4.4 Modeling and Optimization, p. 219 AP Calculus AB/BC

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.

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?

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.

Inscribing Rectangles – A rectangle is to be inscribed in the parabola y = 4 – x 2 in [-2, 2]. What is the largest area the rectangle can have, and what dimensions give that area?  Day 1

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? 24” 18” x x x xx x x x

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? 24” 18” x x x xx x x x

Minimizing Perimeter – What is the smallest perimeter possible for a rectangle whose area is 36 in 2, and what are its dimensions? P = 2l + 2w P = 2 ∙ ∙ 6 P = 24 in.

Example 4: 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 :

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 in 2 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? l w

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.  Day 2