AP Calculus AB
Name ________________________________________________
Semester Exam Review
1.
1
Let ππ −5 = 0, ππ ′ −5 = −10, ππ −5 = 1, and ππ′ −5 = − 5
ππ π₯π₯
(A)
(C)
2.
(A)
(C)
3.
(A)
(C)
4.
(A)
(C)
5.
(A)
(C)
Find β′ (−5) if β π₯π₯ = ππ π₯π₯
10
−10
(B)
(D)
−2
50
Find an equation of the tangent line to the graph of ππ ππ = tan ππ at the point
(B)
4π₯π₯ − 4π¦π¦ = ππ − 4
(D)
Find the limit.
π₯π₯ 2 + 11π₯π₯ + 28
lim
π₯π₯→−4
π₯π₯ + 4
(B)
4 2 π₯π₯ − 4π¦π¦ = ππ − 4
0
3
π₯π₯ + β 3 − π₯π₯ 3
equals
lim
β→0
β
3π₯π₯ 2
0
π₯π₯ 3 − 8
lim
equals
π₯π₯→2 π₯π₯ − 2
0
0
16
© 2023 Jean Adams
(D)
(B)
(D)
(B)
(D)
Flamingo Math.com
4π₯π₯ − 2π¦π¦ = ππ − 2
π¦π¦ = π₯π₯
0
Does not exist
π₯π₯ 3
3π₯π₯ 2 β
4
12
ππ
,1
4
6.
lim +
π₯π₯→−7
(A)
(C)
7.
(A)
(C)
8.
(A)
(C)
9.
(A)
(C)
10.
(A)
(C)
−1
π₯π₯ + 7
equals
π₯π₯ + 7
(B)
(D)
0
Find the limit.
π₯π₯ + 9 − 3
lim
π₯π₯→0
π₯π₯
(B)
∞
(D)
1
3π₯π₯ + sin π₯π₯
π₯π₯→0
π₯π₯
1
∞
0
1
6
lim
(B)
4
(D)
1
3π₯π₯ 2 + 10π₯π₯ − 8
equals
π₯π₯→−4 π₯π₯ 2 + π₯π₯ − 12
3
0
lim
−
2
14
3
(B)
(D)
14
3
∞
π₯π₯−3
At which value(s) of π₯π₯ is ππ π₯π₯ = π₯π₯ 2−2π₯π₯ −8 discontinuous?
3
−4, −2, 3, 4
© 2023 Jean Adams
(B)
(D)
−2, 3,4
−2, 4
Flamingo Math.com
11.
(A)
(C)
12.
(A)
(B)
(C)
(D)
13.
Determine the value of ππ so that ππ π₯π₯ is continuous on the entire real number line when
π₯π₯ − 2, ; π₯π₯ ≤ 5
ππ π₯π₯ = οΏ½
ππππ − 3 ; π₯π₯ > 5
(B)
0
5
6
(D)
(C)
14.
(A)
(C)
1
Which of the following statements is not true of ππ π₯π₯ = π₯π₯ 2 − 49 ?
ππ is continuous on the interval (−∞, −7]
ππ is continuous on the interval [−7, 7]
ππ is continuous at π₯π₯ = 14
ππ is continuous on the interval [7, ∞)
Find the limit: lim
π₯π₯→1
(A)
6
5
5
π₯π₯ − 1 2
0
1
(B)
(D)
6π₯π₯ 2 − 13π₯π₯ − 5
equals
π₯π₯→−∞ 4π₯π₯ 2 + 19π₯π₯ − 63
−∞
∞
lim
0
3
2
© 2023 Jean Adams
(B)
(D)
2
3
5
63
Flamingo Math.com
15.
(A)
(C)
16.
(A)
(C)
17.
(A)
(C)
18.
(A)
(C)
19.
