Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine whether the improper integral is convergent or divergent.
1)
0
–
e8x dx
1)
A)
Divergent
B)
Convergent
Let x be a continuous random variable over [a, b] with probability density function f. Then the median of the x–values is
that number m for which
m
af(x) dx =1
2.
Find the median.
2)
f(x) =3e–3x, [0, )
2)
A)
1
3ln 2
B)
1
C)
1
2ln 3
D)
1
3
Find the consumer surplus at the equilibrium point.
3)
D(x) =ex + 1; x = 0
3)
A)
$3
B)
$7
C)
$1
D)
$0
Determine whether the integral is convergent or divergent.
4)
9e–4x dx
4)
A)
Convergent
B)
Divergent
Solve the problem.
5)
The time of a telephone call (in minutes) to a certain town is a continuous random variable with a
probability density function defined by f(x) =3x–4 for [1, ). Find the probability: P(3 x 5).
5)
A)
0.0914
B)
0.0290
C)
0.6934
D)
0.2780
6)
Capital value is defined as
0R(t)e–ktdt where k is the annual rate of interest compounded
continuously. Find the capital value of an asset that produces $5000 yearly income at 4%
compounded continuously. Round to the nearest dollar.
6)
A)
$125,000
B)
$150,000
C)
$100,000
D)
$130,000
7)
In 1980 the world use of oil was 5,000,000,000 barrels and the demand for oil was growing
exponentially at the rate of 15% per year. If the demand for oil continues at this rate, how many
barrels of oil will the world use from 1980 to 2000?
7)
A)
636 billion barrels
B)
428 billion barrels
C)
98 billion barrels
D)
267 billion barrels
8)
Find the demand function q = D(x), given that E(x) =7 for all x > 0.
8)
A)
q =C1ex7, where ln C1= C
B)
q =C1
x7, where ln C1= C
C)
q =C1(ex7– 1), where ln C1= C
D)
q =C1
x1/7, where ln C1= C
Let x be a continuous random variable over [a, b] with probability density function f. Then the median of the x–values is
that number m for which
m
af(x) dx =1
2.
Find the median.
9)
f(x) =3
16x2, [–2, –2]
9)
A)
0
B)
1
2
C)
1
D)
3
32
Solve the problem.
10)
Find the amount of the following continuous money flow:
R(t) =t2 , k =6%, T =30 years
10)
A)
$48,422.66
B)
$78,422.66
C)
$24,348.59
D)
$36,673.25
Determine if the function is a solution to the given differential equation.
11)
y = –5e2x + xe2x; y’‘ + 2y’ + 1 = 0.
11)
A)
Yes
B)
No
Determine whether the function is a probability density function over the given interval.
12)
f(x) =3
67x2, [4, 8]
12)
A)
No
B)
Yes
13)
P(1.26 x 2.15)
13)
A)
0.088
B)
0.0862
C)
0.0898
D)
0.588
14)
P0= $1,580,000, t =15 yr, k =3.1%
14)
A)
$199,563,907.55
B)
$76,128,663.25
C)
$2,515,382.42
D)
$1,942,724.78
15)
P(0.11 x 2.11)
15)
A)
0.4383
B)
0.9388
C)
0.4423
D)
0.4388
16)
D(x) =30 – x; x = 5
16)
A)
$1.20
B)
$3.70
C)
$4.80
D)
$2.50
17)
The function f(x) = 80x4 is a probability density function on the interval [–a, a]. What is a?
17)
A)
0.25
B)
1
C)
0.75
D)
0.50
18)
y=54x2– 16x
18)
A)
18x3– 8x2+ C
B)
54x3– 8x2+ C
C)
18x3– 16x2+ C
D)
54x3– 16x2+ C
Approximate the integral.
19)
1
4
1 +x2 dx
19)
A)
–5.5452
B)
6.2832
C)
0.8637
D)
3.1416
20)
Find the accumulated present value of the following continuous money flow:
R(t) =et , k =7%, T =10 years
20)
A)
$314,670.89
B)
$11,760.24
C)
$20,585.95
D)
$314,656.70
21)
1
dx
x2.936
21)
A)
Divergent
B)
1
2.936
C)
1
3.936
D)
1
1.936
22)
f(x) =1
4, [3, 8]
22)
A)
Yes
B)
No
23)
D(x) =2–4x; x = 1
23)
A)
$0
B)
$2
C)
$4
D)
$8
Solve the problem.
24)
What should P0 be so that the amount of a continuous money flow over 10 years at interest rate
8.5%, compounded continuously, will be $40,000?
24)
A)
$1453.21
B)
$298.59
C)
$1018.07
D)
$2537.98
25)
A man takes a job as a record keeper at the age of 21. Assuming retirement at age 65 and annual
salary $3200 that is paid in a continuous money flow, what is the man‘s accumulated present
value? The current interest rate is 7%, compounded continuously.
25)
A)
$2100.99
B)
–$43,613.29
C)
$43,613.29
D)
$47,815.28
26)
y’ =3y–2; y = 3 when x =2
26)
A)
y = –30
B)
y =39x + 9
C)
y =33x + 3
D)
y = –318
27)
f(x) =1
2x, [3, 8]
27)
A)
No
B)
Yes
28)
D(x) =5x +2; x = 0
28)
A)
$2
B)
$5
C)
$7
D)
$0
6
29)
P0= $92,000, t =4 yr, k =3%
29)
A)
$194,764.00
B)
$250,081.93
C)
$103,729.71
D)
$349,017.45
30)
y– 12x2=14
30)
A)
4x3– 7x + C
B)
4x3+ 7x + C
C)
4x3+ 14x + C
D)
– 4x3+ 14x + C
Solve the problem.
31)
A woman accepts a position as president of a company at age 37. Assuming retirement at age 67
and annual salary $14,000 that is paid in a continuous money flow, what is the president’s
accumulated present value? The current interest rate is 7%, compounded continuously.
31)
A)
$175,508.71
B)
–$175,508.71
C)
$224,491.29
D)
$24,491.29
32)
–2
–
(e100x +x3)dx
32)
A)
Convergent
B)
Divergent
Solve the problem.
