Chapter 2 Section 4 Marginal Analysis: Approximation by ... calculus...Marginal analysis is an important example of a general approximation procedure based on the fact that since f

one of the factors is constant. Show that the two rules are consistent. In particular,

use the product rule to show that if c is a constant.

44. Derive the quotient rule. [Hint: Show that the difference quotient for is

Before letting h approach zero, rewrite this quotient using the trick of subtractingand adding g(x)f(x) in the numerator.]

45. Prove the power rule for the case where n � �p is a negative

integer. [Hint: Apply the quotient rule to ]

46. Use a graphing utility to sketch the curve f(x) � x2(x � 1), and on the same setof coordinate axes, draw the tangent line to the graph of f(x) at x � 1. Use thetrace and zoom to find where f �(x) � 0.

47. Use a graphing utility to sketch the curve , and on the same

set of coordinate axes, draw the tangent lines to the graph of f(x) at x � �2 andat x � 0. Use the trace and zoom to find where f �(x) � 0.

48. Use a graphing utility to graph f(x) � x4 � 2x3 � x � 1 using a viewing rec-tangle of [�5, 5]1 by [0, 2].5. Use trace and zoom, or other graphing utility meth-ods, to find the minima and maxima of this function. Find the derivative functionf �(x) algebraically and graph f(x) and f �(x) on the same axes using a viewing rec-tangle of [�5, 5]1 by [�2, 2].5. Use the trace and zoom to find the x interceptsof f�(x). Explain why the maximum or minimum of f(x) occurs at the x interceptsof f �(x).

49. Repeat Problem 48 for the product function f(x) � x3 (x � 3)2.

Marginal analysis is an area of economics concerned with estimating the effect onquantities such as cost, revenue, and profit when the level of production is changedby a unit amount. For instance, if C(x) is the cost of producing x units of a certaincommodity, then the cost of producing the (x0 � 1)st unit is C(x0 � 1) � C(x0). How-ever, since the derivative of the cost function C(x), called marginal cost, is given by

MC(x) � C�(x) � limhfi 0

C(x � h) � C(x)

h

f(x) �3x2 � 4x � 1

x � 1

y � x�p �1

x p

d

dx(xn) � nxn�1

1

h �f(x � h)

g(x � h)�

f(x)

g(x) �g(x)f(x � h) � f(x)g(x � h)

g(x � h)g(x)h

f

g

d

dx(cf) � c

df

dx

Chapter 2 � Section 4 Marginal Analysis: Approximation by Increments 133

MarginalAnalysis:

Approximationby Increments

4

it follows that

so that when h � 1, we can make the approximation

MC(x0) C(x0 � 1) � C(x0)

In other words, at the level of production x � x0, the cost of producing one additionalunit is approximately equal to the marginal cost MC(x0). The geometric relationshipbetween C(x0 � 1) � C(x0) and MC(x0) is shown in Figure 2.9.

For future reference, here is the definition of marginal cost, together with analo-gous definitions for marginal revenue and marginal profit.

FIGURE 2.9 Marginal cost MC(x0) approximates C (x0 � 1) � C (x0).

y

xx0 x0 + 1

C(x0 + 1) – C(x0)

y = C(x)

y

xx0 x0 + 1

1C�(x0)

y = C(x)

MC(x0) �C(x0 � h) � C(x0)

h

134 Chapter 2 Differentiation: Basic Concepts

(a) The marginal cost MC(x0) atx � x0 is C�(x0).

(b) The cost of producing the (x0 � 1)th unit is C(x0 � 1) � C(x0).

A manufacturer estimates that when x units of a particular commodity are produced,

the total cost will be C(x) � x2 � 3x � 98 dollars and that all x units will be sold

when the price is p(x) � 25 � x dollars per unit.

(a) Use the marginal cost function to estimate the cost of producing the ninth unit.What is the actual cost of producing the ninth unit?

(b) Find the revenue function for the commodity. Then use the marginal revenue func-tion to estimate the revenue derived from the sale of the ninth unit. What is theactual revenue derived from the sale of the ninth unit?

(c) Find the profit associated with the production of x units. Sketch the profit func-tion and determine the level of production where profit is maximized. What is themarginal profit at this optimal level of production?

