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Chapter 14

Partial Derivatives

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Problem 1

In Example 2 we considered the function $W=f(T, v),$ where
$W$ is the wind-chill index, $T$ is the actual temperature, and $v$ is
the wind speed. A numerical representation is given in Table 1 .
$\begin{array}{l}{\text { (a) What is the value of } f(-15,40) ? \text { What is its meaning? }} \\ {\text { (b) Describe in words the meaning of the question "For what }} \\ {\text { value of } v \text { is } f(-20, v)=-30 ?^{\circ \prime} \text { Then answer the question. }}\end{array}$
$\begin{array}{l}{\text { (c) Describe in words the meaning of the question "For what }} \\ {\text { value of } T \text { is } f(T, 20)=-49 ?^{\prime \prime} \text { Then answer the question. }} \\ {\text { (d) What is the meaning of the function } W=f(-5, v) ?} \\ {\text { Describe the behavior of this function. }}\end{array}$
$\begin{array}{l}{\text { (e) What is the meaning of the function } W=f(T, 50) ?} \\ {\text { Describe the behavior of this function. }}\end{array}$

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Problem 2

The temperature-humidity index $I$ (or humidex, for short) is the
perceived air temperature when the actual temperature is $T$ and
the relative humidity is $h,$ so we can write $I=f(T, h)$ . The fol-
lowing table of values of $I$ is an excerpt from a table compiled
by the National Oceanic \& Atmospheric Administration.
$$\begin{array}{l}{\text { (a) What is the value of } f(95,70) ? \text { What is its meaning? }} \\ {\text { (b) For what value of } h \text { is } f(90, h)=100 ?} \\ {\text { (c) For what value of } T \text { is } f(T, 50)=88 ?}\end{array}$$
$$\begin{array}{l}{\text { (d) What are the meanings of the functions } I=f(80, h)} \\ {\text { and } I=f(100, h) ? \text { Compare the behavior of these two }} \\ {\text { functions of } h .}\end{array}$$

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Problem 3

A manufacturer has modeled its yearly production function $P$
(the monetary value of its entire production in millions of
dollars) as a Cobb-Douglas function
$$P(L, K)=1.47 L^{0.65} K^{0.35}$$
where $L$ is the number of labor hours (in thousands) and $K$ is
the invested capital (in millions of dollars). Find $P(120,20)$
and interpret it.

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Problem 4

Verify for the Cobb-Douglas production function
$$P(L, K)=1.01 L^{\mathrm{a} \pi} K^{\mathrm{azs}}$$
discussed in Example 3 that the production will be doubled
if both the amount of labor and the amount of capital are
doubled. Determine whether this is also true for the general
production function
$$P(L, K)=b L^{\alpha} K^{1-\alpha}$$

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Problem 5

A model for the surface area of a human body is given by the
function
$$S=f(w, h)=0.1091 w^{0.425} h^{0.725}$$
where $w$ is the weight (in pounds), $h$ is the height (in inches),
and $S$ is measured in square feet.
$\begin{array}{l}{\text { (a) Find } f(160,70) \text { and interpret it. }} \\ {\text { (b) What is your own surface area? }}\end{array}$

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Problem 6

The wind-chill index $W$ discussed in Example 2 has been
modeled by the following function:
$$W(T, v)=13.12+0.6215 T-11.37 v^{0.16}+0.3965 T v^{0.16}$$
Check to see how closely this model agrees with the values in
Table 1 for a few values of $T$ and $v$ .

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Problem 7

The wave heights $h$ in the open sea depend on the speed $v$
of the wind and the length of time $t$ that the wind has been
blowing at that speed. Values of the function $h=f(v, t)$ are
recorded in feet in Table 4 .
$$\begin{array}{l}{\text { (a) What is the value of } f(40,15) ? \text { What is its meaning? }} \\ {\text { (b) What is the meaning of the function } h=f(30, \text { t)? Describe }} \\ {\text { the behavior of this function. }} \\ {\text { (c) What is the meaning of the function } h=f(v, 30) ? \text { Describe }} \\ {\text { the behavior of this function. }}\end{array}$$