33)
The marginal cost for a product is given by C'(x) =2.5 –0.68x. Find the total cost function C(x),
assuming the fixed costs are $120, that is C(0) = $120.
33)
A)
C(x) =2.5x –0.68x2+120
B)
C(x) =120 –0.68x
C)
C(x) =2.5x –0.34x2+120
D)
C(x) =2.5x –x2+120
34)
dy
dx =7
y
34)
A)
y =7x + C, or y = – 7x + C
B)
y =14x + C
C)
y =14x + C
D)
y =14x + C, or y = – 14x + C
Determine if the function is a solution to the given differential equation.
35)
y =5x ln x + 7x – 6; y’‘ –1
x= 0.
35)
A)
Yes
B)
No
Solve the problem.
36)
Plutonium has a decay rate of 0.003% per year. Suppose that a nuclear accident causes plutonium
to be released into the atmosphere each year perpetually at the rate of 2 lb per year. What is the
limiting value of the radioactive buildup? Round to the nearest pound.
36)
A)
666,667 lb
B)
66,667 lb
C)
6667 lb
D)
667 lb
37)
In 1995, 2.3 million cubic feet of methane gas were added to the atmosphere, and the amount was
growing at a rate of 1% per year. How much methane will be in the atmosphere in the year 2000?
37)
A)
7.5 million cubic feet
B)
14 million cubic feet
C)
10.6 million cubic feet
D)
11.8 million cubic feet
38)
Find the amount of a continuous money flow in which $700 per year is being invested at 8.5%,
compounded continuously for 30 years.
38)
A)
$97,234.97
B)
$105,470.27
C)
$113,705.56
D)
$826,497.26
39)
Find the present value of $50,000 due 10 years later at 6.8%, compounded continuously.
39)
A)
$6879.29
B)
$25,330.85
C)
$172,249.78
D)
$25,897.48
Find the equilibrium point.
40)
D(x) = –3x +6, S(x) =3x +2
40)
A)
–2
3 , $0.03
B)
3, $–3
C)
2
3, $4
D)
–3, $15
For the given probability density function, over the stated interval, find the requested value.
41)
f(x) =1
ln 2·1
x, over the interval [1.5,4.9]. Find E(x2).
41)
A)
11.255
B)
10.88
ln 2
C)
10.605
ln 2
D)
2.65
ln 2
42)
f(x) =1
3x2, [–1, 2]
42)
A)
µ=5
4; 2=33
15; =355
15
B)
µ=7
12; 2=8
15; =230
15
C)
µ=1
2; 2=1
3; =3
3
D)
µ=5
4; 2=51
80; =255
20
43)
y =8, x =1, x =6
43)
A)
320
B)
448
C)
80
D)
40
44)
f(x) = 4, [2.25, 2.50]
44)
A)
µ= 2.375; 2= 0.021; = 0.144
B)
µ= 2.500; 2= 0.0048; = 0.069
C)
µ= 2.375; 2= 0.0052; = 0.072
D)
µ= 2.500; 2= 0.0049; = 0.070
45)
dy
dx =6y
45)
A)
y =6eCx
B)
y = Cex
C)
y = Ce6x
D)
y = Ce–6x
Let x be a continuous random variable over [a, b] with probability density function f. Then the median of the x–values is
that number m for which
m
af(x) dx =1
2.
Find the median.
46)
f(x) =1
8x, [0, 4]
46)
A)
3
2
B)
2
C)
4
D)
2 2
47)
023ex dx
47)
A)
Convergent
B)
Divergent
The capitalized cost c of an asset is computed by the formula
c =c0+L
0m(t)e–rt dt ,
where c0 is the initial cost of the asset, L is the lifetime (in years), r is the interest rate (compounded continuously), and
m(t) is the annual cost of maintenance. Find the capitalized cost under the given assumptions.
48)
c0=$800,000, r = 4%, m(t) =60,000, L =5
48)
A)
$1,071,903.87
B)
$1,311,903.87
C)
$1,231,903.87
D)
–$271,903.87
49)
P(–1.2 x 0.5)
49)
A)
0.5784
B)
0.1934
C)
0.5601
D)
0.5764
50)
The time of a telephone call (in minutes) to a certain town is a continuous random variable with a
probability density function defined by f(x) =3x–4 for [1, ). Find the probability: P(x 2).
50)
A)
0.2500
B)
0.1250
C)
0.8409
D)
0.8914
51)
Plutonium has a decay rate of 0.003% per year. Suppose Plutonium is released into the atmosphere
each year for 20 years at the rate of 2 pounds per year. What is the total amount of radioactive
buildup?
51)
A)
38.94 pounds
B)
42.55 pounds
C)
39.98 pounds
D)
39.14 pounds
Determine the domain of the probability function.
52)
The function f(x) = –32x + 8 is a probability density function on the interval
[0, b]. What is b?
52)
A)
1
B)
0.50
C)
0.38
D)
0.25
Solve the problem.
53)
The life span of a certain insect in days is uniformly distributed over the interval [20, 36]. What is
the standard deviation?
53)
A)
5.774
B)
4.33
C)
4.619
D)
10.392
54)
f(x) =1
2(e2– 1) e2x,[0, 1]
54)
A)
No
B)
Yes
Solve the problem.
55)
The time of a telephone call (in minutes) to a certain town is a continuous random variable with a
probability density function defined by f(x) = 3x–4 for [1, ). Find the probability: P(1 x 3).
55)
A)
0.8027
B)
0.6934
C)
0.3066
D)
0.9630
56)
A company installs 5,000 light bulbs. Each bulb has an average life of 500 hours with a standard
deviation of 100 hours. The life of each bulb is approximated by a normal curve. Find the number
of bulbs that can be expected to last between 290 hours and 540 hours.
