1. Optimization Buffalo Bill’s Ranch, North Platte, Nebraska
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1999

2. Optimization Practical Problems such as:
What’s the most profit you can make?
What’s the least amount of materials needed?
What’s the shortest distance?

3. Steps in Solving Optimization Problems
Understand the Problem….Read carefully!
What is the unknown?
What are the given quantities?
What are the given conditions?
Draw a Picture
Identify the givens and the unknowns on the diagram

4. Steps in Solving Optimization Problems
Introduce Notation
It may help to use letters that are suggestive
of the quantities they represent
A for area, h for height, t for time, etc.
Use derivatives for rates of change

5. Steps in Solving Optimization Problems
Express what you want to optimize in terms of the other symbols
You may have to find relationships (equations)
among these variables
Use these equations to eliminate all but one variable
in the optimization equation

6. Steps in Solving Optimization Problems
Find the absolute maximum or minimum value of the optimization function
If the function is on a closed interval, remember to check the endpoints!

7. A Classic Problem
You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?
There must be a local maximum here, since the endpoints are minimums.

8. A Classic Problem
You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

9. We can minimize the material by minimizing the area.
Example 5:
What dimensions for a one liter cylindrical can will use the least amount of material?
Motor
Oil
We can minimize the material by minimizing the area.
We need another equation that relates r and h:
area of
ends
lateral
area

10. Example 5:
What dimensions for a one liter cylindrical can will use the least amount of material?
area of
ends
lateral
area

11. Notes:
If the function that you want to optimize has more than one variable, use substitution to rewrite the function.
If you are not sure that the extreme you’ve found is a maximum or a minimum, you have to check.
If the end points could be the maximum or minimum, you have to check. p