Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
1)
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.
1)
2)
Find the producers’ surplus at a price level of p= $30 for the price–supply equation
p = S(x) = 14 + 0.0004x2.
2)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
3)
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.)
A)
10
B)
8.67
C)
– 8.67
D)
8
4)
xe2xdx
A)
1
2xe2x –1
4e2x + C
B)
1
2xe2x –1
2e2x + C
C)
1
2xe2x +e2x + C
D)
1
2xe2x +1
2e2x + C
Solve the problem.
5)
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)
6.25
B)
6.14
C)
0.23
D)
0.38
Find the indefinite integral using a table of integration formulas.
6)
x x4+ 81 dx
A)
1
4(x x4+ 81 + ln x +x4+ 81 ) + C
B)
1
4(x x4+ 81 + 81 ln x +x4+ 81 ) + C
C)
1
2(x x4+ 81 + ln x +x4+ 81 ) + C
D)
1
4(x2x4+ 81 + 81 ln x2+x4+ 81 ) + C
7)
1
x2–36 dx
A)
1
12 ln 6+ x
6– x + C
B)
ln x +x2–36 + C
C)
1
12 ln x –6
x +6+ C
D)
ln x +x2+36 + C
Use the Trapezoidal Rule to approximate the integral using the indicated value of n.
8)
1
01 +x3dx; n = 4, Round to two decimal places.
A)
1.02
B)
1.62
C)
1.12
D)
1.42
Solve the problem.
9)
Find the equilibrium quantity if the price–demand equation is p = D(x) = 23 –1
20x, and the
price–supply equation is p = S(x) = 8 +1
8,000x2.
A)
–600, 200
B)
200
C)
–600
D)
13
10)
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.)
10)
A)
$2743
B)
$2706
C)
$14,286
D)
$4375
Find the consumer’s surplus for the following demand function at the given point.
11)
D(x) =(x – 3)2; x =3
2
11)
A)
$3.25
B)
$4.33
C)
$7.28
D)
$4.50
Use Simpson’s rule to approximate the integral using the indicated value of n (so there are 2n subintervals).
12)
2
0(x4+6) dx ; n = 4, Write answer as whole number or reduced fraction.
12)
A)
361
24
B)
221
24
C)
221
12
D)
305
16
13)
1
01 +x3dx; n = 4, Round to two decimal places.
13)
A)
1.13
B)
1.11
C)
1.21
D)
1.03
Find the producer’s surplus for the following supply function at the given point.
14)
S(x) =x2+1; x = 1
14)
A)
$0.67
B)
$1
C)
–$1
D)
–$0.33
Use Simpson’s rule to approximate the integral using the indicated value of n (so there are 2n subintervals).
15)
3
1(8x +2) dx ; n = 4, Write answer as whole number or reduced fraction.
15)
A)
36
B)
72
C)
18
D)
30
Solve the problem.
16)
Find the total income produced by a continuous income stream in the first nine years if the rate of
flow is f(t) =5000.
16)
A)
$90,000
B)
$45,000
C)
$22.500
D)
$4500
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 ten days of operation.
17)
A)
$2397.90
B)
$2843.18
C)
$239.79
D)
$58,874.14
Evaluate using integration by parts.
18)
4
3ln 5x dx
18)
A)
1.4597
B)
5.0213
C)
14.2939
D)
2.8588
Set up a definite integral that represents the shaded area.
19)
y = f(x)
19)
A)
5
0f(x)dx
B)
3
1f(x)dx
C)
3
0f(x)dx
D)
20
1f(x)dx
Solve the problem.
20)
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.
20)
A)
0.474
B)
0.35714
C)
0.143
D)
0.263
Use Simpson’s rule to approximate the integral using the indicated value of n (so there are 2n subintervals).
21)
3
1
1
x dx; n = 4, Round to two decimal places.
21)
A)
1.24
B)
0.89
C)
1.10
D)
1.33
Find the consumer’s surplus for the following demand function at the given point.
22)
Find the consumers’ surplus at a price level of p= $7 for the price–demand equation
p = D(x) = 25 – 0.4x.
22)
A)
$720
B)
$29,250
C)
$4050
D)
$405
23)
The Lorenz curve for the income distribution in a certain country is given by f(x) =3
4x2+1
4x.
I) Find the Gini index of income concentration.
II) Use the answer found in I) to determine if the income of this country is more equally
distributed, less equally distributed, or distributed the same as a second country having an index of
income concentration of 0.2.
23)
A)
I) 0.25
II) less equally distributed
B)
I) 0.25
II) more equally distributed
C)
I) 0.33
II) more equally distributed
D)
I) 0.33
II) less equally distributed
24)
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.)
