4.4

Optimization Optimization 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.

Optimization We have actually done an example of optimization in the past Given find where the height is at a maximum.

Optimization Steps 1: 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.

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

Example A rectangle is inscribed between the function and the x axis. Find the dimensions of the rectangle which produce the maximum area. What is that area?

Example A 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.

Example Find the dimensions of the rectangle with the smallest perimeter with an area of 49 ft 2.

Homework Pg 226 #1-6, 9, 10, 13