1. MARGINAL ANALYSIS APPROXIMATIONS by INCREMEMENTS DIFFERENTIALS

2. MARGINAL ANALYSIS Definition: The use of the derivative to approximate the change in a quantity that results from a 1-unit increase in production

3. MARGINAL REVENUE MARGINAL PROFIT MARGINAL COST

4. 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) x 2 + 3x + 98 dollars, that all units will sell when the price per unit is P(x) = (1/3) (75-x) dollars.

5. Marginal Analysis 1.Find the marginal cost. 2.Use marginal cost to estimate the cost of producing the 9 th unit. 3.What is the actual cost of producing the 9 th unit?

6. Answers 1.C’(x) = (1/4) x + 3 2.C’(8) = $5 3.C(9) –C(8) = $5.13

7. Quick discussion of Analysis of Results of 2.5.2

8. Approximation by Increments Definition If f(x) is differentiable at x = x 0 and ∆x is a small change in x, then: ∆f ≈ f’(x 0 ) ∆x

9. An Example of the Approximation Formula Suppose the total cost in $ of manufacturing q units of a certain commodity is C (q) = 3q 2 + 5q + 10. If the current level of production is 40 units, estimate how the total cost will change if 40.5 units are produced.

10. ∆C ≈ C’(40) ∆x ∆x = 0.5 C’(40) = 245 ∆C = 245 (0.5) ∆C ≈ $122.50

11. Analysis of the approximation The actual change X Change using the approximation Formula Q1= Is the approximation a good one?

12. 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)

13. An Example of percentage change The GDP of a certain country was N(t) = t 2 + 5t + 200 billions of dollars t years after 1997. Estimate the percentage of change in the GDP during the first quarter of 2005.

14. 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%

15. 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

16. df ≈ f’(x) dx ∆f ≈ f’(x 0 ) ∆x

17. An Example of Differentials Find the differential of f(x) = x 3 – 7x 2 +2 Using the formula, dy = f’(x) dx Answer: dy = (3x 2 – 14x) dx