13.1 Introduction to Functions of Several Variables
768
13.1 Introduction to Functions of Several Variables
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Find and simplify the function
at the given value
.
a.
–56
b.
–26
c.
–14
d.
–44
e.
–27
____
2.
Find and simplify the function
at the given value
.
a.
b.
c.
d.
e.
____
3.
Find and simplify the function
at the given value
.
a.
b.
c.
d.
e.
769
____ 4.
Find and simplify the function
at the given value
.
a.
–42
b.
–14
c.
42
d.
14
e.
____
5.
Find and simplify the function
at the given value
.
a.
b.
c.
d.
e.
____
6.
Find and simplify
for the given function
.
a.
b.
c.
d.
e.
____
7.
Describe the domain of the function
.
a.
b.
c.
d.
e.
____
8.
Describe the domain of the function
.
a.
b.
c.
d.
e.
13.1 Introduction to Functions of Several Variables
770
____ 9.
Sketch the surface given by the function
.
a.
d.
b. e.
c.
771
Chapter 13: Functions of Several Variables
____
10.
Sketch the surface given by the function.
a. d.
b. e. none of the above
c.
13.1 Introduction to Functions of Several Variables
772
____ 11.
Describe the level curves of the function. Sketch the level curves for the given
c-values.
a. d.
b. e.
c.
773 Chapter 13: Functions of Several Variables
____ 12. Sketch the level curves of the function for the given c-values
.
a. d.
b. e.
c.
13.1 Introduction to Functions of Several Variables
774
____ 13.
Sketch the level curves for the function
for the given c-values
.
a.
d.
b. e.
c.
775 Chapter 13: Functions of Several Variables
____ 14. Assume in 2009, an investment of $2400 was made in a bond earning 9%
compounded annually. Assume that the buyer pays at rate R and the annual rate of inflation is
I. Suppose in the year 2019, the value V of the investment in constant 2009 dollars is
. Calculate the function for and .
Round your answer to the nearest cent.
$5,148.94
$3,319.05
$2,493.01
$2,479.09
$4,219.43
____ 15. Assume a rule that is one of several methods used to determine the lumber yield of a log
(in board-feet) in terms of its diameter d (in inches) and its length L (in feet). The number of
board-feet is . Find the number of board-feet of lumber in a log 28 inches in
diameter and feet in length.
7,290 board-feet
1,690 board-feet
270 board-feet
3,380 board-feet
6,760 board-feet
____ 16. The average length of time that a customer waits in line for service is
where y is the average arrival rate, written as the number of customers per
unit of time, and x is the average service rate, written in the same units. Evaluate . Note: x
and y are given as the number of customers per hour.
7 min
5 min
3 min
4 min
6 min
____ 17. A rectangular box with an open top has a length of x feet, a width of y feet, and a height of
z feet. It costs $2.10 per square foot to build the base and $0.75 per square foot to build the sides.
Write the cost C of constructing the box as a function of x, y, and z.
a.
b.
c.
d.
e.
13.1 Introduction to Functions of Several Variables
776
____ 18.
According to the Ideal Gas Law,
, where P is pressure, V is volume, T is
temperature (in Kelvins), and k is a constant of proportionality. A tank contains 2000 cubic inches of
nitrogen at a pressure of 26 pounds per square inch and a temperature of 600 K. Determine k.
a.
b.
c.
d.
e.
____ 19. According to the Ideal Gas Law, where P is pressure, V is volume, T is
temperature (in Kelvins), and k is a constant of proportionality. A tank contains 3500 cubic inches of
nitrogen at a pressure of 34 pounds per square inch and a temperature of 300 K. Write P as a function
of V and T after evaluating k.
a.
b.
c.
d.
e.
777 Chapter 13: Functions of Several Variables
13.1 Introduction to Functions of Several
Variables Answer Section
13.2 Limits and Continuity
778
13.2 Limits and Continuity
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find by using the limits and
.
a.
b.
c.
d.
e.
____ 2. Find by using the limits and
.
a.
b.
c.
d.
e.
____ 3. Find the limit.
a.
b.
c.
d.
e.
779
Chapter 13: Functions of Several Variable
____
4.
Determine the continuity of the function
.
continuous except at
continuous except at
continuous except at
continuous except at
continuous everywhere
____ 5. Find the limit .
3
2
48
37
____ 6. Determine the continuity of the function .
continuous except at
continuous except at
continuous except at
continuous except at
continuous everywhere
13.2 Limits and Continuity
780
____ 7. Find the limit.
a.
b.
c.
d.
e.
____
8.
Determine the continuity of the function
.
a.
continuous except at
b.
continuous except at
c.
continuous for all
d.
continuous for all
e.
continuous everywhere
____
9.
Find the limit.
a.
b.
c.
d.
e.
781 Chapter 13: Functions of Several Variable
____ 10. Find the limit.
a.
b.
c.
d.
e.
____ 11.
Use polar coordinates to find the limit. [Hint: Let
and
, and
note that
implies
.]
a.
b.
c.
limit does not exist
____ 12. Use polar coordinates to find the limit . [Hint: Let
and , and note that implies .]
a.
b.
c.
d.
e.
13.2 Limits and Continuity
782
____ 13.
Use polar coordinates and L’Hopital’s Rule to find the limit.
a.
b.
c.
d.
e.
____ 14. Determine the continuity of the function .
continuous except at
continuous for
continuous except at
continuous except at
continuous for
____ 15. Discuss the continuity of the function.
is continuous everywhere except at those points such that .
is continuous everywhere except at those points such that .
783 Chapter 13: Functions of Several Variable
is continuous at those points such that .
is continuous only when .
is continuous everywhere in the xy-plane.
____ 16. Determine the continuity of the composite function where and
.
continuous except at
continuous for all
continuous for all
continuous except at
continuous everywhere
____ 17. Find the limit for the function .
a. –5y
b.
5y
8x
____ 18. Find the limit for the function .
–6
16x
6y
13.2 Limits and Continuity 784
____ 19. Find the limit for the function .
a.
b.
c.
–5y
____
20.
Find the limit
for the function
.
a.
b.
c.
d.
e.
____
21.
Use spherical coordinates to find the limit
[Hint: Let
and
, and note that
implies
.]
a.
6
b.
9
c.
1
d.
7
e.
785 Chapter 13: Functions of Several Variable
13.2 Limits and Continuity
Answer Section
13.3 Partial Derivatives
786
13.3 Partial Derivatives
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Find the partial derivative
for the function
.
a.
b.
c.
d.
e.
____
2.
Find the partial derivative
for the function
.
a.
b.
c.
d.
e.
____
3.
Find the partial derivative
for the function
.
a.
b.
c.
d.
e.
787
Chapter 13: Functions of Several Variables
____
4.
Find the partial derivative
for the function
.
a.
b.
c.
d.
e.
____
5.
Find the partial derivative
for the function
.
a.
b.
c.
d.
e.