(A)
(C)
π₯π₯−2
Find all vertical asymptotes of the graph of ππ π₯π₯ = π₯π₯ 2−3π₯π₯ −10
(B)
π₯π₯ = −2, π₯π₯ = 5
(D)
π¦π¦ = 0
π¦π¦ = 1
π₯π₯ = 2, π¦π¦ = 5
If ππ 2 = 3 and ππ ′ 2 = −1, find an equation of the tangent line when π₯π₯ = 2.
(B)
π¦π¦ = 2(π₯π₯ + 1) + 3
π¦π¦ = −1 π₯π₯ − 2 + 3
π¦π¦ = 3 π₯π₯ + 1 + 2
(D)
π¦π¦ = 2 π₯π₯ − 2 − 1
(B)
12π₯π₯ 3 − 18π₯π₯ 2 + 3
Find ππ ′ π₯π₯ : ππ π₯π₯ = 3π₯π₯ 4 − 6π₯π₯ 3 + 3π₯π₯ − 2
3π₯π₯ 3 − 6π₯π₯ 2 + 3
3π₯π₯ 4 − 6π₯π₯ 3 + 3π₯π₯
Find ππ ′ π₯π₯ : ππ π₯π₯ =
(D)
12π₯π₯ 3 − 18π₯π₯ 2 + 3π₯π₯ − 2
π₯π₯ 2 − 4π₯π₯
π₯π₯
3π₯π₯ − 4
2π₯π₯ 1/2
2π₯π₯ − 4
1
2 π₯π₯
(B)
2π₯π₯ − 4
π₯π₯
(D)
π₯π₯ 3/2 − 4π₯π₯1/2
(B)
−5 sin π₯π₯ − 3 cos π₯π₯ + 4
ππππ
Find ππππ : π¦π¦ = 5 cos π₯π₯ − 3 sin π₯π₯ + 4π₯π₯
5 sin π₯π₯ − 3 cos π₯π₯ + 4
−5 sin π₯π₯ − 3 cos π₯π₯
© 2023 Jean Adams
(D)
Flamingo Math.com
−5 cos π₯π₯ + 3 sin π₯π₯
20.
(A)
(C)
21.
(A)
(C)
22.
(A)
(C)
23.
(A)
Find an equation for the tangent line to the graph of ππ π₯π₯ = 2π₯π₯ 2 − 2π₯π₯ + 3 at the point
where π₯π₯ = 1.
(B)
π¦π¦ = 2π₯π₯ − 2
π¦π¦ = 4π₯π₯ 2 − 6π₯π₯ + 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 π π ′ π‘π‘ = −15π‘π‘ 2 + 8, where π π is measured in feet and
π‘π‘ is measured in seconds. What is the instantaneous velocity when π‘π‘ = 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
π π π‘π‘ = −16π‘π‘ 2 − 46π‘π‘ + 225
π π π‘π‘ = 16π‘π‘ 2 − 46π‘π‘ + 225
24.
Differentiate: π¦π¦ = π₯π₯ 2
(C)
π¦π¦ = 2π₯π₯ + 1
The position function for an object is given by π π π‘π‘ = 6π‘π‘ 2 + 240 π‘π‘, where π π is measured in feet
and π‘π‘ is measured in seconds. Find the velocity of the object when π‘π‘ = 2 seconds.
(C)
(A)
π¦π¦ = 4π₯π₯ 2 − 6π₯π₯ + 5
(B)
(D)
π π π‘π‘ = −9.8π‘π‘ 2 + 46π‘π‘ + 225
π π π‘π‘ = 9.8π‘π‘ 2 − 46π‘π‘ + 225
5π₯π₯
+1
5
2π₯π₯
5
1 + π₯π₯ 2
© 2023 Jean Adams
(B)
(D)
Flamingo Math.com
5(1 − π₯π₯ 2 )
1 + π₯π₯ 2 2
5π₯π₯ 2 − 5
1 + π₯π₯ 2 3
25.
(A)
(C)
26.
