Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
0
–
1
(x+4)2dx =
1)
A)
–1
2
B)
1
4
C)
1
2
D)
–1
4
E)
integral diverges
2)
1
0
4x– 11
x2–x– 20 dx =
2)
A)
2 ln(5) – 2 ln(4)
B)
3 ln(5) – 2 ln(5)
C)
4 ln(5) + 3 ln(4)
D)
3 ln(5) – 2 ln(4)
E)
2 ln(5) – ln(4)
3)
2
1
xln(x) dx =
3)
A)
ln(2) – 4
B)
2 ln(2) –3
4
C)
4 ln 2
D)
4 ln(2) –1
4
E)
ln(2)2
4
4)
Use the formula du
u u2+a2
= – 1
a ln u2+a2+a
u+C to find dx
x4x2+ 1
.
4)
A)
–4 ln 4x2+ 1 +1
4x+C
B)
–1
4 ln 4x2+ 1 +1
4x+C
C)
–2 ln 4x2+ 1 +1
2x+C
D)
–ln 4x2+ 1 +1
2x+C
E)
–1
2 ln 4x2+ 1 +1
2x+C
D
5)
0
1
2x+ 1 dx =
5)
A)
1
B)
1
2
C)
–1
D)
0
E)
integral diverges
E
B
6)
Suppose 40% of a radioactive substance remains after 2 years. Assume that ln 0.4 = – 0.9 and find
the decay constant.
6)
A)
1.8
B)
0.225
C)
0.45
D)
3.6
E)
0.6
7)
1
0
1
x2– 4 dx =
7)
A)
1
4 ln(2) –1
4 ln(3)
B)
–1
4 ln(3)
C)
3 ln(3) –1
8 ln(2)
D)
3 ln(3)
E)
4 ln(2) – 2 ln(3)
8)
If dy
dx =ex
y2, then y=
8)
A)
2ex+C.
B)
33
ex
+
C
.
C)
ex+C.
D)
ex+C
3.
E)
3ex+C
3.
9)
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
9)
A)
N= 10,000e0.7t.
B)
N= 20,000e0.07t.
C)
N= 10,000e0.07t.
D)
N= 0.7e1/2.
E)
N= 20,000e0.7t.
10)
If dy
dx =1
xy where x> 0 and y> 0, then y=
10)
A)
ln(x) +C.
B)
2 ln(x) +C.
C)
ln x
C.
D)
ln(x) +C.
E)
1
2 ln(x) +C.
11)
2
–1
x
x+ 2 dx =
11)
A)
4
B)
1
3
C)
2
3
D)
0
E)
3
12)
0
1
(x+2)5dx =
12)
A)
0
B)
integral diverges
C)
1
16
D)
1
64
E)
–1
16
13)
0
–1
x
(x+2)2dx =
13)
A)
4 ln(2)
B)
ln(2) – 1
C)
ln(2)
D)
2 ln(2) + 3
E)
3 ln(2) – 4
B
14)
If dy
dx = 3x2y and y> 0, y(0) = 4, then y=
14)
A)
x3+ 4.
B)
ln(x3).
C)
2 ln(x3).
D)
4ex3.
E)
ex3+ 3.
D
D
15)
2
1
2x+ 15
x2+ 5xdx =
15)
A)
2 ln(7) – 3 ln(2) + ln(6)
B)
2 ln(3) – 3 ln(7) – ln(8)
C)
3 ln(2) – ln(7) + ln(6)
D)
ln(7) – 3 ln(3) + 2 ln(6)
E)
ln(7) – 2 ln(3) + ln(8)
16)
3
0
3xex/3 dx =
16)
A)
9
B)
3
C)
6
D)
0
E)
27
17)
5
4
x x – 4 dx =
17)
A)
18
5
B)
16
3
C)
4
D)
15
4
E)
46
15
18)
The average value of f(x) =x– 4 over the interval 5, 13 is
18)
A)
13
4.
B)
2.
C)
1
6.
D)
13
6.
E)
52
3.
19)
–
1
(x+1)2dx =
19)
A)
2
B)
integral diverges
C)
0
D)
1
E)
–1
20)
The average value of f(x) =x2+ 1 over the interval 0, 4 is
20)
A)
19
3.
B)
17
3.
C)
17.
D)
17
4.
E)
76
3.
A
21)
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
21)
A)
N= 20,000e50t.
B)
N=
200
1 + 50 3
2
t.
C)
N=
20,000
1 + 399 99
399
t.
D)
N=20,000
1 + 50e–200t.
E)
none of the above
C
C
22)
Use the formula du
u2+a2
= ln u+u2+a2+C to find dx
9x2+ 4
.
22)
A)
1
3 ln 3x+9x2+ 4 +C
B)
ln 3x+9x2+ 4 +C
C)
ln x+9x2+ 4 +C
D)
ln 9x+9x2+ 4 +C
E)
3 ln 3x+9x2+ 4 +C
23)
0
–1
xe–xdx =
23)
A)
1 +
e
B)
–1
C)
1 –
e
D)
1
E)
–1 –
e
24)
2
1
x2– 4
x(x2+ 4) dx =
24)
A)
ln(8) – 2 ln(5)
B)
4 ln(2)
C)
ln(8) – ln(2) – ln(5)
D)
2 ln(8) – 3 ln(5)
E)
ln(5) – ln(2)
25)
3
2
x+ 1
x(x–1)2dx =
25)
A)
4 ln(3)
B)
4 – 2 ln(2)
C)
0
D)
ln(3) – 2 ln(2) + 1
E)
2 ln(3) – ln(2) + 3
26)
If dy
dx =ex+2y2 and y(–2) = – 1
2, then
26)
A)
y=ex+2–1
2.
