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Differentiation

Differentiation. Differentiation is all about measuring change! If the function is linear , such as: y = x. y = m x + c c = intercept m = constant slope i.e. the impact of a unit change in x on the level of y m = =. If the function is non-linear: y = x 2.

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Differentiation

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  1. Differentiation

  2. Differentiation is all about measuring change! If the function is linear, such as: y = x y = mx + c c = intercept m = constant slope i.e. the impact of a unit change in x on the level of y m = =

  3. If the function is non-linear: y = x2

  4. The slope of any curve is calculated by considering the tangent line that touches the curve at that point The slope varies along the curve line

  5. Example:A firms cost function is Y = X2

  6. The slope of the graph of a function is called the derivative of the function The process of differentiation involves letting the change in x become arbitrarily small, i.e. letting  x  0 e.g if = 2X+X and X 0  = 2X in the limit as X 0

  7. the slope tells us the change in y that results from a very small change in X We see the slope varies with X e.g. the curve at X = 2 has a slope = 4 and the curve at X = 4 has a slope = 8 In this example, the slope is steeper at higher values of X The slope of the non-linear function Y = X2 is 2X

  8. Rules for Differentiation

  9. 3. The Power Function Rule

  10. 4. The Sum-Difference Rule

  11. 5. The Product Rule

  12. Examples

  13. 6. The Quotient Rule If y = u/v where u and v are functions of x (u = f(x) and v = g(x) ) Then

  14. Example 1

  15. 7. The Chain Rule (Implicit Function Rule) If y is a function of v, and v is a function of x, then y is a function of x and

  16. Examples

  17. Examples 8. The Inverse Function Rule

  18. Differentiation in EconomicsApplication I • Total Costs = TC = FC + VC • Total Revenue = TR = P * Q •  = Profit = TR – TC • Break even:  = 0, or TR = TC • Profit Maximisation: MR = MC

  19. Calculating Marginal Functions Application I: Marginal Functions (Revenue, Costs and Profit)

  20. Example 1 • A firm faces the demand curve P=17-3Q • (i) Find an expression for TR in terms of Q • (ii) Find an expression for MR in terms of Q Solution: TR = P.Q = 17Q – 3Q2

  21. Example 2 A firms total cost curve is given by TC=Q3- 4Q2+12Q (i) Find an expression for AC in terms of Q (ii) Find an expression for MC in terms of Q (iii) When does AC=MC? (iv) When does the slope of AC=0? (v) Plot MC and AC curves and comment on the economic significance of their relationship

  22. Solution

  23. Solution continued….

  24. 9. Differentiating Exponential Functions

  25. Examples

  26. 10. Differentiating Natural Logs

  27. Proof

  28. Examples

  29. how does demand change with a change in price…… ed= Applications II

  30. Point elasticity of demand

  31. Example 1

  32. Example 2

  33. Application III: Differentiation of Natural Logs to find Proportional Changes

  34. Solution Continued…

  35. Example 2: If Price level at time t is P(t) = a+bt+ct2 Calculate the rate of inflation.

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