1. 5.4 Modeling and Optimization
AP Calculus AB
5.4 Modeling and Optimization

2. 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.

3. Fabricating A Box – An open-top box is to made by cutting congruent squares of side length x from the corners of a 18- by 24-inch sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible? What is the resulting maximum volume? x x x x
18” x x x x
24”

4. Fabricating A Box – An open-top box is to made by cutting congruent squares of side length x from the corners of a 18- by 24-inch sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible? What is the resulting maximum volume? x x x x
18” x x x x
24”

5. We can minimize the material by minimizing the area.
Example 4:
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

6. Example 4:
What dimensions for a one liter cylindrical can will use the least amount of material?
area of
ends
lateral
area

7. Designing a Poster – You are designing a rectangular poster to contain 72 in2 of printing with a 1-in. margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the amount of paper used? 1 w 2 2 l 1

8. 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.
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