Solve the problem.
44)
Test marketing for a new health–food snack product in a selected area suggests that sales (in
thousands of dollars) will increase at a rate given by S‘(t) = 40 – 40e–0.16t, t months after an
aggressive national advertising campaign is begun. Find total sales during the second 12 months of
the campaign. (Round to the nearest thousand dollars.)
44)
A)
$511,000
B)
$449,000
C)
$267,000
D)
$693,000
45)
Divide the interval [0, 8] into four equal subintervals and draw in the corresponding left
rectangles.
45)
A)
B)
20
C)
D)
46)
(5 +x3)(4 –x2) dx
46)
A)
20 –5
3x3+x4–1
6x6+ C
B)
20x –5
3x3+5
3x4–1
6x6+ C
C)
20x –5
3x3+x4–1
6x6+ C
D)
20x –5
3x3+x4–1
6+ C
Provide an appropriate response.
47)
Given
3
1
f(x) dx = 4 and
3
1
g(x) dx = 2, use properties of definite integrals to evaluate
3
1
[2f(x) + 5g(x)] dx.
47)
A)
18
B)
24
C)
13
D)
54
Solve the problem.
48)
At the beginning of an advertising campaign for a new product in a city of 500,000 people, no one is
aware of the product. After 10 days, 100,000 people are aware of the product. If N = N(t) is the
number of people (in thousands) who are aware of the product t days after the beginning of the
advertising campaign, solve the following differential equation for N(t):
dN
dt = k(500 – N); N(0) = 0; N(10) = 100.
48)
A)
N(t) = 500(1 –e–0.022t)
B)
N(t) = 500(e0.022t – 1)
C)
N(t) = 500(1 –e0.022t)
D)
N(t) = 500(e–0.022t – 1)
49)
3
1
(2x3– 6x–2) dx
49)
A)
64
B)
36
C)
48.5
D)
52
Solve the problem.
50)
A single injection of a drug is administered to a patient. The amount Q in the body then decreases
at a rate proportional to the amount present, and for this particular drug the rate is 3% per hour.
Thus, dQ
dt = – 0.03Q with Q(0) =Q0, where t is time in hours. If the initial injection is 4 milliliters
[Q(0) = 4], about how many hours after the drug is given will there be 2 milliliters of the drug
remaining in the body? (Round answer to the nearest tenth of an hour.)
50)
A)
23.1 hours
B)
11.3 hours
C)
21.3 hours
D)
69.3 hours
Provide an appropriate response.
51)
Find the general solution for the differential equation y’ =7e3x
51)
A)
7
3 e3x + C
B)
7e3x + C
C)
21e3x + C
D)
1
3 e3x + C
52)
Find the particular solution for the differential equation y‘ = 4xe2x ; y(0) = 20.
52)
A)
y = 2xe2x –e2x + 21
B)
y = 4xe2x –e2x + 21
C)
y = 4xe2x –e2x + 22
D)
y = 2xe2x + 20
53)
7x6dx
(6 +x7)5
53)
A)
–1
6(6 +x7)6+ C
B)
–1
4(6 +x7)4+ C
C)
1
6(6 +x7)6+ C
D)
–7x6
(6 +x7)4+ C
54)
e
1
16x –5
x dx
(Express your answer in terms for e.)
54)
A)
8e2– 13
B)
8e2– 8
C)
16e2– 5
D)
8e2– 5
55)
The management of an oil company estimates that oil will be pumped from a producing field at a
rate given by R(t) =56
t + 7 for 0 t 20, where R(t) is the rate of production in thousands of barrels
per year, t years after pumping begins. How many barrels of oil, Q(t), will be produced the first five
years? (Round answer to the nearest thousand barrels.)
55)
A)
148 thousand barrels
B)
46 thousand barrels
C)
92 thousand barrels
D)
296 thousand barrels
Identify the rectangles shown in the graph as left rectangles, right rectangles, or neither.
56)
56)
A)
neither
B)
left rectangles
C)
right rectangles
57)
57)
A)
neither
B)
right rectangles
C)
left rectangles
58)
Find the average value of the function y = 2x4 over the interval [– 2, 2].
58)
A)
32
5
B)
128
5
C)
0
D)
16
5
59)
23x4–x–3 dx
59)
A)
4
3x7/3 – 3x–2+ C
B)
6
7x6/7 +1
2x2+ C
C)
6
7x7/3 – 2x–2+ C
D)
6
7x7/3 +1
2x2+ C
60)
Find the average value of the function y =e–x over the interval [ 0, 3]. Express an exact answer in
terms of e.
