86)
Solve the differential equation y’ =ex
y if y> 0.
86)
87)
Evaluate the following integral by using partial fractions x2
(x–1)2dx
87)
88)
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).
88)
89)
The marginal revenue from the sales of x blenders per week is given by
R’(x) = 30 – 2ln(x+ 1), R(0) = 150, where R(x) is the revenue in dollars. Find the revenue
function of R(x). Hint: x
x+ 1 dx =1 –1
x+ 1 dx
89)
90)
Determine if
–
x
(x2+2)4dx converges or diverges. If it converges, to what number does
it converge?
90)
91)
Use the formula du
u a2–u2
= – 1
aln a+a2–u2
u+C to find 2dx
x1 – 4x2.
91)
92)
Solve the differential equation: 4y’ + 3x2= 0, y(1) =3
4
92)
19
93)
Either find the value of
3
1
x+ 7 dx or show that it diverges.
93)
94)
The monthly sales of a new tennis racket (in hundreds) are estimated to increase at the rate
of S’(t) = 16t3ln t rackets per month, where t is time in months and S(t) is the number of
rackets sold each month (in hundreds). If there are two hundred rackets sold in the first
month (S(1) = 2), find S(t).
94)
95)
The total amount of nuclear waste that will accumulate at a nuclear plant is given by
0
400te–0.03(b –t)dt. Find
0
400te–0.03(b –t)dt.
95)
96)
Find the average value of the function f(x) = 2x+ 3 over the interval 2, 5 .
96)
97)
The number of weekly sales of a product was found to be given by S(t) = 100te–t2+ 100,
where t is the number of days after the start of advertising on the radio. Find the average
weekly sales during the first 10 weeks of the advertising.
97)
98)
Use the tables to find:
2
1
dx
x29 –x2
98)
99)
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?
99)
20
100)
Determine:
3
1
(x+ 1) ln xdx
100)
101)
Use integration by parts to find: x
1 + 3xdx.
101)
102)
The amount A of glucose present in the bloodstream at any time t is governed by the
differential equation dA
dt =kR
k– A , where R and k are constants. Find A as a function of t.
102)
103)
Determine: xexdx
103)
104)
Determine if
0
(x2+ 1) dx converges or diverges. If it converges, to what number does it
converge?
104)
105)
Determine: ln(2x) dx
105)
106)
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.
106)
21
107)
A company performed a marketing study and concluded that there is a potential of 100,000
customers for its product. At the beginning of the study there were 40,000 customers that
used the product, and one year later there were 50,000 customers. Determine the number N
of customers t years after the beginning of the study if N follows logistic growth.
107)
108)
If 65% of the initial amount of a radioactive sample has decayed after 5 hours, find the
decay constant and the half life of the element.
108)
109)
The marginal cost for producing x thousand gallons of milk is given by C’(x) = 10x2e–x.
Find the cost function C(x).
109)
110)
Use the table below to determine x
1 –xdx. 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
110)
111)
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?
111)
22
112)
Determine: (2x+ 7) ln (2x) dx
112)
113)
Use the tables to find: 16 – 5x2
7xdx
113)
114)
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).
114)
115)
Use the formula du
(a2–u2)3/2 =u
a2a2–u2
+C to find dx
(5 – 9x2)3/2 .
115)
116)
The growth rate of a tumor satisfies dV
dt = 0.2Ve–0.1t, where t is the time in months and V is
the volume in cubic millimeters. If the initial volume is 1.86 millimeters, find V as a
function of t.
116)
117)
Determine: z
4z+ 9 dz
117)
23
118)
Use the table below to determine x
2x+ 3 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
118)
119)
Determine: 5x2+ 4
x3+ 4xdx
119)
120)
Suppose a bacteria population grows at a rate of P’(t) =t2e0.5t. Find P(t).
120)
121)
Solve the differential equation: (x2+ 3x+ 1) dy
dx =y(2x+ 3), y(0) = 2
121)
122)
Find the average value of f(x) = 9x2+ 1 over the interval –2, 6 .
122)
123)
The amount of a certain drug in a patient’s body t days after it has been administered is C(t)
= 6e–0.2t units. Determine the average amount of the drug present in the patient’s body for
the first five days after the drug has been administered.
123)
124)
Find the average value of f(x) = 5 –x3 over the interval –4, 2 .
