Chapter 12, Functions of Several Variables Video Solutions, Calculus, Early Transcendentals

Section 1

Functions of Two or More Variables and Their Graphs

00:47

Problem 1

For the function $z=f(x, y)=\sqrt{16-x^{2}-y^{2}}$ $f(2,3)=$ _____.

Lucas F.

Numerade Educator

02:01

Problem 2

True or False The domain of a function of two variables is a nonempty set of points $(x, y)$ in the $x y$ -plane.

Lucas F.

Numerade Educator

00:47

Problem 3

For a function $w=f(x, y, z),$ the independent variable(s) is(are) _____ and the dependent variable(s) is(are) _____.

Lucas F.

Numerade Educator

01:06

Problem 4

True or False A level curve of the graph of $z=f(x, y)$ is the set of points in the $x y$ -plane for which $f(x, y)=c$.

Lucas F.

Numerade Educator

01:52

Problem 5

Evaluate each function.
$f(x, y)=3 x+2 y+x y$
(a) $f(0,1)$
(b) $f(2,1)$
(c) $f(x+\Delta x, y)$
(d) $f(x, y+\Delta y)$

Lucas F.

Numerade Educator

01:47

Problem 6

Evaluate each function.
$f(x, y)=x^{2} y+x+1$
(a) $f(0,1)$
(b) $f(2,1)$
(c) $f(x+\Delta x, y)$
(d) $f(x, y+\Delta y)$

Lucas F.

Numerade Educator

01:48

Problem 7

Evaluate each function.
$f(x, y)=\sqrt{x y}+x$
(a) $f(0,1)$
(b) $f\left(a^{2}, t^{2}\right) ; \quad a>0, t>0$
(c) $f(x+\Delta x, y)$
(d) $f(x, y+\Delta y)$

Lucas F.

Numerade Educator

01:10

Problem 8

Evaluate each function.
$f(x, y)=e^{x+y}$
(a) $f(1,-1)$
(b) $f(x+\Delta x, y)$
(c) $f(x, y+\Delta y)$

Lucas F.

Numerade Educator

01:58

Problem 9

Evaluate each function.
$F(x, y, z)=\frac{3 x y+z}{x^{2}+y^{2}+z^{2}}$
(a) $F(0,0,1)$
(b) $F(0,1,0)$
(c) $F(\sin t, \cos t, 0)$

Lucas F.

Numerade Educator

01:28

Problem 10

Evaluate each function.
$F(x, y, z)=\frac{x y z}{x^{2}+y^{2}+z^{2}}$
(a) $F(1,-1,1)$
(b) $F(a, a, a), a \neq 0$
(c) $F(\sin t, \cos t, a)$

Lucas F.

Numerade Educator

02:17

Problem 11

$f(x, y)=3 x y-x^{2},$ where $x=x(t)=\sqrt{t}$ and $y=y(t)=t^{2}$
Find:
(a) $f(x(0), y(0))$
(b) $f(x(1), y(1))$
(c) $f(x(4), y(4))$

Lucas F.

Numerade Educator

02:05

Problem 12

$f(x, y)=x^{2}+x y+y^{2},$ where $x=x(t)=t$ and $y=y(t)=t^{2}$ Find:
(a) $f(x(0), y(0))$
(b) $f(x(1), y(1))$
(c) $f(x(2), y(2))$

Lucas F.

Numerade Educator

01:15

Find the domain of each function and graph the domain. Use a solid curve to indicate that the domain includes the boundary and a dashed curve to indicate that the domain excludes the boundary.
$$
f(x, y)=\frac{\sqrt{x}}{\sqrt{y}}
$$

Lucas F.

Numerade Educator

00:34

Problem 14

Lucas F.

Numerade Educator

01:18

Problem 15

Lucas F.

Numerade Educator

01:15

Problem 16

Lucas F.

Numerade Educator

00:35

Problem 17

Lucas F.

Numerade Educator

01:18

Problem 18

Lucas F.

Numerade Educator

01:59

Problem 19

Lucas F.

Numerade Educator

01:36

Problem 20

Lucas F.

Numerade Educator

01:11

Problem 21

Lucas F.

Numerade Educator

01:20

Problem 22

Lucas F.

