Deck 8: Functions of Several Variables

Question 1

The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a function of the number of tickets sold, t, and the number of snacks sold, f. Thus, . Assuming there is only one price for tickets, what is that price?

Answer 1

$8.00

Question 2

The following table gives the number f(x, y) of grape vines, in thousands, of age x in year y. In one year a fungal disease killed most of the older grapevines, and in the following year a long freeze killed most of the young vines. Which are these years?

Answer 2

1982 and 1983

Question 3

A company sells two products. The fixed costs for the company are $3300, the variable costs for the first product are $8 per unit, and the variable costs for the second product are $10 per unit.. If the company produces units of the first product and units of the second product, and the total cost is given by , is C an increasing or decreasing function of ?

Answer 3

increasing

Question 4

The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a function of the number of tickets sold, t, and the number of food items sold, f. Thus, . Find .

Question 5

A company sells two products. The fixed costs for the company are $3300, the variable costs for the first product are $7 per unit, and the variable costs for the second product are $9 per unit.. If the company produces units of the first product and units of the second product, which of the following is a formula for the total cost, C, as a function of and ?

A)
B)
C)
D)

Question 6

The following figure shows contours for the function . Is z an increasing or decreasing function of x?

Question 7

The profit, P, from producing a product is expressed as a function of the cost, C, of producing the product and the revenue, R, from selling the product. Do you expect P to be an increasing or decreasing function of R?

Question 8

The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a function of the number of tickets sold, t, and the number of food items sold, f. Thus, . What is the revenue when ticket sales are 100 and food sales are 200?

Question 9

The following table shows the revenue R, (in hundreds of dollars) at a movie theater as a function of the number of tickets sold, t, and the number of food items sold, f. Thus, . What is the revenue when ticket sales are 300 and food sales are 600?

Question 10

You are in a nicely heated cabin in the winter. Deciding that it's too warm, you open a small window. Let T be the temperature in the room, t minutes after the window was opened, x feet from the window. Is T an increasing or decreasing function of x?

A)
B)
C)Neither

Question 11

The cost, C, of renting a car is $45 per day and $0.20 per mile. Write a cost function, , for the total cost of renting a car where d is the number of days and m is the number of miles.

Question 12

The fuel cost, C, for a 3000 mile trip depends on the price, p, per gallon of gasoline and fuel economy in miles per gallon, m, of the car according to the formula . On the same set of axes, sketch a graph of C as a function of m with fixed gas prices of $2.00, $2.25, $2.50, and $2.75. What do you observe as gas prices increase?

A)The graphs are all straight lines from the origin with slopes that get progressively less steep.
B)The graphs are similarly shaped curves that get progressively higher.
C)The graphs are all straight lines from the origin with slopes that get progressively steeper.
D)The graphs are similarly shaped curves that get progressively lower.

Question 13

A certain piece of electronic surveying equipment is designed to operate in temperatures ranging from 0° C to 30° C. Its performance index, p(t, h), measured on a scale from 0 to 1, depends on both the temperature t and the humidity h of its surrounding environment. Values of the function p = f(t, h) are given in the following table. (The higher the value of p, the better the performance.) What is the value of p(20, 25)?

Question 14

Your monthly payment, C(s, t), on a car loan depends on the amount, s, of the loan (in thousands of dollars), and the time, t, required to pay it back (in months). What is the meaning of C(6, 24) = 200?

A)If you borrow $6,000 from the bank for 24 months (2 year loan), your monthly car loan payment is $200.
B)If you borrow $2,000 from the bank for 24 months (6 year loan), your monthly car loan payment is $200.
C)If you borrow $200 from the bank for 24 months (2 year loan), your monthly car loan payment is $6.
D)If you borrow $6 from the bank for 24 months (2 year loan), your monthly car loan payment is $200.

Question 15

The fuel cost, C, for a 3000 mile trip depends on the price, p, per gallon of gasoline and fuel economy in miles per gallon, m, of the car according to the formula . What is the cost when p = $2.00 and m = 20?

Question 16

A company sells two products. The fixed costs for the company are $2500, the variable costs for the first product are $5 per unit, and the variable costs for the second product are $9 per unit.. If the company produces units of the first product and units of the second product, and the total cost is given by , what is ?

