83)
If z=(x2+y2)10 where x= 4r2s3 and y=e2r+3s–3, then by means of the chain rule, (a) find
z
r;
(b) evaluate z
r when r= 0 and s= 1.
83)
84)
Determine all of the critical points of f(x, y) =x3+ 3x2– 9x+y3– 12y. Also use the second
derivative test to determine, if possible, whether a maximum, minimum or saddle point
occurs at each of these critical points.
84)
85)
For the production function P= 5.4l0.741k0.517, find the marginal productivity functions
P
l and P
k.
85)
86)
If f(x, y) =e–7xy find fx(x, y)
86)
87)
The Cobb–Douglas production function for a company is given by P(l, k) = 70l1/4k3/4,
where P is the monthly production value when k is the amount of the company’s capital
investment (in dollars per month) and l is the size of the labor force (in work hours per
month). Find 2P
lk(2401, 10,000) and 2P
k2(2401, 10,000) .
87)
88)
Use the method of Lagrange multipliers to determine the critical points of f(x, y, z) =x2– 3
y2–z2+ 6 subject to the constraint 5x– 3y+z= 21.
88)
89)
A manufacturer produces products A and B for which the average costs of production are
constant at 3 and 5 (dollars per unit), respectively. The quantities qA, qB of A and B that can
be sold each week are given by the joint–demand functions qA= 10 –pA+pB and qB= 12
+pA– 3pB where pA and pB are the prices (in dollars per unit) of A and B, respectively.
Determine the prices of A and B at which the manufacturer can maximize profit.
89)
90)
An empirical formula relating the surface area A (in square inches) of an average human
body to the weight w (in pounds) and the height h (in inches) of the person is A(w, h) =
15.64w0.425h0.725. Find Ah(w, h). Then find and interpret Ah(155, 66)
90)
91)
If f(x, y) =
33x2– 5y3find fx(x, y)
91)
92)
Determine the critical points of f(x, y) =x2+ 2xy +2y2– 4y and also determine by the
second–derivative test whether each point corresponds to a relative maximum, to a relative
minimum, to neither, or whether the test gives no information.
92)
93)
Let f(x, y) = 3xy3+ 5e3xy. Find: 2f
x2, 2f
y2, 2f
xy
93)
94)
Evaluate:
2
1
x2
0
(x+y) dy dx
94)
21
95)
The demand function for product A is qA= 500 – 25pA+pB, and the demand function for
product B is qB= 250 + 2pA– 10pB, where qA and qB are the quantities demanded for A
and B, respectively, and pA and pB are their respective prices. Determine:
(a) the marginal demand for A with respect to pB
(b) the marginal demand for B with respect to pA
(c) whether A and B are competitive, complementary, or neither
95)
96)
If z=x2+ 1
y, find (a) z
x and (b) z
y.
96)
97)
The Cobb–Douglas production function for a company is given by P(l, k) = 20l1/3k2/3,
where P is the monthly production value when k is the amount of the company’s capital
investment (in dollars per month) and l is the size of the labor force (in work hours per
month). Find 2P
lk(1728, 27,000) and 2P
l2(1728, 27,000) .
97)
98)
If z– ln (2x+ 3y) and x=re5; y=ser find
(a) z
r
(b) z
s
98)
99)
A television manufacturing company makes two types of TV’s. The cost of producing x
units of type A and y units of type B is given by the function C(x, y) = 120 +x3+ 8y3– 24xy.
How many units of type A and type B televisions should the company produce to
minimize its cost?
99)
100)
Find the equation of the least squares linear regression line of y on x for the data table
below.
x0 2 4
y6 5 1
100)
101)
Evaluate:
2
1
y2
0
xdx dy
101)
102)
An empirical formula relating the surface area A (in square inches) of an average human
body to the weight w (in pounds) and the height h (in inches) of the person is A(w, h) =
15.64w0.425h0.725.
Find 2A
wh(155, 66) and 2A
h2(155, 66) .
102)
103)
A sporting goods store determines that the optimal quantity of athletic shoes (in pairs) to
order each month is given by the Wilson lot size formula: Q(C, M, s) =2CM
s, where C is
the cost (in dollars) of placing an order, M is the number of pairs sold each month, and s is
the monthly storage cost (in dollars) per pair of shoes. Find Q
s. Then find and interpret
Q
s(100, 500, 3) .
103)
23
104)
If z=exy and x=rs; y= ln (r+s) find:
(a) z
r
(b) z
s
104)
105)
An empirical formula relating the surface area A (in square inches) of an average human
body to the weight w (in pounds) and the height h (in inches) of the person is A(w, h) =
15.64w0.425h0.725.
Find 2A
hw(105, 64) and 2A
w2(105, 64) .
105)
Explanation:
106)
Evaluate
1
0
1
0
x2
0
(x2+y2) dz dy dx
106)
Explanation:
107)
Evaluate:
1
0
y2
y
1
0
9x2yz2dz dx dy
107)
Explanation:
108)
For the production function P= 6l3+ 5l2k+ 6lk2+k3, find the marginal productivity
functions P
l and P
k.
108)
Explanation:
24
Explanation:
109)
For the joint–cost function c= 3xy + 5x+ 2y+ 6000 (in $), determine the marginal costs c
c
and c
y when x= 15 and y= 20.
109)
110)
If z=exy
2x+ 3y, find z
x.
110)
111)
Determine the critical points of f(x, y) = 2xy – 3x–y–x2– 3y2 and also determine by the
second–derivative test whether each point corresponds to a relative maximum, to a relative
minimum, to neither, or whether the test gives no information.
111)
112)
An empirical formula relating the surface area A (in square inches) of an average human
body to the weight w (in pounds) and the height h (in inches) of the person is A(w, h) =
15.64w0.425h0.725. Find Aw(w, h). Then find and interpret Aw(155, 66)
112)
113)
If z= 2x2y+ 3xy +y2 where x=r2+ 2rs and y= 2r– 4s, then by means of the chain rule, (a)
find z
s;
(b) evaluate when r= 1 and s= 0.
113)
114)
Find f
x and f
y where f(x, y, z) =4y3
x3+y2.
114)
25
115)
A company’s production function is given by P= 40Lk – 3L2– 2k2+ 500, where P is the
total output generated by L units of labor and k units of capital. Determine:
(a) the marginal production function with respect to L
(b) the marginal production function with respect to k
115)
116)
Evaluate:
1
0
2
0
(2ex– 5ey) dx dy
116)
117)
For the joint–cost function c= 5xy(x+y)2+ 8000 (in $), determine the marginal costs c
c and
c
y when x= 10 and y= 5.
117)
118)
 
