Deck 18: Multivariable Calculus

Question 1

If z = y ,then
=

A)
B)
C) (2x + 6)y
D) +
E) +

Answer 1

D

Question 2

Find
and
where f(x,y)=

+
ln 3x and evaluate both derivatives at

Answer 2

3 + ; 2 + 2yln 3x; 3; 2

Question 3

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)= 15.64

.Find
Then find and interpret
(155,66)

Answer 3

(w,h)= 6.647 ; (155,66)≈ 7.63; the surface area of a 155-pound person who is 66 inches tall increases by about 7.63 square inches for every 1-pound increase in weight.

Question 4

If f(x,y,z)=

,find (a)
(x,y,z),(b)
(x,y,z),and (c)
(x,y,z).

Question 5

If z = ,then
=

A)
B)
C)
D)
E) -

Question 6

If z = (2x - y)
,find (a)
and (b)

.

Question 7

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)= 15.64

.Find
Then find and interpret
(105,64).

Question 8

Find
and
where f(x,y)=
.

Question 9

If f(x,y)= ,then
(x,y)=

A)
B)
C)
D)
E) none of the above

Question 10

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)= 15.64

.Find
Then find and interpret
(155,66)

Question 11

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)=
,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
.Then find and interpret

.

Question 12

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)=
,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
.Then find and interpret

.

Question 13

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)=
,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
.Then find and interpret

.

Question 14

Find
and
where f(x,y,z)=
.

Question 15

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)=
,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
.Then find and interpret

.

Question 16

If z =
,find (a)
and (b)
.

Question 17

If f(x,y)= 4

+ 3

- 7
+ 4x - 3y + 2,find (a)
(x,y)and (b)
(x,y).

Question 18

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)= 15.64

.Find
Then find and interpret
(105,64).

Question 19

If f(x,y,z)=
+
Z + xy,then
(1,2,3)=

A) 36.
B) 48.
C) 50.
D) 55.
E) none of the above

Question 20

If z =
,find
.

Question 21

If f(x,y)=
find

Question 22

For the joint-cost function c = 5xy
+ 8000 (in $),determine the marginal costs
and
when
and y = 5.

Question 23

If f(x,y)=
find

Question 24

The demand function for product A is
=
,and the demand function for product B is
where
and
are the quantities demanded for A and B,respectively,and
and
are their respective prices.Determine:
(a)the marginal demand for A with respect to
(b)the marginal demand for B with respect to
(c)whether A and B are competitive,complementary,or neither

Question 25

If z = 4xy ln (3x + 9y)find

Question 26

For the joint-cost function c = 3xy + 5x + 2y + 6000 (in $),determine the marginal costs
and
when
and y = 20.

Question 27

If z = 4xy ln (3x + 9y)find

Question 28

If f(x,y)=
find

Question 29

If z = 3

- 4

find

Question 30

If f(x,y)=
find

Question 31

A company manufactures two products,X and Y,and the joint-cost function for these products is given by
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
and

Question 32

If f(x,y)=
find

Question 33

A company's production function is given by P = 2.1

,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

Question 34

If z = 3

- 4

find

Question 35

A manufacturer of widgets has determined that the production function for a weekly production of p thousand gross of widgets is p = 1000 + 20

- 5
- 3
,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.

Question 36

If f(x,y)=
find

Question 37

Let
= 50 - 5
+ 6
and
= 20
×
be demand functions,where
and
are prices for products A and B,respectively.Find all four marginal demand functions.

Question 38

The demand function for product A is
,and the demand function for product B is
where
and
are the quantities demanded for A and B,respectively,and
and
are their respective prices.Determine:
(a)the marginal demand for A with respect to
(b)the marginal demand for B with respect to
(c)whether A and B are competitive,complementary,or neither

Question 39

A company's production function is given by P = 40Lk - 3
- 2
+ 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

Question 40

A company manufactures two products,X and Y,and the joint-cost function for these products is given by
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.

Question 41

For the production function P = 6
+ 5
k + 6l
+
,find the marginal productivity functions
and
.

Question 42

Let f(x,y,z)= ln(
+ 6
)- 2


+

.Find
.

