Find the specific function values.

$$f(x, y)=x^{2}+x y^{3}$$

a. $f(0,0)$

b. $f(-1,1)$

c. $f(2,3)$

d. $f(-3,-2)$

Thomas E.

Numerade Educator

Find the specific function values.

$$f(x, y)=\sin (x y)$$

a. $f\left(2, \frac{\pi}{6}\right)$

b. $f\left(-3, \frac{\pi}{12}\right)$

c. $f\left(\pi, \frac{1}{4}\right)$

d. $f\left(-\frac{\pi}{2},-7\right)$

Thomas E.

Numerade Educator

Find the specific function values.

$$f(x, y, z)=\frac{x-y}{y^{2}+z^{2}}$$

a. $f(3,-1,2)$

b. $f\left(1, \frac{1}{2},-\frac{1}{4}\right)$

c. $f\left(0,-\frac{1}{3}, 0\right)$

d. $f(2,2,100)$

Thomas E.

Numerade Educator

Find the specific function values.

$$f(x, y, z)=\sqrt{49-x^{2}-y^{2}-z^{2}}$$

a. $f(0,0,0)$

b. $f(2,-3,6)$

c. $f(-1,2,3)$

d. $f\left(\frac{4}{\sqrt{2}}, \frac{5}{\sqrt{2}}, \frac{6}{\sqrt{2}}\right)$

Thomas E.

Numerade Educator

Find and sketch the domain for each function.

$$f(x, y)=\ln \left(x^{2}+y^{2}-4\right)$$

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Find and sketch the domain for each function.

$$f(x, y)=\frac{(x-1)(y+2)}{(y-x)\left(y-x^{3}\right)}$$

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Find and sketch the domain for each function.

$$f(x, y)=\frac{\sin (x y)}{x^{2}+y^{2}-25}$$

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Find and sketch the domain for each function.

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

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Find and sketch the domain for each function.

$$f(x, y)=\ln (x y+x-y-1)$$

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Find and sketch the domain for each function.

$$f(x, y)=\sqrt{\left(x^{2}-4\right)\left(y^{2}-9\right)}$$

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Find and sketch the domain for each function.

$$f(x, y)=\frac{1}{\ln \left(4-x^{2}-y^{2}\right)}$$

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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.

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

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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.

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

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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.

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

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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.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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In Exercise (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 whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.

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

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Show level curves for six functions. The graphs of these functions are given on the next page (items a-f), as are their equations (items g $-1$). Match each set of level curves with the appropriate graph and the appropriate equation.

(Graph cant copy)

a. (Graph cant copy)

b. (Graph cant copy)

c. (Graph cant copy)

d. (Graph cant copy)

e. (Graph cant copy)

f. (Graph cant copy)

g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$

h. $z=y^{2}-y^{4}-x^{2}$

i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$

j. $z=e^{-y} \cos x$

k. $z=\frac{1}{4 x^{2}+y^{2}}$

l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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Show level curves for six functions. The graphs of these functions are given on the next page (items a-f), as are their equations (items g $-1$). Match each set of level curves with the appropriate graph and the appropriate equation.

(Graph cant copy)

a. (Graph cant copy)

b. (Graph cant copy)

c. (Graph cant copy)

d. (Graph cant copy)

e. (Graph cant copy)

f. (Graph cant copy)

g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$

h. $z=y^{2}-y^{4}-x^{2}$

i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$

j. $z=e^{-y} \cos x$

k. $z=\frac{1}{4 x^{2}+y^{2}}$

l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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Show level curves for six functions. The graphs of these functions are given on the next page (items a-f), as are their equations (items g $-1$). Match each set of level curves with the appropriate graph and the appropriate equation.

(Graph cant copy)

a. (Graph cant copy)

b. (Graph cant copy)

c. (Graph cant copy)

d. (Graph cant copy)

e. (Graph cant copy)

f. (Graph cant copy)

g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$

h. $z=y^{2}-y^{4}-x^{2}$

i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$

j. $z=e^{-y} \cos x$

k. $z=\frac{1}{4 x^{2}+y^{2}}$

l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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(Graph cant copy)

b. (Graph cant copy)

c. (Graph cant copy)

d. (Graph cant copy)

e. (Graph cant copy)

f. (Graph cant copy)

g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$

h. $z=y^{2}-y^{4}-x^{2}$

i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$

j. $z=e^{-y} \cos x$

k. $z=\frac{1}{4 x^{2}+y^{2}}$

l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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(Graph cant copy)

b. (Graph cant copy)

c. (Graph cant copy)

d. (Graph cant copy)

e. (Graph cant copy)

f. (Graph cant copy)

g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$

h. $z=y^{2}-y^{4}-x^{2}$

i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$

j. $z=e^{-y} \cos x$

k. $z=\frac{1}{4 x^{2}+y^{2}}$

l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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(Graph cant copy)

b. (Graph cant copy)

c. (Graph cant copy)

d. (Graph cant copy)

e. (Graph cant copy)

f. (Graph cant copy)

g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$

h. $z=y^{2}-y^{4}-x^{2}$

i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$

j. $z=e^{-y} \cos x$

k. $z=\frac{1}{4 x^{2}+y^{2}}$

l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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Display the values of the functions 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.

