Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Choose the surface that the function describes.
1)
f(x, y) = 1 – x – 2y
1)
A)
B)
C)
D)
2)
f(x, y) = 3 –x2
2)
A)
B)
C)
D)
3)
f(x, y) = 5
3)
A)
2
B)
C)
D)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
4)
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
4)
5)
Find fxy for the function f(x, y) =3x2y + 5 .
5)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
6)
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.]
6)
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
Provide an appropriate response.
7)
Use Lagrange multipliers to maximize f(x, y, z) = 24x + 12y + 24z subject to x2+y2+z2= 324.
7)
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
4
Find the partial derivative.
8)
Find fx(2, –1) for f(x, y) =4x3–2x2+3y2– 3.
8)
A)
16
B)
8
C)
–48
D)
40
Provide an appropriate response.
9)
Find the least squares line for the following data:
n = 12, x = 38, y = 64, x2= 764, xy = 86
9)
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:
D
Explanation:
Evaluate.
10)
(9x3y + 3y2)dx
10)
A)
9
4x4y + xy2+ C(y)
B)
9
4x4+ 3xy2+ C(y)
C)
9
4x4+ xy2+ C(y)
D)
9
4x4y + 3xy2+ C(y)
Answer:
D
Explanation:
Solve the problem.
11)
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 =48MT – 6M – 4T + 1, where
P, M, and T are in hundreds of thousands. Find the maximum profit.
11)
A)
$100,000
B)
$120,000
C)
$50,000
D)
$60,000
Answer:
C
Explanation:
Answer:
D
Explanation:
A)
B)
C)
D)
Provide an appropriate response.
12)
Find the double integral over the rectangular region R with the given boundaries.
(x4y + y) dx dy;
R: 0 x 2, 0 y 3
12)
A)
144
5
B)
128
5
C)
126
5
D)
189
5
Solve the problem.
13)
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?
13)
A)
30 red players and 30 green players
B)
59 red players and 1 green players
C)
0 red players and 60 green players
D)
60 red players and 0 green players
Provide an appropriate response.
14)
Find the local extrema for f(x, y) =x3– 12x +y2 .
14)
A)
f(0, 2) = 4 is a maximum
B)
f(0, 0) = 0 is a minimum
C)
f(2, 0) = –16 is a minimum
D)
f(0, 0) = 0 is a maximum
Solve the problem.
15)
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.
15)
A)
6 x5y5
B)
12 x5y5
C)
10 x4y6
D)
5 x4y6
Provide an appropriate response.
16)
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}.
16)
A)
55
3
B)
16
3
C)
–12
D)
91
3
17)
Use Lagrange multiplier to maximize f(x, y, z) = xy + z subject to x2+y2+z2= 1.
17)
A)
f(1, 1, 1) = 1
B)
f(0, 1, 0) = 1
C)
f(0, 0, 1) = 1
D)
f(1, 1, 0) = 1
Find the value.
18)
Find f(2, –1, 4) for f(x, y, z) = 3x2– 4y4+ z – 7.
18)
A)
5
B)
–5
C)
13
D)
27
7
Solve the problem.
19)
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.
19)
A)
1
10,000
3500
2500
55
45 0.00002xy2 dy dx = 150.50 feet
B)
1
10,000
3500
2500
55
45 0.00002xy2 dy dx = 15,050 feet
C)
1
10,000
3500
2500
55
45 0.00002xy2 dy dx = 200 feet
D)
1
10,000
3500
2500
55
45 0.00002xy2 dy dx = 1505 feet
Provide an appropriate response.
20)
Find fxx +fyy for f(x, y) =5x3–2x2y2–y3+ 1 .
20)
A)
12x2–4xy2–4x2y
B)
30y –4x2–4y2– 6y
C)
30x –4y2–4x2
D)
30x2–4xy2–4x2y
8
Solve the problem.
21)
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.
21)
A)
about 3.2
B)
about 2.6
C)
about 2.8
D)
about 3.0
Provide an appropriate response.
22)
Maximize the product of two numbers if their sum must be 26.
22)
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
23)
Find the average value of f(x, y) = 2 – 4x + 2y over the rectangle R = {(x, y) 0 x 1, 0 y 2}.
23)
A)
8
B)
2
C)
1
D)
4
Find the partial derivative.
