39)
Use Lagrange multipliers to maximize f(x, y) = 5xy subject to x + y = – 6.
39)
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
40)
Find z
x for z = f(x, y) = 4x2– 11xy + 4y3 .
40)
A)
8x – 11y
B)
–11x – 12y
C)
– 11x + 12y2
D)
8x + 11y2
Provide an appropriate response.
41)
Use Lagrange multipliers to minimize f(x, y) =x2+y2– xy subject to x – y = 10.
41)
A)
f(5, –5) = 75
B)
f(2, –1) = 7
C)
f(1, 2) = 3
D)
f(5, 5) = 25
Solve the problem.
42)
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.]
42)
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
Find the value.
43)
Find f(2, –1, 4) for f(x, y, z) = 3x2– 4y4+ z – 7.
43)
A)
5
B)
13
C)
–5
D)
27
Provide an appropriate response.
44)
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
44)
A)
8
B)
6
C)
20
3
D)
10
3
45)
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.
45)
A)
1
10,000
3500
2500
55
45
0.00002xy2 dy dx = 1505 feet
B)
1
10,000
3500
2500
55
45
0.00002xy2 dy dx = 150.50 feet
C)
1
10,000
3500
2500
55
45
0.00002xy2 dy dx = 15,050 feet
D)
1
10,000
3500
2500
55
45
0.00002xy2 dy dx = 200 feet
Provide an appropriate response.
46)
Find the local extrema for f(x, y) =x3– 12xy + 8y3 .
46)
A)
(2, 1) = – 8 is a minimum
B)
(2, 1) = – 8 is a maximum
C)
(– 2, – 1) = 9 is a maximum
D)
(– 2, 1) = 9 is a minimum
Solve the problem.
47)
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.
47)
A)
$49,975
B)
$50,195
C)
$29,415
D)
$20,565
A
Provide an appropriate response.
48)
Find fxy for f(x, y) = 8x3 y – 7y2+ 2x .
48)
A)
24x2
B)
48xy
C)
–14
D)
–28
A
49)
For f(x, y) = 6x2+ 7xy4– 5y2+ 8, find fxx(x, y) +fyx(x, y).
49)
A)
12 + 28xy3
B)
12 + 28y3
C)
28y3
D)
12
B
16
A
50)
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.
50)
A)
105,074 units
B)
117,517 units
C)
175,174 units
D)
150,000 units
Provide an appropriate response.
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)
36
B)
10
C)
18
D)
27
Find the partial derivative.
52)
Find fy(–2, –3) for the function f(x, y) =7y2+5x3–4x5y.
52)
A)
–170
B)
86
C)
– 86
D)
170
Provide an appropriate response.
53)
Use Lagrange multiplier to maximize f(x, y, z) = xy + z subject to x2+y2+z2= 1.
53)
A)
f(1, 1, 1) = 1
B)
f(1, 1, 0) = 1
C)
f(0, 0, 1) = 1
D)
f(0, 1, 0) = 1
17
54)
Evaluate.
1
0
1
9y
dx dy
54)
A)
5
B)
–7
2
C)
11
2
D)
–4
Solve the problem.
55)
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 .
55)
A)
8.1 x
y
0.7
B)
8.1 y
x
0.7
C)
8.1x y0.7
D)
8.1 y
x
1.3
56)
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.
56)
A)
327 in.3
B)
1523 in.3
C)
779 in.3
D)
790 in.3
Provide an appropriate response.
57)
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.
57)
A)
y =1.50 x + 0.7
B)
y = 13x + 8.19
C)
y = – 2.10x + 0.2
D)
y = 1.33x + 0.21
58)
Evaluate 4x2y2dA for R = {(x, y) 0 x 3, 0 y 1}.
R
58)
A)
24
B)
12
C)
–12
D)
15
59)
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.
