calculus of several variables - lamar university calculus/hoffman...502 chapter 7 calculus of...

Chapter� 1. Functions of Several

Variables

� 2. Partial Derivatives

� 3. OptimizingFunctions of TwoVariables

� 4. ConstrainedOptimization: TheMethod of LagrangeMultipliers

� 5. Double Integralsover RectangularRegions

� Chapter Summary andReview Problems

Calculus of Several

Variables

502 Chapter 7 Calculus of Several Variables

In business, if a manufacturer determines that x units of a particular commodity canbe sold domestically for $90 per unit, and y units can be sold to foreign markets for$110 per unit, then the total revenue obtained from all sales is given by

R � 90x � 110y

In psychology, a person’s intelligence quotient IQ is measured by the ratio

where a and m are the person’s actual age and mental age, respectively. A carpenterconstructing a storage box x feet long, y feet wide, and z feet deep knows that thebox will have volume V and surface area S, where

V � xyz and S � 2xy � 2xz � 2yz

These are typical of practical situations in which a quantity of interest dependson the values of two or more variables. Other examples include the volume of waterin a community’s reservoir, which may depend on the amount of rainfall as well asthe population of the community, and the output of a factory, which may depend onthe amount of capital invested in the plant, the size of the labor force, and the costof raw materials.

In this chapter, we will extend the methods of calculus to include functions oftwo or more independent variables. Most of our work will be with functions of twovariables, which you shall find can be represented geometrically by surfaces in three-dimensional space instead of curves in the plane. We begin with a definition and someterminology.

Domain Convention: Unless otherwise stated, we assume that the domainof f is the set of all (x, y) for which the expression f(x, y) is defined.

As in the case of a function of one variable, a function of two variables f(x, y)can be thought of as a “machine” in which there is a unique “output” f(x, y) for each“input” (x, y), as illustrated in Figure 7.1. The domain of f is the set of all possibleinputs, and the set of all possible corresponding outputs is the range of f. Functionsof three independent variables f(x, y, z) or four independent variables f(x, y, z, t), and so on can be defined in a similar fashion.

Function of Two Variables � A function f of the two independentvariables x and y is a rule that assigns to each ordered pair (x, y) in a given setD (the domain of f ) exactly one real number, denoted by f(x, y).

IQ �100m

a

Functions of SeveralVariables

1

Note

FIGURE 7.1 A function of two variables as a “machine.”

Suppose .

(a) Find the domain of f.(b) Compute f(1, �2).

Solution

(a) Since division by any real number except zero is possible, the expression f(x, y) canbe evaluated for all ordered pairs (x, y) with x � y � 0 or x � y. Geometrically,this is the set of all points in the xy-plane except for those on the line y � x.

(b) .

Suppose f(x, y) � xey � ln x.(a) Find the domain of f.(b) Compute f(e2, ln 2)

Solution

(a) Since xey is defined for all real numbers x and y and since ln x is defined only forx � 0, the domain of f consists of all ordered pairs (x, y) of real numbers forwhich x � 0.

(b) f(e2, ln 2) � e2eln 2 � ln e2 � 2e2 � 2 � 2(e2 � 1) � 16.78

f(1, �2) �3(1)2 � 5(�2)

1 � (�2)�

3 � 10

1 � 2� �

7

3

f(x, y) �3x2 � 5y

x � y

f machine

x inputx x x x

y inputy y y y

outputf(x, y)

Chapter 7 � Section 1 Functions of Several Variables 503

EXAMPLE 1 .1EXAMPLE 1 .1


E x p l o r e !E x p l o r e !Standard graphing calculators

usually only graph functions

of one independent variable,

namely, x. One method to dis-

play multivariable functions is

to graph one variable at differ-

ent values of the other variable.

For example, store the function

f(x, y) � x3 � x2y2 � xy3 � y4

into Y1 as X^3 � X^2*L1^2 �

X*L1^3 � L1^4, where L1

takes on the values {�1, 0, 1.5,

2, 2.25, 2.5}. Graph using the

window [�9.4, 9.4]1 by

[�150, 100]20. How does

varying the y values in L1 affect

the shape of the graph?