56)
A)
1641 bulbs
B)
1639 bulbs
C)
3190 bulbs
D)
3188 bulbs
57)
f(x) =7x, [0, 6]
57)
A)
Yes
B)
No
58)
f(x) =1
8, [12, 20]
58)
A)
µ= 1.6; 2= 5.24; = 2.29
B)
µ= 15; 2= 5.34; = 2.31
C)
µ= 15.5; 2= 5.20; = 2.28
D)
µ= 16; 2= 5.33; = 2.31
59)
6
dx
x9/5
59)
A)
Convergent
B)
Divergent
60)
The mean clotting time of blood is 7.35 seconds, with a standard deviation of 0.35 seconds. What is
the probability that blood clotting time will be less than 7 seconds? Assume the distribution is
normal.
60)
A)
0.14
B)
0.16
C)
0.15
D)
0.84
61)
dy
dx =x
8y
61)
A)
y = ± x2
8+ C
B)
y =1
4x + C
C)
y = ± x2
4+ C
D)
y =1
2x + C
62)
D(x) = 4 –x, 0 x 4; S(x) =x– 2
62)
A)
(6, $3)
B)
(3, $6)
C)
(3, $1)
D)
(1, $18)
63)
f(x) =1
2x, [0, 2]
63)
A)
µ=1
2; 2=1
3; =3
3
B)
µ=4
3; 2=2
9; =2
3
C)
µ=8
3; 2=4
3; =2 3
3
D)
µ=2
3; 2=1
3; =3
3
64)
y =25 –x2, x = 0, x =5
64)
A)
250
3
B)
500
3
C)
10
D)
100
65)
y = x +3, x = –3, x =2
65)
A)
8
B)
5
2
C)
25
D)
125
3
66)
The capitalized cost c of an asset for unlimited lifetimes is computed by the formula
c =c0+
0m(t)e–rt dt ,
where c0 is the initial cost of the asset, r is the interest rate (compounded continuously), and m(t) is
the annual cost of maintenance. Find the capitalized cost under the given c0=$800,000,r = 5%,
m(t) =50,000. Round to the nearest dollar.
66)
A)
$1,550,000
B)
$1,800,000
C)
$1,300,000
D)
$2,050,000
67)
Find the area, if it exists, of the region bounded by the graph of y =3e–3x and the lines y = 0 and
x = 0.
67)
A)
1
B)
3
C)
Divergent
D)
3e–3
68)
y =1
x, x = 1, x =9
68)
A)
1
9
B)
8
9
C)
4
9
D)
ln 9
69)
A number x is selected at random from the interval [18, 57]. The probability density function for x is
given by
f(x) =1
39, for 18 x 57.
Find the probability that a number selected is in the subinterval [25, 41].
69)
A)
352
13
B)
16
39
C)
4
15
D)
33
70)
–
x5e–x6 dx
70)
A)
0
B)
Divergent
C)
1
6
D)
–1
3
71)
Find the interest earned when $22,082 is invested at 6% compounded continuously for 5 years.
71)
A)
$15,367.97
B)
$7725.58
C)
$14,322.61
D)
$1515.86
72)
Find the area, if it exists, of the region bounded by y =3xe–x2 and the lines x = 0 and y = 0.
72)
A)
3
2
B)
e–2
C)
3
D)
e–9
73)
P = $10,000, t =10 yr, k =9%
73)
A)
$24,596.03
B)
$9139.31
C)
$4065.70
D)
$4106.46
74)
y=33x2
74)
A)
x3
3+ C
B)
11x3+ C
C)
x3+ C
D)
33x3+ C
75)
0
–
5
(x – 1)2 dx
75)
A)
Divergent
B)
15
C)
5
D)
10
76)
y =4e2x + xe2x; y’‘ – 2y’ + 1 = 0.
76)
A)
No
B)
Yes
77)
9ln x dx
77)
A)
Divergent
B)
0
C)
9 ln 9
D)
1
9
78)
D(x) =4x, S(x) =x
78)
A)
(4, $2)
B)
(4, $4)
C)
(2, $2)
D)
(2, $4)
Solve the problem.
79)
The life (in months) of an automobile battery has a probability density function defined by
f(x) =1
4e–x/4 for x in [0, ). Find the probability that the life of a randomly selected battery is
greater than 6 years.
79)
A)
0.0558
B)
0.2231
C)
0.7769
D)
0.1942
80)
S(x) =5–4x; x = 1
80)
A)
$1
B)
$0.80
C)
$1.25
D)
–$2
Solve the problem.
81)
A machine produces bolts with an average diameter of 0.30 inches and a standard deviation of 0.01
inches. What is the probability that a bolt will have a diameter greater than 0.32 inches? Assume
the distribution is normal.
81)
A)
0.01
B)
0.98
C)
0.03
D)
0.02
82)
f(x) =3
8x2,[0, 2]
82)
A)
Yes
B)
No
The capitalized cost c of an asset is computed by the formula
c =c0+L
0m(t)e–rt dt ,
where c0 is the initial cost of the asset, L is the lifetime (in years), r is the interest rate (compounded continuously), and
m(t) is the annual cost of maintenance. Find the capitalized cost under the given assumptions.
83)
c0=$700,000, r = 4%, m(t) = 25,000 + 100t, L = 10
83)
A)
$915,921
B)
$909,897
C)
$898,625
D)
$906,874
84)
Find the area, if it exists, of the region under the graph of y =13e–x over (–, e].
84)
A)
–197.002
B)
Divergent
C)
0.858
D)
–0.858
85)
y’ = 4x + 24; y = –16 when x = 0
85)
A)
y = 4x2+ 24x – 8
B)
y = 2x2+ 24x – 8
C)
y = 4x2+ 24x – 16
D)
y = 2x2+ 24x – 16
86)
S(x) = –3x2; x = 1
86)
A)
–$1
B)
–$2
C)
–$6
D)
$6
87)
P(x 0.97)
87)
A)
0.1660
B)
0.8315
C)
0.8078
D)
0.8340
Solve the problem.
88)
In 1990, the world usage of natural gas was 72.137 billion cubic feet, and the demand for natural
gas was growing exponentially at the rate of 4% per year. If the demand continues to grow at this
rate, how many cubic feet of gas will the world use from 1990 to 2000?
