Deck 7: Multivariable Calculus

Question 1

Choose the surface that the function describes.

A)
B)
C)
D)

Answer 1

Question 2

Find the value.
Find f(2, -1, 4) for f(x, y, z) =

A) -5
B) 13
C) 5
D) 27

Answer 2

5

Question 3

Choose the surface that the function describes.
f(x, y) = 5

A)
B)
C)
D)

Answer 3

Question 4

Find the partial derivative.
For

A) 12
B) 36
C) 24
D) -12

Question 5

Find the partial derivative.
Let z =

A)
B) -12x - 18y
C) 10x - 12y
D)

Question 6

Find the value.
Let Find f(4, -7).

A) 198
B) (4, -7) is not in the domain of f.
C) -24
D) -119

Question 7

Solve the problem.
The number of cows that can graze on a ranch is approximated by C(x,y) = 9x + 5y - 4, where x is the number of acres of grass and y the number of acres of alfalfa. If the ranch has 30 acres of alfalfa and 25 acres of grass, how
Many cows may graze?

A) 395 cows
B) 371 cows
C) 375 cows
D) 391 cows

Question 8

Find the partial derivative.
Find

A) -32
B) 20
C) 92
D) -20

Question 9

Find the value.
Find f(-3, 1, 5) for f(x, y, z) =

A) -5
B) 1
C) 10
D) -4

Question 10

Solve the problem.
The productivity of a petroleum company is given approximately by the function where x is the utilization of labor and y is the utilization of capital. If the company uses 1200 units of labor and 2100 units
Of capital, how many units of petroleum will be produced? Round to the nearest whole unit.

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

Question 11

Choose the surface that the function describes.
f(x, y) = 1 - x - 2y

A)
B)
C)
D)

Question 12

Solve the problem.
The Cobb-Douglas production function for a steel company is given by f(x, y) = 78x0.3y0.7 where x is the utilization of labor and y is the utilization of capital. If the company uses 1500 units of labor and 2200 units of
Capital, how many units of steel will be produced? Round to the nearest whole unit.

A) 5,405,400,000 units
B) 656,640 units
C) 54,054,000 units
D) 152,974 units

Question 13

Find the partial derivative.
Find

A) - 86
B) 170
C) 86
D) -170

Question 14

Find the value.
Let Find f(4, 8).

A) 40
B) 258
C) 36
D) 136

Question 15

Solve the problem.
The volume of a flower pot is given by where is the major radius and is the minor radius and h is the height of the pot (see figure below). If the dimensions of the pot are = 5 inches, = 3 inches and h = 6 inches, find the volume of potting soil
Required to fill the pot to the top. Round to the nearest cubic inch.

A)
B)
C)
D)

Question 16

Solve the problem.
Poiseuille's law states that the resistance, R, for blood in a blood vessel varies directly as the length of the vessel, L, and inversely as the fourth power of its diameter, d. This can be written as an equation where
K is a constant. Find R(5, 0.3). Round your answer to the nearest whole number.

A) about 61.7k
B) about 617k
C) about 16.67k
D) about 630k

Question 17

Solve the problem.
The marketing research department of a large manufacturing company has determined that the demand equations for two major items it produces are given by p = 2,000 - 5x + 8y and q = 4,000 + 9x - 7y where p is the
Price of item A, q is the price of item B, x is the monthly demand for item A, and y is the monthly demand for
Item B. Find the total monthly revenue from items A and B when x = 15 and y = 5.

A) $20,565
B) $50,195
C) $49,975
D) $29,415

Question 18

Solve the problem.
The surface area of a human body (in square meters) is approximated by A = 0.202W(.425)H(.725), where W is the weight of the person in kilograms and H is the height in meters. Find A if W = 74 and H = 1.61.

A)
B)
C)
D)

Question 19

Find the value.
Let . Find f(6, 6).

A) 1
B) 37
C) 72
D) 35

Question 20

Find the value.
Find f(-8, 8) when f(x, y) = 3x + 5y - 8

A) 5
B) 16
C) 8
D) 3

Question 21

Find the partial derivative.
Find

A) 8
B) -48
C) 16
D) 40

Question 22

Find the partial derivative.
Find fx for f(x, y) = 3 .

