AP Calculus Chapter 2 Practice Test
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Assume that a watermelon dropped from a tall building falls y = 16t2 ft in t sec. Find the
watermelon's average speed during the first 5 sec of fall.
A) 40 ft/sec
B) 81 ft/sec
C) 160 ft/sec
D) 80 ft/sec
Determine the limit by substitution.
2) lim (x2 - 5)
x→0
A) Does not exist
3) lim
x→0
4) lim
x→3
2)
B) 0
C) -5
D) 5
x3 - 6x + 8
x- 2
A) 4
3)
B) -4
C) 0
D) Does not exist
x2 + 12x + 36
A) Does not exist
1)
4)
B) 9
C) ± 9
1
D) 81
Complete the table and state the lim f(x). Round table values to six decimal places when necessary.
x→0
2
5) f(x) = x + 5x + 12
x +4
x
f(x)
-0.1
?
-0.01
?
5)
-0.001 -0.0001 0.0001
?
?
?
0.001
?
0.01
?
0.1
?
A)
x
-0.1
-0.01
-0.001 -0.0001 0.0001
0.001
0.01
0.1
f(x) 2.951282 2.995013 2.9995 2.99995 3.05122 3.005012 3.0005 3.00005
lim = 3
x→0
B)
x
-0.1
-0.01
-0.001 -0.0001 0.0001 0.001
0.01
0.1
f(x) 3.951282 3.995013 3.9995 3.99995 4.00005 4.0005 4.005012 4.05122
lim = 4
x→0
C) none of these
D)
x
-0.1
-0.01
-0.001 -0.0001 0.0001 0.001
0.01
0.1
f(x) 2.951282 2.995013 2.9995 2.99995 3.00005 3.0005 3.005012 3.05122
lim = 3
x→0
Determine the limit algebraically, if it exists.
2
6) lim x - 16
x→ -4 x + 4
A) -4
6)
B) Does not exist
C) -8
D) 1
2
7) lim x + 3x - 10
x-2
x→2
A) 3
8) lim
x→0
7)
B) Does not exist
C) 0
D) 7
1 -1
x +6 6
A) 0
8)
x
B) - 1
36
C) 1
36
2
D) Does not exist
Determine the limit graphically, if it exists.
9) lim f(x)
x→ -1/2
A) Does not exist
9)
B) -2
C) 0
D) -1
Find the indicated limit.
10) lim - int x
x→7
10)
A) 6
B) -6
C) 0
D) 7
11) lim + 9x
x→0 x
11)
A) 0
B) Does not exist
C) -9
D) 9
Find the limit.
[f(x)]2 .
12) Let lim f(x) = 8 and lim g(x) = 8. Find lim
x → -1
x → -1
x → -1 2 + g(x)
A) 16
25
B) 32
5
12)
D) 4
5
C) -1
Evaluate or determine that the limit does not exist for each of the limits (a)
lim f(x), (b) lim f(x), and
x→dx→d+
(c) lim f(x) for the given function f and number d.
x→d
13)
13)
f(x) =
7x - 10,
3x - 6,
for x ≤ 1,
for x > 1
d= 1
A) (a) -10
(b) -6
(c) Does not exist
C) (a) -3
(b) -3
(c) Does not exist
B) (a) -3
(b) -3
(c) -3
D) (a) -6
(b) -10
(c) Does not exist
3
Provide an appropriate response.
14) If x3 ≤ f(x) ≤ x for x in [-1, 1], find lim f(x) if it exists.
x→0
A) -1
B) 1
14)
C) 0
D) Does not exist
Find the limit.
15) lim 6x + 1
x→∞ 16x - 7
A) 3
8
15)
B) - 1
7
C) ∞
D) 0
Find the indicated limit.
16) lim 1- cos x
x→∞
x2
A) ∞
16)
B) -∞
C) 1
D) 0
Find the limit.
17)
lim + 1
x→(-2) x + 2
A)
18)
1
2
B) -∞
C) ∞
D) - 1
2
lim + tan x
x→(π/2)
A) 0
19)
17)
18)
B) 1
C) ∞
D) -∞
lim + (1 + csc x)
x→0
A) -∞
19)
B) ∞
C) 1
D) 0
Find the vertical asymptotes of the graph of f(x).