Given the function of three variables f(x, y, z) � xy � xz � yz, evaluate f(�1, 2, 5).

Solution

f(�1, 2, 5) � (�1)(2) � (�1)(5) � (2)(5) � 3

Here are two elementary applications of functions of two variables to business andeconomics.

A sports store in St. Louis carries two kinds of tennis rackets, the Venus Williams andthe Martina Hingis autograph brands. The consumer demand for each brand dependsnot only on its own price, but also on the price of the competing brand. Sales figuresindicate that if the Williams brand sells for x dollars per racket and the Hingis brandfor y dollars per racket, the demand for Williams rackets will be D1 � 300 � 20x �30y rackets per year and the demand for Hingis rackets will be D2 � 200 � 40x �10y rackets per year. Express the store’s total annual revenue from the sale of theserackets as a function of the prices x and y.

SolutionLet R denote the total monthly revenue. Then

R � (number of Williams rackets sold)(price per Williams racket) � (number of Hingis rackets sold)(price per Hingis racket)

Hence,

R(x, y) � (300 � 20x � 30y)(x) � (200 � 40x � 10y)(y)

� 300x � 200y � 70xy � 20x2 � 10y2

Output Q at a factory is often regarded as a function of the amount K of capitalinvestment and the size L of the labor force. Output functions of the form

Q(K, L) � AK�L�

where A, �, and � are positive constants, have proved to be especially useful in eco-nomic analysis and are known as Cobb-Douglas production functions.* Here is anexample involving such a function.

APPLICATIONS




* For instance, see Dominick Salvatore, Managerial Economics, McGraw-Hill, Inc., New York, 1989,pages 332–336.


Suppose that at a certain factory, output is given by the Cobb-Douglas productionfunction Q(K, L) � 60K1/3L2/3 units, where K is the capital investment measured inunits of $1,000 and L the size of the labor force measured in worker-hours.(a) Compute the output if the capital investment is $512,000 and 1,000 worker-hours

of labor are used.(b) Show that the output in part (a) will double if both the capital investment and the

size of the labor force are doubled.

Solution

(a) Evaluate Q(K, L) with K � 512 (thousand) and L � 1,000 to get

Q(512, 1,000) � 60(512)1/3(1,000)2/3

� 60(8)(100) � 48,000 units

(b) Evaluate Q(K, L) with K � 2(512) and L � 2(1,000) as follows to get

Q[2(512), 2(1,000)] � 60[2(512)]1/3[2(1,000)]2/3

� 60(2)1/3(512)1/3(2)2/3(1,000)2/3� 96,000 units

which is twice the output when K � 512 and L � 1,000.

The graph of a function of two variables f(x, y) is the set of all triples (x, y, z) suchthat (x, y) is in the domain of f and z � f(x, y). To “picture” such graphs, we need toconstruct a three-dimensional coordinate system. The first step in this constructionis to add a third coordinate axis (the z axis) perpendicular to the familiar xy coordi-nate plane, as shown in Figure 7.2. Note that the xy plane is taken to be horizontal,and the positive z axis is “up”.

FIGURE 7.2 A three-dimensional coordinate system.

y

z

x(2, –1, –3)

(2, –1, 0)(1, 2, 0)

(1, 2, 4)4

GRAPHS OF FUNCTIONSOF TWO VARIABLES

E x p l o r e !E x p l o r e !Refer to Example 1.5. Store the

equation

0 � Q � 60K^(1/3)*L^(2/3)

in the equation solver of the

graphing calculator. Calculate

Q for K � 512 and L � 1,000.

What happens to Q if the labor

force L is doubled?



You can describe the location of a point in three-dimensional space by specify-ing three coordinates. For example, the point that is 4 units above the xy plane andlies directly above the point with xy coordinates (x, y) � (1, 2) is represented by theordered triple (x, y, z) � (1, 2, 4). Similarly, the ordered triple (2, �1, �3) representsthe point that is 3 units directly below the point (2, �1) in the xy plane. These pointsare shown in Figure 7.2.

