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2.6


Marginals and DifferentialsOBJECTIVE• Find marginal cost, revenue, and profit.• Find ∆y and dy.• Use differentials for approximations.


DEFINITIONS:

Let C(x), R(x), and P(x) represent, respectively, the total cost, revenue, and profit from the production and sale of x items.

The marginal cost at x, given by C (x), is the approximate cost of the (x + 1)th item:

C (x) ≈ C(x + 1) – C(x), or C(x + 1) ≈ C(x) + C (x).

2.6 Marginals and Differentials


DEFINITIONS (concluded):

The marginal revenue at x, given by R (x), is the approximate revenue from the (x + 1)th item:

R (x) ≈ R(x + 1) – R(x), or R(x + 1) ≈ R(x) + R (x).

The marginal profit at x, given by P (x), is the approximate profit from the (x + 1)th item:

P (x) ≈ P(x + 1) – P(x), or P(x + 1) ≈ P(x) + P (x).



Example 1: Given

find each of the following:a) Total profit, P(x).b) Total cost, revenue, and profit from the

production and sale of 50 units of the product.c) The marginal cost, revenue, and profit when 50

units are produced and sold.



Example 1 (continued):a)

b)


( )P x ( ) ( )R x C x3 2 212 40 10 (62 27,500)x x x x 3 274 40 10x x x

( )P x ( )P x

(50)C(50)R(50)P

262(50) 27,5003 2(50) 12(50) 40(50) 10 3 2(50) 74(50) 40(50) 10

$182,500$97,010$85,490


Example 1 (concluded):c)

So, when 50 units have been made, the approximate cost of the 51st unit will be $6200, and the approximate revenue from the sale of the 51st unit will be $6340 for an approximate profit on the 51st unit of $140.


( )C x

(50)C

( )R x

(50)R

( )P x

(50)P

124x124(50)23 24 40x x

23(50) 24(50) 40 23 148 40x x

23(50) 148(50) 40

$6200

$6340

$140


Example 2: For x = 4, and ∆x = 0.1, find ∆y.


y ( ) ( )f x x f x 2 2(4 0.1) (4)

2 24.1 416.81 16

0.81

y y y y


2.6 Marginals and DifferentialsQuick Check 1

For , and , find .42 , 2y x x x 0.05x y

( ) ( )y f x x f x 4 4[2(2 0.05) (2 0.05)] [2(2) 2]y

4 4[2(1.95) 1.95] [2(2) 2]y

[2(14.45900625) 1.95] [2(16) 2]y

30.8680125 34y

3.1319875y


For f a continuous, differentiable function, and small ∆x.



Example 3: Approximate using

Let and x equal a number close to 27 and A number for which the square root is easy tocompute. So, here let x = 25 with ∆x = 2. Then,


( )f x

y

1 and

2 x(25) 2f 1 2

2 25

0.2

y

y


Example 3 (concluded):Now we can approximate


27 25 y

5 y

5 0.2

5.2

27

27

27



Quick Check 2

Approximate using . To five decimal places,

. How close is your approximation?

Let and equal a number close to 98 and a number for which the square root is easy to compute. So here let with . Then,

98 ( )y f x x

98 9.89949

( )f x x x100,x

2x 1( )

2f x

x and

(100) 2y f 1 2

2 100y

0.1y



Quick Check 2 Concluded

Now we can approximate :

This is within 0.001 of the actual value of

98

98 100 y

98 10 y

98 10 0.1

98 9.9

98.


DEFINITION:

For y = f (x), we define

dx, called the differential of x, by dx = ∆xand

dy, called the differential of y, by dy = f (x)dx.



Example 4: For a) Find dy.b) Find dy when x = 5 and dx = 0.2.

a)


dydx

2 33(4 ) 1 (4 ) 1x x x

2(4 ) ( 3 4 )x x x

2(4 ) ( 4 4)x x

24(4 ) ( 1)x x

24(4 ) ( 1)x x dx

dydx

dydx

dydx

dydx


Example 4 (concluded):

b)


dy 24(4 5) (4 1) 0.2 3.2dy



Section Summary• If represents the cost for producing items, then marginal

cost is its derivative, and . Thus, the cost to produce the can be approximated by

• If represents the revenue from selling items, then marginal revenue is its derivative, and . Thus, the revenue from the item can be approximated by

( )C x x( )C x ( ) ( 1) ( )C x C x C x

( 1)stx

( 1) ( ) ( ).C x C x C x

( )R x x( )R x ( ) ( 1) ( )R x R x R x

( 1)stx

( 1) ( ) ( ).R x R x R x


2.6 Marginals and DifferentialsSection Summary Continued

• If represents profit from selling items, then marginal profit a is its derivative, and . Thus, the profit from the item can be approximated by

• In general, profit = revenue – cost, or• In delta notation, , and .

For small values of , we have which is equivalent to

• The differential of is . Since , we have . In general, , and the approximation can be very

close for sufficiently small .

( )P x x( )P x ( ) ( 1) ( )P x P x P x

( 1)stx ( 1) ( ) ( ).P x P x P x

( ) ( ) ( ).P x R x C x

( )x x h x h ( ) ( )y f x h f x x ( )y f x

x

( ) .y f x x

x dx x ( )dy f xdx

( )dy f x dx dy ydx

copyright © 2016, 2012 pearson education, inc. 2.6 - 1

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