MARGINAL ANALYSIS APPROXIMATIONS by INCREMEMENTS DIFFERENTIALS

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MARGINAL ANALYSISAPPROXIMATIONS by
INCREMEMENTSDIFFERENTIALS
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MARGINAL ANALYSIS

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

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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.

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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?

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Answers

1. C(x) (1/4) x 3
2. C(8) 5
3. C(9) C(8) 5.13

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• Quick discussion of Analysis of Results of 2.5.2

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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

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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.

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?C C(40) ?x

• ?x 0.5
• C(40) 245
• ?C 245 (0.5)
• ?C 122.50

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Analysis of the approximation

• The actual change
• X
• Change using the approximation Formula
• Q1 Is the approximation a good one?

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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)

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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.

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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

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Differentials

• Definitions
• The differential of x is dx ?x
• If y f(x) is a differentiable function of x,
  then
• dy f(x) dx is the differential of y

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• df f(x) dx
• ?f f(x0) ?x

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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