Chapter 6
1.
Suppose
( )
‘ 4 6F x x=+
and
( )
0 –1F=
. Find the value of
( )
–1F
.
2.
Suppose
( )
‘4
x
Fx=
and
( )
0 –1F=
. Estimate the value of
( )
3F
to 4 decimal
places.
3.
Suppose
and
( )
0 –1F=
. Estimate the value of
( )
3F
to 4 decimal
places.
4.
Suppose
( )
‘ 2 6F x x=+
and
( )
6 77F=
. Find the value of
( )
3F
.
derivative. difficulty: medium section: 6.1
Chapter 6
5.
This figure shows the rate of change of F.
Given that F(0) = 2, sketch the graph of F.
-5
-4
-3
-2
-1
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
x
y
Chapter 6
6.
This figure shows the rate of change of F.
Given that F(0) = 2, sketch the graph of F.
7.
Given the graph:
Estimate the value of
5
0()f x dx
to three decimal places.
-5
-4
-3
-2
-1
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
x
y
-5
-4
-3
-2
-1
1
2
3
4
5
1 2 3 4 5 6 7
x
y
Chapter 6
8.
The graph of the derivative F’ of a function F is shown. Assuming that F(20) = 10,
estimate the maximum value attained by F.
10
20
30
10 20 30 40 50 60 70
x
y
Chapter 6
Page 5
9.
Choose the function that would correspond to this graph of F’.
A)
B)
-4
-3
-2
-1
1
2
3
4
1 2 3 4 5 6 7
x
y
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
1 2 3 4 5
x
y
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
1 2 3 4 5
x
y
Chapter 6
C)
D)
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
1 2 3 4 5
x
y
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
1 2 3 4 5
x
y
Chapter 6
Page 7
10.
Choose the function that would correspond to this graph of F’.
A)
B)
-4
-3
-2
-1
1
2
3
4
1 2 3 4 5 6 7
x
y
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
1 2 3 4 5
x
y
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
1 2 3 4 5
x
y
Chapter 6
C)
D)
11.
What is the antiderivative of
2
( ) –6 7f x x=+
?
A)
3
–2 7x x C++
B)
3
–6 7x x C++
C)
–12 7xC++
D)
–12xC+
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
1 2 3 4 5
x
y
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
1 2 3 4 5
x
y
Chapter 6
12.
What is the antiderivative of
6
()ht t
=
?
A)
2
6C
t
−+
B)
6ln tC+
C)
6 lnt t C+
D)
6
ln
tC
t+
13.
Find an antiderivative
()Gz
with
‘( ) ( )G z g z=
and
(0) 6G=
, given that
()g z z z=−
.
A)
2 3/ 2 6zz−+
B)
1
52z
−
C)
2 3/ 2
12 6
23
zz−+
D)
2 3/ 2
12
23
zz−
14.
Find an antiderivative
()Fx
of
( ) sinf x x=
such that
(0) 5F=
.
A)
cos x
B)
cos 4x−+
C)
cos 5x+
D)
cos 6x−+
15.
Find an antiderivative
()Fx
of
( ) 1
x
f x e=+
such that
(0) 0F=
.
A)
–1
x
ex+
B)
+1
x
ex+
C)
x
ex+
D)
x
xe x+
Chapter 6
16.
Find an antiderivative
()Fx
of
2
1
( ) 6fx x
=+
such that
(1)Fa=
, for some constant a.
A)
165xa
x
− + + −
B)
16xa
x
− + +
C)
3
265xa
x
−+ + −
D)
3
26xa
x
−++
17.
Evaluate
2
( 8 8)x x dx+−
.
A)
3
2
48
3
xx x C+ − +
B)
32
48x x x C+ − +
C)
3
2
88
3
xx x C
+ − +
D)
32
88x x x C+ − +
18.
Evaluate
2
13
dx
xx

+


.
A)
3
ln xC
x
++
B)
3
ln xC
x
−+
C)
23
29
C
xx
−+
D)
2
23
C
xx
++
Chapter 6
19.
Evaluate
18 xdx
.
A)
3/ 2
18xC+
B)
3/ 2
2
3xC+
C)
1/ 2
36xC+
D)
3/ 2
12xC+
20.
Find the indefinite integral
2
( 3 5)p p dp++
.
A)
32
35
32
pppC+ + +
B)
32
5
32
pp pC+ + +
C)
32
35p p p C+ + +
D)
3
2
35
2
pp p C+ + +
21.
Find the indefinite integral
cos d

.
A)
sin C
−+
B)
sin C
+
C)
cos C
+
D)
2
cos
2C
+
Chapter 6
22.
Find the indefinite integral
8t
e dt
.
A)
81
1
9
t
eC
++
B)
8t
eC+
C)
8
1
8
t
eC+
D)
81
1
81
t
eC
t
++
+
notation. difficulty: medium section: 6.2
23.
3 2 4 3 2
(12 9 6 3) 3 3 3 3x x x dx x x x x C+ − + = + − + +
A)
True
B)
False
functions. difficulty: easy section: 6.2
24.
(cos 3sin ) sin 3cost t dt t t C+ = − + +
A)
True
B)
False
functions. difficulty: medium section: 6.2
25.
12dx x C
x=+
A)
True
B)
False
functions. difficulty: medium section: 6.2
26.
(4 2) 2
4
x
xe
e dx x C+ = + +
A)
True
B)
False
functions. difficulty: medium section: 6.2
Chapter 6
27.
3
2
2
11
3
t
t dt C
tt

