Deck 7: Functions of Several Variables

Question 1

Let f(x, y,) = xy + 5. Compute f(1, 2 + k) - f(1, 2).
Enter a polynomial in k in standard form.

Answer 1

k

Question 2

Let f(x, y) = - 2xy. Compute f .
Enter your answer as a polynomial in h in standard form.

Answer 2

h + 1

Question 3

A petroleum company has a Cobb-Douglas production function where x is the utilization of labor and y is the utilization of capital. Determine the number of units of petroleum produced when 1200 units of labor and 2100 units of capital are used.

A) 175,174 units
B) 150,000 units
C) 105,074 units
D) 117,517 units

Answer 3

117,517 units

Question 4

Let f(x, y) = 3 . Compute f(4a, 4b).
Enter your answer as c .

Question 5

Let f(x, y) = . Compute f(3, 4).
Enter just an integer.

Question 6

Let f (x, y, z) = . Compute f(1, -1, -2).
Enter just an integer.

Question 7

A company has a Cobb-Douglas production function where x is the utilization of labor and y is the utilization of capital. Determine the number of units of product produced when 1728 units of labor and 27,000 units of capital are used.

A) 29,292 units
B) 46,605 units
C) 217,988 units
D) 85,612 units

Question 8

A rectangular garden is to be surrounded on three sides by a fence costing $5 per foot and on one side by a stone wall costing $15 per foot. Let x be the length of the side with the stone wall, and let y be the length of each of the other three sides. Express the cost of enclosing the garden as a function of two variables.

A) C(x, y) = (15x)(5y)
B) C(x, y) = 15x + 5y
C) C(x, y) = (20x)(10y)
D) C(x, y) = 15x + 15y
E) none of these

Question 9

Let f(x, y, z) = xyz(1 + ). Find .

A) xy(1 + ) +
B) xy(1 + )
C) xy(1 + ) + xy(1 + y )
D) xy(1 + )
E) none of these

Question 10

Let f(x, y) = 8x + 9y - 6. Compute f(-6, -5).

A) 12
B) -93
C) -108
D) -107

Question 11

Are these the level curves of heights 0, 1, 2, and 3 for ?

Question 12

Suppose the number of leather bags produced by a certain firm, utilizing x units of raw materials and y units of labor, is given by the formula f(x, y) = 10 . What happens to the production of bags if supplies of both raw materials and labor are tripled?

A) Production is increased by a factor of 6.
B) Production is increased by a factor of 9.
C) Production is increased by a factor of 3.
D) Production is increased by a factor of .
E) none of these

Question 13

Compute for k ≠ 0.
Enter your answer as a polynomial in k in standard form.

Question 14

Let f(x, y) = ln(y + 2) + . Compute f(0, -1).

A) e
B) 1
C) 1 + e
D) 0

Question 15

Let Compute f(1, -2, -1).
Enter your answer as a .

Question 16

A company makes cylindrical cans of radius r and height h at a cost of a cents per unit area for the top and bottom and b cents per unit area for the side. Express the cost of producing a can as a function of the two variables r, and h.
Enter your answer in the form:

Question 17

Let f(x, y) = x + . Simplify for h ≠ 0.
Enter your answer exactly in the form:

Question 18

Let g(x, y) = 9 - 2xy. Compute g(-7, -3).

A) 46
B) 399
C) 39
D) 396

Question 19

Let g(x, y) = . Compute g(3, 4).

A) -
B) -
C) -
D) -

Question 20

Let f(x, y) = ∙ . Find .

A)
B) - 1
C) -
D)
E) none of these

Question 21

Let f(x, y) = + 2 + 3xy. Find .
Enter just a polynomial in y plus or minus a polynomial in x both in standard form (no label, no parentheses).

Question 22

Let H(x, y) = . Find .

A)
B)
C)
D)
E) none of these

Question 23

Let f(x, y) = . Find .
Enter just ± where P is a polynomial in standard form (do not label).

