Calculus

Earl W. Swokowski

Chapter 16

Partial Differentation - all with Video Answers

Educators

Section 1

Functions of Several Variables

Problem 1

Describe the domain of $f,$ and find the indicated function values.
$$
\begin{aligned}
f(x, y)=2 x-y^{2} & \\
& f(-2,5), \quad f(5,-2), \quad f(0,-2)
\end{aligned}
$$

Claire Rochford

Claire Rochford

Numerade Educator

Problem 2

Describe the domain of $f,$ and find the indicated function values.
$$
f(x, y)=\frac{y+2}{x}
f(3,1), \quad f(1,3), \quad f(2,0)
$$

Claire Rochford

Claire Rochford

Numerade Educator

Problem 3

Describe the domain of $f,$ and find the indicated function values.
$$
\begin{aligned}
f(u, t)=\frac{u r}{u-2 v}: & \\
& f(2,3), \quad f(-1,4)
\end{aligned}
$$

Claire Rochford

Claire Rochford

Numerade Educator

Problem 4

Describe the domain of $f,$ and find the indicated function values.
$$
\begin{aligned}
f(r, s)=\sqrt{1-r}-e^{r s} & \\
f(1,1), & f(0,4), & f(-3,3)
\end{aligned}
$$

Claire Rochford

Claire Rochford

Numerade Educator

Problem 5

Describe the domain of $f,$ and find the indicated function values.
$$
\begin{aligned}
f(x, y, z)=\sqrt{25-x^{2}-y^{2}-z^{2}} & \\
\qquad f(1,-2,2), & f(-3,0,2)
\end{aligned}
$$

Claire Rochford

Claire Rochford

Numerade Educator

Problem 6

Describe the domain of $f,$ and find the indicated function values.
$$
\begin{aligned}
f(x, y, z)=2+\tan x+y \sin z & \\
f(\pi / 4,4, \pi / 6), & f(0,0,0)
\end{aligned}
$$

Sanu Kumar

Numerade Educator

Problem 7

Sketch the graph of $f$.
$$
f(x, y)=\sqrt{1-x^{2}-y^{2}}
$$

Abigail Martyr

Numerade Educator

Problem 8

Sketch the graph of $f$.
$$
f(x, y)=4-x^{2}-4 y^{2}
$$

Abigail Martyr

Numerade Educator

Problem 9

Sketch the graph of $f$.
$$
f(x, y)=x^{2}+y^{2}-1
$$

Abigail Martyr

Numerade Educator

Problem 10

Sketch the graph of $f$.
$$
f(x, y)=\frac{1}{6} \sqrt{9 x^{2}+4 y^{2}}
$$

Vikash Ranjan

Numerade Educator

Problem 11

Sketch the graph of $f$.
$$
f(x, y)=6-2 x-3 y
$$

Lucas Finney

Numerade Educator

Problem 12

Sketch the graph of $f$.
$$
f(x, y)=\sqrt{72}+4 x^{2}-9 y^{2}
$$

Nick Johnson

Numerade Educator

Problem 13

Sketch the graph of $f$.
$$
f(x, y)=\sqrt{y^{2}-4 x^{2}-16}
$$

Lucas Finney

Numerade Educator

Problem 14

Sketch the graph of $f$.
$$
f(x, y)=\sqrt{x^{2}+4 y^{2}+25}
$$

Vikash Ranjan

Numerade Educator

Problem 15

Sketch the level curves of $f$ for the given values of $k$.
$$
f(x, y)=y^{2}-x^{2} ; \quad k=-4,0,9
$$

William Semus

Numerade Educator

Problem 16

Sketch the level curves of $f$ for the given values of $k$.
$$
f(x, y)=3 x-2 y: \quad k=-4,0,6
$$

William Semus

Numerade Educator

Problem 17

Sketch the level curves of $f$ for the given values of $k$.
$$
f(x, y)=x^{2}-y: \quad k=-2,0,3
$$

William Semus

Numerade Educator

Problem 18

Sketch the level curves of $f$ for the given values of $k$.
$$
f(x, y)=x y: \quad k=-4,1,4
$$

William Semus

Numerade Educator

Problem 19

Sketch the level curves of $f$ for the given values of $k$.
$$
f(x, y)=(x-2)^{2}+(y+3)^{2}: \quad k=1,4,9
$$

William Semus

Numerade Educator

Problem 20

Sketch the level curves of $f$ for the given values of $k$.
$$
f(x, y)=4 x^{2}+y^{2}: \quad k=4,9,16
$$

Lucas Finney

Numerade Educator

Problem 21

Find an equation of the level curve of $f$ that contains the point $P$.
$$
f(x, y)=y \arctan x: \quad P(1,4)
$$

Kaylee Mcclellan

Kaylee Mcclellan

Numerade Educator

Problem 22

Find an equation of the level curve of $f$ that contains the point $P$.
$$
f(x, y)=\left(2 x+y^{2}\right) e^{x y}: \quad P(0,2)
$$

