83)
Use the tables to find: 1
x3+ 9x2dx
83)
84)
Solve the differential equation: y’ =5
y(x+ 1) , y > 0
84)
85)
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.
85)
86)
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
86)
87)
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
87)
88)
Use the tables to find: dx
x2(5 + 4x)2
88)
89)
Solve the differential equation y’ =y2(2x+ 1) subject to the condition that y(–1) =1
89)
19
90)
Use the tables to find: x
2x2–x– 6 dx
90)
91)
Determine:
4
0
(3x– 7)exdx
91)
92)
Determine: x4 ln(x) dx
92)
93)
Solve the differential equation: dy
dx =e2x+1
y, y > 0
93)
94)
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.
94)
95)
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.
95)
96)
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.
96)
97)
Use the tables to find: 16 – 5x2
7xdx
97)
20
98)
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
98)
99)
Find the average value of the function f(x) = (x+1)2 over the interval 0, 3 .
99)
100)
Find y if dy
dx =y
x+ 4 and y(1) = 10, x> 0, y> 0.
100)
101)
Solve the differential equation dy
dx =x3
y2.
101)
102)
The marginal cost for producing x thousand gallons of milk is given by C’(x) = 10x2e–x.
Find the cost function C(x).
102)
103)
Determine: x
3x+ 4
dx
103)
21
104)
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.
104)
105)
Determine: ln(2x) dx
105)
106)
Determine: 5x2+ 17x– 6
x3+ 6x2dx
106)
107)
Use the tables to find: 5 +x
3 +xdx
107)
108)
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).
108)
109)
Determine: x5 ln xdx
109)
110)
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.
110)
111)
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.
111)
22
112)
Determine if
0
–
e2xdx converges or diverges. If it converges, to what number does it
converge?
112)
113)
Determine if
4
1
xdx converges or diverges. If it converges, to what number does it
converge?
113)
114)
Suppose a bacteria population grows at a rate of P’(t) =t2e0.5t. Find P(t).
114)
115)
Use the tables to find:
2
1
dx
x29 –x2
115)
116)
Solve the differential equation: y’ = 3x2y
116)
117)
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).
117)
118)
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.
118)
23
119)
Use the formula du
(a2–u2)3/2 =u
a2a2–u2
+C to find dx
(5 – 9x2)3/2 .
119)
120)
Determine if
1
1
x7dx converges or diverges. If it converges, to what number does it
converge?
120)
121)
Determine if
–
x
x2+ 1 dx converges or diverges. If it converges, to what number does it
converge?
121)
122)
Solve the differential equation dy
dx =(x+3)5
y subject to y(–2) = 1, y> 0.
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)
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).
124)
125)
Determine: x
x– 5 dx
125)
126)
Use the formula du
u2±a2
= ln u+u2±a2+C to find dx
9x2– 5
.
126)
127)
Use the formula du
u a2–u2
= – 1
aln a+a2–u2
u+C to find 2dx
x1 – 4x2.
127)
128)
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).
128)
129)
Evaluate the following integral by using partial fractions 2x3+ 9x2+ 21x+ 18
(x2+ 3x+5)2dx
129)
130)
Determine: 3
x2– 9 dx
130)
131)
Determine: xexdx
131)
132)
Use integration by parts to find: ln x
x4dx.
132)
25
133)
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?
133)
134)
Determine: x– 4
x2+ 5x+ 6 dx
134)
135)
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.
135)
136)
Use integration by parts to find: x
1 + 3xdx.
136)
137)
Determine: 5x2+ 4
x3+ 4xdx
137)
138)
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.
138)
139)
Use integration by parts to find: x2e3x+5dx.
139)
26
140)
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.
140)
141)
Solve the differential equation dy
dx =x3y if y> 0 and y= 2 when x= 0.
141)
142)
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.
142)
143)
Either find the value of
2
1
(x+7)3dx or show that it diverges.
143)
144)
Determine the value of the constant k so that
0
k
(x+4)2dx = 1.
144)
145)
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?
145)
146)
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?
146)
147)
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).
147)
148)
Use the formula ueaudu =eau
a2(au – 1) +C to find
1
0
xe2xdx
148)
149)
Find the average value of f(x) = 5 –x3 over the interval –4, 2 .
149)
150)
4
1
2x– 4
x2(x+ 1) dx
150)
151)
Either find the value of
3
e–4xdx or show that it diverges.
151)
152)
Evaluate the following integral by using partial fractions x2
(x–1)2dx
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)
Determine the value of the constant k so that
1
k
x3dx = 1.
154)
155)
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
155)
156)
Use the tables to find: x2+ 4
x2dx
156)
157)
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.
157)
158)
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.
158)
159)
Use the tables to find: 3x
2x2– 9x– 5 dx
159)
29
160)
Either find the value of
4
1
(x+5)3dx or show that it diverges.
160)
161)
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
161)
162)
Determine if
–2
–
3
x– 4 dx converges or diverges. If it converges, to what number does it
converge?
162)
163)
Solve the differential equation dy
dx =2y
x if x> 0 and y> 0.
163)
164)
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
164)
30
Answer Key
Testname: C15
3(4 +x2)(3/2) +C
31
Answer Key
Testname: C15
Answer Key
Testname: C15
33
Answer Key
Testname: C15
34
Answer Key
Testname: C15
Answer Key
Testname: C15