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

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Display the values of the functions 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.

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

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Display the values of the functions 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.

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

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Display the values of the functions 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.

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

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Display the values of the functions 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.

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

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Display the values of the functions 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.

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

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Display the values of the functions 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.

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

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Display the values of the functions 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.

$$f(x, y)=6-2 x-3 y$$

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Display the values of the functions 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.

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

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Display the values of the functions 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.

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

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Display the values of the functions 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.

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

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Display the values of the functions 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.

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

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Find an equation for, and sketch the graph of, the level curve of the function $f(x, y)$ that passes through the given point.

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

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Find an equation for, and sketch the graph of, the level curve of the function $f(x, y)$ that passes through the given point.

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

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Find an equation for, and sketch the graph of, the level curve of the function $f(x, y)$ that passes through the given point.

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

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Find an equation for, and sketch the graph of, the level curve of the function $f(x, y)$ that passes through the given point.

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

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Sketch a typical level surface for the function.

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

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Sketch a typical level surface for the function.

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

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Sketch a typical level surface for the function.

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

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Sketch a typical level surface for the function.

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

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Sketch a typical level surface for the function.

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

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Sketch a typical level surface for the function.

$$f(x, y, z)=\left(x^{2} / 25\right)+\left(y^{2} / 16\right)+\left(z^{2} / 9\right)$$

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Find an equation for the level surface of the function through the given point.

$$f(x, y, z)=\sqrt{x-y}-\ln z, \quad(3,-1,1)$$

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Find an equation for the level surface of the function through the given point.

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

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Find an equation for the level surface of the function through the given point.

$$g(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}, \quad(1,-1, \sqrt{2})$$

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Find an equation for the level surface of the function through the given point.

$$g(x, y, z)=\frac{x-y+z}{2 x+y-z}, \quad(1,0,-2)$$

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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.

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

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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.

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

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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.

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

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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.

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

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Use a CAS to perform the following steps for each of the functions.

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.

$$\begin{aligned}

&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, \quad P(3 \pi, 3 \pi)

\end{aligned}$$

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Use a CAS to perform the following steps for each of the functions.

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.

$$\begin{array}{l}

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

0 \leq y \leq 5 \pi, \quad P(4 \pi, 4 \pi)

\end{array}$$

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Use a CAS to perform the following steps for each of the functions.

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.

$$\begin{array}{l} f(x, y)=\sin (x+2 \cos y), \quad-2 \pi \leq x \leq 2 \pi, \quad -2 \pi \leq y \leq 2 \pi, \quad P(\pi, \pi) \end{array}$$

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Use a CAS to perform the following steps for each of the functions.

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.

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

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Use a CAS to plot the implicitly defined level surfaces.

$$4 \ln \left(x^{2}+y^{2}+z^{2}\right)=1$$

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Use a CAS to plot the implicitly defined level surfaces.

$$x^{2}+z^{2}=1$$

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Use a CAS to plot the implicitly defined level surfaces.

$$x+y^{2}-3 z^{2}=1$$

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Use a CAS to plot the implicitly defined level surfaces.

$$\sin \left(\frac{x}{2}\right)-(\cos y) \sqrt{x^{2}+z^{2}}=2$$

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Just as you describe curves in the plane para metrically 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 15.5.) Use a CAS to plot the surfaces. Also plot several level curves in the $x y$ -plane.

$$\begin{array}{l}

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

\end{array}$$

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Just as you describe curves in the plane para metrically 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 15.5.) Use a CAS to plot the surfaces. Also plot several level curves in the $x y$ -plane.

$$\begin{array}{l}x=u \cos v, \quad y=u \sin v, \quad z=v, \quad 0 \leq u \leq 2, \quad

0 \leq v \leq 2 \pi

\end{array}$$

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Just as you describe curves in the plane para metrically 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 15.5.) Use a CAS to plot the surfaces. Also plot several level curves in the $x y$ -plane.

$$\begin{array}{l}

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

\end{array}$$

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Just as you describe curves in the plane para metrically 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 15.5.) Use a CAS to plot the surfaces. Also plot several level curves in the $x y$ -plane.

$$\begin{array}{l}

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

\end{array}$$

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