Download Problem Sheet IV (Multivariable Functions)

Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the value.
1) Find f(-2, -8) when f(x, y) = 3x + 8y - 8
A) -78
B) -86
2) Let f(x, y) = xy2 +
A) 136
C) -81
x. Find f(16, 8).
B) 144
3) Find f(2, -1, 4) for f(x, y, z) = 3x2 - 4y4 + z - 7.
A) 13
B) -5
D) -70
C) 1028
D) 2056
C) 27
D) 5
Solve the problem.
4) 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 40 acres of grass, how many cows may graze?
A) 477 cows
B) 417 cows
C) 425 cows
D) 485 cows
5) 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) $49,975
C) $29,415
D) $20,565
Find the partial derivative.
6) Let z = f(x,y) = 6x2 - 18xy + 9y3. Find
A) 12x - 18y
z
.
x
C) -18x + 27y2
B) -18x - 27y
C) -3
B) 86
C) - 86
9) For f(x, y) = 3x4 - 4x3 y + 5y3 - 4, find fx(1, -2).
4)
5)
D) 12x + 18y2
D) 87
8)
D) -170
9)
B) 12
C) 24
10) Find fx (2, -1) for f(x, y) = 4x3 - 2x2 + 3y2 - 3.
A) 16
3)
7)
8) Find fy(-2, -3) for the function f(x, y) = 7y2 + 5x3 - 4x5 y.
A) 36
2)
6)
7) Find fx (3, -5) when f(x,y) = 7x2 - 9xy.
A) -72
B) 3
A) 170
1)
D) -12
10)
B) -48
C) 8
1
D) 40
11) Find
z
for z = f(x, y) = 4x 2 - 11xy + 4y3 .
x
A) 8x + 11y2
11)
B) -11x - 12y
C) 8x - 11y
12) Find fx for f(x, y) = x3 + 9x2 y + 4xy3 .
A) 3x2 + 2xy + 4y3
D) - 11x + 12y2
12)
B) 3x2 + 18xy + 4y3
D) x2 + 9xy + 4y3
C) 3x2
Provide an appropriate response.
13) 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.
13)
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) = 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.
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) = 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
14) 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) 32 units
B) 2 units
C) 217 units
D) 129 units
14)
15) 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 .
x 0.7
y 1.3
A) 8.1
B) 8.1
C) 8.1x y0.7
y
x
15)
D) 8.1
y 0.7
x
16) The production function z for an industrial country was estimated as z = x5 y6 , where x is the
amount of labor and y, the amount of capital. Find the marginal productivity of labor.
A) 12 x5 y5
B) 6 x5 y5
C) 10 x4 y6
D) 5 x4 y6
16)
17) Find fxy for f(x, y) = 8x3 y - 7y2 + 2x .
17)
A) 48xy
C) 24x2
B) -28
2
D) -14
18) Find fxy for f(x, y) = 10 x2 y4 - 7 x3 y5 .
A) 80x y3 - 21x2 y4
18)
B) 160xy3 - 21x2 y4
D) 160x y3 -105 x2 y4
C) 80x y3 - 105x2 y4
19) For f(x, y) = 6x2 + 7xy4 - 5y2 + 8, find fxx(x, y) + fyx(x, y).
A) 12 + 28y3
B) 28y3
C) 12 + 28xy3
19)
D) 12
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
20) Find fxy for the function f(x, y) =
3x2y + 5 .
20)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
21) Find fxx + fyy for f(x, y) = 5x3 - 2x2 y2 - y3 + 1 .
A) 30x2 - 4xy2 - 4x2y
21)
B) 30y - 4x2 - 4y2 - 6y
D) 12x2 - 4xy2 - 4x2y
C) 30x - 4y2 - 4x2
22) 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)
23) 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)
22)
23)
24) Find the local extrema for f(x, y) = x2 - 2xy + 4y2 - 6x - 6y + 8.
A) f(5, 2) = -13 is a minimum
B) f(-5, 2) = 55 is a maximum
C) f(0, 0) = 1 is a minimum
D) f(1, 1) = -1 is a minimum
24)
25) Find the local extrema for f(x, y) = x3 - 12xy + 8y3 .
A) (- 2, 1) = 9 is a minimum
C) (2, 1) = -8 is a minimum
B) (- 2, - 1) = 9 is a maximum
D) (2, 1) = -8 is a maximum
25)
26) Find the local extrema for f(x, y) = x3 + y3 + 6xy + 1.
A) f(-2, -2) = 9 is a local maximum
C) f(1, 1) = -1 is a local minimum
B) f(0, 0) = 1 is a local minimum
D) f(2, 2) = 41 is a local maximum
27) Find the local extrema for f(x, y) = x3 - 12x + y2 .
A) f(2, 0) = -16 is a minimum
C) f(0, 0) = 0 is a minimum
B) f(0, 0) = 0 is a maximum
D) f(0, 2) = 4 is a maximum
Solve the problem.
28) Suppose that the labor cost for a building is approximated by
C(x,y) = 4x2 + 6y2 - 280x - 360y + 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.
A) x = 90, y = 90
B) x = 35, y = 30
C) x = 70, y = 60
D) x = 30, y = 90
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26)
27)
28)
29) 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 - 3M - 2T + 2, where
P, M, and T are in hundreds of thousands. Find the maximum profit.
A) $185,000
B) $175,000
C) $350,000
D) $370,000
Provide an appropriate response.
30) 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
29)
30)
31) Maximize the product of two numbers if their sum must be 26.
A) f(x, y) = f(13, 13) = 26
B) f(x, y) = f(-13, -13) = 169
C) f(x, y) = f(13, 13) = 169
D) f(x, y) = f(-13, -13) = 26
31)
32) Use Lagrange multipliers to minimize f(x, y) = x2 + y2 - xy subject to x - y = 10.
A) f(2, -1) = 7
B) f(5, 5) = 25
C) f(1, 2) = 3
D) f(5, -5) = 75
32)
33) 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 2000 units of capital
max N(x,y) = N(2000, 2000) 30,195 units
D) 2000 units of labor and 6000 units of capital
max N(x,y) = N(2000, 6000) 46,555 units
33)
34) 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.
34)
If a total of 60 players must be made, how should production be alloctated so that the total cost is
minimized?
A) 30 red players and 30 green players
B) 0 red players and 60 green players
C) 59 red players and 1 green players
D) 60 red players and 0 green players
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