Section 1
Functions of Several Variables from the Numerical, Algebraic, and Graphical Viewpoints
For each function, evaluate (a) $f(0,0)$; (b) $f(1,0) ;$ (c) $f(0,-1)$; (d) $f(a, 2) ;$ (e) $f(y, x)$;(f) $f(x+h, y+k)$ HINT [See Quick Examples page 1080.]$$f(x, y)=x^{2}+y^{2}-x+1$$
For each function, evaluate (a) $f(0,0)$; (b) $f(1,0) ;$ (c) $f(0,-1)$; (d) $f(a, 2) ;$ (e) $f(y, x)$;(f) $f(x+h, y+k)$ HINT [See Quick Examples page 1080.]$$f(x, y)=x^{2}-y-x y+1$$
For each function, evaluate (a) $f(0,0)$; (b) $f(1,0) ;$ (c) $f(0,-1)$; (d) $f(a, 2) ;$ (e) $f(y, x)$;(f) $f(x+h, y+k)$ HINT [See Quick Examples page 1080.]$$f(x, y)=0.2 x+0.1 y-0.01 x y$$
For each function, evaluate (a) $f(0,0)$; (b) $f(1,0) ;$ (c) $f(0,-1)$; (d) $f(a, 2) ;$ (e) $f(y, x)$;(f) $f(x+h, y+k)$ HINT [See Quick Examples page 1080.]$$f(x, y)=0.4 x-0.5 y-0.05 x y$$
For each function, evaluate (a) $g(0,0,0)$; (b) $g(1,0,0) ;$ (c) $g(0,1,0) ;$ (d) $g(z, x, y)$; (e) $g(x+h, y+k, z+l)$, provided such a value exists.$$g(x, y, z)=e^{x+y+z}$$
For each function, evaluate (a) $g(0,0,0)$; (b) $g(1,0,0) ;$ (c) $g(0,1,0) ;$ (d) $g(z, x, y)$; (e) $g(x+h, y+k, z+l)$, provided such a value exists.$$g(x, y, z)=\ln (x+y+z)$$
For each function, evaluate (a) $g(0,0,0)$; (b) $g(1,0,0) ;$ (c) $g(0,1,0) ;$ (d) $g(z, x, y)$; (e) $g(x+h, y+k, z+l)$, provided such a value exists.$$g(x, y, z)=\frac{x y z}{x^{2}+y^{2}+z^{2}}$$
For each function, evaluate (a) $g(0,0,0)$; (b) $g(1,0,0) ;$ (c) $g(0,1,0) ;$ (d) $g(z, x, y)$; (e) $g(x+h, y+k, z+l)$, provided such a value exists.$$g(x, y, z)=\frac{e^{x y z}}{x+y+z}$$
Let $f(x, y, z)=1.5+2.3 x-1.4 y-2.5 z$. Complete the following sentences. HINT [See Example 1.]a. $f$ ______ by ______ units for every 1 unit of increase in $x$.b. $f$ ______ by ______ units for every 1 unit of increase in $y$.c. ______ by $2.5$ units for every ______
Let $g(x, y, z)=0.01 x+0.02 y-0.03 z-0.05 .$ Complete the following sentences.a. g ___ by ___ units for every 1 unit of increase in z.b. g ___ by ___ units for every 1 unit of increase in x.c. ______ by 0.02 units for every _______.
Classify each function as linear, interaction, or neither. HINT [See Quick Examples page 1083.]$$L(x, y)=3 x-2 y+6 x y-4 y^{2}$$
Classify each function as linear, interaction, or neither. HINT [See Quick Examples page 1083.]$$L(x, y, z)=3 x-2 y+6 x z$$
Classify each function as linear, interaction, or neither. HINT [See Quick Examples page 1083.]$$P\left(x_{1}, x_{2}, x_{3}\right)=0.4+2 x_{1}-x_{3}$$
Classify each function as linear, interaction, or neither. HINT [See Quick Examples page 1083.]$$Q\left(x_{1}, x_{2}\right)=4 x_{2}-0.5 x_{1}-x_{1}^{2}$$
Classify each function as linear, interaction, or neither. HINT [See Quick Examples page 1083.]$$f(x, y, z)=\frac{x+y-z}{3}$$
Classify each function as linear, interaction, or neither. HINT [See Quick Examples page 1083.]$$g(x, y, z)=\frac{x z-3 y z+z^{2}}{4 z} \quad(z \neq 0)$$
Classify each function as linear, interaction, or neither. HINT [See Quick Examples page 1083.]$$g(x, y, z)=\frac{x z-3 y z+z^{2} y}{4 z} \quad(z \neq 0)$$
Classify each function as linear, interaction, or neither. HINT [See Quick Examples page 1083.]$$f(x, y)=x+y+x y+x^{2} y$$