Linh Vu

Numerade Educator

Problem 23

Find an equation of the level surface of $f$ that contains the point $P$.
$$
f(x, y, z)=x^{2}+4 y^{2}-z^{2}: \quad P(2,-1,3)
$$

Tyler Moulton

Numerade Educator

Problem 24

Find an equation of the level surface of $f$ that contains the point $P$.
$$
f(x, y, z)=z^{2} y+x: \quad P(1,4,-2)
$$

Tyler Moulton

Numerade Educator

Problem 25

Describe the level curves, whenever they exist, for the surface in the indicated figure if
(a) $k>0,$
(b) $k=0,$ and $|c| k<0.$
Figure 16.17 (iii)

Lucas Finney

Numerade Educator

Problem 26

Describe the level curves, whenever they exist, for the surface in the indicated figure if
(a) $k>0,$
(b) $k=0,$ and $|c| k<0.$
Figure $16.17(\mathrm{xi})$

Lucas Finney

Numerade Educator

Problem 27

Describe the level curves, whenever they exist, for the surface in the indicated figure if
(a) $k>0,$
(b) $k=0,$ and $|c| k<0.$
Figure $16.17(\mathrm{vi})$

Lucas Finney

Numerade Educator

Problem 28

Describe the level curves, whenever they exist, for the surface in the indicated figure if
(a) $k>0,$
(b) $k=0,$ and $|c| k<0.$
Figure $16.17(x$ ii $)$

Lucas Finney

Numerade Educator

Problem 29

Correctly match one of the graphs shown in (a)-(f) to the system of level curves of the function given by $z=f(x, y)$
a.
b.
c.
d.
e.
f.
$$
z=\frac{\sin x y}{x y}
$$

Linh Vu

Numerade Educator

Problem 30

Correctly match one of the graphs shown in (a)-(f) to the system of level curves of the function given by $z=f(x, y)$
a.
b.
c.
d.
e.
f.
$$
z=\frac{15 x^{2} y^{2} e^{-x^{2}-y^{2}}}{x^{2}+y^{2}}
$$

Lucas Finney

Numerade Educator

Problem 31

Correctly match one of the graphs shown in (a)-(f) to the system of level curves of the function given by $z=f(x, y)$
a.
b.
c.
d.
e.
f.
$$
\begin{array}{l}
z=\sin \sqrt{x^{2}+y^{2}} \\
-7 \leq x \leq 7,-7 \leq y \leq 7
\end{array}
$$

Linh Vu

Numerade Educator

Problem 32

Correctly match one of the graphs shown in (a)-(f) to the system of level curves of the function given by $z=f(x, y)$
a.
b.
c.
d.
e.
f.
$$
\begin{array}{l}
z=e^{-x^{2}}+e^{-4 y^{2}} \\
-2 \leq x \leq 2,-2 \leq y \leq 2
\end{array}
$$

AG

Ankit Gupta

Numerade Educator

Problem 33

Correctly match one of the graphs shown in (a)-(f) to the system of level curves of the function given by $z=f(x, y)$
a.
b.
c.
d.
e.
f.
$$
\begin{array}{l}
z=\frac{x y^{3}-x^{3} y}{2} \\
-2 \leq x \leq 2,-2 \leq y \leq 2
\end{array}
$$

Nick Johnson

Numerade Educator

Problem 34

Correctly match one of the graphs shown in (a)-(f) to the system of level curves of the function given by $z=f(x, y)$
a.
b.
c.
d.
e.
f.
$$
\begin{array}{l}
z=y^{4}-8 y^{2}-4 x^{2} \\
-3 \leq x \leq 3,-3 \leq y \leq 3
\end{array}
$$

AG

Ankit Gupta

Numerade Educator

Problem 35

Graph, on the same coordinate plane, the level curves for $k=1,2,$ and $3,$ using polar coordinates.
$$
f(x, y)=x^{4}+8 x^{2} y^{2}+y^{4}
$$

Norman Atentar

Numerade Educator

Problem 36

Graph, on the same coordinate plane, the level curves for $k=1,2,$ and $3,$ using polar coordinates.
$$
f(x, y)=3 x^{4}+x^{2} y^{2}+y^{4}
$$

Norman Atentar

Numerade Educator

Problem 37

Describe the level surfaces of $f$ for the given values of $k$.
$$
f(x, y, z)=x^{2}+y^{2}+z^{2} ; \quad k=-1,0,4
$$

ag

Alan Ghazarians

Numerade Educator

Problem 38

Describe the level surfaces of $f$ for the given values of $k$.
$$
f(x, y, z)=z+x^{2}+4 y^{2} ; \quad k=-1,0,2
$$

ag

Alan Ghazarians

Numerade Educator

Problem 39

Describe the level surfaces of $f$ for the given values of $k$.
$$
f(x, y, z)=x+2 y+3 z ; \quad k=-6,6,12
$$

