Slide 1

Modeling and
Optimization
Section 4.4

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Slide 2

Quick Review Solutions
1. U se the first derivative test to iden tify the local extrem a of
y  x  6 x  12 x  9.
3

2

none

2. U se the second derivative test to ide ntify the local extrem a of
y  2 x  3 x  12 x  1 L ocal m axim um : (-2, 19); L ocal m inim um (1, -8)
3

2

3. Find the volum e of a cone w ith radius 4 cm and height 7 cm .

112 

cm

3

3
R ew rite the expression as a trigonom etri c function of the angle  .
4. sin (   )  sin 

5. cos(   )

cos 

6. U se substitution to find the exact so lution of the follow ing system
of equat ion s.

1, 3  ;   1 , 

3



x  y  4

 y  3x
2

2

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Slide 3

What you’ll learn about
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Examples from Mathematics
Examples from Business and Industry
Examples from Economics
Modeling Discrete Phenomena with Differentiable
Functions

…and why
Historically, optimization problems were among the
earliest applications of what we now call differential
calculus.
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Slide 4

Strategy for Solving Max-Min Problems
1. U n d erstan d th e P rob lem R ead the problem carefully. Identify th e inform ation
you need to solve the problem .
2. D evelop a M ath em atical M od el of th e P rob lem D raw pictures and label the
parts that are im portant to the problem . Introduce a var iable to represent the
quantity to be m axim ized or m inim ized. U sing that variable, w rite a function
w hose extrem e value gives the inform atio n sought.
3. G rap h t h e fu n ction F ind the dom ain of the function. D eterm ine w hat values
of the variable m ake sense in the proble m .
4. Id en tify th e C ritical P oin ts an d E n d p oin ts F ind w here the derivative is zero or
fails to exist.
5. S olve th e M ath em atical M od el If unsure of the result, support or con firm your
solution w ith another m ethod.
6. In terp ret th e S olu tion T ranslate your m athem atical result into the problem setti n g
and decide w hether the result m akes sens e.
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Slide 5

Example Inscribing Rectangles
A rectangle is to be inscribed under one arch of the sine curve. W hat is the la rgest
area the rectangle can have, and w hat di m ensions give that area?

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Slide 6

Maximum Profit
M axim im profit (if any) occurs at a prod uction level at w hich m arginal
revenue equals m arginal cost.

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Slide 7

Example Maximizing Profit
S uppose that r ( x )  9 x and c ( x )  x  6 x  15 x , w here x repre sents thousands
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2

of units. Is there a production level th at m axim izes profit? If so, w hat is it?

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Slide 8

Minimizing Average Cost
T he production level (if any) at w hich a verage cost is sm allest is a
level at w hich the average cost equals t he m arginal cost.

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Slide 9

Page 214 (1-10)
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Pages 215-216
(13-17, 19, 25, 43, 49)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

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