139)
Which of the following pairs of values (x, y) maximizes the function f(x, y) = x + 3y subject to the
constraint x2+ 9y2= 72 assuming x and y are positive?
(I) (–6, –2) (II) 8, 8
9(III) (2, –6) (IV) (6, 2)
139)
A)
IV only
B)
I only
C)
III only
D)
I and II
E)
none of these
140)
Calculate the iterated integral
2
–1
1
–2
32x3y3 dy dx.
140)
A)
B)
C)
–480
D)
A
141)
Find the formula that gives the least squares error for the points (–2, –3), (1, –1), (3, 4).
141)
A)
(–2A + B + 3)2+(A + B + 1)2+(3A + B – 4)2
B)
(–2A + B + 3)2+(A + B + 1)2+(A + B – 4)2
C)
(–A + B + 3)2+(A + B + 1)2+(A + B – 4)2
D)
(–2A + B + 1)2+(A + B + 3)2+(3A + B – 4)2
A
A
142)
A rectangular garden is to be surrounded on three sides by a fence costing $5 per foot and on one
side by a stone wall costing $15 per foot. Let x be the length of the side with the stone wall, and let
y be the length of each of the other three sides. Express the cost of enclosing the garden as a
function of two variables.
142)
A)
C(x, y) = (20x)(10y)
B)
C(x, y) = 15x + 15y
C)
C(x, y) = (15x)(5y)
D)
C(x, y) = 15x + 5y
E)
none of these
143)
Let f(x, y, z) = xyz(1 +eyz). Find f
z.
143)
A)
xy(1 +yeyz)
B)
xy(1 +eyz) +xy2zeyz
C)
xy(1 +eyz) + xy(1 + yeyz)
D)
xy(1 +ey)
E)
none of these
144)
Let H(x, y) =3xy
x2– y . Find 2H
y2.
144)
A)
3x3
(x2– y)2
B)
6x3y –6x5
(x2– y)4
C)
6x3
(x2– y)3
D)
9x4–6x3–9x2y
(x2– y)3
E)
none of these
Solve the problem.
145)
Find two positive numbers x and y such that x + y =60 and xy2 is maximized.
145)
A)
x =15 and y =45
B)
x =20 and y =40
C)
x =30 and y =30
D)
x = 1 and y =59
146)
A company has the following production function for a certain product:
p(x, y) =22x0.4y0.6 .
Find the marginal productivity with fixed capital, p
x.
146)
A)
B)
C)
8.8 y
x
0.6
D)
147)
Suppose the partial derivatives of a Lagrange function F(x, y, ) are F
x= 2 – 8x, F
y= 1 –2y,
F
= 32 – 4x2–y2.What values of x and y minimize F(x, y, )? (Assume x and y are positive.)
147)
A)
(4, 2)
B)
(2, 4)
C)
(2, 2)
D)
(2 2, 4)
E)
none of these
148)
A tennis racket manufacturer produces two types of rackets, standard and competition. The weekly
revenue function, in dollars, for x standard rackets and y competition rackets is given by
R(x, y) = 54x + 2xy + 398y – 2x2– 9y2
i) How many of each type of racket must be produced each week to maximize revenue?
ii) What is the maximum weekly revenue?
148)
A)
i) 25 standard rackets and 25 competition rackets;
ii) $5675
B)
i) 26 standard rackets and 25 competition rackets;
ii) $5677
C)
i) 26 standard rackets and 26 competition rackets;
ii) $5668
D)
i) 25 standard rackets and 26 competition rackets;
ii) $5664
Evaluate the function.
149)
Find f(0, 1, –1) when f(x, y, z) =9x–6yz +8x.
149)
A)
B)
C)
–5
D)
Solve the problem.
150)
The production level P of a factory during one time period is modeled by P(x, y) = Kx1/2y1/2 where
K is a positive integer, x is the number of units of labor scheduled and y is the number of units of
capital invested. If labor costs $1200/unit, capital costs $700/unit and the owner has $1,700,000
available for one time period, what amount of labor and capital would maximize production?
