Warmup- no calculator 1) 2)

4.4: Modeling and Optimization

ex. Find two numbers who’s sum is 20 and product is as large as possible Steps: 1)find a primary (what your optimizing) and secondary equation (concrete info in problem) 2) solve the secondary for one variable 3) substitute it into the primary 4) find extrema of the function check endpoints and critical #’s Keys to Optimization primary: f(x,y) = xy secondary: x + y = 20 y = -x+20  f(x)= x(-x+20)  f(x) = -x 2 +20x =0 x = 10, so y = 10

Find your: primary equation (idea your optimizing) secondary equation (additional info in problem).

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.

Ex. A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if none is needed along the river?

Four feet of wire is to be used to form a square and a circle. How much wire should be used for the square and how much should be used for the circle to enclose a maximum area?

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

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. 

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