1 Derivatives Difference quotients are used in many business situations, other than marginal...

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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

h

hxfhxflimxfmh 20

xf

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Derivatives

• Ex. Evaluate if

62 xxf

5452.5002.0

0110903561.0002.0

99445674.130055471.14001.02

62623

999.2001.3

f

3f

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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.xlsIncrement

x f (x ) f ' (x ) a b h s

= = #VALUE! 0.000001 t

u

v

w

29303132333435

ConstantsDefinition Plot Interval

Formula for f (x )

Computation

FUNCTION

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

x

f (x )

DERIVATIVE

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

x

f ' (x )

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Derivatives• Use Differentiating.xls to graph the derivative of

on the interval [-2, 8]. Then evaluate .

62 xxf 3f DERIVATIVE

0

50

100

150

200

-5 0 5 10

x

f ' (x )

5452.53 f

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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

xfaxg xfaxg

xgxfxh xgxfxh

bmxxf mxf

cxf 0 xf

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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

62 xxf

5452.53 f

3f The slope of the graph of f at the point (3,14)

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Derivatives• Project (Marginal Revenue)

- Typically

- In project,

-

qRqMR

qRqMR 1000

h

hqRhqRqR

2

Why ?

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Recall:Revenue function-R(q)• Revenue in million dollars R(q)

• Why do this conversion?Marginal Revenue in dollars per drive

qRqMR

qR

qRqMR

10001000

1000000

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Derivatives• Project (Marginal Cost)

- Typically

- In project,

-

qCqMC

qCqMC 1000

h

hqChqCqC

2

14


- 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)

-

h

hqDhqhqDhqh

h

hqRhqR

qRqMR

hqDhqhqDhq

2

21000

21000

1000

10001000

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- 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

q D (q ) R (q ) C (q ) P (q ) MR (q ) MC (q ) MP (q )575.644 167.70$ 96.53$ 86.66$ 9.87$ 100.000$ 100.000$ 0.000$

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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%



q D (q ) R (q ) C (q ) P (q ) MR (q ) MC (q ) MP (q )585.644 166.51$ 97.52$ 87.66$ 9.85$ 96.287$ 100.000$ (3.713)$


q D (q ) R (q ) C (q ) P (q ) MR (q ) MC (q ) MP (q )625.644 161.53$ 101.06$ 91.66$ 9.40$ 80.745$ 100.000$ (19.255)$

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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

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