Use the given tabular representation of the function fto compute the quantities asked for. HINT [See Example 3.]$$\begin{array}{|c|c|c|c|c|}\hline & \mathbf{1 0} & \mathbf{2 0} & \mathbf{3 0} & \mathbf{4 0} \\\hline \mathbf{1 0} & -1 & 107 & 162 & -3 \\\hline \mathbf{2 0} & -6 & 194 & 294 & -14 \\\hline \mathbf{3 0} & -11 & 281 & 426 & -25 \\\hline \mathbf{4 0} & -16 & 368 & 558 & -36 \\\hline\end{array}$$a. $f(20,10) \quad$ b. $f(40,20) \quad$ c. $f(10,20)-f(20,10)$
Use the given tabular representation of the function fto compute the quantities asked for. HINT [See Example 3.]$$\begin{array}{|c|c|c|c|c|}\hline & \mathbf{1 0} & \mathbf{2 0} & \mathbf{3 0} & \mathbf{4 0} \\\hline \mathbf{1 0} & 162 & 107 & -5 & -7 \\\hline \mathbf{2 0} & 294 & 194 & -22 & -30 \\\hline \mathbf{3 0} & 426 & 281 & -39 & -53 \\\hline \mathbf{4 0} & 558 & 368 & -56 & -76 \\\hline\end{array}$$a. $f(10,30) \quad$ b. $f(20,10) \quad$ c. $f(10,40)+f(10,20)$
Use a spreadsheet or some other method to complete the given tables.$$\begin{gathered}P(x, y)=x-0.3 y+0.45 x y \\\boldsymbol{x} \rightarrow\end{gathered}$$$$\begin{array}{|l|l|l|l|l|}\hline & \mathbf{1 0} & \mathbf{2 0} & \mathbf{3 0} & \mathbf{4 0} \\\hline \mathbf{1 0} & & & & \\\hline \mathbf{2 0} & & & & \\\hline \mathbf{3 0} & & & & \\\hline \mathbf{4 0} & & & & \\\hline\end{array}$$
Use a spreadsheet or some other method to complete the given tables.$Q(x, y)=0.4 x+0.1 y-0.06 x y$$$\begin{array}{|l|l|l|l|l|}\hline & \mathbf{1 0} & \mathbf{2 0} & \mathbf{3 0} & \mathbf{4 0} \\\hline \mathbf{1 0} & & & & \\\hline \mathbf{2 0} & & & & \\\hline \mathbf{3 0} & & & & \\\hline \mathbf{4 0} & & & & \\\hline\end{array}$$
The following statistical table lists some values of the "Inverse F distribution" $(\alpha=0.5)$ :$$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|}\hline & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} & \mathbf{1 0} \\\hline \mathbf{1} & 161.4 & 199.5 & 215.7 & 224.6 & 230.2 & 234.0 & 236.8 & 238.9 & 240.5 & 241.9 \\\hline \mathbf{2} & 18.51 & 19.00 & 19.16 & 19.25 & 19.30 & 19.33 & 19.35 & 19.37 & 19.39 & 19.40 \\\hline \mathbf{3} & 10.13 & 9.552 & 9.277 & 9.117 & 9.013 & 8.941 & 8.887 & 8.812 & 8.812 & 8.785 \\\hline \mathbf{4} & 7.709 & 6.944 & 6.591 & 6.388 & 6.256 & 6.163 & 6.094 & 5.999 & 5.999 & 5.964 \\\hline \mathbf{5} & 6.608 & 5.786 & 5.409 & 5.192 & 5.050 & 4.950 & 4.876 & 4.772 & 4.772 & 4.735 \\\hline \mathbf{6} & 5.987 & 5.143 & 4.757 & 4.534 & 4.387 & 4.284 & 4.207 & 4.099 & 4.099 & 4.060 \\\hline \mathbf{7} & 5.591 & 4.737 & 4.347 & 4.120 & 3.972 & 3.866 & 3.787 & 3.677 & 3.677 & 3.637 \\\hline \mathbf{8} & 5.318 & 4.459 & 4.066 & 3.838 & 3.688 & 3.581 & 3.500 & 3.388 & 3.388 & 3.347 \\\hline \mathbf{9} & 5.117 & 4.256 & 3.863 & 3.633 & 3.482 & 3.374 & 3.293 & 3.179 & 3.179 & 3.137 \\\hline \mathbf{1 0} & 4.965 & 4.103 & 3.708 & 3.478 & 3.326 & 3.217 & 3.135 & 3.020 & 3.020 & 2.978 \\\hline\end{array}$$In Excel, you can compute the value of this function at $(n, d)$ by the formula$=$ FINV $(0.05, n$, d) The $0.05$ is the value of alpha $(\alpha)$.Use Excel to re-create this table.
