Chapter 2 ■ Section 4
Marginal Analysis: Approximation by Increments
133
one of the factors is constant. Show that the two rules are consistent. In particular,
d
df
use the product rule to show that (cf) c if c is a constant.
dx
dx
44. Derive the quotient rule. [Hint: Show that the difference quotient for
f
is
g
f(x)
g(x)f(x h) f(x)g(x h)
1 f(x h)
h g(x h) g(x)
g(x h)g(x)h
Before letting h approach zero, rewrite this quotient using the trick of subtracting
and adding g(x)f(x) in the numerator.]
d n
(x ) nxn1 for the case where n p is a negative
dx
1
integer. [Hint: Apply the quotient rule to y xp p]
x
45. Prove the power rule
46. Use a graphing utility to sketch the curve f(x) x2(x 1), and on the same set
of coordinate axes, draw the tangent line to the graph of f(x) at x 1. Use the
trace and zoom to find where f(x) 0.
3x2 4x 1
, and on the same
x1
set of coordinate axes, draw the tangent lines to the graph of f(x) at x 2 and
at x 0. Use the trace and zoom to find where f (x) 0.
47. Use a graphing utility to sketch the curve f(x) 48. Use a graphing utility to graph f(x) x4 2x3 x 1 using a viewing rectangle of [5, 5]1 by [0, 2].5. Use trace and zoom, or other graphing utility methods, to find the minima and maxima of this function. Find the derivative function
f(x) algebraically and graph f(x) and f(x) on the same axes using a viewing rectangle of [5, 5]1 by [2, 2].5. Use the trace and zoom to find the x intercepts
of f(x). Explain why the maximum or minimum of f(x) occurs at the x intercepts
of f (x).
49. Repeat Problem 48 for the product function f(x) x3 (x 3)2.
4
Marginal
Analysis:
Approximation
by Increments
Marginal analysis is an area of economics concerned with estimating the effect on
quantities such as cost, revenue, and profit when the level of production is changed
by a unit amount. For instance, if C(x) is the cost of producing x units of a certain
commodity, then the cost of producing the (x0 1)st unit is C(x0 1) C(x0). However, since the derivative of the cost function C(x), called marginal cost, is given by
MC(x) C(x) lim
hfi 0
C(x h) C(x)
h
134
Chapter 2
Differentiation: Basic Concepts
it follows that
MC(x0) C(x0 h) C(x0)
h
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 additional
unit is approximately equal to the marginal cost MC(x0). The geometric relationship
between C(x0 1) C(x0) and MC(x0) is shown in Figure 2.9.
(a) The marginal cost MC(x0) at
x x0 is C(x0).
y
y = C(x)
C(x0)
1
x
x0
(b) The cost of producing the
(x0 1)th unit is
C(x0 1) C(x0).
x0 + 1
y
y = C(x)
C(x0 + 1) – C(x0)
x
x0
x0 + 1
FIGURE 2.9 Marginal cost MC(x0) approximates C(x0 1) C(x0).
For future reference, here is the definition of marginal cost, together with analogous definitions for marginal revenue and marginal profit.
Chapter 2 ■ Section 4
Marginal Analysis: Approximation by Increments
135
Marginal Cost, Revenue, and Profit
■ If C(x) is the total cost
of producing x units of a commodity, and R(x) and P(x) R(x) C(x) are the
corresponding 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)
Explore!
Refer to Example 4.1. Graph
C(x) and R(x) on the same coordinate 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 tangent line on the same set of coordinate axes. Then change the
viewing rectangle to [6, 11]1 by
[120, 140]1 to see why the marginal cost is a good approximation to the actual change in
C(x). Continue finding an equation 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 tracing R(x). Move the cursor back
and forth between R(x) and the
tangent line to see why the marginal revenue is a good approximation to the actual change in
R(x).
EXAMPLE 4.1
A manufacturer estimates that when x units of a particular commodity are produced,
1
the total cost will be C(x) x2 3x 98 dollars and that all x units will be sold
8
1
when the price is p(x) 25 x dollars per unit.
3
(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 function to estimate the revenue derived from the sale of the ninth unit. What is the
actual revenue derived from the sale of the ninth unit?