Question 24

Let f(x, y) = y(x + ). Compute (2, 3).
Enter your answer exactly as just a(b ± e).

Question 25

Let G(x, r, t) = 2 t + - . Find .

A) 4xt - -
B) 4x -
C)
D) 4xt + -
E) none of these

Question 26

Let f(x, y) = 4 - 2 + 5 . Find .
Enter just a polynomial in x plus or minus a polynomial in y both in standard form (do not label, no parentheses).

Question 27

Let f(x, y) = $x ^ { 3 }$ y + $e ^ { x + 3 y }$ . Compute $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }$ (1, 0).

A) 6
B) 3 + 3e
C) 6 + e
D) 3 + e
E) none of these

Question 28

Let P(x, y, z) = xy + 2 . Compute (2, 1, 3).

A) 2
B) 0
C) 3
D) 1
E) none of these

Question 29

Enter just a polynomial in x plus or minus a polynomial in y plus or minus two, both polynomials in standard form (no label, no parentheses).

Question 30

Let F(x, y, z) = - . Compute (-1, 0, 1).

A)
B) 2
C) -
D) 0
E) none of these

Question 31

Let f(x, y, z) = ln( ). Find .
Enter your answer in the unlabeled form

Question 32

Let f(x, y) = + 2 + . Find .
Enter just a power function in y in standard form (do not label).

Question 33

Let g(x, t, z) = (x - t). Find .

A) -2zt
B) 0
C) - + 1
D) -2zt +
E) none of these

Question 34

Let f(x, y) = 3 + 2xy. Find .
Enter your answer exactly as just a polynomial in x plus or minus a polynomial in y both in standard form (do not label, no parentheses).

Question 35

Let f(x, y, z) = z - y + - . Find .

A) - 2x + 1
B) z - +
C) z - +
D) z - + +
E) none of these

Question 36

Let f(x, y) = . Find .
Enter your answer exactly as just (P(x)) where P is a polynomial in x in standard form (do not label).

Question 37

Let f(x, y) = + 2xy + . Find .
Enter just a polynomial in x plus or minus a polynomial in both in standard form (do not label, no parentheses).

Question 38

Let f(x, y) = ( + x) . Find .
Enter your answer as just (P(x)) where P is a polynomial in x in standard form.

Question 39

Let f(x, y) = + y. Find .
Enter just an integer.

Question 40

Let f(x, y) = + 4 + 2xy. Find .
Enter just a polynomial in y plus or minus a polynomial in x both in standard form (no label, no parentheses).

Question 41

Let f(x, y) = - . Find . Is the correct answer?

Question 42

Let f(x, y, z) = . Find . Is the correct answer?

Question 43

The demand for a certain energy-efficient home is given by f( , ), where p1 is the price of the home and p2 is the price of electricity. Which of the following explains why > 0?

A) The price of electricity keeps going up due to increased demand.
B) Homes are too expensive during energy crises.
C) As the price of electricity goes up, demand for the home goes up.
D) As the price of electricity increases, demand for the home depends more on the price of the home than on the price of electricity.
E) As the price of electricity goes up, demand for the home goes down.

Question 44

Assume that a manufacturer has productivity function P(l, c) where l and c are the amounts of labor and capital utilized. Which of the following indicates that a slight increase in the amount of labor utilized will result in an increase in productivity of 3 units.

A) (100, 75) = 3
B) P(10, 75) = 3
C) P(1, 75) = 3
D) (100, 75) = 3
E) none of these

Question 45

Find all points (x, y) where f(x, y) = 3xy - - - 2x - y + 3 has a possible relative maximum or minimum.
Enter your answer as just (a, b) where a, b are reduced fractions of form .

Question 46

Let f(x, y) = xy. Find and . Is correct in the corresponding order?

Question 47

A certain manufacturer can produce f(x, y) = 10(6 + ) units of goods by utilizing x units of labor and y units of capital. What is the marginal productivity of labor when and

A) 22,000 units
B) 2200 units
C) 18,000 units
D) 1800 units
E) none of these

Question 48

Let f(x, y) = ( + . Find . Is the correct answer?