The formula for body mass index $M(w, h)$, if $w$ is given in kilograms and $h$ is given in meters, is$$M(w, h)=\frac{w}{h^{2}} \quad \text { See Example } 3$$Use this formula to complete the following table in Excel:$$\begin{array}{|l|l|l|l|l|l|l|l|}\hline & \mathbf{7 0} & \mathbf{8 0} & \mathbf{9 0} & \mathbf{1 0 0} & \mathbf{1 1 0} & \mathbf{1 2 0} & \mathbf{1 3 0} \\\hline \mathbf{1 . 8} & & & & & & & \\\hline \mathbf{1 . 8 5} & & & & & & & \\\hline \mathbf{1 . 9} & & & & & & & \\\hline \mathbf{1 . 9 5} & & & & & & & \\\hline \mathbf{2} & & & & & & & \\\hline \mathbf{2 . 0 5} & & & & & & & \\\hline \mathbf{2 . 1} & & & & & & & \\\hline \mathbf{2 . 1 5} & & & & & & & \\\hline \mathbf{2 . 2} & & & & & & & \\\hline \mathbf{2 . 2 5} & & & & & & & \\\hline \mathbf{2 . 3} & & & & & & & \\\hline\end{array}$$
Use either a graphing calculator or a spreadsheet to complete each table. Express all your answers as decimals rounded to four decimal places.$$\begin{array}{|c|c|c|}\hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{f}(\boldsymbol{x}, \boldsymbol{y})=\boldsymbol{x}^{\boldsymbol{2}} \sqrt{\mathbf{1}+\boldsymbol{x y}} \\\hline 3 & 1 & \\\hline 1 & 15 & \\\hline 0.3 & 0.5 & \\\hline 56 & 4 & \\\hline\end{array}$$
Use either a graphing calculator or a spreadsheet to complete each table. Express all your answers as decimals rounded to four decimal places.$$\begin{array}{|c|c|c|}\hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{f}(\boldsymbol{x}, \boldsymbol{y})=\boldsymbol{x}^{2} \boldsymbol{e}^{y} \\\hline 0 & 2 & \\\hline-1 & 5 & \\\hline 1.4 & 2.5 & \\\hline 11 & 9 & \\\hline\end{array}$$
Use either a graphing calculator or a spreadsheet to complete each table. Express all your answers as decimals rounded to four decimal places.$$\begin{array}{|c|c|c|}\hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{f}(\boldsymbol{x}, \boldsymbol{y})=\boldsymbol{x} \ln \left(\boldsymbol{x}^{\mathbf{2}}+\boldsymbol{y}^{\mathbf{2}}\right) \\\hline 3 & 1 & \\\hline 1.4 & -1 & \\\hline e & 0 & \\\hline 0 & e & \\\hline\end{array}$$
Use either a graphing calculator or a spreadsheet to complete each table. Express all your answers as decimals rounded to four decimal places.$$\begin{array}{|c|c|c|}\hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{f}(\boldsymbol{x}, \boldsymbol{y})=\frac{\boldsymbol{x}}{\boldsymbol{x}^{2}-\boldsymbol{y}^{2}} \\\hline-1 & 2 & \\\hline 0 & 0.2 & \\\hline 0.4 & 2.5 & \\\hline 10 & 0 & \\\hline\end{array}$$
Brand Z's annual sales are affected by the sales of related products $\mathrm{X}$ and $\mathrm{Y}$ as follows: Each $$\$ 1$$ million increase in sales of brand $X$ causes a $$\$ 2.1$$ million decline in sales of brand $Z$, whereas each $$\$ 1$$ million increase in sales of brand Y results in an increase of $$\$ 0.4$$ million in sales of brand $Z$. Currently, brands X, Y, and $\mathrm{Z}$ are each selling $$\$ 6$$ million per year. Model the sales of brand $\mathrm{Z}$ using a linear function.
Let $f(x, y, z)=43.2-2.3 x+11.3 y-4.5 z .$ Complete the following: An increase of 1 in the value of $y$ causes the value of $f$ to ______ by ______, whereas increasing the value of $x$ by 1 and ______ the value of $z$ by causes a decrease of $11.3$ in the value of $f$.
Sketch the cube with vertices $(0,0,0),(1,0,0),(0,1,0)$, $(0,0,1),(1,1,0),(1,0,1),(0,1,1)$, and $(1,1,1) .$ HINT $[$ See Example 4.]
Sketch the cube with vertices $(-1,-1,-1),(1,-1,-1)$ $(-1,1,-1),(-1,-1,1),(1,1,-1),(1,-1,1),(-1,1,1)$ and $(1,1,1)$. HINT [See Example 4.]