60)
A)
1 –e–3
3
B)
e– 1.5
C)
1 –e–3
D)
e– 3 – 1
3
61)
6 +x2
xdx
61)
A)
6
x2+x2+ C
B)
3
x2+x2+ C
C)
6 ln x+1
2x2+ C
D)
6 ln x+1
3x3+ C
Provide an appropriate response.
62)
Find f(x) if f'(x) =7
x4 and f(1) = 4.
62)
A)
f(x) = – 7
3x–3+19
3
B)
–28x–5+ 32
C)
–7
3x–3– 3
D)
–28x–5– 3
Find the integral.
63)
9e 0.2x dx.
63)
A)
18e0.2x + C
B)
45e0.2x + C
C)
9e0.2x + C
D)
9e0.2x + 1
0.2x + 1 + C
B
64)
1
3 – 2x dx
64)
A)
2 ln 3 – 2x + C
B)
1
2 ln 3 – 2x + C
C)
–1
2 ln 3 – 2x + C
D)
– 2 ln 3 – 2x + C
C
Evaluate the integral.
65)
16
3
3 x dx .
65)
A)
128
B)
288
C)
24
D)
192
A
A
Solve the problem.
66)
Find the amount A in an account (to the nearest dollar) after 5 years if dA
dt = rA, A(0) = 800, and
A(10) = 1800.
66)
A)
$1100
B)
$1000
C)
$919
D)
$1200
Find the integral.
67)
(2 + 2x)e(4x + 2x2) dx
67)
A)
1
2e(4x + 2x2)+ C
B)
e[2(4x + 2x2)] + C
C)
2e(4x + 2x2)+ C
D)
e[(1/2)(4x + 2x2)] + C
68)
x2
5x3+ 8 dx
68)
A)
15 ln (5x3– 8) + C
B)
1
5 ln 5x3+ 8 + C
C)
1
15 ln 5x3+ 8 + C
D)
15 ln 5x3+ 8 + C
69)
Find the general solution for the differential equation dx
dt = – 3x
69)
A)
x = Ce–3t
B)
x =e–3t + C
C)
t = Ce–3x
D)
t =e–3x + C
Solve the problem.
70)
The rate of expenditure for maintenance of a particular machine is given by M'(x) = 12x x2+ 5,
where x is time measured in years. Total maintenance costs through the second year are $105. Find
the total maintenance function.
70)
A)
M(x) = 12(x2+ 5)3/2 – 93
B)
M(x) = 4(x2+ 5)3/2 – 93
C)
M(x) = 4(x2+ 5)3/2 – 3
D)
M(x) = 12(x2+ 5)3/2 – 3
Provide an appropriate response.
71)
How large should n (n an integer) be chosen for Ln and Rn for the approximation of
4
1
(ln x + 1) dx to be within 0.05 of the true value?
71)
A)
n 5
B)
n 28
C)
n 90
D)
n 42
Approximate the area under the graph of f(x) and above the x–axis using n rectangles.
72)
f(x) =x2+ 2; interval [0, 5]; n = 5; compute L5
72)
A)
65
B)
40
C)
32
D)
66
Find the integral.
73)
2
t–7et dt
73)
A)
ln t– 7 + C
B)
2– 7et+ C
C)
2 ln t– 7 et+ C
D)
2 ln t–7et+ C
74)
9x– 5 dx .
74)
A)
–9
4x– 4 + C
B)
9
4x6+ C
C)
36
x4+ C
D)
– 45x– 6 + C
Evaluate the integral.
75)
b
0
9x8 dx
75)
A)
b7
B)
9b9
C)
b9
D)
1
9b9
Provide an appropriate response.
76)
Divide the interval [0, 8] into four equal subintervals and draw in the corresponding right
rectangles.
76)
A)
B)
29
C)
D)
77)
Find f(x) if f'(x) =3
x5 and f 1
2= 1.
77)
A)
–3
4x–4+ C
B)
–3
4x–4+ 13
C)
–4
5x–4+ 13
D)
–3
4x–4+5
4
Evaluate the integral.
78)
b
0
2exdx
78)
A)
2eb– 1
B)
2eb + 1
b + 1 –e
2
C)
2eb
D)
2eb–2
79)
The number of cheeseburgers (in thousands) sold each day by a chain of restaurants t days after the
end of an advertising campaign is given by S(t) = 9 – 10e–0.3t. What is the average number of
cheeseburgers sold each day during the first 7 days after the end of the advertising campaign?