Question 43

If g(u,v,w)= ( + 3
(4w - 5),then
(-1,1,3)=

A) 16.
B) -32.
C) 32.
D) 64.
E) -64.

Question 44

If w = z(
+ 3
,find:
(a)
(b)
(c)
(d)
(e)

Question 45

The Cobb-Douglas production function for a company is given by P = 20

,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
and
.

Question 46

If w = f (x,y,z)= 2
y + 3

+ 4
,find:
(a)
(b)
(c)
(d)
(e)

Question 47

If f(x,y,z)= (2x + +
,then
=

A) 0.
B) 3 + 2y
C) 24y
D) 2x + + z
E) 6(2x + + z)

Question 48

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

.Find
and
.

Question 49

The Cobb-Douglas production function for a company is given by P(l,k)= 70

,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

and

.

Question 50

If f(x,y)=
,find:
(a)
(x,y)
(b)
(x,y)
(c)
(x,y)

Question 51

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

.Find
and
.

Question 52

The Cobb-Douglas production function for a company is given by P(l,k)= 20

,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

and

.

Question 53

Let f(x,y)=

+
ln 2x.Find:
,
,

Question 54

If w = f (x,y,z)=
yz -
+
,find:
(a)
(b)
(c)
(d)
(e)

Question 55

If f(x,y)= y,then
(1,0)=

A) 1.
B) 2.
C) 3.
D) 4.
E) 0.

Question 56

The Cobb-Douglas production function for a company is given by P = 70

,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
and
.

Question 57

For the production function P = 5.4

,find the marginal productivity functions
and
.

Question 58

Let f(x,y,z)=

+


+ y ln(
- 1).Find
.

Question 59

Let f(x,y)= 3
+ 5
.Find:
,
,

Question 60

If f(x,y)= 2

- 3

+ 4xy - x + 2y + 4,find:
(a)
(x,y)
(b)
(x,y)
(c)
(x,y)
(d)
(-1,1)
(e)
(x,y)

Question 61

Use the method of Lagrange multipliers to determine the critical points of f(x,y)= 4
+ 2
+ 3 subject to the constraint x+ 2y = 9.

Question 62

Determine the critical points of f(x,y)=
+ xy +
- 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.

Question 63

The number of critical points of f(x,y)= + 3
+ 3
- 15x + 2 is

A) 0.
B) 1.
C) 2.
D) 3.
E) 4.

Question 64

Determine the critical points of f(x,y)= 2xy - 3x - y -
- 3
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.

Question 65

Determine the critical points of f(x,y)=
+

- 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.

Question 66

The number of critical points of f(x,y)= +
Y +
- 2y + 2 is

A) 0.
B) 1.
C) 2.
D) 3.
E) 4.

Question 67

The function f(x,y)=
+

+ xy - 6x + 3 has a relative minimum at

A) (3, -3).
B) (-3, 3).
C) (2, -2).
D) (-2, 2).
E) (2, 2).

Question 68

Determine the critical points of f(x,y)=
+ 2xy + 2
- 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.

Question 69

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.

Question 70

Determine all of the critical points of f(x,y)=
+ 3
- 9x +
- 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.

Question 71

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 +
+ 8
- 24xy.How many units of type A and type B televisions should the company produce to minimize its cost?

Question 72

Determine the critical points of f(x,y)= 4
+ 2x -
+ 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.

Question 73

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 +
+ 64
- 96xy.How many units of type A and type B televisions should the company produce to minimize its cost?

Question 74

Determine all of the critical points of f(x,y)=

+
- 3x +

- 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.

Question 75

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
,
of A and B that can be sold each week are given by the joint-demand functions
= 10 -
+
and
= 12 +
- 3
where
and
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.

Question 76

How many critical points does the function f(x,y)= + xy - 2
- ln x - ln y have?

A) 0
B) 1
C) 2
D) 3
E) 4

Question 77

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)= 15.64

.
Find

and

.

Question 78

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)= 15.64

.
Find

and

.

Question 79

Determine the critical points of f(x,y)= 3
+ 4
- 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.

Question 80

The function f(x,y)= +

+ 2xy - 8y + 6 has a relative minimum at

A) (3, 6).
B) (6, -6).
C) (-3, 3).
D) (2, -2).
E) (-4, 4).