Sketch the pyramid with vertices $(1,1,0),(1,-1,0)$, $(-1,1,0),(-1,-1,0)$, and $(0,0,2)$
Sketch the solid with vertices $(1,1,0),(1,-1,0),(-1,1,0)$, $(-1,-1,0),(0,0,-1)$, and $(0,0,1)$.
Sketch the planes.$$z=-2$$
Sketch the planes.$$z=4$$
Sketch the planes.$$y=2$$
Sketch the planes.$$y=-3$$
Sketch the planes.$$x=-3$$
Sketch the planes.$$x=2$$
Match each equation with one of the graphs below. (If necessary, use technology to render the surfaces.) HINT [See Examples 5,6, and $7 .]$$$f(x, y)=1-3 x+2 y$$
Match each equation with one of the graphs below. (If necessary, use technology to render the surfaces.) HINT [See Examples 5,6, and $7 .]$$$f(x, y)=1-\sqrt{x^{2}+y^{2}}$$
Match each equation with one of the graphs below. (If necessary, use technology to render the surfaces.) HINT [See Examples 5,6, and $7 .]$$$f(x, y)=1-\left(x^{2}+y^{2}\right)$$
Match each equation with one of the graphs below. (If necessary, use technology to render the surfaces.) HINT [See Examples 5,6, and $7 .]$$$f(x, y)=y^{2}-x^{2}$$
Match each equation with one of the graphs below. (If necessary, use technology to render the surfaces.) HINT [See Examples 5,6, and $7 .]$$$f(x, y)=-\sqrt{1-\left(x^{2}+y^{2}\right)}$$
Match each equation with one of the graphs below. (If necessary, use technology to render the surfaces.) HINT [See Examples 5,6, and $7 .]$$$f(x, y)=1+\left(x^{2}+y^{2}\right)$$
Match each equation with one of the graphs below. (If necessary, use technology to render the surfaces.) HINT [See Examples 5,6, and $7 .]$$$f(x, y)=\frac{1}{x^{2}+y^{2}}$$
Match each equation with one of the graphs below. (If necessary, use technology to render the surfaces.) HINT [See Examples 5,6, and $7 .]$$$f(x, y)=3 x-2 y+1$$
Sketch the level curves $f(x, y)=c$ for the given function and values of c. HINT [See Example 5.]$$f(x, y)=2 x^{2}+2 y^{2} ; c=0,2,18$$
Sketch the level curves $f(x, y)=c$ for the given function and values of c. HINT [See Example 5.]$$f(x, y)=3 x^{2}+3 y^{2} ; c=0,3,27$$
Sketch the level curves $f(x, y)=c$ for the given function and values of c. HINT [See Example 5.]$$f(x, y)=y+2 x^{2} ; c=-2,0,2$$
Sketch the level curves $f(x, y)=c$ for the given function and values of c. HINT [See Example 5.]$$f(x, y)=2 y-x^{2} ; c=-2,0,2$$
Sketch the level curves $f(x, y)=c$ for the given function and values of c. HINT [See Example 5.]$$f(x, y)=2 x y-1 ; c=-1,0,1$$
Sketch the level curves $f(x, y)=c$ for the given function and values of c. HINT [See Example 5.]$$f(x, y)=2+x y ; c=-2,0,2$$
Sketch the graphs of the functions. HINT [See Example 7.]$$f(x, y)=1-x-y$$
Sketch the graphs of the functions. HINT [See Example 7.]$$f(x, y)=x+y-2$$
Sketch the graphs of the functions. HINT [See Example 7.]$$g(x, y)=2 x+y-2$$
Sketch the graphs of the functions. HINT [See Example 7.]$$g(x, y)=3-x+2 y$$
Sketch the graphs of the functions. HINT [See Example 7.]$$h(x, y)=x+2$$
Sketch the graphs of the functions. HINT [See Example 7.]$$h(x, y)=3-y$$
Sketch the graphs of the functions. HINT [See Example 7.]$$r(x, y)=x+y$$
Sketch the graphs of the functions. HINT [See Example 7.]$$r(x, y)=x-y$$
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$s(x, y)=2 x^{2}+2 y^{2}$. Show cross sections at $z=1$ and $z=2$.
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$s(x, y)=-\left(x^{2}+y^{2}\right) .$ Show cross sections at $z=-1$ and $z=-2$
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$t(x, y)=x^{2}+2 y^{2} .$ Show cross sections at $x=0$ and $z=1$.
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$t(x, y)=\frac{1}{2} x^{2}+y^{2} .$ Show cross sections at $x=0$ and $z=1$.
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$f(x, y)=2+\sqrt{x^{2}+y^{2}}$. Show cross sections at $z=3$ and $y=0 .$
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$f(x, y)=2-\sqrt{x^{2}+y^{2}}$. Show cross sections at $z=0$ and $y=0 .$
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$f(x, y)=-2 \sqrt{x^{2}+y^{2}} .$ Show cross sections at $z=-4$ and $y=1$.
