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Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

Chapter 13

Functions of Several Variables - all with Video Answers

Educators

Section 1

Introduction to Functions of Several Variables

00:49

Problem 1

Use the graph to determine whether $z$ is a function of $x$ and $y .$ Explain.

Lucas Finney
Lucas Finney
Numerade Educator
00:33

Problem 2

Use the graph to determine whether $z$ is a function of $x$ and $y .$ Explain.

Lucas Finney
Lucas Finney
Numerade Educator
00:46

Problem 3

Determine whether $z$ is a function of $x$ and $y$.
$$x^{2} z+3 y^{2}-x y=10$$

Lucas Finney
Lucas Finney
Numerade Educator
00:45

Problem 4

Determine whether $z$ is a function of $x$ and $y$.
$$x z^{2}+2 x y-y^{2}=4$$

Lucas Finney
Lucas Finney
Numerade Educator
00:54

Problem 5

Determine whether $z$ is a function of $x$ and $y$.
$$\frac{x^{2}}{4}+\frac{y^{2}}{9}+z^{2}=1$$

Lucas Finney
Lucas Finney
Numerade Educator
00:57

Problem 6

Determine whether $z$ is a function of $x$ and $y$.
$$z+x \ln y-8 y z=0$$

Lucas Finney
Lucas Finney
Numerade Educator
00:51

Problem 7

Find and simplify the function values.
$f(x, y)=x y$
(a) (3,2)$\quad$
(b) (-1,4)
(c) (30,5)
(d) $(5, y)$
(e) $(x, 2)$
(f) $(5, t)$

Lucas Finney
Lucas Finney
Numerade Educator
01:25

Problem 8

Find and simplify the function values.
$f(x, y)=4-x^{2}-4 y^{2}$
(a) (0,0)$\quad$
(b) (0,1)
(c) (2,3)
(d) $(1, y)$
(e) $(x, 0)$
(f) $(t, 1)$

Lucas Finney
Lucas Finney
Numerade Educator
00:51

Problem 9

Find and simplify the function values.
$f(x, y)=x e^{y}$
(a) (5,0)
(b) (3,2)
(c) (2,-1)
(d) $(5, y)$
(e) $(x, 2)$
(f) $(t, t)$

Lucas Finney
Lucas Finney
Numerade Educator
01:23

Problem 10

Find and simplify the function values.
$g(x, y)=\ln |x+y|$
(a) (1,0)
(b) (0,-1)
(c) $(0, e)$
(d) (1,1)
(e) $(e, e / 2)$
(f) (2,5)

Lucas Finney
Lucas Finney
Numerade Educator
00:58

Problem 11

Find and simplify the function values.
$h(x, y, z)=\frac{x y}{z}$
(c) (-2,3,4)
(a) (2,3,9)
(b) (1,0,1)
(d) (5,4,-6)

Lucas Finney
Lucas Finney
Numerade Educator
01:07

Problem 12

Find and simplify the function values.
$f(x, y, z)=\sqrt{x+y+z}$
(a) (0,5,4)$\quad$
(b) (6,8,-3)
(c) (4,6,2)
(d) (10,-4,-3)

Lucas Finney
Lucas Finney
Numerade Educator
00:50

Problem 13

Find and simplify the function values.
$f(x, y)=x \sin y$
(a) $(2, \pi / 4)$
(b) (3,1)
(c) $(-3, \pi / 3)$
(d) $(4, \pi / 2)$

Lucas Finney
Lucas Finney
Numerade Educator
01:12

Problem 14

Find and simplify the function values.
$V(r, h)=\pi r^{2} h$
(a) (3,10)
(b) (5,2)
(c) (4,8)
(d) (6,4)

Lucas Finney
Lucas Finney
Numerade Educator
01:39

Problem 15

Find and simplify the function values.
$g(x, y)=\int_{x}^{y}(2 t-3) d t$
(a) (4,0)
(b) (4,1)
(c) $\left(4, \frac{3}{2}\right)$
(d) $\left(\frac{3}{2}, 0\right)$

