Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
2
1
2x+ 15
x2+ 5xdx =
1)
A)
2 ln(3) – 3 ln(7) – ln(8)
B)
ln(7) – 2 ln(3) + ln(8)
C)
2 ln(7) – 3 ln(2) + ln(6)
D)
3 ln(2) – ln(7) + ln(6)
E)
ln(7) – 3 ln(3) + 2 ln(6)
2)
1
0
1
x2– 4 dx =
2)
A)
4 ln(2) – 2 ln(3)
B)
3 ln(3)
C)
–1
4 ln(3)
D)
3 ln(3) –1
8 ln(2)
E)
1
4 ln(2) –1
4 ln(3)
3)
2
–1
x
x+ 2 dx =
3)
A)
2
3
B)
1
3
C)
4
D)
0
E)
3
4)
0
–
1
(x+4)2dx =
4)
A)
1
2
B)
–1
4
C)
integral diverges
D)
–1
2
E)
1
4
5)
–
1
(x+1)2dx =
5)
A)
2
B)
–1
C)
1
D)
integral diverges
E)
0
E
B
6)
Use the formula du
u u2+a2
= – 1
a ln u2+a2+a
u+C to find dx
x4x2+ 1
.
6)
A)
–4 ln 4x2+ 1 +1
4x+C
B)
–1
2 ln 4x2+ 1 +1
2x+C
C)
–1
4 ln 4x2+ 1 +1
4x+C
D)
–ln 4x2+ 1 +1
2x+C
E)
–2 ln 4x2+ 1 +1
2x+C
7)
The average value of f(x) =x2+ 1 over the interval 0, 4 is
7)
A)
17
3.
B)
17
4.
C)
76
3.
D)
17.
E)
19
3.
8)
The population of a town has exponential growth. In 1976 the population was 10,000 and in 1986 it
was 20,000. If N is the population at time t, where t is the number of years after 1976, and if you
assume than ln 2 = 0.7, then
8)
A)
N= 20,000e0.7t.
B)
N= 0.7e1/2.
C)
N= 10,000e0.07t.
D)
N= 10,000e0.7t.
E)
N= 20,000e0.07t.
9)
If dy
dx =ex+2y2 and y(–2) = – 1
2, then
9)
A)
y=ex+3– 2
2.
B)
y= – 1
ex+2+ 1
.
C)
y=ex+3
5–1
2.
D)
y=ex+2.
E)
y=ex+2–1
2.
10)
A rumor is circulating in a city whose population is 20,000. At the beginning of the month 50
people know of the rumor, and one week later 200 know of it. Assuming logistic growth, the
number N of people who know the rumor t weeks after the beginning of the month is given by
10)
A)
N=
20,000
1 + 399 99
399
t.
B)
N=
200
1 + 50 3
2
t.
C)
N= 20,000e50t.
D)
N=20,000
1 + 50e–200t.
E)
none of the above
11)
The average value of f(x) =x– 4 over the interval 5, 13 is
11)
A)
13
4.
B)
13
6.
C)
1
6.
D)
2.
E)
52
3.
12)
If dy
dx = 3x2y and y> 0, y(0) = 4, then y=
12)
A)
2 ln(x3).
B)
ex3+ 3.
C)
ln(x3).
D)
4ex3.
E)
x3+ 4.
D
13)
If dy
dx =ex
y2, then y=
13)
A)
33
ex
+
C
.
B)
ex+C.
C)
ex+C
3.
D)
3ex+C
3.
E)
2ex+C.
A
14)
3
2
x+ 1
x(x–1)2dx =
14)
A)
0
B)
4 – 2 ln(2)
C)
4 ln(3)
D)
ln(3) – 2 ln(2) + 1
E)
2 ln(3) – ln(2) + 3
D
B
15)
2
1
x2– 4
x(x2+ 4) dx =
15)
A)
ln(8) – ln(2) – ln(5)
B)
2 ln(8) – 3 ln(5)
C)
4 ln(2)
D)
ln(8) – 2 ln(5)
E)
ln(5) – ln(2)
16)
Use the formula du
u2+a2
= ln u+u2+a2+C to find dx
9x2+ 4
.