Numerade Educator

01:28

Problem 23

Lucas F.

Numerade Educator

00:44

Problem 24

Lucas F.

Numerade Educator

00:43

Problem 25

Find the domain of each function.
$$
f(x, y, z)=\frac{x^{2}+y^{2}}{z^{2}}
$$

Lucas F.

Numerade Educator

01:20

Problem 26

Find the domain of each function.
$$
f(x, y, z)=e^{z} \ln \left(x^{2}+y^{2}\right)
$$

Lucas F.

Numerade Educator

00:57

Problem 27

Find the domain of each function.
$$
f(x, y, z)=\frac{z \sin x}{\cos y}
$$

Lucas F.

Numerade Educator

00:57

Problem 28

Find the domain of each function.
$$
f(x, y, z)=\frac{x y z}{\sqrt{x^{2}+y^{2}+z^{2}}}
$$

Lucas F.

Numerade Educator

01:25

Problem 29

Graph each surface.
$$
z=f(x, y)=3-x-y
$$

Lucas F.

Numerade Educator

01:35

Problem 30

Graph each surface.
$$
z=f(x, y)=2+x-y
$$

Lucas F.

Numerade Educator

01:31

Problem 31

Graph each surface.
$$
z=f(x, y)=x^{2}+y^{2}
$$

Lucas F.

Numerade Educator

01:34

Problem 32

Graph each surface.
$$
z=f(x, y)=x^{2}-y^{2}
$$

Lucas F.

Numerade Educator

00:54

Problem 33

Graph each surface.
$$
z=f(x, y)=\sqrt{4-x^{2}-y^{2}}
$$

Lucas F.

Numerade Educator

01:12

Problem 34

Graph each surface.
$$
z=f(x, y)=\sqrt{x^{2}+y^{2}-4}
$$

Lucas F.

Numerade Educator

01:25

Problem 35

Graph each surface.
$$
z=f(x, y)=\sin y
$$

Lucas F.

Numerade Educator

01:04

Problem 36

Graph each surface.
$$
z=f(x, y)=\cos x
$$

Lucas F.

Numerade Educator

01:11

Problem 37

Graph each surface.
$$
z=f(x, y)=4-x^{2}-y^{2}
$$

Lucas F.

Numerade Educator

01:01

Problem 38

Graph each surface.
$$
z=f(x, y)=x^{2}+y^{2}-4
$$

Lucas F.

Numerade Educator

02:15

Problem 39

For each function
(a) Graph the level curves corresponding to the given values of $c$.
(b) Use a CAS to graph the surface.
$z=f(x, y)=x^{2}-y^{2}$ at $c=0,1,4,9$

Lucas F.

Numerade Educator

00:33

Problem 40

For each function
(a) Graph the level curves corresponding to the given values of $c$.
(b) Use a CAS to graph the surface.
$z=f(x, y)=2 x^{2}+y^{2}$ at $c=0,1,4,9$

Lucas F.

Numerade Educator

00:33

Problem 41

For each function
(a) Graph the level curves corresponding to the given values of $c$.
(b) Use a CAS to graph the surface.
$z=f(x, y)=\sqrt{9-x^{2}-y^{2}}$ at $c=0,1,3$

Lucas F.

Numerade Educator

00:33

Problem 42

For each function
(a) Graph the level curves corresponding to the given values of $c$.
(b) Use a CAS to graph the surface.
$z=f(x, y)=x^{2}-2 y$ at $c=-4,-1,0,1,4$

Lucas F.

Numerade Educator

00:49

Problem 43

For each function
(a) Graph the level curves corresponding to the given values of $c$.
(b) Use a CAS to graph the surface.
$z=f(x, y)=x^{2}-2 y$ at $c=-4,-1,0,1,4$

Lucas F.

Numerade Educator

00:32

Problem 44

For each function
(a) Graph the level curves corresponding to the given values of $c$.
(b) Use a CAS to graph the surface.
$z=f(x, y)=y^{2}-x$ at $c=-4,-1,0,1,4$

Lucas F.

Numerade Educator

00:41

Problem 45

For each function
(a) Graph the level curves corresponding to the given values of $c$.
(b) Use a CAS to graph the surface.
$z=f(x, y)=x+\sin y$ at $c=0,2,4,8$

Lucas F.