Linda Hand

Numerade Educator

Problem 40

Describe the level surfaces of $f$ for the given values of $k$.
$$
f(x, y, z)=x^{2}+y^{2}-z^{2} ; \quad k=-1,0,1
$$

ag

Alan Ghazarians

Numerade Educator

Problem 41

Describe the level surfaces of $f$ for the given values of $k$.
$$
f(x, y, z)=x^{2}+y^{2} ; \quad k=-1,0,4
$$

ag

Alan Ghazarians

Numerade Educator

Problem 42

Describe the level surfaces of $f$ for the given values of $k$.
$$
f(x, y, z)=z ; \quad k=-2,0,3
$$

Carson Merrill

Numerade Educator

Problem 43

A flat metal plate is situated in an $x y$ -plane such that the temperature $T$ (in $C$ ) at the point $(x, y)$ is inversely proportional to the distance from the origin.
(a) Describe the isotherms.
(b) If the temperature at the point $P(4,3)$ is $40 \mathrm{C}$, find an equation of the isotherm for a temperature of $20^{\circ} \mathrm{C}$

Suzanne W.

Numerade Educator

Problem 44

If the voltage $V$ at the point $P(x, y, z)$ is given by $V=6 /\left(x^{2}+4 y^{2}+9 z^{2}\right)^{1 / 2}$
(a) describe the equipotential surfaces
(b) find an equation of the equipotential surface $V=120$

Lucas Finney

Numerade Educator

Problem 45

According to Newton's law of universal gravitation, if a particle of mass $m_{0}$ is at the origin of an xyz-coordinate system, then the magnitude $F$ of the force exerted on a particle of mass $m$ located at the point $(x, y, z)$ is given by
$$
F=\frac{G m_{0} m}{x^{2}+y^{2}+z^{2}}
$$
where $G$ is the universal gravitational constant. How many independent variables are present? If $m_{0}$ and $m$ are constant. describe the level surfaces of the resulting function of $x, y,$ and $z$. What is the physical significance of these level surfaces?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 46

According to the ideal gas law, the pressure $P$, volume $V$. and temperature $T$ of a confined gas are related by the formula $P V=k T$ for a constant $k$. Express $P$ as a function of $V$ and $T$, and describe the level curves associated with this function. What is the physical significance of these level curves?

Lucas Finney

Numerade Educator

Problem 47

The power $P$ generated by a wind rotor is proportional to the product of the area $A$ swept out by the blades and the third power of the wind velocity $t$.
(a) Express $P$ as a function of $A$ and $v$.
(b) Describe the level curves of $P$ and explain their physical significance.
(c) When the diameter of the circular area swept out by the blades is 10 feet and the wind velocity is $20 \mathrm{mi} / \mathrm{hr}$ then $P=3000$ watts. Find an equation of the level curve $P=4000$.

Calin Lupas

Numerade Educator

Problem 48

If $x$ is wind velocity (in $\mathrm{m} / \mathrm{sec}$ ) and $y$ is temperature (in
C), then the windchill factor $F$ (in $\left(\mathrm{k} \mathrm{cal} / \mathrm{m}^{2}\right) \mathrm{hr}$ ) is given by $F=(33-y)(10 \sqrt{x}-x+10.5)$
(a) Find the velocities and temperatures for which the windchill factor is $0 .$ (Assume that $0 \leq x \leq 50$ and $-50 \leq y \leq 50 .)$
(b) If $F \geq 1400$, frostbite will occur on exposed human skin. Sketch the graph of the level curve $F=1400$.

AG

Ankit Gupta

Numerade Educator

Problem 49

Shown in the figure are level curves for the surface area of the human body (in tenths of a square meter) as a function of weight $x$ (in kilograms) and height $y$ (in centimeters). Use this system of level curves to estimate your own body surface area. Compare this estimate to the more exact estimate given by the DuBois and DuBois formula
$$
S=0.007184 x^{0.425} y^{0.725}
$$

Kayla Robinson

Numerade Educator

Problem 50

The upper boundary of the semicircular region shown in the figure is kept at a temperature of $10{ }^{\circ} \mathrm{C}$, while the lower boundary is maintained at 0 C. The long-range, or steady state, temperature $T$ at a point $(x, y)$ inside the region is given by
$$
T(x, y)=\frac{20}{\pi} \tan ^{-1}\left(\frac{2 y}{1-x^{2}-y^{2}}\right)
$$
Show that the isotherms are arcs of circles that have centers on the negative $y$ -axis and pass through the points (-1,0) and (1,0) . Sketch the isotherm corresponding to a temperature of $5 \mathrm{C}$

Raj Bala

Numerade Educator

Problem 51

A garbage incineration plant will be built to service two cities. Each city would like to maximize its distance from the plant. but for economic reasons the sum of the distances from each city to the plant cannot exceed $M$ miles. Show that the level curves for the location of the plant are ellipses.

Sarah Parrigin

Numerade Educator

Problem 52

The atmospheric pressure near ground level in a certain region is given by
$$
p(x, y)=a x^{2}+b y^{2}+c
$$
where $a, b,$ and $c$ are positive constants.
(a) Describe the isobars in this region for pressures greater than $c$.
(b) Is this a region of high or low pressure?

Nicole Wood

Numerade Educator