Books(current) Courses (current) Earn 💰 Log in(current)

Thomas Calculus 12

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Chapter 14

Partial Derivatives

Educators

IZ

Problem 1

In Exercises $1-4,$ find the specific function values.
$$f(x, y)=x^{2}+x y^{3}$$
$$\begin{array}{ll}{\text { a. } f(0,0)} & {\text { b. } f(-1,1)} \\ {\text { c. } f(2,3)} & {\text { d. } f(-3,-2)}\end{array}$$

Check back soon!

Problem 2

In Exercises $1-4,$ find the specific function values.
$$f(x, y)=\sin (x y)$$
$$\begin{array}{ll}{\text { a. } f\left(2, \frac{\pi}{6}\right)} & {\text { b. } f\left(-3, \frac{\pi}{12}\right)} \\ {\text { c. } f\left(\pi, \frac{1}{4}\right)} & {\text { d. } f\left(-\frac{\pi}{2},-7\right)}\end{array}$$

Check back soon!

Problem 3

In Exercises $1-4,$ find the specific function values.
$$f(x, y, z)=\frac{x-y}{y^{2}+z^{2}}$$
$$\begin{array}{ll}{\text { a. } f(3,-1,2)} & {\text { b. } f\left(1, \frac{1}{2},-\frac{1}{4}\right)} \\ {\text { c. } f\left(0,-\frac{1}{3}, 0\right)} & {\text { d. } f(2,2,100)}\end{array}$$

Check back soon!

Problem 4

In Exercises $1-4,$ find the specific function values.
$$f(x, y, z)=\sqrt{49-x^{2}-y^{2}-z^{2}}$$
$$\begin{array}{ll}{\text { a. } f(0,0,0)} & {\text { b. } f(2,-3,6)} \\ {\text { c. } f(-1,2,3)} & {\text { d. } f\left(\frac{4}{\sqrt{2}}, \frac{5}{\sqrt{2}}, \frac{6}{\sqrt{2}}\right)}\end{array}$$

Check back soon!

Problem 5

In Exercises $5-12,$ find and sketch the domain for each function.
$f(x, y)=\sqrt{y-x-2}$

Check back soon!

Problem 6

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\ln \left(x^{2}+y^{2}-4\right)$$

Check back soon!

Problem 7

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\frac{(x-1)(y+2)}{(y-x)\left(y-x^{3}\right)}$$

Check back soon!

Problem 8

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\frac{\sin (x y)}{x^{2}+y^{2}-25}$$

IZ
Ivy Z.
Numerade Educator

Problem 9

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\cos ^{-1}\left(y-x^{2}\right)$$

Check back soon!

Problem 10

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\ln (x y+x-y-1)$$

Check back soon!

Problem 11

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\sqrt{\left(x^{2}-4\right)\left(y^{2}-9\right)}$$

Check back soon!

Problem 12

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\frac{1}{\ln \left(4-x^{2}-y^{2}\right)}$$

Check back soon!

Problem 13

In Exercises $13-16,$ find and sketch the level curves $f(x, y)=c$ on
the same set of coordinate axes for the given values of $c .$ We refer to
these level curves as a contour map.
$f(x, y)=x+y-1, \quad c=-3,-2,-1,0,1,2,3$

Check back soon!

Problem 14

In Exercises $13-16,$ find and sketch the level curves $f(x, y)=c$ on the same set of coordinate axes for the given values of $c .$ We refer to these level curves as a contour map.
$$f(x, y)=x^{2}+y^{2}, \quad c=0,1,4,9,16,25$$

Check back soon!

Problem 15

In Exercises $13-16,$ find and sketch the level curves $f(x, y)=c$ on the same set of coordinate axes for the given values of $c .$ We refer to these level curves as a contour map.
$$f(x, y)=x y, \quad c=-9,-4,-1,0,1,4,9$$

Check back soon!

Problem 16

In Exercises $13-16,$ find and sketch the level curves $f(x, y)=c$ on the same set of coordinate axes for the given values of $c .$ We refer to these level curves as a contour map.
$$f(x, y)=\sqrt{25-x^{2}-y^{2}}, \quad c=0,1,2,3,4$$

Check back soon!

Problem 17

In Exercises $17-30,($ a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$f(x, y)=y-x$

Check back soon!