Solution

(a) The marginal cost function is MC(x) � C�(x) � x � 3, and the change in cost

as x increases from 8 to 9 (the ninth unit) is approximately MC(8) � (8) � 3

� $5. The actual cost of the ninth unit is C(9) � C(8) � $5.13.(b) The revenue function is

and the marginal revenue function is MR(x) � R�(x) � 25 � x. The revenue

derived from the sale of the ninth unit is approximately MR(8) � 25 � (8) �

$19.67, and the actual revenue is R(9) � R(8) � $19.33.

2

3

2

3

R(x) � xp(x) � x�25 � 13x� � 25x � 13x2

1

4

1

4

1

3

1

8

Marginal Cost, Revenue, and Profit � If C(x) is the total costof producing x units of a commodity, and R(x) and P(x) � R(x) � C(x) are thecorresponding revenue and profit functions, respectively, then

the marginal cost function is MC(x) � C�(x)the marginal revenue function is MR(x) � R�(x)the marginal profit function is MP(x) � P�(x)


EXAMPLE 4 .1EXAMPLE 4 .1E x p l o r e !E x p l o r e !Refer to Example 4.1. Graph

C(x) and R(x) on the same coor-

dinate axes using a viewing

rectangle of [0, 80]10 by

[0, 500]50. Find the tangent line

to C(x) at x � 8. Graph the tan-

gent line on the same set of co-

ordinate axes. Then change the

viewing rectangle to [6, 11]1 by

[120, 140]1 to see why the mar-

ginal cost is a good approxima-

tion to the actual change in

C(x). Continue finding an equa-

tion of the tangent line to R(x)

at x � 8. Graph the tangent line

and R(x) on the same coordinate

axes. Use trace and move the

cursor close to x � 8 while trac-

ing R(x). Move the cursor back

and forth between R(x) and the

tangent line to see why the mar-

ginal revenue is a good approx-

imation to the actual change in

R(x).

(c) The profit is

and the graph of y � x2 � 22x � 98 is a downward opening parabola with

its highest point (vertex) above

(see Figure 2.10). Thus, profit is maximized when x � 24 units are sold and the

price is p � 25 � (24) � $17 per unit. The marginal profit function is MP(x) �

P�(x) � x � 22, and at the optimal level of production x � 24, the marginal

profit is P�(24) � (24) � 22 � 0.

Cost per unit of production is also important in economics. This function is calledaverage cost and its derivative is marginal average cost.

Similar definitions apply to average revenue and average profit. Here is an exampleinvolving average cost.

Let C(x) � x2 � 3x � 98 be the total cost function for the commodity in Example 4.1.1

8

�11

12

�11

12

1

3

x ��B

2A�

�22

2��1124 �� 24

�11

24

P(x) � R(x) � C(x) � 25x �1

3x2 � �18 x2 � 3x � 98� � �1124 x2 � 22x � 98


Average Cost and Marginal Average Cost � If C(x) is thetotal cost associated with the production of x units of a particular commodity,then

the average cost is AC(x) �

and

marginal average cost is MAC � (AC)�(x)

C(x)

x

y

x43.034.97 24

FIGURE 2.10 The graph of theprofit function

P (x) � x2 � 22x � 98.�1124

EXAMPLE 4 .2EXAMPLE 4 .2

(a) Find the average cost and the marginal average cost for the commodity.(b) For what level of production is marginal average cost equal to 0?(c) For what level of production does marginal cost equal average cost?

Solution

(a) The average cost is

and the marginal average cost is

(b) Marginal average cost is 0 when

(c) The marginal cost is MC � C�(x) � x � 3, so marginal cost equals average cost

when

In Example 4.1, the profit is maximized at the level of production where mar-ginal profit is zero, and in Example 4.2, average cost is minimized whenaverage cost equals marginal cost. In Chapter 3, we use calculus to show thatboth these results are consequences of general rules of economics.

