Optimization4.4

OptimizationOptimization is one of the most useful

applications of the derivative. It is the process of finding when something is at a maximum or minimum value.

Example: If in production I can fit my profit to a function, If I find the point where that function is at a maximum, I can find how exactly to make the most money.

OptimizationWe have actually done an example of

optimization in the pastGiven find where the

height is at a maximum.

2( ) 16 96 112s t t t

OptimizationSteps1: Identify what it is that you are trying to

maximize or minimize. (Draw a picture whenever appropriate)

2: Find an equation for that value in terms of ONE variable.

3: Perform the first derivative test to identify the appropriate point.

4: Answer the question that was asked.

ExampleA rectangle is inscribed between the Function and the x axis.What dimensions give the maximum area of the

rectangle? What is the maximum area?

6

5

4

3

2

1

-1

-4 -2 2 4 6 8 102( ) 5f x x x

ExampleA rectangle is inscribed betweenthe function and the x axis. Find the

dimensions of the rectangle which produce the maximum area. What is that area?

3

2.5

2

1.5

1

0.5

-0.5

-1

-4 -3 -2 -1 1 2 3 4

2( ) 9f x x

ExampleA farmer has 2400 ft of fencing to build a

rectangular field that boarders a straight river. No fencing is needed along the river. Find the dimensions of the field with the largest area.

ExampleFind the dimensions of the rectangle with the

smallest perimeter with an area of 49 ft2.

HomeworkPg 226 #1-6, 9, 10, 13