Lucas Finney
Lucas Finney
Numerade Educator
01:37

Problem 16

Find and simplify the function values.
$g(x, y)=\int_{x}^{y} \frac{1}{t} d t$
(a) (4,1)
(b) (6,3)
(c) (2,5)
(d) $\left(\frac{1}{2}, 7\right)$

Lucas Finney
Lucas Finney
Numerade Educator
01:31

Problem 17

Find and simplify the function values.
$f(x, y)=2 x+y^{2}$
(a) $\frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}$ (b) $\frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}$

Lucas Finney
Lucas Finney
Numerade Educator
01:58

Problem 18

Find and simplify the function values.
$f(x, y)=3 x^{2}-2 y$
(a) $\frac{f(x+\Delta x, y)-f(x, y)}{\Delta x}$ (b) $\frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}$

Lucas Finney
Lucas Finney
Numerade Educator
01:00

Problem 19

Find the domain and range of the function.
$$f(x, y)=x^{2}+y^{2}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:11

Problem 20

Find the domain and range of the function.
$$f(x, y)=e^{x y}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:04

Problem 21

Find the domain and range of the function.
$$g(x, y)=x \sqrt{y}$$

Lucas Finney
Lucas Finney
Numerade Educator
00:49

Problem 22

Find the domain and range of the function.
$$g(x, y)=\frac{y}{\sqrt{x}}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:00

Problem 23

Find the domain and range of the function.
$$z=\frac{x+y}{x y}$$

Lucas Finney
Lucas Finney
Numerade Educator
00:57

Problem 24

Find the domain and range of the function.
$$z=\frac{x y}{x-y}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:10

Problem 25

Find the domain and range of the function.
$$f(x, y)=\sqrt{4-x^{2}-y^{2}}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:23

Problem 26

Find the domain and range of the function.
$$f(x, y)=\sqrt{4-x^{2}-4 y^{2}}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:51

Problem 27

Find the domain and range of the function.
$$f(x, y)=\arccos (x+y)$$

Lucas Finney
Lucas Finney
Numerade Educator
01:21

Problem 28

Find the domain and range of the function.
$$f(x, y)=\arcsin (y / x)$$

Lucas Finney
Lucas Finney
Numerade Educator
02:08

Problem 29

Find the domain and range of the function.
$$f(x, y)=\ln (4-x-y)$$

Lucas Finney
Lucas Finney
Numerade Educator
01:46

Problem 30

Find the domain and range of the function.
$$f(x, y)=\ln (x y-6)$$

Lucas Finney
Lucas Finney
Numerade Educator
02:42

Problem 31

It The graphs labeled (a), (b), (c), and (d) are graphs of the function $f(x, y)=-4 x /\left(x^{2}+y^{2}+1\right) .$ Match the four graphs with the points in space from which the surface is viewed. The four points are (20,15,25),(-15,10,20) $(20,20,0),$ and (20,0,0). (GRAPH CAN'T COPY)

Lucas Finney
Lucas Finney
Numerade Educator
05:35

Problem 32

It Use the function given in Exercise $31 .$
(a) Find the domain and range of the function.
(b) Identify the points in the $x y$ -plane at which the function value is 0
(c) Does the surface pass through all the octants of the rectangular coordinate system? Give reasons for your answer.

