Derivatives Difference quotients are used in many business

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

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

Derivatives • Ex. Evaluate if 3

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 4

Derivatives • Differentiating. xls 5

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

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)=10 x in [-2, 8] 7

Derivatives • Properties If then If If then 8

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

Derivatives • Ex. Determine the equation of the tangent line to at x = 3. • Recall and we have the point (3, 14) The slope of the graph of f at the point (3, 14) • Tangent line is y = 5. 5452 x – 2. 6356 10

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

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

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

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

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 15

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

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

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 18

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

Derivatives • Project (Answering Questions 1 -3) 1. What price? $167. 70 2. What quantity? 575, 644 units 3. What profit? $9. 87 million 20

Derivatives • Project (Answering Question 4) 4. How sensitive? Somewhat sensitive -0. 2% -4. 7% 21

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 22