A)
B)
C)
D)

Question 23

Find the partial derivative.
Find

A) 8x - 11y
B) -11x - 12y
C)
D)

Question 24

Provide an appropriate response.
Find fxy for the function

Question 25

Provide an appropriate response.
Find the local extrema for

A) f(0, 0) = 0 is a minimum
B) f(0, 2) = 4 is a maximum
C) f(2, 0) = -16 is a minimum
D) f(0, 0) = 0 is a maximum

Question 26

Provide an appropriate response.
Find the local extrema for

A) f(5, 2) = -13 is a minimum
B) f(1, 1) = -1 is a minimum
C) f(-5, 2) = 55 is a maximum
D) f(0, 0) = 1 is a minimum

Question 27

Provide an appropriate response.
Find critical points for

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

Question 28

Provide an appropriate response.
Find the critical points for

A) (-2, -1)
B) (-2, 1)
C) (2, -1)
D) (2, 1)

Question 29

Provide an appropriate response.
Find

A) -28
B)
C) -14
D)

Question 30

Provide an appropriate response.
Find the local extrema for

A) f(1, 1) = -1 is a local minimum
B) f(2, 2) = 41 is a local maximum
C) f(-2, -2) = 9 is a local maximum
D) f(0, 0) = 1 is a local minimum

Question 31

Solve the problem.
The productivity of a major manufacturer of microwave ovens is given approximately by the Cobb-Douglas production function with the utilization of x units of labor and y units of capital. If the
Company is currently utilizing 4500 units of labor and 2000 units of capital, find the marginal productivity of
Labor to the nearest unit.

A) 129 units
B) 2 units
C) 217 units
D) 32 units

Question 32

Provide an appropriate response.
Find fxy for

A)
B)
C)
D)

Question 33

Provide an appropriate response.
Find the local extrema for

A) (2, 1) = -8 is a minimum
B) (- 2, - 1) = 9 is a maximum
C) (- 2, 1) = 9 is a minimum
D) (2, 1) = -8 is a maximum

Question 34

Solve the problem.
A company has the following production function for a certain product Find the marginal productivity with fixed capital, Px .

A)
B)
C)
D)

Question 35

Solve the problem.
Suppose that the labor cost for a building is approximated by where x is the number of days of skilled labor and y is the number of days of semiskilled labor required. Find the x and
Y that minimize cost C.

A) x = 8, y = 210
B) x = 105, y = 42
C) x = 4, y = 105
D) x = 42, y = 315

Question 36

Solve the problem.
The production function z for an industrial country was estimated as where x is the amount of labor and y, the amount of capital. Find the marginal productivity of labor.

A)
B)
C)
D)

Question 37

Solve the problem.
The profit function for sales of two models of television sets at a chain discount store is given by where x is the number of sales per week of model A, and y is the number of sales per week of model B. Find Px(10, 15) and interpret the result.

A) At a sales level of 10 units of model A and 15 units of model B, increasing sales of model A by one unit and holding
Sales of model B at 15 units will increase profit by approximately $60
B) At a sales level of 10 units of model A and 15 units of model B, increasing sales of model A by one unit and holding
Sales of model B at 15 units will increase profit by approximately $80.
C) At a sales level of 10 units of model A and 15 units of model B, increasing sales of model A by one unit and holding
Sales of model B at 15 units will increase profit by approximately $120
D) At a sales level of 10 units of model A and 15 units of model B, increasing sales of model A by one unit and holding
Sales of model B at 15 units will increase profit by approximately $140

Question 38

Provide an appropriate response.
For

A)
B)
C) 12
D)

Question 39

Solve the problem.
A company uses TV and magazines for advertising. They know that profit P is related to the amounts T spent on TV and M spent on magazines by the equation where P, M, and T are in hundreds
Of thousands. Find the maximum profit.

A) $120,000
B) $130,000
C) $260,000
D) $240,000

Question 40

Provide an appropriate response.
Find .