88)
A)
973 billion cubic feet
B)
741 billion cubic feet
C)
923 billion cubic feet
D)
887 billion cubic feet
Find k such that the function is a probability density function over the given interval. Then write the probability density
function.
89)
f(x) = kx; [0, 5]
89)
A)
2
5; f(x) =2
5x
B)
1
25; f(x) =1
25x
C)
1
5; f(x) =1
5x
D)
2
25; f(x) =2
25x
90)
f(x) = kx; [2, 4]
90)
A)
1
8; f(x) =1
8x
B)
1
16; f(x) =1
16x
C)
1
6; f(x) =1
6x
D)
1
12; f(x) =1
12x
91)
f(x) =1
2x, [1, e2]
91)
A)
No
B)
Yes
92)
y =x2, x = 0, x =4
92)
A)
64
B)
64
3
C)
1024
5
D)
256
Solve the problem.
93)
A dart is thrown at a number line in such a way that it always lands in the interval [0, 7]. Let x be
the number the dart hits. Suppose the probability density function for x is given by
f(x) =3
343x2, for 0 x 7.
Find P(2 x 5), the probability that it lands in [2, 5].
93)
A)
0.03
B)
0.43
C)
0.34
D)
0.06
94)
0
–
11e5x dx
94)
A)
Divergent
B)
Convergent
Solve the problem.
95)
The world reserves of oil are about 923 thousand million barrels. In 1990 the world use of oil was
6,600 million barrels and the growth rate for the use of oil was 10%. Assuming that this growth rate
continues and that no new reserves are discovered, in what year will the world’s reserves of oil be
exhausted?
95)
A)
2012
B)
2010
C)
2017
D)
2022
96)
f(x) = kx2; [0, 3]
96)
A)
1
27; f(x) =1
27x2
B)
1
9; f(x) =1
9x2
C)
2
9; f(x) =2
9x2
D)
3
26; f(x) =3
26x2
97)
f(x) =1
9, [5, 14]
97)
A)
µ=7; 2=142
27 ; =2.29
B)
µ=19
2; 2=142
27 ; =2.29
C)
µ=19
2; 2=27
4; =2.60
D)
µ=7; 2=563
3; =13.70
98)
0
–
19
(x – 1)3 dx
98)
A)
–19
2
B)
19
2
C)
Divergent
D)
–38
99)
The marginal revenue for a certain product is given by R'(x) =600–4x. Find the total–revenue
function R(x), assuming that R(0) = 0.
99)
A)
R(x) = – 2x2
B)
R(x) =600x –2x2
C)
R(x) =600x –4x2
D)
R(x) =600–2x2
100)
S(x) =x+ 3, 0 x 3; x= 3
100)
A)
$0.80
B)
$2.45
C)
–$9.00
D)
$1.01
22
101)
P0= $77,000, t =5 yr, k =5.7%
101)
A)
$73,990.00
B)
$102,391.68
C)
$89,950.00
D)
$102,491.68
102)
Find the amount in a savings account after 11 yr from an initial investment of $120 at interest rate
8% compounded continuously.
102)
A)
$248.33
B)
$289.31
C)
$183,656.98
D)
$474.61
103)
The time between major earthquakes in a particular region of the Mediterranean is a random
variable with probability density function f(x) =1
1600e–x/1600 for x in [0, ), where x is measured
in days. Find the expected value and the standard deviation of this probability density function.
103)
A)
µ=1600 days; =3200 days
B)
µ=1600 days; =1600 days
C)
µ=1600 days; =2262.4 days
D)
µ=3200 days; =2262.4 days
104)
Find the amount of a continuous money flow in which $2000 per year is being invested at 5%,
compounded continuously for 20 years.
104)
A)
$148,731.27
B)
$108,731.27
C)
$68,731.27
D)
$343,656.37
105)
f(x) =1
7, [6, 13]
105)
A)
µ= 3.50; 2= 4.080; = 2.02
B)
µ= 9.50; 2= 4.083; = 2.02
C)
µ= 3.45; 2= 4.080; = 2.02
D)
µ= 9.50; 2= 2.993; = 1.73
106)
y =1x, x = 1, x =7
106)
A)
1
7
B)
1
2 ln 7
C)
ln 7
D)
7
107)
Following the birth of a child, a parent wants to make an initial investment P0 that will grow to
$70,000 by the child’s 20th birthday. Interest is compounded continuously at 8%. What should the
initial investment be?
107)
A)
$14,132.76
B)
$113,062.05
C)
$507.32
D)
$15,018.37
108)
f(x) =1
3x, over the interval [0,4]. Find E(x2).
108)
A)
85
4
B)
64
3
C)
127
6
D)
257
12
Solve the problem.
109)
An artist has designed a curved planter that has a flat top and bottom. The shape of the curve for
the planter (laying on its side) can be modeled by the function f(x) =36 1 –x2
6400, for –50 x 75,
where x is in centimeters. Find the volume of the planter.
109)
A)
98,566 cm3
B)
392,969 cm3
C)
197,132 cm3
D)
785,938 cm3
110)
D(x) =(x – 6)2; x = 3
110)
A)
$12
B)
$36
C)
$18
D)
$25
111)
f(x) = k(16 – x); [0, 16]
111)
A)
256; f(x) =256(16 – x)
B)
1
256; f(x) =1
256(16 – x)
C)
16; f(x) =16(16 – x)
D)
1
128; f(x) =1
128(16 – x)
Solve the problem.
112)
What should P0 be so that the amount of a continuous money flow over 40 years at interest rate 7%,
compounded continuously, will be $60,000?
112)
A)
$240.76
B)
$255.40
C)
$271.94
D)
$38.85
113)
Capital value is defined as
0R(t)e–ktdt where k is the annual rate of interest compounded
continuously. Find the capital value of an asset that produces $5000 yearly income at 6%
compounded continuously. Round to the nearest dollar.
113)
A)
$85,000
B)
$80,000
C)
$83,333
D)
$100,000
114)
D(x) =(x –2)2, S(x) =x2
114)
A)
2, $0
B)
0, $4
C)
4, $4
D)
1, $1
115)
The mean is µ= 15.2 and the standard deviation is = 0.9.
