Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the value.
1)
Find f(5, –5) when f(x, y) =6x + 8y – 2
A)
–12
B)
–18
C)
–10
D)
–20
Answer:
A
2)
Let f(x, y) = xy2+x. Find f(9, 8).
A)
656
B)
579
C)
81
D)
80
Answer:
B
3)
Let f(x, y) =x
y+ xy. Find f(–3, –3).
A)
18
B)
1
C)
8
D)
10
Answer:
D
4)
Let f(x, y) = xy2+x. Find f(16, –5).
A)
–64
B)
(16, –5) is not in the domain of f.
C)
–1285
D)
404
Answer:
D
5)
Find f(2, –1, 4) for f(x, y, z) = 3x2– 4y4+ z – 7.
A)
5
B)
13
C)
–5
D)
27
Answer:
A
6)
Find f(–3, 1, 5) for f(x, y, z) =1
3x2– 8y5+ z – 4.
A)
–5
B)
1
C)
10
D)
–4
Answer:
D
1
Choose the surface that the function describes.
7)
f(x, y) = 5
A)
B)
C)
D)
Answer:
D
2
8)
f(x, y) = 1 – x – 2y
A)
B)
C)
D)
Answer:
C
3
9)
f(x, y) = 3 –x2
A)
B)
C)
D)
Answer:
A
Solve the problem.
10)
The number of cows that can graze on a ranch is approximated by C(x,y) = 9x + 5y – 8, where x is the number of
acres of grass and y the number of acres of alfalfa. If the ranch has 25 acres of alfalfa and 65 acres of grass, how
many cows may graze?
A)
702 cows
B)
542 cows
C)
710 cows
D)
550 cows
Answer:
A
11)
Poiseuille’s law states that the resistance, R, for blood in a blood vessel varies directly as the length of the vessel,
L, and inversely as the fourth power of its diameter, d. This can be written as an equation R(L, d) = k L
d4 where
k is a constant. Find R(5, 0.3). Round your answer to the nearest whole number.
A)
about 16.67k
B)
about 630k
C)
about 617k
D)
about 61.7k
Answer:
C
12)
The marketing research department of a large manufacturing company has determined that the demand
equations for two major items it produces are given by p = 2,000 – 5x + 8y and q = 4,000 + 9x – 7y where p is the
price of item A, q is the price of item B, x is the monthly demand for item A, and y is the monthly demand for
item B. Find the total monthly revenue from items A and B when x = 15 and y = 5.
A)
$50,195
B)
$29,415
C)
$49,975
D)
$20,565
Answer:
C
4
13)
The surface area of a human body (in square meters) is approximated by A = 0.202W(.425)H(.725), where W is
the weight of the person in kilograms and H is the height in meters. Find A if W =59 and H =1.42.
A)
1.62 m2
B)
1.50 m2
C)
1.38 m2
D)
1.47 m2
Answer:
D
14)
The productivity of a petroleum company is given approximately by the function f(x, y) = 70x0.4y0.6, where x is
the utilization of labor and y is the utilization of capital. If the company uses 1200 units of labor and 2100 units
of capital, how many units of petroleum will be produced? Round to the nearest whole unit.
A)
105,074 units
B)
117,517 units
C)
175,174 units
D)
150,000 units
Answer:
B
15)
The Cobb–Douglas production function for a steel company is given by f(x, y) =78x0.3y0.7 where x is the
utilization of labor and y is the utilization of capital. If the company uses 1500 units of labor and 2200 units of
capital, how many units of steel will be produced? Round to the nearest whole unit.
A)
656,640 units
B)
152,974 units
C)
5,405,400,000 units
D)
54,054,000 units
Answer:
B
16)
The volume of a flower pot is given by V =1
3h r12+r22+r1r2 where r1 is the major radius and r2 is the
minor radius and h is the height of the pot (see figure below).
If the dimensions of the pot are r1=7 inches, r2=4 inches and h =6 inches, find the volume of potting soil
required to fill the pot to the top. Round to the nearest cubic inch.
A)
595in.3
B)
1142 in.3
C)
245in.3
D)
584in.3
Answer:
D
Find the partial derivative.
17)
Let z = f(x,y) =9x2– 15xy + 9y3. Find z
x.
A)
18x + 15y2
B)
–15x + 27y2
C)
18x – 15y
D)
–15x – 27y
Answer:
C
18)
Find fx(6, –8) when f(x,y) = 7x2– 9xy.
