2 Differentiation is all about measuring change Measuring change in a linear function y a bx a intercept b constant slope ie the impact of a unit change in x on the level of y ID: 999223
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1. 1DifferentiationMona Kapoor
2. 2 Differentiation is all about measuring change! Measuring change in a linear function: y = a + bxa = interceptb = constant slope i.e. the impact of a unit change in x on the level of yb = =
3. 3If the function is non-linear: e.g. if y = x2
4. 4The slope of a curve is equal to the slope of the line (or tangent) that touches the curve at that pointwhich is different for different values of x
5. 5Example:A firms cost function is Y = X2
6. 6The slope of the graph of a function is called the derivative of the functionThe process of differentiation involves letting the change in x become arbitrarily small, i.e. letting x 0e.g if = 2X+X and X 0 = 2X in the limit as X 0
7. 7the slope of the non-linear function Y = X2 is 2Xthe slope tells us the change in y that results from a very small change in XWe 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 = 8In this example, the slope is steeper at higher values of X
8. 8Rules for Differentiation (section 4.3)
9. 9
10. 103. The Power Function Rule
11. 114. The Sum-Difference Rule
12. 125. The Product Rule
13. 13Examples
14. 146. The Quotient Rule If y = u/v where u and v are functions of x (u = f(x) and v = g(x) ) Then
15. 15Example 1
16. 167. 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
17. 17Examples
18. 188. The Inverse Function Rule Examples
19. 19 Differentiation in Economics Application ITotal Costs = TC = FC + VCTotal Revenue = TR = P * Q = Profit = TR – TCBreak even: = 0, or TR = TCProfit Maximisation: MR = MC
20. 20Application I: Marginal Functions (Revenue, Costs and Profit)Calculating Marginal Functions
21. 21Example 1A 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
22. 22Example 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
23. 23Solution
24. 24Solution continued….
25. 259. Differentiating Exponential Functions
26. 26Examples
27. 2710. Differentiating Natural Logs
28. 28
29. 29Proof
30. 30Examples
31. 31
32. 32Applications II how does demand change with a change in price……ed=
33. 33Point elasticity of demand
34. 34Example 1
35. 35Example 2
36. 36Application III: Differentiation of Natural Logs to find Proportional Changes
37. 37
38. 38Solution Continued…
39. 39Example 2: If Price level at time t is P(t) = a+bt+ct2 Calculate the rate of inflation.