(A)
(C)
27.
(A)
(C)
28.
(A)
(C)
Differentiate: ππ π₯π₯ = π₯π₯ 3 + 3 cot π₯π₯
3π₯π₯ 2 + 3 cot π₯π₯
3π₯π₯ 2 − 3 csc 2 π₯π₯
(D)
3π₯π₯ 2 + 3 csc 2 π₯π₯
3π₯π₯ − 3 csc 2 π₯π₯
Find ππ ′ (π₯π₯) for ππ π₯π₯ = 2π₯π₯ 2 + 5 7
7 4π₯π₯ 6
Find the derivative: ππ π₯π₯ = sec
4π₯π₯ 7
(B)
7 2π₯π₯ 2 + 5 6
(D)
28π₯π₯ 2π₯π₯ 2 + 5 6
(B)
sec
π₯π₯
2
1
π₯π₯
π₯π₯
sec
tan
2
2
2
1
π₯π₯
π₯π₯
tan
− sec
2
2
2
(D)
−
π₯π₯
π₯π₯
tan
2
2
1
π₯π₯
csc 2
2
2
Differentiate: π¦π¦ = sin2 π‘π‘ − cos2 π‘π‘
2 sin(2π‘π‘)
−4 sin π‘π‘ cos π‘π‘
29.
Find ππ ′′ (π₯π₯) if ππ π₯π₯ = sin π₯π₯ 2
(A)
2(cos π₯π₯ 2 − 2π₯π₯ 2 sin π₯π₯ 2 )
(C)
(B)
2π₯π₯ cos π₯π₯ 2
© 2023 Jean Adams
(B)
(D)
(B)
(D)
4 sin π‘π‘ cos π‘π‘
1
−4π₯π₯ sin π₯π₯ 2
2(cos π₯π₯ 2 − π₯π₯ sin π₯π₯ 2 )
Flamingo Math.com
30.
(A)
(C)
31.
(A)
(C)
32.
(A)
(C)
33.
(A)
(C)
34.
(A)
(C)
ππππ
Find ππππ for 2π₯π₯ 2 + π₯π₯π₯π₯ + 3π¦π¦ 2 = 0
−4π₯π₯ − π¦π¦
6π¦π¦
4π₯π₯ + π¦π¦ + 6π¦π¦
(B)
(D)
− 4π₯π₯ + π¦π¦
π₯π₯ + 6π¦π¦
4π₯π₯ + 6π¦π¦
−π₯π₯
ππππ
Find ππππ, 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 ππ π₯π₯ = 5 − 6π₯π₯ 2 − 2π₯π₯ 3 on [−3, 1] equals
5
1
© 2023 Jean Adams
(B)
(D)
Flamingo Math.com
−3
3
35.
(A)
(C)
36.
(A)
(C)
37.
(A)
(C)
38.
(A)
(C)
39.
(A)
(C)
ππ π₯π₯ = 6π₯π₯ 2 − 9π₯π₯ + 5 has a local minimum at
π₯π₯ =
π₯π₯ =
1
2
3
4
(B)
(D)
π₯π₯ = 1
π₯π₯ =
3
2
The set of all critical numbers of ππ π₯π₯ = π₯π₯ 2 − π₯π₯ − 2 1/3 equals
{−1, 2}
{0, 2}
(B)
(D)
{0, −1}
1
−1, 2 , 2
The set of all numbers ππ in (0, 4) satisfying the conclusion of the Mean Value Theorem for the
function ππ π₯π₯ = 3π₯π₯ 2 − 12π₯π₯ + 11 on the interval [0, 4] equals
{2}
{1, 3}
(B)
(D)
{1}
{1, 2}
The set of all numbers ππ in (0, 4) satisfying the conclusion of Rolle’s Theorem for the function
ππ π₯π₯ = 3π₯π₯ 2 − 12π₯π₯ + 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 ππ π₯π₯ = π₯π₯ 3 − 3π₯π₯ 2 − π₯π₯ + 7
(0,0) and (1, 4)
(1, 4)
© 2023 Jean Adams
(B)
(D)
Flamingo Math.com
(0,0)
(2, 1)
40.