Problem 18

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=\sqrt{y-x}$$

Check back soon!

Problem 19

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=4 x^{2}+9 y^{2}$$

Check back soon!

Problem 20

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=x^{2}-y^{2}$$

Check back soon!

Problem 21

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=x y$$

Check back soon!

Problem 22

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=y / x^{2}$$

Check back soon!

Problem 23

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$f(x, y)=\frac{1}{\sqrt{16-x^{2}-y^{2}}}$

Check back soon!

Problem 24

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=\sqrt{9-x^{2}-y^{2}}$$

Check back soon!

Problem 25

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=\ln \left(x^{2}+y^{2}\right)$$

Check back soon!

Problem 26

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=e^{-\left(x^{2}+y^{2}\right)}$$

Check back soon!

Problem 27

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=\sin ^{-1}(y-x)$$

Check back soon!

Problem 28

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=\tan ^{-1}\left(\frac{y}{x}\right)$$

Check back soon!

Problem 29

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=\ln \left(x^{2}+y^{2}-1\right)$$

Check back soon!

Problem 30

In Exercises $17-30,($ a) find the function's domain, $($ b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded.
$$f(x, y)=\ln \left(9-x^{2}-y^{2}\right)$$

Check back soon!

Problem 31

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the appropriate function.

Check back soon!

Problem 32

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the ap-
propriate function.

Check back soon!

Problem 33

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the ap-
propriate function.

Check back soon!

Problem 34

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the ap-
propriate function.

Check back soon!

Problem 35

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the ap-
propriate function.

Check back soon!

Problem 36

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the ap-
propriate function.

Check back soon!

Problem 37

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=y^{2}$$

Check back soon!

Problem 38

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=\sqrt{x}$$

Check back soon!

Problem 39

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=x^{2}+y^{2}$$

Check back soon!

Problem 40

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=\sqrt{x^{2}+y^{2}}$$

Check back soon!

Problem 41

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=x^{2}-y$$

Check back soon!

Problem 42

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=4-x^{2}-y^{2}$$

Check back soon!

Problem 43

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=4 x^{2}+y^{2}$$

Check back soon!

Problem 44

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=1-|x|-|y|$$

Check back soon!

Problem 45

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value..
$$f(x, y)=1-|y|$$

Check back soon!

Problem 46

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=1-|x|-|y|$$

Check back soon!

Problem 47

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=\sqrt{x^{2}+y^{2}+4}$$

Check back soon!

Problem 48

Display the values of the functions in Exercises $37-48$ in two ways: (a) by sketching the surface $z=f(x, y)$ and $(b)$ by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=\sqrt{x^{2}+y^{2}-4}$$

Check back soon!

Problem 49

In Exercises $49-52,$ find an equation for and sketch the graph of the level curve of the function $f(x, y)$ that passes through the given point.
$$f(x, y)=16-x^{2}-y^{2}, \quad(2 \sqrt{2}, \sqrt{2})$$

Check back soon!

Problem 50

In Exercises $49-52,$ find an equation for and sketch the graph of the level curve of the function $f(x, y)$ that passes through the given point.
$$f(x, y)=\sqrt{x^{2}-1}, \quad(1,0)$$

Check back soon!

Problem 51

In Exercises $49-52,$ find an equation for and sketch the graph of the level curve of the function $f(x, y)$ that passes through the given point.
$$f(x, y)=\sqrt{x+y^{2}-3}, \quad(3,-1)$$

Check back soon!

Problem 52

In Exercises $49-52,$ find an equation for and sketch the graph of the level curve of the function $f(x, y)$ that passes through the given point.
$$f(x, y)=\frac{2 y-x}{x+y+1}, \quad(-1,1)$$

Check back soon!

Problem 53

In Exercises $53-60,$ sketch a typical level surface for the function.
$$f(x, y, z)=x^{2}+y^{2}+z^{2}$$

Check back soon!

Problem 54

In Exercises $53-60,$ sketch a typical level surface for the function.
$$f(x, y, z)=\ln \left(x^{2}+y^{2}+z^{2}\right)$$

Check back soon!

Problem 55

In Exercises $53-60,$ sketch a typical level surface for the function.
$$f(x, y, z)=x+z$$

Check back soon!

Problem 56

In Exercises $53-60,$ sketch a typical level surface for the function.
$$f(x, y, z)=z$$

Check back soon!

