑
12
46
© 2023 Jean Adams
II only
I and II only
𝑑𝑑𝑑𝑑 =
(B)
(D)
Flamingo Math.com
40
55
44.
(A)
(C)
45.
(A)
(C)
46.
4
If ∫0 𝑥𝑥 2 − 6𝑥𝑥 + 9 𝑑𝑑𝑑𝑑 is approximated by 4 inscribed rectangles of equal width on the x-axis,
then the approximation is
(B)
4
5
(D)
6
10
𝑥𝑥 2
𝑑𝑑
� [cos(𝑡𝑡 3 )] 𝑑𝑑𝑑𝑑 =
𝑑𝑑𝑑𝑑
0
− sin 𝑥𝑥 6
cos 𝑥𝑥 6
(B)
2𝑥𝑥 cos(𝑥𝑥 3 )
2𝑥𝑥 cos(𝑥𝑥 6 )
(D)
Choose the correct statement, given that
5
5
0
2
� 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑 = 7 and � 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑 = −1
(A)
(C)
2
� 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑 = 6
(B)
� 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑 = 8
(D)
0
0
(A)
(C)
� 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑 = −1
5
2
� 𝑓𝑓 𝑥𝑥 𝑑𝑑𝑑𝑑 = 8
0
2
47.
2
Use the Fundamental Theorem of Calculus to evaluate the integral:
1
� 1 − 2𝑥𝑥 𝑑𝑑𝑑𝑑
−2
6
−6
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(B)
(D)
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−2
2
48.
(A)
(C)
49.
(A)
(C)
50.
(A)
(C)
Evaluate the definite integral.
1
�(3 − 2𝑥𝑥) 𝑑𝑑𝑑𝑑
−2
−12
−3
Evaluate the integral:
𝑥𝑥 4 − 𝑥𝑥 3
�
𝑑𝑑𝑑𝑑
𝑥𝑥 2
2𝑥𝑥 − 1 + 𝐶𝐶
𝑥𝑥 3
𝑥𝑥 2
−
+ 𝐶𝐶
3
2
(B)
12
15
2
(D)
(B)
(D)
2𝑥𝑥 3 − 3𝑥𝑥 2 + 𝐶𝐶
3
20
4𝑥𝑥 2 − 5𝑥𝑥 + 𝐶𝐶
Two equal rectangular lots are to be made by fencing in a rectangular lot and putting a fence
across the middle. If each lot is to contain 1875 square feet, what size lots require the minimum
amount of fence?
60 feet by 35 feet
50 feet by 50 feet
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(B)
(D)
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50 feet by 37.5 feet
52 feet by 36 feet