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)
Find fxy for the function f(x, y) =3x2y + 5 .
4)
5)
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
5)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
6)
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.
6)
A)
217 units
B)
129 units
C)
32 units
D)
2 units
Evaluate the integral.
7)
Evaluate the integral with the order reversed.
1
0
2
0
dy dx
7)
A)
1
2
B)
2
C)
x
D)
1
Provide an appropriate response.
4
8)
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.
8)
A)
y = 0.17x + 6
B)
y = 1.07x – 0.51
C)
y = 0.43x + 1.71
D)
y = 0.51x + 1.07
9)
Find critical points for f(x, y) = 5x2– 5y2+ 2xy + 34x + 38y + 12.
9)
A)
(–4, –3)
B)
(–4, 3)
C)
(4, 3)
D)
(4, –3)
10)
Find the double integral over the rectangular region R with the given boundaries.
(x4y + y) dx dy; R: 0 x 2, 0 y 3
10)
A)
128
5
B)
189
5
C)
144
5
D)
126
5
11)
Find the least squares line for the following data:
n = 12, x = 38, y = 64, x2= 764, xy = 86
11)
A)
y = – 0.18x – 5.90
B)
y = 0.18x – 5.90
C)
y = – 0.18x + 5.90
D)
y = 0.18x + 5.90
12)
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
12)
A)
36
B)
26
C)
28
D)
38
Find the partial derivative.
13)
Find fx for f(x, y) =x3+ 9x2y + 4xy3 .
13)
A)
3x2+ 18xy + 4y3
B)
3x2+ 2xy + 4y3
C)
3x2
D)
x2+ 9xy + 4y3
6
Provide an appropriate response.
14)
Maximize the product of two numbers if their sum must be 26.
14)
A)
f(x, y) = f(–13, –13) = 169
B)
f(x, y) = f(–13, –13) = 26
C)
f(x, y) = f(13, 13) = 169
D)
f(x, y) = f(13, 13) = 26
15)
Find the average value of the function over the shaded region. f(x, y) = 9x + y
15)
A)
3
4
B)
12
17
C)
15
8
D)
2
5
Find the partial derivative.
16)
Let z = f(x,y) =3x2– 19xy + 4y3. Find z
x.
16)
A)
6x + 19y2
B)
–19x + 12y2
C)
–19x – 12y
D)
6x – 19y
7
Provide an appropriate response.
17)
Find the least squares line for the following data:
x y
2–1
4 1
6 0
8 1
10 2
17)
A)
y = – 0.3x + 1.2
B)
y = 0.13x – 1.2
C)
y = 0.3x – 1.2
D)
y = – 0.13x – 1.2
18)
Find the critical points for f(x, y) =x2+ xy +y2– 3x + 2.
18)
A)
(–2, –1)
B)
(–2, 1)
C)
(2, 1)
D)
(2, –1)
Solve the problem.
19)
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 – 3T + 4, where
P, M, and T are in hundreds of thousands. Find the maximum profit.
19)
A)
$362,500
B)
$745,000
C)
$372,500
D)
$725,000
20)
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.
20)
A)
2 inches by 2 inches by 1 inch
B)
3 inches by 3 inches by 1 inch
C)
2 inches by 3 inches by 1 inch
D)
3 inches by 2 inches by 1 inch
Provide an appropriate response.
21)
Find the average value of f(x, y) = 2 – 4x + 2y over the rectangle R = {(x, y) 0 x 1, 0 y 2}.
21)
A)
2
B)
4
C)
1
D)
8
Solve the problem.
22)
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.
22)
A)
1
24
2
–2
3
–3
[120 – 15(x2+y2)] dy dx = 55 parts per million
B)
1
12
2
–2
3
–3
[120 – 15(x2+y2)] dy dx = 145 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 = 55 parts per million
Provide an appropriate response.
23)
Evaluate
2
–1
1
–2
32x3y3dy dx
23)
A)
30
B)
–480
C)
450
D)
–450
Solve the problem.
24)
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.
24)
A)
about 2.6
B)
about 3.0
C)
about 3.2
D)
about 2.8
Find the value.
25)
Let f(x, y) = xy2+x. Find f(4, -7).
25)
A)
198
B)
-119
C)
(4, -7) is not in the domain of f.
D)
-24
A
26)
Let f(x, y) = xy2+x. Find f(9, 8).
26)
A)
B)
C)
D)
D
Solve the problem.
27)
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.
27)
A)
656,640 units
B)
54,054,000 units
C)
152,974 units
D)
5,405,400,000 units
C
B
Provide an appropriate response.
28)
Find the local extrema for f(x, y) =x2– 2xy + 4y2– 6x – 6y + 8.
28)
A)
f(1, 1) = – 1 is a minimum
B)
f(0, 0) = 1 is a minimum
C)
f(–5, 2) = 55 is a maximum
D)
f(5, 2) = – 13 is a minimum
Find the value.
29)
Let f(x, y) =x
y+ xy. Find f(7, 7).
29)
A)
1
B)
48
C)
50
D)
98
Provide an appropriate response.
30)
Find fxy for f(x, y) = 10 x2y4– 7 x3y5 .
30)
A)
80x y3– 105x2y4
B)
160xy3– 21x2y4
C)
160x y3–105 x2y4
D)
80x y3– 21x2y4
31)
Evaluate yexy dA for R = {(x, y) 0 x 1, 1 y 2}.
R
31)
A)
e2+ 2e – 1
B)
e2+ e – 1
C)
e2+ e + 1
D)
e2– e – 1
32)
Find the double integral over the rectangular region R with the given boundaries.
(x4y + y) dx dy;
R: 0 x 2, 0 y 3
32)
A)
126
5
B)
189
5
C)
144
5
D)
128
5
Solve the problem.
33)
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.
33)
A)
y = 8.3x + 22.1
B)
y = 0.37x – 21.8
C)
y = 0.15x + 60.9
D)
y = – 0.11.2x + 23.1
Evaluate the integral.
34)
Evaluate the integral with the order reversed.
2
–2
4 –y2
0
dx dy
34)
A)
4
3
B)
2
32
C)
4
D)
32
3
Evaluate.
35)
(9x3y + 3y2)dx
35)
A)
9
4x4+ 3xy2+ C(y)
B)
9
4x4y + xy2+ C(y)
C)
9
4x4+ xy2+ C(y)
D)
9
4x4y + 3xy2+ C(y)
Solve the problem.
36)
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 75 acres of alfalfa
and 35 acres of grass, how many cows may graze?
36)
A)
847 cows
B)
690 cows
C)
850 cows
D)
687 cows
37)
Suppose that the labor cost for a building is approximated by
C(x,y) =2x2+ 6y2– 80x – 480y + 12,000, 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.
37)
A)
x =20, y =40
B)
x =40, y =240
C)
x =240, y =120
D)
x =40, y =80
Provide an appropriate response.
38)
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.
38)
A)
R consists of the points on or inside the rectangle with corners (±3, ±6); both
B)
R consists of the points on or inside the rectangle with corners (±6, ±3); neither
C)
R consists of the points on or inside the rectangle with corners (±6, ±3); both
D)
R consists of the points on or inside the rectangle with corners (±6, ±3); regular x region