Question 17

Lift ticket sales, S, at a ski resort are a function of the price, p, of a ticket and the current number, n, of inches of snowpack on the mountain. Thus, . Do you expect f to be an increasing or a decreasing function of n?

Question 18

Yummy Potato Chip Company has manufacturing plants in N.Y. and N.J. The cost of manufacturing depends on the quantities (in thousand of bags), q₁ and q₂, produced in the N.Y. and N.J. factories respectively. Suppose the cost function is given by Find .

Question 19

The following figure is a contour diagram for the demand for pork as a function of the price of pork and the price of beef. Which axis corresponds to the price of beef?

A)The x-axis
B)The y-axis

Question 20

Your monthly payment, C(s, t), on a car loan depends on the amount, s, of the loan (in thousands of dollars), and the time, t, required to pay it back (in months). Is C an increasing or decreasing function of t?

A)Decreasing
B)Increasing

Question 21

Which of the following describes the contour diagram for the function (with m on the vertical axis and d on the horizontal axis)?

A)A set of parallel lines, all with slopes -250
B)A set of parallel lines, all with slopes 10
C)A set of parallel lines, all with slopes -0.004
D)A set of parallel lines, all with slopes 50

Question 22

You build a campfire while up in the mountains. It is 45 F when you start the fire. Let be the temperature x feet from the fire t minutes after you start it. The following figure shows the contour diagram for H. How many F is it 8 feet from the fire after 25 minutes?

Question 23

Which of the following describes the contour diagram of ?

A)A set of lines extending from the point (0,1).
B)A set of lines extending from the origin.
C)A set of parallel lines, all with slope 5.
D)A set of parallel lines, all with slope .

Question 24

For a function , we are given , , and . Estimate .

Question 25

A table of values for is given below. Is f_x positive or negative?

Question 26

The amount, , of your paycheck is a function of your hourly wage, w, and the number of hours, h, you work. What does the statement mean in terms of your paycheck?

A)An increase of 1 hour worked from 33 hours results in a wage increase of $9.
B)An increase of 1 hour worked from 9 hours results in a wage increase of $33.
C)An wage increase of $1 per hour from $33 per hour results in a wage increase of $9.
D)An wage increase of $1 per hour from $9 per hour results in a wage increase of $33.

Question 27

The heat index tells you how hot it feels as a result of the combination of temperature and humidity. The figure below gives a contour diagram for the heat index as a function of temperature and humidity. Is the heat index an increasing or decreasing function of temperature?

Question 28

Plants need varying amount of water and sunlight to grow. The following contour diagrams each show the growth of a different plant as a function of the amount of water and sunlight. Which diagram would represent a plant that does best in a temperate climate (moderate water and sunshine)?

Question 29

Below is a contour diagram depicting D, the average fox population density as a function of , kilometers east of the western end of England, and , kilometers north of the same point. Is D increasing or decreasing at the point (120, 25) in the northern direction?

Question 30

The demand, D, for gasoline at Station A is a function of the price, a, of gas at Station A and the price, b, of gas at Station B across the street. Thus, . Do you expect b to be positive or negative?

Question 31

The number of tons, N, of a product produced at a factory in a day is a function of the number of workers, w, and the number of hours worked, h. We have . Which of the following equations tells us that when the factory has 9 workers who work for 12 hours, adding one additional hour will cause the amount produced to go up by about 44 tons?

A)f_w

B)f_h

C)f_w

D)f_h

Question 32

The heat index tells you how hot it feels as a result of the combination of temperature and humidity. The figure below gives a contour diagram for the heat index as a function of temperature and humidity. Heat exhaustion is likely to occur when the heat index is 105 or higher. If the temperature is about 95 , at about what humidity level does heat exhaustion become likely?

A)39%
B)43%
C)47%
D)51%

Question 33

Estimate f_y (3,2) from the contour diagram shown below. Use the next higher point to make your estimate.

Question 34

The heat index tells you how hot it feels as a result of the combination of temperature and humidity. The figure below gives a contour diagram for the heat index as a function of temperature and humidity. Fix the humidity level at 50%. As the temperature rises, does the heat index rise more rapidly at milder temperatures (70 to 90 ) or at very hot temperatures (90 to 110 )?

A)milder temperatures
B)very hot temperatures

Question 35

You build a campfire while up in the mountains. It is 45 F when you start the fire. Let be the temperature x feet from the fire t minutes after you start it. The following figure shows the contour diagram for H. Is H an increasing or decreasing function of x?

Question 36

The following is a contour diagram for . Is f_x

positive or negative?

Question 37

The heat index tells you how hot it feels as a result of the combination of temperature and humidity. The figure below gives a contour diagram for the heat index as a function of temperature and humidity. If the humidity is 30%, what temperature feels like 90 ?

Question 38

The following is a contour diagram for . Estimate f_x
.

A)-3
B)0
C)3
D)6

Question 39

The amount, , of your paycheck is a function of your hourly wage, w, and the number of hours, h, you work. Would you expect f_h(w,h) to be positive or negative?

Question 40

A table of values for is given below. Estimate f_y (8,10). Use the next higher point to make your estimate.

Question 41

Find if .

A)
B)
C)
D)

Question 42

If , find f_x(1, 3).

Question 43

Given the following contour diagram of , estimate f_x (1,2). Use the next higher point to make your estimate.

Question 44

The monthly mortgage payment in dollars, P, for a house is a function of three variables P = f(A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years before the mortgage is paid off. It is given that: Estimate the value of

Question 45

The consumption of beef, C (in pounds per week per household) is given by the function C = f(I, p), where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound. Do you expect to be positive or negative?

A)Negative
B)Positive

Question 46

If , .

Question 47

If , .

Question 48

The amount of principal, $P, needed to obtain a balance of $B after t years at r % interest, compounded continuously, is given by the formula . Which of the following gives the amount principal can be changed by and still maintain the same balance if there is an increase of 1% in the interest rate?

A)-Br
B)
C)-Bt
D)

Question 49

Suppose the Cobb-Douglas production function for a company is given by , where P is production in tons, L is the number of workers, and K is the capital investment, in thousands of dollars. How many tons are produced if the company has an initial investment of 20 thousand dollars and the company employs 16 workers? Round to the nearest ton.

Question 50

The table below gives values of a function f(x, y) near x = 1, y = 2. Estimate .

Question 51

The monthly mortgage payment in dollars, P, for a house is a function of three variables P = f(A, r, N), where A is the amount borrowed in dollars, r is the interest rate, and N is the number of years before the mortgage is paid off. It is given that:

Estimate the value of and interpret your answer in terms of a mortgage payment. Select all answers that apply.

A)We are currently borrowing $100,000 at 7% interest rate on a 20-year mortgage.
B)The monthly payment will go up by approximately $0.007753 for each extra percentage point charged.
C)The monthly payment will go up by approximately $0.007753 for each extra dollar we borrow.
D)The monthly payment will go up by approximately $0.007753 for each extra year of the mortgage.
E)The monthly payment will go down by approximately $0.007753 for each extra dollar we borrow.

Question 52

An airline's revenue, , is a function of the number of full price tickets, x, and the number of discount tickets, y, sold. When 300 full price and 600 discount tickets are sold, R = $225,000. Use partial derivatives and to estimate revenue when x = 304 and y = 603.

Question 53

Find if .

A)
B)
C)
D)

Question 54

The amount in dollars, A, spent weekly by a household on food is a function of the household's weekly income in dollars, I, and the number of people in the household, n. Thus, . If A_I (1000, 4) = a > 0, then which of the following is true?

A)At a weekly income of $1000 and a household size of 4, an increase of 1 person in the household produces a decrease of $a in the amount spent weekly on food.
B)At a weekly income of $1000 and a household size of 4, an increase of $1 in weekly income produces a decrease of $a in the amount spent weekly on food.
C)At a weekly income of $1000 and a household size of 4, an increase of 1 person in the household produces an increase of $a in the amount spent weekly on food.
D)At a weekly income of $1000 and a household size of 4, an increase of $1 in weekly income produces an increase of $a in the amount spent weekly on food.