Problem 57

In Exercises $53-60,$ sketch a typical level surface for the function.
$$f(x, y, z)=x^{2}+y^{2}$$

Check back soon!

Problem 58

In Exercises $53-60,$ sketch a typical level surface for the function.
$$f(x, y, z)=y^{2}+z^{2}$$

Check back soon!

Problem 59

In Exercises $53-60,$ sketch a typical level surface for the function.
$$f(x, y, z)=z-x^{2}-y^{2}$$

Check back soon!

Problem 60

In Exercises $53-60,$ sketch a typical level surface for the function.
$$f(x, y, z)=\left(x^{2} / 25\right)+\left(y^{2} / 16\right)+\left(z^{2} / 9\right)$$

Check back soon!

Problem 61

In Exercises $61-64$ , find an equation for the level surface of the function through the given point.
$$f(x, y, z)=\sqrt{x-y}-\ln z, \quad(3,-1,1)$$

Check back soon!

Problem 62

In Exercises $61-64$ , find an equation for the level surface of the function through the given point.
$$f(x, y, z)=\ln \left(x^{2}+y+z^{2}\right), \quad(-1,2,1)$$

Check back soon!

Problem 63

In Exercises $61-64$ , find an equation for the level surface of the function through the given point.
$$g(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}, \quad(1,-1, \sqrt{2})$$

Check back soon!

Problem 64

In Exercises $61-64$ , find an equation for the level surface of the function through the given point.
$$g(x, y, z)=\frac{x-y+z}{2 x+y-z}, \quad(1,0,-2)$$

Check back soon!

Problem 65

In Exercises $65-68,$ find and sketch the domain of $f .$ Then find an equation for the level curve or surface of the function passing through the given point.
$$f(x, y)=\sum_{n=0}^{\infty}\left(\frac{x}{y}\right)^{n}, \quad(1,2)$$

Check back soon!

Problem 66

In Exercises $65-68,$ find and sketch the domain of $f .$ Then find an equation for the level curve or surface of the function passing through the given point.
$$g(x, y, z)=\sum_{n=0}^{\infty} \frac{(x+y)^{n}}{n ! z^{n}}, \quad(\ln 4, \ln 9,2)$$

Check back soon!

Problem 67

In Exercises $65-68,$ find and sketch the domain of $f .$ Then find an equation for the level curve or surface of the function passing through the given point.
$$f(x, y)=\int_{x}^{y} \frac{d \theta}{\sqrt{1-\theta^{2}}}, \quad(0,1)$$

Check back soon!

Problem 68

In Exercises $65-68,$ find and sketch the domain of $f .$ Then find an equation for the level curve or surface of the function passing through the given point.
$$g(x, y, z)=\int_{x}^{y} \frac{d t}{1+t^{2}}+\int_{0}^{z} \frac{d \theta}{\sqrt{4-\theta^{2}}}, \quad(0,1, \sqrt{3})$$

Check back soon!

Problem 69

Use a CAS to perform the following steps for each of the functions in Exercises $69-72 .$
a. Plot the surface over the given rectangle.
b. Plot several level curves in the rectangle.
c. Plot the level curve of $f$ through the given point.
$$f(x, y)=x \sin \frac{y}{2}+y \sin 2 x, \quad 0 \leq x \leq 5 \pi, \quad 0 \leq y \leq 5 \pi$ $P(3 \pi, 3 \pi)$$

Check back soon!

Problem 70

Use a CAS to perform the following steps for each of the functions in Exercises $69-72 .$
a. Plot the surface over the given rectangle.
b. Plot several level curves in the rectangle.
c. Plot the level curve of $f$ through the given point.
$$f(x, y)=(\sin x)(\cos y) e^{\sqrt{x^{2}+y^{2}} / 8}, \quad 0 \leq x \leq 5 \pi$$
$$0 \leq y \leq 5 \pi, \quad P(4 \pi, 4 \pi)$$

Check back soon!

Problem 71

Use a CAS to perform the following steps for each of the functions in Exercises $69-72 .$
a. Plot the surface over the given rectangle.
b. Plot several level curves in the rectangle.
c. Plot the level curve of $f$ through the given point.
$$f(x, y)=\sin (x+2 \cos y), \quad-2 \pi \leq x \leq 2 \pi$$
$$2 \pi \leq y \leq 2 \pi, \quad P(\pi, \pi)$$

Check back soon!