20) f(x) = csc x
20)
A) x = nπ, n is any integer
B) no vertical asymptotes
C) x = 0
D) x = π + nπ, n is any integer
2
Find a power function end behavior model.
2
21) y = 8x + x - 1
x3 - 5x2
A) y = 8x
21)
B) y = 8x + 1
x2 - 5x
C) y = x
8
4
D) y = 8
x
Find the points of discontinuity. Identify each type of discontinuity.
1
22) y =
(x + 3)2 + 6
A) x = -3, jump discontinuity
C) None
23) y =
22)
B) x = -3, infinite discontinuity
D) x = 15
5x + 9
23)
A) x > - 9 , all points not in the domain
5
B) x = - 9 , infinite discontinuity
5
C) x < - 9 , all points not in the domain
5
D) x = - 9 , jump discontinuity
5
Provide an appropriate response.
3
24) Given f(x) = 3x and g(x) = x - 7, where is the function f(x)/g(x) continuous?
A) The function f(x)/g(x) is continuous for all x except x = 7.
B) The function f(x)/g(x) is continuous for all x.
C) The function f(x)/g(x) is continuous for all x except x < 0 and x = -7.
D) The function f(x)/g(x) is continuous for all x except x = -7.
Solve the problem.
25) The graph below shows the amount of income tax that a single person must pay on his or her
income when claiming the standard deduction. Identify the income levels where discontinuities
occur and explain the meaning of the discontinuities.
Income Tax, 1000's of dollars
40
30
20
10
20
40
60
80
100 120 140
Income, 1000's of dollars
A) Discontinuities at x = $22,000, x = $44,000, and x = $60,000. Discontinuities represent tax
cheating on the part of high-income earners.
B) Discontinuities at x = $44,000 and x = $60,000. Discontinuities represent boundaries
between tax brackets.
C) Discontinuities at x = $22,000, x = $44,000, and x = $60,000. Discontinuities represent
boundaries between tax brackets.
D) Discontinuities at x = $44,000 and x = $60,000. Discontinuities represent tax shelters.
5
24)
25)
Find the average rate of change of the function over the given interval.
26) f(x) = x 2 + 2x, [2, 6]
B) 20
3
A) 8
C) 10
26)
D) 12
27) f(x) = 8 + cos x, [0, π]
A) 0
27)
B)
1 ≈ 0.318
π
C)
8 ≈ 2.546
π
D) - 2 ≈ -0.637
π
Find the slope of the line tangent to the curve at the given value of x.
28) f(x) = -10x2 + 12x; x = 8
A) -148
B) -192
C) -106
28)
D) -64
29) f(x) = -4 at x = 11
x
A) 121
4
29)
B) 4
121
C) 4
11
D) 11
4
Find the instantaneous rate of change of the position function y = f(t) in feet at the given time t in seconds.
30) f(t) = 5t3 - 4t2 + 2, t = 3
30)
A) 21 ft/sec
B) 66 ft/sec
C) 113 ft/sec
D) 111 ft/sec
Solve the problem.
31) Find the points where the graph of the function has horizontal tangents.
f(x) = 5x2 + 3x - 4
A)
3 , - 25
10
2
B) - 3 , - 89
10
20
C) (0, 4)
31)
D) (-13, 1582)
32) For a motorcycle traveling at speed v (in mph) when the brakes are applied, the distance d (in
feet) required to stop the motorcycle may be approximated by the formula d = 0.05v 2 + v. Find
32)
the instantaneous rate of change of distance with respect to velocity when the speed is 41 mph.
A) 10.2 mph
B) 4.1 mph
C) 5.1 mph
D) 42 mph
33) A cubic salt crystal expands by accumulation on all sides. As it expands outward find the rate of
change of its volume with respect to the length of an edge when the edge is 0.31 millimeter.
A) 0.09 mm 3/mm
B) 28.83 mm 3/mm
C) 2.88 mm 3/mm
D) 0.2883 mm 3/mm
6
33)
Answer Key
Testname: CHAPTER 2 PRACTICE
1) D
2) C
3) B
4) B
5) D
6) C
7) D
8) B
9) D
10) A
11) D
12) B
13) B
14) C
15) A
16) D
17) C
18) D
19) B
20) A
21) D
22) C
23) C
24) A
25) C
26) C
27) D
28) A
29) B
30) D
31) B
32) C
33) D
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