lecture9-09[1] - Math 18.02(Spring 2009 Lecture 9 Max/min...

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Math 18.02 (Spring 2009): Lecture 9 Max/min problems. Least squares approximation February 24 Last time: Partial derivatives. Tangent planes. Tangent plane approximation Today: Max/min problems. Least squares approximation Reading Material: From Simmos 19.7. From the Lecture Notes LS. In this lecture we will learn how to find the local maximum and minimum values of multivariable functions. To get an idea of a typical application let’s consider the following example: Exercise 1. [Post Office Problem] Suppose you are moving to Europe for a year and you are allowed to use the post office to send only ONE box. The post office has some restrictions on the size of boxes that one can send: Size limit: length + girth 9 feet. Here L = Length W = Width D = Depth and 2( W + D ) = girth . Of course you want to use the largest possible box, so you want to maximize the volume function by using the largest possible value given by the size limit: V = LWD L + 2( W + D ) = 9 From the second equation we write L in terms of W and D . L = 9 - 2( W + D ) and we replace it in V : V = LWD = [9 - 2( W + D )] WD = 9 WD - 2 W 2 D - 2 WD 2 maximize . So the final question is finding the maximum value of V ( W, D ) and the corresponding maximum point ( W 0 , D 0 ) (and as a consequence L 0 ) where this maximum is achieved.
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