William Semus
William Semus
Numerade Educator
00:38

Problem 33

Sketch the surface given by the function.
$$f(x, y)=4$$

Lucas Finney
Lucas Finney
Numerade Educator
02:02

Problem 34

Sketch the surface given by the function.
$$f(x, y)=6-2 x-3 y$$

Lucas Finney
Lucas Finney
Numerade Educator
01:32

Problem 35

Sketch the surface given by the function.
$$f(x, y)=y^{2}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:39

Problem 36

Sketch the surface given by the function.
$$g(x, y)=\frac{1}{2} y$$

Lucas Finney
Lucas Finney
Numerade Educator
00:46

Problem 37

Sketch the surface given by the function.
$$z=-x^{2}-y^{2}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:23

Problem 38

Sketch the surface given by the function.
$$z=\frac{1}{2} \sqrt{x^{2}+y^{2}}$$

Lucas Finney
Lucas Finney
Numerade Educator
00:58

Problem 39

Sketch the surface given by the function.
$$f(x, y)=e^{-x}$$

Lucas Finney
Lucas Finney
Numerade Educator
02:07

Problem 40

Sketch the surface given by the function.
$$f(x, y)=\left\{\begin{array}{cc}
x y, & x \geq 0, y \geq 0 \\
0, & x<0 \text { or } y<0
\end{array}\right.$$

Lucas Finney
Lucas Finney
Numerade Educator
00:40

Problem 41

Use a computer algebra system to graph the function.
$$z=y^{2}-x^{2}+1$$

Lucas Finney
Lucas Finney
Numerade Educator
00:32

Problem 42

Use a computer algebra system to graph the function.
$$z=\frac{1}{12} \sqrt{144-16 x^{2}-9 y^{2}}$$

Lucas Finney
Lucas Finney
Numerade Educator
00:32

Problem 43

Use a computer algebra system to graph the function.
$$f(x, y)=x^{2} e^{(-x y / 2)}$$

Lucas Finney
Lucas Finney
Numerade Educator
00:35

Problem 44

Use a computer algebra system to graph the function.
$$f(x, y)=x \sin y$$

Lucas Finney
Lucas Finney
Numerade Educator
01:08

Problem 45

Match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).] (GRAPH CAN'T COPY)
$$f(x, y)=e^{1-x^{2}-y^{2}}$$

Lucas Finney
Lucas Finney
Numerade Educator
00:29

Problem 46

Match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).] (GRAPH CAN'T COPY)
$$f(x, y)=e^{1-x^{2}+y^{2}}$$(GRAPH CAN'T COPY)

Lucas Finney
Lucas Finney
Numerade Educator
00:52

Problem 47

Match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).] (GRAPH CAN'T COPY)
$$f(x, y)=\ln \left|y-x^{2}\right|$$(GRAPH CAN'T COPY)

Lucas Finney
Lucas Finney
Numerade Educator
00:37

Problem 48

Match the graph of the surface with one of the contour maps. [The contour maps are labeled (a), (b), (c), and (d).] (GRAPH CAN'T COPY)
$$f(x, y)=\cos \left(\frac{x^{2}+2 y^{2}}{4}\right)$$(GRAPH CAN'T COPY)

Lucas Finney
Lucas Finney
Numerade Educator
01:24

Problem 49

Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given $c$ -values.
$$z=x+y, \quad c=-1,0,2,4$$

Lucas Finney
Lucas Finney
Numerade Educator
02:55

Problem 50

Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given $c$ -values.
$$z=6-2 x-3 y, \quad c=0,2,4,6,8,10$$

Lucas Finney
Lucas Finney
Numerade Educator
02:17

Problem 51

Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given $c$ -values.
$$z=x^{2}+4 y^{2}, \quad c=0,1,2,3,4$$

Lucas Finney
Lucas Finney
Numerade Educator
01:37

Problem 52

Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given $c$ -values.
$$f(x, y)=\sqrt{9-x^{2}-y^{2}}, \quad c=0,1,2,3$$

Lucas Finney
Lucas Finney
Numerade Educator
00:34

Problem 53

Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given $c$ -values.
$$f(x, y)=x y, \quad c=\pm 1,\pm 2, \ldots,\pm 6$$

Lucas Finney
Lucas Finney
Numerade Educator
00:47

Problem 54

Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given $c$ -values.
$$f(x, y)=e^{x y / 2}, \quad c=2,3,4, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:10

