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
MARGINAL
COST
MARGINAL
REVENUE
MARGINAL
PROFIT
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 change
X
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 = .25
N’(t) = 2t + 5
N
≈ 100 (2t + 5)(.25) / N(8)
N ≈ 1.73%
Differentials
Definitions:
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