Thomas Calculus 12

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

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

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

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

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

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

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

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

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

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

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

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

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.
$f(x, y)=x+y-1, \quad c=-3,-2,-1,0,1,2,3$

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.
$$f(x, y)=x^{2}+y^{2}, \quad c=0,1,4,9,16,25$$

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.
$$f(x, y)=x y, \quad c=-9,-4,-1,0,1,4,9$$

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.
$$f(x, y)=\sqrt{25-x^{2}-y^{2}}, \quad c=0,1,2,3,4$$

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.
$f(x, y)=y-x$

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.
$$f(x, y)=\sqrt{y-x}$$

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.
$$f(x, y)=4 x^{2}+9 y^{2}$$

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.
$$f(x, y)=x^{2}-y^{2}$$

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.
$$f(x, y)=x y$$

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.
$$f(x, y)=y / x^{2}$$

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.
$f(x, y)=\frac{1}{\sqrt{16-x^{2}-y^{2}}}$

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.
$$f(x, y)=\sqrt{9-x^{2}-y^{2}}$$

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.
$$f(x, y)=\ln \left(x^{2}+y^{2}\right)$$

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.
$$f(x, y)=e^{-\left(x^{2}+y^{2}\right)}$$

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.
$$f(x, y)=\sin ^{-1}(y-x)$$

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.
$$f(x, y)=\tan ^{-1}\left(\frac{y}{x}\right)$$

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.
$$f(x, y)=\ln \left(x^{2}+y^{2}-1\right)$$

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.
$$f(x, y)=\ln \left(9-x^{2}-y^{2}\right)$$

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.

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.

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.

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.

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.

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.

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.
$$f(x, y)=y^{2}$$

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.
$$f(x, y)=\sqrt{x}$$

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.
$$f(x, y)=x^{2}+y^{2}$$

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.
$$f(x, y)=\sqrt{x^{2}+y^{2}}$$

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.
$$f(x, y)=x^{2}-y$$

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.
$$f(x, y)=4-x^{2}-y^{2}$$

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.
$$f(x, y)=4 x^{2}+y^{2}$$

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.
$$f(x, y)=1-|x|-|y|$$

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..
$$f(x, y)=1-|y|$$

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.
$$f(x, y)=1-|x|-|y|$$

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.
$$f(x, y)=\sqrt{x^{2}+y^{2}+4}$$

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

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.
$$f(x, y)=16-x^{2}-y^{2}, \quad(2 \sqrt{2}, \sqrt{2})$$

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.
$$f(x, y)=\sqrt{x^{2}-1}, \quad(1,0)$$

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.
$$f(x, y)=\sqrt{x+y^{2}-3}, \quad(3,-1)$$

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.
$$f(x, y)=\frac{2 y-x}{x+y+1}, \quad(-1,1)$$

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

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

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

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

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

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

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

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

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

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

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

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

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.
$$f(x, y)=\sum_{n=0}^{\infty}\left(\frac{x}{y}\right)^{n}, \quad(1,2)$$

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.
$$g(x, y, z)=\sum_{n=0}^{\infty} \frac{(x+y)^{n}}{n ! z^{n}}, \quad(\ln 4, \ln 9,2)$$

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.
$$f(x, y)=\int_{x}^{y} \frac{d \theta}{\sqrt{1-\theta^{2}}}, \quad(0,1)$$

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.
$$g(x, y, z)=\int_{x}^{y} \frac{d t}{1+t^{2}}+\int_{0}^{z} \frac{d \theta}{\sqrt{4-\theta^{2}}}, \quad(0,1, \sqrt{3})$$

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.
$$f(x, y)=x \sin \frac{y}{2}+y \sin 2 x, \quad 0 \leq x \leq 5 \pi, \quad 0 \leq y \leq 5 \pi$ $P(3 \pi, 3 \pi)$$

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.
$$f(x, y)=(\sin x)(\cos y) e^{\sqrt{x^{2}+y^{2}} / 8}, \quad 0 \leq x \leq 5 \pi$$
$$0 \leq y \leq 5 \pi, \quad P(4 \pi, 4 \pi)$$

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.
$$f(x, y)=\sin (x+2 \cos y), \quad-2 \pi \leq x \leq 2 \pi$$
$$2 \pi \leq y \leq 2 \pi, \quad P(\pi, \pi)$$

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.
$$f(x, y)=e^{\left(x^{01}-y\right)} \sin \left(x^{2}+y^{2}\right), \quad 0 \leq x \leq 2 \pi$$
$$-2 \pi \leq y \leq \pi, \quad P(\pi,-\pi)$$

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

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

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

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

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.
$$x=u \cos v, \quad y=u \sin v, \quad z=u, \quad 0 \leq u \leq 2$$
$$0 \leq v \leq 2 \pi$$

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.
$$x=u \cos v, \quad y=u \sin v, \quad z=v, \quad 0 \leq u \leq 2$$
$$0 \leq v \leq 2 \pi$$

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.
$$x=(2+\cos u) \cos v, \quad y=(2+\cos u) \sin v, \quad z=\sin u$$
$$0 \leq u \leq 2 \pi, \quad 0 \leq v \leq 2 \pi$$

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.
$$x=2 \cos u \cos v, \quad y=2 \cos u \sin v, \quad z=2 \sin u$$
$$0 \leq u \leq 2 \pi, \quad 0 \leq v \leq \pi$$