Find P(14.3 x 16.1).
115)
A)
0.8413
B)
0.6826
C)
0.1587
D)
0.3413
Solve the problem.
116)
A company installs 5,000 light bulbs. Each bulb has an average life of 500 hours with a standard
deviation of 100 hours. The life of each bulb is approximated by a normal curve. Find the number
of bulbs that can be expected to last less than 500 hours.
116)
A)
2500 bulbs
B)
3000 bulbs
C)
2400 bulbs
D)
1000 bulbs
117)
0
dx
x2/5
117)
A)
Divergent
B)
Convergent
118)
Find the area, if it exists, of the region bounded by the graph of y =1
(8x – 4)3 and the lines y = 0
and x = 6.
118)
A)
Divergent
B)
1
24
C)
1
48
D)
1
12
119)
c0=$800,000, r = 5%, m(t) =60,000 + 2000e0.01t, L =20
119)
A)
$1,466,078.22
B)
$1,746,078.22
C)
$1,666,078.22
D)
$1,586,078.22
27
120)
y =x, x = 0, x =6
120)
A)
12
B)
3
C)
18
D)
6
121)
A company determines that its marginal profit, in dollars, from producing x units is given by
P'(x) =342x–1.05, where x 1. Suppose that it would be possible for the company to make infinitely
many units of this item. Find the total profit. Round to the nearest dollar.
121)
A)
$6830
B)
$6850
C)
$6860
D)
$6840
122)
dP
dt =3P
122)
A)
P = Cet
B)
P = Ce–3t
C)
P =3eCt
D)
P = Ce3t
123)
0
dx
(x + 1)5/2
123)
A)
2
3
B)
1
2
C)
Divergent
D)
2
7
124)
P(–2.34 x –1.1)
124)
A)
0.1249
B)
0.1261
C)
0.6261
D)
0.1263
Solve the problem.
125)
A firm’s marginal profit P as a function of its total cost C is given by dP
dC =–100
(C + 5)2. Find P(C), if
P(95) = 45.
125)
A)
P(C) =100
C + 5 + 44
B)
P(C) =–100
C + 5 + 46
C)
P(C) =–300
(C + 5)3+ 48
D)
P(C) =–100
(C + 5)3+ 5
126)
P = $100,000, t =11 yr, k =9%
126)
A)
$91,393.12
B)
$37,567.65
C)
$37,157.67
D)
$269,123.45
127)
S(x) =x2+4; x = 1
127)
A)
$0.67
B)
–$4
C)
–$1.33
D)
$4
128)
2
7
(x + 1)2 dx
128)
A)
Divergent
B)
Convergent
Find the particular solution determined by the given condition.
129)
f'(x) = 4xe2x; f(0) =17
129)
A)
f(x) = 4xe2x – 2e2x + 19
B)
f(x) = 4xe2x – e2x + 18
C)
f(x) = 2xe2x – e2x + 18
D)
f(x) = 2xe2x + 17
130)
P(–3.1 x 2.65)
130)
A)
0.4956
B)
0.9961
C)
0.9947
D)
0.0037
131)
The growth rate of a certain stock is modeled by dV
dt = k(36 – V), V = $23 when t = 0, where V = the
value of the stock, per share, after time t (in months), and k = a constant. Find the solution to the
differential equation in terms of t and k.
131)
A)
V =23 –13e–kt
B)
V =36 –13e–kt
C)
V =36 –36e–kt
D)
V =36 –13ekt
132)
f(x) = k; [–5, 0]
132)
A)
5; f(x) =5
B)
–1
5; f(x) = – 1
5
C)
–5; f(x) = –5
D)
1
5; f(x) =1
5
133)
f(x) =1
9x2, [0, 3]
133)
A)
312
2
B)
334
2
C)
1
D)
1
27
134)
P(–0.75 x 0.75)
134)
A)
0.2734
B)
0.5528
C)
0.5468
D)
0.7734
135)
y = x ln x + 6x + 5; y’‘ –1
x= 0.
135)
A)
No
B)
Yes
136)
–2
–
2
x4 dx
136)
A)
1
12
B)
Divergent
C)
–1
4
D)
1
48
137)
f(x) =1
6x2, over the interval [–2,1]. Find E(x2).
137)
A)
–31
30
B)
11
10
C)
7
6
D)
–5
8
138)
The time of a telephone call (in minutes) to a certain town is a continuous random variable with a
probability density function defined by f(x) =3x–4 for [1, ). Find the probability: P(x 100).
138)
A)
0.000683
B)
0.000316
C)
0.000074
D)
0.000001
139)
The mean is µ= 137.0 and the standard deviation is = 5.3.
Find P(134.4 x 140.1).
139)
A)
0.4069
B)
0.9069
C)
0.0311
D)
1.0311
140)
The function f(x) = 96x2 is a probability density function on the interval [–a, a]. What is a?
140)
A)
0.50
B)
1.00
C)
0.38
D)
0.25
141)
The time needed to repair a machine follows an exponential distribution with mean 33 minutes.
What is the probability that the next repair of this machine will take 8 minutes or less?
141)
A)
0.1615
B)
0.2583
C)
0.2153
D)
0.1722
142)
In 1990 the world reserves of aluminum ore stood at 75,000,000,000 tons. World use of aluminum
that year was 100,000,000 tons, and the demand was growing exponentially at 10% per year.
Assuming this growth rate continues, and no new deposits of ore are discovered, in what year will
the world reserves of aluminum ore be exhausted?
142)
A)
2033
B)
2079
C)
2110
D)
2058
143)
Find the accumulated present value of the following continuous money flow:
R(t) = t , k =6%, T =20 years
143)
A)
$694.78
B)
$139.22
C)
$55.56
D)
$93.71
144)
dy
dx =6x5y
144)
A)
y =6Cx5ex5
B)
y =6Cx6ex6
C)
y =6Cx6ex5
D)
y = Cex6
145)
D(x) =(x –10)2, S(x) =x2+ 2x + 1
145)
A)
50, $1600
B)
20, $100
C)
2
11, $96.40
D)
9
2, $30.25
33
146)
1
20
8x(x + 1)2 dx
146)
A)
–1.569
B)
1.569
C)
Divergent
D)
0.482
147)
S(x) =1– x; x = 0
147)
A)
$0
B)
$1.75
C)
$1
D)
$2.30
148)
Find the area, if it exists, of the region under the graph of y =1
x2.8 on the interval (1, ).
