Derivatives

Derivatives

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

Derivatives

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

Derivatives

• Ex. Evaluate if

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

Derivatives

• Differentiating.xls

Derivatives

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

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]

Derivatives

• Properties
• If then
• If then
• If then
• If then

Derivatives

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

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)

Derivatives

• Project (Marginal Revenue)
• - Typically
• - In project,
• -

• Why ?

Recall:Revenue function-R(q)

• Revenue in million dollars R(q)
• Why do this conversion?
• Marginal Revenue in dollars per drive

Derivatives

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

Derivatives

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

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

Derivatives

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

Derivatives

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

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

Derivatives

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

Derivatives

• Project (Answering Questions 1-3)
• 1. What price? $167.70
• 2. What quantity? 575,644 units
• 3. What profit? $9.87 million

Derivatives

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

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