Marginal analysis is an important example of a general approximation procedurebased on the fact that since

f�(x) � limhfi 0

f(x0 � h) � f(x0)

h

APPROXIMATION BYINCREMENTS

x � 28

x2 � 98(8)

1

8x �

98

x

1

4x � 3 �

1

8x � 3 �

98

x

1

4

1

8�

98

x2� 0; x2 � 8(98); x � 28

MAC � MC�(x) �1

8�

98

x2

AC(x) �C(x)

x�

1

8x2 � 3x � 98

x�

1

8x � 3 �

98

x


Note

Then for small h, the derivative f �(x) is approximately equal to the difference quo-tient. That is,

so

f(x0 � h) f(x0) � f �(x0)h

or equivalently,

f(x0 � h) � f(x0) f �(x0)h

To emphasize that the incremental change is in the variable x, we write h � �x andsummarize the incremental approximation formula as follows.

Here is an example of how this approximation formula can be used in economics.

Suppose the total cost in dollars of manufacturing q units of a certain commodity isC(q) � 3q2 � 5q � 10. If the current level of production is 40 units, estimate howthe total cost will change if 40.5 units are produced.

SolutionIn this problem, the current value of production is q � 40 and the change in pro-duction is �q � 0.5. By the approximation formula, the corresponding change in costis

�C � C(40.5) � C(40) C�(40)�q � C�(40)(0.5)

Since

C�(q) � 6q � 5 and C�(40) � 6(40) � 5 � 245

it follows that

�C C�(40)(0.5) � 245(0.5) � $122.50

Approximation by Increments � If f(x) is differentiable at x �x0 and �x is a small change in x, then

f(x0 � �x) f(x0) � f �(x0)�x

or, equivalently, if �f � f(x0 � �x) � f(x0), then

�f f �(x0)�x

f�(x0)

f(x0 � h) � f(x0)

h



For practice, compute the actual change in cost caused by the increase in the levelof production from 40 to 40.5 and compare your answer with the approximation. Isthe approximation a good one?

In the next example, the approximation formula is used to study propagation oferror. In particular, the derivative is used to estimate the maximum error in a calcu-lation that is based on figures obtained through imperfect measurement.

You measure the side of a cube to be 12 centimeters long and conclude that thevolume of the cube is 123 � 1,728 cubic centimeters. If your measurement of theside is accurate to within 2%, approximately how accurate is your calculation ofthe volume?

SolutionThe volume of the cube is V(x) � x3, where x is the length of a side. The error youmake in computing the volume if you take the length of the side to be 12 when it isreally 12 � �x is

�V � V(12 � �x) � V(12) V�(12)�x

Your measurement of the side can be off by as much as 2%, that is, by as muchas 0.02(12) � 0.24 centimeter in either direction. Hence, the maximum error in yourmeasurement of the side is �x � 0.24, and the corresponding maximum error inyour calculation of the volume is

Maximum error in volume � �V V�(12)(0.24)

Since

V�(x) � 3x2 and V�(12) � 3(12)2 � 432

it follows that

Maximum error in volume 432(0.24) � 103.68

This says that, at worst, your calculation of the volume as 1,728 cubic centimeters isoff by approximately 103.68 cubic centimeters.

In the next example, the desired change in the function is given, and the goal isto estimate the necessary corresponding change in the variable.



E x p l o r e !E x p l o r e !Refer to Example 4.4. Conjec-

ture the accuracy of your calcu-

lation of the volume if you can

measure accurately to 1%. Is

it half of what was found in

Example 4.4? Graph V(x) using

a viewing rectangle of [11.8,

12.5].1 by [1700, 1875]100.

Conjecture what the value of

V(x � �x) � V(x) and V�(x)

will be using the graph. Check

your conjecture by calculating

these values using x � 12 and

�x � 0.12.

The daily output at a certain factory is Q(L) � 900L1/3 units, where L denotes thesize of the labor force measured in worker-hours. Currently, 1,000 worker-hours oflabor are used each day. Use calculus to estimate the number of additional worker-hours of labor that will be needed to increase daily output by 15 units.

SolutionSolve for �L using the approximation formula

�Q Q�(L)�L

with �Q � 15 L � 1,000 and Q�(L) � 300L�2/3

to get 15 300(1,000)�2/3 �L

or worker-hours

The percentage change of a quantity expresses the change in that quantity as a per-centage of its size prior to the change. In particular,

This formula can be combined with the approximation formula and written in func-tional notation as follows.

