# Calculus Early Transcendentals

## Educators

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 $1 .$
(a) What is the value of $f(-15,40) ?$ What is its meaning?
(b) Describe in words the meaning of the question "For what
value of $v$ is $f(-20, v)=-30 ?^{\prime \prime}$ Then answer the question.
(c) Describe in words the meaning of the question "For what
value of $T$ is $f(T, 20)=-49 ?^{\circ}$ Then answer the question.
(d) What is the meaning of the function $W=f(-5, v) ?$
Describe the behavior of this function.
(e) What is the meaning of the function $W=f(T, 50) ?$
Describe the behavior of this function.

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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 following table of values of $I$ is an excerpt from a table compiled
by the National Oceanic $\&$ Atmospheric Administration.
(a) What is the value of $f(95,70)$ ? What is its meaning?
(b) For what value of $h$ is $f(90,$ h) $=100 ?$
(c) For what value of $T$ is $f(T, 50)=88 ?$
(d) What are the meanings of the functions $I=f(80, h)$
and $I=f(100, h) ?$ Compare the behavior of these two
functions of $h .$

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Problem 3

Verify for the Cobb-Douglas production function
$$P(L, K)=1.01 L^{0.75} K^{025}$$
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 4

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 5

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 .
(a) What is the value of $f(40,15) ?$ What is its meaning?
(b) What is the meaning of the function $h=f(30, t) ?$ Describe
the behavior of this function.
(c) What is the meaning of the function $h=f(v, 30) ?$ Describe
the behavior of this function.

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Problem 6

Let $$f(x, y)=\ln (x+y-1)$$
(a) Evaluate $f(1,1), \quad$ (b) Evaluate $f(e, 1)$
(c) Find and sketch the domain of $f$ .
(d) Find the range of $f .$

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Problem 7

Let $$f(x, y)=x^{2} e^{3 x y}$$
(a) Evaluate $f(2,0) . \quad$ (b) Find the domain of $f$
(c) Find the range of $f .$

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Problem 8

Find and sketch the domain of the function
$f(x, y)=\sqrt{1+x-y^{2}} .$ What is the range of $f ?$

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Problem 9

Let $$f(x, y, z)=e^{\sqrt{1-x^{2}-y^{2}}}$$
(a) Evaluate $f(2,-1,6), \quad$ (b) Find the domain of $f$
(c) Find the range of $f$

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Problem 10

Let $$g(x, y, z)=\ln \left(25-x^{2}-y^{2}-z^{2}\right)$$
(a) Evaluate $g(2,-2,4)$. (b) Find the domain of $g$.
(c) Find the range of $g$.

