Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Find the producer’s surplus for the following supply function at the given point.
1)
Find the producers’ surplus at a price level of p= $30 for the price–supply equation
p = S(x) = 14 + 0.0004x2.
1)
2)
The Lorenz curve for the income distribution in a certain country is given by the function
f(x) =x3.2 . Find the Gini index of income concentration. Round the answer to three
decimal places and interpret the results.
2)
3)
The Lorenz curve for the income distribution in a small country is given by the function f(x) =x3.8.
Find the Gini index of income concentration. Round the answer to three decimal places.
A)
0.584
B)
–0.584
C)
0.684
D)
0.292
4)
After a person takes a pill, the drug contained in the pill is assimilated into the bloodstream. The
rate of assimilation minutes after taking the pill is R(t) =te–0.4t. Find the total amount of the drug
that is assimilated into the bloodstream during the first 15 minutes after the pill is taken. Round
your answer to 2 decimal places.
A)
0.38
B)
6.14
C)
0.23
D)
6.25
Use a graphing calculator to graph the equation over the indicated interval and find the area between the curve and the x
axis over that interval. Find the answer to two decimal places.
5)
y = x – 4 – ln x; 1 x 8
A)
2.14
B)
–6.14
C)
8.28
D)
10.42
Solve the problem.
6)
The length of telephone calls (in minutes) in a public telephone booth has the probability density
function:
f(t) =
1
6e–t/6 t 0
0otherwise
Determine the probability that a call selected at random will last longer than seven minutes.
(Round answer to two decimal places.)
A)
0.54
B)
0.42
C)
0.39
D)
0.31
7)
1
x2–49
dx
A)
1
14 ln 7+ x
7– x + C
B)
1
14 ln x –7
x +7+ C
C)
ln x +x2–49 + C
D)
ln x +x2+49 + C
8)
5
1
e3(2 – t) dt
A)
27.29
B)
400.71
C)
145.69
D)
6.70
9)
The length of telephone calls (in minutes) in a public telephone booth has the probability density
function:
f(t) =
1
6e–t/6 t 0
0otherwise
Determine the probability that a call selected at random will last between 2 and 6 minutes. (Round
answer to two decimal places.)
A)
0.65
B)
0.95
C)
0.35
D)
0.70
10)
y = g(x)
10)
A)
2
0
g(x)dx
B)
9
0
g(x)dx
C)
7
3
g(x)dx
D)
2
1
g(x)dx
Use the Trapezoidal Rule to approximate the integral using the indicated value of n.
11)
3
1
(6x +2) dx ; n = 4, Write answer as a whole number or reduced fraction.
11)
A)
28
B)
35
2
C)
14
D)
56
Provide an appropriate response.
12)
Find the area bounded by f(x) =x2– 3x + 7and g(x) = 2x + 7. (Round answer to two decimal places.)
12)
A)
20.83
B)
55.83
C)
16.13
D)
32.65
Solve the problem.
13)
Find the total income produced by a continuous income stream in the first nine years if the rate of
flow is f(t) = 3300.
13)
A)
$18,000
B)
$29,700
C)
$9900
D)
$27,000
Use Simpson’s rule to approximate the integral using the indicated value of n (so there are 2n subintervals).
14)
1
0
x4dx; n = 4, Round to two decimal places.
14)
A)
0.34
B)
0.30
C)
0.20
D)
0.24
Provide an appropriate response.
15)
Find the area bounded by f(x) = 3x2– 4 and y = 0 for 0 x 1.
15)
A)
2
B)
1
C)
3
D)
6
Solve the problem.
16)
The rate of water usage for a business, in gallons per day, is given by W(t) =648te–t, where t = the
number of hours since midnight. Approximately how many gallons of water does the business use
in the first 4 hours of the day?
16)
A)
707 gallons
B)
612 gallons
C)
589 gallons
D)
59 gallons
17)
The rate of flow of a continuous income stream (in thousands of dollars per day) is given by
f(t) =1
t + 1 . Find the total income produced during the first thirty days of operation.
17)
A)
$34,339.87
B)
$343.40
C)
$34.34
D)
$3433.99
18)
Find the area bounded by the parabolas y = 6x –x2 and y =x2– 2x. (Round answer to three
decimal places.)
18)
A)
21.667
B)
22
C)
21.333
D)
– 21.333
Use Simpson’s rule to approximate the integral using the indicated value of n (so there are 2n subintervals).