Numerade Educator

00:24

Problem 46

For each function
(a) Graph the level curves corresponding to the given values of $c$.
(b) Use a CAS to graph the surface.
$z=f(x, y)=y-\ln x$ at $c=1,2,4$

Lucas F.

Numerade Educator

Problem 47

(a) Describe in words the level surfaces of each function $f$.
(b) Graph a level surface.
(c) Does the graph match your verbal description?
$$
w=f(x, y, z)=x^{2}+z^{2}
$$

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Problem 48

(a) Describe in words the level surfaces of each function $f$.
(b) Graph a level surface.
(c) Does the graph match your verbal description?
$$
w=f(x, y, z)=x^{2}+y^{2}
$$

Check back soon!

Problem 49

(a) Describe in words the level surfaces of each function $f$.
(b) Graph a level surface.
(c) Does the graph match your verbal description?
$$
w=f(x, y, z)=z-2 x-2 y
$$

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Problem 50

(a) Describe in words the level surfaces of each function $f$.
(b) Graph a level surface.
(c) Does the graph match your verbal description?
$$
w=f(x, y, z)=x+y-z
$$

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Problem 51

(a) Describe in words the level surfaces of each function $f$.
(b) Graph a level surface.
(c) Does the graph match your verbal description?
$$
w=f(x, y, z)=4-x^{2}-y^{2}
$$

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Problem 52

(a) Describe in words the level surfaces of each function $f$.
(b) Graph a level surface.
(c) Does the graph match your verbal description?
$$
w=f(x, y, z)=z
$$

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00:54

Problem 53

Match each surface to its corresponding level curves shown in $(A)-(F).$
$$
z=f(x, y)=2 x-y-2
$$

Lucas F.

Numerade Educator

00:28

Problem 54

Match each surface to its corresponding level curves shown in $(A)-(F).$
$$
z=f(x, y)=x+y-3
$$

Lucas F.

Numerade Educator

00:35

Problem 55

Match each surface to its corresponding level curves shown in $(A)-(F).$
$$
z=f(x, y)=4 x^{2}+y^{2}
$$

Lucas F.

Numerade Educator

00:35

Problem 56

Match each surface to its corresponding level curves shown in $(A)-(F).$
$$
z=f(x, y)=\sqrt{\frac{x^{2}}{4^{2}}+\frac{y^{2}}{4^{2}}}
$$

Lucas F.

Numerade Educator

00:53

Problem 57

Match each surface to its corresponding level curves shown in $(A)-(F).$
$$
z=f(x, y)=\frac{x^{2}}{2^{2}}-\frac{y^{2}}{1^{2}}
$$

Lucas F.

Numerade Educator

01:03

Problem 58

Match each surface to its corresponding level curves shown in $(A)-(F).$
$$
z=f(x, y)=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}
$$

Lucas F.

Numerade Educator

02:32

Problem 59

A contour map of the temperatures in ${ }^{\circ} \mathrm{F}$ of the western United States made on a given day in December 2011 is shown. The level curves are called isotherms.
(a) Estimate the temperature at the Four Corners, the point where Utah, Colorado, Arizona, and New Mexico meet.
(b) Describe the change in temperature as you move south from the Four Corners.
(c) Describe the direction you would travel if you were at the Four Corners and wanted to move as quickly as possible to the next warmer temperature zone. Comment on how your path intersects the isotherm.
(d) A weather front is moving in if the isotherms are close together. Do you think a front is approaching the Four Corners? From what direction? Explain.

Lucas F.

Numerade Educator

02:40

Problem 60

A topographical map of Mount Washington in New Hampshire is shown. Suppose a climber is at the base of the mountain and
wants to climb to the summit. If she can begin at point $A, B, C,$ or $D$ and climb straight up the
(a) At which point should she start to climb so the distance climbed is a minimum?
(b) At which point is the climb initially the steepest?
(c) Suppose someone is on any level curve of the mountain. Describe how he would walk so that no change in altitude occurs.
(d) Comment on how the paths in (a)-(c) intersect the contour lines.

Lucas F.

