Applied Mathematics for Business and Economics

Lecture Note Differentiation

Δ= + − ≈R Rx() 1 Rx R x( ) ′( )

That is, the marginal cost Cx′() is an approximation to the cost Cx( +− (^1) )(Cx) of
producing the()x+1st unit, and similarly, the marginal revenue R′(x) is an
approximation to the revenue derived from the sale of the (x+ (^1) ). To
summarize:
stunit
Approximation by Marginal cost and Marginal Revenue
If Cx( )is the total cost of producing x units and R(x)is the total revenue
derived from the sale of x units, then
Marginal cost =Cx′()≈ the cost of producing the (x+1st) unit
Marginal revenue =R′(x)≈the revenue derived from the sale of the ()x+1st
unit
xample 5
re estimates that when x units of a particular commodity are produced,

E

A manufactu

the total cost will be ()^2

1

Cx=++x 398 x dollars, and that ()

8

()

1

Px 75 x

3

= − dollars

per unit is the price at l be sold.
a. Find the marginal cost and the marginal reve

which all x units wil

nue.

g the 9th unit.

he sale of the

Solut

The marginal cost is

b. Use marginal cost to estimate the cost of producin

c. What is the actual cost of producing the 9th unit?

d. Use the marginal revenue to estimate the revenue derived from t

9 th unit.

e. What is the actual revenue derived from the sale of the 9th unit?

ion

()

1

3

4

Cx′ = x+. Since x units of the commodity are

sold at a price of () (

1

75

3

Px x

a.

= − ) dollars per unit, the total revenue is

Rx( )=×(n ld) (price per unit)

() ()^2

umber of units so

11

75 25

33

== −=−xP x x⎡⎤⎢⎥x x x

⎣⎦

The marginal revenue is ()

2

25

3

R′ xx=−

b. The cost of producing the 9 th unit is the change in cost as x increase from 8

to 9 and can be estimated by the marginal cost

()

1

83 $5

4

Cx′ =×+=

c. The actual cost of producing the 9 th unit is

Δ= − =CC( (^9) ) C( (^8) ) $5.13
which is reasonable well approximated by the marginal cost
d. ated by the
arginal revenue:

C′()8$5=

The revenue obtained from the sale of the 9th unit is approxim

m