The GDP of a certain country was N(t) � t2 � 5t � 200 billion dollars t years after1994. Use calculus to estimate the percentage change in the GDP during the first quar-ter of 2002.

Percentage change � 100change in quantity

size of quantity

APPROXIMATION OFPERCENTAGE CHANGE

�L

15

300(1,000)2/3 �

15

300(10)2 � 5


Approximation Formula for Percentage Change � If �xis a (small) change in x, the corresponding percentage change in the functionf(x) is

Percentage change in f � 100�f

f(x)

100

f�(x)�x

f(x)



SolutionUse the formula

with t � 8 �t � 0.25 and N�(t) � 2t � 5

to get

The next example illustrates how the percentage change can sometimes be esti-mated even though the numerical value of the variable is not known.

At a certain factory, the daily output is Q(K) � 4,000K1/2 units, where K denotes thefirm’s capital investment. Use calculus to estimate the percentage increase in outputthat will result from a 1% increase in capital investment.

SolutionThe derivative is Q�(K) � 2,000K�1/2. The fact that K increases by 1% means that�K � 0.01K. Hence,

Sometimes the increment �x is referred to as the differential of x and is denoted by dx,and then our approximation formula can be written as �f f�(x) dx. If y � f(x), thedifferential of y is defined to be dy � f�(x) dx. The following summarizes this concept.

DIFFERENTIALS

�2,000

4,000� 0.5%

�2,000K1/2

4,000K1/2 since K�1/2K � K1/2

� 1002,000K�1/2(0.01K)

4,000K1/2

Percentage change in Q 100Q�(K)�K

Q(K)

1.73%

� 100[2(8) � 5](0.25)

(8)2 � 5(8) � 200

Percentage change in N 100N�(8)0.25

N(8)

Percentage change in N 100N�(t)�t

N(t)



In each case, find the differential of y � f(x).(a) f(x) � x3 � 7x2 � 2(b) f(x) � (x2 � 5)(3 � x � 2x2)

Solution

(a) dy � f�(x) dx � [3x2 � 7(2x)] dx � (3x2 � 14x) dx(b) By the product rule,

A geometric interpretation of the approximation of �y by the differential dy isshown in Figure 2.11. Note that since the slope of the tangent line at P(x, f(x)) isf �(x), the differential dy � f�(x) dx is the change in the height of the tangent that cor-responds to a change from x to x � �x. On the other hand, �y is the change in theheight of the curve corresponding to this change in x. Hence, approximating �y bythe differential dy is the same as approximating the change in the height of a curveby the change in height of the tangent line. If �x is small, it is reasonable to expectthis to be a good approximation.

FIGURE 2.11 Approximation of �y by the differential dy.

x

y

x + ∆xx

Tangentline

dy

∆x

∆y

y = f (x)

dy � f�(x) dx � [(x2 � 5)(�1 � 4x) � (2x)(3 � x � 2x2)] dx

Differentials � The differential of x is dx � �x, and if y � f(x) is adifferentiable function of x, then dy � f �(x) dx is the differential of y.



MARGINAL ANALYSIS 1. A manufacturer’s total cost is C(q) � 0.1q3 � 0.5q2 � 500q � 200 dollars, whereq is the number of units produced.(a) Use marginal analysis to estimate the cost of manufacturing the fourth unit.(b) Compute the actual cost of manufacturing the fourth unit.

MARGINAL ANALYSIS 2. A manufacturer’s total monthly revenue is R(q) � 240q � 0.05q2 dollars when qunits are produced and sold during the month. Currently, the manufacturer is pro-ducing 80 units a month and is planning to increase the monthly output by 1 unit.(a) Use marginal analysis to estimate the additional revenue that will be gener-

ated by the production and sale of the 81st unit.(b) Use the revenue function to compute the actual additional revenue that will be

generated by the production and sale of the 81st unit.

MARGINAL ANALYSIS In Problems 3 through 8, C(x) is the total cost of producing x units of a particularcommodity and p(x) is the price at which all x units will be sold.

(a) Find the marginal cost and the marginal revenue.(b) Use marginal cost to estimate the cost of producing the fourth unit.(c) Find the actual cost of producing the fourth unit.(d) Use marginal revenue to estimate the revenue derived from the sale of the

fourth unit.(e) Find the actual revenue derived from the sale of the fourth unit.

