Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
1) If f(x, y) = 4x 3 y2 + 3x 2 y4 - 7xy2 + 4x - 3y + 2, find (a) f x (x, y) and (b) f y (x, y).
1)
2) If f(x, y, z) = x 2 y2 + z, find (a) f x (x, y, z), (b) f y(x, y, z), and (c) f z(x, y, z).
2)
z
z
and (b)
3) If z = (2x - y)e7x 2, find (a)
y
x
3)
(1, 0)
.
4) If z =
x2 + 1
z
z
, find (a)
and (b)
.
y
x
y
4)
5) If z =
exy
z
, find
.
2x + 3y
x
5)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
6) If f(x, y, z) = x 2 yz2 + xy2 z + xy, then fx (1, 2, 3) =
6)
A) 50.
B) 55.
C) 48.
D) 36.
E) none of the above
7) If z = y x 2 + 6y, then
A)
3y
+
x 2 + 6y
B)
x 2 + 6y.
z
=
y
7)
x2 + 6y.
C) (2x + 6)y x2 + 6y.
y
+ x2 + 6y.
D)
2
x + 6y
E)
3y
.
2
x + 6y
8) If z = ex/y, then
A)
y x/y
e .
x
z
=
y
8)
B)
x 2 x/y
e .
y
C)
1 x/y
e .
y
1
D) -
x
y2
ex/y .
E)
x x/y
e .
y
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
9) Find
10) Find
f
f
and
where f(x, y) = x 3e2y + y 2 ln 3x and evaluate both derivatives at (1, 0).
x
y
9)
f
f
5xy2
and
where f(x, y) =
.
x
y
3
(x + y3 )
10)
11) If
z = 3x 2 y 3 - 4x5 y 2
find
z
y
11)
12) If
z = 4xy ln (3x + 9y) find
z
x
12)
13) If
f(x, y) = e-7xy
find fx(x, y)
13)
14) 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
14)
15) 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.
15)
16) For the joint-cost function c = 3xy + 5x + 2y + 6000 (in $), determine the marginal costs
and
c
c
16)
c
when x = 15 and y = 20.
y
17) For the production function P = 5.4l 0.741k 0.517, find the marginal productivity functions
P
P
and
.
l
k
17)
18) For the production function P = 6l 3 + 5l 2 k + 6lk 2 + k3 , find the marginal productivity
P
P
functions
and
.
l
k
18)
2
Answer Key
Testname: UNTITLED1
(b) 8x 3 y + 12x 2 y3 - 14xy - 3
1) (a) 12x 2y 2 + 6xy4 - 7y 2 + 4
x2y
x2
; (c)
2) (a) 2x y 2 + z ; (b)
y2 + z
2 y2 + z
3) (a) -e7x2 ; (b) 30e7
2x
(x 2 + 1)
; (b) 4) (a)
y
y2
5)
exy(3y 2 + 2xy - 2)
(2x + 3y)2
6) A
7) A
8) D
y2
; 2x3e2y + 2yln 3x; 3; 2
9) 3x 2 e2y +
x
10)
5y 5 - 10x 3y 2 10x4 y - 5xy4
;
(x 3 + y 3) 2
(x 3 + y3 ) 2
11)
z
= 9x2 y2 - 8x 5y
y
12)
z
4xy
=
+ 4y ln (3x + 9y)
x x + 3y
13) fx(x, y) = -7ye-7xy
14) (a) 40k - 6L
(b) 40L - 4k
15) 5
c
c
|
= $65/unit;
|
= $47/unit
16)
x (15,20)
y (15,20)
17)
P
k0.517 P
l 0.741
= 4.0014
;
= 2.7918
l
l 0.259 k
k0.483
18)
P
P
= 18l2 + 10lk + 6k2 ;
= 5l 2 + 12lk + 3k2
l
k
3