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Problem 11

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

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Problem 12

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

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Problem 13

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 14

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

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Problem 15

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

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Problem 16

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

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Problem 17

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

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Problem 18

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

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Problem 19

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 20

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 21

Sketch the graph of the function.
$$f(x, y)=3$$

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Problem 22

Sketch the graph of the function.
$$f(x, y)=y$$

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Problem 23

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

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Problem 24

Sketch the graph of the function.
$$f(x, y)=\cos x$$

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Problem 25

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

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Problem 26

Sketch the graph of the function.
$$f(x, y)=3-x^{2}-y^{2}$$

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Problem 27

Sketch the graph of the function.
$$f(x, y)=4 x^{2}+y^{2}+1$$

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Problem 28

Sketch the graph of the function.
$$f(x, y)=\sqrt{16-x^{2}-16 y^{2}}$$

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Problem 29

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

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Problem 30

Match the function with its graph (labeled I-VI).Give reasons
(a) $f(x, y)=|x|+|y|$ (b) f(x, y)=|x y|$(c) f(x, y)=\frac{1}{1+x^{2}+y^{2}}$ (d) $f(x, y)=\left(x^{2}-y^{2}\right)^{2}$ (e) $f(x, y)=(x-y)^{2}$ (f) f(x, y)=\sin (|x|+|y|)$Check back soon! Problem 31 A contour map for a function$f$is shown. Use it to estimate the values of$f(-3,3)$and$f(3,-2) .$What can you say about the shape of the graph? Check back soon! Problem 32 Two contour maps are shown. One is for a function$f$whose graph is a cone. The other is for a function$g$whose graph is a paraboloid. Which is which, and why? Check back soon! Problem 33 Locate the points$A$and$B$in the map of Lonesome Mountain (Figure$12 ) .$How would you describe the terrain near$A$? Near$B$? Check back soon! Problem 34 Make a rough sketch of a contour map for the function whose graph is shown. Check back soon! Problem 35 A contour map of a function is shown. Use it to make a rough sketch of the graph of$f$. Check back soon! Problem 36 A contour map of a function is shown. Use it to make a rough sketch of the graph of$f$. Check back soon! Problem 37 A contour map of a function is shown. Use it to make a rough sketch of the graph of$f .$Check back soon! Problem 38 A contour map of a function is shown. Use it to make a rough sketch of the graph of$f .$Check back soon! Problem 39 Draw a contour map of the function showing several level curves. $$f(x, y)=(y-2 x)^{2}$$ Check back soon! Problem 40 Draw a contour map of the function showing several level curves. $$f(x, y)=x^{3}-y$$ Check back soon! Problem 41 Draw a contour map of the function showing several level curves. $$f(x, y)=x^{3}-y$$ Check back soon! Problem 42 Draw a contour map of the function showing several level curves. $$f(x, y)=e^{y / x}$$ Check back soon! Problem 43 Draw a contour map of the function showing several level curves. $$f(x, y)=y e^{x}$$ Check back soon! Problem 44 Draw a contour map of the function showing several level curves. $$f(x, y)=y \sec x$$ Check back soon! Problem 45 Draw a contour map of the function showing several level curves. $$f(x, y)=\sqrt{y^{2}-x^{2}}$$ Check back soon! Problem 46 Draw a contour map of the function showing several level curves. $$f(x, y)=y /\left(x^{2}+y^{2}\right)$$ Check back soon! Problem 47 Sketch both a contour map and a graph of the function and compare them. $$f(x, y)=x^{2}+9 y^{2}$$ Check back soon! Problem 48 Sketch both a contour map and a graph of the function and compare them. $$f(x, y)=\sqrt{36-9 x^{2}-4 y^{2}}$$ Check back soon! Problem 49 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 an isothermal the temperature is the same. Sketch some isothermals if the temperature function is given by $$T(x, y)=100 /\left(1+x^{2}+2 y^{2}\right)$$ Check back soon! Problem 50 If$V(x, y)$is the electric potential at a point$(x, y)$in the xy-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. Check back soon! Problem 51 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, 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. $$f(x, y)=e^{-x^{2}}+e^{-2 y^{2}}$$ Check back soon! Problem 52 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, 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. $$f(x, y)=\left(1-3 x^{2}+y^{2}\right) e^{1-x^{2}-y^{2}}$$ Check back soon! Problem 53 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, 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} \quad $(monkey saddle) Check back soon! Problem 54 Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.$f(x, y)=x y^{3}-y x^{3} \quad($dog saddle$)$Check back soon! Problem 55 Match the function (a) with its graph (labeled$A-F$on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices. $$z=\sin (x y)$$ Check back soon! Problem 56 Match the function (a) with its graph (labeled$A-F$on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices. $$z=e^{x} \cos y$$ Check back soon! Problem 56 Match the function (a) with its graph (labeled$A-F$on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices. $$z=e^{x} \cos y$$ Check back soon! Problem 57 Match the function (a) with its graph (labeled$A-F$on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices. $$z=\sin (x-y)$$ Check back soon! Problem 58 Match the function (a) with its graph (labeled$A-F$on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices. $$z=\sin x-\sin y$$ Check back soon! Problem 59 Match the function (a) with its graph (labeled$A-F$on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices. $$z=\left(1-x^{2}\right)\left(1-y^{2}\right)$$ Check back soon! Problem 60 Match the function (a) with its graph (labeled$A-F$on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices. $$z=\frac{x-y}{1+x^{2}+y^{2}}$$ Check back soon! Problem 61 Describe the level surfaces of the function. $$f(x, y, z)=x+3 y+5 z$$ Check back soon! Problem 62 Describe the level surfaces of the function. $$f(x, y, z)=x^{2}+3 y^{2}+5 z^{2}$$ Check back soon! Problem 63 Describe the level surfaces of the function. $$f(x, y, z)=x^{2}-y^{2}+z^{2}$$ Check back soon! Problem 64 Describe the level surfaces of the function. $$f(x, y, z)=x^{2}-y^{2}$$ Check back soon! Problem 65 Describe how the graph of$g$is obtained from the graph of$f .$$$g(x, y)=f(x, y)+2 \quad \text { (b) } g(x, y)=2 f(x, y)$$ $$g(x, y)=-f(x, y) \quad \text { (d) } g(x, y)=2-f(x, y)$$ Check back soon! Problem 66 Describe how the graph of$g$is obtained from the graph of$f .$(a)$g(x, y)=f(x-2, y) \quad$(b)$g(x, y)=f(x, y+2)$(c)$g(x, y)=f(x+3, y-4)$Check back soon! Problem 66 Describe how the graph of$g$is obtained from the graph of$f .$(a)$g(x, y)=f(x-2, y) \quad$(b)$g(x, y)=f(x, y+2)$(c)$g(x, y)=f(x+3, y-4)$Check back soon! Problem 67 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 maximum value? Can you identify any points on the graph that you might consider to be "local maximum points"? What about "local minimum points"? $$f(x, y)=3 x-x^{4}-4 y^{2}-10 x y$$ Check back soon! Problem 68 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 maximum value? Can you identify any points on the graph that you might consider to be "local maximum points"?What about "local minimum points"? $$f(x, y)=x y e^{-x^{2}-y^{2}}$$ Check back soon! Problem 69 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? $$f(x, y)=\frac{x+y}{x^{2}+y^{2}}$$ Check back soon! Problem 70 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? $$f(x, y)=\frac{x y}{x^{2}+y^{2}}$$ Check back soon! Problem 71 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$c ?$Check back soon! Problem 72 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$b ?$Check back soon! Problem 73 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. Check back soon! Problem 74 Graph the functions 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. $$f(x, y)=\sqrt{x^{2}+y^{2}} \quad f(x, y)=e^{\sqrt{x^{2}+y^{2}}}$$ $$f(x, y)=\ln \sqrt{x^{2}+y^{2}} \quad f(x, y)=\sin \left(\sqrt{x^{2}+y^{2}}\right)$$ 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$g ?$Check back soon! Problem 75 (a) Show that, by taking logarithms, the general Cobb- Douglas function$P=b L^{\alpha} K^{1-\alpha}$can be expressed as $$\ln \frac{P}{K}=\ln b+\alpha \ln \frac{L}{K}$$ (b) If we let$x=\ln (L / K)$and$y=\ln (P / K),$the equation in part (a) becomes the linear equation$y=\alpha X+\ln b .$Use Table 2 (in Example 3 ) to make a table of values of$\ln (L / K)$and$\ln (P / K)$for the years$1899-1922 .$Then use a graphing calculator or computer to find the least squares regression line through the points$(\ln (L / K), \ln (P / K))$(c) Deduce that the Cobb-Douglas production function is$P=1.01 L^{0.75} K^{0.5}\$