 
 
 
Find an equation of the least squares linear regression line of y on x for the data in the table
below. Predict the value of y corresponding to x= 1.5.
x 0 1 2
y 0 2 1
Recall a=
n
i=1
x2
i
n
i=1
yi–
n
i=1
xi
n
i=1
xiyi
n
n
i=1
x2
i–
n
i=1
xi
2 and
b=
n
n
i=1
xiyi–
n
i=1
xi
n
i=1
yi
n
n
i=1
x2
i–
n
i=1
xi
2
118)
26
119)
 
 
 
 
Find an equation of the least squares linear regression line of y on x for the data in the table
below. Predict the value of y corresponding to x= 4.
x 1 2 3
y 2 1 4
Recall a=
n
i=1
x2
i
n
i=1
yi–
n
i=1
xi
n
i=1
xiyi
n
n
i=1
x2
i–
n
i=1
xi
2 and
b=
n
n
i=1
xiyi–
n
i=1
xi
n
i=1
yi
n
n
i=1
x2
i–
n
i=1
xi
2
119)
120)
The Cobb–Douglas production function for a company is given by P(k, l) = 163k1/5l4/5
where P is the monthly production value when k is the number of units of capital and l is
the number of units of labor. Suppose that capital costs $105 per unit, labor costs $70 per
unit, and the total cost of capital and labor is limited to $152,250. Use Lagrange Multiplier’s
to write the system of equations you would use to find the number of units of capital and
labor that maximize production.
120)
121)
The production function for a company’s product is P= 100L+ 50k–L2–k2, where P is the
output that results from L units of labor and k units of capital. The unit costs of labor and
capital are 6 and 3, respectively. If the company wants the total cost of inputs to be 30,
determine the greatest output possible subject to this budget constraint.
121)
122)
The Cobb–Douglas production function for a company is given by P= 70l1/4k3/4, where P
is the monthly production value when k is the amount of the company’s capital investment
(in dollars per month) and l is the size of the labor force (in work hours per month). Find
Pkk and Pkl.
122)
27
123)
If f(x, y, z) =x2y2+z, find (a) fx(x, y, z), (b) fy(x, y, z), and (c) fz(x, y, z).
123)
124)
For exy + 7x3+ 8z– 18 = 0, the partial derivative z
y evaluated at x= – 1, y= 0, z= 3 is
124)
125)
For ln(xyz) +e =ey+ 1, the partial derivative z
x evaluated at x=e–2, y= 1, z =e3
125)
126)
If z= 3x2y3– 4x5y2find z
y
126)
127)
Let f(x, y, z) =xz3ez2+1+x2y3z4+y ln(e2– 1). Find 3f
xyz.
127)
128)
If z= (2x–y)e7x2, find (a) z
y and (b) z
x(1, 0) .
128)
129)
A company manufactures two products, X and Y, and the joint–cost function for these
products is given by c=x x + 4y, where c is the total cost of producing x units of X and y
units of Y. Determine the marginal cost with respect to x when x= 36 and y= 16.
129)
130)
Evaluate:
1
0
2
0
xy2dx dy
130)
28
131)
Determine the critical points of f(x, y) = 4x2+ 2x–y2+ 2y and also determine by the
second–derivative test whether each point corresponds to a relative maximum, to a relative
minimum, to neither, or whether the test gives no information.
131)
132)
The Cobb–Douglas production function for a company is given by P(k, l) = 70k3/4l1/4
where P is the monthly production value when k is the number of units of capital and l is
the number of units of labor. Suppose that capital costs $450 per unit, labor costs $75 per
unit, and the total cost of capital and labor is limited to $60,000. Use Lagrange multipliers
to write the system of equations you would use to find the number of units of capital and
labor that maximize production.
132)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
133)
133)
A)
^
y= 11.7 + 1.02x
B)
^
y= 92.3 – 0.669x
C)
^
y= 2.81 + 1.35x
D)
^
y= – 47.3 + 2.02x
134)
134)
A)
8.675
B)
9.064
C)
8.669
D)
8.874
Answer Key
Testname: C17
Answer Key
Testname: C17
Answer Key
Testname: C17
Answer Key
Testname: C17
Answer Key
Testname: C17
Answer Key
Testname: C17
36
Answer Key
Testname: C17