244 chapter 3 additional applications of the derivative calculus... · 2010. 8. 24. · 248 chapter...

(b) The critical volume Vc is the volume for which P�(Vc) � 0 and P�(Vc) � 0.Show that Vc � 3b.

(c) Find the critical pressure Pc � P(Vc) and then Tc in terms of a, b, n, and R.

58. Evaluate the limit

for constants a0, a1, . . . , an and b0, b1, . . . , bm in each of the following cases:(a) n � m(b) n � m(c) n � m (Note: There are two possible answers, depending on the signs of an

and bm.)

59. Let f(x) � x1/3(x � 4).(a) Find f �(x) and determine intervals of increase and decrease for f(x). Locate all

relative extrema on the graph of f(x).(b) Find f �(x) and determine intervals of concavity for f(x). Find all inflection

points on the graph of f(x).(c) Find all intercepts for the graph of f(x). Does the graph have any asymptotes?(d) Sketch the graph of f(x).

60. Repeat Problem 59 for the function f(x) � x2/3(2x � 5).

61. Repeat Problem 59 for the function

62. Let f(x) � and let g(x) � .

(a) Use a graphing utility to sketch the graph of f(x). What happens at x � 1?(b) Sketch the graph of g(x). Now what happens at x � 1?

You have already seen several situations where the methods of calculus were used todetermine the largest or smallest value of a function of interest (for example, maxi-mum profit or minimum cost). In most such optimization problems, the goal is to findthe absolute maximum or absolute minimum of a particular function on some rele-vant interval. The absolute maximum of a function on an interval is the largest valueof the function on that interval and the absolute minimum is the smallest value. Hereis a definition of absolute extrema.

x � 1.01

x2 � 1

x � 1

x2 � 1

f(x) �x � 9.4

25 � 1.1x � x2

limxfi ��

anxn � an�1xn�1 � . . . � a1x � a0

bmxm � bm�1xm�1 � . . . � b1x � b0

244 Chapter 3 Additional Applications of the Derivative

Optimization

4

Absolute extrema often coincide with relative extrema but not always. For exam-ple, in Figure 3.28 the absolute maximum and relative maximum on the interval a x b are the same, but the absolute minimum occurs at the left endpoint, x � a.

In this section, you will learn how to find absolute extrema of functions on inter-vals. We begin by considering intervals a x b that are “closed” in the sense theyinclude both endpoints, a and b. It can be shown that a function continuous on suchan interval has both an absolute maximum and an absolute minimum on the interval.Moreover, each absolute extremum can occur either at an endpoint of the interval (ata or b) or at a critical number c between a and b (that is, a � c � b) (Figure 3.29).

The Extreme Value Property � A function f(x) that is continuouson the closed interval a x b attains its absolute extrema on the intervaleither at an endpoint of the interval (a or b) or at a critical number c such thata � c � b.

FIGURE 3.28 Absolute extrema.

x

y

a b

Absolutemaximum

Absoluteminimum

Absolute Maxima and Minima of a Function � Let f be afunction defined on an interval I that contains the number c. Then

f(c) is the absolute maximum of f on I if f(c) f(x) for all x in If(c) is the absolute minimum of f on I if f(c) f(x) for all x in I

Collectively, absolute maxima and minima are called absolute extrema.

Chapter 3 � Section 4 Optimization 245

E x p l o r e !E x p l o r e !Use a graphing calculator to

graph f(x) � with the

modified decimal window [0,

4.7]1 by [0, 60]5. Trace or use

other utility methods to find the

absolute maximum and ab-

solute minimum of f(x) over the

interval [1, 3].

x3

�(x � 2)

Thanks to the extreme value property, you can find the absolute extrema of a con-tinuous function on a closed interval a x b by using the following straightfor-ward procedure.

How to Find the Absolute Extrema of a ContinuousFunction f on a Closed Interval a # x # b

Step 1. Find the x coordinates of all of the first-order critical points of f in theinterval a x b.

Step 2. Compute f(x) at these critical values and at the endpoints x � a and x � b.

Step 3. Select the largest and smallest values of f(x) obtained in step 2. Theseare the absolute maximum and absolute minimum, respectively.