Find all the interval(s) on which the graph of the function ππ π₯π₯ = π₯π₯ 4 − 4π₯π₯ 3 + 2 is concave
downward.
(A)
(C)
41.
(A)
(D)
(0, 2) and (2, ∞)
(−∞, 0) and (0, 2)
(0, 2)
Let ππ be a continuous function whose derivative is given by ππ ′ π₯π₯
= οΏ½1
For what values of π₯π₯ does the graph of ππ have a point of inflection?
(B)
0 only
(C)
42.
(B)
(−∞, 0) and (2, ∞)
(D)
4 only
0 and 2 only
Let ππ be a function that is differentiable on the open interval (ππ, ππ). If ππ has a relative minimum
at ππ, ππ ππ and ππ < ππ < ππ, which of the following must be true?
II. ππ ′′ ππ must exist
III. If ππ ′′ (ππ) exists, the ππ ′′ ππ > 0
(A)
(C)
(B)
I only
(C)
(A)
π₯π₯ + 3 ; π₯π₯ > 2
2 only
I. ππ ′ ππ = 0
43.
2
π₯π₯ 2 ; π₯π₯ ≤ 2
(D)
III only
4
οΏ½
2
ππ
3π‘π‘ 2 + 2π‘π‘ − 1
ππππ
12
46
© 2023 Jean Adams
II only
I and II only
ππππ =
(B)
(D)
Flamingo Math.com
40
55
44.
(A)
(C)
45.
(A)
(C)
46.
4
If ∫0 π₯π₯ 2 − 6π₯π₯ + 9 ππππ is approximated by 4 inscribed rectangles of equal width on the x-axis,
then the approximation is
(B)
4
5
(D)
6
10
π₯π₯ 2
ππ
οΏ½ [cos(π‘π‘ 3 )] ππππ =
ππππ
0
− sin π₯π₯ 6
cos π₯π₯ 6
(B)
2π₯π₯ cos(π₯π₯ 3 )
2π₯π₯ cos(π₯π₯ 6 )
(D)
Choose the correct statement, given that
5
5
0
2
οΏ½ ππ π₯π₯ ππππ = 7 and οΏ½ ππ π₯π₯ ππππ = −1
(A)
(C)
2
οΏ½ ππ π₯π₯ ππππ = 6
(B)
οΏ½ ππ π₯π₯ ππππ = 8
(D)
0
0
(A)
(C)
οΏ½ ππ π₯π₯ ππππ = −1
5
2
οΏ½ ππ π₯π₯ ππππ = 8
0
2
47.
2
Use the Fundamental Theorem of Calculus to evaluate the integral:
1
οΏ½ 1 − 2π₯π₯ ππππ
−2
6
−6
© 2023 Jean Adams
(B)
(D)
Flamingo Math.com
−2
2
48.
(A)
(C)
49.
(A)
(C)
50.
(A)
(C)
Evaluate the definite integral.
1
οΏ½(3 − 2π₯π₯) ππππ
−2
−12
−3
Evaluate the integral:
π₯π₯ 4 − π₯π₯ 3
οΏ½
ππππ
π₯π₯ 2
2π₯π₯ − 1 + πΆπΆ
π₯π₯ 3
π₯π₯ 2
−
+ πΆπΆ
3
2
(B)
12
15
2
(D)
(B)
(D)
2π₯π₯ 3 − 3π₯π₯ 2 + πΆπΆ
3
20
4π₯π₯ 2 − 5π₯π₯ + πΆπΆ
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
© 2023 Jean Adams
(B)
(D)
Flamingo Math.com
50 feet by 37.5 feet
52 feet by 36 feet
0
