Derivatives

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# Derivatives - PowerPoint PPT Presentation

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 .

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### 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

1. What price? \$167.70

2. What quantity? 575,644 units

3. What profit? \$9.87 million

Derivatives

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