FIGURE 3.29 Absolute extrema of a continuous function on a closed interval: (a) the absolutemaximum coincides with a relative maximum, (b) the absolute maximum occurs at an endpoint,(c) the absolute minimum coincides with a relative minimum, and (d) the absolute minimum oc-curs at an endpoint.

x

y

x

y

x

y

x

y

a b a b

a ba b

(a) (b)

(c) (d)

max

min

max

min

max

min

max

min


The procedure is illustrated in the following examples.

Find the absolute maximum and absolute minimum of the function

f(x) � 2x3 � 3x2 � 12x � 7

on the interval �3 x 0.

SolutionFrom the derivative

f�(x) � 6x2 � 6x � 12 � 6(x � 2)(x � 1)

you see that the first-order critical points occur when x � �2 and x � 1. Of these,only x � �2 lies in the interval �3 x 0. Compute f(x) at x � �2 and at theendpoints x � �3 and x � 0.

f(�2) � 13 f(�3) � 2 f(0) � �7

Compare these values to conclude that the absolute maximum of f on the interval �3 x 0 is f(�2) � 13 and the absolute minimum is f(0) � �7.

Notice that you did not have to classify the critical points or draw the graph tolocate the absolute extrema. The sketch in Figure 3.30 is presented only for the sakeof illustration.

For several weeks, the highway department has been recording the speed of freewaytraffic flowing past a certain downtown exit. The data suggest that between 1:00 and6:00 P.M. on a normal weekday, the speed of the traffic at the exit is approximatelyS(t) � t3 � 10.5t2 � 30t � 20 miles per hour, where t is the number of hours pastnoon. At what time between 1:00 and 6:00 P.M. is the traffic moving the fastest, andat what time is it moving the slowest?

SolutionThe goal is to find the absolute maximum and absolute minimum of the function S(t)on the interval 1 t 6. From the derivative

S�(t) � 3t2 � 21t � 30 � 3(t2 � 7t � 10) � 3(t � 2)(t � 5)

you get the t coordinates t � 2 and t � 5 of the first-order critical points, both ofwhich lie in the interval 1 t 6.


EXAMPLE 4 .1EXAMPLE 4 .1


y

x

(–2, 13)

Absolute maximum

Absoluteminimum

(–3, 2)

–3 0

(0, –7)

FIGURE 3.30 The absoluteextrema of y � 2x3 � 3x2 �12x � 7 on �3 x 0.

E x p l o r e !E x p l o r e !Refer to Example 4.2. Due to an

increase in the speed limit the

speed past the certain exit is

now S1(t) � t3 � 10.5t2 �

30t � 25. Graph both S1(t) and

S(t) using the window [0, 6]1

by [20, 60]5. At what time be-

tween 1 P.M. and 6 P.M. is the

maximum speed achieved using

S1(t)? At what time is the mini-

mum speed attained?

Compute S(t) for these values of t and at the endpoints t � 1 and t � 6 to get

S(1) � 40.5 S(2) � 46 S(5) � 32.5 S(6) � 38

Since the largest of these values is S(2) � 46 and the smallest is S(5) � 32.5, youcan conclude that the traffic is moving fastest at 2:00 P.M., when its speed is 46 milesper hour, and slowest at 5:00 P.M., when its speed is 32.5 miles per hour. For refer-ence, the graph of S is sketched in Figure 3.31.

When you cough, the radius of your trachea (windpipe) decreases, affecting the speedof the air in the trachea. If r0 is the normal radius of the trachea, the relationshipbetween the speed S of the air and the radius r of the trachea during a cough is givenby a function of the form S(r) � ar2(r0 � r), where a is a positive constant.* Findthe radius r for which the speed of the air is greatest.

FIGURE 3.31 Traffic speed S(t) � t3 � 10.5t2 � 30t � 20.

S(t)

50

45

40

35

30

(1, 40.5)

Maximum speed (2, 46)

(6, 38)

(5, 32.5)Minimum

speed

t1 2 3 4 5 6


* Philip M. Tuchinsky, “The Human Cough,” UMAP Modules 1976: Tools for Teaching, Consortium forMathematics and Its Application, Inc., Lexington, MA, 1977.


