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 on page 889.

(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 $?" Then answer the question.

(c) Describe in words the meaning of the question "For what value of $ T $ is $ f(T, 20) = -49 $?" 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.

Jacquelyn T.

Numerade Educator

The $ \textit{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 $.

Alan G.

Numerade Educator

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

$$ P(L, K) = 1.47L^{0.65}K^{0.35} $$

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.

Carson M.

Numerade Educator

Verify for the Cobb-Douglas production function $$ P(L, K) = 1.01L^{0.75}K^{0.25} $$ 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 $$ P(L, K) = bL^{\alpha}K^{1 - \alpha} $$

Jun F.

Numerade Educator

A model for the surface area of a human body is given by the function $$ S = f(w, h) = 0.1091w^{0.425}h^{0.725} $$ where $ w $ is the weight (in pounds), $ h $ is the height (in inches), and $ S $ is measured in square feet.

(a) Find $ f(160, 70) $ and interpret it.

(b) What is your own surface area?

Carson M.

Numerade Educator

The wind-chill index $ W $ discussed in Example 2 has been modeled by the following function:

$$ W(T, v) = 13.12 + 0.6215T - 11.37v^{0.16} + 0.3965Tv^{0.16} $$

Check to see how closely this model agrees with the values in Table 1 for a few values of $ T $ and $ v $.

Alan G.

Numerade Educator

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.

Alan G.

Numerade Educator

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 $.

(a) Express the cost of making $ x $ small boxes, $ y $ medium boxes, and $ z $ large boxes as a function of three variables: $ C = f(x, y, z) $.

(b) Find $ f(3000, 5000, 4000) $ and interpret it.

(c) What is the domain of $ f $?

Alan G.

Numerade Educator

Let $ g(x, y) = \cos(x + 2y) $.

(a) Evaluate $ g(2, -1) $.

(b) Find the domain of $ g $.

(c) Find the range of $ g $.

Bobby B.

University of North Texas

Let $ F(x, y) = 1 + \sqrt{4 - y^2} $.

(a) Evaluate $ F(3, 1) $.

(b) Find and sketch the domain of $ F $.

(c) Find the range of $ F $.

Jacquelyn T.

Numerade Educator

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

(a) Evaluate $ f(1, 1, 1) $.

(b) Find and describe the domain of $ f $.

Alan G.

Numerade Educator

Let $ g(x, y, z) = x^3y^2z\sqrt{10 - x - y - z} $.

(a) Evaluate $ g(1, 2, 3) $.

(b) Find and describe the domain of $ g $.

Alan G.

Numerade Educator

Find and sketch the domain of the function.

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

Alan G.

Numerade Educator

Find and sketch the domain of the function.

$ f(x, y) = \sqrt[4]{x - 3y} $

Alan G.

Numerade Educator

Find and sketch the domain of the function.

$ f(x, y) = \ln (9 - x^2 - 9y^2) $

Alan G.

Numerade Educator

Find and sketch the domain of the function.

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

Alan G.

Numerade Educator

Find and sketch the domain of the function.

$ g(x, y) = \dfrac{x - y}{x + y} $

Alan G.

Numerade Educator

Find and sketch the domain of the function.

$ g(x, y) = \dfrac{\ln(2 - x)}{1 - x^2 - y^2} $

Alan G.

Numerade Educator

Find and sketch the domain of the function.

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

Alan G.

Numerade Educator

Find and sketch the domain of the function.

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

Jacquelyn T.

Numerade Educator

Find and sketch the domain of the function.

$ f(x, y, z) = \sqrt{4 - x^2} + \sqrt{9 - y^2} + \sqrt{1 - z^2} $

Alan G.

Numerade Educator

Find and sketch the domain of the function.

$ f(x, y, z) = \ln(16 - 4x^2 - 4y^2 - z^2) $

Jacquelyn T.

Numerade Educator

Sketch the graph of the function.

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

Carson M.

Numerade Educator

Sketch the graph of the function.

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

Carson M.

Numerade Educator

Sketch the graph of the function.

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

Carson M.

Numerade Educator

Match the function with its graph (labeled I - VI). Give reasons for your choices.

(a) $ f(x, y) = \dfrac{1}{1 + x^2 + y^2} $

(b) $ f(x, y) = \dfrac{1}{1 + x^2y^2} $

(c) $ f(x, y) = \ln(x^2 + y^2) $

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

(e) $ f(x, y) = | xy | $

(f) $ f(x, y) = \cos(xy) $

Alan G.

Numerade Educator

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?

Alan G.

Numerade Educator

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).

(a) Estimate the pressure at $ C $ (Chicago), $ N $ (Nashville), $ S $ (San Francisco), and $ V $ (Vancouver).

(b) At which of these locations were the winds strongest?

Alan G.

Numerade Educator

Level curves (isothermals) are shown for the typical water temperature (in $^{\circ} $C) in Long Lake (Minnesota) 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 m and on June 29 (day 180) at a depth of 5 m.

Alan G.

Numerade Educator

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?

Jacquelyn T.

Numerade Educator

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

Alan G.

Numerade Educator

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

Alan G.

Numerade Educator

The $ \textit{body mass index} $ (BMI) of a person is defined by $$ B(m, h) = \dfrac{m}{h^2} $$

where $ m $ is the person's mass (in kilograms) and $ h $ is the height (in meters). Draw the level curves $ B(m, h) = 18.5 $, $ B(m, h) = 25 $, $ B(m, h) = 30 $, and $ B(m, h) = 40 $. A rough guideline is that a person is underweight if the BMI is less than 18.5; optimal if the BMI lies between 18.5 and 25; overweight if the BMI lies between 25 and 30; and obese if the BMI exceeds 30. Shade the region corresponding to optimal BMI. Does someone who weighs 62 kg and is 152 cm tall fall into this category?