Numerade Educator

02:23

Problem 61

In psychology, the intelligence quotient (IQ) is measured by $I Q=f(M, C)=100 \frac{M}{C},$ where $M$ is a person's mental age and $C$ is the person's chronological or actual age, $0<C \leq 16$
(a) Find the IQ of a 12 -year-old child whose mental age is $10 .$
(b) Find the IQ of a 10 -year-old child whose mental age is 12 .
(c) If a 10-year-old girl has an IQ of 139 , what is her mental age? Round answers to the nearest integer.

Lucas F.

Numerade Educator

01:17

Problem 62

One version of the AT\&T DataConnect Pass plan for tablets has a monthly cost (in dollars) of $C(x, y, z)=35.00+0.1 x+0.015 y+0.0195 z,$ where $x$ is the number of megabytes (mb) used domestically in excess of 200 , $y$ is the number of kilobytes $(\mathrm{kb})$ of data sent to or received from Canada, and $z$ is the number of kilobytes of data used outside the United States and Canada. Find the total monthly bill (in dollars) for $210 \mathrm{mb}$ domestic, $250 \mathrm{~kb}$ Canadian, and $15 \mathrm{~kb}$ international usage.

Lucas F.

Numerade Educator

01:15

Problem 63

A pitcher's earned run average is calculated using the function $A(N, I)=9\left(\frac{N}{I}\right),$ where $N$ is the total number of earned runs given up in $I$ innings pitched. Find:
(a) $A(3,4)$
(b) $A(6,3)$
(c) $A(2,9)$
(d) $A(3,18)$

Lucas F.

Numerade Educator

01:36

Problem 64

In the National Basketball Association (NBA), the adjusted field goal percentage is modeled by the function $f(x, y, s)=\frac{x+1.5 y}{s},$ where $x$ is the number of two-point field goals made, $y$ is the number of three-point field goals made, and $s$ is the total of all field goals attempted.
(a) During the 2013 season, DeAndre Jordan of the Los Angeles Clippers led the league in field goal shooting. He made 234 out of 488 two-point attempts and no three-point attempts. Find his adjusted field goal percentage.
(b) During the 2013 season, Gordon Hayward of the Utah Jazz led the league in three-point shooting. He made 234 out of 527 two-point attempts and 234 out of 246 three-point attempts. Find his adjusted field goal percentage.

Lucas F.

Numerade Educator

02:10

Problem 65

The apparent temperature (in degrees Fahrenheit) is measured by the heat index $H$ according to the formula
$$ \begin{aligned} H=& H(t, r)=-42.379+2.04901523 t \\ &+10.1333127 r-0.22475541 \mathrm{tr}-0.00683783 t^{2} \\ &-0.05481717 r^{2}+0.00122874 t^{2} r+0.00085282 \mathrm{tr}^{2} \\&-0.00000199 t^{2} r^{2} \end{aligned} $$ where $H=$ the heat index, $t=$ the air temperature, and $r=$ the percent relative humidity (for example, $r=75$ when the relative humidity is $75 \%$ ).
(b) If the air temperature is $97^{\circ},$ what is the lowest relative humidity that will result in a heat index of $105^{\circ} ?$
(a) What is the heat index when the air temperature is $95^{\circ}$ and the relative humidity is $50 \% ?$

Lucas F.

Numerade Educator

01:02

Problem 66

The production function for a toy manufacturer is given by the equation $Q(L, M)=400 L^{0.3} M^{0.7},$ where $Q$ is the output in units, $L$ is the labor in hours, and $M$ is the number of machine hours. Find:
(a) $Q(19,21)$
(b) $Q(21,20)$

Lucas F.

Numerade Educator

01:50

Problem 67

Write the equation for the surface area $S$ of an open box as a function of its length $x$, width $y,$ and depth $z$.

Lucas F.

Numerade Educator

01:30

Problem 68

The cost $\mathrm{C}$ of the bottom and top of a cylindrical tank is $\$ 300$ per square meter and the cost of the sides is $\$ 500$ per square meter. Find a function that models the total cost of constructing such a tank as a function of the radius $R$ and height $h,$ both in meters.

Lucas F.