SolutionThe radius r of the contracted trachea cannot be greater than the normal radius r0 orless than zero. Hence, the goal is to find the absolute maximum of S(r) on the inter-val 0 r r0.

First differentiate S(r) with respect to r using the product rule and factor the deriv-ative as follows (note that a and r0 are constants):

S�(r) � �ar2 � (r0 � r)(2ar) � ar[�r � 2(r0 � r)] � ar(2r0 � 3r)

Then set the factored derivative equal to zero and solve to get the r coordinates ofthe first-order critical points:

ar(2r0 � 3r) � 0

r � 0 or r � r0

Both of these values of r lie in the interval 0 r r0, and one is actually an end-point of the interval. Compute S(r) for these two values of r and for the other end-point r � r0 to get

S(0) � 0 S(r0) � 0

Compare these values and conclude that the speed of the air is greatest when the

radius of the contracted trachea is r0; that is, when it is two thirds the radius of the

uncontracted trachea.

A graph of the function S(r) is given in Figure 3.32. Note that the r intercepts ofthe graph are obvious from the factored function S(r) � ar2(r0 � r). Notice also thatthe graph has a horizontal tangent when r � 0, reflecting the fact that S�(0) � 0.

When the interval on which you wish to maximize or minimize a continuous func-tion is not of the form a x b, the procedure illustrated in the preceding exam-ples no longer applies. This is because there is no longer any guarantee that the func-tion actually has an absolute maximum or minimum on the interval in question. Onthe other hand, if an absolute extremum does exist and the function is continuous onthe interval, the absolute extremum will still occur at a relative extremum or endpointcontained in the interval. Two of the possibilities for functions on unbounded inter-vals are illustrated in Figure 3.33.

To find the absolute extrema of a continuous function on an interval that is notof the form a x b, you still evaluate the function at all the critical points andendpoints that are contained in the interval. However, before you can draw any finalconclusions, you must find out if the function actually has relative extrema on the

MORE GENERAL OPTIMIZATION

2

3

S�2

3r0� �

4a

27r3

0

2

3


S(r)Greatest

speed

r

4a27

r03

13

r023

r0r0

FIGURE 3.32 The speed of airduring a cough S(r) � ar2(r0 � r).

interval. One way to do this is to use the first derivative to determine where the func-tion is increasing and where it is decreasing and then to sketch the graph. The techniqueis illustrated in the next example.

Find the absolute maximum and absolute minimum (if any) of the function f(x) �

x2 � on the interval x � 0.

SolutionThe function is continuous on the interval x � 0 since its only discontinuity occursat x � 0. The derivative is

f �(x) � 2x �

which is zero when

x3 � 8 � 0 x3 � 8 or x � 2

Since f �(x) � 0 for 0 � x � 2 and f �(x) � 0 for x � 2, the graph of f is decreas-ing for 0 � x � 2 and increasing for x � 2, as shown in Figure 3.34. It follows that

f(2) � 22 � � 12

is the absolute minimum of f on the interval x � 0 and that there is no absolutemaximum.

16

2

16

x2 �2x3 � 16

x2 �2(x3 � 8)

x2

16

x

FIGURE 3.33 Extrema of functions on unbounded intervals: (a) no absolute maximum for x � 0and (b) no absolute minimum for x 0.

x x

y y

Absoluteminimum

Absolutemaximum

(a) (b)



x

y

Absolute minimum

12

2

FIGURE 3.34 The function f(x) �

x2 � on the interval x � 0.16x

The procedure illustrated in Example 4.4 can be used whenever we wish to findthe largest or smallest value of a function f that is continuous on an interval I onwhich it has exactly one critical number c. In particular, if this condition is satisfiedand f(x) has a relative maximum (minimum) at x � c, it also has an absolute maxi-mum (minimum) there. To see why, suppose the graph has a relative maximum at x � c. Then the graph is always rising before c and always falling after c, since tochange direction would require the presence of a second critical point (see Figure3.35). Thus, the relative maximum is also an absolute maximum. These observationssuggest that any test for relative extrema becomes a test for absolute extrema in thisspecial case. Here is a statement of the second derivative test for absolute extrema.