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Problem 8

A company makes three sizes of cardboard boxes: small,
medium, and large. It costs $\$ 2.50$ to make a small box, $\$ 4.00$
for a medium box, and $\$ 4.50$ for a large box. Fixed costs
are $\$ 8000 .$
$\begin{array}{l}{\text { (a) Express the cost of making } x \text { small boxes, } y \text { medium }} \\ {\text { boxes, and } z \text { large boxes as a function of three variables: }} \\ {C=f(x, y, z) .} \\ {\text { (b) Find } f(3000,5000,4000) \text { and interpret it. }} \\ {\text { (c) What is the domain of } f ?}\end{array}$

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Problem 9

$Let $ g(x, y)=\cos (x+2 y)$$
$\begin{array}{l}{\text { (a) Evaluate } g(2,-1) \text { . }} \\ {\text { (b) Find the domain of } g .} \\ {\text { (c) Find the range of } g .}\end{array}$

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Problem 10

Let $F(x, y)=1+\sqrt{4-y^{2}}$
$$\begin{array}{l}{\text { (a) Evaluate } F(3,1) \text { . }} \\ {\text { (b) Find and sketch the domain of } F .} \\ {\text { (c) Find the range of } F .}\end{array}$$

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Problem 11

$Let f(x, y, z)=\sqrt{x}+\sqrt{y}+\sqrt{z}+\ln \left(4-x^{2}-y^{2}-z^{2}\right)$
$\begin{array}{l}{\text { (a) Evaluate } f(1,1,1) \text { . }} \\ {\text { (b) Find and describe the domain of } f}\end{array}$

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Problem 12

$Let g(x, y, z)=x^{3} y^{2} z \sqrt{10-x-y-z}.$
$$\begin{array}{l}{\text { (a) Evaluate } g(1,2,3) .} \\ {\text { (b) Find and describe the domain of } g .}\end{array}$$

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Problem 13

$13-22$ Find and sketch the domain of the function.
$$f(x, y)=\sqrt{2 x-y}$$

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Problem 14

$13-22$ Find and sketch the domain of the function.
$$f(x, y)=\sqrt{x y}$$

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Problem 15

$13-22$ Find and sketch the domain of the function.
$$f(x, y)=\ln \left(9-x^{2}-9 y^{2}\right)$$

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Problem 16

$13-22$ Find and sketch the domain of the function.
$$f(x, y)=\sqrt{x^{2}-y^{2}}$$

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Problem 17

$13-22$ Find and sketch the domain of the function.
$$f(x, y)=\sqrt{1-x^{2}}-\sqrt{1-y^{2}}$$

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Problem 18

$13-22$ Find and sketch the domain of the function.
$$f(x, y)=\sqrt{y}+\sqrt{25-x^{2}-y^{2}}$$

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Problem 19

$13-22$ Find and sketch the domain of the function.
$$f(x, y)=\frac{\sqrt{y-x^{2}}}{1-x^{2}}$$

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Problem 20

$13-22$ Find and sketch the domain of the function.
$$f(x, y)=\arcsin \left(x^{2}+y^{2}-2\right)$$

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Problem 21

$13-22$ Find and sketch the domain of the function.
$$f(x, y, z)=\sqrt{1-x^{2}-y^{2}-z^{2}}$$

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Problem 22

$13-22$ Find and sketch the domain of the function.
$$f(x, y, z)=\ln \left(16-4 x^{2}-4 y^{2}-z^{2}\right)$$