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$f(x, y)=2+2 \sqrt{x^{2}+y^{2}}$. Show cross sections at $z=4$ and $y=1 .$
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$f(x, y)=y^{2}$
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$g(x, y)=x^{2}$
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$h(x, y)=\frac{1}{y}$
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$k(x, y)=e^{y}$
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$f(x, y)=e^{-\left(x^{2}+y^{2}\right)}$
Sketch the graphs of the functions. HINT [See Example 7.]Use of technology is suggested. HINT [See Example 6.]$$g(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}}}$$
Your weekly cost (in dollars) to manufacture $x$ cars and $y$ trucks is$$C(x, y)=240,000+6,000 x+4,000 y$$a. What is the marginal cost of a car? Of a truck? HINT [See Example 1.]b. Describe the graph of the cost function C. HINT [See Example 7.]c. Describe the slice $x=10$. What cost function does this slice describe?d. Describe the level curve $z=480,000$. What does this curve tell you about costs?
Your weekly cost (in dollars) to manufacture $x$ bicycles and $y$ tricycles is$$C(x, y)=24,000+60 x+20 y$$a. What is the marginal cost of a bicycle? Of a tricycle? HINT [See Example 1.]b. Describe the graph of the cost function $C$. HINT [See Example 7.]c. Describe the slice by $y=100 .$ What cost function does this slice describe?d. Describe the level curve $z=72,000$. What does this curve tell you about costs?
Your sales of online video and audio clips are booming. Your Internet provider, Moneydrain.com, wants to get in on the action and has offered you unlimited technical assistance and consulting if you agree to pay Moneydrain $3 \&$ for every video clip and $4 \&$ for every audio clip you sell on the site. Further, Moneydrain agrees to charge you only $$\$ 10$$ per month to host your site. Set up a (monthly) cost function for the scenario, and describe each variable.
Your Cabaret nightspot "Jazz on Jupiter" has become an expensive proposition: You are paying monthly costs of $$\$ 50,000$$ just to keep the place running. On top of that, your regular cabaret artist is charging you $$\$ 3,000$$ per performance, and your jazz ensemble is charging $$\$ 1,000$$ per hour. Set up a (monthly) cost function for the scenario, and describe each variable.
In each year from 1983 to 2003, the percentage $y$ of research articles in Physical Review written by researchers in the United States can be approximated by$y=82-0.78 t-1.02 x$ percentage points $\quad(0 \leq t \leq 20)$where $t$ is the year since 1983 and $x$ is the percentage of articles written by researchers in Europe. ${ }^{1}$a. In 2003, researchers in Europe wrote $38 \%$ of the articles published by the journal that year. What percentage was written by researchers in the United States?b. In 1983 , researchers in the United States wrote $61 \%$ of the articles published that year. What percentage was written by researchers in Europe?c. What are the units of measurement of the coefficient of $t$ ?
The number $z$ of research articles in Physical Review that were written by researchers in the United States from 1993 through 2003 can be approximated by$$z=5,960-0.71 x+0.50 y \quad(3,000 \leq x, y \leq 6,000)$$articles each year, where $x$ is the number of articles written by researchers in Europe and $y$ is the number written by researchers in other countries (excluding Europe and the United States). ${ }^{2}$a. In the year 2000 , approximately 5,500 articles were written by researchers in Europe, and 4,500 by researchers in other countries. How many (to the nearest 100 ) were written by researchers in the United States?b. According to the model, if 5,000 articles were written in Europe and an equal number by researchers in the United States and other countries, what would that number be?c. What is the significance of the fact that the coefficient of $x$ is negative?
In the late 1900 s, the relationship between the domestic market shares of three major U.S. manufacturers of cars and light trucks could be modeled by$$x_{3}=0.66-2.2 x_{1}-0.02 x_{2}$$where $x_{1}, x_{2}$, and $x_{3}$ are, respectively, the fractions of the market held by Chrysler, Ford, and General Motors. $^{3}$ Thinking of General Motors' market share as a function of the shares of the other two manufacturers, describe the graph of the resulting function. How are the different slices by $x_{1}=$ constant related to one another? What does this say about market share?
In the late 1900s, the relationship among the domestic market shares of three major manufacturers of breakfast cereal was$$x_{1}=-0.4+1.2 x_{2}+2 x_{3}$$where $x_{1}, x_{2}$, and $x_{3}$ are, respectively, the fractions of the market held by Kellogg, General Mills, and General Foods. ${ }^{4}$ Thinking of Kellogg's market share as a function of shares of the other two manufacturers, describe the graph of the resulting function. How are the different slices by $x_{2}=$ constant related to one another? What does this say about market share?
