1 Derivatives Difference quotients are used in many business situations, other than marginal analysis (as in the previous section)

2 Derivatives Difference quotients Called the derivative of f(x) Computing Called differentiation

3 Derivatives Ex. Evaluate if

4 Derivatives Numerical differentiation is used to avoid tedious difference quotient calculations Differentiating.xls file (Numerical differentiation utility) Graphs both function and derivative Can evaluate function and derivative

5 Derivatives Differentiating.xls

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

7 Important If f '(x) is constant, the displayed plot will be distorted. To correct this, format the y-axis to have fixed minimum and maximum values. Eg: Lets try to plot g(x)=10x in [-2,8]

8 Derivatives Properties If then

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

10 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 = x – The slope of the graph of f at the point (3,14)

11 Derivatives Project (Marginal Revenue) - Typically - In project, - Why ?

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

13 Derivatives Project (Marginal Cost) - Typically - In project, -

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

15 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

16 Derivatives Project (Marginal Revenue) - Calculate MR(q) -

17 Derivatives Project (Marginal Cost) - Calculate MC(q) - IF(q<=500,115,IF(q<=1100,100,90))

18 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

19 Derivatives Project (Maximum Profit) - Create table for calculations

20 Derivatives Project (Answering Questions 1-3) 1. What price? $ What quantity? 575,644 units 3. What profit? $9.87 million

21 Derivatives Project (Answering Question 4) 4. How sensitive? Somewhat sensitive -0.2% -4.7%

22 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- 4