24)
A)
0.36
B)
0.33
C)
0.57
D)
0.21
25)
Find the consumers’ surplus and producers’ surplus for p = D(x) = 71 –1
10x and
p = S(x) = 35 +1
20x.
25)
A)
CS = $2880
PS = $1440
B)
CS = $2880
PS = $1660
C)
CS = $15,160
PS = $1440
D)
CS = $14,160
PS = $1440
Illustrate the integral graphically and describe what the integral represents in terms of areas.
26)
4
2ln 2x dx
26)
7
A)
The integral represents the area
between the graph of y = ln 2x and
the x axis from x = 2 to x = 4.
B)
The integral represents the negative of the
area between the graph of y = ln 2x and
the x axis from x = 2 to x = 4.
C)
The integral represents the area
between the graph of y = ln 2x and
the x axis from x = 2 to x = 4.
8
D)
The integral represents the area
between the graph of y = ln 2x and
the x axis from x = 2 to x = 4.
27)
1
0(x –5)exdx
27)
A)
The integral represents the negative of the
area between the graph of y = (x –5)ex and
the x axis from x = 0 to x = 1.
9
B)
The integral represents the negative of the
area between the graph of y = (x –5)ex and
the x axis from x = 0 to x = 1.
C)
The integral represents the area between the
graph of y = (x –5)ex and the x axis from
x = 0 to x = 1.
D)
The integral represents the area between
the graph of y = (x –5)ex and the x axis
from x = 0 to x = 1.
Solve the problem.
28)
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.)
28)
A)
0.39
B)
0.42
C)
0.54
D)
0.31
Use the Trapezoidal Rule to approximate the integral using the indicated value of n.
29)
3
1x2+2 dx ; n = 4, Round to three decimal places.
29)
A)
3.108
B)
2.477
C)
4.954
D)
9.908
Evaluate using integration by parts.
30)
x 3 – x dx
30)
A)
2
3x(3 – x)3/2 +4
15(3 – x)5/2 + C
B)
–2
3x(3 – x)3/2 +4
15(3 – x)5/2 + C
C)
–2
3x(3 – x)3/2 –2
5(3 – x)5/2 + C
D)
–2
3x(3 – x)3/2 –4
15(3 – x)5/2 + C
31)
x2e2xdx
31)
A)
x2
2e2x – 2xe2x + 1 + C
B)
x2
2e2x – 2xe2x + C
C)
x2
2e2x –x
2e2x +1
4e2x + C
D)
x2
2e2x – xe2x + C
Evaluate the definite integral to two decimal places.
32)
5
1e3(2 – t) dt
32)
A)
400.71
B)
6.70
C)
145.69
D)
27.29
Solve the problem.
33)
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.)
33)
A)
0.95
B)
0.65
C)
0.70
D)
0.35
Provide an appropriate response.
34)
Find the area bounded by f(x) = 3x2– 4 and y = 0 for 0 x 1.
34)
A)
1
B)
2
C)
3
D)
6
Set up a definite integral that represents the shaded area.
35)
y = g(x)
35)
A)
9
0g(x)dx
B)
7
3g(x)dx
C)
2
1g(x)dx
D)
2
0g(x)dx
Find the indefinite integral using a table of integration formulas.
36)
x
(4 + 5x)(5 + x) dx
36)
A)
–4
105 ln 4 + 5x +5
21 ln 5 + x + C
B)
–4
21 ln 4 + 5x +5
105 ln 5 + x + C
C)
4
21 ln 4 + 5x +5
105 ln 5 + x + C
D)
–4
21 ln 4 + 5x + C
Use Simpson’s rule to approximate the integral using the indicated value of n (so there are 2n subintervals).
37)
1
0x4dx; n = 4, Round to two decimal places.
37)
A)
0.34
B)
0.24
C)
0.20
D)
0.30
Find the indefinite integral using a table of integration formulas.
38)
25x2+75 dx
38)
A)
5
2x x2+75 +75 ln x +x2+75 + C
B)
1
2x x2+3+3 ln x +x2+3+ C
C)
5
2x x2+3+3 ln x +x2+3+ C
D)
1
2x25x2+75 +75 ln x +25x2+75 + C
Evaluate using integration by parts.
39)
xe8xdx
39)
A)
e8x(8x – 1) + C
B)
xe8x + C
C)
xe8x –e8x
8+ C
D)
xe8x
8–e8x
64 + C
Set up a definite integral that represents the shaded area.
40)
y = h(x)
40)
A)
9
0h(x)dx
B)
7
3h(x)dx
C)
7
0h(x)dx
D)
2
0h(x)dx
Evaluate the definite integral to two decimal places.