The number of prisoners in federal prisons in the United States can be approximated by$N(x, y)=27-0.08 x+0.08 y+0.0002 x y$ thousand inmateswhere $x$ is the number, in thousands, in state prisons, and $y$ is the number, in thousands, in local jails. $^{5}$a. In 2007 there were approximately $1.3$ million in state prisons and 781 thousand in local jails. Estimate, to the nearest thousand, the number of prisoners in federal prisons that year.b. Obtain $N$ as a function of $x$ for $y=300$, and again for $y=500 .$ Interpret the slopes of the resulting linear functions.
The number of prisoners in state prisons in the United States can be approximated by$$N(x, y)=-260+7 x+2 y-0.009 x y \text { thousand inmates }$$where $x$ is the number, in thousands, in federal prisons, and $y$ is the number, in thousands, in local jails. ${ }^{6}$a. In 2007 there were approximately 189 thousand in federal prisons and 781 thousand in local jails. Estimate, to the nearest $0.1$ million, the number of prisoners in state prisons that year.b. Obtain $N$ as a function of $y$ for $x=80$, and again for $x=100$. Interpret the slopes of the resulting linear functions.
Your weekly cost (in dollars) to manufacture $x$ cars and $y$ trucks is$$C(x, y)=240,000+6,000 x+4,000 y-20 x y$$(Compare with Exercise 77.)a. Describe the slices $x=$ constant and $y=$ constant.b. Is the graph of the cost function a plane? How does your answer relate to part (a)?c. What are the slopes of the slices $x=10$ and $x=20 ?$ What does this say about cost?
Repeat the preceding exercise using the weekly cost to manufacture $x$ bicycles and $y$ tricycles given by$$C(x, y)=24,000+60 x+20 y+0.3 x y$$(Compare with Exercise $78 .$ )
Let us look once again at the example we used to introduce the chapter. Your major online bookstore is in direct competition with Amazon.com, BN.com, and Borders.com. Your company's daily revenue in dollars is given by $R(x, y, z)=10,000-0.01 x-0.02 y-0.01 z+0.00001 y z$where $x, y$, and $z$ are the online daily revenues of Amazon.com, BN.com, and Borders.com respectively.a. If, on a certain day, Amazon.com shows revenue of $$\$ 12,000$$, while BN.com and Borders.com each show $$\$ 5,000$$, what does the model predict for your company's revenue that day?b. If Amazon.com and BN.com each show daily revenue of $$\$ 5,000$$, give an equation showing how your daily revenue depends on that of Borders.com.
Repeat the preceding exercise, using the revised revenue function$R(x, y, z)=20,000-0.02 x-0.04 y-0.01 z+0.00001 y z .$
The following table shows the approximate net earnings, in billions of dollars, of Walmart and Target in 2000,2004, and $2008 .^{7}$$$\begin{array}{|r|c|c|c|}\hline & \mathbf{2 0 0 0} & \mathbf{2 0 0 4} & \mathbf{2 0 0 8} \\\hline \text { Walmart } & 160 & 250 & 370 \\\hline \text { Target } & 27 & 42 & 62 \\\hline\end{array}$$Model Walmart's net earnings as a function of Target's net earnings and time, using a linear function of the form$$f(x, t)=A x+B t+C \quad(A, B, C \text { constants })$$where $f$ is Walmart's net earnings (in billions of dollars), $x$ is Target's net earnings (in billions of dollars), and $t$ is time in years since 2000 . In 2006 Target's net earnings were about $\$ 52.5$ billion. What does your model estimate as Walmart's net earnings that year?
The following table shows the approximate net earnings of Nintendo (in billions of yen) and Nokia (in billions of euro) in 2000,2004 , and $2008 .{ }^{8}$$$\begin{array}{|r|c|c|c|}\hline & \mathbf{2 0 0 0} & \mathbf{2 0 0 4} & \mathbf{2 0 0 8} \\\hline \text { Nintendo } & 530 & 510 & 1700 \\\hline \text { Nokia } & 30 & 30 & 52 \\\hline\end{array}$$Model Nintendo's net earnings as a function of Nokia's net earnings and time, using a linear function of the form$$f(x, t)=A x+B t+C \quad(A, B, C \text { constants })$$where $f$ is Nintendo's net earnings (in billions of yen), $x$ is Nokia's net earnings (in billions of euro), and $t$ is time in years since 2000 . In 2007 Nokia's net earnings were about $€ 50$ billion. What does your model estimate as Nokia's net earnings that year?
Suppose your newspaper is trying to decide between two competing desktop publishing software packages, Macro Publish and Turbo Publish. You estimate that if you purchase $x$ copies of Macro Publish and $y$ copies of Turbo Publish, your company's daily productivity will be$$U(x, y)=6 x^{0.8} y^{0.2}+x$$where $U(x, y)$ is measured in pages per day $(U$ is called a utility function). If $x=y=10$, calculate the effect of increasing $x$ by one unit, and interpret the result.