59)
A)
about 630k
B)
about 617k
C)
about 61.7k
D)
about 16.67k
60)
Find fx(2, -6) when f(x,y) = 7x2– 9xy.
60)
A)
-80
B)
26
C)
-26
D)
82
Evaluate.
61)
(6x2y4– 7x3y) dx
61)
A)
3xy4– 21x2y + C(y)
B)
2x3y4+4
7x4y + C(y)
C)
2x3y4–7
4x4y + C(y)
D)
6
5y5x2+7
2x3y2+ C(y)
Provide an appropriate response.
62)
Find the local extrema for f(x, y) =x3+y3+ 6xy + 1.
62)
A)
f(1, 1) = – 1 is a local minimum
B)
f(2, 2) = 41 is a local maximum
C)
f(0, 0) = 1 is a local minimum
D)
f(–2, –2) = 9 is a local maximum
Solve the problem.
63)
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.
63)
A)
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
B)
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
C)
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
D)
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.
Provide an appropriate response.
64)
Use Lagrange multipliers to maximize f(x, y, z) = 24x + 12y + 24z subject to x2+y2+z2= 324.
64)
A)
max f(x, y, z) = f(12, 6, 12) = 648
B)
max f(x, y, z) = f(6, 12, 12) = 576
C)
max f(x, y, z) = f(12, 12, 12) = 720
D)
max f(x, y, z) = f(12, 12, 6) = 576
Solve the problem.
65)
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?
65)
A)
60 red players and 0 green players
B)
59 red players and 1 green players
C)
0 red players and 60 green players
D)
30 red players and 30 green players
66)
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.
66)
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
Provide an appropriate response.
67)
Find the local extrema for f(x, y) =x3– 12x +y2 .
67)
A)
f(2, 0) = – 16 is a minimum
B)
f(0, 2) = 4 is a maximum
C)
f(0, 0) = 0 is a maximum
D)
f(0, 0) = 0 is a minimum
Find the value.
68)
Find f(7, 5) when f(x, y) =8x + 9y – 6
68)
A)
101
B)
87
C)
95
D)
86
Solve the problem.
69)
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 =62
and H =1.49.
69)
A)
1.48 m2
B)
1.74 m2
C)
1.59 m2
D)
1.56 m2
Provide an appropriate response.
70)
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}.
70)
A)
55
3
B)
16
3
C)
91
3
D)
–12
Solve the problem.
71)
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.
71)
A)
10 x4y6
B)
12 x5y5
C)
5 x4y6
D)
6 x5y5
Provide an appropriate response.
72)
Find the least squares line for the following data:
x y
49 61
67 72
78 77
85 87
91 93
72)
A)
y = 1.29x – 26.83
B)
y = – 0.74x – 23.04
C)
y = 13.46x + 0.04
D)
y = 0.74x + 23.04
Evaluate.
73)
1
0
(6x2y2+ x + 2y) dy
73)
A)
2x2+ x + 1
B)
x2+ x + 1
C)
x2+ x + 2
D)
x2+ 2x + 1
Find the partial derivative.
74)
For f(x, y) = 3x4– 4x3y + 5y3– 4, find fx(1, –2).
74)
A)
12
B)
36
C)
24
D)
–12
Find the value.
75)
Find f(–3, 1, 5) for f(x, y, z) =1
3x2– 8y5+ z – 4.
75)
A)
1
B)
–5
C)
–4
D)
10
Provide an appropriate response.
76)
Find fxx +fyy for f(x, y) =5x3–2x2y2–y3+ 1 .
76)
A)
30x –4y2–4x2
B)
30x2–4xy2–4x2y
C)
12x2–4xy2–4x2y
D)
30y –4x2–4y2– 6y
77)
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.
77)
A)
R is a regular x 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 y 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}
Find the partial derivative.
78)
Find fx(2, –1) for f(x, y) =4x3–2x2+3y2– 3.
78)
A)
16
B)
8
C)
40
D)
–48
24
Answer Key