1. 4.7 Optimization Buffalo Bill’s Ranch, North Platte, Nebraska Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1999

2. 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.

3. 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?

4. To find the maximum (or minimum) value of a function: 1 Write it in terms of one variable. 2 Find the first derivative and set it equal to zero. 3 Check the end points if necessary.

5. Example 2: What dimensions for a one liter cylindrical can will use the least amount of material? We can minimize the material by minimizing the area. area of ends lateral area We need another equation that relates r and h :

6. Example 2: What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area

7. Example 3: A closed rectangular container with a square base is to have a volume of 2250 cubic inches. Cost of materials: Top and Bottom: $2 per square inch Sides: $3 per square inch. a) Find the dimensions of the container with the least cost. b) What is the cost?

8. If the end points could be the maximum or minimum, you have to check. 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. 