(c) Find the profit associated with the production of x units. Sketch the profit function and determine the level of production where profit is maximized. What is the
marginal profit at this optimal level of production?
Solution
1
(a) The marginal cost function is MC(x) C(x) x 3, and the change in cost
4
1
as x increases from 8 to 9 (the ninth unit) is approximately MC(8) (8) 3
4
$5. The actual cost of the ninth unit is C(9) C(8) $5.13.
(b) The revenue function is
1
1
R(x) xp(x) x 25 x 25x x2
3
3
2
x. The revenue
3
2
derived from the sale of the ninth unit is approximately MR(8) 25 (8) 3
$19.67, and the actual revenue is R(9) R(8) $19.33.
and the marginal revenue function is MR(x) R(x) 25 136
Chapter 2
Differentiation: Basic Concepts
(c) The profit is
y
x
4.97
24
43.03
x
FIGURE 2.10 The graph of the
profit function
11 2
P (x) x 22x 98.
24
1
11 2
1
P(x) R(x) C(x) 25x x2 x2 3x 98 x 22x 98
3
8
24
11 2
and the graph of y x 22x 98 is a downward opening parabola with
24
its highest point (vertex) above
B
2A
22
24
11
2
24
(see Figure 2.10). Thus, profit is maximized when x 24 units are sold and the
1
price is p 25 (24) $17 per unit. The marginal profit function is MP(x) 3
11
P(x) x 22, and at the optimal level of production x 24, the marginal
12
11
profit is P(24) (24) 22 0.
12
Cost per unit of production is also important in economics. This function is called
average cost and its derivative is marginal average cost.
Average Cost and Marginal Average Cost
■ If C(x) is the
total cost associated with the production of x units of a particular commodity,
then
the average cost is AC(x) C(x)
x
and
marginal average cost is MAC (AC)(x)
Similar definitions apply to average revenue and average profit. Here is an example
involving average cost.
EXAMPLE 4.2
1
Let C(x) x2 3x 98 be the total cost function for the commodity in Example 4.1.
8
Chapter 2 ■ Section 4
Marginal Analysis: Approximation by Increments
137
(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
1 2
x 3x 98
C(x) 8
1
98
AC(x) x3
x
x
8
x
and the marginal average cost is
MAC MC(x) 1 98
2
8
x
(b) Marginal average cost is 0 when
1 98
2 0;
8
x
x2 8(98);
x 28
1
(c) The marginal cost is MC C(x) x 3, so marginal cost equals average cost
4
when
Note
1
1
98
x3 x3
4
8
x
1
98
x
8
x
2
x 98(8)
x 28
APPROXIMATION BY
INCREMENTS
In Example 4.1, the profit is maximized at the level of production where marginal profit is zero, and in Example 4.2, average cost is minimized when
average cost equals marginal cost. In Chapter 3, we use calculus to show that
both these results are consequences of general rules of economics.
Marginal analysis is an important example of a general approximation procedure
based on the fact that since
f(x0 h) f(x0)
hfi 0
h
f(x) lim
138
Chapter 2
Differentiation: Basic Concepts
Then for small h, the derivative f(x) is approximately equal to the difference quotient. That is,
f(x0) f(x0 h) f(x0)
h
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 and
summarize the incremental approximation formula as follows.
Approximation by Increments
x0 and x is a small change in x, then
■
If f(x) is differentiable at x f(x0 x) f(x0) f(x0)x
or, equivalently, if f f(x0 x) f(x0), then
f f(x0)x
Here is an example of how this approximation formula can be used in economics.
EXAMPLE 4.3
Suppose the total cost in dollars of manufacturing q units of a certain commodity is
C(q) 3q2 5q 10. If the current level of production is 40 units, estimate how
the total cost will change if 40.5 units are produced.
Solution
In this problem, the current value of production is q 40 and the change in production is q 0.5. By the approximation formula, the corresponding change in cost
is
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
Chapter 2 ■ Section 4
Explore!
Refer to Example 4.4. Conjecture the accuracy of your calculation 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.
Marginal Analysis: Approximation by Increments
139
For practice, compute the actual change in cost caused by the increase in the level
of production from 40 to 40.5 and compare your answer with the approximation. Is
the approximation a good one?
