Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
For x2+xy +yz +z2= 6, the partial derivative z
y evaluated at x= 1, y= 2, z= 1 is
1)
A)
–3
5
B)
–3
2
C)
–1
D)
–5
2
E)
–1
2
2)
Evaluate:
1
0
2x
0
xdy dx
2)
A)
4
5
B)
6
5
C)
2
D)
2
5
E)
8
5
3)
The number of critical points of f(x, y) =x3+ 3xy2+ 3y2– 15x+ 2 is
3)
A)
0.
B)
1.
C)
2.
D)
3.
E)
4.
1
4)
If f(x, y) =3xy +y2
x2y3+ 9 , then fx(x, y) =
4)
A)
3y
2xy3
B)
3x
(x2y3+9)2
C)
y(27 – 3x2y3– 2xy4)
(x2y3+9)2
D)
y(27 + 9x2y3+ 2xy4)
(x2y3+9)2
E)
none of the above
5)
For s3t3+ 2r2–s2= 2, the partial derivative t
r evaluated at r = 1, s= 1, t= 1 is
5)
A)
–2.
B)
4.
C)
–4
3.
D)
0.
E)
none of the above
6)
If f(x, y, z) =x2yz2+xy2z+xy, then fx(1, 2, 3) =
6)
A)
50.
B)
48.
C)
36.
D)
55.
E)
none of the above
2
7)
If x2+xy +yz +z2= 6, then z
y=
7)
A)
–x+z
2z
B)
–x+ 2z
y
C)
–y+ 3z
y
D)
–x+ 2z
x+ 2y
E)
–x+z
y+ 2z
8)
The number of critical points of f(x, y) =x2+x2y+y2– 2y+ 2 is
8)
A)
0.
B)
1.
C)
2.
D)
3.
E)
4.
B)
E)
9)
If s3t3+ 2r2–s2= 1, then t
r=
9)
A)
r– 3s2
t.
B)
–4r
3s3t2.
C)
31 – 2r2+s2
s3.
D)
4r.
E)
none of the above
B)
E)
10)
A critical point of f(x, y) = 2x2+ 3y2+ 7 subject to the constraint 2x– 5y= 31 is
10)
A)
7
6, 4
3
B)
(–4, 1)
C)
(0, 0)
D)
(3, –5)
E)
(2, 3)
B)
E)
3
B)
E)
11)
The function f(x, y) =x2+1
3y3+ 2xy – 8y+ 6 has a relative minimum at
11)
A)
(–3, 3).
B)
(3, 6).
C)
(–4, 4).
D)
(2, –2).
E)
(6, –6).
12)
If z= (2x+ 3y)3 and x=r2– 2s and y= 2s–r, then z
s when r= 2 and s= 1 is
12)
A)
104.
B)
84.
C)
96.
D)
88.
E)
92.
13)
A company must fill an order for 200 units of its product. It wishes to distribute the production
between its two plants, plant A and plant B. The total cost function is
c= 300qA+ 200qB+q2
B+ 8000, where qA and qB are the number of units produced at plant A and
plant B, respectively. To minimize costs, plant A and plant B should produce
13)
A)
50 units and 150 units, respectively.
B)
25 units and 175 units, respectively.
C)
150 units and 50 units, respectively.
D)
100 units and 100 units, respectively.
E)
175 units and 25 units, respectively.
14)
If g(u, v, w) = (u2+ 3v)2(4w– 5), then gwu(–1, 1, 3) =
14)
A)
–32.
B)
64.
C)
16.
D)
32.
E)
–64.
4
15)
If z=y x2+ 6y, then z
y=
15)
A)
3y
x2+ 6y
+x2+ 6y.
B)
y
x2+ 6y
+x2+ 6y.
C)
(2x+ 6)y x2+ 6y.
D)
3y
x2+ 6y
.
E)
x2+ 6y.
16)
Evaluate:
1
–1
y2
y
(2x+ 3y) dx dy
16)
A)
–34
15
B)
–4
15
C)
–9
16
D)
–11
3
E)
–4
3
17)
The function f(x, y) =1
3x3+1
2y2+xy – 6x+ 3 has a relative minimum at
17)
A)
(–3, 3).