16)
A)
ln x+9x2+ 4 +C
B)
ln 9x+9x2+ 4 +C
C)
ln 3x+9x2+ 4 +C
D)
1
3 ln 3x+9x2+ 4 +C
E)
3 ln 3x+9x2+ 4 +C
17)
If dy
dx =1
xy where x> 0 and y> 0, then y=
17)
A)
ln(x) +C.
B)
ln(x) +C.
C)
ln x
C.
D)
2 ln(x) +C.
E)
1
2 ln(x) +C.
18)
0
1
(x+2)5dx =
18)
A)
–1
16
B)
0
C)
1
64
D)
integral diverges
E)
1
16
19)
2
1
xln(x) dx =
19)
A)
ln(2) – 4
B)
4 ln 2
C)
4 ln(2) –1
4
D)
ln(2)2
4
E)
2 ln(2) –3
4
E
20)
0
1
2x+ 1 dx =
20)
A)
1
2
B)
–1
C)
1
D)
integral diverges
E)
0
D
C
21)
3
0
3xex/3 dx =
21)
A)
3
B)
27
C)
0
D)
9
E)
6
22)
0
–1
xe–xdx =
22)
A)
1 +
e
B)
–1 –
e
C)
1 –
e
D)
–1
E)
1
23)
1
0
4x– 11
x2–x– 20 dx =
23)
A)
2 ln(5) – 2 ln(4)
B)
2 ln(5) – ln(4)
C)
3 ln(5) – 2 ln(4)
D)
4 ln(5) + 3 ln(4)
E)
3 ln(5) – 2 ln(5)
24)
0
–1
x
(x+2)2dx =
24)
A)
4 ln(2)
B)
2 ln(2) + 3
C)
ln(2)
D)
3 ln(2) – 4
E)
ln(2) – 1
25)
5
4
x x – 4 dx =
25)
A)
15
4
B)
18
5
C)
16
3
D)
4
E)
46
15
26)
Suppose 40% of a radioactive substance remains after 2 years. Assume that ln 0.4 = – 0.9 and find
the decay constant.
26)
A)
1.8
B)
0.225
C)
0.6
D)
3.6
E)
0.45
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
27)
Determine:
6
2
x2x– 3 dx
27)
28)
Determine if
–2
–
3
x– 4 dx converges or diverges. If it converges, to what number does it
converge?
28)
29)
Determine if
–
e2xdx converges or diverges. If it converges, to what number does it
converge?
29)
30)
Determine: x
x– 5 dx
30)
31)
Suppose the infant mortality rate in a given country is given by f’(t) =6(t+ 2)
(t+4)2, where t is
the number of years after 1950, and f(t) is the number of deaths per 10,000 births. Find f(t).
31)
32)
Suppose cash reserves (in hundred–thousands of dollars) are approximated by R(x) = 1 +
12x–x2, where x is the number of months since the reserve was started. What is the
average cash reserve for the first four months?
32)
33)
Use the tables to find: x
2x2–x– 6 dx
33)
34)
Determine: x5 ln xdx
34)
35)
The probability density function for the life span of an electronics part is f(t) = 0.08e–0.08t,
where t is the number of months in service. Find
6
0.08e–0.08tdt, the probability that
the part lasts longer than 6 months.
35)
36)
0
–3
3x
(x+ 4)2dx
36)
37)
Determine the value of the constant k so that
1
k
x3dx = 1.
37)
38)
Use the table below to determine
4
0
x
9 + 4xdx. Indicate the formula number
employed.