148)
A)
Divergent
B)
5
19
C)
14
19
D)
5
9
149)
f'(x) = 3x2 – 2x; f(0) =9
149)
A)
f(x) = 3x3+ x2+ 9
B)
f(x) = x3+ 2x2+ 9
C)
f(x) = x3– x2+ 9
D)
f(x) = 3x3+ 2x2+ 9
150)
7y6dy
dx =8x
150)
A)
y =74x2+ C
B)
y = –78x + C
C)
y = –74x2+ C
D)
y =78x2+ C
151)
Find the area, if it exists, of the region bounded by the graph of y =5
x on the interval (–, –1).
151)
A)
1
B)
Divergent
C)
ln 5
D)
1
5
152)
S(x) =x– 2, 0 x 4; x= 2
152)
A)
$0
B)
$2
C)
$8
D)
$5
153)
Scores on an English test are normally distributed with a mean of 32.7 and a standard deviation of
6.9. Find the 41st percentile.
153)
A)
28.6
B)
34.3
C)
31.1
D)
36.8
154)
Find the demand function q = D(x), given that E(x) =6
x and q = e when x =6.
154)
A)
q =e6/x
B)
q =ln 6
x
C)
q =6 lnx
D)
q = ln 6
x
155)
f(x) = kx1/2; [1, 9]
155)
A)
3
17; f(x) =3x1/2
17
B)
3
54; f(x) =3x1/2
54
C)
3
52; f(x) =3x1/2
52
D)
1
56; f(x) =x1/2
56
C
156)
P(0 x 2.5)
156)
A)
0.4938
B)
0.9940
C)
0.4062
D)
0.9938
A
Solve.
157)
y’ =5x + xy; y =2 when x = 0
157)
A)
y =7ex2/2
B)
y =7ex2/2 –5
C)
y =2ex2/2
D)
y = –3ex2/2 –2
B
Solve the problem.
158)
Find the accumulated present value of an investment for which there is a perpetual continuous
money flow of $1400 per year. The current interest rate is 5%. Round to the nearest dollar.
158)
A)
$28,000
B)
$14,000
C)
$1400
D)
$56,000
A
159)
Find the present value of $12,500 due 9 yr later at 12.9% compounded continuously.
159)
A)
$3914.66
B)
$38,664.03
C)
$39,914.03
D)
$2664.66
A
160)
f(x) = kx2; [–1, 4]
160)
A)
3
64; f(x) =3
64x2
B)
1
16; f(x) =1
16x2
C)
1
21; f(x) =1
21x2
D)
3
65; f(x) =3
65x2
Solve the problem.
161)
Suppose that replacement times for washing machines are normally distributed with a mean of
10.6 years and a standard deviation of 1.7 years. Find the 82nd percentile.
161)
A)
9.0 years
B)
10.9 years
C)
11.5 years
D)
12.2 years
162)
Capital value is defined as
0R(t)e–ktdt where k is the annual rate of interest compounded
continuously. Find the capital value of an asset that produces $5000 yearly income at 8%
compounded continuously. Round to the nearest dollar.
162)
A)
$71,429
B)
$60,500
C)
$65,000
D)
$62,500
163)
c0=$500,000, r = 5%, m(t) =50,000, L =20
163)
A)
–$632,120.56
B)
$1,132,120.56
C)
–$1,118,281.83
D)
$1,282,120.56
164)
P(x 0.59)
164)
A)
0.2776
B)
0.2190
C)
0.7224
D)
0.2224
165)
D(x) =10 –8x, S(x) =3+8x
165)
A)
7
16, $31.00
B)
7
2, $31.00
C)
–7
16, –$0.50
D)
7
16, $6.50
166)
f'(x) =x2/5 + x; f(1) = –6
166)
A)
f(x) =5
7x1/5 +1
2x2–101
12
B)
f(x) =7
5x7/5 + 2x2+13
2
C)
f(x) =5
7x7/5 +1
2x2–101
14
D)
f(x) =2
5x7/5 –1
2x2+43
7
167)
f(x) =1
10x, over the interval [1,4]. Find E(x).
167)
A)
21
10
B)
32
15
C)
3
5
D)
31
15
Solve the problem.
168)
Capital value is defined as
0R(t)e–ktdt where k is the annual rate of interest compounded
continuously. Find the capital value of an asset that produces $5000 yearly income at 7%
compounded continuously. Round to the nearest dollar.
168)
A)
$70,000
B)
$71,429
C)
$83,333
D)
$75,000
169)
1
17
x dx
169)
A)
Divergent
B)
Convergent
Solve the problem.
170)
Find the amount of the following continuous money flow:
R(t) =2500t +5, k =8%, T =30 years
170)
A)
$3,759,679.61
B)
$6,419,550.88
C)
$4,306,617.11
D)
$1,328,187.50
171)
D(x) =(x – 3)2; x =3
2
171)
A)
$7.28
B)
$4.33
C)
$4.50
D)
$3.25
172)
f(x) =1
5x2, over the interval [–2, 3]. Find E(x).
172)
A)
71
20
B)
13
4
C)
81
20
D)
73
20
173)
y =ex+ 4xex; y’‘ – 2y’ + y = 0.
173)
A)
Yes
B)
No
174)
The mean is µ= 22.0 and the standard deviation is = 2.4.
Find P(19.7 x 25.3).
174)
A)
0.3370
B)
0.7477
C)
0.4107
D)
1.0847
175)
S(x) =x2; x =2
175)
A)
$1.33
B)
$6
C)
$5.33
D)
$8
176)
S(x) =e2x + 1; x= 0
176)
A)
$0
B)
$2.33
C)
$1
D)
–$1.20
40
177)
The function f(x) =x2 is a probability density function on the interval [0, b]. What is b?
