Calculus Early Transcendentals: Pearson New International Edition

Dale Varberg, Edwin J. Purcell, Steve E. Rigdon

Chapter 12

Derivatives for Functions of Two or More Variables - all with Video Answers

Educators

Section 1

Functions of Two or More Variables

Problem 1

Let $f(x, y)=x^{2} y+\sqrt{y}$. Find each value.
(a) $f(2,1)$
(b) $f(3,0)$
(c) $f(1,4)$
(d) $f\left(a, a^{4}\right)$
(e) $f\left(1 / x, x^{4}\right)$
(f) $f(2,-4)$
What is the natural domain for this function?

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Let $f(x, y)=y / x+x y$. Find each value.
(a) $f(1,2)$
(b) $f\left(\frac{1}{4}, 4\right)$
(c) $f\left(4, \frac{1}{4}\right)$
(d) $f(a, a)$
(e) $f\left(1 / x, x^{2}\right)$
(f) $f(0,0)$
What is the natural domain for this function?

Lucas Finney

Numerade Educator

Problem 3

Let $g(x, y, z)=x^{2} \sin v z$. Find each value.
(a) $g(1, \pi, 2)$
(b) $g(2,1, \pi / 6)$
(c) $g(4,2, \pi / 4)$
(d) $g(\pi, \pi, \pi)$

Lucas Finney

Numerade Educator

Problem 4

Let $g(x, y, z)=\sqrt{x \cos y}+z^{2} .$ Find each value.
(a) $g(4,0,2)$
(b) $g(-9, \pi, 3)$
(c) $g(2, \pi / 3,-1)$
(d) $g(3,6,1.2)$

Lucas Finney

Numerade Educator

Problem 5

Find $F(f(t), g(t))$ if $F(x, y)=x^{2} y$ and $f(t)=t \cos t$, $g(t)=\sec ^{2} t$

Lucas Finney

Numerade Educator

Problem 6

Find $F(f(t), g(t))$ if $F(x, y)=e^{x}+y^{2}$ and $f(t)=\ln t^{2}$ $g(t)=e^{t / 2} .$

Lucas Finney

Numerade Educator

Problem 7

Sketch the graph of $\bar{f}$.
$$
f(x, y)=6
$$

Lucas Finney

Numerade Educator

Problem 8

Sketch the graph of $\bar{f}$.
$$
f(x, y)=6-x
$$

Lucas Finney

Numerade Educator

Problem 9

Sketch the graph of $\bar{f}$.
$$
f(x, y)=6-x-2 y
$$

Lucas Finney

Numerade Educator

Problem 10

Sketch the graph of $\bar{f}$.
$$
f(x, y)=6-x^{2}
$$

Lucas Finney

Numerade Educator

Problem 11

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

Lucas Finney

Numerade Educator

Problem 12

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

Lucas Finney

Numerade Educator

Problem 13

Sketch the graph of $\bar{f}$.
$$
f(x, y)=3-x^{2}-y^{2}
$$

Lucas Finney

Numerade Educator

Problem 14

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

Abigail Martyr

Numerade Educator

Problem 15

Sketch the graph of $\bar{f}$.
$$
f(x, y)=e^{-\left(x^{2}+y^{2}\right)}
$$

Abigail Martyr

Numerade Educator

Problem 16

Sketch the graph of $\bar{f}$.
$$
f(x, y)=x^{2} / y, y>0
$$

Lucas Finney

Numerade Educator

Problem 17

Sketch the level curve $z=k$ for the indicated values of $k$.
$z=\frac{1}{2}\left(x^{2}+y^{2}\right), k=0,2,4,6,8$

Lucas Finney

Numerade Educator

Problem 18

Sketch the level curve $z=k$ for the indicated values of $k$.
$z=\frac{x}{y}, k=-2,-1,0,1,2$

Lucas Finney

Numerade Educator

Problem 19

Sketch the level curve $z=k$ for the indicated values of $k$.
$z=\frac{x^{2}}{y}, k=-4,-1,0,1,4$

Lucas Finney

Numerade Educator

Problem 20

Sketch the level curve $z=k$ for the indicated values of $k$.
$z=x^{2}+y, k=-4,-1,0,1,4$

Lucas Finney

Numerade Educator

Problem 21

Sketch the level curve $z=k$ for the indicated values of $k$.
$z=\frac{x^{2}+1}{x^{2}+y^{2}}, k=1,2,4$

Lucas Finney

Numerade Educator

Problem 22

Sketch the level curve $z=k$ for the indicated values of $k$.
$z=y-\sin x, k=-2,-1,0,1,2$

Lucas Finney

Numerade Educator

Problem 23

Let $T(x, y)$ be the temperature at a point $(x, y)$ in the plane. Draw the isothermal curves corresponding to $T=\frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 0$ if
$$
T(x, y)=\frac{x^{2}}{x^{2}+y^{2}}
$$

Manik Pulyani

Numerade Educator

Problem 24

If $V(x, y)$ is the voltage at a point $(x, y)$ in the plane, the level curves of $V$ are called equipotential curves. Draw the equipotential curves corresponding to $V=\frac{1}{2}, 1,2,4$ for
$$
V(x, y)=\frac{4}{\sqrt{(x-2)^{2}+(y+3)^{2}}}
$$

Lucas Finney

Numerade Educator

Problem 25

Figure 20 shows isotherms for the United States.
(a) Which of San Francisco, Denver, and New York had approximately the same temperature as St. Louis?
(b) If you were in Kansas City and wanted to drive toward cooler weather as quickly as possible, in which direction would you travel? What if you wanted to drive toward warmer weather?
(c) If you were leaving Kansas City, in which directions could you go and stay at approximately the same temperature?