I. u du
a+bu
=2(bu – 2a)a+bu
3b2+C
II. u2 du
a+bu
=2(3b2u2– 4abu + 8a2)a+bu
15b3+C
III. u2±a2du =1
2u u2±a2±a2 ln u+u2±a2+C
IV. u2u2±a2du =u
8(2u2±a2) u2±a2–a4
8 ln u+u2±a2+C
38)
39)
Solve the differential equation dy
dx =x3
y2.
39)
40)
Use integration by parts to find: ln x
x4dx.
40)
41)
Evaluate the following integral by using partial fractions 2x2+ 4x– 4
x2– 1 dx
41)
11
42)
Use the table below to determine x2x2– 5 dx. Indicate the formula number employed.
I. u du
a+bu
=2(bu – 2a)a+bu
3b2+C
II. u2 du
a+bu
=2(3b2u2– 4abu + 8a2)a+bu
15b3+C
III. u2±a2du =1
2u u2±a2+a2 ln u +u2±a2+C
IV. u2u2±a2du =u
8(2u2±a2) u2±a2–a4
8ln u+u2±a2+C
42)
43)
The marginal revenue from the sales of x blenders per week is given by
R’(x) = 100 – 10 ln(x+ 5), R(0) = 50, where R(x) is the revenue in dollars. Find the revenue
function of R(x). Hint: x
x+ 5 dx =1 –3
x+ 5 dx
43)
44)
Use the table below to determine 4x2+ 25 dx. Indicate the formula number employed.
I. u du
a+bu
=2(bu – 2a)a+bu
3b2+C
II. u2 du
a+bu
=2(3b2u2– 4abu + 8a2)a+bu
15b3+C
III. u2±a2du =1
2u u2±a2±a2ln u+u2±a2+C
IV. u2u2±a2du =u
8(2u2±a2) u2±a2–a4
8 ln u+u2±a2+C
44)
45)
Find the average value of the function f(x) =x2+ 4 over the interval 1, 3 .
45)
46)
Use the tables to find: 3x
2x2– 9x– 5 dx
46)
12
47)
Either find the value of
2
1
(x+7)3dx or show that it diverges.
47)
48)
Determine: 5x2+ 17x– 6
x3+ 6x2dx
48)
49)
For a company, the annual sales S (in millions of dollars) t years from now are predicted to
be given by S= 4 2t+ 1, 0 t 4. Determine the average value of annual sales over the
next four years.
49)
50)
Determine if
1
1
x7dx converges or diverges. If it converges, to what number does it
converge?
50)
51)
4
1
2x– 4
x2(x+ 1) dx
51)
52)
Determine: 2x2– 5
(x– 1)(x+ 2)2dx
52)
53)
Solve the differential equation: dy
dx =e2x+1
y, y > 0
53)
13
54)
The rate of seepage from a toxic waste dump is R(t) =600
(t+1)2, where t is the time in years
and R(t) is the number of gallons of toxic waste. Find
0
600
(t+1)2dt , the total amount of
waste that will seep from the dump.
54)
55)
Solve the differential equation dy
dx =2y
x if x> 0 and y> 0.
55)
56)
If 55% of the initial amount of a radioactive sample remains after 10 hours, find the decay
constant and the half–life of the element.
56)
57)
Find the average value of the function f(x) = (x+1)2 over the interval 0, 3 .
57)
58)
Determine:
6
2
p p – 2 dp
58)
59)
Use the tables to find: x2dx
7 + 4x
59)
60)
Use integration by parts to find: x3
4 +x2dx.
60)
14
61)
Use the table below to determine 3x2
2 + 5xdx. Indicate the formula number employed.
I. u du
a+bu
=2(bu – 2a)a+bu
3b2+C
II. u2 du
a+bu
=2(3b2u2– 4abu + 8a2)a+bu
15b3+C
III. u2±a2du =1
2u u2±a2+a2ln u+u2±a2+C
IV. u2u2±a2du =u
8(2u2±a2) u2±a2–a4
8 ln u+u2±a2+C
61)
62)
Find the average value of the function y=f(x) =ex+ 1 on the interval –1, 2 . Use your
graphing calculator to graph both y=f(x) =ex+ 1 and y=
the average value of f(x) that you found on the same set of axes. Estimate the value of x
when these two curves intersect.