Numerade Educator

01:42

Problem 69

Find a function that models the total cost $\mathrm{C}$ of constructing an open rectangular box if the cost per square centimeter of the material to be used for the bottom is $\$ 4,$ for two of the opposite sides is $\$ 3,$ and for the remaining pair of opposite sides is $\$ 2$

Lucas F.

Numerade Educator

00:57

Problem 70

Repeat Problem 69 for a closed rectangular box that has a top made of material costing $\$ 5$ per square centimeter.

Lucas F.

Numerade Educator

Problem 71

The formula $$ V(x, y)=\frac{9}{\sqrt{4-\left(x^{2}+y^{2}\right)}} $$ gives the electrical potential $V$ (in volts) at a point $(x, y)$ in the $x y$ -plane. Draw the equipotential curves (level curves) for $V=18,9,$ and 6 volts. Describe the surface $z=V(x, y)$.

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Problem 72

The temperature $T$ in degrees Celsius at any point $(x, y)$ of a flat plate in the $x y$ -plane, is $T=60-2 x^{2}-3 y^{2}$ Draw the isothermal curves (level curves) for $T=60^{\circ} \mathrm{C}, 54^{\circ} \mathrm{C}$ $48^{\circ} \mathrm{C}, 6^{\circ} \mathrm{C}, 0^{\circ} \mathrm{C} .$ Describe the surface $z=T(x, y)$.

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Problem 73

The strength $E$ of an electric field at a point $(x, y, z)$ resulting from an infinitely long charged wire lying along the $x$ -axis is $$ E(x, y, z)=\frac{3}{\sqrt{y^{2}+z^{2}}} $$ Describe the level surfaces of $E$.

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Problem 74

The magnitude $F$ of the force of attraction between two objects, one located at the origin and the other at the point $(x, y, z) \neq(0,0,0),$ of masses $m$ and $M$ is given by $$ F=\frac{G m M}{x^{2}+y^{2}+z^{2}} $$ where $G=6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^{2} / \mathrm{kg}^{2}$ is the gravitational constant. Describe the level surfaces of $F$.

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01:48

Problem 75

The Ideal Gas Law, $P V=n R T,$ is used to describe the relationship among the pressure $P,$ volume $V,$ and temperature $T$ of an ideal gas, where $n$ is the number of moles of gas and $R$ is the universal gas constant. Describe the level curves for each of the following thermodynamic processes on an ideal gas:
(a) An isothermal process, that is, a process in which the temperature $T$ is held constant
(b) An isobaric process, that is, a process in which the pressure $P$ is held constant
(c) An isochoric process, that is, a process in which the volume $V$ is held constant

Pawan Y.

Numerade Educator

Problem 76

The electrostatic potential $V$ (in volts) from a point charge $Q$ at the origin is given by $$ V=\frac{k Q}{\sqrt{x^{2}+y^{2}+z^{2}}} $$ where $k$ is a constant.
(a) Describe the level surfaces of $V$. In electronics these are called the equipotential surfaces.
(b) What happens as the potential $V$ gets close to $0 ?$

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Problem 77

The magnitude of the magnetic field $B$ produced by a very long wire carrying a current $I$ along the $z$ -axis is given by $B=\frac{\mu_{0} I}{2 \pi \sqrt{x^{2}+y^{2}}},$ where $\mu_{0}$ is a constant.
(a) Describe the level surfaces for $B$.
(b) What happens as the magnitude of $B$ increases?

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Problem 78

The gravitational potential energy $U$ of a satellite in orbit around Earth is $U=-\frac{G M_{\text {Earth }} m_{\text {satellite }}}{\sqrt{x^{2}+y^{2}+z^{2}}},$ where the origin is placed at the center of Earth and $G$ is the universal gravitation constant.
(a) Describe the level surfaces for $U$.
(b) What happens as the gravitational potential energy gets close to $0 ?$

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Problem 79

(a) Graph the surface $z=f(x, y)=5 e^{-\left(x^{2}+y^{2}\right)}$.
(b) Experiment by changing the perspectives and plotting options until a clear picture of the graph is obtained.
(c) Change the style of the graph to show contour lines.
(d) Plot the level curves $f(x, y)=c \quad c=1,2,3,4$.
(e) Explain how the curves obtained in (d) relate to the graph from (a).

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