96)
The productivity of a certain country is P(x, y) = 1600x1/4y3/4 units, where x and y are the amounts
of labor and capital utilized. What is the marginal productivity of capital when x = 16 and y = 625?
96)
A)
480
B)
2500
C)
1600
D)
5000
E)
none of these
Solve the problem.
97)
A farmer has 420 m of fencing. Find the dimensions of the rectangular field of maximum area that
can be enclosed by this amount of fencing.
97)
A)
42 m by 168 m
B)
95 m by 115 m
C)
105 m by 315 m
D)
105 m by 105 m
98)
Let g(x, y) =2y2– 6xy. Compute g(–4, –8).
98)
A)
–64
B)
–168
C)
–60
D)
–160
Solve the problem.
99)
The chart shows the amount of money in a student’s bank account for four consecutive months.
Year Dollars
1$1075
21070
31089
41134
Use the method of least squares to find the line that best fits these data. Then, use the least–squares
line to predict how much will be in the bank account after 7 years.
99)
A)
y = 19.6x +1043; $1244.20
B)
y = 19.6x +1107; $1244.20
C)
y = 39.2x +1043; $1317.40
D)
y = 19.6x +1043; $1180.20
100)
Find all points (x, y) where f(x, y) =x3– 12y +y2 has a possible relative maximum or minimum.
Use the second–derivative test to determine, if possible, the nature of f(x, y) at each of these points.
100)
A)
(0, 0) gives a relative maximum
B)
Test Inconclusive
C)
(0, 6) gives a relative maximum
D)
(0, 0) gives a relative minimum
Solve the problem.
101)
Assuming that a cylindrical container can be mailed only if the sum of its height and circumference
do not exceed 300 centimeters, what are the dimensions of the cylinder with the largest volume that
can be mailed?
101)
A)
Height 100 centimeters and radius 300/ centimeters
B)
Height 200 centimeters and radius 100 centimeters
C)
Height 300 centimeters and radius 100/ centimeters
D)
Height 100 centimeters and radius 100/ centimeters
Find the partial derivative.
102)
f(x, y) =e–2xy. Find f
x.
102)
A)
fx(x, y) = – 2ye–2x
B)
fx(x, y) = – 2(x + y)e–2xy
C)
fx(x, y) = – 2ye–2xy
D)
fx(x, y) = – 2e–2xy
103)
Let f(x, y) =1
2x2·ex/y. Find f
y.
103)
A)
x3
yex/y – 1
B)
1
2x2ex/y
C)
–x3
2y2ex/y
D)
x2
yex/y
E)
none of these
104)
Suppose the number of leather bags produced by a certain firm, utilizing x units of raw materials
and y units of labor, is given by the formula f(x, y) = 10x1/2y1/2. What happens to the production
of bags if supplies of both raw materials and labor are tripled?
104)
A)
Production is increased by a factor of 3.
B)
Production is increased by a factor of 6.
C)
Production is increased by a factor of 3.
D)
Production is increased by a factor of 9.
E)
none of these
Solve the problem.
105)
A company has the following production function for a certain product:
p(x, y) =25x0.4y0.6 .
Find the marginal productivity with fixed labor, p
y.
105)
A)
15 y
x
0.4
B)
15yx0.4
C)
15 y
x
0.6
D)
15 x
y
0.4
106)
A petroleum company has a Cobb–Douglas production function f(x, y) = 70x2/5y3/5 where x is the
utilization of labor and y is the utilization of capital. Determine the number of units of petroleum
produced when 1200 units of labor and 2100 units of capital are used.
106)
A)
105,074 units
B)
150,000 units
C)
117,517 units
D)
175,174 units
Find the double integral over the rectangular region R with the given boundaries.
107)
(x2+y2) dx dy
R
0 x 4, –2 y 1
107)
A)
112
B)
67
C)
76
D)
73
3
C
108)
Find all points (x, y) where f(x, y) =x2+ xy +y2– 3x + 2 has a possible relative maximum or
minimum. Use the second–derivative test to determine, if possible, the nature of f(x, y) at each of
these points.
108)
A)
(2, –1) is a relative minimum
B)
(–2, 1) is a relative minimum
C)
(2, –1) is a relative maximum
D)
(–2, 1) is a relative maximum
A
Solve the problem.