To graph a function f(x, y) of the two independent variables x and y, it is custom-ary to introduce the letter z to stand for the dependent variable and to write z � f(x, y)(Figure 7.3). The ordered pairs (x, y) in the domain of f are thought of as points in thexy plane, and the function f assigns a “height” z to each such point (“depth” if z is neg-ative). Thus, if f(1, 2) � 4, you would express this fact geometrically by plotting thepoint (1, 2, 4) in a three-dimensional coordinate space. The function may assign dif-ferent heights to different points in its domain, and in general, its graph will be a sur-face in three-dimensional space.

FIGURE 7.4 Several surfaces in three-dimensional space.

y

z

x

(b) A paraboloid z = x2 + y2

y

x

z

x

y

(d) A saddle surface z = y2 – x2(c) An ellipsoid z = √9 – 3x2 – 2y2

(a) A cone z = √x2 + y2

z

y

x

z

√3

3√22

3

x

y

z

(x, y, 0)

(x, y, f (x, y))

FIGURE 7.3 The graph of z �f (x, y).


Four such surfaces are shown in Figure 7.4. The surface in Figure 7.4a is a cone,Figure 7.4b shows a paraboloid, Figure 7.4c shows an ellipsoid, and Figure 7.4dshows what is commonly called a saddle surface. Surfaces such as these play animportant role in examples and exercises in this chapter.

It is usually not easy to sketch the graph of a function of two variables. One way tovisualize a surface is shown in Figure 7.5. Notice that when the plane z � C inter-sects the surface z � f(x, y), the result is a curve in space. The corresponding set ofpoints (x, y) in the xy plane that satisfy f(x, y) � C is called the level curve of f atC, and an entire family of level curves is generated as C varies over a set of num-bers. By sketching members of this family in the xy plane, you can obtain a usefulrepresentation of the surface z � f(x, y).

FIGURE 7.5 A level curve of the surface z � f (x, y).

For instance, imagine that the surface z � f(x, y) is a “mountain” whose “eleva-tion” at the point (x, y) is given by f(x, y), as shown in Figure 7.6a. The level curvef(x, y) � C lies directly below a path on the mountain where the elevation is alwaysC. To graph the mountain, you can indicate the paths of constant elevation by sketch-ing the family of level curves in the plane and pinning a “flag” to each curve to showthe elevation to which it corresponds (Figure 7.6b). This “flat” figure is called a topo-graphical map of the surface z � f(x, y).

x y

z

z = f (x, y)

f (x, y) = C

z = C

Level curve of f at C

LEVEL CURVES

FIGURE 7.6 (a) The surface z � f (x, y) as a mountain and (b) level curves provide a topo-graphical map of z � f (x, y).

Discuss the level curves of the function f(x, y) � x2 � y2.

SolutionThe level curve f(x, y) � C has the equation x2 � y2 � C. If C � 0, this is the point(0, 0), and if C � 0, it is a circle of radius . If C 0, there are no points thatsatisfy x2 � y2 � C.

The graph of the surface z � x2 � y2 is shown in Figure 7.7. The level curvesyou have just found correspond to cross sections perpendicular to the z axis. It canbe shown that cross sections perpendicular to the x axis and the y axis are parabolas.(Try to see why this is true.) For this reason, the surface is shaped like a bowl. It iscalled a circular paraboloid or a paraboloid of revolution.

�C

x

y

z

z = 1,500

z = 1,000

z = 300

z = 100

(a) (b)

300

1,000100

1,500



FIGURE 7.7 (a) Level curves of f(x, y) � x2 � y2 are the circles x2 � y2 � C and (b) the sur-face z � x2 � y2 is a bowl.

Level curves appear in many different applications. For instance, in economics, if theoutput Q(x, y) of a production process is determined by two inputs x and y (say, hoursof labor and capital investment), then the level curve Q(x, y) � C is called the curveof constant product C or, more briefly, an isoquant (“iso” means “equal”).