19)
1
0
1 +x3dx; n = 4, Round to two decimal places.
19)
A)
1.11
B)
1.21
C)
1.03
D)
1.13
20)
3
1
1
x dx; n = 4, Round to two decimal places.
20)
A)
1.33
B)
1.24
C)
1.10
D)
0.89
21)
Find the area lying above the x–axis and under the parabola y = 4x –x2. (Round answers to three
decimal places.)
21)
A)
11
B)
10.667
C)
32
D)
– 10.667
Use the Trapezoidal Rule to approximate the integral using the indicated value of n.
22)
3
1
x2+8 dx ; n = 4, Round to three decimal places.
22)
A)
4.391
B)
3.501
C)
7.001
D)
14.002
Provide an appropriate response.
23)
Use an integral table to find 9x6 ln x dx .
23)
A)
x7
9
ln x
7–1
49 + C
B)
x7ln x
7–1
49 + C
C)
9x7ln x
7–1
49 + C
D)
9x ln x – 9x + C
24)
Find the area between the graph of f(x) =x2– 4x and the x–axis over the interval –3 x 2.
(Round answer to two decimal places.)
24)
A)
32.33
B)
21.67
C)
27
D)
5.33
Provide an appropriate response.
Find the indefinite integral using a table of integration formulas.
25)
x x4+ 81 dx
25)
A)
1
2(x x4+ 81 + ln x +x4+ 81 ) + C
B)
1
4(x x4+ 81 + 81 ln x +x4+ 81 ) + C
C)
1
4(x x4+ 81 + ln x +x4+ 81 ) + C
D)
1
4(x2x4+ 81 + 81 ln x2+x4+ 81 ) + C
Evaluate using integration by parts.
26)
4
2
6x ln x dx
26)
A)
55.2
B)
9.48
C)
6.70
D)
40.2
Use Simpson’s rule to approximate the integral using the indicated value of n (so there are 2n subintervals).
27)
2
0
(x4+3) dx ; n = 4, Write answer as whole number or reduced fraction.
27)
A)
149
12
B)
149
24
C)
241
24
D)
209
16
Find the indefinite integral using a table of integration formulas.
28)
x
(4 + 5x)(5 + x) dx
28)
A)
–4
21 ln 4 + 5x +5
105 ln 5 + x + C
B)
–4
21 ln 4 + 5x + C
C)
–4
105 ln 4 + 5x +5
21 ln 5 + x + C
D)
4
21 ln 4 + 5x +5
105 ln 5 + x + C
Evaluate using integration by parts.
29)
1
0
x
x + 1 dx
29)
A)
0.39
B)
–2.27
C)
0.94
D)
–1.33
Solve the problem.
30)
Find the interest earned at 5% compounded continuously for two years by a continuous income
stream with rate flow of f(t) = 1250. (Round answer to the nearest dollar.)
30)
A)
$2629
B)
$22,621
C)
$2379
D)
$129
Set up a definite integral that represents the shaded area.
31)
y = h(x)
31)
A)
7
0
h(x)dx
B)
7
3
h(x)dx
C)
2
0
h(x)dx
D)
9
0
h(x)dx
Solve the problem.
32)
Find the future value at 8% interest compounded continuously for five years for the continuous
income stream with rate of flow f(t) = 560. (Round answer to the nearest dollar.)
32)
A)
$2308
B)
$3443
C)
$2750
D)
$835
Evaluate using integration by parts.
33)
x 8 – x dx
33)
A)
–2
3x(8 – x)3/2 +4
15 (8 – x)5/2 + C
B)
–2
3x(8 – x)3/2 –4
15 (8 – x)5/2 + C
C)
2
3x(8 – x)3/2 +4
15 (8 – x)5/2 + C
D)
–2
3x(8 – x)3/2 –2
5(8 – x)5/2 + C
Use a graphing calculator to graph the equation over the indicated interval and find the area between the curve and the x
axis over that interval. Find the answer to two decimal places.
34)
y = 2 – xex, 0 x 3
34)
A)
37.27
B)
–36.22
C)
1.05
D)
–35.17
Find the indefinite integral using a table of integration formulas.
35)
16x2+96 dx
35)
A)
2x x2+6+6 ln x +x2+6+ C
B)
1
2x x2+6+6 ln x +x2+6+ C
C)
2x x2+96 +96 ln x +x2+96 + C
D)
1
2x16x2+96 +96 ln x +16x2+96 + C
Find the consumer’s surplus for the following demand function at the given point.
