2.5. Marginal Analysis and Approximation using Increments

2.5. Marginal Analysis and
Increments

Marginal Cost

If C(x) is the total cost of produc
then the marginal cost of produc
C (x0), which approximate the a
incurred when the level of produ
from x0 to x0 + 1.

Marginal Revenue and Margin

The marginal revenue is R (x0).
R(x0 + 1) − R(x0), the additiona
producing one more unit.
The marginal profit is P (x0). It a
the additional profit obtained by

Approximation using

cing x units of a commodity,
cing x0 units is the derivative
additional cost C(x0 + 1) − C(x0)
uction is increased by one unit,

nal Profit

It approximates
al revenue generated by

approximates P(x0 + 1) − P(x0),
producing one more unit.

Marginal Analysis

Example

C(x) = 1 x2 + 3x + 67 is the tot
4

p(x) = 1 (45 − x) is the price at
5

(a) Find the marginal cost and t
(b) Use marginal cost to estima

fourth unit.
(c) Find the actual cost of produ
(d) Use marginal revenue to est

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

unit.

tal cost of producing x units and
which all x units will be sold.
the marginal revenue.
ate the cost of producing the

ucing the fourth unit.
timate the revenue derived from

ived from the sale of the fourth

Approximation by Incremen

If f (x) is differentiable at x = x0
then

f (x0 + ∆x) ≈ f (
or, equivalently, if ∆f = f (x0 + ∆

∆f ≈ f

nts

and ∆x is a small change in x,
(x0) + f (x0)∆x
∆x) − f (x0), then
(x )∆x

Approximation by Incremen

Example

A 5-year projection of population
from now, the population of certa
P(t) = −t3 + 9t2 + 48t + 200 th

(a) Find the rate of change of po
respect to time t.

(b) At what rate does the popula
with respect to time?

(c) Use increments to estimate
the first month of the fourth y
in R(t) during this time perio

nts

n trends suggests that t years
ain community will be
housand.
opulation R(t) = P (t) with

ation growth rate R(t) change

how much R(t) changes during
year. What is the actual change
od?

Approximation by Incremen

Approximation formula for Pe

If ∆x is a (small) change in x, th
change in the function f (x) is

Percentage change in f =

Example

Use increments to estimate the
2

function f (x) = 3x + x as x dec

nts

ercentage Change

he corresponding percentage

= 100 ∆f ≈ 100 f (x)∆x
f (x) f (x)

percentage change in the
creases from 5 to 4.6.