A)
B)
C)
D)

Question 41

Solve the problem.
The market research department for a drug store chain arrived at the demand table below, where y is the number of bottles of multivitamins purchased per month (in thousands) at x dollars per bottle. I) Find a demand equation using the method of least squares.
II) If each bottle of multivitamins costs the drug store chain $4, how should it be priced to achieve a maximum monthly
Profit? [Hint: Use the result from I) with C = 4y, R = xy, and P = R - C.]

A)
B)
C)
D)

Question 42

Evaluate.

A)
B)
C)
D)

Question 43

Solve the problem.
The rectangular box below, with an open top and one partition, is to be constructed from 18 square inches of cardboard. Find the dimensions that will result in a box with the largest possible volume.

A) 2 inches by 3 inches by 1 inch
B) 2 inches by 2 inches by 1 inch
C) 3 inches by 2 inches by 1 inch
D) 3 inches by 3 inches by 1 inch

Question 44

Provide an appropriate response.
Find the least squares line for the following data:

A) y = 0.74x + 23.04
B) y = - 0.74x - 23.04
C) y = 1.29x - 26.83
D) y = 13.46x + 0.04

Question 45

Provide an appropriate response.
Find the least squares line for the points (4, 3), (6, 6), (8, 0), and (9, 9). Graph the data and the least squares line on the same axes.

A)
B)
C)
D)

Question 46

Provide an appropriate response.
Consider the following data on the growth of peach grafts under controlled conditions. Find the regression line y = ax + b.

A) y = 13x + 8.19
B) y = 1.33x + 0.21
C) y =1.50 x + 0.7
D) y = -2.10x + 0.2

Question 47

Solve the problem.
The Cobb-Douglas function for a new product is given by where x is the number of units of labor and y is the number of units of capital required to produce N(x, y) units of the product. Each unit of
Labor costs $40, and each unit of capital costs $80. If $400,000 has been budgeted for the production of this
Product, determine how this amount should be allocated in order to maximize production, and find the
Maximum production.

A)
B)
C)
D)

Question 48

Provide an appropriate response.
Use Lagrange multipliers to maximize f(x, y, z) = 24x + 12y + 24z subject to

A) max f(x, y, z) = f(12, 12, 12) = 720
B) max f(x, y, z) = f(6, 12, 12) = 576
C) max f(x, y, z) = f(12, 12, 6) = 576
D) max f(x, y, z) = f(12, 6, 12) = 648

Question 49

Provide an appropriate response.
Refer to the table given below. Find the least squares line and use it to estimate y when x = 15.

Question 50

Solve the problem.
The total cost to produce MP3 players in 2 models is given by where red model is x and the green one is y. If a total of 60 players must be made, how should production be allocated so that the total cost is minimized?

A) 60 red players and 0 green players
B) 30 red players and 30 green players
C) 0 red players and 60 green players
D) 59 red players and 1 green players

Question 51

Provide an appropriate response.
Use Lagrange multipliers to minimize subject to x - y = 10.

A) f(5, -5) = 75
B) f(5, 5) = 25
C) f(1, 2) = 3
D) f(2, -1) = 7

Question 52

Provide an appropriate response.
Use Lagrange multipliers to maximize f(x, y) = 5xy subject to x + y = - 6.

A) max f(x, y) = f(-3, -3) = 45
B) max f(x, y) = f(-3, 3) = -45
C) max f(x, y) = f(3, -3) = -45
D) max f(x, y) = f(3, 3) = 45

Question 53

Provide an appropriate response.
Find the least squares line for the following data:

A) y = 0.18x + 5.90
B) y = 0.18x - 5.90
C) y = - 0.18x + 5.90
D) y = - 0.18x - 5.90

Question 54

Provide an appropriate response.
Maximize the product of two numbers if their sum must be 26.

A) f(x, y) = f(-13, -13) = 26
B) f(x, y) = f(13, 13) = 26
C) f(x, y) = f(-13, -13) = 169
D) f(x, y) = f(13, 13) = 169

Question 55

Evaluate.

A)
B)
C)
D)

Question 56

Solve the problem.
The table lists the high school grade-point averages of six students and their college grade-point averages after one year of college. Use the least-squares line to estimate the college GPA for a student with a high school GPA of 3.5.

A) about 2.6
B) about 3.0
C) about 3.2
D) about 2.8

Question 57

Evaluate.