3.

4.

5.

6.

7.

8.

9. Estimate how much the function f(x) � x2 � 3x � 5 will change as x increasesfrom 5 to 5.3.

C(x) �2

7x2 � 65; p(x) �

12 � 2x

3 � x

C(x) �1

4x2 � 43; p(x) �

3 � 2x

1 � x

C(x) �5

9x2 � 5x � 73; p(x) � �x2 � 2x � 13

C(x) �1

3x2 � 2x � 39; p(x) � �x2 � 4x � 10

C(x) �1

4x2 � 3x � 67; p(x) �

1

5(45 � x)

C(x) �1

5x2 � 4x � 57; p(x) �

1

4(36 � x)


P . R . O . B . L . E . M . S 2.4P . R . O . B . L . E . M . S 2.4

10. Estimate how much the function f(x) � � 3 will change as x decreasesfrom 4 to 3.8.

11. Estimate the percentage change in the function f(x) � x2 � 2x � 9 as x increasesfrom 4 to 4.3.

12. Estimate the percentage change in the function f(x) � 3x � as x decreases from

5 to 4.6.

In each of the following problems, use calculus to obtain the required estimate.

MANUFACTURING 13. A manufacturer’s total cost is C(q) � 0.1q3 � 0.5q2 � 500q � 200 dollars whenthe level of production is q units. The current level of production is 4 units, andthe manufacturer is planning to increase this to 4.1 units. Estimate how the totalcost will change as a result.

MANUFACTURING 14. A manufacturer’s total monthly revenue is R(q) � 240q � 0.05q2 dollars when qunits are produced during the month. Currently, the manufacturer is producing 80units a month and is planning to decrease the monthly output by 0.65 unit. Esti-mate how the total monthly revenue will change as a result.

PRODUCTION 15. The daily output at a certain factory is Q(L) � 300L2/3 units, where L denotes thesize of the labor force measured in worker-hours. Currently, 512 worker-hours oflabor are used each day. Estimate the number of additional worker-hours of laborthat will be needed to increase daily output by 12.5 units.

MANUFACTURING 16. A manufacturer’s total cost is C(q) � q3 � 642q � 400 dollars when q units

are produced. The current level of production is 4 units. Estimate the amount bywhich the manufacturer should decrease production to reduce the total cost by$130.

PROPERTY TAX 17. Records indicate that x years after 1997, the average property tax on a three-bed-room home in a certain community was T(x) � 60x3/2 � 40x � 1,200 dollars.Estimate the percentage by which the property tax increased during the first halfof 2001.

ANNUAL EARNINGS 18. The gross annual earnings of a company were A(t) � 0.1t2 � 10t � 20 thousanddollars t years after its formation in 1996. Estimate the percentage change in thegross annual earnings during the third quarter of 2000.

EFFICIENCY 19. An efficiency study of the morning shift at a certain factory indicates that an aver-age worker arriving on the job at 8:00 A.M. will have assembled f(x) � �x3 �6x2 � 15x transistor radios x hours later. Approximately how many radios will theworker assemble between 9:00 and 9:15 A.M.?

PRODUCTION 20. At a certain factory, the daily output is Q(K) � 600K1/2 units, where K denotesthe capital investment measured in units of $1,000. The current capital investmentis $900,000. Estimate the effect that an additional capital investment of $800 willhave on the daily output.

1

6

2

x

x

x � 1


PRODUCTION 21. At a certain factory, the daily output is Q(L) � 60,000L1/3 units, where L denotesthe size of the labor force measured in worker-hours. Currently 1,000 worker-hours of labor are used each day. Estimate the effect on output that will be pro-duced if the labor force is cut to 940 worker-hours.

MARGINAL ANALYSIS 22. At a certain factory, the daily output is Q � 3,000K1/2L1/3 units, where K denotesthe firm’s capital investment measured in units of $1,000 and L denotes the sizeof the labor force measured in worker-hours. Suppose that the current capitalinvestment is $400,000 and that 1,331 worker-hours of labor are used each day.Use marginal analysis to estimate the effect that an additional capital investmentof $1,000 will have on the daily output if the size of the labor force is notchanged.