124)
125)
The marginal revenue for a company manufacturing x computers per week is given by
R‘(x) =11(x+ 7)
x2+ 9x+ 8 where R(x) is the revenue in ten thousands of dollars. Find the equation
for R(x).
125)
126)
Evaluate the following integral by using partial fractions x2+ 2x– 1
x2–xdx
126)
127)
The marginal revenue for a company manufacturing x discs per week is given by
R‘(x) =6(x+ 5)
x2+ 3x+ 2 where R(x) is the revenue in thousands of dollars. Find the equation
for R(x).
127)
128)
Use the tables to find: 1
x3+ 9x2dx
128)
129)
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
129)
25
130)
Determine:
0
(3x– 7)exdx
130)
131)
Use the formula du
u2±a2
= ln u+u2±a2+C to find dx
9x2– 5
.
131)
132)
If 25% of the initial amount of a radioactive sample remains after 2.5 years, find the decay
constant and the half–life of the element.
132)
133)
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).
133)
134)
Use the tables to find:
6
5
dx
x2x2–9
134)
135)
The population of a certain city was 1,500,000 in the year 1580. The population grew by 5%
each year of the next 15 years. The population declined by 10% each year for the rest of the
century. What was the population of the city in the year 1600?
135)
136)
Solve the differential equation dy
dx =(x+3)5
y subject to y(–2) = 1, y> 0.
136)
26
137)
Use integration by parts to find: ln x
x3dx.
137)
138)
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.
138)
139)
Use the table below to determine 9x2– 4 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
139)
140)
Find y if dy
dx =y
x+ 4 and y(1) = 10, x> 0, y> 0.
140)
141)
Determine: x
3x+ 4
dx
141)
142)
Use the tables to find: x2+ 4
x2dx
142)
27
143)
Determine: x+ 2
x(x– 1)2dx
143)
144)
Determine: x– 4
x2+ 5x+ 6 dx
144)
145)
Solve the differential equation y’ =y2(2x+ 1) subject to the condition that y(–1) =1
5.
145)
146)
Suppose the infant mortality rate in a given country is given by f’(t) =8(t+ 7)
(t+1)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).
146)
147)
Solve the differential equation: y’ =5
y(x+ 1) , y > 0
147)
148)
Either find the value of
e–4xdx or show that it diverges.
148)
149)
Use the formula ueaudu =eau
a2(au – 1) +C to find
1
0
xe2xdx
149)
150)
A car moves along a straight road in such a way that its velocity in feet/sec is given by v(t)
= 3t16 –t2, where t is in seconds. Find the average velocity over the first 3 seconds.
150)
151)
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).
151)
152)
Determine if
–
x
x2+ 1 dx converges or diverges. If it converges, to what number does it
converge?
152)
153)
The marginal revenue from the sales of x vacuums per week is given by
R’(x) = 50 – 4 ln(x+ 2), R(0) = 100, where R(x) is the revenue in dollars. Find the revenue
function of R(x). Hint: x
x+ 2 dx =1 –2
x+ 2 dx
153)
154)
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).
154)
155)
Use the formula du
u(a+bu)=1
aln u
a+bu +C to find dx
x(2x– 3) .
155)
156)
Determine: 3
x2– 9 dx
156)
157)
Use the tables to find: dx
x2(5 + 4x)2
157)
29
158)
Determine if
4
1
xdx converges or diverges. If it converges, to what number does it
converge?
158)
159)
Evaluate the following integral by using partial fractions 2x3+ 9x2+ 21x+ 18
(x2+ 3x+5)2dx
159)
160)
Solve the differential equation: y’ = 3x2y
160)
161)
Determine the value of the constant k so that
0
k
(x+4)2dx = 1.
161)
162)
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
162)
163)
The marginal revenue from the sales of x blenders per week is given by
R’(x) = 80 – 7ln(x+ 3), R(0) = 300, where R(x) is the revenue in dollars. Find the revenue
function of R(x). Hint: x
x+ 3 dx =1 –3
x+ 3 dx
163)
164)
The growth rate of a cell satisfies dV
dt =kV2/3, where t is the time in months, V is the
volume in cubic millimeters, and k is a constant. If V= 0 when t= 0, find V as a function of
t.
164)
Answer Key
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Answer Key
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Answer Key
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Answer Key
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Answer Key
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Answer Key
Testname: C15