In the next example, the approximation formula is used to study propagation of
error. In particular, the derivative is used to estimate the maximum error in a calculation that is based on figures obtained through imperfect measurement.
EXAMPLE 4.4
You measure the side of a cube to be 12 centimeters long and conclude that the
volume of the cube is 123 1,728 cubic centimeters. If your measurement of the
side is accurate to within 2%, approximately how accurate is your calculation of
the volume?
Solution
The volume of the cube is V(x) x3, where x is the length of a side. The error you
make in computing the volume if you take the length of the side to be 12 when it is
really 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 much
as 0.02(12) 0.24 centimeter in either direction. Hence, the maximum error in your
measurement of the side is x 0.24, and the corresponding maximum error in
your 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 is
off by approximately 103.68 cubic centimeters.
In the next example, the desired change in the function is given, and the goal is
to estimate the necessary corresponding change in the variable.
140 Chapter 2
Differentiation: Basic Concepts
EXAMPLE 4.5
The daily output at a certain factory is Q(L) 900L1/3 units, where L denotes the
size of the labor force measured in worker-hours. Currently, 1,000 worker-hours of
labor are used each day. Use calculus to estimate the number of additional workerhours of labor that will be needed to increase daily output by 15 units.
Solution
Solve for L using the approximation formula
Q Q(L)L
Q 15
with
L 1,000
APPROXIMATION OF
PERCENTAGE CHANGE
Q(L) 300L2/3
15 300(1,000)2/3 L
to get
or
and
L 15
15
(1,000)2/3 (10)2 5 worker-hours
300
300
The percentage change of a quantity expresses the change in that quantity as a percentage of its size prior to the change. In particular,
Percentage change 100
change in quantity
size of quantity
This formula can be combined with the approximation formula and written in functional notation as follows.
■ If x
is a (small) change in x, the corresponding percentage change in the function
f(x) is
Approximation Formula for Percentage Change
Percentage change in f 100
f
f(x)x
100
f(x)
f(x)
EXAMPLE 4.6
The GDP of a certain country was N(t) t2 5t 200 billion dollars t years after
1994. Use calculus to estimate the percentage change in the GDP during the first quarter of 2002.
Chapter 2 ■ Section 4
Marginal Analysis: Approximation by Increments
141
Solution
Use the formula
Percentage change in N 100
with
t8
t 0.25
to get
Percentage change in N 100
and
N(t)t
N(t)
N(t) 2t 5
N(8)0.25
N(8)
[2(8) 5](0.25)
100 2
(8) 5(8) 200
1.73%
The next example illustrates how the percentage change can sometimes be estimated even though the numerical value of the variable is not known.
EXAMPLE 4.7
At a certain factory, the daily output is Q(K) 4,000K1/2 units, where K denotes the
firm’s capital investment. Use calculus to estimate the percentage increase in output
that will result from a 1% increase in capital investment.
Solution
The derivative is Q(K) 2,000K1/2. The fact that K increases by 1% means that
K 0.01K. Hence,
Q(K)K
Q(K)
2,000K1/2(0.01K)
100
4,000K1/2
2,000K1/2
since K1/2K K1/2
4,000K1/2
2,000
0.5%
4,000
Percentage change in Q 100
DIFFERENTIALS
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), the
differential of y is defined to be dy f(x) dx. The following summarizes this concept.
142
Chapter 2
Differentiation: Basic Concepts
Differentials ■ The differential of x is dx x, and if y f(x) is a
differentiable function of x, then dy f (x) dx is the differential of y.