41)
18
0e0.06t e0.14(15 – t)dt
41)
A)
23.19
B)
77.89
C)
70.04
D)
53.64
Provide an appropriate response.
42)
Find the area bounded by f(x) =x2– 4x – 5 and y = x + 1.
42)
A)
23
6
B)
343
6
C)
301
6
D)
54
15
Solve the problem.
43)
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.
43)
A)
$3433.99
B)
$343.40
C)
$34,339.87
D)
$34.34
Find the indefinite integral using a table of integration formulas.
44)
x2+ 9dx
44)
A)
1
4(x x2+ 9 + ln x +x2+ 9 ) + C
B)
1
2x x2+ 9 + 9 ln x +x2+ 9 + C
C)
1
2(x x2+ 9 + ln x +x2+ 9 ) + C
D)
1
4(x x2+ 9 + 9 ln x +x2+ 9 ) + C
45)
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.)
45)
A)
21.67
B)
27
C)
32.33
D)
5.33
Solve the problem.
46)
The rate of growth of a microbe population is given by m'(x) = 30 x e2x, where x is time in days.
What is the net growth between day 3 and day 7?
46)
A)
222,613,544
B)
117,238,789
C)
222,613,533
D)
111,306,789
Evaluate using integration by parts.
47)
x2ln 4x dx
47)
A)
4
3ln34x –1
9x3+ C
B)
4
3ln34x –1
3x3+ C
C)
1
3ln34x –1
3x3+ C
D)
1
3ln34x –1
9x3+ C
Solve the problem.
48)
Find the total income produced by a continuous income stream in the first four years if the rate of
flow is f(t) = 500e0.03t. (Round answer to the nearest dollar.)
48)
A)
$2125
B)
$564
C)
$2486
D)
$18,792
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.
49)
y = x – 4 – ln x; 1 x 8
49)
A)
8.28
B)
2.14
C)
10.42
D)
–6.14
Use the Trapezoidal Rule to approximate the integral using the indicated value of n.
50)
2
04x2 dx ; n = 4, Write answer as a whole number or reduced fraction.
50)
A)
22
B)
15
C)
11
D)
32
3
Evaluate using integration by parts.
51)
1
0
x
x + 1 dx
51)
A)
–2.27
B)
–1.33
C)
0.94
D)
0.39
Provide an appropriate response.
52)
Find the area bounded by f(x) =x2– 3x + 7and g(x) = 2x + 7. (Round answer to two decimal places.)
52)
A)
16.13
B)
55.83
C)
20.83
D)
32.65
Solve the problem.
53)
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.
53)
A)
–0.584
B)
0.584
C)
0.292
D)
0.684
54)
The rate of water usage for a business, in gallons per day, is given by W(t) =697te–t, where t = the
number of hours since midnight. Approximately how many gallons of water does the business use
in the first 6 hours of the day?
54)
A)
688 gallons
B)
709 gallons
C)
12 gallons
D)
685 gallons
Provide an appropriate response.
55)
Find the area bounded by the parabolas y = 6x –x2 and y =x2– 2x. (Round answer to three
decimal places.)
55)
A)
– 21.333
B)
21.333
C)
21.667
D)
22
56)
Find the area between the graph of f(x) = 100 – 4x2 and the x–axis over the interval [–5, 5]. (Round
answer to two decimal places.)
56)
A)
66.67
B)
33.33
C)
666.67
D)
333.33
57)
Use the integral table to find x e3x dx .
57)
A)
x e3x
3+e3x
9+ C
B)
x e3x –e3x
3+ C
C)
x e3x
3–e3x
3+ C
D)
x e3x
3–e3x
9+ C
Evaluate using integration by parts.
58)
(x + 4) ln x dx
58)
A)
ln x –1
4x2– 4x + C
B)
1
2x2 ln x –1
4x2+ C
C)
1
2x2 ln x –1
4x2– 4x + C
D)
1
2x2 ln x –1
4x2+4x + C
59)
xx + 3 dx
59)
A)
2
3x(x + 3)3/2 –4
15(x + 3)5/2 + C
B)
2x(x + 3)3/2 – 4(x + 3)3/2 + C
C)
2
5x(x + 3)1/2 –4
5(x + 3)1/2 + C
D)
2
5x(x + 3)3/2 –4
5(x + 3)3/2 + C
Solve the problem.
60)
Find the total income produced by a continuous income stream in the first nine years if the rate of
flow is f(t) = 3300.
60)
A)
$18,000
B)
$9900
C)
$27,000
D)
$29,700
Find the indefinite integral using a table of integration formulas.