B)
y=ex+2.
C)
y= – 1
ex+2+ 1
.
D)
y=ex+3
5–1
2.
E)
y=ex+3– 2
2.
Answer:
C
Explanation:
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
27)
Suppose the infant mortality rate in a given country is given by f’(t) =9(t+ 6)
(t+5)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).
27)
Answer:
f(t) = – 9
t + 5 + 9 ln t+ 5 +C
Explanation:
9
Answer:
D
Explanation:
28)
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
28)
29)
Solve the differential equation y’ =ex
y if y> 0.
29)
30)
Determine:
2
–1
x
x2+ 7x+ 12 dx
30)
31)
Evaluate the following integral by using partial fractions x2+ 2x– 1
x2–xdx
31)
32)
Determine if
0
(x2+ 1) dx converges or diverges. If it converges, to what number does it
converge?
32)
33)
0
–3
3x
(x+ 4)2dx
33)
34)
Determine:
3
1
(x+ 1) ln xdx
34)
35)
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
35)
36)
Solve the differential equation: 4y’ + 3x2= 0, y(1) =3
4
36)
37)
Find the present value of a continuous annuity at an annual rate of 6% for two years if the
payment at time t is at the annual rate of f(t) = 200 dollars. Express your answer in terms of
e.
37)
38)
The rate of change of the voting population of a city with respect to time t (in years) is
estimated to be V‘(t) =150 t3
t2+ 9 . Find V(t).
38)
39)
Use integration by parts to find: x3
4 +x2dx.
39)
11
40)
If 80% of the initial amount of a radioactive substance remains after 2 hours, find the decay
constant and the half–life of the substance. Express your answers in terms of natural
logarithms.
40)
41)
Use the table below to determine x24x2+ 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
41)
42)
Use the tables to find:
6
5
dx
x2x2–9
42)
43)
Determine: 2x2+ 6x– 1
(x+ 2)(x2+ 1) dx
43)
44)
Determine: x+ 2
x(x– 1)2dx
44)
12
45)
Determine:
6
2
x2x– 3 dx
45)
46)
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.
46)
47)
Determine: (x+ 1)exdx
47)
48)
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.
48)
49)
The marginal revenue for a company manufacturing x engines per week is given by
R‘(x) =4(x+ 9)
x2+ 6x+ 8 where R(x) is the revenue in hundred thousands of dollars. Find the
equation for R(x).
49)
50)
Evaluate the following integral by using partial fractions 2x2+ 4x– 4
x2– 1 dx
50)
13
51)
Find the present value of a continuous annuity at an annual rate of 9% for five years if the
payment at time t is at the annual rate of f(t) = 3000 dollars. Express your answer in terms
of e.
51)
52)
Determine: x
e2xdx
52)
53)
Determine if
–
x
(x2+2)4dx converges or diverges. If it converges, to what number does
it converge?
53)
54)
Determine: x+ 3
x2+ 3x+ 2 dx
54)
55)
Find the average value of the function f(x) = 2x+ 3 over the interval 2, 5 .
55)
56)
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
56)
57)
Find the average value of f(x) = 9x2+ 1 over the interval –2, 6 .
57)
14
58)
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
58)
59)
Systolic blood pressure is the blood pressure in a body when the heart contracts. An animal
undergoing laboratory testing was injected with an experimental drug. The animal’s
systolic blood pressure P (in millimeters of mercury) was found to be given by P=21
(t+1)2
+ 100; 0 t 4, where t was the number of hours after the injection. What was the average
systolic blood pressure over the first two hours after the injection?
59)
60)
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?
60)
61)
Use integration by parts to find: ln x
x3dx.
61)
62)
The monthly sales of a car are estimated to increase at the rate of S’(t) =ln t
t2cars per month,
where t is time in months and S(t) is the number of cars sold each month. If there are 80
cars sold in the first month (S(1) = 80), find S(t).
62)
63)
Suppose the infant mortality rate in a given country is given by f’(t) =4(t+ 1)
(t+3)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).
63)
15
64)
Use the tables to find: x2dx
7 + 4x
64)
65)
Use the formula du
u u2+a2
= – 1
a ln u2+a2+a
u+C to find dx
x4x2+1
.
65)
66)
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?
66)
67)
Determine: xe5xdx
67)
68)
Suppose the membership in a new country club is to be a maximum of 1,000 persons. One
year ago, the initial membership was 100 persons and now there are 500. Suppose the
enrollment follows a logistic function, i.e., N=M
1 +be–Ct . How many members will there
be three years from now?
68)
69)
Find the average value of the function f(x) =x2+ 4 over the interval 1, 3 .
69)
70)
Use the formula du
u(a+bu)=1
aln u
a+bu +C to find dx
x(2x– 3) .
70)
16
71)
Determine: 2x2– 5
(x– 1)(x+ 2)2dx
71)
72)
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).
72)
73)
Either find the value of
3
1
x+ 7 dx or show that it diverges.
73)
74)
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).
74)
75)
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.
75)
76)
Determine:
6
2
p p – 2 dp
76)
17
77)
Determine: (2x+ 7) ln (2x) dx
77)
78)
Determine:
4
1
6 –x
x(x+ 3) dx
78)
79)
Solve the differential equation: (x2+ 3x+ 1) dy
dx =y(2x+ 3), y(0) = 2
79)
80)
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.
80)
81)
Determine: z
4z+ 9 dz
81)
82)
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
82)
18