24)
Let z = f(x,y) =6x2– 10xy + 2y3. Find z
x.
24)
A)
–10x – 6y
B)
12x – 10y
C)
–10x + 6y2
D)
12x + 10y2
9
Solve the problem.
25)
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.
25)
A)
Px(10, 15) = 60
At a sales level of 10 units of model A and 15 units of model B, increasing sales of model A by
one unit and holding sales of model B at 15 units will increase profit by approximately $60
B)
Px(10, 15) = 120
At a sales level of 10 units of model A and 15 units of model B, increasing sales of model A by
one unit and holding sales of model B at 15 units will increase profit by approximately $120
C)
Px(10, 15) = 80
At a sales level of 10 units of model A and 15 units of model B, increasing sales of model A by
one unit and holding sales of model B at 15 units will increase profit by approximately $80.
D)
Px(10, 15) = 140
At a sales level of 10 units of model A and 15 units of model B, increasing sales of model A by
one unit and holding sales of model B at 15 units will increase profit by approximately $140
Find the partial derivative.
26)
Find fx for f(x, y) =x3+ 9x2y + 4xy3 .
26)
A)
3x2
B)
3x2+ 18xy + 4y3
C)
x2+ 9xy + 4y3
D)
3x2+ 2xy + 4y3
Provide an appropriate response.
27)
Find the least squares line for the following data:
x y
2–1
4 1
6 0
8 1
10 2
27)
A)
y = 0.13x – 1.2
B)
y = –0.13x – 1.2
C)
y = 0.3x – 1.2
D)
y = –0.3x + 1.2
Find the value.
28)
Let f(x, y) = xy2+x. Find f(4, -7).
28)
A)
198
B)
-119
C)
-24
D)
(4, -7) is not in the domain of f.
Solve the problem.
29)
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.
29)
A)
1
12
2
–2
3
–3[120 – 15(x2+y2)] dy dx = 55 parts per million
B)
1
24
2
–2
3
–3[120 – 15(x2+y2)] dy dx = 55 parts per million
C)
1
24
2
–2
3
–3[120 – 15(x2+y2)] dy dx = 145 parts per million
D)
1
12
2
–2
3
–3[120 – 15(x2+y2)] dy dx = 145 parts per million
B
Provide an appropriate response.
30)
Use Lagrange multipliers to minimize f(x, y) =x2+y2– xy subject to x – y = 10.
30)
A)
f(2, –1) = 7
B)
f(5, –5) = 75
C)
f(1, 2) = 3
D)
f(5, 5) = 25
B
11
A
Find the partial derivative.
31)
Find fy(–2, –3) for the function f(x, y) =7y2+5x3–4x5y.
31)
A)
170
B)
86
C)
–170
D)
– 86
Solve the problem.
32)
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.
32)
A)
150,000 units
B)
175,174 units
C)
105,074 units
D)
117,517 units
33)
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.
33)
A)
32 units
B)
217 units
C)
2 units
D)
129 units
Provide an appropriate response.
34)
Find fxy for f(x, y) = 10 x2y4– 7 x3y5 .
34)
A)
160x y3–105 x2y4
B)
80x y3– 21x2y4
C)
80x y3– 105x2y4
D)
160xy3– 21x2y4
35)
Use Lagrange multipliers to maximize f(x, y) = 5xy subject to x + y = – 6.
35)
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
36)
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.
36)
A)
y = 1.07x – 0.51
B)
y = 0.51x + 1.07
13
C)
y = 0.43x + 1.71
D)
y = 0.17x + 6
Find the partial derivative.
37)
Find fx(3, -7) when f(x,y) = 7x2– 9xy.
37)
A)
105
B)
21
C)
-126
D)
-21
38)
For f(x, y) = 3x4– 4x3y + 5y3– 4, find fx(1, –2).
38)
A)
24
B)
–12
C)
36
D)
12
Solve the problem.
39)
The number of cows that can graze on a ranch is approximated by C(x,y) = 9x + 5y – 3, where x is
the number of acres of grass and y the number of acres of alfalfa. If the ranch has 40 acres of alfalfa
and 80 acres of grass, how many cows may graze?
39)
A)
757 cows
B)
760 cows
C)
920 cows
D)
917 cows
14
Provide an appropriate response.