150)
A)
708.3 units of labor and 1214.3 units of capital
B)
1416.7 units of labor and 2428.6 units of capital
C)
653.8 units of labor and 1062.5 units of capital
D)
1214.3 units of labor and 708.3 units of capital
151)
A company has a Cobb–Douglas production function f(x, y) = 20x0.33y0.67 where x is the
utilization of labor and y is the utilization of capital. Determine the number of units of product
produced when 1728 units of labor and 27,000 units of capital are used.
151)
A)
B)
C)
46,605 units
D)
Evaluate the iterated integral.
152)
8
0
3
0
(2x + 8y) dx dy
152)
A)
B)
C)
35
D)
153)
Let R be the rectangle consisting of all points (x, y) such that 1 x 4, 1 y 9. Calculate
R
4xy dx dy.
153)
A)
B)
C)
4800
D)
Evaluate the iterated integral.
154)
 
5
0
4
0
(1 + x + y) dx dy
154)
A)
B)
C)
81
2
D)
38
155)
Let g(x, y) =x – 6y
x2+y2. Compute g(3, 4).
155)
A)
B)
C)
–25
21
D)
156)
Let f(x, y) =5x + 7y – 7. Compute f(–8, –3).
156)
A)
B)
C)
–73
D)
D
Find the partial derivative.
157)
Find f
x(9, –8) when f(x,y) = 7x2– 9xy.
157)
A)
B)
C)
–81
D)
B
158)
Let f(x, y) =x4–y2+3x2+ 2y – 7. The first partial derivatives of f(x, y) are zero at the points (0, 1)
and (–1, 1). Use the second derivative test to determine the nature of f(x, y) at each of these points.
158)
A)
(0, 1) no conclusion possible, (–1, 1) minimum
B)
(0,1) neither relative maximum nor minimum, (–1, 1) maximum
C)
(0, 1) relative maximum, (–1, 1) relative minimum
D)
(–1, 1) relative maximum, (0, 1) neither relative maximum nor minimum
E)
none of these
D
159)
Let f(x, y) = ln(y + 2) +e2xy. Compute f(0, –1).
159)
A)
B)
C)
1 + e
D)
D
39
B
160)
The table below gives the height and weight of four randomly selected University undergraduate
women. What is the least squares error E for these data points?
height (cm) 150 155 165 170
weight (kg) 60 55 70 62
160)
A)
E = 150(A + B – 60)2+ 155(A + B – 55)2+ 165(A + B – 70)2+ 170(A + B – 62)2
B)
E = (150A + 60B)2+ (155A + 55B)2+ (165A + 70B)2+ (170A + 62B)2
C)
E = (150A – 60B)2+ (155A – 55B)2+ (165A – 70B)2+ (170A – 62B)2
D)
E = (150A + B – 60)2+ (155A + B – 55)2+ (165A + B – 70)2+ (170A + B – 62)2
E)
none of these
Solve the problem.
161)
The profit (in thousands of dollars) that a company earns from producing x tons of
brass and y tons of steel can be approximated by P(x, y) =80xy – 8x3–y3. Find the amount of brass
and steel that maximize profit and find the value of the maximum profit.
161)
A)
40
7 tons of brass and 16 tons of steel; maximum profit is $1,733,632
B)
8 tons of brass and 10 tons of steel; maximum profit is $1,304,064
C)
20
3 tons of brass and 40
3 tons of steel; maximum profit is $2,370,370
D)
40
3 tons of brass and 20
3 tons of steel; maximum profit is $2,370,370
162)
Calculate the iterated integral
ln 2
0
2
1
xy + yexy dx dy.
162)
A)
3
4(ln 2) +1
2
B)
3
2ln –e2 ln 2 –3
2
C)
3
2ln 2 –1
2ex2–5
2
D)
3
4(ln 2)2+1
2
E)
none of these
41
Answer Key
Testname: C7
42
Answer Key
Testname: C7
43
Answer Key
Testname: C7
Answer Key
Testname: C7