Another application of level curves in economics involves the concept of indif-ference curves. A consumer who is considering the purchase of a number of units ofeach of two commodities is associated with a utility function U(x, y), which mea-sures the total satisfaction (or utility) the consumer derives from having x units ofthe first commodity and y units of the second. A level curve U(x, y) � C of the util-ity function is called an indifference curve and gives all of the combinations of xand y that lead to the same level of consumer satisfaction. These terms are illustratedin Example 1.7.

Suppose the utility derived by a consumer from x units of one commodity and y unitsof a second commodity is given by the utility function U(x, y) � x3/2y. If the con-sumer currently owns x � 16 units of the first commodity and y � 20 units of thesecond, find the consumer’s current level of utility and sketch the corresponding indif-ference curve.

LEVEL CURVES INECONOMICS: ISOQUANTS

AND INDIFFERENCE CURVES

x

y

x

y

z

C = 9

C = 4

C = 1

(a) (b)

321

Plane z = C

(0, 0, 0)

x2 + y2 = C



SolutionThe current level of utility is

U(16, 20) � (16)3/2(20) � 1,280

and the corresponding indifference curve is

x3/2y � 1,280

or y � 1,280x�3/2. This curve consists of all points (x, y) where the level of utilityU(x, y) is 1,280. The curve x3/2y � 1,280 and several other curves of the family x3/2y � C are shown in Figure 7.8.

FIGURE 7.8 Indifference curves for the utility function U(x, y) � x3/2y.

In practical work in the social, managerial, or life sciences, you will rarely, if ever,have to graph a function of two variables. Hence, we will spend no more time devel-oping graphing procedures for such functions.

Computer software is now available for graphing functions of two variables. Suchsoftware often allows you to choose different scales along each coordinate axis andmay also enable you to visualize a given surface from different viewpoints. Thesefeatures permit you to obtain a detailed picture of the graph. A variety of computer-generated graphs are displayed in Figure 7.9 on page 511.

COMPUTER GRAPHICS

x

y

0

(16, 20)

x3/2 y = 1,280


E x p l o r e !E x p l o r e !Refer to Example 1.7. Repre-

sent the indifference curves

U(x, y) � C � x3/2y by storing y

� L1*x^(3/2) into Y1, where the

list L1 assumes the following

values for C: L1 � {800, 1,280,

2,000, 3,000}. Use the viewing

window [0, 30]5 by [0, 150]10.

What effect does the change in

the C constant have on the

graph? Locate the point (16, 20)

for which U � 1,280.


FIGURE 7.9 Some computer-generated surfaces.

Three views of the surface z � �xye� 1�2 (x2�y2)

Hyperboloid ofone sheet

x2 � y2 � 0.2z2 � 1Hyperboloid of

two sheets�10x2 � 10y2 � 5z2 � 1

Ellipsoid3x2 � y2 � z2 � 1

z � �e�3(x2�y2)

z � x4 � y4 � 2.3(x2 � y2)

z � (0.8x2 � y2)e(1�1.4x2�y2)


In Problems 1 through 12, compute the indicated function value.

1. f(x, y) � (x � 1)2 � 2xy3; f(2, �1), f(1, 2)

2. f(x, y) � ; f(1, 2), f(�4, 6)

3. g(x, y) � ; g(4, 5), g(�1, 2)

4. g(u, v) � 10u1/2v2/3; g(16, 27), g(4, �1331)

5. f(r, s) � ; f(e2, 3), f(ln 9, e3)

6. f(x, y) � xyexy; f(1, ln 2), f(ln 3, ln 4)

7. g(x, y) � ; g(1, 2), g(2, �3)

8. f(s, t) � ; f(1, 0), f(ln 2, 2)

9. f(x, y, z) � xyz; f(1, 2, 3), f(3, 2, 1)

10. g(x, y, z) � (x � y)eyz; g(1, 0, �1), g(1, 1, 2)

11. F(r, s, t) � ; F(1, 1, 1), F(0, e2, 3e2)

12. f(x, y, z) � xyez � xzey � yzex; f(1, 1, 1), f(ln 2, ln 3, ln 4)

In Problems 13 through 18, describe the domain of the given function.