Question 49

The productivity of a certain country is P(x, y) = 1600 units, where x and y are the amounts of labor and capital utilized. What is the marginal productivity of capital when x = 16 and y = 625?

A) 1600
B) 5000
C) 480
D) 2500
E) none of these

Question 50

Let f(x, y) = . Find . Is the correct answer?

Question 51

Let f(x, y) = . Find . Is the correct answer?

Question 52

Let f(x, y, z) be the amount of heat lost each day by a rectangular building x feet wide, y feet long, and z feet high. Suppose that (50, 80, 15) = 45. Which one of the following conclusions can be drawn?

A) A building with dimensions 51 × 81 × 16 will lose about 45 more units of heat each day than a building with dimensions 50 × 80 × 15 .
B) A building with dimensions 50 × 80 × 15 loses about 45 units of heat each day.
C) A building with dimensions 50 × 80 × 16 will lose about 45 more units of heat each day than a building with dimensions 50 × 80 × 15 .
D) A building with dimensions 50 × 80 × 15 loses about 45 units of heat from the top of the building each day
E) The marginal heat loss per day is 45 units of heat per square foot of surface area of the roof.

Question 53

Suppose that a retailer sells f(p, a) units of an item, where p is the price per unit of the item and a is the amount of money spent on advertising that item. Which of the following indicates that as the amount spent on advertising is decreased, demand for the item also decreases?

A) $\frac { \partial f } { \partial a }$ (50, 75) = -10
B) $\frac { \partial f } { \partial p }$ (50, 25) = -1
C) $\frac { \partial f } { \partial p }$ (50, -1) = -5
D) $\frac { \partial f } { \partial a }$ (50, 25) = 10

Question 54

Let f(x, y) = + x . At which point does f(x, y) have a possible maximum or minimum value?

A)
B) (0, 0)
C)
D)
E) none of these

Question 55

Let f(x, y) = + 2xy + + 2x + 10y - 3. At which point(s) does f(x, y) have possible maximum/minimum values?

A) (-1, 17) and (0, 5)
B) (-1, 0) and (0, 1)
C) (0, -1)
D) (1, 0)
E) none of these

Question 56

Let f(x, y) = . Find . Is the correct answer?

Question 57

.Is the correct answer?

Question 58

Let f(x, y) = 5 - 5 + 2xy + 34x + 38y + 12. At which point does f(x, y) have a possible maximum or minimum value?

A) (4, 3)
B) (-4, -3)
C) (-4, 3)
D) (4, -3)

Question 59

Suppose that a company can produce P(x, y) = 50 items using x units of labor and y units of capital. What is the productivity of capital when x = 10 and ?

A) 3000
B) 100
C) 500
D) 2500
E) none of these

Question 60

Let f(x, y) = ln(x + 2y). Find . Is the correct answer?

Question 61

Let f(x, y, z) = xyz. Find the point(s) where f(x, y, z) may have a possible relative maximum or minimum, subject to the constraint x + 6y + 3z = 36 and where x > 0, y > 0, z > 0. Use the method of Lagrange multipliers.
Enter your answer as just (a, b, c) where a, b, c are all integers.

Question 62

Find all points (x, y) where has a possible relative maximum or minimum.
Enter your answer exactly as just (a, b), (c, d) with a > c and where a, b, c, d are either integers or reduced fractions of form .

Question 63

Let f(x, y, z) = y + . Find .
Enter your answer as a polynomial in x in standard form (unlabeled).

Question 64

The function H(x, y) = $x ^ { 4 }$ - 9 $y ^ { 2 }$ - 2 $x ^ { 2 }$ y + 20y + 4 has

A) neither a relative maximum nor minimum at (1, 4)
B) a relative minimum at the point (0, 0)
C) a relative maximum at the point (-1, 1)
D) a relative maximum at the point (0, 1)
E) none of these

Question 65

A rectangular box of length x, width y, and height z with no top is to be constructed having a volume of 32 cubic inches. Determine the dimensions that will require the least amount of material to construct the box.