The following example illustrates how the second derivative test for absoluteextrema can be used in practice.

A manufacturer estimates that when q units of a particular commodity are producedeach month, the total cost will be C(q) � 0.4q2 � 3q � 40 dollars, and all q unitscan be sold at a price of p(q) � 0.2(45 � 0.5q) dollars per unit.(a) Determine the level of production that results in maximum profit. What is the max-

imum profit?

(b) At what level of production is the average cost per unit A(q) � minimized?

(c) At what level of production is the average cost equal to the marginal cost C�(q)?

Solution

(a) The revenue is

R(q) � qp(q) � q[0.2(45 � 0.5q)] � 9q � 0.1q2

so the profit is

P(q) � R(q) � C(q) � (9q � 0.1q2) � (0.4q2 � 3q � 40)

� �0.5q2 � 6q � 40

The derivative

P�(q) � �q � 6

is 0 only when q � 6, and since P�(q) � �1 � 0, the second derivative test forabsolute extrema tells us that maximum profit occurs when q � 6 units are produced.

C(q)

q

The Second Derivative Test for Absolute Extrema � Sup-pose that f(x) is continuous on an interval I where x � c is the only criticalnumber and that f�(c) � 0. Then,

if f�(c) � 0, the absolute minimum of f(x) on I is f(c)if f�(c) � 0, the absolute maximum of f(x) on I is f(c)



x

y

Relativeminimum

Curve “turns around”and starts down again

c1 c2

FIGURE 3.35 The relative mini-mum is not the absolute minimumbecause of the effect of anothercritical point.

The graph of the profit function is shown in Figure 3.36a.(b) The average cost is

A(q) � � 0.4q � 3 �

for q � 0 (the level of production cannot be negative or zero). You find

A�(q) � 0.4 �

which is 0 for q � 0 only when q � 10. Since

A�(q) � � 0 when q � 0

it follows from the second derivative test for absolute extrema that average cost A(q) is minimized when q � 10 units. The minimal average cost is

A(10) � 0.4(10) � 3 � � 11 dollars/unit.

(c) The marginal cost is C�(q) � 0.8q � 3, and it equals average cost when

which equals the optimal level of production in part (b). The graphs of the mar-

ginal cost C�(q) and average cost A(q) � are shown in Figure 3.36b.

FIGURE 3.36 Graphs of profit, average cost, and marginal cost for Example 4.5

q

y

q

y

11

3

11

6 10

(a) The profit function (b) Average and marginal cost

y = –0.5q2 + 6q – 40y = C'(q)

y = A(q)

C(q)

q

0.4q2 � 40; q � 10

0.4q �40

q

0.8q � 3 � 0.4q � 3 �40

q

40

10

80

q3

40

q2 �0.4q2 � 40

q2

40

q

C(q)

q�

0.4q2 � 3q � 40

q


In general, if the revenue derived from the sale of q units is R(q) and the cost of pro-ducing those units is C(q), then the profit is P(q) � R(q) � C(q). Since

P�(q) � [R(q) � C(q)]� � R�(q) � C�(q)

it follows that P�(q) � 0 when R�(q) � C�(q). If it is also true that P�(q) � 0, orequivalently, that R�(q) � C�(q), then the profit will be maximized.

For instance, in Example 4.5, the revenue function is R(q) � 9q � 0.1q2, and thecost is C(q) � 0.4q2 � 3q � 40, so the marginal revenue is R�(q) � 9 � 0.2q andmarginal cost is C�(q) � 0.8q � 3. Thus, marginal revenue equals marginal cost when

which is the optimal production level found in part (a) of the example.

In part (c) of Example 4.5, you found that marginal cost equals average cost atthe level of production where average cost is minimized. This, too, is no accident. Tosee why, let C(q) be the cost of producing q units of a commodity. Then the average

cost per unit is A(q) � , and by applying the quotient rule, you get

A�(q) �

Thus, A�(q) � 0 when

qC�(q) � C(q)

or equivalently, when

C�(q) � � A(q)

marginal averagecost cost

To show that average cost is actually minimized where average cost equals marginalcost, it is necessary to make a few reasonable assumptions about total cost (see Prob-lem 43).

