Thursday, February 15, 2007

4.4 Optimization I

Optimization
Optimization is the process of finding the ABSOLUTE extrema in a given interval.



















The absolute maximimum and minimum are the largest and smallest y-values, respectively, within the interval. As seen in the picture, the absolute mimimum is the lowest point on the graph and the absolute minimum is the highest point on the graph.

At a given point "c":
If f(c) is the smallest value of f(x), then the point (c, f(c)) is the absolute minimum.
If f(c) is the largest value of f(x), then the point (c, f(c)) is the absolute maximum.

3 Steps to Finding Absolute Extrema
1. Find the critical points of f(x) that are within the given interval (a,b)
2. Compute the f(x) values for each of the critical points AND f(a) and f(b)
3. The point with the largest f(x) value is the absolute maximum and the point with the smallest f(x) value is the absolute minimum

Sample Problem:
Find the absolute extrema of the function on the interval [0,6]:



ANSWER:
In order to find the critical points we must use the derivative of the function and find its zeros. The zeros are -1.09 and .76, but because -1.09 is not within the interval, that number is discarded. Then we plug .76, 0, and 6 into the original equation. Once we find those values for f(x), we determine the absolute minimum and maximum.
















We find that the absolute maximum is at the point (6, 439) and the absolue minimum is at the point (.76, -1.34).

Another source for Optimization:
http://tutorial.math.lamar.edu/AllBrowsers/2413/AbsExtrema.asp

KJ, you're up next.

After that last quiz, I have a good quote for everyone:
"Don't worry about your difficulties in mathematics. I assure you mine are greater." - Albert Einstein

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