1. Optimization Chapter 4.4

2. Optimization
One of the most common applications of calculus involves the determination of minimum and maximum values. We know that the absolute maximum and minimum values of a function can occur at critical points or at endpoints of the domain.

3. Fencing Example
A rectangular fence enclosing 450 ft2 is to be built alongside a barn. What dimensions of the fence will minimize the amount of fencing required?

4. Guidelines for Optimization Problems
If there’s a picture to draw, draw it! Determine what the variables are and how they are related.
Identify the quantity that needs to be maximized or minimized, and write it as a function of the variables in the problem. This is called the objective function.
Identify the constraint(s), and write them in terms of the variables in the problem.
Use the constraint(s) to eliminate all but one independent variable in the objective function.
Determine the minimum and maximum allowable values (if any) of the independent variable. This is the domain.
Use calculus to solve the problem and be sure to answer the question that is asked.

5. Page Design Example
A rectangular page is to contain 36 square inches of print. The margins on each side are 1 ½ inches. Find the dimensions of the page so that the least amount of paper is used.
36 in2

6. Ladder Example
A 6-ft tall fence runs parallel to the wall of a house at a distance of 4 ft from the house. Find the length of the shortest ladder that extends from the ground, over the fence, to the house.

7. USPS Example
In order to mail a package with the U.S. Postal Service, the Box with a sum of length, width and height of the package must be no more than 108 inches. What are the dimensions of a square-based box of maximum volume that can be mailed under these restrictions?

8. Topless Box Example
Squares with sides x are cut out of each corner of a rectangular piece of cardboard measuring 25 inches by 20 inches. The sides of the resulting piece of cardboard are then folded up to create a box with no top. Find the volume of the largest box that can be formed this way.