36)
Find the consumers’ surplus at a price level of p= $7 for the price–demand equation
p = D(x) = 25 – 0.4x.
36)
A)
$405
B)
$720
C)
$29,250
D)
$4050
37)
Find the future value at 9% interest compounded continuously for five years for the continuous
income stream with rate of flow f(t) = 750. (Round answer to the nearest dollar.)
37)
A)
$4736
B)
$750
C)
$47,359
D)
$474
Provide an appropriate response.
38)
Use an integral table to find x3e2x dx.
38)
A)
x3e2x
2–3x2e2x
4–3xe2x
4+ C
B)
x3e2x
2–3x2e2x
4–3xe2x
4–3e2x
8+ C
C)
x3e2x
2+3x2e2x
4–3xe2x
4–3e2x
8+ C
D)
x3e2x
2–3x2e2x
4–3e2x
8+ C
39)
Find the area between the graph of f(x) =x2 and the x–axis over the interval [1, 3]. (Round answer
to two decimal places.)
39)
A)
8
B)
8.67
C)
10
D)
– 8.67
Solve the problem.
40)
The rate of flow of a continuous income stream (in thousands of dollars per day) is given by
f(t) =1
t + 1 . Find the total income produced during the first ten days of operation.
40)
A)
$239.79
B)
$58,874.14
C)
$2843.18
D)
$2397.90
41)
xe8x dx
41)
A)
xe8x + C
B)
xe8x –e8x
8+ C
C)
e8x(8x – 1) + C
D)
xe8x
8–e8x
64 + C
Solve the problem.
42)
The life expectancy (in years) of a certain type of computer chip is a continuous random variable
with probability density function:
f(x) =
4
(x +4)2x 0
0otherwise
Find the probability that a randomly selected chip will last from three to seven years. (Round
answer to two decimal places.)
42)
A)
0.57
B)
0.36
C)
0.21
D)
0.33
Use Simpson’s rule to approximate the integral using the indicated value of n (so there are 2n subintervals).
43)
3
1
(10x +3) dx ; n = 4, Write answer as whole number or reduced fraction.
43)
A)
46
B)
92
C)
115
3
D)
23
11
Provide an appropriate response.
44)
Find the area (to three decimal places) bounded by f(x) =x2ex and q(x) = 4 –x2.
44)
A)
7.333
B)
7.0
C)
7.676
D)
7.555
Solve the problem.
45)
The Lorenz curve for the income distribution in a small country is given by the function f(x) =x2.8.
Find the Gini index of income concentration. Round the answer to three decimal places.
45)
A)
0.474
B)
0.263
C)
0.143
D)
0.35714
Provide an appropriate response.
46)
Find the area bounded by f(x) =x2– 4x – 5 and y = x + 1.
46)
A)
23
6
B)
54
C)
301
6
D)
343
6
Solve the problem.
47)
Find the total income produced by a continuous income stream in the first nine years if the rate of
flow is f(t) =5000.
47)
A)
$22.500
B)
$45,000
C)
$4500
D)
$90,000
Find the consumer’s surplus for the following demand function at the given point.
48)
D(x) =(x – 3)2; x =3
2
48)
A)
$4.33
B)
$3.25
C)
$4.50
D)
$7.28
Solve the problem.
49)
The rate of flow of income from a continuous income stream is given by f(t) = 500e0.045t. Find the
future value of this income stream at 8% compounded continuously for six years. (Round answer
to the nearest dollar.)
49)
A)
$4375
B)
$2743
C)
$2706
D)
$14,286
Evaluate the definite integral to two decimal places.
50)
5
1
e–tdt
50)
A)
145.69
B)
0.36
C)
147.41
D)
0.99
51)
x2+ 9 dx
51)
A)
1
4(x x2+ 9 + ln x +x2+ 9 ) + C
B)
1
2(x x2+ 9 + ln x +x2+ 9 ) + C
C)
1
4(x x2+ 9 + 9 ln x +x2+ 9 ) + C
D)
1
2x x2+ 9 + 9 ln x +x2+ 9 + C
52)
(x + 4) ln x dx
52)
A)
1
2x2 ln x –1
4x2+4x + C
B)
1
2x2 ln x –1
4x2– 4x + C
C)
1
2x2 ln x –1
4x2+ C
D)
ln x –1
4x2– 4x + C