Problem 55

Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given $c$ -values.
$$f(x, y)=x /\left(x^{2}+y^{2}\right), \quad c=\pm \frac{1}{2},\pm 1, \pm \frac{3}{2},\pm 2$$

Lucas Finney
Lucas Finney
Numerade Educator
00:53

Problem 56

Describe the level curves of the function. Sketch a contour map of the surface using level curves for the given $c$ -values.
$$f(x, y)=\ln (x-y), \quad c=0, \pm \frac{1}{2},\pm 1, \pm \frac{3}{2},\pm 2$$

Lucas Finney
Lucas Finney
Numerade Educator
00:57

Problem 57

Use a graphing utility to graph six level curves of the function.
$$f(x, y)=x^{2}-y^{2}+2$$

Lucas Finney
Lucas Finney
Numerade Educator
00:46

Problem 58

Use a graphing utility to graph six level curves of the function.
$$f(x, y)=|x y|$$

Lucas Finney
Lucas Finney
Numerade Educator
00:25

Problem 59

Use a graphing utility to graph six level curves of the function.
$$g(x, y)=\frac{8}{1+x^{2}+y^{2}}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:07

Problem 60

Use a graphing utility to graph six level curves of the function.
$$h(x, y)=3 \sin (|x|+|y|)$$

Lucas Finney
Lucas Finney
Numerade Educator
01:32

Problem 61

What is a graph of a function of two variables? How is it interpreted geometrically? Describe level curves.

Lucas Finney
Lucas Finney
Numerade Educator
01:10

Problem 62

All of the level curves of the surface given by $z=f(x, y)$ are concentric circles. Does this imply that the graph of $f$ is a hemisphere? Illustrate your answer with an example.

Lucas Finney
Lucas Finney
Numerade Educator
01:18

Problem 63

Construct a function whose level curves are lines passing through the origin.

Lucas Finney
Lucas Finney
Numerade Educator
07:07

Problem 64

Consider the function $f(x, y)=x y,$ for $x \geq 0$ and $y \geq 0$.
(a) Sketch the graph of the surface given by $f$.
(b) Make a conjecture about the relationship between the graphs of $f$ and $g(x, y)=f(x, y)-3 .$ Explain your reasoning.
(c) Make a conjecture about the relationship between the graphs of $f$ and $g(x, y)=-f(x, y) .$ Explain your reasoning.
(d) Make a conjecture about the relationship between the graphs of $f$ and $g(x, y)=\frac{1}{2} f(x, y) .$ Explain your reasoning.
(e) On the surface in part (a), sketch the graph of $z=f(x, x)$.

William Semus
William Semus
Numerade Educator
03:46

Problem 65

Use the graphs of the level curves ( $c$ -values evenly spaced) of the function $f$ to write a description of a possible graph of $f .$ Is the graph of $f$ unique? Explain. (GRAPH CAN'T COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
03:46

Problem 66

Use the graphs of the level curves ( $c$ -values evenly spaced) of the function $f$ to write a description of a possible graph of $f .$ Is the graph of $f$ unique? Explain. (GRAPH CAN'T COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:17

Problem 67

In $2012,$ an investment of $\$ 1000$ was made in a bond earning $6 \%$ compounded annually. Assume that the buyer pays tax at rate $R$ and the annual rate of inflation is $I .$ In the year $2022,$ the value $V$ of the investment in constant 2012 dollars is $V(I, R)=1000\left[\frac{1+0.06(1-R)}{1+I}\right]^{10}$. Use this function of two variables to complete the table. $$\begin{array}{|l|l|l|l|}\hline & \text { Inflation Rate } \\\hline \text { Tax Rate } & 0 & 0.03 & 0.05 \\\hline 0 & & & \\\hline 0.28 & & & \\\hline 0.35 & & & \\\hline\end{array}$$

