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4.4 Modeling and Optimization, p. 219

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

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4.4 Modeling and Optimization, p. 219

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

  2. 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?

  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. 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. To find the maximum (or minimum) value of a function:

  5. Inscribing Rectangles – A rectangle is to be inscribed in the parabola y = 4 – x2 in [-2, 2]. What is the largest area the rectangle can have, and what dimensions give that area? p Day 1

  6. 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”

  7. 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”

  8. Minimizing Perimeter – What is the smallest perimeter possible for a rectangle whose area is 36 in2, and what are its dimensions? P = 2l + 2w P = 2 ∙ 6 + 2 ∙ 6 P = 2 ∙ 6 + 2 ∙ 6 P = 24 in.

  9. 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

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

  11. 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

  12. 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

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