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L20 - Lecture 20 Optimization We have now learned two main...

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Lecture 20 — Optimization We have now learned two main techniques to find the absolute maxima and minima of functions that are PROVIDED to us. One is the strategy for con- tinuous functions on closed, bounded intervals; we also can use graphical techniques to see the range of the function. In this lecture we must first apply modeling techniques to provide the function our- selves, and then use calculus to find the extrema. This will NOT generally be on a closed, bounded interval. A simple technique that is often used in optimization problems is the following: If c is the only critical number of f and there is a relative extremum at c (the derivative changes sign at c ), then f ( c ) is an absolute extreme value. Example: f ( x ) = xe - x on ( 0 , ) .
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Strategy for optimization problems: (1) Identify the quantity that is to be maximized or minimized. Assign some convenient symbol to represent it ( Q , for example). (2) Modeling!! Write an equation that represents your quantity from step one as a function of some other variables in the problem.
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