A)
156
B)
–12
C)
12
D)
–180
Answer:
A
5
19)
Find fy(–2, –3) for the function f(x, y) =7y2+5x3–4x5y.
A)
–170
B)
– 86
C)
86
D)
170
Answer:
C
20)
For f(x, y) = 3x4– 4x3y + 5y3– 4, find fx(1, –2).
A)
–12
B)
12
C)
24
D)
36
Answer:
D
21)
Find fx(2, –1) for f(x, y) =4x3–2x2+3y2– 3.
A)
40
B)
8
C)
16
D)
–48
Answer:
A
22)
Find z
x for z = f(x, y) = 4x2– 11xy + 4y3 .
A)
– 11x + 12y2
B)
–11x – 12y
C)
8x – 11y
D)
8x + 11y2
Answer:
C
23)
Find fx for f(x, y) =x3+ 9x2y + 4xy3 .
A)
3x2
B)
x2+ 9xy + 4y3
C)
3x2+ 18xy + 4y3
D)
3x2+ 2xy + 4y3
Answer:
C
Provide an appropriate response.
24)
Find fxy for f(x, y) = 8x3 y – 7y2+ 2x .
A)
48xy
B)
–28
C)
24x2
D)
–14
Answer:
C
25)
Find fxy for f(x, y) = 10 x2y4– 7 x3y5 .
A)
160x y3–105 x2y4
B)
160xy3– 21x2y4
C)
80x y3– 105x2y4
D)
80x y3– 21x2y4
Answer:
C
26)
For f(x, y) = 6x2+ 7xy4– 5y2+ 8, find fxx(x, y) +fyx(x, y).
A)
12 + 28xy3
B)
12
C)
12 + 28y3
D)
28y3
Answer:
C
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
27)
Find fxy for the function f(x, y) =3x2y + 5 .
Answer:
3x
23x2+ 5
6
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
28)
Find fxx +fyy for f(x, y) =5x3–2x2y2–y3+ 1 .
A)
30y –4x2–4y2– 6y
B)
12x2–4xy2–4x2y
C)
30x2–4xy2–4x2y
D)
30x –4y2–4x2
Answer:
D
Solve the problem.
29)
The profit function for sales of two models of television sets at a chain discount store is given by
P(x, y) = 140x + 160y – 6x2+ 4xy – 8y2– 500, where x is the number of sales per week of model A, and y is the
number of sales per week of model B. Find Px(10, 15) and interpret the result.
A)
B)
C)
D)
Answer:
D
30)
The productivity of a major manufacturer of microwave ovens is given approximately by the Cobb–Douglas
production function f(x, y) = 45x0.1y0.9 with the utilization of x units of labor and y units of capital. If the
company is currently utilizing 4500 units of labor and 2000 units of capital, find the marginal productivity of
labor to the nearest unit.
A)
217 units
B)
32 units
C)
129 units
D)
2 units
Answer:
D
31)
A company has the following production function for a certain product
P(x, y) = 27 x0.3 y0.7 .
Find the marginal productivity with fixed capital, Px .
A)
8.1 y
x0.7
B)
8.1 x
y0.7
C)
8.1x y0.7
D)
8.1 y
x1.3
Answer:
A
32)
The production function z for an industrial country was estimated as z =x5y6 , where x is the amount of labor
and y, the amount of capital. Find the marginal productivity of labor.
A)
10 x4y6
B)
6 x5y5
C)
5 x4y6
D)
12 x5y5
Answer:
C
7
Provide an appropriate response.
33)
Find critical points for f(x, y) = 5x2– 5y2+ 2xy + 34x + 38y + 12.
A)
(–4, 3)
B)
(–4, –3)
C)
(4, –3)
D)
(4, 3)
Answer:
A
34)
Find the critical points for f(x, y) =x2+ xy +y2– 3x + 2.
A)
(2, –1)
B)
(2, 1)
C)
(–2, –1)
D)
(–2, 1)
Answer:
A
35)
Find the local extrema for f(x, y) =x2– 2xy + 4y2– 6x – 6y + 8.
A)
f(1, 1) = –1 is a minimum
B)
f(0, 0) = 1 is a minimum
C)
f(5, 2) = –13 is a minimum
D)
f(–5, 2) = 55 is a maximum
Answer:
C
36)
Find the local extrema for f(x, y) =x3– 12xy + 8y3 .