The cost $C$ (in dollars) of building a house is related to the number $k$ of carpenters used and the number $e$ of electricians used by$$C(k, e)=15,000+50 k^{2}+60 e^{2} .$$If $k=e=10$, compare the effects of increasing $k$ by one unit and of increasing $e$ by one unit. Interpret the result.
The volume of an ellipsoid with cross-sectional radii $a, b$, and $c$ is $V(a, b, c)=\frac{4}{3} \pi a b c$.a. Find at least two sets of values for $a, b$, and $c$ such that $V(a, b, c)=1$b. Find the value of $a$ such that $V(a, a, a)=1$, and describe the resulting ellipsoid.
The volume of a right elliptical cone with height $h$ and radii $a$ and $b$ of its base is $V(a, b, h)=\frac{1}{3} \pi a b h$.a. Find at least two sets of values for $a, b$, and $h$ such that $V(a, b, h)=1 .$b. Find the value of $a$ such that $V(a, a, a)=1$, and describe the resulting cone.
Involve "Cobb-Douglas" productivity functions. These functions have the form$$P(x, y)=K x^{a} y^{1-a}$$where P stands for the number of items produced per year, $x$ is the number of employees, and $y$ is the annual operating budget. (The numbers $K$ and a are constants that depend on the situation we are looking $a t$, with $0 \leq a \leq 1 .)$Productivity How many items will be produced per year by a company with 100 employees and an annual operating budget of $\$ 500,000$ if $K=1,000$ and $a=0.5 ?$ (Round your answer to one significant digit.)
Involve "Cobb-Douglas" productivity functions. These functions have the form$$P(x, y)=K x^{a} y^{1-a}$$where P stands for the number of items produced per year, $x$ is the number of employees, and $y$ is the annual operating budget. (The numbers $K$ and a are constants that depend on the situation we are looking $a t$, with $0 \leq a \leq 1 .)$Productivity How many items will be produced per year by a company with 50 employees and an annual operating budget of $\$ 1,000,000$ if $K=1,000$ and $a=0.5$ ? (Round your answer to one significant digit.)
Involve "Cobb-Douglas" productivity functions. These functions have the form$$P(x, y)=K x^{a} y^{1-a}$$where P stands for the number of items produced per year, $x$ is the number of employees, and $y$ is the annual operating budget. (The numbers $K$ and a are constants that depend on the situation we are looking $a t$, with $0 \leq a \leq 1 .)$Two years ago my piano manufacturing plant employed 1,000 workers, had an operating budget of $$\$ 1$$ million, and turned out 100 pianos. Last year, I slashed the operating budget to $$\$ 10,000$$, and production dropped to 10 pianos.a. Use the data for each of the two years and the CobbDouglas formula to obtain two equations in $K$ and $a$.b. Take logs of both sides in each equation and obtain two linear equations in $a$ and $\log K$.c. Solve these equations to obtain values for $a$ and $K$.d. Use these values in the Cobb-Douglas formula to predict production if I increase the operating budget back to $\$ 1$ million but lay off half the work force.
Involve "Cobb-Douglas" productivity functions. These functions have the form$$P(x, y)=K x^{a} y^{1-a}$$where P stands for the number of items produced per year, $x$ is the number of employees, and $y$ is the annual operating budget. (The numbers $K$ and a are constants that depend on the situation we are looking $a t$, with $0 \leq a \leq 1 .)$Repeat the preceding exercise using the following data: Two years ago- 1,000 employees, $$\$ 1$$ million operating budget, 100 pianos; last year $-1,000$ employees, $$\$ 100,000$$ operating budget, 10 pianos.
The burden of man-made aerosol sulfate in the earth's atmosphere, in grams per square meter, is$$B(x, n)=\frac{x n}{A}$$where $x$ is the total weight of aerosol sulfate emitted into the atmosphere per year and $n$ is the number of years it remains in the atmosphere. $A$ is the surface area of the earth, approximately $5.1 \times 10^{14}$ square meters. ${ }^{10}$a. Calculate the burden, given the 1995 estimated values of $x=1.5 \times 10^{14}$ grams per year, and $n=5$ days.b. What does the function $W(x, n)=x n$ measure?
The amount of aerosol sulfate (in grams) was approximately $45 \times 10^{12}$ grams in 1940 and has been increasing exponentially ever since, with a doubling time of approximately 20 years. $^{11}$ Use the model from the preceding exercise to give a formula for the atmospheric burden of aerosol sulfate as a function of the time $t$ in years since 1940 and the number of years $n$ it remains in the atmosphere.