109)
Suppose that the manufacturing cost of a precision instrument is approximated by
M(x,y) =30x2+ 10y2– 8xy, where x is the cost of materials and y is the cost of labor. Find M
x(3, 7).
109)
A)
102
B)
322
C)
124
D)
116
C
C
Find the partial derivative.
110)
f(x, y) =10x –6y2–8. Find f
x.
110)
A)
10x
B)
–12y
C)
10
D)
2
Solve the problem.
111)
Find two positive numbers whose sum is 36 and whose product is a maximum.
111)
A)
27 and 27
B)
18 and 27
C)
9 and 27
D)
18 and 18
D
112)
Let F(x, y, z) =xz
y2z + x
–5x2y3. Compute F
y(–1, 0, 1).
112)
A)
–1
2
B)
3
5
C)
0
D)
2
E)
none of these
C
113)
What is the least squares error E for the points (1, –3), (–2, 5), and (0, 10) and the line y = Ax + B?
113)
A)
E =(A – 3B + 3)2+(–2A + 5B – 5)2+ (10B – 10)2
B)
E =(A – 3B)2+ (–2A + 5B)2+ (10B)2
C)
E =(A + B + 3)2+(–2A + B – 5)2+ (B – 10)2
D)
E =(A + B – 3)2+(–2A + B – 5)2+ (B – 10)2
C
25
C
Find the partial derivative.
114)
f(x, y) =8(x +7y –8)2. Find f
y.
114)
A)
112x +784y –896
B)
56x +392y –448
C)
112x +784y +896
D)
56x +392y
Solve the problem.
115)
A company’s monthly sales, in thousands, is given by S(x, y) =5x0.4y0.7, where x is the amount
spent on newspaper advertising per month in thousands of dollars and y is the amount spent on
radio advertising per month in thousands of dollars. Suppose the company currently spends $5000
on newspaper advertising per month and $3000 on radio advertising per month. What would be
the effect on sales if the company increases the amount spent on newspaper advertising to $6000,
while the amount spent on radio advertising remains constant?
115)
A)
Sales would increase by $1472.73.
B)
Sales would increase by $3100.08.
C)
Sales would increase by $4792.03.
D)
Sales would decrease by $147.27.
Find the partial derivative.
116)
f(x,y) =2x2– 19xy + 4y3. Find f
x.
116)
A)
–19x + 12y2
B)
4x – 19y + 12y2
C)
4x – 19y
D)
4x2– 19x
26
117)
Calculate the iterated integral
2
1
4
2
x
ydy dx.
117)
A)
3
2ln 2
B)
1
2ln 2
C)
6 ln 2
D)
3
2ln 4
E)
none of these
118)
Find all points (x, y) where f(x, y) =e(x2+y2) has a possible relative maximum or minimum. Use
the second–derivative test to determine, if possible, the nature of f(x, y) at each of these points.
118)
A)
(0, 0) gives a relative maximum point
B)
(0, 1) gives a relative maximum point
C)
(0, 1) gives a relative minimum point
D)
(0, 0) gives a relative minimum point
Evaluate the iterated integral.
119)
1
0
10
10x
y dy dx
119)
A)
100
3
B)
500
C)
1000
3
D)
50
120)
Suppose that a company can produce P(x, y) = 50 (x3+y3)
10 items using x units of labor and y
units of capital. What is the productivity of capital when x = 10 and y = 20?
120)
A)
500
B)
100
C)
2500
D)
3000
E)
none of these
121)
Calculate the iterated integral
1
0
x
x2
(x – 1) dy dx.
121)
A)
–3
4
B)
–1
2
C)
1
2
D)
–1
12
E)
none of these
Solve the problem.
122)
The production function f(x, y) for an industrial country was estimated as f(x, y) = x4y8, where x is
the amount of labor and y the amount of capital. Find the marginal productivity of labor.
122)
A)
8x4y7
B)
4x3y8
C)
16x4y7
D)
8x3y8
Evaluate the function.
123)
Find f(5, 0, 7) when f(x, y, z) =5x2+5y2–z2.
123)
A)
76
B)
–24
C)
85
D)
174
Find the partial derivative.
124)
Find f
y(–2, 5) when f(x, y) =4xy –6y.
124)
A)
–8
B)
–14
C)
20
D)
0
Solve the problem.