EXAMPLE 4.8
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,
dy f(x) dx [(x2 5)(1 4x) (2x)(3 x 2x2)] dx
A geometric interpretation of the approximation of y by the differential dy is
shown in Figure 2.11. Note that since the slope of the tangent line at P(x, f(x)) is
f (x), the differential dy f(x) dx is the change in the height of the tangent that corresponds to a change from x to x x. On the other hand, y is the change in the
height of the curve corresponding to this change in x. Hence, approximating y by
the differential dy is the same as approximating the change in the height of a curve
by the change in height of the tangent line. If x is small, it is reasonable to expect
this to be a good approximation.
y
y = f (x)
dy
∆y
∆x
Tangent
line
x
x + ∆x
FIGURE 2.11 Approximation of y by the differential dy.
x
Chapter 2 ■ Section 4
Marginal Analysis: Approximation by Increments
P . R . O . B . L . E . M . S
143
2.4
MARGINAL ANALYSIS
1. A manufacturer’s total cost is C(q) 0.1q3 0.5q2 500q 200 dollars, where
q 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 q
units are produced and sold during the month. Currently, the manufacturer is producing 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 generated 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 particular
commodity 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.
1
1
3. C(x) x2 4x 57; p(x) (36 x)
5
4
1
1
4. C(x) x2 3x 67; p(x) (45 x)
4
5
1
5. C(x) x2 2x 39; p(x) x2 4x 10
3
5
6. C(x) x2 5x 73; p(x) x2 2x 13
9
3 2x
1
7. C(x) x2 43; p(x) 4
1x
12 2x
2
8. C(x) x2 65; p(x) 7
3x
9. Estimate how much the function f(x) x2 3x 5 will change as x increases
from 5 to 5.3.
144
Chapter 2
Differentiation: Basic Concepts
x
10. Estimate how much the function f(x) 3 will change as x decreases
x1
from 4 to 3.8.
11. Estimate the percentage change in the function f(x) x2 2x 9 as x increases
from 4 to 4.3.
12. Estimate the percentage change in the function f(x) 3x 2
as x decreases from
x
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 when
the level of production is q units. The current level of production is 4 units, and
the manufacturer is planning to increase this to 4.1 units. Estimate how the total
cost will change as a result.
MANUFACTURING
14. A manufacturer’s total monthly revenue is R(q) 240q 0.05q2 dollars when q
units are produced during the month. Currently, the manufacturer is producing 80
units a month and is planning to decrease the monthly output by 0.65 unit. Estimate 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 the
size of the labor force measured in worker-hours. Currently, 512 worker-hours of
labor are used each day. Estimate the number of additional worker-hours of labor
that will be needed to increase daily output by 12.5 units.
MANUFACTURING
1
16. A manufacturer’s total cost is C(q) q3 642q 400 dollars when q units
6
are produced. The current level of production is 4 units. Estimate the amount by
which 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-bedroom 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 half
of 2001.
ANNUAL EARNINGS
18. The gross annual earnings of a company were A(t) 0.1t2 10t 20 thousand
dollars t years after its formation in 1996. Estimate the percentage change in the
gross annual earnings during the third quarter of 2000.
EFFICIENCY
19. An efficiency study of the morning shift at a certain factory indicates that an average 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 the
worker 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 denotes
the capital investment measured in units of $1,000. The current capital investment
is $900,000. Estimate the effect that an additional capital investment of $800 will
have on the daily output.
Chapter 2 ■ Section 4
Marginal Analysis: Approximation by Increments
145
PRODUCTION
21. At a certain factory, the daily output is Q(L) 60,000L1/3 units, where L denotes
the size of the labor force measured in worker-hours. Currently 1,000 workerhours of labor are used each day. Estimate the effect on output that will be produced 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 denotes
the firm’s capital investment measured in units of $1,000 and L denotes the size
of the labor force measured in worker-hours. Suppose that the current capital
investment 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 investment
of $1,000 will have on the daily output if the size of the labor force is not
changed.
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 be
C(t) 100t2 400t 5,000. Estimate the amount by which the circulation will
increase 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
6
community will be P(t) 20 thousand. By approximately how much
t1
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 level
change during the coming 6 months?
AREA
27. You measure the radius of a circle to be 12 cm and use the formula A r2 to
calculate 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 is
decreased by 2%. Express your answer as a percentage and verify that your result
is 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 capital investment and L is the size of the labor force. Estimate the percentage
increase in output that will result from a 2% increase in the size of the labor force
if capital investment is not changed.
PRODUCTION
30. At a certain factory, the daily output is Q(K) 1,200K1/2 units, where K denotes
the firm’s capital investment. Estimate the percentage increase in capital investment that is needed to produce a 1.2% increase in output.