B)
(2, 2).
C)
(3, –3).
D)
(–2, 2).
E)
(2, –2).
18)
If f(x, y, z) = (2x+y2+z)3, then 3f
zyx=
18)
A)
3 + 2y
B)
24y
C)
2x+y2+z
D)
0.
E)
6(2x+y2+z)
19)
If f(x, y) =x2y, then fyx(1, 0) =
19)
A)
1.
B)
2.
C)
3.
D)
4.
E)
0.
20)
A monopolist produces two products, A, and B. The joint–cost function is c= 5qA+ 3qB+ 5000
where c is the total cost of producing qA units of A and qB units of B. the demand functions for these
products are given by pA= 205 – 2qA–qBand pB= 153 –qA–qB, where pA and pB are the prices of
A and B, respectively. The number of units of A and the number of units B that should be sold to
maximize the monopolist’s profit is
20)
A)
10 units of A and 15 units of B.
B)
75 units of A and 100 units of B.
C)
25 units of A and 50 units of B.
D)
50 units of A and 75 units of B.
E)
15 units of A and 25 units of B.
6
21)
Evaluate:
2
0
1
0
x2y3 dx dy
21)
A)
2
3
B)
1
3
C)
1
D)
5
3
E)
4
3
22)
If z=ex/y, then z
y=
22)
A)
y
xex/y.
B)
1
yex/y.
C)
x2
yex/y.
D)
–x
y2ex/y.
E)
x
yex/y.
23)
If z=x3y2+x2y3– 3xy and x= 2r+ 3s and y=r–s, then z
s when r= 1 and s= 0 is
23)
A)
8.
B)
35.
C)
1.
D)
17.
E)
5.
24)
The critical point of f(x, y) =x2– 4y2+ 3z2 subject to the constraint x+ 16y– 9z= 36 is
24)
A)
(3, 1, –4)
B)
(1, –3, 4)
C)
(–1, 4, 3)
D)
(4, 1, –3)
E)
(4, –3, 1)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
25)
If 2x2+ 3y2+ 2z2= 16, find z
y.
25)
7
26)
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.
26)
27)
If f(x, y) = 4x3y2+ 3x2y4– 7xy2+ 4x– 3y+ 2, find (a) fx(x, y) and (b) fy(x, y).
27)
28)
Use the method of Lagrange multipliers to find the critical points of f(x, y, z) = 2x+ 4y– 4z
subject to the constraint x2+y2+z2= 9.
28)
29)
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.
29)
30)
If z=x2ey+y2ex where x= 2rs2 and y= ln r2+ ln s2, find z
s.
30)
31)
A manufacturer of widgets has determined that the production function for a weekly
production of p thousand gross of widgets is p= 1000 + 20l2k3– 5l3– 3k4, where l is the
number of labor hours per week in thousands and k is the amount of capital in thousands
of dollars per week. Determine both of the marginal productivity functions.
31)
32)
Use the method of Lagrange multipliers to determine the critical points of f(x, y) = 4x2+ 2
y2+ 3 subject to the constraint x+ 2y= 9.
32)
8
33)
The Cobb–Douglas production function for a company is given by P= 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
Pll and Plk.
33)
34)
If x2y+xz +z2= 4, find z
x.
34)
35)
Determine the critical points of f(x, y) =x2+xy +y3–y 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.
35)
36)
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.
36)
37)
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.
37)
38)
The Cobb–Douglas production function for a company is given by P(k, l) = 65k0.3l0.7
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 $60 per unit, labor costs $140 per
unit, and the total cost of capital and labor is limited to $70,000. 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.
38)
39)
Evaluate:
1
0
x3+1
1
x2ydy dx
39)
40)
An empirical formula relating the surface area A (in square inches) of an average human
body to the weight (in pounds) and the height h (in inches) of the person is A(w, h) = 15.64
w0.425h0.725. Find Awh and Aww.
