Chapter 2 Section 4 Marginal Analysis: Approximation by Increments133

Chapter 2 I Section 4

Marginal Analysis: Approximation by Increments

133

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

use the product rule to show that d (cf) c df if c is a constant.

dx

dx

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

g

1

h

f(x g(x

h) h)

f(x) g(x)

g(x)f(x

h) f(x)g(x g(x h)g(x)h

h)

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

45. Prove the power rule d (xn) nxn1 for the case where n p is a negative dx

integer.

[Hint:

Apply

the

quotient

rule

to

y

xp

1 x p]

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 47. Use a graphing utility to sketch the curve f(x) x 1 , and on the same

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.

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

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

by Increments

MC(x) C(x) lim C(x h) C(x)

hfi 0

h

134 Chapter 2

Differentiation: Basic Concepts

it follows that

MC(x0)

C(x0

h) h

C(x0)

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)

(b) The cost of producing the

(x0 1)th unit is C(x0 1) C(x0).

C(x0) 1

x 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 I Section 4

Marginal Analysis: Approximation by Increments

135

Marginal Cost, Revenue, and Profit I 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)

E x p l o r e ! EXAMPLE 4.1

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).

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

the total cost will be C(x) 1x2 3x 98 dollars and that all x units will be sold 8

when

the

price

is

p(x)

25

1 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 func-

tion 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 func-

tion and determine the level of production where profit is maximized. What is the

marginal profit at this optimal level of production?

Solution

(a)

The

marginal

cost

function

is

MC(x)

C(x)

1 x

3,

and

the

change

in

cost

4

as

x

increases

from

8

to

9

(the

ninth

unit)

is

approximately

MC(8)

1 (8)

3

4

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

(b) The revenue function is

R(x) xp(x) x 25 1 x 25x 1 x2

3

3

and

the

marginal

revenue

function

is

MR(x)

R(x)

25

2 x.

The

revenue

3

derived

from

the

sale

of

the

ninth

unit

is

approximately

MR(8)

25

2 (8)

3

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

136 Chapter 2

Differentiation: Basic Concepts

y

x 4.97 24 43.03

FIGURE 2.10 The graph of the

profit function

P (x)

11 24

x2

22x

98.

(c) The profit is

P(x) R(x) C(x) 25x 1 x2 1 x2 3x 98 11 x2 22x 98

3

8

24

and the graph of y 11x2 22x 98 is a downward opening parabola with 24

its highest point (vertex) above

x B 2A

22 11

24

2

24

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

price

is

p

25

1 (24)

$17

per

unit.

The

marginal

profit

function

is

MP(x)

3

P(x)

11 x

22,

and

at

the

optimal

level

of

production

x

24,

the

marginal

12

profit

is

P(24)

11 (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 I 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

Let C(x) 1x2 3x 98 be the total cost function for the commodity in Example 4.1. 8

Chapter 2 I 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 x2 3x 98

AC(x) C(x) 8

1 x 3 98

x

x

8

x

and the marginal average cost is

MAC

MC (x)

1 8

98 x2

(b) Marginal average cost is 0 when

1 8

98 x2

0;

x2 8(98);

x 28

(c)

The

marginal

cost

is

MC

C(x)

1 x

3,

so

marginal

cost

equals

average

cost

4

when

1 x 3 1 x 3 98

4

8

x

1 x

98

8x

x2 98(8)

x 28

Note

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(x) lim f(x0 h) f(x0)

hfi 0

h

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