A)
(– 2, 1) = 9 is a minimum
B)
(– 2, – 1) = 9 is a maximum
C)
(2, 1) = –8 is a minimum
D)
(2, 1) = –8 is a maximum
Answer:
C
37)
Find the local extrema for f(x, y) =x3+y3+ 6xy + 1.
A)
f(2, 2) = 41 is a local maximum
B)
f(1, 1) = –1 is a local minimum
C)
f(0, 0) = 1 is a local minimum
D)
f(–2, –2) = 9 is a local maximum
Answer:
D
38)
Find the local extrema for f(x, y) =x3– 12x +y2 .
A)
f(0, 2) = 4 is a maximum
B)
f(2, 0) = –16 is a minimum
C)
f(0, 0) = 0 is a minimum
D)
f(0, 0) = 0 is a maximum
Answer:
B
Solve the problem.
39)
Suppose that the labor cost for a building is approximated by C(x,y) =10x2+ 3y2– 280x – 360y + 16,000, 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.
A)
x =36, y =180
B)
x =60, y =36
C)
x =28, y =120
D)
x =14, y =60
Answer:
D
40)
A company uses TV and magazines for advertising. They know that profit P is related to the amounts T spent
on TV and M spent on magazines by the equation P =24MT – 6M – 3T + 4, where P, M, and T are in hundreds
of thousands. Find the maximum profit.
A)
$650,000
B)
$670,000
C)
$335,000
D)
$325,000
Answer:
D
Provide an appropriate response.
41)
Use Lagrange multipliers to maximize f(x, y) = 5xy subject to x + y = – 6.
A)
max f(x, y) = f(3, –3) = –45
B)
max f(x, y) = f(–3, –3) = 45
C)
max f(x, y) = f(3, 3) = 45
D)
max f(x, y) = f(–3, 3) = –45
Answer:
B
8
42)
Maximize the product of two numbers if their sum must be 26.
A)
f(x, y) = f(–13, –13) = 169
B)
f(x, y) = f(13, 13) = 169
C)
f(x, y) = f(–13, –13) = 26
D)
f(x, y) = f(13, 13) = 26
Answer:
B
43)
Use Lagrange multipliers to minimize f(x, y) =x2+y2– xy subject to x – y = 10.
A)
f(1, 2) = 3
B)
f(5, 5) = 25
C)
f(5, –5) = 75
D)
f(2, –1) = 7
Answer:
C
44)
Use Lagrange multipliers to maximize f(x, y, z) = 24x + 12y + 24z subject to x2+y2+z2= 324.
A)
max f(x, y, z) = f(6, 12, 12) = 576
B)
max f(x, y, z) = f(12, 6, 12) = 648
C)
max f(x, y, z) = f(12, 12, 6) = 576
D)
max f(x, y, z) = f(12, 12, 12) = 720
Answer:
B
45)
Use Lagrange multiplier to maximize f(x, y, z) = xy + z subject to x2+y2+z2= 1.
A)
f(1, 1, 0) = 1
B)
f(0, 0, 1) = 1
C)
f(1, 1, 1) = 1
D)
f(0, 1, 0) = 1
Answer:
B
Solve the problem.
46)
The Cobb–Douglas function for a new product is given by N(x, y) = 15x0.6y0.4 where x is the number of units
of labor and y is the number of units of capital required to produce N(x, y) units of the product. Each unit of
labor costs $40, and each unit of capital costs $80. If $400,000 has been budgeted for the production of this
product, determine how this amount should be allocated in order to maximize production, and find the
maximum production.
A)
6000 units of labor and 6000 units of capital
max N(x,y) = N(6000, 6000)
89,995 units
B)
6000 units of labor and 2000 units of capital
max N(x, y) = N(6000, 2000)
57,995 units
C)
2000 units of labor and 6000 units of capital
max N(x,y) = N(2000, 6000)
46,555 units
D)
2000 units of labor and 2000 units of capital
max N(x,y) = N(2000, 2000)
30,195 units
Answer:
B
47)
The total cost to produce MP3 players in 2 models is given by
C(x, y) = 2x2+ 4y2+ 4xy + 60, where red model is x and the green one is y.
If a total of 60 players must be made, how should production be allocated so that the total cost is minimized?
A)
60 red players and 0 green players
B)
30 red players and 30 green players
C)
59 red players and 1 green players
D)
0 red players and 60 green players
Answer:
A
48)
The rectangular box below, with an open top and one partition, is to be constructed from 18 square inches of
cardboard. Find the dimensions that will result in a box with the largest possible volume.