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Problem 23

$23-31$ Sketch the graph of the function.
$$f(x, y)=1+y$$

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Problem 24

$23-31$ Sketch the graph of the function.
$$f(x, y)=2-x$$

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Problem 25

$23-31$ Sketch the graph of the function.
$$f(x, y)=10-4 x-5 y$$

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Problem 26

$23-31$ Sketch the graph of the function.
$$f(x, y)=e^{-y}$$

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Problem 27

$23-31$ Sketch the graph of the function.
$$f(x, y)=y^{2}+1$$

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Problem 28

$23-31$ Sketch the graph of the function.
$$f(x, y)=1+2 x^{2}+2 y^{2}$$

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Problem 29

$23-31$ Sketch the graph of the function.
$$f(x, y)=9-x^{2}-9 y^{2}$$

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Problem 30

$23-31$ Sketch the graph of the function.
$$f(x, y)=\sqrt{4 x^{2}+y^{2}}$$

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Problem 31

$23-31$ Sketch the graph of the function.
$$f(x, y)=\sqrt{4-4 x^{2}-y^{2}}$$

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Problem 32

Match the function with it graph (labeled I-VI). Give reasons
for your choices.
$${ (a) } f(x, y)=|x|+|y| \quad \text { (b) } f(x, y)=|x y|$$
$${ (c) } ff(x, y)=\frac{1}{1+x^{2}+y^{2}} \quad \text { (d) } f(x, y)=\left(x^{2}-y^{2}\right)^{2}$$
$${ (e) } ff(x, y)=(x-y)^{2} \quad \text { (f) } f(x, y)=\sin (|x|+|y|)$$

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Problem 33

A contour map for a function $f$ is shown. Use it to estimate the
values of $f(-3,3)$ and $f(3,-2)$ . What can you say about the
shape of the graph?

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Problem 34

Shown is a contour map of atmospheric pressure in North
America on August 12, 2008. On the level curves (called
isobars) the pressure is indicated in millibars (mb).
$\begin{array}{l}{\text { (a) Estimate the pressure at } C \text { (Chicago), } N \text { (Nashville), }} \\ {S(\text { San Francisco), and } V(\text { Vancouver) }} \\ {\text { (b) At which of these locations were the winds strongest? }}\end{array}$

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Problem 35

Level curves (isothermals) are shown for the water temperature
$\left(\text { in }^{\circ} \mathrm{C}\right)$ in Long Lake (Minnesota) in 1998 as a function of
depth and time of year. Estimate the temperature in the lake on
June 9$($ day 160$)$ at a depth of 10 $\mathrm{m}$ and on June 29$(\mathrm{day} 180)$
at a depth of 5 $\mathrm{m} .$

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Problem 36

Two contour maps are shown. One is for a function $f$ whose
graph is a cone. The other is for a function $g$ whose graph is a
paraboloid. Which is which, and why?

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Problem 37

Locate the points $A$ and $B$ on the map of Lonesome Mountain
(Figure $12 ) .$ How would you describe the terrain near $A ?$
Near $B ?$

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Problem 38

Make a rough sketch of a contour map for the function whose
graph is shown.

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Problem 39

$39-42$ A contour map of a function is shown. Use it to make a
rough sketch of the graph of $f .$
GRAPH

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Problem 40

$39-42$ A contour map of a function is shown. Use it to make a
rough sketch of the graph of $f .$
GRAPH

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Problem 41

$39-42$ A contour map of a function is shown. Use it to make a
rough sketch of the graph of $f .$
GRAPH

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Problem 42

$39-42$ A contour map of a function is shown. Use it to make a
rough sketch of the graph of $f .$
GRAPH

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Problem 43

$43-50$ Draw a contour map of the function showing several level
curves.
$$f(x, y)=(y-2 x)^{2}$$

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Problem 44

$43-50$ Draw a contour map of the function showing several level
curves.
$$f(x, y)=x^{3}-y$$

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Problem 45

$43-50$ Draw a contour map of the function showing several level
curves.
$$f(x, y)=\sqrt{x}+y$$

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Problem 46

$43-50$ Draw a contour map of the function showing several level
curves.
$$f(x, y)=\ln \left(x^{2}+4 y^{2}\right)$$

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Problem 47

$43-50$ Draw a contour map of the function showing several level
curves.
$$f(x, y)=y e^{x}$$

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Problem 48

$43-50$ Draw a contour map of the function showing several level
curves.
$$f(x, y)=y \sec x$$

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Problem 49

$43-50$ Draw a contour map of the function showing several level
curves.
$$f(x, y)=\sqrt{y^{2}-x^{2}}$$

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Problem 50

$43-50$ Draw a contour map of the function showing several level
curves.
$$f(x, y)=y /\left(x^{2}+y^{2}\right)$$

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Problem 51

$51-52$ Sketch both a contour map and a graph of the function and
compare them.
$$f(x, y)=x^{2}+9 y^{2}$$

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Problem 52

$51-52$ Sketch both a contour map and a graph of the function and
compare them.
$$f(x, y)=\sqrt{36-9 x^{2}-4 y^{2}}$$

