1. Differentiation Purpose- to determine instantaneous rate of change Eg: instantaneous rate of change in total cost per unit of the good We will learn Marginal Demand, Marginal Revenue, Marginal Cost, and Marginal Profit

2. Marginal Cost : MC(q) What is Marginal cost? The cost per unit at a given level of production EX: Recall Dinner problem C(q) = C 0 + VC(q). = 9000+177*q 0.633 MC(q)- the marginal cost at q dinners MC(100)- gives us the marginal cost at 100 dinners This means the cost per unit at 100 dinners How to find MC(q) ? We will learn 3 plans

3. Marginal Analysis First Plan Cost of one more unit

4. Marginal Analysis Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500).

5. Marginal Analysis Second Plan Average cost of one more and one less unit

6. Marginal Analysis Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500).

7. Marginal Analysis Final Plan Average cost of fractionally more and fractionally less units difference quotients Typically use with h = 0.001

8. Marginal Analysis Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500). In terms of money, the marginal cost at the production level of 500, $6.71 per unit

9. Marginal Analysis Use “Final Plan” to determine answers All marginal functions defined similarly

10. Marginal Analysis Graphs D(q) is always decreasing All the difference quotients for marginal demand are negative MD(q) is always negative

11. Marginal Analysis Graphs R(q) is increasing difference quotients for marginal revenue are positive MR(q) is positive R(q) is decreasing difference quotients for marginal revenue are negative MR(q) is negative Maximum revenue Marginal revenue 0 q1q1 q1q1

12. Marginal Analysis Graphs

13. Derivatives-part 2 Difference quotients Called the derivative of f(x)

14. Derivatives Ex1. Evaluate if

15. Derivatives Differentiating.xls file Graphs both function and derivative Can evaluate function and derivative

16. Derivatives Differentiating.xls

17. Derivatives Use Differentiating.xls to graph the derivative of on the interval [-2, 8]. Then evaluate.

19. Differentiating.xslm Key points 1)Formula for the function in x 2)Plot Interval is ESSENTIAL 3)You can use the computation cells to evaluate the f(x) & f’(x) at different values 4)If using Office 2007, save it as a macro enabled file

20. Derivatives Properties If (c is a constant) then If (m is a constant) then If then

21. Derivatives Tangent line approximations Useful for easy approximations to complicated functions Need a point and a slope (a derivative) Use y = mx +b

22. Derivatives Ex. Determine the equation of the tangent line to at x = 3. Recall and we have the point (3, 14) Tangent line is y = 5.5452x – 2.6356

23. Derivatives Project (Marginal Revenue) - Typically - In project, - units are

24. 24 Recall:Revenue function-R(q) Revenue in million dollars R(q) Why do this conversion? Marginal Revenue in dollars per drive

25. Derivatives Project (Marginal Cost) - Typically - In project, - units are

26. Derivatives Project (Marginal Cost) - Marginal Cost is given in original data - Cost per unit at different production levels - Use IF function in Excel

27. Derivatives Project (Marginal Profit) MP(q) = MR(q) – MC(q) - If MP(q) > 0, profit is increasing - If MR(q) > MC(q), profit is increasing - If MP(q) < 0, profit is decreasing - If MR(q) < MC(q), profit is decreasing

28. Derivatives Project (Marginal Cost) - Calculate MC(q) Nested If function, the if function using values for Q1-4 & 6 - IF(q<=800,160,IF(q<=1200,128,72)) In the GOLDEN sheet need to use cell referencing for IF function because we will make copies of it, and do other project questions =IF(B30<$E$20,$D$20,IF(B30<$E$22,$D$21,$D$22))

29. Recall -Production cost estimates Fixed overhead cost - $ 135,000,000 Variable cost (Used for the MC(q) function) 1)First 800,000 - $ 160 per drive 2)Next 400,000- $ 128 per drive 3)All drives after the first 1,200,000- $ 72 per drive

30. Derivatives Project (Maximum Profit) - Maximum profit occurs when MP(q) = 0 - Max profit occurs when MR(q) = MC(q) - Estimate quantity from graph of Profit - Estimate quantity from graph of Marginal Profit

31. Derivatives-change Project (Answering Questions 1-3) 1. What price? $285.88 2. What quantity? 1262(K’s) units 3. What profit? $42.17 million

32. Derivatives Project (What to do) - Create one graph showing MR and MC - Create one graph showing MP - Prepare computational cells answering your team’s questions 1- 3

33. Marginal Analysis-  where h = 0.000001  MR(q) = R′(q) ∙ 1,000  Marketing Project

34. Marginal Analysis-  where h = 0.000001  In Excel we use derivative of R(q)  R(q)=aq^3+bq^2+cq  R’(q)=(a*3*q^2+b*2*q+c)/1000  Marketing Project

35. Marginal Analysis (continued)- Marketing Project

36. Marginal Analysis  MP(q) = MR(q) – MC(q)  We will use Solver to find the exact value of q for which MP(q) = 0. Here we estimate from the graph Marketing Project

37. Profit Function  The profit function, P(q), gives the relationship between the profit and quantity produced and sold.  P(q) = R(q) – C(q)

38. 38 Goals 1. What price should Card Tech put on the drives, in order to achieve the maximum profit? 2. How many drives might they expect to sell at the optimal price? 3. What maximum profit can be expected from sales of the 12-GB? 4. How sensitive is profit to changes from the optimal quantity of drives, as found in Question 2? 5. What is the consumer surplus if profit is maximized?

39. 39 Goals-Contd. 6. What profit could Card Tech expect, if they price the drives at $299.99? 7. How much should Card Tech pay for an advertising campaign that would increase demand for the 12-GB drives by 10% at all price levels? 8. How would the 10% increase in demand effect the optimal price of the drives? 9. Would it be wise for Card Tech to put $15,000,000 into training and streamlining which would reduce the variable production costs by 7% for the coming year?