Lucas Finney
Lucas Finney
Numerade Educator
02:49

Problem 68

A principal of $\$ 5000$ is deposited in a savings account that earns interest at a rate of $r$ (written as a decimal), compounded continuously. The amount $A(r, t)$ after $t$ years is
$A(r, t)=5000 e^{r t}$ Use this function of two variables to complete the table. $$\begin{array}{|l|l|l|l|l|}\hline&\text { Number of Years } \\\hline \text { Rate } & 5 & 10 & 15 & 20 \\\hline 0.02 & & & & \\\hline 0.03 & & & & \\\hline 0.04 & & & & \\\hline 0.05 & & & & \\\hline\end{array}$$

Lucas Finney
Lucas Finney
Numerade Educator
01:25

Problem 69

Sketch the graph of the level surface $f(x, y, z)=c$ at the given value of $c$.
$$f(x, y, z)=x-y+z, \quad c=1$$

Lucas Finney
Lucas Finney
Numerade Educator
01:04

Problem 70

Sketch the graph of the level surface $f(x, y, z)=c$ at the given value of $c$.
$$f(x, y, z)=4 x+y+2 z, \quad c=4$$

Lucas Finney
Lucas Finney
Numerade Educator
01:07

Problem 71

Sketch the graph of the level surface $f(x, y, z)=c$ at the given value of $c$.
$$f(x, y, z)=x^{2}+y^{2}+z^{2}, \quad c=9$$

Lucas Finney
Lucas Finney
Numerade Educator
01:31

Problem 72

Sketch the graph of the level surface $f(x, y, z)=c$ at the given value of $c$.
$$f(x, y, z)=x^{2}+\frac{1}{4} y^{2}-z, \quad c=1$$

Lucas Finney
Lucas Finney
Numerade Educator
01:39

Problem 73

Sketch the graph of the level surface $f(x, y, z)=c$ at the given value of $c$.
$$f(x, y, z)=4 x^{2}+4 y^{2}-z^{2}, \quad c=0$$

Lucas Finney
Lucas Finney
Numerade Educator
01:22

Problem 74

Sketch the graph of the level surface $f(x, y, z)=c$ at the given value of $c$.
$$f(x, y, z)=\sin x-z, \quad c=0$$

Lucas Finney
Lucas Finney
Numerade Educator
01:46

Problem 75

The Doyle Log Rule 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 $N(d, L)=\left(\frac{d-4}{4}\right)^{2} L$
(a) Find the number of board-feet of lumber in a log 22 inches in diameter and 12 feet in length.
(b) Find $N(30,12)$.

Lucas Finney
Lucas Finney
Numerade Educator
01:14

Problem 76

The average length of time that a customer waits in line for service is $W(x, y)=\frac{1}{x-y}, \quad x>y$
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 each of the following.
(a) $W(15,9)$
(b) $W(15,13)$
(c) $W(12,7)$
(d) $W(5,2)$

Lucas Finney
Lucas Finney
Numerade Educator
01:47

Problem 77

The temperature $T$ (in degrees Celsius) at any point $(x, y)$ in a circular steel plate of radius 10 meters is $T=600-0.75 x^{2}-0.75 y^{2}$ where $x$ and $y$ are measured in meters. Sketch some of the isothermal curves.

Lucas Finney
Lucas Finney
Numerade Educator
01:25

Problem 78

The electric potential $V$ at any point $(x, y)$ is $V(x, y)=\frac{5}{\sqrt{25+x^{2}+y^{2}}}$ .Sketch the equipotential curves for $V=\frac{1}{2}, V=\frac{1}{3},$ and $V=\frac{1}{4}$.

Lucas Finney
Lucas Finney
Numerade Educator
01:36

Problem 79

Use the CobbDouglas production function (see Example 5 ) to show that when the number of units of labor and the number of units of capital are doubled, the production level is also doubled.

Lucas Finney
Lucas Finney
Numerade Educator
02:07

Problem 80

Show that the Cobb-Douglas production function $z=C x^{a} y^{1-a}$ can be rewritten as $\ln \frac{z}{y}=\ln C+a \ln \frac{x}{y}$.