Frank Drake, an astronomer at the University of California at Santa Cruz, devised the following equation to estimate the number of planet-based civilizations in our Milky Way galaxy willing and able to communicate with Earth: $^{12}$$$N\left(R, f_{p}, n_{e}, f_{l}, f_{i}, f_{c}, L\right)=R f_{p} n_{e} f_{l} f_{i} f_{c} L$$$R=$ the number of new stars formed in our galaxy each year $f_{p}=$ the fraction of those stars that have planetary systems $n_{e}=$ the average number of planets in each such system that can support life $f_{l}=$ the fraction of such planets on which life actually evolves $f_{i}=$ the fraction of life-sustaining planets on which intelligent life evolves $f_{c}=$ the fraction of intelligent-life-bearing planets on which the intelligent beings develop the means and the will to communicate over interstellar distances $L=$ the average lifetime of such technological civilizations (in years)a. What would be the effect on $N$ if any one of the variables were doubled?b. How would you modify the formula if you were interested only in the number of intelligent-life-bearing planets in the galaxy?c. How could one convert this function into a linear function?d. (For discussion) Try to come up with an estimate of $N$
The formula given in the preceding exercise restricts attention to planet-based civilizations in our galaxy. Give a formula that includes intelligent planetbased aliens from the galaxy Andromeda. (Assume that all the variables used in the formula for the Milky Way have the same values for Andromeda.)
Let $f(x, y)=\frac{x}{y} .$ How are $f(x, y)$ and $f(y, x)$ related?
Let $f(x, y)=x^{2} y^{3} .$ How are $f(x, y)$ and $f(-x,-y)$ related?
Give an example of a function of the two variables $x$ and $y$ with the property that interchanging $x$ and $y$ has no effect.
Give an example of a function $f$ of the two variables $x$ and $y$ with the property that $f(x, y)=-f(y, x)$.
Give an example of a function $f$ of the three variables $x, y$, and $z$ with the property that $f(x, y, z)=f(y, x, z)$ and $f(-x,-y,-z)=-f(x, y, z)$
Give an example of a function $f$ of the three variables $x, y$, and $z$ with the property that $f(x, y, z)=f(y, x, z)$ and $f(-x,-y,-z)=f(x, y, z)$.
Illustrate by means of an example how a real-valued function of the two variables $x$ and $y$ gives different real-valued functions of one variable when we restrict $y$ to be different constants.
Illustrate by means of an example how a real-valued function of one variable $x$ gives different real-valued functions of the two variables $y$ and $z$ when we substitute for $x$ suitable functions of $y$ and $z$.
If $f$ is a linear function of $x$ and $y$, show that if we restrict $y$ to be a fixed constant, then the resulting function of $x$ is linear. Does the slope of this linear function depend on the choice of $y$ ?
If $f$ is an interaction function of $x$ and $y$, show that if we restrict $y$ to be a fixed constant, then the resulting function of $x$ is linear. Does the slope of this linear function depend on the choice of $y$ ?
Suppose that $C(x, y)$ represents the cost of $x$ CDs and $y$ cassettes. If $C(x, y+1)<C(x+1, y)$ for every $x \geq 0$ and $y \geq 0$, what does this tell you about the cost of CDs and cassettes?
Suppose that $C(x, y)$ represents the cost of renting $x$ DVDs and $y$ video games. If $C(x+2, y)<C(x, y+1)$ for every $x \geq 0$ and $y \geq 0$, what does this tell you about the cost of renting DVDs and video games?
Complete the following: The graph of a linear function of two variables is a ____ .
Complete the following: The level curves of a linear function of two variables are ___ .
The following diagram shows some level curves of the temperature, in degrees Fahrenheit, of a region in space, as well as the location, on the 100 -degree curve, of a heat-seeking missile moving through the region. (These level curves are called isotherms.) In which of the three directions shown should the missile be traveling so as to experience the fastest rate of increase in temperature at the given point? Explain your answer.
The following diagram shows some level curves of the altitude of a mountain valley, as well as the location, on the 2,000 -ft curve, of a hiker. The hiker is currently moving at the greatest possible rate of descent. In which of the three directions shown is he moving? Explain your answer.
Your study partner Slim claims that because the surface $z=f(x, y)$ you have been studying is a plane, it follows that all the slices $x=$ constant and $y=$ constant are straight lines. Do you agree or disagree? Explain.
Your other study partner Shady just told you that the surface $z=x y$ you have been trying to graph must be a plane because you've already found that the slices $x=$ constant and $y=$ constant are all straight lines. Do you agree or disagree? Explain.
Why do we not sketch the graphs of functions of three or more variables?
The surface of a mountain can be thought of as the graph of what function?
Why is three-dimensional space used to represent the graph of a function of two variables?
Why is it that we can sketch the graphs of functions of two variables on the two-dimensional flat surfaces of these pages?