13. f(x, y) � 14. f(x, y) �

15. f(x, y) � 16. f(x, y) �

17. f(x, y) � ln(x � y � 4) 18. f(x, y) �

In Problems 19 through 26, sketch the indicated level curve f(x, y) � C for eachchoice of constant C.

19. f(x, y) � x � 2y; C � 1, C � 2, C � �3

20. f(x, y) � x2 � y; C � 0, C � 4, C � 9

21. f(x, y) � x2 � 4x � y; C � �4, C � 5

22. f(x, y) � ; C � �2, C � 2x

y

exy

�x � 2y

x

ln(x � y)�x2 � y

�9 � x2 � y25x � 2y

4x � 3y

ln (r � t)

r � s � t

est

2 � est

y

x�

x

y

s

ln r

�y2 � x2

3x � 2y

2x � 3y

P . R . O . B . L . E . M . S 7.1P . R . O . B . L . E . M . S 7.1

23. f(x, y) � xy; C � 1, C � �1, C � 2, C � �2

24. f(x, y) � yex; C � 0, C � 1

25. f(x, y) � xey; C � 1, C � e

26. f(x, y) � ln (x2 � y2); C � 4, C � ln 4

27. Using x skilled workers and y unskilled workers, a manufacturer can produce Q(x, y) � 10x2y units per day. Currently there are 20 skilled workers and 40unskilled workers on the job.(a) How many units are currently being produced each day?(b) By how much will the daily production level change if 1 more skilled worker

is added to the current workforce?(c) By how much will the daily production level change if 1 more unskilled

worker is added to the current workforce?(d) By how much will the daily production level change if 1 more skilled worker

and 1 more unskilled worker are added to the current workforce?

28. A manufacturer can produce scientific graphing calculators at a cost of $40 apieceand business calculators for $20 apiece.(a) Express the manufacturer’s total monthly production cost as a function of the

number of graphing calculators and the number of business calculators produced.(b) Compute the total monthly cost if 500 scientific and 800 business calculators

are produced.(c) The manufacturer wants to increase the output of scientific calculators by 50

a month from the level in part (b). What corresponding change should be madein the monthly output of business calculators so the total monthly cost willnot change?

29. A paint store carries two brands of latex paint. Sales figures indicate that if the firstbrand is sold for x1 dollars per gallon and the second for x2 dollars per gallon, thedemand for the first brand will be D1(x1, x2) � 200 � 10x1 � 20x2 gallons per monthand the demand for the second brand will be D2(x1, x2) � 100 � 5x1 � 10x2 gallonsper month.(a) Express the paint store’s total monthly revenue from the sale of the paint as

a function of the prices x1 and x2.(b) Compute the revenue in part (a) if the first brand is sold for $6 per gallon and

the second for $5 per gallon.

30. The output at a certain factory is Q(K, L) � 120K2/3L1/3 units, where K is the capitalinvestment measured in units of $1,000 and L the size of the labor force measured inworker-hours.(a) Compute the output if the capital investment is $125,000 and the size of the

labor force is 1,331 worker-hours.(b) What will happen to the output in part (a) if both the level of capital invest-

ment and the size of the labor force are cut in half?

PRODUCTION

RETAIL SALES

PRODUCTION COST

PRODUCTION



31. The Easy-Gro agricultural company estimates that when 100x worker-hours of labor are employed on y acres of land, the number of bushels of wheat produced isf(x, y) � Axayb, where A, a, and b are positive constants. Suppose the companydecides to double the production factors x and y. Determine how this decisionaffects the production of wheat in each of these cases:(a) a � b � 1(b) a � b 1(c) a � b � 1

32. Suppose that when x machines and y worker-hours are used each day, a certainfactory will produce Q(x, y) � 10xy cell phones. Describe the relationship betweenthe inputs x and y that results in an output of 1,000 phones each day. (Note that youare finding a level curve of Q.)