Lucas Finney
Lucas Finney
Numerade Educator
02:09

Problem 81

According to the Ideal Gas Law, $P V=k T$ 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 $300 \mathrm{K}$.
(a) Determine $k$.
(b) Write $P$ as a function of $V$ and $T$ and describe the level curves.

Lucas Finney
Lucas Finney
Numerade Educator
03:28

Problem 82

The table shows the net sales $x$ (in billions of dollars), the total assets $y$ (in billions of dollars), and the shareholder's equity $z$ (in billions of dollars) for Apple for the years 2006 through 2011 . (Source: Apple Inc.) $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Year } & 2006 & 2007 & 2008 & 2009 & 2010 & 2011 \\\hline x & 19.3 & 24.6 & 37.5 & 42.9 & 65.2 & 108.2 \\\hline y & 17.2 & 24.9 & 36.2 & 47.5 & 75.2 & 116.4 \\\hline z & 10.0 & 14.5 & 22.3 & 31.6 & 47.8 & 76.6 \\\hline\end{array}$$
A model for these data is $z=f(x, y)=0.035 x+0.640 y-1.77$
(a) Use a graphing utility and the model to approximate $z$ for the given values of $x$ and $y$
(b) Which of the two variables in this model has the greater influence on shareholder's equity? Explain.
(c) Simplify the expression for $f(x, 150)$ and interpret its meaning in the context of the problem.

Lucas Finney
Lucas Finney
Numerade Educator
01:21

Problem 83

Meteorologists measure the atmospheric pressure in millibars. From these observations, they create weather maps on which the curves of equal atmospheric pressure (isobars), are drawn (see figure). On the map, the closer the isobars, the higher the wind speed. Match points $A$ $B,$ and $C$ with (a) highest pressure, (b) lowest pressure, and (c) highest wind velocity.

Lucas Finney
Lucas Finney
Numerade Educator
03:25

Problem 84

The acidity of rainwater is measured in units called pH. A pH of 7 is neutral, smaller values are increasingly acidic, and larger values are increasingly alkaline. The map shows curves of equal pH and gives evidence that downwind of heavily industrialized areas, the acidity has been increasing. Using the level curves on the map, determine the direction of the prevailing winds in the northeastern United States..

James Kiss
James Kiss
Numerade Educator
03:22

Problem 85

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 $\$ 1.20$ 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 $z$.

William Semus
William Semus
Numerade Educator
01:08

Problem 86

The contour map shown in the figure was computer generated using data collected by satellite instrumentation. Color is used to show the "ozone hole" in Earth's atmosphere. The purple and blue areas represent the lowest levels of ozone, and the green areas represent the highest levels. (Source: National Aeronautics and Space Administration) (IMAGE CAN'T COPY)
(a) Do the level curves correspond to equally spaced ozone levels? Explain.
(b) Describe how to obtain a more detailed contour map.

Lucas Finney
Lucas Finney
Numerade Educator
00:54

Problem 87

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $f\left(x_{0}, y_{0}\right)=f\left(x_{1}, y_{1}\right),$ then $x_{0}=x_{1}$ and $y_{0}=y_{1}$.

Lucas Finney
Lucas Finney
Numerade Educator
00:51

Problem 88

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $f$ is a function, then $f(a x, a y)=a^{2} f(x, y)$.

Lucas Finney
Lucas Finney
Numerade Educator
00:54

Problem 89

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
A vertical line can intersect the graph of $z=f(x, y)$ at most once.

Lucas Finney
Lucas Finney
Numerade Educator
00:48

Problem 90

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
Two different level curves of the graph of $z=f(x, y)$ can intersect.

Lucas Finney
Lucas Finney
Numerade Educator
02:04

Problem 91

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function such that $f(x, y)+f(y, z)+f(z, x)=0$ for all real numbers $x, y,$ and $z .$ Prove that there exists a function $g: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x, y)=g(x)-g(y)$ for all real numbers $x$ and $y$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator