Chapter 8
1.
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?
2.
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,
( , )S f p n=
. Do you expect
f to be an increasing or a decreasing function of n?
3.
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,
( , )R g t f=
. Find
(500,600)g
.
Chapter 8
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,
( , )R g t f=
. What is the revenue when ticket sales are 300 and food sales are 600?
5.
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,
( , )R g t f=
. What is the revenue when ticket sales are 100 and food sales are 200?
Chapter 8
6.
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,
( , )R g t f=
. Assuming there is only one price for tickets, what is that price?
7.
The cost, C, of renting a car is $45 per day and $0.20 per mile. Write a cost function,
( , )C f d m=
, for the total cost of renting a car where d is the number of days and m is
the number of miles.
8.
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
1
q
units of the first product and
2
q
units of
the second product, which of the following is a formula for the total cost, C, as a
function of
1
q
and
2
q
?
A)
12
3300 7 9C q q= + +
B)
12
3300 7 9C q q= − −
C)
1 2 1 2
3300( ) 7 9C q q q q= + + +
D)
12
3300 7 9C q q= − + +
Chapter 8
9.
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
1
q
units of the first product and
2
q
units of
the second product, and the total cost is given by
12
( , )C f q q=
, what is
(500,400)f
?
Ans:
$8600
Learning Objectives: Evaluate and interpret functions of two variables represented by a
formula, a table, or words. difficulty: easy section: 8.1
10.
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
1
q
units of the first product and
2
q
units
of the second product, and the total cost is given by
12
( , )C f q q=
, is C an increasing or
decreasing function of
2
q
?
Ans:
increasing
Learning Objectives: Evaluate and interpret functions of two variables represented by a
11.
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
3000 p
Cm
=
.
What is the cost when p = $2.00 and m = 20?
Ans:
$300.00
Learning Objectives: Evaluate and interpret functions of two variables represented by a
formula, a table, or words. difficulty: easy section: 8.1
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
3000 p
Cm
=
.
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.
Ans: B Learning Objectives: Graph and interpret cross-sections of functions of
Chapter 8
13.
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)
Increasing
B)
Decreasing
C)
Neither
two variables. difficulty: easy section: 8.1
14.
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?
Ans:
1982 and 1983
formula, a table, or words. difficulty: medium section: 8.1
Chapter 8
15.
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)?
Ans:
0.99
Learning Objectives: Evaluate and interpret functions of two variables represented by a
formula, a table, or words. difficulty: easy section: 8.1
16.
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
22
1 2 1 1 2 2
( , ) 4 330C q q q q q q
= + + +
Find
(15,75)C
.
Ans:
7980
Learning Objectives: Evaluate and interpret functions of two variables represented by a
formula, a table, or words. difficulty: easy section: 8.1
17.
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.
Ans: A Learning Objectives: Evaluate and interpret functions of two variables
represented by a formula, a table, or words. difficulty: easy section: 8.1
Chapter 8
18.
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
represented by a formula, a table, or words. difficulty: easy section: 8.1
19.
The following figure shows contours for the function
( , )z f x y=
. Is z an increasing or
decreasing function of x?
Ans:
increasing
a contour diagram. difficulty: easy section: 8.2
Chapter 8
Page 8
20.
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
21.
Which of the following describes the contour diagram of
( , ) 5 1f x y x y= − +
?
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
1
5
.
variables using a contour diagram. difficulty: medium section: 8.2
variables using a contour diagram. difficulty: easy section: 8.2
Chapter 8
22.
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 ?
Chapter 8
23.
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%
variables using a contour diagram. difficulty: easy section: 8.2
Chapter 8
24.
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
variables using a contour diagram. difficulty: easy section: 8.2
Chapter 8
Page 12
25.
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?
26.
Which of the following describes the contour diagram for the function
50 0.2C d m=+
(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
section: 8.2
Ans:
increasing
a contour diagram. difficulty: easy section: 8.2
Chapter 8
27.
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)?
28.
You build a campfire while up in the mountains. It is 45 F when you start the fire.
Let
( , )H x t
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?
Chapter 8
29.
You build a campfire while up in the mountains. It is 45 F when you start the fire.
Let
( , )H x t
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?
Chapter 8
30.
Below is a contour diagram depicting D, the average fox population density as a
function of
E
x
, kilometers east of the western end of England, and
N
x
, kilometers
north of the same point.
Is D increasing or decreasing at the point (120, 25) in the northern direction?
31.
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,
( , )D f a b=
. Do you
expect
/D
b to be positive or negative?
32.
The amount,
( , )A f w h=
, of your paycheck is a function of your hourly wage, w, and
the number of hours, h, you work. Would you expect fh(w,h) to be positive or
negative?
Chapter 8
33.
The amount,
( , )A f w h=
, of your paycheck is a function of your hourly wage, w, and
the number of hours, h, you work. What does the statement
(9,33) 9
h
f=
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.
Ans: A Learning Objectives: Estimate and interpret partial derivatives.
difficulty: medium section: 8.3
34.
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
( , )N f w h=
.
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)
fw
(9,12) 44=
B)
fh
(9,12) 44=
C)
fw
(9,12) 44=−
D)
fh
(9,12) 44=−
Chapter 8
35.
The following is a contour diagram for
( , )z f x y=
. Is fx
( , )xy
positive or negative?
Chapter 8
36.
The following is a contour diagram for
( , )z f x y=
. Estimate fx
(1,4)
.
A)
–3
B)
0
C)
3
D)
6
difficulty: easy section: 8.3
37.
A table of values for
( , )f x y
is given below. Is fx positive or negative?
Ans:
positive
section: 8.3
Chapter 8
38.
A table of values for
( , )f x y
is given below. Estimate fy (8,10). Use the next higher
point to make your estimate.
39.
Estimate fy (3,2) from the contour diagram shown below. Use the next higher point to
make your estimate.
40.
For a function
( , )F n m
, we are given
(20,5) 3.3f=
,
(20,5) 0.7
n
f=
, and
(20,5) –0.3
m
f=
. Estimate
(23,6)f
.
Chapter 8
41.
An airline’s revenue,
( , )R f x y=
, 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
(300,600) 350
x
f=
and
(300,600) 200
y
f=
to estimate revenue when x = 304 and y = 603.
Ans:
$227,000
Learning Objectives: Estimate and interpret partial derivatives. difficulty: medium
section: 8.3
42.
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,
( , )A f I n=
. If AI (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.
Ans: D Learning Objectives: Estimate and interpret partial derivatives.
difficulty: medium section: 8.3
43.
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
?
P
A
A)
Negative
B)
Positive
Ans: A Learning Objectives: Estimate and interpret partial derivatives.
difficulty: easy section: 8.3
44.
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
fp
to be positive or negative?
A)
Negative
B)
Positive
Ans: A Learning Objectives: Estimate and interpret partial derivatives.
difficulty: easy section: 8.3