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Problem 53

A thin metal plate, located in the $x y$ -plane, has temperature
$T(x, y)$ at the point $(x, y) .$ The level curves of $T$ are called
isothermals because at all points on such a curve the tempera-
ture is the same. Sketch some isothermals if the temperature
function is given by
$$T(x, y)=\frac{100}{1+x^{2}+2 y^{2}}$$

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Problem 54

If $V(x, y)$ is the electric potential at a point $(x, y)$ in the
$x y$ -plane, then the level curves of $V$ are called equipotential
curves because at all points on such a curve the electric
potential is the same. Sketch some equipotential curves if
$V(x, y)=c / \sqrt{r^{2}-x^{2}-y^{2}},$ where $c$ is a positive constant.

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Problem 55

$55-58$ Use a computer to graph the function using various
domains and viewpoints. Get a printout of one that, in your opin-
ion, gives a good view. If your software also produces level
curves, then plot some contour lines of the same function and
compare with the graph.
$$f(x, y)=x y^{2}-x^{3}(monkey saddle)$$

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Problem 56

$55-58$ Use a computer to graph the function using various
domains and viewpoints. Get a printout of one that, in your opin-
ion, gives a good view. If your software also produces level
curves, then plot some contour lines of the same function and
compare with the graph.
$$f(x, y)=x y^{3}-y x^{3} (dog saddle)$$

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Problem 57

$55-58$ Use a computer to graph the function using various
domains and viewpoints. Get a printout of one that, in your opin-
ion, gives a good view. If your software also produces level
curves, then plot some contour lines of the same function and
compare with the graph.
$$f(x, y)=e^{-\left(x^{2}+y^{2}\right) / 3}\left(\sin \left(x^{2}\right)+\cos \left(y^{2}\right)\right)$$

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Problem 58

$55-58$ Use a computer to graph the function using various
domains and viewpoints. Get a printout of one that, in your opin-
ion, gives a good view. If your software also produces level
curves, then plot some contour lines of the same function and
compare with the graph.
$$f(x, y)=\cos x \cos y$$

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Problem 59

$59-64$ Match the function (a) with its graph (labeled $A-F$ below)
and (b) with its contour map (labeled I-VI). Give reasons for
your choices.
$$z=\sin (x y)$$

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Problem 60

$59-64$ Match the function (a) with its graph (labeled $A-F$ below)
and (b) with its contour map (labeled I-VI). Give reasons for
your choices.
$$z=e^{x} \cos y$$

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Problem 61

$59-64$ Match the function (a) with its graph (labeled $A-F$ below)
and (b) with its contour map (labeled I-VI). Give reasons for
your choices.
$$z=\sin (x-y)$$

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Problem 62

$59-64$ Match the function (a) with its graph (labeled $A-F$ below)
and (b) with its contour map (labeled I-VI). Give reasons for
your choices.
$$z=\sin x-\sin y$$

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Problem 63

$59-64$ Match the function (a) with its graph (labeled $A-F$ below)
and (b) with its contour map (labeled I-VI). Give reasons for
your choices.
$$z=\left(1-x^{2}\right)\left(1-y^{2}\right)$$

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Problem 64

$59-64$ Match the function (a) with its graph (labeled $A-F$ below)
and (b) with its contour map (labeled I-VI). Give reasons for
your choices.
$$z=\frac{x-y}{1+x^{2}+y^{2}}$$

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Problem 65

$65-68$ Describe the level surfaces of the function.
$$f(x, y, z)=x+3 y+5 z$$

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Problem 66

$65-68$ Describe the level surfaces of the function.
$$f(x, y, z)=x^{2}+3 y^{2}+5 z^{2}$$

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Problem 67

$65-68$ Describe the level surfaces of the function.
$$f(x, y, z)=y^{2}+z^{2}$$

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Problem 68

$65-68$ Describe the level surfaces of the function.
$$f(x, y, z)=x^{2}-y^{2}-z^{2}$$

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Problem 69

$69-70$ Describe how the graph of $g$ is obtained from the graph
of $f .$
$$\begin{array}{l}{\text { (a) } g(x, y)=f(x, y)+2} \\ {\text { (b) } g(x, y)=2 f(x, y)} \\ {\text { (c) } g(x, y)=-f(x, y)} \\ {\text { (d) } g(x, y)=2-f(x, y)}\end{array}$$