125)
Suppose that the labor cost for a building is approximated by
C(x,y) =2x2+ 6y2– 200x – 420y + 24,000, where x is the number of days of skilled labor and y is the
number of days of semiskilled labor required. Find the x and y that minimize cost C.
125)
A)
x =100, y =70
B)
x =210, y =105
C)
x =35, y =210
D)
x =50, y =35
126)
An ecologist is studying pollution in a local river. He takes water samples from various distances
away from a factory on the river. He then measures the amount of a certain chemical in each
sample.
Distance from Factory
(hundreds of yards)
Amount of Chemical
(parts per thousand)
0308
5221
10 165
15 144
20 119
Use the method of least squares to find the line that best fits these data. Using the least–squares, how
much of the chemical should the ecologist expect to find 6 yards from the factory?
126)
A)
y = – 9.1x +259.4; 204.8 parts per thousand
B)
y = – 9.1x +282.4; 227.8 parts per thousand
C)
y = 9.1x +281.6; 336.2 parts per thousand
D)
y = – 9.1x +219.6; 165 parts per thousand
Find the second–order partial derivative.
127)
Find 2f
x2, where f(x, y) =x4y5+x5y.
127)
A)
4x3y5+5x4y
B)
20y3
C)
12x2y5+20x3y
D)
12x2y5+20x3
128)
A certain manufacturer can produce f(x, y) = 10(6x3+y2) units of goods by utilizing x units of
labor and y units of capital. What is the marginal productivity of labor when x = 10 and y = 20?
128)
A)
2200 units
B)
1800 units
C)
18,000 units
D)
22,000 units
E)
none of these
129)
Let R be the rectangle consisting of all points (x, y) such that 0 x 3, 0 y 1. Calculate
R
4x2y2dy dx.
129)
A)
24
B)
15
C)
–12
D)
12
Find the least–squares error E for the least–squares line fit to the points in the figure.
130)
y = 1.8x +0.5
130)
A)
E = 5.2
B)
E = 5.05
C)
E = 4.7
D)
E = 5.8
Find the partial derivative.
131)
f(x, y) =(x + y)4. Find f
x.
131)
A)
4x(x + y)3
B)
4y(x + y)3
C)
4(x + y)
D)
4(x + y)3
132)
Find the formula that gives the least squares error for the points (–1, 5), (2, 2), (5, –1).
132)
A)
(–2A + B – 5)2+(A + B – 2)2+(5A + B + 1)2
B)
(A – B + 5)2+(2A + B – 2)2+(A + B + 1)2
C)
(–A + B – 5)2+(2A + B – 2)2+(A + B + 5)2
D)
(–A + B – 5)2+(2A + B – 2)2+(5A + B + 1)2
133)
Let f(x, y, z) =y3z –x2y +xyz –1
x. Find f
y.
133)
A)
3y2z –x2+xz
2xyz
B)
3y2z –x2+xz
C)
3y2– 2x + 1
D)
3y2z –x2+xz
xyz
+1
x2
E)
none of these
Find the partial derivative.
134)
Find f
x(–4, 2) when f(x, y) =(5x +4y)2.
134)
A)
–200
B)
18
C)
–60
D)
–120
135)
A rectangular box of length x, width y, and height z with no top is to be constructed having a
volume of 32 cubic inches. Determine the dimensions that will require the least amount of material
to construct the box.
135)
A)
4 inches by 4 inches by 2 inches
B)
2 inches by 2 inches by 4 inches
C)
4 inches by 2 inches by 4 inches
D)
4 inches by 4 inches by 4 inches
136)
Let G(x, r, t) = 2x2t +1
3t2–r3xt. Find G
x.
136)
A)
4xt –1
3t2–3r2t
2xt
B)
4xt +1
3t2–r31
2xt
C)
4xt –r3t
2xt
D)
4x –r3
2xt
E)
none of these
Evaluate the iterated integral.
137)
2
0
5
0
3xy dx dy
137)
A)
25
B)
150
C)
75
D)
300
Find the partial derivative.
138)
f(x, y) =2x
y–y
2x . Find f
x.
138)
A)
fx(x, y) =2x2
y+y
2x2
B)
fx(x, y) =2
y+y
2x2
C)
fx(x, y) =2
y–y
2x2
D)
fx(x, y) = – 2
y2–2
y