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Chapter 5 Review
AP Calculus – Meinke
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1. Let f be a function that is continuous on the closed interval [–2, 3] such that f '(0) does not exist,
f '(2) = 0, and f "(x) < 0 for all x except x = 0. Which of the following could be the graph of f ?
(A)
(B)
(C)
2. If the definite integral
2 x2
0 e
(D)
(E)
dx is first approximated by using two inscribed rectangles of equal
width and then approximated by using the trapezoidal rule with n = 2, the difference between the two
appoximations is
(A) 53.60
(B) 30.51
(C) 27.80
(D) 26.80
(E) 12.78
3. If f and g are continuous functions, and if f (x)  0 for all real numbers x, which of the following must
be true?
b
b








f
(
x
)

g
(
x
)
d
x

f
(
x
)

d
x
)
g
(
x
)

d
x
)
I. 






a
a
a




b
II.
b
b
b
a
a
a
f
(
x
)g

(
x
)
d
xf

(
x
)
d
xg

x
)
d
x




(
b
III.
b
)d
x
f
(
x
)
d
x
f(x
a
a
(A) I only
(B) II only
(C) III only
(D) II and III only
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(E) I, II, and III
4. If f is the continuous, strictly increasing function on the interval a  x  b as shown below, which of
the following must be true?
a
b
b
I.
)d
x
<
f
(
b
)(
ba
)
f(x
a
b
II.
)d
 x>
f
(
a
)(
ba
)
f(x
a
b
III.
)d
 x=
f
(
c
)(
ba
)for some number c such that a < c < b
f(x
a
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
x
5. An antiderivative of f (x) = e x  e is
(A)
ex e
x
1  ex
x
(B) (1ex)exe
x
(C) e1 e
(D) e x  e
(E) ee
x
x
x
6. Suppose g(x) < 0 for all x  0 and F(x) =
 t g'(t) dt for all x  0.
Which of the following
0
statements is false?
(A) F takes on negative values.
(B) F is continuous for all x > 0
x
(C) F(x) = x  g(x) – 0 g(t) dt
(D) F(x) exists for all x > 0.
(E) F is an increasing function.
2
x
7. If F(x) =

t3 1 dt , then F(2) =
0
(A) –3
(B) –2
(C) 2
(D) 3
(E) 18
x
8. If f (x) =

0
1
3
t 2
dt , which of the following is false?
(A) f (0) = 0
(B) f is continuous at x for all x  0.
(C) f (1) > 0
(D) f (1) =
1
3
(E) f (–1) > 0
9. Let f and g have continuous first and second derivatives everywhere. If f (x)  g(x) for all real x,
which of the following must be true?
I. f (x)  g(x) for all real x
II. f (x)  g(x) for all real x
III.
1
1
0
0
 f(x)dx   g(x)dx
(A) None
(B) I only
(C) III only
(D) I and II only
(E) I, II, and III
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Show your symbolic set-up where appropriate! Round decimal approximations to three places.
10. A water tank at Camp Newton holds 1200 gallons of water at time t = 0. During the time interval
t
0  t  18 hours, water is pumped into the tank at the rate W(t) = 95 t sin2 ( ) gallons per hour.
6
t
During the same time interval, water is removed from the tank at the rate R(t) = 275sin2 ( ) gallons
3
per hour.
a. Is the amount of water in the tank increasing at time t = 15? Why or why not?
b. To the nearest whole number, how many gallons of water are in the tank at time t = 18?
c. At what time t, for 0  t  18, is the amount of water in the tank at an absolute minimum? Show the
work that leads to your conclusion.
d. For t > 18, no water is pumped into the tank, but water continues to be removed at the rate R(t) until
the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not
solve, an equation involving an integral expression that can be used to find the value of k.
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