61)
2
5x (7x + 7) dx
61)
A)
1
7 ln x
7x + 7 + C
B)
1
7+x
7–1
7 ln 7x + 7 + C
C)
2
7 ln x
7x + 7 + C
D)
2
35 ln x
7x + 7 + C
Use the Trapezoidal Rule to approximate the integral using the indicated value of n.
62)
3
1(6x +3) dx ; n = 4, Write answer as a whole number or reduced fraction.
62)
A)
75
4
B)
60
C)
15
D)
30
Evaluate using integration by parts.
63)
5xex dx
63)
A)
5xex– 5ex+ C
B)
5ex– ex+ C
C)
5ex– 5xex+ C
D)
xex– 5ex+ C
Solve the problem.
64)
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.)
64)
A)
$3443
B)
$835
C)
$2308
D)
$2750
Provide an appropriate response.
65)
Use an integral table to find x3e2x dx.
65)
A)
x3e2x
2–3x2e2x
4–3e2x
8+ 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
4–3xe2x
4+ 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.
66)
y = 2 – xex, 0 x 3
66)
A)
–36.22
B)
37.27
C)
1.05
D)
–35.17
67)
5
1e–tdt
67)
A)
0.36
B)
0.99
C)
145.69
D)
147.41
Provide an appropriate response.
68)
Find the area (to three decimal places) bounded by f(x) =x2ex and q(x) = 4 –x2.
68)
A)
7.0
B)
7.333
C)
7.555
D)
7.676
69)
Find the area lying above the x–axis and under the parabola y = 4x –x2. (Round answers to three
decimal places.)
69)
A)
– 10.667
B)
10.667
C)
32
D)
11
Solve the problem.
70)
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.)
70)
A)
$129
B)
$2629
C)
$22,621
D)
$2379
Use Simpson’s rule to approximate the integral using the indicated value of n (so there are 2n subintervals).
71)
4
0
1
x2+ 1dx; n = 2, Round to two decimal places.
71)
A)
1.46,
B)
1.29
C)
1.20
D)
1.33
Provide an appropriate response.
72)
Find the area between the graph of f(x) = 50 +3x2 and the x–axis over the interval [–2, 4].
72)
A)
269
B)
372
C)
–92
D)
92
22
Use Simpson’s rule to approximate the integral using the indicated value of n (so there are 2n subintervals).
73)
6
2
1
x2+ 2 dx ; n = 4, Round to three decimal places.
73)
A)
1.006
B)
1.106
C)
1.059
D)
0.904
Solve the problem.
74)
Find the equilibrium price and quantity,producers’ surplus for p = D(x) = 71 –1
10x and
p = S(x) = 35 +1
20x.
74)
A)
p = 47
q = 180
B)
p = 47
q = 240
C)
p = 50
q = 240
D)
p = 47
q = 288
75)
Find the equilibrium price if the price–demand equation is p = D(x) = 23 –1
20x, and the
price–supply equation is p = S(x) = 8 +1
8,000x2.
75)
A)
$13.00
B)
$20.00
C)
$7.00
D)
$60.00
Provide an appropriate response.
76)
Find the area between the graph of f(x) =e0.2x + 2 and the x–axis over the interval 2 x 5.
(Round answer to two decimal places, if necessary.)
76)
A)
3.22
B)
8.97
C)
12.13
D)
20.13
23
Evaluate using integration by parts.
77)
4
26x ln x dx
77)
A)
9.48
B)
40.2
C)
55.2
D)
6.70
Find the equilibrium point.
78)
D(x) =(x –6)2, S(x) =x2+ 2x + 1
78)
A)
2
7, $32.65
B)
12, $36
C)
5
2, $12.25
D)
18, $144
Find the indefinite integral using a table of integration formulas.
79)
1
x2– 49dx
79)
A)
ln x +x2– 49 + C
B)
1
14 ln 7 + x
7 – x + C
C)
1
14ln x – 7
x + 7 + C
D)
ln x +x2+ 49 + C
Evaluate using integration by parts.
80)
x2ln 3x dx
80)
A)
1
3 x3 ln 3x –1
9 x3+ C
B)
ln 3x –1
3 x3+ C
C)
1
3x3 ln 3x –1
12 x4+ C
D)
1
3x3 ln 3x +1
9 x3+ C
Solve the problem.
81)
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.)
81)
A)
$47,359
B)
$4736
C)
$750
D)
$474
Provide an appropriate response.
82)
Use an integral table to find 9x6 ln x dx.
82)
A)
x7ln x
7–1
49 + C
B)
x7
9ln x
7–1
49 + C
C)
9x ln x – 9x + C
D)
9x7ln x
7–1
49 + C
Answer Key
Testname: C6
Answer Key
Testname: C6