Chapter 14

Partial Derivatives

Educators

ag
JF

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 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.

ag
Alan G.
Numerade Educator

Problem 2

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 $.

ag
Alan G.
Numerade Educator

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

Problem 4

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 $$ P(L, K) = bL^{\alpha}K^{1 - \alpha} $$

JF
Jun F.
Numerade Educator

Problem 5

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

Problem 6

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 $.

ag
Alan G.
Numerade Educator

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.
(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.

ag
Alan G.
Numerade Educator

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 $.
(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 $?

ag
Alan G.
Numerade Educator

Problem 9

Let $ g(x, y) = \cos(x + 2y) $.
(a) Evaluate $ g(2, -1) $.
(b) Find the domain of $ g $.
(c) Find the range of $ g $.

ag
Alan G.
Numerade Educator

Problem 10

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 $.

ag
Alan G.
Numerade Educator

Problem 11

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 $.

ag
Alan G.
Numerade Educator

Problem 12

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 $.

ag
Alan G.
Numerade Educator

Problem 13

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

ag
Alan G.
Numerade Educator

Problem 14

Find and sketch the domain of the function.
$ f(x, y) = \sqrt[4]{x - 3y} $

ag
Alan G.
Numerade Educator

Problem 15

Find and sketch the domain of the function.
$ f(x, y) = \ln (9 - x^2 - 9y^2) $

ag
Alan G.
Numerade Educator

Problem 16

Find and sketch the domain of the function.
$ f(x, y) = \sqrt{x^2 + y^2 - 4} $

ag
Alan G.
Numerade Educator

Problem 17

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

ag
Alan G.
Numerade Educator

Problem 18

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

ag
Alan G.
Numerade Educator

Problem 19

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

ag
Alan G.
Numerade Educator

Problem 20

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

ag
Alan G.
Numerade Educator

Problem 21

Find and sketch the domain of the function.
$ f(x, y, z) = \sqrt{4 - x^2} + \sqrt{9 - y^2} + \sqrt{1 - z^2} $

ag
Alan G.
Numerade Educator

Problem 22

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

ag
Alan G.
Numerade Educator

Problem 23

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

Amrita B.
Numerade Educator

Problem 24

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

Amrita B.
Numerade Educator

Problem 25

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

Amrita B.
Numerade Educator

Problem 26

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

Amrita B.
Numerade Educator

Problem 27

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

Amrita B.
Numerade Educator

Problem 28

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

Carson M.
Numerade Educator

Problem 29

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

ag
Alan G.
Numerade Educator

Problem 30

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

Carson M.
Numerade Educator

Problem 31

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

Carson M.
Numerade Educator

Problem 32

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) $

ag
Alan G.
Numerade Educator

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?

ag
Alan G.
Numerade Educator

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).
(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?

ag
Alan G.
Numerade Educator

Problem 35

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.

ag
Alan G.
Numerade Educator

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?

ag
Alan G.
Numerade Educator

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 $?

ag
Alan G.
Numerade Educator

Problem 38

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

ag
Alan G.
Numerade Educator

Problem 39

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?

ag
Alan G.
Numerade Educator

Problem 40

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

Problem 41

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

ag
Alan G.
Numerade Educator

Problem 42

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

ag
Alan G.
Numerade Educator

Problem 43

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

Carson M.
Numerade Educator

Problem 44

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

ag
Alan G.
Numerade Educator

Problem 45

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

ag
Alan G.
Numerade Educator

Problem 46

Draw a contour map of the function showing several level curves.
$ f(x, y) = xy $

ag
Alan G.
Numerade Educator

Problem 47

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

ag
Alan G.
Numerade Educator

Problem 48

Draw a contour map of the function showing several level curves.
$ f(x, y) = \ln(x^2 + 4y^2) $

ag
Alan G.
Numerade Educator

Problem 49

Draw a contour map of the function showing several level curves.
$ f(x, y) = ye^x $

ag
Alan G.
Numerade Educator

Problem 50

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

ag
Alan G.
Numerade Educator

Problem 51

Draw a contour map of the function showing several level curves.
$ f(x, y) = \sqrt[3]{x^2 + y^2} $

ag
Alan G.
Numerade Educator

Problem 52

Draw a contour map of the function showing several level curves.
$ f(x, y) = y/(x^2 + y^2) $

Carson M.
Numerade Educator

Problem 53

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

ag
Alan G.
Numerade Educator

Problem 54

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

ag
Alan G.
Numerade Educator

Problem 55

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 $$ T(x, y) = \dfrac{100}{1 + x^2 + 2y^2} $$

ag
Alan G.
Numerade Educator

Problem 56

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.

ag
Alan G.
Numerade Educator

Problem 57

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)

ag
Alan G.
Numerade Educator

Problem 58

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)

ag
Alan G.
Numerade Educator

Problem 59

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)) $

ag
Alan G.
Numerade Educator

Problem 60

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) = \cos x \cos y $

ag
Alan G.
Numerade Educator

Problem 61

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) $

ag
Alan G.
Numerade Educator

Problem 62

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 $

ag
Alan G.
Numerade Educator

Problem 63

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) $

ag
Alan G.
Numerade Educator

Problem 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 $

ag
Alan G.
Numerade Educator

Problem 65

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) $

ag
Alan G.
Numerade Educator

Problem 66

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} $

ag
Alan G.
Numerade Educator

Problem 67

Describe the level surfaces of the function.

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

ag
Alan G.
Numerade Educator

Problem 68

Describe the level surfaces of the function.

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

ag
Alan G.
Numerade Educator

Problem 69

Describe the level surfaces of the function.

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

ag
Alan G.
Numerade Educator

Problem 70

Describe the level surfaces of the function.

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

ag
Alan G.
Numerade Educator

Problem 71

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) $

ag
Alan G.
Numerade Educator

Problem 72

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) $

ag
Alan G.
Numerade Educator

Problem 73

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 $

ag
Alan G.
Numerade Educator

Problem 74

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} $

ag
Alan G.
Numerade Educator

Problem 75

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} $

ag
Alan G.
Numerade Educator

Problem 76

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} $

ag
Alan G.
Numerade Educator

Problem 77

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

ag
Alan G.
Numerade Educator

Problem 78

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 $?

ag
Alan G.
Numerade Educator

Problem 79

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.

ag
Alan G.
Numerade Educator

Problem 80

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 $?

ag
Alan G.
Numerade Educator

Problem 81

(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} $.

ag
Alan G.
Numerade Educator