+ = − +


A)
True
B)
False
functions. difficulty: medium section: 6.2
28.
Find the indefinite integral
2kxdx
, where k is a constant.
A)
2
kx C+
B)
22
k x C+
C)
2
2
kx C+
D)
2
2kx C+
notation. difficulty: easy section: 6.2
29.
Find the indefinite integral
sin kd

, where k is a constant.
A)
1cos kC
k
+
B)
1cos kC
k
−+
C)
D)
cos kC
+
notation. difficulty: medium section: 6.2
Chapter 6
Page 14
30.
Find the antiderivative of
() a
f x b
x
=+
, where a and b are constants.
A)
2
2abx C
x++
B)
2
abx C
x
− + +
C)
lna x bx C++
D)
1ln x bx C
a++
31.
Find a possible antiderivative of
() bx
f x ae−
=
, where a and b are constants.
A)
1
1
bx
ae
bx
−+
−+
B)
1
1
bx
ae
b
−−
−+
C)
bx
ae
b
−
D)
bx
ae
b
−
−
32.
Find an antiderivative F of
( ) 2cos sinf x x x=+
satisfying
(0) 8F=
.
A)
( ) 2sin cos 9F x x x= − +
B)
( ) 2sin cosF x x x=−
C)
( ) 2sin cos 7F x x x= − + +
D)
2
2sin
( ) cos 7
2
x
F x x
= + +
Chapter 6
33.
Evaluate
11
10x dx
.
A)
12
5
6xC+
B)
12
xC+
C)
12
10
11 xC+
D)
12
11
12 xC+
34.
Evaluate
22xx dx
x
−+
.
A)
32
ln 2
32
xx
x x C

− + +



B)
2
2ln
2
xx x C− + +
C)
24
3
xC
x
++
D)
2
2
4
2
xxC
x
− + +
Chapter 6
35.
Find an antiderivative of
2
3
36
xxx
−+
.
A)
3
24
6 24
3
xC
xx
−−+
B)
3
2
36
3
xC
xx
− + +
C)
3
2
6
3ln
3
xxC
x
− + +
D)
3
2
3
3ln
3
xxC
− − +
36.
Evaluate
32
13x dx
.
37.
Use a definite integral to find the area under the graph of
2
2 3 5y x x= − + +
between
0x=
and
2x=
. Round to 2 decimal places.
38.
Find the average value of
2
( ) 3f x x=+
on the interval
0x=
to
3x=
.
39.
Evaluate
32
0( 4 4)x x dx++
.
Chapter 6
40.
Evaluate
22
2(3 4 2)x x dx
−−+
.
Ans:
24
Learning Objectives: Use the Fundamental Theorem to calculate definite integrals
41.
Evaluate
Ans:
0.44
Learning Objectives: Use the Fundamental Theorem to calculate definite integrals
exactly. difficulty: medium section: 6.3
3
3
1
1dx
. Round to 2 decimal places.
42.
Evaluate
/4
/4sin d
− 
. Round to 2 decimal places.
Ans:
0
Learning Objectives: Use the Fundamental Theorem to calculate definite integrals
43.
Evaluate
4
0
3
3dx
x+
. Round to 2 decimal places.
Ans:
2.54
44.
Propellant is leaking out from the pressurized fuel tanks of the space shuttle, causing the
pressure to decrease at a rate of
0.1
( ) 15 t
r t e−
=
psi per second at time t in seconds. By
how many psi has the pressure dropped during the first 30 seconds? Round to 2
decimal places.
Ans:
142.53
45.
Use the Fundamental Theorem of Calculus to determine the value of b if the area under
the graph of
2
( ) 3f x x=
between
0x=
and
xb=
is 8. Assume
0b
.
Ans:
2
Learning Objectives: Use the Fundamental Theorem to calculate definite integrals
exactly. difficulty: medium section: 6.3
Chapter 6
46.
A. Use the Fundamental Theorem to find
0
bx
e dx
−
for a constant
0b
.
B. Take the limit of your answer to part (A) as
b→
to find
0
x
e dx
−
.
47.
At time t hours after taking medication, the rate at which the medication is being
eliminated from the body is given by
0.2 0.3
( ) 30( )
tt
r t e e
−−
=−
mg/hr. Assuming that all
of the medication is eventually eliminated, how many mg was the original dose?
48.
Evaluate
2
38
0
x
xe dx
.
A)
72 1
16
e−
B)
72
16
e
C)
72
8
e
D)
72
e
Chapter 6
49.
Evaluate
5
7
4
( 4)x dx−
.
A)
1
7
B)
1
8
C)
88
54
88
−
D)
1
32
−
50.
Find
8
1
1dx
x
.
A)
1
7
B)
1
8
C)
1
7
−
D)
1
E)
This improper integral diverges.
section: 6.3
51.
The improper integral
12x
e dx
−
diverges.
A)
True
B)
False
Chapter 6
52.
The improper integral
1
1
pdx
x
converges for all positive values of p.
A)
True
B)
False
integrals exactly. difficulty: medium section: 6.3
53.
The improper integral
9
0
6dx
x
converges.
A)
True
B)
False
integrals exactly. difficulty: medium section: 6.3
54.
Compute
2
0
1
(3 )
R
dx
x
+
.
A)
1
3R+
B)
11
33R
−+
C)
3
1
3(3 )R+
D)
33
33
(3 )RR
−+
integrals exactly. difficulty: hard section: 6.3
55.
Fuel pressure in the fuel tanks of the space shuttle is decreasing at a rate of
0.1
( ) 17 t
r t e−
=
psi per second at time t in seconds. By how many total psi has the
pressure decreased during the first minute? Round to 2 decimal places.
Ans:
169.58
exactly. difficulty: medium section: 6.3