5.4 Modeling and Optimization

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4.4 Modeling and Optimization

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Presentation transcript:

5.4 Modeling and Optimization AP Calculus AB 5.4 Modeling and Optimization

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.

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”

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”

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

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

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

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 Day 2