33. A manufacturer with exclusive rights to a sophisticated new industrial machine isplanning to sell a limited number of the machines to both foreign and domesticfirms. The price the manufacturer can expect to receive for the machines will dependon the number of machines made available. It is estimated that if the manufacturersupplies x machines to the domestic market and y machines to the foreign market,

the machines will sell for 60 � thousand dollars apiece at home and 50 �

thousand dollars apiece abroad. Express the manufacturer’s revenue R as a

function of x and y.

34. A manufacturer is planning to sell a new product at the price of A dollars per unit andestimates that if x thousand dollars is spent on development and y thousand dollars

on promotion, consumers will buy approximately units of the product.

If manufacturing costs are $50 per unit, express the total profit as a function of x and y.

35. Pediatricians and medical researchers sometimes use the following empiricalformula* relating the surface area S (m2) of a person to the person’s weight W (kg)and height H (cm):

(a) Find S(15.83, 87.11). Sketch the level curve of S(W, H) that passes throughthe point (15.83, 87.11). Sketch several additional level curves of S(W, H).What do these level curves represent?

(b) If Marc weighs 18.37 kg and has surface area 0.648 m2, approximately howtall would you expect him to be?

S(W, H) � 0.0072W 0.425H 0.725

SURFACE AREA OF THE HUMAN BODY

320y

y � 2�

160x

x � 4

RETAIL SALES

y

10�

x

20

x

5�

y

20

RETAIL SALES

PRODUCTION

PRODUCTION

* J. Routh, Mathematical Preparation for Laboratory Technicians, Saunders Co., Philadelphia, PA, 1971,page 92.


(c) Suppose at some time in her life, Jenny weighs twice as much and is threetimes as tall as she was at birth. What is the corresponding percentage changein the surface area of her body?

(d) Ask your parents what your birth weight and length were (they will know!).Then buy or borrow a doll that is approximately the same length you were atbirth, and measure the doll’s surface area. Does the empirical formula accu-rately predict the result you obtained? Write a paragraph on any conclusionsyou draw from this “experiment.”

36. A person’s intelligence quotient (IQ) is measured by the function

I(m, a) �

where a is the person’s actual age and m is his or her mental age.(a) Find I(12, 11) and I(16, 17).(b) Sketch the graphs of several level curves of I(m, a). How would you describe

these curves?

37. Using x skilled and y unskilled workers, a manufacturer can produce Q(x, y) � 3x � 2yunits per day. Currently the workforce consists of 10 skilled workers and 20unskilled workers.(a) Compute the current daily output.(b) Find an equation relating the levels of skilled and unskilled labor if the daily

output is to remain at its current level.(c) On a two-dimensional coordinate system, draw the isoquant (constant pro-

duction curve) that corresponds to the current level of output.(d) What change should be made in the level of unskilled labor y to offset an

increase in skilled labor x of two workers so that the output will remain at itscurrent level?

38. Suppose the utility derived by a consumer from x units of one commodity and yunits of a second commodity is given by the utility function U(x, y) � 2x3y2. Theconsumer currently owns x � 5 units of the first commodity and y � 4 units of thesecond. Find the consumer’s current level of utility and sketch the correspondingindifference curve.

39. The utility derived by a consumer from x units of one commodity and y units of asecond is given by the utility function U(x, y) � (x � 1)(y � 2). The consumercurrently owns x � 25 units of the first commodity and y � 8 units of the second.Find the consumer’s current level of utility and sketch the correspondingindifference curve.