146
GROWTH OF A CELL
Chapter 2
Differentiation: Basic Concepts
4
31. A certain cell has the shape of a sphere. If the formulas S 4r2 and V r3
3
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 the
radius of a sphere if you want the error in the calculation of its volume using the
4
formula V r3 to be no greater than 8%.
3
VOLUME
33. A soccer ball made of leather
VOLUME
34. A melon in the form of a sphere has a rind
BLOOD CIRCULATION
EXPANSION OF MATERIAL
1
1
inch thick has an inner diameter of 8 inches.
8
2
Estimate the volume of its leather shell. [Hint: Think of the volume of the shell
as a certain change V in volume.]
1
inch thick and an inner diameter of
5
8 inches. Estimate what percentage of the total volume of the melon is rind.
35. In Problem 53, Section 1 of Chapter 1, we introduced an important law attributed
to the French physician, Jean Poiseuille. Another law discovered by Poiseuille says
that the volume of the fluid flowing through a small tube in unit time under fixed
pressure is given by the formula V kR4, where k is a positive constant and R
is the radius of the tube. This formula is used in medicine to determine how wide
a 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.*
36. The (linear) thermal expansion coefficient of an object is defined to be
L(T)
L(T)
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 1.4 105 per degree centigrade. Approximately how much will the length change during a year when the
temperature 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 the
formula R(T) kT 4, where R is the amount of energy emitted from a surface
whose temperature is T (in degrees Kelvin) and k is a positive constant. Estimate
the percentage change in R that results from a 2% increase in T.
* You may wish to begin your research by consulting such textbooks as Elaine N. Marieb, Human
Anatomy 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.
Chapter 2 ■ Section 4
Marginal Analysis: Approximation by Increments
147
Newton’s Method
■ Tangent line approximations can be used in a
variety 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 to
an actual root c, we can often obtain an improved guess by finding the x intercept x1 of the tangent line to the curve y f(x) at x x0 (see the figure). The
process can then be repeated until a desired degree of accuracy is attained. In
practice, it is usually easier and faster to use the graphing utility, zoom, and
trace features of your calculator to find roots, but the ideas behind Newton’s
method are still important. Problems 38 through 42 involve Newton’s method.
y
y = f(x)
c
x1 x
0
x
38. Show that when Newton’s method is applied repeatedly, the nth approximation is
obtained from the (n 1)st approximation by the formula
xn xn1 f(xn1)
f(xn1)
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 consecutive approximations agree to four decimal places.
(c) Take the root you found graphically in part (a) and the root you found by Newton’s method in part (b) and substitute each into the equation f(x) 0. Which
is more accurate?
40. Let f(x) x4 4x3 10. Use your graphing utility to graph f(x). Note that there
are two roots of the equation f(x) 0. Estimate each root using Newton’s method
and then check your results using zoom and trace or other utility features.
148
Chapter 2
Differentiation: Basic Concepts
41. The ancient Babylonians (circa 1700
formula
xn1 1
N
xn 2
xn
approximated N by applying the
B.C.)
for n 1, 2, 3, . . .
(a) Show that this formula can be derived from the formula for Newton’s method
in Problem 38, and then use it to estimate 1,265. Repeat the formula until
two consecutive approximations agree to four decimal places. Use your calculator to check your result.
(b) The spy wakes up one morning in Babylonia and finds that his calculator has
been stolen. Create a spy story problem based on using the ancient formula to
compute a square root.
42. Sometimes Newton’s method fails no matter what initial value x0 is chosen (unless
3
we are lucky enough to choose the root itself). Let f(x) x and choose x0 arbitrarily (x0 0).
(a) Show that xn1 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?
5
The Chain Rule
In many practical situations, you find that the rate at which one quantity is changing
can be expressed as the product of other rates. For example, suppose a car is traveling 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 would
multiply 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 the
number of units produced, which in turn is a function of the number of hours the factory has been operating. If C, q, and t denote the cost, units produced, and time,
respectively, then
dC
rate of change of cost
with respect to output
dq
(dollars per unit)
and
rate of change of output
dq
with respect to time
dt
(units per hour)
The product of these two rates is the rate of change of cost with respect to time; that
is,