A)
2 inches by 3 inches by 1 inch
B)
2 inches by 2 inches by 1 inch
C)
3 inches by 3 inches by 1 inch
D)
3 inches by 2 inches by 1 inch
Answer:
D
9
Provide an appropriate response.
49)
Consider the following data on the growth of peach grafts under controlled conditions.
weeks after
grafting
x
height
(inches)
y
1
2
4
5
2
2.4
5.1
7.3
Find the regression line y = ax + b.
A)
y = 1.33x + 0.21
B)
y =1.50 x + 0.7
C)
y = –2.10x + 0.2
D)
y = 13x + 8.19
Answer:
A
50)
Find the least squares line for the following data:
x y
2–1
4 1
6 0
8 1
10 2
A)
y = –0.13x – 1.2
B)
y = 0.3x – 1.2
C)
y = –0.3x + 1.2
D)
y = 0.13x – 1.2
Answer:
B
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
51)
Refer to the table given below. Find the least squares line and use it to estimate y when x = 15.
x y
–215
111
4 4
7–1
10 –7
Answer:
39.92; y = –1.87x + 11.87
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
52)
Find the least squares line for the following data:
x y
49 61
67 72
78 77
85 87
91 93
A)
y = – 0.74x – 23.04
B)
y = 13.46x + 0.04
C)
y = 0.74x + 23.04
D)
y = 1.29x – 26.83
Answer:
C
53)
Find the least squares line for the following data:
n = 12, x = 38, y = 64, x2= 764, xy = 86
A)
y = – 0.18x + 5.90
B)
y = 0.18x + 5.90
C)
y = – 0.18x – 5.90
D)
y = 0.18x – 5.90
Answer:
A
10
54)
Find the least squares line for the points (4, 3), (6, 6), (8, 0), and (9, 9). Graph the data and the least squares line
on the same axes.
A)
y = 0.51x + 1.07
B)
y = 0.17x + 6
C)
y = 0.43x + 1.71
D)
y = 1.07x – 0.51
Answer:
A
11
Solve the problem.
55)
The market research department for a drug store chain arrived at the demand table below, where y is the
number of bottles of multivitamins purchased per month (in thousands) at x dollars per bottle.
x 5.0 5.5 6.0 6.5 7.0
y 2.5 2.4 2.3 2.2 2.1
I) Find a demand equation using the method of least squares.
II) If each bottle of multivitamins costs the drug store chain $4, how should it be priced to achieve a maximum monthly
profit? [Hint: Use the result from I) with C = 4y, R = xy, and P = R – C.]
A)
I) y = –0.20x + 3.50
II) $10.75
B)
I) y = –0.20x – 3.50
II) $10.75
C)
I) y = 0.20x + 3.50
II) $10.75
D)
I) y = 0.20x – 3.50
II) $10.75
Answer:
A
56)
Consider the data showing the average life expectancy of woman in various years.
year life
expectancy
1900 61.2
1910 62.9
1920 63.75
1930 65.0
1940 66.5
1950 68.1
1960 69.9
1970 70.95
1980 73.7
Find the regression line. Let the year 1900 represent x = 0.
A)
y = 0.37x – 21.8
B)
y = 0.15x + 60.9
C)
y = – 0.11.2x + 23.1
D)
y = 8.3x + 22.1
Answer:
B
57)
The table lists the high school grade–point averages of six students and their college grade–point averages after
one year of college.
High School GPA College GPA
2.1 1.6
2.4 1.9
2.7 2.3
3.0 2.5
3.3 2.7
3.8 3.4
Use the least–squares line to estimate the college GPA for a student with a high school GPA of 3.5.
A)
about 2.8
B)
about 2.6
C)
about 3.0
D)
about 3.2
Answer:
C
12
Evaluate.
58)
(9x3y + 3y2)dx
A)
9
4x4+ 3xy2+ C(y)
B)
9
4x4+ xy2+ C(y)
C)
9
4x4y + xy2+ C(y)
D)
9
4x4y + 3xy2+ C(y)
Answer:
D
59)
(6x2y4– 7x3y)dx
A)
6
5y5x2+7
2x3y2+ C(y)
B)
3xy4– 21x2y + C(y)
C)
2x3y4+4
7x4y + C(y)
D)
2x3y4–7
4x4y + C(y)
Answer:
D
60)
1
0(6x2y2+ x + 2y) dy
A)
x2+ x + 1
B)
2x2+ x + 1
C)
x2+ 2x + 1
D)
x2+ x + 2
Answer:
B
Provide an appropriate response.