MARGINAL ANALYSIS 23. Suppose the total cost in dollars of manufacturing q units is C(q) � 3q2 � q �500.(a) Use marginal analysis to estimate the cost of manufacturing the 41st unit.(b) Compute the actual cost of manufacturing the 41st unit.

NEWSPAPER CIRCULATION 24. It is projected that t years from now, the circulation of a local newspaper will beC(t) � 100t2 � 400t � 5,000. Estimate the amount by which the circulation willincrease during the next 6 months. [Hint: The current value of the variable is t � 0.]

POPULATION GROWTH 25. It is projected that t years from now, the population of a certain suburban

community will be P(t) � 20 � thousand. By approximately how much

will the population increase during the next quarter year?

AIR POLLUTION 26. An environmental study of a certain community suggests that t years from now,the average level of carbon monoxide in the air will be Q(t) � 0.05t2 � 0.1t �3.4 parts per million. By approximately how much will the carbon monoxide levelchange during the coming 6 months?

AREA 27. You measure the radius of a circle to be 12 cm and use the formula A � �r2 tocalculate the area. If your measurement of the radius is accurate to within 3%,approximately how accurate is your calculation of the area?

VOLUME 28. Estimate what will happen to the volume of a cube if the length of each side isdecreased by 2%. Express your answer as a percentage and verify that your resultis consistent with the calculation in Example 4.4.

PRODUCTION 29. The output at a certain factory is Q � 600K1/2L1/3 units, where K denotes the cap-ital investment and L is the size of the labor force. Estimate the percentageincrease in output that will result from a 2% increase in the size of the labor forceif capital investment is not changed.

PRODUCTION 30. At a certain factory, the daily output is Q(K) � 1,200K1/2 units, where K denotesthe firm’s capital investment. Estimate the percentage increase in capital invest-ment that is needed to produce a 1.2% increase in output.

6

t � 1


GROWTH OF A CELL 31. A certain cell has the shape of a sphere. If the formulas S � 4�r2 and V � �r3

are used to compute the surface area and volume of the cell, respectively. Estimate the effect on S and V produced by a 1% increase in the radius r.

VOLUME 32. Estimate the largest percentage error you can allow in the measurement of theradius of a sphere if you want the error in the calculation of its volume using the

formula V � �r3 to be no greater than 8%.

VOLUME 33. A soccer ball made of leather inch thick has an inner diameter of 8 inches.

Estimate the volume of its leather shell. [Hint: Think of the volume of the shell as a certain change �V in volume.]

VOLUME 34. A melon in the form of a sphere has a rind inch thick and an inner diameter of

8 inches. Estimate what percentage of the total volume of the melon is rind.

BLOOD CIRCULATION 35. In Problem 53, Section 1 of Chapter 1, we introduced an important law attributedto the French physician, Jean Poiseuille. Another law discovered by Poiseuille saysthat the volume of the fluid flowing through a small tube in unit time under fixedpressure is given by the formula V � kR4, where k is a positive constant and Ris the radius of the tube. This formula is used in medicine to determine how widea clogged artery must be opened to restore a healthy flow of blood.(a) Suppose the radius of a certain artery is increased by 5%. Approximately what

effect does this have on the volume of blood flowing through the artery?(b) Read an article on the cardiovascular system and write a paragraph on the flow

of blood.*

EXPANSION OF MATERIAL 36. The (linear) thermal expansion coefficient of an object is defined to be

where L(T) is the length of the object when the temperature is T. Suppose a 50-ft span of a bridge is built of steel with � � 1.4 � 10�5 per degree centi-grade. Approximately how much will the length change during a year when thetemperature varies from �20°C (winter) to 35°C (summer)?

RADIATION 37. Stefan’s law in physics states that a body emits radiant energy according to theformula R(T) � kT 4, where R is the amount of energy emitted from a surfacewhose temperature is T (in degrees Kelvin) and k is a positive constant. Estimatethe percentage change in R that results from a 2% increase in T.

