85)
If z= (2x–y)e7x2, find (a) z
y and (b) z
x(1, 0) .
85)
86)
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.
86)
87)
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) .
87)
88)
Evaluate:
5
3
(y+ 1) dx dy
88)
89)
Evaluate:
1
0
2
0
(2ex– 5ey) dx dy
89)
90)
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.
90)
19
91)
If z= 4xy ln (3x+ 9y)find z
y
91)
92)
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.
92)
93)
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) .
93)
94)
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.
94)
95)
Determine the critical points of f(x, y) = 3x2+ 4y2– 2x+ 8y 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.
95)
96)
A company manufactures two products, X and Y, and the joint–cost function for these
products is given by c= 0.002(x+y)2+x+ 0.25y+ 8000, 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= 450 and y= 550.
96)
97)
For x2y+xz +z2= 4, evaluate z
x when x= – 1, y= 2, z= – 1.
97)
98)
If z=x y and x=2r
s; y= 4r2s find:
(a) z
r
(b) z
s
98)
99)
If z= 10x+ 5y and x= 2rs; y= 3r+ 5s find:
(a) z
r
(b) z
s
99)
100)
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(105, 64).
100)
21
101)
A firm has an order of 10,000 units of its product and has two plants at which to
manufacture these units. Let q1 be the number of units to be produced at the first plant and
q2 denote the number to be manufactured at the second plant. It is known that the cost
function is given by
C= 48 q3
1+ 3 q3
2+ 25,000. Use the method of Lagrange multipliers to determine how many
units should be produced at each plant to minimize this cost function.
101)
102)
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) .
102)
103)
Find f
x and f
y where f(x, y) =5xy2
(x3+y3).
103)
104)
If w=f (x, y, z) =x2yz –yz2+xz2, find:
(a) w
x
(b) w
y
(c) w
z
(d) 2w
y2
(e) 2w
xz
104)
22
105)
If w=f (x, y, z) = 2x2y+ 3xy2z2+ 4xz3, find:
(a) w
x
(b) w
y
(c) w
z
(d) 2w
y2
(e) 2w
xz
105)
106)
A company’s production function is given by P= 2.1L0.6k0.4, 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
106)
107)
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.
107)
108)
If f(x, y) =x+ 9
xy2+ 5 find fy(x, y)
108)
109)
Use the method of Lagrange multipliers to determine the critical points of f(x, y, z) =x2+
4y–z2subject to the constraint x +2y– 4z= 3.
109)
23
110)
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, 1000, 2) .
110)
111)
Let qA= 50 – 5pA+ 6 p2
B and qB= 20 pA×p
–1
B be demand functions, where pA and pB
are prices for products A and B, respectively. Find all four marginal demand functions.
111)
112)
 
 
 
 
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
112)
24
113)
An open rectangular cardboard box is to have a volume of 4 cubic feet. Find the
dimensions of the box so that the amount of cardboard is minimized.
113)
114)
For exy + 7x3+ 8z– 18 = 0, the partial derivative z
y evaluated at x= – 1, y= 0, z= 3 is
114)
115)
If f(x, y) =
33x2– 5y3find fx(x, y)
115)
116)
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)
116)
117)
If z= 3x2y3– 4x5y2find z
y
117)
118)
For 2x2+ 3y2+ 2z2= 16, evaluate z
y when x= 1, y= 2, z= – 1.
118)
119)
Find f
x and f
y where f(x, y) =x3e2y+y2 ln 3x and evaluate both derivatives at (1, 0).
119)
25
120)
If f(x, y) =e–7xy find fx(x, y)
120)
121)
Evaluate:
1
0
2
0
xy2dx dy
121)
122)
Let f(x, y, z) = ln(x4+ 6y2) – 2z4x2e3y+x20y3. Find 3f
xyz.
122)
123)
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
M. Then find and interpret
Q
M(100, 500, 3) .
123)
124)
Determine all of the critical points of f(x, y) =1
3x3+x2– 3x+1
3y3– 4y. Also use the
second derivative test to determine, if possible, whether a maximum, minimum or saddle
point occurs at each of these critical points.
124)
125)
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.
125)
26
126)
Evaluate
1
0
1
0
x2
0
(x2+y2) dz dy dx
126)
127)
Evaluate:
1
0
x3+1
1
x2ydy dx
127)
128)
Determine the critical points of f(x, y) =x3+1
2y2– 3xy – 4y+ 2 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.
128)
129)
If f(x, y) = 2x4y3– 3x3y3+ 4xy –x+ 2y+ 4, find:
(a) fx(x, y)
(b) fy(x, y)
(c) fxy(x, y)
(d) fxy(–1, 1)
(e) fyyx(x, y)
129)
130)
Let f(x, y, z) =xz3ez2+1+x2y3z4+y ln(e2– 1). Find 3f
xyz.
130)
27
131)
Find the equation of the least squares linear regression line of y on x for the data table
below.
x0 1 2
y3 4 5.5
131)
132)
If f(x, y) =
33x2– 5y3find fy(x, y)
132)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
133)
133)
A)
9.064
B)
8.669
C)
8.874
D)
8.675
134)
134)
A)
^
y= – 47.3 + 2.02x
B)
^
y= 11.7 + 1.02x
C)
^
y= 92.3 – 0.669x
D)
^
y= 2.81 + 1.35x
Answer Key
Testname: C17
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Answer Key
Testname: C17
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Answer Key
Testname: C17
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Answer Key
Testname: C17
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Answer Key
Testname: C17
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Answer Key
Testname: C17
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Answer Key
Testname: C17