Problem 72

Use a CAS to perform the following steps for each of the functions in Exercises $69-72 .$
a. Plot the surface over the given rectangle.
b. Plot several level curves in the rectangle.
c. Plot the level curve of $f$ through the given point.
$$f(x, y)=e^{\left(x^{01}-y\right)} \sin \left(x^{2}+y^{2}\right), \quad 0 \leq x \leq 2 \pi$$
$$-2 \pi \leq y \leq \pi, \quad P(\pi,-\pi)$$

Check back soon!

Problem 73

Use a CAS to plot the implicitly defined level surfaces in Exercises $73-76 .$
$$4 \ln \left(x^{2}+y^{2}+z^{2}\right)=1 \quad$$

Check back soon!

Problem 74

Use a CAS to plot the implicitly defined level surfaces in Exercises $73-76 .$
$$x^{2}+z^{2}=1$$

Check back soon!

Problem 75

Use a CAS to plot the implicitly defined level surfaces in Exercises $73-76 .$
$$x+y^{2}-3 z^{2}=1$$

Check back soon!

Problem 76

Use a CAS to plot the implicitly defined level surfaces in Exercises $73-76 .$
$$\sin \left(\frac{x}{2}\right)-(\cos y) \sqrt{x^{2}+z^{2}}=2$$

Check back soon!

Problem 77

Parametrized Surfaces Just as you describe curves in the plane parametrized with a pair of equations $x=f(t), y=g(t)$ defined on some parameter interval $I,$ you can sometimes describe surfaces in space with a triple of equations $x=f(u, v), y=g(u, v), z=h(u, v)$ defined on some parameter rectangle $a \leq u \leq b, c \leq v \leq d .$ Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section $16.5 .$ Use a CAS to plot the surfaces in Exercises $77-80 .$ Also plot several level curves in the xy-plane.
$$x=u \cos v, \quad y=u \sin v, \quad z=u, \quad 0 \leq u \leq 2$$
$$0 \leq v \leq 2 \pi$$

Check back soon!

Problem 78

Parametrized Surfaces Just as you describe curves in the plane parametrized with a pair of equations $x=f(t), y=g(t)$ defined on some parameter interval $I,$ you can sometimes describe surfaces in space with a triple of equations $x=f(u, v), y=g(u, v), z=h(u, v)$ defined on some parameter rectangle $a \leq u \leq b, c \leq v \leq d .$ Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section $16.5 .$ Use a CAS to plot the surfaces in Exercises $77-80 .$ Also plot several level curves in the xy-plane.
$$x=u \cos v, \quad y=u \sin v, \quad z=v, \quad 0 \leq u \leq 2$$
$$0 \leq v \leq 2 \pi$$

Check back soon!

Problem 79

Parametrized Surfaces Just as you describe curves in the plane parametrized with a pair of equations $x=f(t), y=g(t)$ defined on some parameter interval $I,$ you can sometimes describe surfaces in space with a triple of equations $x=f(u, v), y=g(u, v), z=h(u, v)$ defined on some parameter rectangle $a \leq u \leq b, c \leq v \leq d .$ Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section $16.5 .$ Use a CAS to plot the surfaces in Exercises $77-80 .$ Also plot several level curves in the xy-plane.
$$x=(2+\cos u) \cos v, \quad y=(2+\cos u) \sin v, \quad z=\sin u$$
$$0 \leq u \leq 2 \pi, \quad 0 \leq v \leq 2 \pi$$

Check back soon!

Problem 80

Parametrized Surfaces Just as you describe curves in the plane parametrized with a pair of equations $x=f(t), y=g(t)$ defined on some parameter interval $I,$ you can sometimes describe surfaces in space with a triple of equations $x=f(u, v), y=g(u, v), z=h(u, v)$ defined on some parameter rectangle $a \leq u \leq b, c \leq v \leq d .$ Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section $16.5 .$ Use a CAS to plot the surfaces in Exercises $77-80 .$ Also plot several level curves in the xy-plane.
$$x=2 \cos u \cos v, \quad y=2 \cos u \sin v, \quad z=2 \sin u$$
$$0 \leq u \leq 2 \pi, \quad 0 \leq v \leq \pi$$

Check back soon!