40)
41)
If z= 4xy ln (3x+ 9y)find z
y
41)
42)
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
42)
43)
Find f
x and f
y where f(x, y) =x3e2y+y2 ln 3x and evaluate both derivatives at (1, 0).
43)
10
44)
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) .
44)
45)
If f(x, y) =x+ 9
xy2+ 5 find fy(x, y)
45)
46)
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) = 100 +x3+ 64y3–
96xy. How many units of type A and type B televisions should the company produce to
minimize its cost?
46)
47)
If f(x, y) =e–7xy find fy(x, y)
47)
48)
Use the method of Lagrange multipliers to determine the critical points of f(x, y) =x+ 2y
subject to the constraint xy = 8.
48)
11
49)
Let f(x, y) =x2e3y+y3ln 2x. Find: 2f
x2, 2f
y2, 2f
yx
49)
50)
An empirical formula relating the surface area A (in square inches) of an average human
body to the weight (in pounds) and the height h (in inches) of the person is A(w, h) = 15.64
w0.425h0.725. Find Ahw and Ahh.
50)
51)
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).
51)
52)
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
52)
53)
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
53)
54)
If f(x, y) =exy, find:
(a) fx(x, y)
(b) fxx(x, y)
(c) fxy(x, y)
54)
55)
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.
55)
56)
If w= z(x2+ 3xy)3, find:
(a) w
x
(b) w
y
(c) w
z
(d) 2w
z2
(e) 2w
xy
56)
57)
The demand function for product A is qA=
10 2pB
pA
, and the demand function for product
B is qB= 20 + 3pA–2pB, 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
57)
58)
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.
58)
59)
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
59)
60)
If f(x, y) =
33x2– 5y3find fy(x, y)
60)
61)
If z= 4xy ln (3x+ 9y)find z
x
61)
62)
To fill an order for 100 units of a product, a firm wishes to distribute the production
between its two plants, Plant 1 and Plant 2. The total cost function is given by
c=f(q1, q2) = 0.5 q2
1+ 2q1+ 32q2+ 500, where q1 and q2 are the number of units produced
at Plants 1
and 2, respectively. How should the output be distributed in order to minimize costs?
62)
63)
Use the method of Lagrange multipliers to find the critical points of f(x, y, z) = 4x+ 2y– 4z
subject to the constraint x2+y2+z2=1.
63)
15
64)
An empirical formula relating the surface are 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. FindAw(w, h). Then find and interpret Aw(105, 64).
64)
65)
If z=x y and x=2r
s; y= 4r2s find:
(a) z
r
(b) z
s
65)
66)
Let f(x, y, z) = ln(x4+ 6y2) – 2z4x2e3y+x20y3. Find 3f
xyz.
66)
67)
Use implicit partial differentiation to find z
x from ln(xyz) =ey+ 79.
67)
68)
If z= 10x+ 5y and x= 2rs; y= 3r+ 5s find:
(a) z
r
(b) z
s
68)
16
69)
If f(x, y) =x+ 9
xy2+ 5 find fx(x, y)
69)
70)
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
C. Then find and interpret
Q
C(100, 500, 3) .
70)
71)
Find f
x and f
y where f(x, y) =5xy2
(x3+y3).
71)
72)
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.
72)
17
73)
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)
73)
74)
Evaluate:
1
0
lnx
0
eydy dx
74)
75)
Use implicit partial differentiation to find z
y from exy + 7x3+ 8z– 19 = 0.
75)
76)
If z= 3x2y3– 4x5y2find z
x
76)
77)
For 2x2+ 3y2+ 2z2= 16, evaluate z
y when x= 1, y= 2, z= – 1.
77)
18
78)
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) .
78)
79)
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.
79)
80)
For x2y+xz +z2= 4, evaluate z
x when x= – 1, y= 2, z= – 1.
80)
81)
Evaluate:
5
–1
3
1
(y+ 1) dx dy
81)
82)
The Cobb–Douglas production function for a company is given by P(k, l) = 20k2/3l1/3
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 $150 per unit, labor costs $225 per
unit, and the total cost of capital and labor is limited to $270,000. 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.
82)
19