177)
A)
3
B)
33
C)
2
2
D)
2
178)
–3
–
x7dx
178)
A)
Convergent
B)
Divergent
179)
Cesium – 137 has a decay rate of 2.3% per year. Suppose that Cesium – 137 is released into the
atmosphere each year for 10 years at the rate of 0.5 pounds per year. What is the total amount of
radioactive buildup?
179)
A)
4.47 pounds
B)
3.89 pounds
C)
4.88 pounds
D)
4.04 pounds
180)
5
dx
(x +3)9/8
180)
A)
Divergent
B)
Convergent
Solve the problem.
181)
The time to failure t, in hours, of a certain machine can often be assumed to be exponentially
distributed with probability density function f(t) = ke–kt, 0 t <, where k =1
a and a is the average
amount of time that will pass before a failure occurs. Suppose the average amount of time that will
pass before a failure occurs is 87 hours. What is the probability that a failure will occur in 43 hours
or less?
181)
A)
0.4680
B)
0.2925
C)
0.3120
D)
0.3900
182)
3
dx
x2.5
182)
A)
Divergent
B)
Convergent
Solve the problem.
183)
The time of a telephone call (in minutes) to a certain town is a continuous random variable with a
probability density function defined by f(x) =3x–4 for [1, ). Find the probability: P(1 x 99).
183)
A)
0.3170
B)
0.2170
C)
0.7830
D)
0.9999
184)
A company determines that its marginal cost, in dollars, for producing x units of a product is given
by C'(x) =5300x–1.4, where x 1. Suppose that it would be possible for the company to make
infinitely many units of this item. Find the total cost. Round to the nearest dollar.
184)
A)
$13,230
B)
$13,300
C)
$13,150
D)
$13,250
185)
y= x – 22
185)
A)
x2
2– x + C
B)
x3– 22x + C
C)
2x2– 22 + C
D)
x2
2– 22x + C
186)
y’ =2
x–x3+x6
186)
A)
y =2 ln x –1
4x4+1
7x7+ C
B)
y =2
ln x –1
4x3+1
7x6+ C
C)
y = –41
x3–3x2+6x5+ C
D)
y =2 ln x –1
5x5+1
8x8+ C
187)
1
2
5 +ex dx
187)
A)
0.4172
B)
2.3147
C)
1.5708
D)
3.1416
188)
f(x) = kex; [0, 4]
188)
A)
2
e4; f(x) =2ex
e4
B)
1
e4– 1; f(x) =ex
e4– 1
C)
e4– 1; f(x) =e4x – 1
D)
1
e4+ 1; f(x) =ex
e4+ 1
189)
1
dx
1x
189)
A)
Divergent
B)
Convergent
190)
y’ =2x – xy; y =5 when x = 0
190)
A)
y =3e–x2/2 +2
B)
y = –3e–x2/2 + 8
C)
y =5e–x2/2
D)
y =7e–x2/2 +5
191)
Find the area, if it exists, under the curve y =1
(x + 1)3/2 bounded on the left by x =15.
191)
A)
8
B)
2
15
C)
1
4
D)
1
2
192)
Capital value is defined as
0R(t)e–ktdt where k is the annual rate of interest compounded
continuously. Find the capital value of an asset that produces $5000 yearly income at 5%
compounded continuously. Round to the nearest dollar.
192)
A)
$83,333
B)
$100,000
C)
$125,000
D)
$95,000
193)
Find the interest earned when $9367 is invested at 8% compounded continuously for 4 years.
193)
A)
$3532.10
B)
$4058.44
C)
$3532.56
D)
$2540.52
194)
y=5e3x
194)
A)
15e3x + C
B)
1
3 e3x + C
C)
5e3x + C
D)
5
3 e3x + C
195)
f(x) = kx3; [0, 2]
195)
A)
4
15; f(x) =4x3
15
B)
1
8; f(x) =x3
8
C)
1
15; f(x) =x3
15
D)
1
4; f(x) =x3
4
196)
Solve the differential equation model of radioactive decay:
dQ
dt = –0.6Q.
196)
A)
Q(t) =Q0e–0.6t
B)
Q(t) =–1
0.6t+Q0
C)
Q(t) =Q0e–t
D)
Q(t) = –Q0ln 0.6t + c
197)
y =e–x+ 5xe–x; y” + 2y’ + y = 0.
197)
A)
No
B)
Yes
45
198)
P = $4,000,000, t =20 yr, k =5%
198)
A)
$1,480,667.15
B)
$1,471,517.76
C)
$10,873,127.31
D)
$3,804,917.70
199)
f(x) = 1 – x, [0, 2]
199)
A)
1
B)
1.5
C)
0.5
D)
0.75
200)
f(x) =1
6, over the interval [2,8]. Find E(x).
200)
A)
59
12
B)
5
C)
10
D)
28
201)
f'(x) = 6x2– 4x + 22; f(1) =19
201)
A)
f(x) = 2x3– 2x2+ 22x – 3
B)
f(x) = 6x3– 4x2+ 22x – 5
C)
f(x) = 2x3– 4x2+ 22x – 1
D)
f(x) = 2x3– 2x2+ 22x + 3
46
Solve the problem.
202)
Find the accumulated present value of an investment over a 40–year period if there is a continuous
money flow of $2300 per year and the current interest rate is 5%, compounded continuously.
202)
A)
$6225.42
B)
$39,774.58
C)
$52,225.42
D)
–$39,774.58
203)
The storage tanks at a beverage company have a shape formed by rotating a parabola around an
axis. The function y = –0.264x2+3.5, for –8.5 x 8.5, where x and y are in feet, describes the shape
of such a tank (laying on its side). Determine the volume of the tank to the nearest cubic foot.
203)
A)
1082 ft3
B)
1946 ft3
C)
2163 ft3
D)
973ft3
204)
Find the area, if it exists, of the region bounded by the graph of y =1
x +5 and the lines y = 0 and
x =5.
