MARGINAL ANALYSIS

APPROXIMATIONS by INCREMEMENTS

DIFFERENTIALS

MARGINAL ANALYSIS

Definition:

The use of the derivative to approximate the change in a quantity that results from a 1-unit increase in production

MARGINALREVENUE

MARGINAL PROFIT

MARGINALCOST

An Example of Marginal Analysis

A manufacturer estimates that when x units of digital cameras are produced, the total cost will be

C(x) = (1/8) x2 + 3x + 98 dollars,

that all units will sell when the price per unit is

P(x) = (1/3) (75-x) dollars.

Marginal Analysis

1. Find the marginal cost.

2. Use marginal cost to estimate the cost of producing the 9th unit.

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

Answers

1.C’(x) = (1/4) x + 3

2.C’(8) = $5

3.C(9) –C(8) = $5.13

• Quick discussion of Analysis of Results of 2.5.2

Approximation by Increments

Definition

If f(x) is differentiable at x = x0 and ∆x is a small change in x, then:

∆f ≈ f’(x0) ∆x

An Example of the Approximation Formula

Suppose the total cost in $ 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.

∆C ≈ C’(40) ∆x

∆x = 0.5

C’(40) = 245

∆C = 245 (0.5)∆C ≈ $122.50

Analysis of the approximation

The actual changeX

Change using the approximation Formula

Q1= Is the approximation a good one?

Percentage Change

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

100 ∆f/f(x) = 100 f’(x)∆x /f(x)

An Example of percentage change

The GDP of a certain country was

N(t) = t2 + 5t + 200 billions of dollars t years after 1997.

Estimate the percentage of change in the GDP during the first quarter of 2005.

Solution

N ≈ 100 N’(t) ∆t / N(t) where t = 8 ∆t = .25N’(t) = 2t + 5

N ≈ 100 (2t + 5)(.25) / N(8)

N ≈ 1.73%

DifferentialsDefinitions:

1. The differential of x is dx = ∆x

2. If y = f(x) is a differentiable function of x, then

dy = f’(x) dx is the differential of y

df ≈ f’(x) dx

∆f ≈ f’(x0) ∆x

An Example of Differentials

Find the differential of f(x) = x3 – 7x2 +2

Using the formula, dy = f’(x) dx

Answer: dy = (3x2 – 14x) dx