Brandon Cleary

Numerade Educator

Problem 26

Figure 23 shows a contour map for barometric pressure in millibars. Level curves for barometric pressure are called isobars.
(a) What part of the country had the lowest barometric pressure? The highest?
(b) If you were in St. Louis, in which direction would you have to travel to move as fast as possible toward lower barometric pressure? Higher barometric pressure?
(c) If you were leaving St. Louis, in which directions could you go in order to remain at approximately the same barometric pressure?

James Kiss

Numerade Educator

Problem 27

Describe geometrically the domain of each of the indicated functions of three variables.
$f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}-16}$

Lucas Finney

Numerade Educator

Problem 28

Describe geometrically the domain of each of the indicated functions of three variables.
$f(x, y, z)=\sqrt{x^{2}+y^{2}-z^{2}-9}$

Lucas Finney

Numerade Educator

Problem 29

Describe geometrically the domain of each of the indicated functions of three variables.
$f(x, y, z)=\sqrt{144-16 x^{2}-9 y^{2}-144 z^{2}}$

Ekaveera Kumar

Numerade Educator

Problem 30

Describe geometrically the domain of each of the indicated functions of three variables.
$f(x, y, z)=\frac{\left(144-16 x^{2}-16 y^{2}+9 z^{2}\right)^{3 / 2}}{x y z}$

Carson Merrill

Numerade Educator

Problem 31

Describe geometrically the domain of each of the indicated functions of three variables.
$f(x, y, z)=\ln \left(x^{2}+y^{2}+z^{2}\right)$

Lucas Finney

Numerade Educator

Problem 32

Describe geometrically the domain of each of the indicated functions of three variables.
$f(x, y, z)=z \ln (x y)$

Lucas Finney

Numerade Educator

Problem 33

Describe geometrically the level surfaces for the functions.
$f(x, y, z)=x^{2}+y^{2}+z^{2} ; k>0$

Lucas Finney

Numerade Educator

Problem 34

Describe geometrically the level surfaces for the functions.
$f(x, y, z)=100 x^{2}+16 y^{2}+25 z^{2} ; k>0$

Lucas Finney

Numerade Educator

Problem 35

Describe geometrically the level surfaces for the functions.
$f(x, y, z)=16 x^{2}+16 y^{2}-9 z^{2}$

Lucas Finney

Numerade Educator

Problem 36

Describe geometrically the level surfaces for the functions.
$f(x, y, z)=9 x^{2}-4 y^{2}-z^{2}$

Lucas Finney

Numerade Educator

Problem 37

Describe geometrically the level surfaces for the functions.
$f(x, y, z)=4 x^{2}-9 y^{2}$

Lucas Finney

Numerade Educator

Problem 38

Describe geometrically the level surfaces for the functions.
$f(x, y, z)=e^{x^{2}+y^{2}+z^{2}}, k>0$

Lucas Finney

Numerade Educator

Problem 39

Find the domain of each function.
(a) $f(w, x, y, z)=\frac{1}{\sqrt{w^{2}+x^{2}+y^{2}+z^{2}}}$
(b) $g\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\exp \left(-x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}\right)$
(c) $h\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\sqrt{1-\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right)}$

Lucas Finney

Numerade Educator

Problem 40

Sketch (as best you can) the graph of the monkey saddle $z=x\left(x^{2}-3 y^{2}\right) .$ Begin by noting where $z=0$

Lucas Finney

Numerade Educator

Problem 41

The contour map in Figure 24 shows level curves for a mountain 3000 feet high.
(a) What is special about the path to the top labeled $A C ?$ What is special about $B C ?$
(b) Make good estimates of the total lengths of path $A C$ and path $B C$.

Lucas Finney

Numerade Educator

Problem 42

Identify the graph of $f(x, y)=x^{2}-x+3 y^{2}+$ $12 y-13$, state where it attains its minimum value, and find this minimum value.

M Hassan Anwar

Numerade Educator

Problem 43

Draw the graph and the corresponding contour plot.
$f(x, y)=\sin \sqrt{2 x^{2}+y^{2}} ;-2 \leq x \leq 2,-2 \leq y \leq 2$

Linh Vu

Numerade Educator

Problem 44

Draw the graph and the corresponding contour plot.
$f(x, y)=\sin \left(x^{2}+y^{2}\right) /\left(x^{2}+y^{2}\right), f(0,0)=1 ;$ $-2 \leq x \leq 2,-2 \leq y \leq 2$

Linh Vu

Numerade Educator

Problem 45

Draw the graph and the corresponding contour plot.
$f(x, y)=\left(2 x-y^{2}\right) \exp \left(-x^{2}-y^{2}\right) ;-2 \leq x \leq 2$ $-2 \leq y \leq 2$

Erika Bustos

Numerade Educator

Problem 46

Draw the graph and the corresponding contour plot.
$f(x, y)=(\sin x \sin y) /\left(1+x^{2}+y^{2}\right) ;-2 \leq x \leq 2$ $-2 \leq y \leq 2$

Linh Vu

Numerade Educator