C(q)

q

qC�(q) � C(q)

q2

C(q)

q

q � 6

9 � 0.2q � 0.8q � 3

Marginal Analysis Criterion for Maximum Profit � ProfitP(q) � R(q) � C(q) is maximized at a level of production q where marginalrevenue equals marginal cost and the rate of change of marginal cost exceedsthe rate of change of marginal revenue; that is

R�(q) � C�(q) and R�(q) � C�(q)

TWO GENERAL PRINCIPLESOF MARGINAL ANALYSIS


Here is an informal explanation of the relationship between average and marginalcost that is often given in economics texts. The marginal cost (MC) is approximatelythe same as the cost of producing one additional unit. If the additional unit costs lessto produce than the average cost (AC) of the existing units (i.e., if MC � AC), thenthis less-expensive unit will cause the average cost per unit to decrease. On the otherhand, if the additional unit costs more than the average cost of the existing units (i.e.,if MC � AC), then this more-expensive unit will cause the average cost per unit toincrease. However, if the cost of the additional unit is equal to the average cost ofthe existing units (i.e., MC � AC), then the average cost will neither increase nordecrease, which means AC� � 0.

The relation between average cost and marginal cost can be generalized to applyto any pair of average and marginal quantities. The only possible modificationinvolves the nature of the first-order critical point that occurs when the average quan-tity equals the marginal quantity. For example, average revenue usually has a relativemaximum (instead of a minimum) when average revenue equals marginal revenue.

In Problems 1 through 16, find the absolute maximum and absolute minimum (ifany) of the given function on the specified interval.

1. f(x) � x2 � 4x � 5; �3 x 1

2. f(x) � x3 � 3x2 � 1; �3 x 2

3. f(x) � x3 � 9x � 2; 0 x 2

4. f(x) � x5 � 5x4 � 1; 0 x 5

5. f(t) � 3t5 � 5t3; �2 t 0

6. f(x) � 10x6 � 24x5 � 15x4 � 3; �1 x 1

7. f(x) � (x2 � 4)5; �3 x 2 8. f(t) � ; �2 t �

9. g(x) � x � ; x 3 10. g(x) � ; 0 x 21

x2 � 9

1

2

1

x

1

2

t2

t � 1

1

3

Marginal Analysis Criterion for Minimal Average Cost �

Average cost is minimized at the level of production where average cost equalsmarginal cost; that is, when A(q) � C�(q).


P . R . O . B . L . E . M . S 3.4P . R . O . B . L . E . M . S 3.4

11. f(u) � u � ; u � 0 12. f(u) � 2u � ; u � 0

13. f(x) � ; x � 0 14. f(x) � ; x � 0

15. f(x) � ; x 0 16. f(x) � ; x 0

In Problems 17 through 22, you are given the price p(q) at which q units of aparticular commodity can be sold and the total cost C(q) of producing the q units.In each case:

(a) Find the profit function P(q), the marginal revenue R�(q), and the marginalcost C�(q). Sketch the graphs of P(q), R�(q), and C�(q) on the same coordi-nate axes and determine the level of production q where P(q) is maximized.

(b) Find the average cost A(q) and sketch the graphs of A(q) and the marginalcost C�(q) on the same coordinate axes. Determine the level of production qwhere A(q) is minimized.

17. p(q) � 49 � q; C(q) � q2 � 4q � 200

18. p(q) � 37 � 2q; C(q) � 3q2 � 5q � 75

19. p(q) � 180 � 2q; C(q) � q3 � 5q � 162

20. p(q) � 710 � 1.1q2; C(q) � 2q3 � 23q2 � 90.7q � 151

21. p(q) � 1.0625 � 0.0025q; C(q) �

22. p(q) � 81 � 3q; C(q) �

AVERAGE REVENUE 23. Suppose the total revenue in dollars from the sale of q units of a certain com-modity is R(q) � �2q2 � 68q � 128.(a) At which level of sales is the average revenue per unit equal to the marginal

revenue?(b) Verify that the average revenue is increasing if the level of sales is less than

the level in part (a) and decreasing if the level of sales is greater than the levelin part (a).