Question 55

The volume of a cylinder is given by , where r is the radius of the base and h is the height, both in centimeters. At a radius of 5 centimeters and a height of 8 centimeters, how much does the volume increase for each 1 cm increase in the height?

A)40 cm³
B)80 cm³
C)50 cm³
D)25 cm³

Question 56

Suppose that the price P (in dollars) to purchase a used car is a function of C, its original cost (in dollars), and its age A (in years). So P = f(C,A). What is the sign of

A)Negative
B)Positive

Question 57

Suppose the Cobb-Douglas production function for a company is given by , where P is production in tons, L is the number of workers, and K is the capital investment, in thousands of dollars. Find L if the company has a capital investment of 25 thousand dollars and the company employs 15 workers. Round to the nearest tenth.

Question 58

The cost of building a new fence is , where f is the number of linear feet of fence and g is the number of gates. Find f.

Question 59

The consumption of beef, C (in pounds per week per household) is given by the function C = f(I, p), where I is the household income in thousands of dollars per year, and p is the price of beef in dollars per pound. Explain the meaning of the statement: , and include units in your answer.

Question 60

Use the values of in the following table to estimate f_yx (3,2). Use the next higher point to make your estimate.

Question 61

Suppose . Then has a local minimum value of 9 at the point (2, 3) when A = _____, B = _____, and C = _____.

Question 62

Find the following partial derivative: f_xy if .

A)
B)
C)
D)
E)

Question 63

The function has a local __________ (maximum / minimum / neither) at the critical point, where x = _____ and y = _____.

Question 64

Is (0, 0) a critical point of the following function?

A)Yes: (global) maximum.
B)Yes: neither a maximum nor a minimum.
C)Yes: (global) minimum.
D)No.

Question 65

The function has a local __________ (maximum / minimum / neither) at the critical point, where x = _____ and y = _____.

Question 66

If $P is invested in a bank account earning r% interest a year, compounded continuously, the balance, $B, at the end of t years is given by Find

Question 67

The function has a critical point which is a local minimum when x = _____ and y = _____.

Question 68

The ideal gas law states that for a fixed amount of gas, called a mole of gas, where P is the pressure (in atmospheres), V is the volume (in cubic meters), T is the temperature (in degrees Kelvin) and R is a positive constant.
Find

Question 69

Let Determine all local critical points. Are the local extrema also global extrema?

Question 70

Find the following partial derivative: (2, 2) if Give your answer to 4 decimal places.

Question 71

Two products are manufactured in quantities and and sold at prices $8 and $12 respectively. The cost of producing them is given by . Find the maximum profit that can be made.

Question 72

For , calculate s² .

Question 73

A company sells two styles of jeans in quantities and at prices and respectively, with and in dollars. The quantities demanded depend on both and according to the formulas and .
Write a formula for total sales revenue as a function of and , and use it to determine what prices the company should charge to maximize sales revenue. They should charge $_____ for style and $_____ for style .

Question 74

A contour diagram of is given in the following figure. Is there a global minimum near the point (5,3)?

Question 75

Find all the critical points of the function Classify these critical points as local maxima, local minima, or neither.

Question 76

The following table gives values for a function . For and , there is a global minimum of at most _____ near the point (_____,_____).

Question 77

For , calculate h s

Question 78

A contour diagram of is given in the following figure. Which one of the following statements is true?

A)The function has a local maximum of about 7 at (1, 3.8), a global maximum of about 15 at (5.2, 4.9), and a global minimum of about -7 at (3, 1.6).
B)The function has a global maximum of about 7 at (1, 3.8), a local maximum of about 15 at (5.2, 4.9), and a global minimum of about -7 at (3, 1.6).
C)The function has a local minimum of about 7 at (1, 3.8), a global minimum of about 15 at (5.2, 4.9), and a global maximum of about -7 at (3, 1.6).
D)The function has a global maximum of about 7 at (1, 3.8), a global maximum of about 15 at (5.2, 4.9), and a global minimum of about -7 at (3,1.6).

Question 79

The function has a local maximum value of _____ when x = _____ and y = _____.

Question 80

For , calculate f_xx (x,y).