A) 4 inches by 4 inches by 4 inches
B) 4 inches by 4 inches by 2 inches
C) 4 inches by 2 inches by 4 inches
D) 2 inches by 2 inches by 4 inches

Question 66

Let f(x, y) = . Compute at (3, 4).
Enter just a reduced fraction .

Question 67

Find the greatest possible volume of a rectangular box that has length plus girth equal to 60 inches. Enter your answer as a single integer (no units).

Question 68

Let f(x, y) = x + y . Compute .
Enter your answer as just an unlabeled polynomial in in standard form.

Question 69

Let f(x, y, z) = . Find .
Enter your answer as a polynomial in in standard form (unlabeled).

Question 70

Find all points (x, y) where has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

A) Test Inconclusive
B) (0, 6) gives a relative maximum
C) (0, 0) gives a relative minimum
D) (0, 0) gives a relative maximum

Question 71

Let Q(x, y) = y + . The point (1, 0) is

A) absolute maximum
B) not a relative extreme point
C) a relative maximum point
D) a relative minimum point

Question 72

Find all points (x, y) where has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

A) (1, 1) gives a relative minimum point
B) (-1, 1) gives a relative maximum point
C) (-1, 1) gives a relative minimum point
D) Neither a relative maximum nor a relative minimum at (-1, 1)

Question 73

Find all points (x, y) where f(x, y) = - 2 + 4x - 6y + 8 has a possible relative maximum or minimum.
Enter your answer as just (a, b) where a, b are either integers or reduced fractions of form .

Question 74

Let f(x, y) = - xy + + 2y - 4. The point is a

A) relative maximum
B) not a relative extreme point
C) absolute maximum
D) relative minimum

Question 75

Find all points (x, y) where f(x, y) = 2 + 2 - x - 6y + 14 has a possible relative maximum or minimum.
Enter your answer exactly as just (a, b), (c, d) with b > d and where a, b, c, d are either integers or reduced fractions of form .

Question 76

Let f(x, y) = - + + 2y - 7. The first partial derivatives of f(x, y) are zero at the points and Use the second derivative test to determine the nature of f(x, y) at each of these points.

A) (0,1) neither relative maximum nor minimum, (-1, 1) maximum
B) (0, 1) relative maximum, (-1, 1) relative minimum
C) (0, 1) no conclusion possible, (-1, 1) minimum
D) (-1, 1) relative maximum, (0, 1) neither relative maximum nor minimum
E) none of these

Question 77

A tennis racket manufacturer produces two types of rackets, standard and competition. The weekly revenue function, in dollars, for x standard rackets and y competition rackets is given by R(x, y) = 54x + 2xy + 398y - 2 $x ^ { 2 }$ - 9 $y ^ { 2 }$ i) How many of each type of racket must be produced each week to maximize revenue?
Ii) What is the maximum weekly revenue?

A) i) 26 standard rackets and 26 competition rackets;
Ii) $5668
B) i) 25 standard rackets and 25 competition rackets;
Ii) $5675
C) i) 25 standard rackets and 26 competition rackets;
Ii) $5664
D) i) 26 standard rackets and 25 competition rackets;
Ii) $5677

Question 78

Find all points (x, y) where has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

A) (2, -1) is a relative maximum
B) (2, -1) is a relative minimum
C) (-2, 1) is a relative maximum
D) (-2, 1) is a relative minimum

Question 79

Find all points (x, y) where has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

A) (0, 1) gives a relative minimum point
B) (0, 0) gives a relative minimum point
C) (0, 1) gives a relative maximum point
D) (0, 0) gives a relative maximum point

Question 80

Find all points (x, y) where f(x, y) = + xy + - x - y + 2 has a possible relative maximum or minimum.
Enter your answer exactly as just (a, b) where a, b are reduced fractions of form or integers.