9.Iff(x) =
2
x
2
1
√
t
3
+ 1
dt
, then
f
′
(1) =
A)0B)13C)23D)√2E)undefined
CHAPTER 4.
INTEGRALS
131
10.If
b
a
f
(
x
)
dx
= 3 and
b
a
g
(
x
)
dx
=
−
2, then which of the following must be true?
I.
f
(
x
)
> g
(
x
) for all
a
≤
x
≤
b
II.
b
a
[
f
(
x
) +
g
(
x
)]
dx
= 1
III.
b
a
[
f
(
x
)
g
(
x
)]
dx
=
−
6
11.The graph offis shown below. Approximate
3
−
3
f
(
x
)
dx
using the trapezoid rule with 3
equal subdivisions.
1
2
3
4
f
12.If
k
0
sec
2
x
1 + tan
x
dx
= ln 2, then the value of
k
is
132
The AP CALCULUS PROBLEM BOOK
13.The graph of the functionfon the interval [−4,4] is shown below.
4
−
4

f
(
x
)

dx
=
A)1B)2C)5D)8E)9
−
4
−
3
−
2
−
1
1
2
3
4
−
3
−
2
−
1
1
2
3
14.The acceleration of a particle moving along thexaxis at timet >0 is given bya(t) =1t2.Whent= 1 second, the particle is atx= 2 and has velocity−1 unit per second. Ifx(t) is theparticle’s position, then the position whent=eseconds is
15.The area enclosed by the two curvesy=x2−4 andy=x−4 is given by
1
0
(
x
−
x
2
)
dx
1
0
(
x
2
−
x
)
dx
2
0
(
x
−
x
2
)
dx
2
0
(
x
2
−
x
)
dx
4
0
(
x
2
−
x
)
dx
CHAPTER 4.
INTEGRALS
133
A.P. Calculus Test Four
Section Two
FreeResponse
Calculators Allowed
Time—45 minutes
Number of Questions—3
Each of the three questions is worth 9 points. The maximum possible points earned on
this section is 27, which represents 50% of the total test score. There is no penalty for
guessing.
•
SHOW ALL YOUR WORK. You will be graded on the methods you use as well as
the accuracy of your answers. Correct answers without supporting work may not
receive full credit.
•
Write all work for each problem in the space provided. Be sure to write clearly and
legibly. Erased or crossed out work will not be graded.
•
Justifications require that you give mathematical (noncalculator) reasons and that
you clearly identify functions, graphs, tables, or other objects that you use.
•
You are permitted to use your calculator to solve an equation or graph a func
tion without showing work. However, you must clearly indicate the setup of your
problem.
•
Your work must be expressed in mathematical notation rather than calculator syn
tax. For example,
5
1
x
2
dx
may not be written as
fnInt(X^2,X,1,5)
.
•
Unless otherwise specified, answers (numeric or algebraic) need not be simplified.
If your answer is given as a decimal approximation, it should be correct to three
places after the decimal point.
Good Luck!
NAME:
134
The AP CALCULUS PROBLEM BOOK
1.
The temperature on New Year’s Day in Bu
ff
alo, New York, is given by
T
(
h
) =
−
A
−
B
cos
π
h
12
,
where
T
is the temperature in degrees Fahrenheit and
h
is the number of hours from midnight
(0
≤
h
≤
24).
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