61)
Evaluate 2
–1
1
–232x3y3dy dx
A)
–480
B)
450
C)
–450
D)
30
Answer:
C
62)
Evaluate yexydA for R = {(x, y)
0 x 1, 1 y 2}.
R
A)
e2+ 2e – 1
B)
e2+ e – 1
C)
e2+ e + 1
D)
e2– e – 1
Answer:
D
63)
Find the double integral over the rectangular region R with the given boundaries.
(x4y + y) dx dy; R: 0 x 2, 0 y 3
A)
144
5
B)
128
5
C)
126
5
D)
189
5
Answer:
B
13
64)
Find the double integral over the rectangular region R with the given boundaries.
(x2+y2) dx dy for R: 0 x 2, – 1 y 1
A)
20
3
B)
10
3
C)
6
D)
8
Answer:
A
65)
Find the average value of f(x, y) = 2 – 4x + 2y over the rectangle R = {(x, y)
0 x 1, 0 y 2}.
A)
8
B)
4
C)
2
D)
1
Answer:
C
66)
Find the average value of the function over the shaded region. f(x, y) = 9x + y
A)
3
4
B)
15
8
C)
12
17
D)
2
5
Answer:
D
67)
Find the volume of the solid under the graph of f(x, y) = 3 + 2x2+ 7y over the rectangle
R = {(x, y) 1 x 3, 0 y 1}.
A)
91
3
B)
16
3
C)
–12
D)
55
3
Answer:
A
14
Solve the problem.
68)
An industrial plant located in the center of a small town emits particulate matter into the atmosphere. Suppose
the concentration of particulate matter in parts per million at a point d miles from the plant is given by
C = 120 – 15d2. If the boundaries of the town form a rectangle four miles long and six miles wide, what is the
average concentration of particulate matter throughout the city? Express C as a function of x and y, set up a
double integral, and evaluate.
A)
B)
C)
D)
Answer:
D
69)
Under ideal conditions, if a person driving a car slams on the brakes and skids to a stop on wet pavement, the
length of the skid marks (in feet) is given by the formula L(x, y) = 0.00002xy2, where x is the weight of the car in
pounds and y is the speed of the car in miles per hour. What is the average length of the skid marks for cars
weighing between 2500 and 3500 pounds and traveling at speeds between 45 and 55 miles per hour? Set up a
double integral and evaluate.
A)
B)
C)
D)
Answer:
B
Provide an appropriate response.
70)
Let R be the region bounded by the graphs of the equations y =x3, y = 33 – 2x, and x = 0. Use set notation and
double inequalities to describe R as a regular x region or regular y region, whichever is simpler.
A)
B)
C)
D)
Answer:
D
15
71)
Give a verbal description of the region R = {(x, y)| x 6, y 3} and determine whether R is a
regular x region, regular y region, both, or neither.
A)
B)
C)
D)
Answer:
C
72)
Evaluate 4x2y2dA for R = {(x, y)
0 x 3, 0 y 1}.
R
A)
24
B)
15
C)
–12
D)
12
Answer:
D
73)
Find the double integral over the rectangular region R with the given boundaries.
(1 + x + y) dx dy; R: 0 x 3, 0 y 3
A)
27
B)
18
C)
36
D)
10
Answer:
C
74)
Find the double integral over the rectangular region R with the given boundaries.
(x4y + y) dx dy;
R: 0 x 2, 0 y 3
A)
189
5
B)
144
5
C)
128
5
D)
126
5
Answer:
A
75)
Evaluate. 1
0
1
9y dx dy
A)
–4
B)
11
2
C)
5
D)
–7
2
Answer:
D
Evaluate the integral.
76)
Evaluate the integral with the order reversed.
2
–2
4 –y2
0
dx dy
A)
32
3
B)
4
3
C)
4
D)
2
32
Answer:
A
16
77)
Evaluate the integral with the order reversed.
1
0
2
0dy dx
A)
1
B)
2
C)
x
D)
1
2
Answer:
B
Provide an appropriate response.
78)
Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.
z = 8x + 4y + 7; 0 x 1, 1 y 3
A)
38
B)
26
C)
28
D)
36
Answer:
A
17