4.4

Modeling and Optimization

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

Differentiable Functions

Essential QuestionsHow can we use differential calculus to solveoptimization problems?

Quick Review

3 2

3 2

1. Use the first derivative test to identify the local extrema of

6 12 9.

2. Use the second derivative test to identify the local extrema of

2 3 12 1

3. Find the volume of a cone with radiu

y x x x

y x x x

s 4 cm and height 7 cm.

Rewrite the expression as a trigonometric function of the angle .

4. sin( ) 5. cos( )

6. Use substitution to find the exact solution of the follo

2 2

wing system

of equations.

4

3

x y

y x

none

,91 ,2Max Local ,8 ,1Min Local 3cm

3

112

sin cos

,3 ,1

3 ,1

Strategy for Solving Max-Min Problems

Read the problem carefully. Identify the information

you need to solve the problem.

Draw pictures and label the

parts that are im

1. Understand the Problem

2. Develop a Mathematical Model of the Problem

portant to the problem. Introduce a variable to represent the

quantity to be maximized or minimized. Using that variable, write a function

whose extreme value gives the information sought.

3. Graph t Find the domain of the function. Determine what values

of the variable make sense in the problem.

Find where the derivative is zero or

fails to

he function

4. Identify the Critical Points and Endpoints

exist.

If unsure of the result, support or confirm your

solution with another method.

Translate your mathematical result into the problem setti

5. Solve the Mathematical Model

6. Interpret the Solution ng

and decide whether the result makes sense.

Example Inscribing Rectangles

1. A rectangle is to be inscribed under one arch of the sine curve. What is the largest area the rectangle can have, and what dimensions give that area?

xxP sin, xxQ sin,

h

xxxA sin2

2/0 x

02/0 AA

xxxA cos2 2sin x 0Graph it to find solutions.

The area of the rectangle is A(0.71) = 1.12.

71.0at 0 xxA

The length is

x2 xsin

1.72 and the height is 0.65.

Example Inscribing Rectangles

2. Two sides of a triangle have length a and b, and the angle between them is . What value of will maximize the triangle’s area?

sin

2

1 :Hint abA

0cos

cos

2

1ab

d

dA

2

A Right Triangle

0

a

b

Example Inscribing Rectangles3. You are a designing a rectangular poster to contain 50 in2 of printing with

a 4-in. margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the amount of paper used?

50

wwA 50

w

2 2

4

44w

8A 8 4w

4

508

A

4A

400 82

4A2

400

0

10 5w188 94 w

Maximum Profit Maximim profit (if any) occurs at a production level at which marginal

revenue equals marginal cost.

Example Maximizing Profit

3 2Suppose that ( ) 9 and ( ) 6 15 , where represents thousands

of units. Is there a production level that maximizes profit? If so, what is it?

r x x c x x x x x 4.

9 xr 15123 2 xxxc

915123 2 xx06123 2 xx0242 xx

22 x586.022 414.322

The maximum profit occurs at about 3.414.

Minimizing Average Cost The production level (if any) at which average cost is smallest is a

level at which the average cost equals the marginal cost.

5. Using the following equation where x represents thousands of units, determine if there is a production level that minimizes cost. If so, what is it? xxxxc 156 23

Marginal cost: xc 15123 2 xx

Average cost: x

xc1562 xx

15615123 22 xxxx062 2 xx

x2 03 x 3x

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