40)
Find the average value of the function over the shaded region. f(x, y) = 9x + y
40)
A)
15
8
B)
12
17
C)
2
5
D)
3
4
Solve the problem.
41)
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 =63
and H =1.7.
41)
A)
2.00 m2
B)
1.59 m2
C)
1.70 m2
D)
1.73 m2
Provide an appropriate response.
42)
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.
42)
A)
y =1.50 x + 0.7
B)
y = –2.10x + 0.2
C)
y = 13x + 8.19
D)
y = 1.33x + 0.21
Evaluate the integral.
43)
Evaluate the integral with the order reversed.
1
0
2
0dy dx
43)
A)
2
B)
1
2
C)
x
D)
1
Provide an appropriate response.
44)
Find critical points for f(x, y) = 5x2– 5y2+ 2xy + 34x + 38y + 12.
44)
A)
(4, 3)
B)
(–4, 3)
C)
(–4, –3)
D)
(4, –3)
B
Evaluate.
45)
(6x2y4– 7x3y)dx
45)
A)
2x3y4–7
4x4y + C(y)
B)
6
5y5x2+7
2x3y2+ C(y)
C)
3xy4– 21x2y + C(y)
D)
2x3y4+4
7x4y + C(y)
A
Solve the problem.
46)
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.
46)
A)
656,640 units
B)
54,054,000 units
C)
152,974 units
D)
5,405,400,000 units
C
A
Provide an appropriate response.
47)
Find the least squares line for the following data:
x y
49 61
67 72
78 77
85 87
91 93
47)
A)
y = 0.74x + 23.04
B)
y = 13.46x + 0.04
C)
y = – 0.74x – 23.04
D)
y = 1.29x – 26.83
48)
Evaluate 2
–1
1
–232x3y3dy dx
48)
A)
450
B)
–450
C)
–480
D)
30
Evaluate.
49)
1
0(6x2y2+ x + 2y) dy
49)
A)
2x2+ x + 1
B)
x2+ 2x + 1
C)
x2+ x + 2
D)
x2+ x + 1
Provide an appropriate response.
50)
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.
50)
A)
R is a regular y region; R = {(x, y)|x3 y 33 – 2x, 0 x 3}
B)
R is a regular x region; R = {(x, y)|x3 y 33 – 2x, 0 x 2}
C)
R is a regular x region; R = {(x, y)|x3 y 33 – 2x, 0 x 3}
D)
R is a regular x region; R = {(x, y)| 33 – 2x y x3, 0 x 3}
51)
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
51)
A)
18
B)
36
C)
10
D)
27
Solve the problem.
52)
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.
52)
A)
about 16.67k
B)
about 61.7k
C)
about 630k
D)
about 617k
53)
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.
53)
A)
2 inches by 2 inches by 1 inch
B)
3 inches by 2 inches by 1 inch
C)
2 inches by 3 inches by 1 inch
D)
3 inches by 3 inches by 1 inch
Provide an appropriate response.
54)
Find the local extrema for f(x, y) =x3+y3+ 6xy + 1.
54)
A)
f(0, 0) = 1 is a local minimum
B)
f(2, 2) = 41 is a local maximum
C)
f(–2, –2) = 9 is a local maximum
D)
f(1, 1) = –1 is a local minimum
18
55)
Find the critical points for f(x, y) =x2+ xy +y2– 3x + 2.
55)
A)
(–2, 1)
B)
(2, 1)
C)
(2, –1)
D)
(–2, –1)
Solve the problem.
56)
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.
56)
A)
$20,565
B)
$49,975
C)
$50,195
D)
$29,415
Find the partial derivative.
57)
Find z
x for z = f(x, y) = 4x2– 11xy + 4y3 .
57)
A)
– 11x + 12y2
B)
8x + 11y2
C)
–11x – 12y
D)
8x – 11y
Provide an appropriate response.
58)
Evaluate 4x2y2dA for R = {(x, y) 0 x 3, 0 y 1}.
R
58)
A)
12
B)
15
C)
–12
D)
24
19
59)
Evaluate yexydA for R = {(x, y) 0 x 1, 1 y 2}.
R
59)
A)
e2+ e + 1
B)
e2+ 2e – 1
C)
e2+ e – 1
D)
e2– e – 1
60)
For f(x, y) = 6x2+ 7xy4– 5y2+ 8, find fxx(x, y) +fyx(x, y).