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Problem 70

$69-70$ Describe how the graph of $g$ is obtained from the graph
of $f .$
$$\begin{array}{l}{\text { (a) } g(x, y)=f(x-2, y)} \\ {\text { (b) } g(x, y)=f(x, y+2)} \\ {\text { (c) } g(x, y)=f(x+3, y-4)}\end{array}$$

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Problem 71

$71-72$ Use a computer to graph the function using various
domains and viewpoints. Get a printout that gives a good view of
the "peaks and valleys." Would you say the function has a maxi-
mum value? Can you identify any points on the graph that you
might consider to be "local maximum points"? What about "local
minimum points"?
$$f(x, y)=3 x-x^{4}-4 y^{2}-10 x y$$

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Problem 72

$71-72$ Use a computer to graph the function using various
domains and viewpoints. Get a printout that gives a good view of
the "peaks and valleys." Would you say the function has a maxi-
mum value? Can you identify any points on the graph that you
might consider to be "local maximum points"? What about "local
minimum points"?
$$f(x, y)=x y e^{-x^{2}-y^{2}}$$

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Problem 73

$73-74$ Use a computer to graph the function using various
domains and viewpoints. Comment on the limiting behavior of
the function. What happens as both $x$ and $y$ become large? What
happens as $(x, y)$ approaches the origin?
$$f(x, y)=\frac{x+y}{x^{2}+y^{2}}$$

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Problem 74

$73-74$ Use a computer to graph the function using various
domains and viewpoints. Comment on the limiting behavior of
the function. What happens as both $x$ and $y$ become large? What
happens as $(x, y)$ approaches the origin?
$$f(x, y)=\frac{x y}{x^{2}+y^{2}}$$

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Problem 75

Use a computer to investigate the family of functions
$f(x, y)=e^{c x^{2}+y^{2}} .$ How does the shape of the graph depend
on $c ?$

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Problem 76

Use a computer to investigate the family of surfaces
$$z=\left(a x^{2}+b y^{2}\right) e^{-x^{2}-y^{2}}$$
How does the shape of the graph depend on the numbers a
and $b ?$

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Problem 77

Use a computer to investigate the family of surfaces
$z=x^{2}+y^{2}+c x y .$ In particular, you should determine the
transitional values of $c$ for which the surface changes from
one type of quadric surface to another.

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Problem 78

Graph the functions
$$f(x, y)=\sqrt{x^{2}+y^{2}}$$
$$f(x, y)=e^{\sqrt{x^{2}+y^{2}}}$$
$$f(x, y)=\ln \sqrt{x^{2}+y^{2}}$$
$$f(x, y)=\sin \left(\sqrt{x^{2}+y^{2}}\right)$$
and
$$f(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}}}$$
In general, if $g$ is a function of one variable, how is the graph
of
$$f(x, y)=g\left(\sqrt{x^{2}+y^{2}}\right)$$
obtained from the graph of $g ?$

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Problem 79

$\begin{array}{l}{\text { (a) Show that, by taking logarithms, the general Cobb- }} \\ {\text { Douglas function } P=b L^{L} K^{1-\alpha} \text { can be expressed as }}\end{array}$
$$\ln \frac{P}{K}=\ln b+\alpha \ln \frac{L}{K}$$
$\begin{array}{l}{\text { (b) If we let } x=\ln (L / K) \text { and } y=\ln (P / K), \text { the equation in }} \\ {\text { part (a) becomes the linear equation } y=\alpha x+\ln b \text { . Use }} \\ {\text { Table } 2 \text { (in Example } 3 \text { ) to make a table of values of }} \\ {\ln (L / K) \text { and } \ln (P / K) \text { for the years } 1899-1922 . \text { Then use a }}\end{array}$
$\begin{array}{l}{\text { graphing calculator or computer to find the least squares }} \\ {\text { regression line through the points }(\ln (L / K), \ln (P / K))}\end{array}$
$\begin{array}{l}{\text { (c) Deduce that the Cobb-Douglas production function is }} \\ {P=1.01 L^{0.75} K^{0.25}}.\end{array}$

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