204)
A)
1
5
B)
1
25
C)
Divergent
D)
ln 5
205)
0
–
13xe3x dx
205)
A)
Divergent
B)
–4.667
C)
0.3333
D)
–1.4444
206)
f(x) =1
4, [6, 10]
206)
A)
No
B)
Yes
207)
f(x) =1
x ln 6, [1, 6]
207)
A)
µ=1
ln 6; 2=34; =34
B)
µ=6
ln 6; 2=2.259; =1.503
C)
µ=5
ln 6; 2=1.980; =1.407
D)
µ=1
ln 6; 2=17; =17
208)
y =ex, x = –4, x =4
208)
A)
(e8–e–8)
B)
2(e4–e–4)
C)
2(e4–e–4)
D)
2(e8–e–8)
209)
f(x) =k
x; [1, 17]
209)
A)
2
ln 17; f(x) =2
x ln 17
B)
1
ln 17; f(x) =1
x ln 17
C)
ln 17; f(x) = x ln 17
D)
1 – ln 17; f(x) =x
1 – ln 17
210)
y = x, x =2, x =6
210)
A)
20
B)
16
C)
4
3
D)
208
3
211)
–
2xe–x2 dx
211)
A)
Divergent
B)
Convergent
212)
P(0 x 0.75)
212)
A)
0.4591
B)
0.9599
C)
0.4599
D)
0.2734
213)
f(x) =1
e – 1 ex,[0, e]
213)
A)
No
B)
Yes
214)
D(x) =(x – 2)2; x = 2
214)
A)
$12.50
B)
$4.75
C)
$2.67
D)
$0
215)
f(x) =6x–7, [1, )
215)
A)
1.07
B)
.90
C)
1.12
D)
3.32
216)
D(x) =e–x +13.2, S(x) =ex –7.8
216)
A)
(21, $221.41)
B)
(10.5, $29.76)
C)
(10.5, $14.88)
D)
(2.7, $0.01)
Find the future value P of the amount P0 invested for time period t at interest rate k, compounded continuously.
217)
P0=$70,000, t =3 yr, k =5%
217)
A)
$81,033.75
B)
$81,328.40
C)
$80,500.00
D)
$81,428.40
218)
Find the area, if it exists, of the region bounded by the curve y =7x–2, the x–axis, and on the left by
x = 1.
218)
A)
7
2
B)
7
C)
14
D)
49
219)
y =2x + 3, x = 0, x = 1
219)
A)
3
2
B)
2
C)
4
D)
220)
f(x) = 5, [5.2, 5.4]
220)
A)
µ= 5.30; 2= 0.0031; = 0.056
B)
µ= 5.30; 2= 0.0034; = 0.058
C)
µ= 5.39; 2= 0.0031; = 0.056
D)
µ= 5.32; 2= 0.0034; = 0.058
221)
f(x) =1
9, over the interval [2,10]. Find E(x2).
221)
A)
16
3
B)
112
3
C)
992
27
D)
991
27
222)
A machine fills quart soda bottles with an average of 32.3 oz per bottle, with a standard deviation
of 1.2 oz. What is the probability that a filled bottle will contain less than 32 oz? Assume the
distribution is normal.
222)
A)
0.60
B)
0.40
C)
0.38
D)
0.41
223)
D(x) = 3 –x, 0 x 3; S(x) =x– 1
223)
A)
(2, $1)
B)
(2, $5)
C)
(1, $2)
D)
(6, $1)
224)
The life span of a certain insect in days is uniformly distributed over the interval [20, 36]. What is
the expected life of this insect?
224)
A)
29 days
B)
28 days
C)
30 days
D)
26 days
225)
09e–9x dx
225)
A)
Divergent
B)
–1
C)
0
D)
1
226)
y’ =7
x ; y =20 when x = 1
226)
A)
y = ln x + 18
B)
y = ln x + 20
C)
y =7 ln x + 20
D)
y =7 ln x + 3.5
227)
1ln x dx
227)
A)
Convergent
B)
Divergent
228)
P = $1,000,000, t =6 yr, k =7%
228)
A)
$1,521,961.56
B)
$659,438.00
C)
$932,393.82
D)
$657,046.82
229)
D(x) =(x –6)2, S(x) =x2+ x +8
229)
A)
13
2, $3.85
B)
28
13, –$3.85
C)
(–2, –$2.15)
D)
28
13, $14.79
230)
f(x) =1
32x, [0, 8]
230)
A)
No
B)
Yes
231)
f(x) =1
ln 2·1
x, over the interval [1.5,9.2]. Find E(x).
231)
A)
41.57
ln 2
B)
7.7
ln 2
C)
7.7
D)
8.2
ln 2
232)
D(x) = 4 – x, 0 x 4; x =3
232)
A)
$1
B)
$12
C)
$4.50
D)
$4
233)
The time between major earthquakes in the Alaska panhandle region is a random variable with
probability density function f(x) =1
620e–x/620 for x in [0, ), where t is measured in days. Find the
probability that the time between a major earthquake and the next one is less than 300 days.
233)
A)
0.0006
B)
0.3836
C)
0.6164
D)
0.0010
234)
0
–
4xe–x2 dx
234)
A)
Divergent
B)
Convergent
235)
2
x2 dx
235)
A)
Divergent
B)
Convergent
Solve the problem.
236)
A dart is thrown at a number line in such a way that it always lands in the interval [0,10]. Let x be
the number the dart hits. Suppose the probability density function for x is given by
f(x) =x
50, for 0 x 10.
Find P(2 x 9), the probability that it lands in [2, 9].
236)
A)
0.07
B)
0.77
C)
1.54
D)
0.49
237)
Find the area, if it exists, of the region under the graph of y = xex2 on the interval (0, ).
237)
A)
1
B)
e2
C)
e–2
D)
Divergent
Solve the problem.
238)
The time of a telephone call (in minutes) to a certain town is a continuous random variable with a
probability density function defined by f(x) =3x–4 for [1, ). Find the probability: P(x 3).
238)
A)
0.0370
B)
0.3066
C)
0.2402
D)
0.0759
Answer Key
Testname: C5
Answer Key
Testname: C5
Answer Key
Testname: C5
Answer Key
Testname: C5
Answer Key
Testname: C5