60)
A)
28y3
B)
12
C)
12 + 28y3
D)
12 + 28xy3
C
Find the value.
61)
Let f(x, y) = xy2+x. Find f(9, 7).
61)
A)
444
B)
574
C)
70
D)
72
A
Provide an appropriate response.
62)
Find fxy for f(x, y) = 8x3 y – 7y2+ 2x .
62)
A)
–28
B)
–14
C)
48xy
D)
24x2
D
Find the value.
63)
Find f(-5, 3) when f(x, y) =9x + 3y – 9
63)
A)
-54
B)
-48
C)
-36
D)
-45
D
20
D
D)
Evaluate the integral.
64)
Evaluate the integral with the order reversed.
2
–2
4 –y2
0
dx dy
64)
A)
4
B)
2
32
C)
4
3
D)
32
3
Provide an appropriate response.
65)
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
65)
A)
36
B)
28
C)
26
D)
38
66)
Find the double integral over the rectangular region R with the given boundaries.
(x4y + y) dx dy; R: 0 x 2, 0 y 3
66)
A)
128
5
B)
126
5
C)
144
5
D)
189
5
Solve the problem.
67)
Suppose that the labor cost for a building is approximated by
C(x,y) =10x2+ 3y2– 240x – 180y + 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.
67)
A)
x =30, y =18
B)
x =24, y =60
C)
x =18, y =90
D)
x =12, y =30
68)
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 =8 inches, find the volume of
potting soil required to fill the pot to the top. Round to the nearest cubic inch.
68)
A)
790in.3
B)
1523 in.3
C)
779in.3
D)
327in.3
69)
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.
69)
A)
6000 units of labor and 2000 units of capital
max N(x, y) = N(6000, 2000)
57,995 units
B)
2000 units of labor and 6000 units of capital
max N(x,y) = N(2000, 6000)
46,555 units
C)
2000 units of labor and 2000 units of capital
max N(x,y) = N(2000, 2000)
30,195 units
D)
6000 units of labor and 6000 units of capital
max N(x,y) = N(6000, 6000)
89,995 units
Find the value.
70)
Find f(–3, 1, 5) for f(x, y, z) =1
3x2– 8y5+ z – 4.
70)
A)
10
B)
–4
C)
–5
D)
1
Provide an appropriate response.
71)
Find the local extrema for f(x, y) =x3– 12xy + 8y3 .
71)
A)
(2, 1) = –8 is a minimum
B)
(– 2, 1) = 9 is a minimum
C)
(– 2, – 1) = 9 is a maximum
D)
(2, 1) = –8 is a maximum
72)
Find the local extrema for f(x, y) =x2– 2xy + 4y2– 6x – 6y + 8.
72)
A)
f(1, 1) = –1 is a minimum
B)
f(–5, 2) = 55 is a maximum
C)
f(5, 2) = –13 is a minimum
D)
f(0, 0) = 1 is a minimum
Answer:
C
Explanation:
73)
Evaluate. 1
0
1
9y dx dy
73)
A)
–7
2
B)
–4
C)
5
D)
11
2
Answer:
A
Explanation:
74)
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
74)
A)
10
3
B)
8
C)
20
3
D)
6
Answer:
C
Explanation:
Answer:
A
Explanation:
A)
B)
C)
D)
75)
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.
75)
A)
R consists of the points on or inside the rectangle with corners (±6, ±3); both
B)
R consists of the points on or inside the rectangle with corners (±3, ±6); both
C)
R consists of the points on or inside the rectangle with corners (±6, ±3); regular x region
D)
R consists of the points on or inside the rectangle with corners (±6, ±3); neither
Find the value.
76)
Let f(x, y) =x
y+ xy. Find f(–5, –5).
76)
A)
24
B)
26
C)
50
D)
1
Solve the problem.
77)
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.
77)
A)
y = 0.15x + 60.9
B)
y = 8.3x + 22.1
C)
y = – 0.11.2x + 23.1
D)
y = 0.37x – 21.8
78)
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 .
78)
A)
8.1 y
x0.7
B)
8.1x y0.7
C)
8.1 x
y0.7
D)
8.1 y
x1.3
Answer Key
Testname: C7
Answer Key
Testname: C7