A)
B)
C)
D)

Question 58

Provide an appropriate response.
Use Lagrange multiplier to maximize f(x, y, z) = xy + z subject to

A) f(1, 1, 1) = 1
B) f(1, 1, 0) = 1
C) f(0, 1, 0) = 1
D) f(0, 0, 1) = 1

Question 59

Solve the problem.
Consider the data showing the average life expectancy of woman in various years. Find the regression line. Let the year 1900 represent x = 0.

A) y = 8.3x + 22.1
B) y = 0.15x + 60.9
C) y = 0.37x - 21.8
D) y = - 0.11.2x + 23.1

Question 60

Provide an appropriate response.
Find the least squares line for the following data:

A) y = -0.3x + 1.2
B) y = 0.3x - 1.2
C) y = -0.13x - 1.2
D) y = 0.13x - 1.2

Question 61

Evaluate the integral.
Evaluate the integral with the order reversed.

A)
B)
C) 4
D)

Question 62

Provide an appropriate response.
Find the volume of the solid under the graph of over the rectangle

A)
B)
C) -12
D)

Question 63

Provide an appropriate response.
Let R be the region bounded by the graphs of the equations Use set notation and double inequalities to describe R as a regular x region or regular y region, whichever is simpler.

A)
B)
C)
D)

Question 64

Provide an appropriate response.
Find the average value of the function over the shaded region. f(x, y) = 9x + y

A)
B)
C)
D)

Question 65

Provide an appropriate response.
Evaluate .

A)
B)
C)
D)

Question 66

Provide an appropriate response.
Evaluate

A) 450
B) 30
C) -450
D) -480

Question 67

Provide an appropriate response.
Find the double integral over the rectangular region R with the given boundaries.

A)
B)
C)
D)

Question 68

Provide an appropriate response.
Find the average value of over the rectangle

A) 8
B) 2
C) 1
D) 4

Question 69

Provide an appropriate response.
Find the double integral over the rectangular region R with the given boundaries.

A)
B)
C)
D)

Question 70

Solve the problem.
Under ideal conditions, if a person driving a car slams on the brakes and skids to a stop on wet pavement, the length of the skid marks (in feet) is given by the formula L(x, y) = 0.00002xy2, where x is the weight of the car in
Pounds and y is the speed of the car in miles per hour. What is the average length of the skid marks for cars
Weighing between 2500 and 3500 pounds and traveling at speeds between 45 and 55 miles per hour? Set up a
Double integral and evaluate.

A)
B)
C)
D)

Question 71

Provide an appropriate response.
Evaluate.

A) 5
B) -4
C)
D)

Question 72

Provide an appropriate response.
Give a verbal description of the region and determine whether R is a regular x region, regular y region, both, or neither.

A) R consists of the points on or inside the rectangle with corners (±3, ±6); both
B) R consists of the points on or inside the rectangle with corners (±6, ±3); regular x region
C) R consists of the points on or inside the rectangle with corners (±6, ±3); both
D) R consists of the points on or inside the rectangle with corners (±6, ±3); neither

Question 73

Solve the problem.
An industrial plant located in the center of a small town emits particulate matter into the atmosphere. Suppose the concentration of particulate matter in parts per million at a point d miles from the plant is given by
C = 120 - 15d2. If the boundaries of the town form a rectangle four miles long and six miles wide, what is the
Average concentration of particulate matter throughout the city? Express C as a function of x and y, set up a
Double integral, and evaluate.

A)
B)
C)
D)

Question 74

Evaluate the integral.
Evaluate the integral with the order reversed.

A)
B) 1
C) 2
D) x

Question 75

Provide an appropriate response.
Find the double integral over the rectangular region R with the given boundaries.

A) 6
B)
C)
D) 8

Question 76

Provide an appropriate response.
Find the volume under the surface z = f(x,y) and above the rectangle with the given boundaries.

A) 26
B) 38
C) 28
D) 36

Question 77

Provide an appropriate response.
Find the double integral over the rectangular region R with the given boundaries.

A) 18
B) 36
C) 10
D) 27

Question 78

Provide an appropriate response.
Evaluate

A) -12
B) 15
C) 12
D) 24