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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 .

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derivatives

Derivatives

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

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

Called differentiation

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Derivatives
  • Ex. Evaluate if
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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
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Derivatives
  • Differentiating.xls
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Derivatives
  • Use Differentiating.xls to graph the derivative of on the interval [-2, 8]. Then evaluate .
important
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]
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Derivatives
  • Properties

If then

If then

If then

If then

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

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Derivatives
  • Project (Marginal Revenue)

- Typically

- In project,

-

Why ?

recall revenue function r q
Recall:Revenue function-R(q)
  • Revenue in million dollars R(q)
  • Why do this conversion?

Marginal Revenue in dollars per drive

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Derivatives
  • Project (Marginal Cost)

- Typically

- In project,

-

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Derivatives
  • Project (Marginal Cost)

- Marginal Cost is given in original data

- Cost per unit at different production levels

- Use IF function in Excel

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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

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Derivatives
  • Project (Marginal Revenue)

- Calculate MR(q)

-

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Derivatives
  • Project (Marginal Cost)

- Calculate MC(q)

- IF(q<=500,115,IF(q<=1100,100,90))

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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

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Derivatives
  • Project (Maximum Profit)

- Create table for calculations

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Derivatives
  • Project (Answering Questions 1-3)

1. What price? $167.70

2. What quantity? 575,644 units

3. What profit? $9.87 million

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Derivatives
  • Project (Answering Question 4)

4. How sensitive? Somewhat sensitive

-0.2%

-4.7%

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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