(c) On the same set of axes, graph the relevant portions of the average and mar-ginal revenue functions.

24. At what point does the tangent to the curve y � 2x3 � 3x2 � 6x have the small-est slope? What is the slope of the tangent at this point?

q � 1

q � 3

q2 � 1

q � 3

1

8

MAXIMUM PROFIT ANDMINIMUM AVERAGE COST

1

(x � 1)2

1

x � 1

1

x2

1

x

32

u

1

u


25. For what value of x in the interval �1 x 4 is the graph of the function

f(x) � 2x2 � x3

steepest? What is the slope of the tangent at this point?

In Problems 26 through 37 solve the practical optimization problem and use one ofthe techniques from this section to verify that you have actually found the desiredabsolute extremum.

BROADCASTING 26. An all-news radio station has made a survey of the listening habits of local resi-dents between the hours of 5:00 P.M. and midnight. The survey indicates that thepercentage of the local adult population that is tuned in to the station x hours after5:00 P.M. is

f(x) � (�2x3 � 27x2 � 108x � 240)

(a) At what time between 5:00 P.M. and midnight are the most people listening tothe station? What percentage of the population is listening at this time?

(b) At what time between 5:00 P.M. and midnight are the fewest people listening?What percentage of the population is listening at this time?

GROUP MEMBERSHIP 27. Suppose that x years after its founding in 1978, a certain national consumers’ asso-ciation had a membership of

f(x) � 100(2x3 � 45x2 � 264x)

(a) At what time between 1978 and 1992 was the membership of the associationlargest? What was the membership at that time?

(b) At what time between 1978 and 1992 was the membership of the associationsmallest? What was the membership at that time?

GROWTH OF A SPECIES 28. More on codling moths* (recall Problem 33, Section 1 of this chapter). Thepercentage of codling moths that survive the pupa stage at a given temperature T(degrees Celsius) is modeled by the formula

P(T ) � �1.42T2 � 68T � 746 for 20 T 30

Find the temperatures at which the greatest and smallest percentage of mothssurvive.

1

8

1

3


* P. L. Shaffer and H. J. Gold, “A Simulation Model of Population Dynamics of the Codling Moth,Cydia pomonelia,” Ecological Modeling, Vol. 30. 1985, pages 247–274.

SPEED OF FLIGHT 29. In a paper published in 1969, C. J. Pennycuick* provided experimental evidenceto show that the power P required by a bird to maintain flight is given by theformula

where v is the relative speed of the bird, w is the weight, is the density of air,and S and A are positive constants associated with the bird’s size and shape. Whatrelative speed v will minimize the power required by the bird?

LEARNING 30. In a certain model, two responses (A and B) are possible for each of the series ofobservations. If there is a probability p of getting response A in any individualobservation, the probability of getting response A exactly n times in a series of mobservations is F(p) � pn(1 � p)m�n. The maximum likelihood estimate is thevalue of p that maximizes F(p) for 0 p 1. For what value of p does thisoccur?

BLOOD CIRCULATION 31. Poiseuille’s law asserts that the speed of blood that is r centimeters from thecentral axis of an artery of radius R is S(r) � c(R2 � r2), where c is a positiveconstant. Where is the speed of the blood greatest?

POLITICS 32. A poll indicates that x months after a particular candidate for public office declaresher candidacy, she will have the support of

S(x) � (�x3 � 6x2 � 63x � 1080) for 0 x 12

percent of the voters. If the election is held in November, when should thepolitician announce her candidacy? Should she expect to win if she needs at least50% of the vote?

ANIMAL BEHAVIOR 33. Recall from Problem 55 in Section 5 of Chapter 2 that according to the results†

of Tucker and Schmidt-Koenig, the energy expended by a certain species of para-keet is given by

E(v) � [0.074(v � 35)2 � 32]

where v is the bird’s velocity (in km/hr).(a) What velocity minimizes energy expenditure?(b) Read an article on how mathematical methods can be used to study animal

behavior and write a paragraph on whether you think such methods are valid.You may wish to begin with the reference cited in this problem.