79)
A)
4770 cheeseburgers
B)
4821 cheeseburgers
C)
5904 cheeseburgers
D)
3740 cheeseburgers
Evaluate the integral.
80)
e
1
4
xdx
80)
A)
–2e2
B)
–4
C)
0
D)
4
Solve the problem.
81)
A population of bacteria grows at a rate of P'(t) = 12 et where t is time in hours. Determine how
much the population increases from t = 0 to t = 3. Round your answer to two decimal places.
81)
A)
470.06
B)
235.03
C)
229.03
D)
241.03
Evaluate the integral.
82)
2
0
4x + 1
4x2+ 2x + 2 dx
(Round to three decimal places.)
82)
A)
1.199
B)
1.040
C)
2.398
D)
1.778
Solve the problem.
Find the integral.
83)
x3– 7
xdx
83)
A)
1
4x4– 7x2+ C
B)
1
3x3–7
x–2+ C
C)
1
3x3– 7 ln x+ C
D)
1
3x3+ 7 ln x+ C
84)
4x + 1
4x2+ 2x + 3 dx
84)
A)
1
2 ln 4x2+ 2x + 3 + C
B)
2
(4x2+ 2x + 3)2+ C
C)
1
2(4x2+ 2x + 3)2+ C
D)
2 ln 4x2+ 2x + 3 + C
Solve the problem.
85)
A newspaper is launching a new advertising campaign in order to increase the number of daily
subscribers. The newspaper currently (t = 0) has 26,000 daily subscribers and management expects
that number, S(t), to grow at the rate of S‘(t) = 80t1/2 subscribers per day, where t is the number of
days since the campaign began. How long (to the nearest day) should the campaign last if the
newspaper wants the number of daily subscribers to grow to 49,000?
85)
A)
33 days
B)
69 days
C)
44 days
D)
57 days
86)
The marginal price for a weekly demand of x bottles of cough medicine in a drug store is given by
p'(x) =
–13,300
(5x +40)2. Find the price–demand equation if the weekly demand is 125 when the price of a
bottle of cough medicine is $4. What is the weekly demand (to the nearest bottle) when the price is
$3?
86)
A)
p(x) = – 2,660
5x + 40 + 8; 98 bottles
B)
p(x) =5,320
5x + 40 ; 347 bottles
C)
p(x) =5,320
5x + 40 – 4; 144 bottles
D)
p(x) =2,660
5x + 40 ; 169 bottles
Provide an appropriate response.
87)
Find the particular solution for the differential equation dy
dx =1
2 + x ; y(0) = 3
87)
A)
y = ln 2 + x +ln 2
3
B)
y = ln 2 + x
C)
y = ln 2 + x + ln 2 + 3
D)
y = ln 2 + x – ln 2 + 3
88)
Find the particular solution for the differential equation y‘ = 4x + 7; y(0) = – 12.
88)
A)
y = 4x2+ 7x – 6
B)
y = 4x2+ 7x – 12
C)
y = 2x2+ 7x – 12
D)
y = 2x2+ 7x – 6
Find the integral.
89)
ln 8x
x dx
89)
A)
(ln 8x)2
2+ C
B)
(ln 8x)2+ C
C)
(ln 8x)2
8+ C
D)
(ln 8x)2
16 + C
90)
5(t2– 5t – 2) dt
90)
A)
5
2t3– 5t2– 2t + C
B)
5
3t3–5
2t2– 2t + C
C)
10t – 5 + C
D)
5t3– 5t2– 2t + C
Solve the problem.
91)
The rate of change in a person’s body temperature, with respect to the dosage of x milligrams of a
drug, is given by D'(x) =9
x + 4 . One milligram raises the temperature 2.1°C. Find the function
giving the total change.
91)
A)
D(x) =9 ln x + 4 + 2.1
B)
D(x) = ln 9
x + 4 – 2.1
C)
D(x) = ln 9
x + 4 – 12.4
D)
D(x) =9 ln x + 4 – 12.4
Identify the rectangles shown in the graph as left rectangles, right rectangles, or neither.
92)
92)
A)
neither
B)
left rectangles
C)
right rectangles
Provide an appropriate response.
93)
Find the average value of the function g(x) =32e0.04x over the interval [10, 30]. Round your
answer to two decimal places.
93)
A)
2.29
B)
1462.63
C)
73.13
D)
45.71
Answer Key
Testname: C13
Answer Key
Testname: C13