� �L�(T)

L(T)

1

5

1

2

1

8

4

3

4

3


* You may wish to begin your research by consulting such textbooks as Elaine N. Marieb, HumanAnatomy and Physiology, 2nd ed., The Benjamin/Cummings Publishing Co., Redwood City, CA, 1992,and Kent M. Van De Graaf and Stuart Ira Fox, Concepts of Human Anatomy and Physiology, 3rd ed.,Wm. C. Brown Publishers, Dubuque, IA, 1992.

38. Show that when Newton’s method is applied repeatedly, the nth approximation isobtained from the (n � 1)st approximation by the formula

n � 1, 2, 3, . . .

[Hint: First find x1 as the x intercept of the tangent line to y � f(x) at x � x0.Then use x1 to find x2 in the same way.]

39. Let f(x) � x3 � x2 � 1.(a) Use your graphing utility to graph f(x). Note that there is only one root located

between 1 and 2. Use zoom and trace or other utility features to find the root.(b) Using x0 � 1, estimate the root by applying Newton’s method until two con-

secutive approximations agree to four decimal places.(c) Take the root you found graphically in part (a) and the root you found by New-

ton’s method in part (b) and substitute each into the equation f(x) � 0. Whichis more accurate?

40. Let f(x) � x4 � 4x3 � 10. Use your graphing utility to graph f(x). Note that thereare two roots of the equation f(x) � 0. Estimate each root using Newton’s methodand then check your results using zoom and trace or other utility features.

xn � xn�1 �f(xn�1)

f�(xn�1)


Newton’s Method � Tangent line approximations can be used in avariety of ways. Newton’s method of approximating the roots of an equation f(x) � 0 is based on the idea that if we start with a “guess” x0 that is close toan actual root c, we can often obtain an improved guess by finding the x inter-cept x1 of the tangent line to the curve y � f(x) at x � x0 (see the figure). Theprocess can then be repeated until a desired degree of accuracy is attained. Inpractice, it is usually easier and faster to use the graphing utility, zoom, andtrace features of your calculator to find roots, but the ideas behind Newton’smethod are still important. Problems 38 through 42 involve Newton’s method.

y

xc

x0x1

y = f(x)

41. The ancient Babylonians (circa 1700 B.C.) approximated by applying theformula

for n � 1, 2, 3, . . .

(a) Show that this formula can be derived from the formula for Newton’s methodin Problem 38, and then use it to estimate . Repeat the formula untiltwo consecutive approximations agree to four decimal places. Use your cal-culator to check your result.

(b) The spy wakes up one morning in Babylonia and finds that his calculator hasbeen stolen. Create a spy story problem based on using the ancient formula tocompute a square root.

42. Sometimes Newton’s method fails no matter what initial value x0 is chosen (unlesswe are lucky enough to choose the root itself). Let and choose x0 arbi-trarily (x0 � 0).(a) Show that xn�1 � �2xn for n � 0, 1, 2, . . . so that the successive “guesses”

generated by Newton’s method are x0, �2x0, 4x0, . . . .(b) Use your graphing utility to graph f(x) and use an appropriate utility to draw

the tangent lines to the graph of y � f(x) at the points that correspond to x0,�2x0, 4x0, . . . . Why do these numbers fail to estimate a root of f(x) � 0?

In many practical situations, you find that the rate at which one quantity is changingcan be expressed as the product of other rates. For example, suppose a car is travel-ing at 50 mph at a particular time when gasoline is being consumed at the rate of 0.1 gal/mile. Then, to find out how much gasoline is being used each hour, you wouldmultiply the rates:

(0.1 gal/mile)(50 miles/hour) � 5 gal/hour

Or, suppose the total manufacturing cost at a certain factory is a function of thenumber of units produced, which in turn is a function of the number of hours the fac-tory has been operating. If C, q, and t denote the cost, units produced, and time,respectively, then

(dollars per unit)

and

(units per hour)

The product of these two rates is the rate of change of cost with respect to time; thatis,

dq

dt� �rate of change of outputwith respect to time

dC

dq� �rate of change of costwith respect to output

f(x) � �3 x

�1,265

xn�1 �1

2�xn � Nxn�

�N


The Chain Rule

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Chapter 2 Section 4 Marginal Analysis: Approximation by ... calculus...Marginal analysis is an important example of a general approximation procedure based on the fact that since f