Alan G.

Numerade Educator

The body mass index is defined in Exercise 39. Draw the level curve of this function corresponding to someone who is 200 cm tall and weighs 80 kg. Find the weights and heights of two other people with that same level curve.

Carson M.

Numerade Educator

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

Alan G.

Numerade Educator

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

Alan G.

Numerade Educator

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

Carson M.

Numerade Educator

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

Alan G.

Numerade Educator

Draw a contour map of the function showing several level curves.

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

Alan G.

Numerade Educator

Draw a contour map of the function showing several level curves.

$ f(x, y) = xy $

Alan G.

Numerade Educator

Draw a contour map of the function showing several level curves.

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

Alan G.

Numerade Educator

Draw a contour map of the function showing several level curves.

$ f(x, y) = \ln(x^2 + 4y^2) $

Jacquelyn T.

Numerade Educator

Draw a contour map of the function showing several level curves.

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

Alan G.

Numerade Educator

Draw a contour map of the function showing several level curves.

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

Alan G.

Numerade Educator

Draw a contour map of the function showing several level curves.

$ f(x, y) = \sqrt[3]{x^2 + y^2} $

Alan G.

Numerade Educator

Draw a contour map of the function showing several level curves.

$ f(x, y) = y/(x^2 + y^2) $

Carson M.

Numerade Educator

Sketch both a contour map and a graph of the function and compare them.

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

Alan G.

Numerade Educator

Sketch both a contour map and a graph of the function and compare them.

$ f(x, y) = \sqrt{36 - 9x^2 - 4y^2} $

Alan G.

Numerade Educator

A thin metal plate, located in the $ xy $-plane, has temperature $ T(x, y) $ at the point $ (x, y) $. Sketch some level curves (isothermals) if the temperature function is given by $$ T(x, y) = \dfrac{100}{1 + x^2 + 2y^2} $$

Jacquelyn T.

Numerade Educator

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 $\textit{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.

Jacquelyn T.

Numerade Educator

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) = xy^2 - x^3 $ (monkey saddle)

Alan G.

Numerade Educator

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) = xy^3 - yx^3 $ (dog saddle)

Alan G.

Numerade Educator

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 + y^2)/3}(\sin(x^2) + \cos(y^2)) $

Alan G.

Numerade Educator

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

Alan G.

Numerade Educator

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(xy) $

Alan G.

Numerade Educator

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 $

Alan G.

Numerade Educator

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) $

Alan G.

Numerade Educator

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 $

Alan G.

Numerade Educator

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 = (1 - x^2)(1 - y^2) $

Alan G.

Numerade Educator

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 = \dfrac{x - y}{1 + x^2 + y^2} $

Alan G.

Numerade Educator

Describe the level surfaces of the function.

$ f(x, y, z) = x + 3y + 5z $

Jacquelyn T.

Numerade Educator

Describe the level surfaces of the function.

$ f(x, y, z) = x^2 + 3y^2 + 5z^2 $

Jacquelyn T.

Numerade Educator

Describe the level surfaces of the function.

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

Alan G.

Numerade Educator

Describe the level surfaces of the function.

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

Alan G.

Numerade Educator

Describe how the graph of $ g $ is obtained from the graph of $ f $.

(a) $ g(x, y) = f(x, y) + 2 $

(b) $ g(x, y) = 2 f(x, y) $

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

(d) $ g(x, y) = 2 - f(x, y) $

Alan G.

Numerade Educator

Describe how the graph of $ g $ is obtained from the graph of $ f $.

(a) $ g(x, y) = f(x - 2, y) $

(b) $ g(x, y) = f(x, y + 2) $

(c) $ g(x, y) = f(x + 3, y - 4) $

Alan G.

Numerade Educator

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) = 3x - x^4 - 4y^2 - 10xy $

Alan G.

Numerade Educator

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) = xye^{-x^2 - y^2} $

Alan G.

Numerade Educator

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) = \dfrac{x + y}{x^2 + y^2} $

Alan G.

Numerade Educator

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) = \dfrac{xy}{x^2 + y^2} $

Alan G.

Numerade Educator

Investigate the family of functions $ f(x, y) = e^{cx^2 + y^2} $. How does the shape of the graph depend on $ c $?

Alan G.

Numerade Educator

Use a computer to investigate the family of surfaces $$ z = \bigl(ax^2 + by^2\bigr) e^{-x^2 - y^2} $$

How does the shape of the graph depend on the numbers $ a $ and $ b $?

Alan G.

Numerade Educator

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

Alan G.

Numerade Educator

Graph the functions

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

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

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

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

and $$ f(x, y) = \dfrac{1}{\sqrt{x^2 + y^2}} $$

In general, if $ g $ is a function of one variable, how is the graph of $$ f(x, y) = g\biggl(\sqrt{x^2 + y^2}\biggr) $$

obtained from the graph of $ g $?

Alan G.

Numerade Educator

(a) Show that, by taking logarithms, the general Cobb-Douglas function $ P = bL^{\alpha}K^{1 - \alpha} $ can be expressed as $$ \ln \dfrac{P}{K} = \ln b + \alpha \ln \dfrac{L}{K} $$

(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 = 1.01L^{0.75}K^{0.25} $.

Jacquelyn T.

Numerade Educator