1

v

1

29

P �w2

2�Sv�

1

2�Av3


* C. J. Pennycuick, “The Mechanics of Bird Migration,” Ibis III, pages 525–556.

† V. A. Tucker and K. Schmidt-Koenig, “Flight Speeds of Birds in Relation to Energetics and WindDirections,” The Auk, Vol. 88 (1971), pages 97–107.

NATIONAL CONSUMPTION 34. Assume that total national consumption is given by a function C(x), where x isthe total national income. The derivative C�(x) is called the marginal propensityto consume, and if S � x � C represents total national savings, then S�(x) iscalled marginal propensity to save. Suppose the consumption function is C(x)� 8 � 0.8x � . Find the marginal propensity to consume and determinethe value of x that results in the smallest total savings.

SENSITIVITY OF DRUGS 35. Body reaction to drugs is often modeled* by an equation of the form

where D is the dosage and C (a constant) is the maximum dosage that can begiven. The rate of change of R(D) with respect to D is called the sensitivity. Findthe value of D for which the sensitivity is the greatest.

AERODYNAMICS 36. In designing airplanes, an important feature is the so-called “drag factor”; that is,the retarding force exerted on the plane by the air. One model measures drag bya function of the form

where A and B are positive constants. It is found experimentally that drag is min-

imized when v � 160 mph. Use this information to find the ratio .

ELECTRICITY 37. When a resistor of R ohms is connected across a battery with electromotive forceE volts and internal resistance r ohms, a current of I amperes will flow thatgenerates P watts of power, where

and

Assuming r is constant, what choice of R results in maximum power?

CRYSTALLOGRAPHY 38. A fundamental problem in crystallography is the determination of the packingfraction of a crystal lattice, which is the fraction of space occupied by the atomsin the lattice, assuming that the atoms are hard spheres. When the lattice containsexactly two different kinds of atoms, it can be shown that the packing fraction isgiven by the formula†

f(x) �K(1 � c2x3)

(1 � x)3

P � I2RI �E

r � R

B

A

F(v) � Av2 �B

v2

R(D) � D2�C

2�

D

3 �

0.8�x


* R. M. Thrall et al., Some Mathematical Models in Biology, U. of Michigan, 1967.

† John C. Lewis and Peter P. Gillis, “Packing Factors in Diatomic Crystals,” American Journal ofPhysics, Vol. 61, No. 5 (1993), pages 434–438.

where x � is the ratio of the radii, r and R, of the two kinds of atoms in the

lattice, and c and K are positive constants.(a) The function f(x) has exactly one critical number. Find it and use the second

derivative test to classify it as a relative maximum or a relative minimum.(b) The numbers c and K and the domain of f(x) depend on the cell structure in

the lattice. For ordinary rock salt: c � 1, K � , and the domain is the inter-

val x 1. Find the largest and smallest values of f(x).

(c) Repeat part (b) for �-cristobalite, for which , , and thedomain is 0 x 1.

(d) Read the article on which this problem is based, and write a paragraph on howpacking factors are computed and used in crystallography.

SURVIVAL OF AQUATIC LIFE 39. It is known that a quantity of water that occupies 1 liter at 0°C will occupy

V(T ) � �6.8 � 10�8 T3 � 8.5 � 10�6 T2 � 6.4 � 10�5 T � 1

liters when the temperature is T°C, for 0 T 30.(a) Use a graphing utility to graph V(T ) for 0 T 10. The density of water is

maximized when V(T ) is minimized. At what temperature does this occur?(b) Does the answer to (a) surprise you? It should. Water is the only common liq-

uid whose maximum density occurs above its freezing point (0°C for water).Read an article on the survival of aquatic life during the winter and then writea paragraph on how the property of water examined in this problem is relatedto such survival.

BLOOD PRODUCTION 40. Recall from Problem 27 in Section 6 of Chapter 2 that a useful model for the pro-duction p(x) of blood cells involves a function of the form

where x is the number of cells present; and A, B, and m are positive constants.*(a) Find the rate of blood production R(x) � p�(x) and determine where R(x) � 0.(b) Find the rate at which R(x) is changing with respect to x and determine where

R�(x) � 0.(c) Sketch the graph of the production function p(x).

