Understanding Derivatives: Applications in Business Analysis and Numerical Differentiation

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 If then If then 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 = 5.5452x – 2.6356 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? $167.70 2. 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