40. Dumping and other materials-handling operations near a landfill may result incontaminated particles being emitted into the surrounding air. To estimate such

AIR POLLUTION

INDIFFERENCE CURVES

INDIFFERENCE CURVES

CONSTANT PRODUCTION CURVES

100m

a

PSYCHOLOGY

particulate emission, the following empirical formula* can be used:

where E is the emission factor (pounds of particles released into the air per tonof soil moved), V is the mean speed of the wind (mph), M is the moisture con-tent of the material (given as a percentage), and k is a constant that depends onthe size of the particles.(a) For a small particle (diameter 5 mm), it turns out that k � 0.2. Find E(10, 13).(b) The emission factor E can be multiplied by the number of tons of material

handled to achieve a measure of total emissions. Suppose 19 tons of the mate-rial in part (a) is handled. How many tons of a second kind of material withk � 0.48 (diameter 15 mm) and moisture content 27% must be handled toachieve the same level of total emissions if the wind velocity stays the same?

(c) Sketch several level curves of E(V, M), assuming the size of the particle staysfixed. What is represented by these curves?

41. One of Poiseuille’s laws† says that the velocity in a blood vessel at a distance r (cm)from the axis of the vessel is given by

where P (dynes/cm2) is the pressure in the vessel, R (cm) is the radius of the ves-sel, and L (cm) is its length.(a) Suppose R � 0.0075 cm. Find V(3875, 1.675, 0.004).(b) If r � 0.5R, V becomes a function of just P and L. Sketch several level curves

of V(P, L).

42. Suppose output Q is given by the Cobb-Douglas production function Q(K, L) �AK�L1��, where A and � are positive constants and 0 � 1. Prove that if K andL are both multiplied by the same positive number m, then the output Q will also bemultiplied by m; that is, show that Q(mK, mL) � mQ(K, L).

43. Van der Waal’s equation of state says that 1 mole of a confined gas satisfies theequation

where T (°C) is the temperature of the gas, V (cm3) is its volume, P (atm) is the

T(P, V) � 0.0122�P �a

V2�(V � b) � 273.15

CHEMISTRY

CONSTANT RETURNS TO SCALE

V(P, L, r) �9.3P

L(R2 � r2)

FLOW OF BLOOD

E(V, M) � k(0.0032)�V

5�1.3

�M

2 ��1.4


* M. D. LaGrega, P. L. Buckingham, and J. C. Evans, Hazardous Waste Management, McGraw-Hill,Inc., New York, 1994, page 140.

† E. Batschelet, Introduction to Mathematics for Life Scientists, 2nd ed., Springer-Verlag, New York,1976, pages 102–103.

pressure of the gas on the walls of its container, and a and b are constants thatdepend on the nature of the gas.(a) Sketch the graphs of several level curves of T. These curves are called curves

of constant temperature or isotherms.(b) If the confined gas is chlorine, experiments show that a � 6.49 106 and

b � 56.2. Find T(1.13, 31.275 103); that is, the temperature that correspondsto 31,275 cm3 of chlorine under 1.13 atmospheres of pressure.

44. The level curves in the land areas of the accompanying figure indicate ice elevationsabove sea level (in meters) during the last major ice age (approximately 18,000years ago). The level curves in the sea areas indicate sea surface temperature. Forinstance, the ice was 3,000 meters thick above the Great Lakes, and the seatemperature near the Hawaiian Islands was about 24°C. Where on earth was the icepack the thickest? What ice-bound land area was adjacent to the warmest sea?

Source: The Cambridge Encyclopedia of Earth Sciences, Crown/Cambridge Press, 1981, page 302.

ICE AGE PATTERNS OF ICEAND TEMPERATURE



45. A machine based on wind energy generally converts the kinetic energy of moving air to mechanical energy by means of a device such as a rotating shaft, as in a windmill. Suppose we have wind of velocity v traveling through a wind-collecting machine with cross-sectional area A. Then, in physics it is shown that the total power generated by the wind is given by a formula* of the form

where b � 1.2 kg/m3 is the density of the air, and a is a positive constant.

(a) Ideally, a � in the formula for P(v, A). How much power would be produced

by an ideal windmill with blade radius 15 meters if the wind speed is 22 m/sec?(b) No wind machine is perfectly efficient. In fact, it has been shown that the best

we can hope for is about 59% of ideal efficiency, and a good empirical formula

for power is obtained by taking a � . Compute P(v, A) using this value of

a if the blade radius of the windmill in part (a) is doubled and the wind veloc-ity is halved.