AMPLITUDE OF OSCILLATION 41. In physics, it can be shown that a particle forced to oscillate in a resisting mediumhas amplitude A(r) given by

p(x) �Ax

B � xm

K ��3�

16c � �2

(�2 � 1)

2�

3

r

R


PROBLEM 38

Na Cl

* M. C. Mackey and L. Glass, “Oscillations and Chaos in Physiological Control Systems,” Science, Vol.197, pages 287–289.

where r is the ratio of the forcing frequency to the natural frequency of oscilla-tion and k is a positive constant that measures the damping effect of the resistingmedium. Show that A(r) has exactly one positive first-order critical number. Doesit correspond to a relative maximum or a relative minimum? Can anything be saidabout the absolute extrema of A(r)?

FIRE FIGHTING 42. If air resistance is neglected, it can be shown that the stream of water emitted bya fire hose will have height

above a point located x feet from the nozzle, where m is the slope of the nozzle,and v is the velocity of the stream of water as it leaves the nozzle. Assume v isconstant.(a) If m is also constant, determine the distance x where the maximum height

occurs.(b) If m is allowed to vary, find the slope that allows a firefighter to spray water

on a fire from the greatest distance.(c) Suppose the firefighter is x � x0 feet from the base of a building. If m is

allowed to vary, what is the highest point on the building that the firefightercan reach with the water from her hose?

43. Suppose q � 0 units of a commodity are produced at a total cost of C(q) dollars

and an average cost of A(q) � . In this section, we showed that q � qc

satisfies A�(qc) � 0 if and only if C�(qc) � A(qc); that is, when marginal costequals average cost. The purpose of this problem is to show that A(q) is mini-mized when q � qc.(a) Generally speaking, the cost of producing a commodity increases at an increas-

ing rate as more and more goods are produced. Using this economic princi-ple, what can be said about the sign of C�(q) as q increases?

C(q)

q

x

y

Slope m

Hose

θ

y � �16(1 � m2)�x

v�2

� mx

A(r) �1

(1 � r2)2 � kr2


(b) Show that A�(qc) � 0 if and only if C�(qc) � 0. Then use part (a) to arguethat average cost A(q) is minimized when q � qc.

In Section 4, you saw a number of applications in which a formula was given and itwas required to determine either a maximum or a minimum value. In practice, thingsare often not that simple, and it is necessary to first gather information about a quan-tity of interest, then formulate and analyze an appropriate mathematical model.

In this section, you will learn how to combine the techniques of model-buildingfrom Section 4 of Chapter 1 with the optimization techniques of Section 4. Here is aprocedure for dealing with such problems.

This procedure is illustrated in the following examples.

The highway department is planning to build a picnic area for motorists along a majorhighway. It is to be rectangular with an area of 5,000 square yards and is to be fencedoff on the three sides not adjacent to the highway. What is the least amount of fenc-ing that will be needed to complete the job?

SolutionAs in Example 4.1 of Chapter 1, Section 4, label the sides of the picnic area as indi-cated in Figure 3.37 and let F denote the amount of fencing required. Then,

F � x � 2y

General Procedure for Analyzing Practical OptimizationProblems

Step 1. Begin by deciding precisely what you want to optimize. Once this hasbeen done, assign names to all variables of interest. It may help to pick let-ters that suggest the nature of the quantity, such as R for revenue or A forarea.

Step 2. Draw a figure, if appropriate, and find an expression for the quantity tobe optimized.

Step 3. Use any equations involving the variables to eliminate all but one vari-able from the quantity to be optimized. Determine any restrictions on theindependent variable.

Step 4. Use the methods of Section 4 to optimize f. Interpret your results interms of appropriate physical, geometric, or economic quantities.

Chapter 3 � Section 5 Practical Optimization 261

PracticalOptimization

5


x

y y

Highway

Picnic area

FIGURE 3.37 Rectangular picnicarea.

244 chapter 3 additional applications of the derivative calculus... · 2010. 8. 24. · 248 chapter...

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