62)
63)
Determine: (x+ 1)exdx
63)
64)
The monthly sales of a dishwasher are estimated to increase at the rate of S’(t) = 5t t + 1
dishwashers per month, where t is time in months and S(t) is the number of dishwashers
sold each month. If there are 400 dishwashers sold in the eighth month (S(8) = 400), find
S(t).
64)
65)
Use the tables to find: 5 +x
3 +xdx
65)
15
66)
Use the table below to determine x2
4x– 1 dx. Indicate the formula number employed.
I. u du
a+bu
=2(bu – 2a)a+bu
3b2+C
II. u2 du
a+bu
=2(3b2u2– 4abu + 8a2)a+bu
15b3+C
III. u2±a2du =1
2u u2±a2±a2ln u+u2±a2+C
IV. u2u2±a2du =u
8(2u2±a2) u2±a2–a4
8 ln u+u2±a2+C
66)
67)
Determine:
4
1
6 –x
x(x+ 3) dx
67)
68)
Determine: x
e2xdx
68)
69)
Determine: 2x2+ 6x– 1
(x+ 2)(x2+ 1) dx
69)
70)
Determine: x4 ln(x) dx
70)
71)
Determine: x+ 3
x2+ 3x+ 2 dx
71)
16
72)
Either find the value of
4
1
(x+5)3dx or show that it diverges.
72)
73)
Solve the differential equation dy
dx =x3y if y> 0 and y= 2 when x= 0.
73)
74)
In a city whose population is 30,000, an outbreak of flu occurs. A researcher studying the
spread of the flu determines from the surrounding hospitals and the health department
that there where 600 infected persons one week ago. This week there are 1,000 infected
persons. Assuming logistic growth, estimate the number of infected people 3 weeks from
now.
74)
75)
Determine: xe5xdx
75)
76)
Use integration by parts to find: x2e3x+5dx.
76)
77)
The population N of a certain city follows the law of exponential growth given by N=N0
ekt where N0 and k are constants and t is the number of years past 1965. In 1965 the
population was 10,000 and in 1985 it was 30,000. What is the expected population in 2005?
77)
78)
An ecologist has determined that a local metro–park has food to support up to 200 rabbits
at any given time. Now, from observations, he estimates that there are 20 rabbits. One
week from now, he estimates that there are 30 rabbits. Assuming logistic growth, find the
logistic function that will model this rabbit population.
78)
17
79)
Suppose that the population of a city in 2000 was 25,400 and in 2005 was 27,300. If the
exponential law of growth is assumed, what is the expected population in 2010?
79)
80)
Determine:
2
–1
x
x2+ 7x+ 12 dx
80)
81)
Health officials are studying the spread of a disease in a city of 50,000 people. At the
beginning of the study, 1000 people were infected. One week later, 2000 people are
infected. Assuming logistic growth, determine the number N of people that are infected t
weeks after the beginning of the study.
81)
82)
Find the average value of the function y=f(x) =x2+ 3 on the interval –2, 5 . Use your
graphing calculator to graph both y=f(x) =x2+ 3 and y=
the average value of f(x) that you found on the same set of axes. Estimate the value of x
when these two curves intersect.
82)
83)
The marginal cost for producing x thousand pounds of cat food is given by C’(x) = (lnx)2, x
1. Find the cost function C(x).
83)
84)
Use the formula du
u u2+a2
= – 1
a ln u2+a2+a
u+C to find dx
x4x2+1
.
84)
85)
Suppose that the population of a city in 1992 was 52,400 and in 1996 was 55,300. If the
exponential law of growth is assumed, what is the expected population in 2007?
85)