(c) Mankind has been trying to harness the wind for at least 4,000 years, some-times with interesting or bizarre consequences. For example, in the sourcequoted in the footnote, the author notes that during World War II, a windmillwas built in Vermont with a blade radius of 175 feet! Read an article on wind-mills and other devices using wind power. Do you feel these devices have anyplace in the modern technological world? Explain.

46. In manufacturing semiconductors, it is necessary to use water with an extremely lowmineral content, and to separate water from contaminants, it is common practice touse a membrane process called reverse osmosis. A key to the effectiveness of sucha process is the osmotic pressure, which may be determined by the van’t Hoffequation.†

P(N, C, T ) � 0.075NC(273.15 � T )

where P is the osmotic pressure (in atmosphere), N is the number of ions in eachmolecule of solute, C is the concentration of solute (gram-mole/liter), and T is thetemperature of the solute (°C). Find the osmotic pressure for a sodium chloridebrine solution with concentration 0.53 gm-mole/liter at a temperature of 23°C.(You will need to know that each molecule of sodium chloride contains two ions:NaCl � Na� � Cl�.)

47. Output in a certain factory is given by the Cobb-Douglas production function Q(K, L) � 57K1/4L3/4, where K is the capital in $1,000 and L is the size of thelabor force, measured in worker-hours.

REVERSE OSMOSIS

8

27

1

2

P(v, A) � abAv3

WIND POWER

* Raymond A. Serway, Physics, 3rd ed., Saunders, Philadelphia, PA, 1992, pages 408–410.† M. D. LaGrega, P. L. Buckingham, and J. C. Evans, Hazardous Waste Management, McGraw-Hill,

Inc., New York, 1994, pages 530–543.

Chapter 7 � Section 2 Partial Derivatives 519

(a) Store the production function as 57K^(1 � 4)*L^(3 � 4). Store 277 as K and743 as L. Evaluate to obtain Q(277, 743); that is, the output when K is$277,000 and L is 743 worker-hours. Repeat to obtain Q(K, L) for the valuesof K and L in the following table:

K($1,000) 277 311 493 554 718

L 743 823 1,221 1,486 3,197

Q(K, L)

(b) Note from the next to last column that the output Q(277, 743) is doubled whenK is doubled from 277 to 554 and L is doubled from 743 to 1,486. In a sim-ilar manner, verify that output is tripled when K and L are both tripled, andthat output is halved when K and L are both halved. Does anything interest-ing happen if K is doubled and L is halved? Verify your response with yourcalculator.

48. Repeat Problem 47 with the production function

Q(K, L) � 83K2/5L3/5

In many problems involving functions of two variables, the goal is to find the rate ofchange of the function with respect to one of its variables when the other is held con-stant. That is, the goal is to differentiate the function with respect to the particularvariable in question while keeping the other variable fixed. This process is known aspartial differentiation, and the resulting derivative is said to be a partial derivativeof the function.

For example, suppose a manufacturer finds that

Q(x, y) � 5x2 � 7xy

units of a certain commodity will be produced when x skilled workers and y unskilledworkers are employed. Then if the number of unskilled workers remains fixed, theproduction rate with respect to the number of skilled workers is found by differenti-ating Q(x, y) with respect to x while holding y constant. We call this the partial deriv-ative of Q with respect to x and denote it by Qx(x, y); thus,

Qx(x, y) � 5(2x) � 7(1)y � 10x � 7y

Similarly, if the number of skilled workers remains fixed, the production rate withrespect to the number of unskilled workers is given by the partial derivative of Qwith respect to y, which is obtained by differentiating Q(x, y) with respect to y,holding x constant; that is, by

Qy(x, y) � (0) � 7x(1) � 7x

Here is a general definition of partial derivatives and some alternative notation.

PartialDerivatives

2

calculus of several variables - lamar university calculus/hoffman...502 chapter 7 calculus of...

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