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Optimization

Optimization. 4.7. 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?. A Classic Problem.

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Optimization

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  1. Optimization 4.7

  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. 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

  6. Example 5: 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

  7. Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area

  8. A rectangular field, bounded on one side by a building, is to be fenced in on the other 3 sides. If 3,000 feet of fence is to be used, find the dimensions of the largest field that can be fenced in. w w 3000-2w CV at w = 750 Max at: w = 750 and l = 1500 Max Area = 1,125,000 sq. ft. Always concave down

  9. A physical fitness room consists of a rectangular region with a semicircle on each end. If the perimeter of the room is to be a 200 meter running track, find the dimensions that will make the area of the rectangular region as large as possible. x 2r We want to maximize the area of the rectangle. 2 variables, so lets solve the above equation for r Concave down

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