# 9.3 -   MATH 1081  Wednesday, March 30

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Unformatted text preview:   MATH 1081  Wednesday, March 30    Chapter 9 – Section 3    MAXIMA AND MINIMA    Homework #10 (due April 4):    Section 9.2 #32, 40, 48, 60                      Section 9.3 #14, 36, 38    Clicker Check­in:  Choose any letter to check in now.  When finding the relative extrema of a function of a single  variable, we began by finding the critical numbers of the  function (points in the domain where the derivative is zero or  the derivative does not exist).  This gave us a list of potential  maxima and minima.    We then had to use some kind of test to classify each of those  critical numbers.  Depending on the result of the test, we could  conclude that there was a maximum, a minimum, or neither at  each of those critical numbers.    In particular, there was the Second Derivative Test for a  function of a single variable.      So, the sign of the second derivative at the critical number  helped us classify it as a maximum or a minimum.      For a function of two variables,  f ( x, y ) , we first find the critical  points.  In this case, the critical points are the points at which  BOTH  f x ( x, y ) = 0  and  f y ( x, y ) = 0 , simultaneously.    We then will test each of these critical numbers that we found  2 by computing the value  D = f xx ⋅ f yy − ⎡ f xy ⎤  at the point.  The  ⎣⎦ sign of  D  (a mix of the second partial derivatives) helps us  classify our critical number as a maximum, a minimum, or  neither.         Example:  Find all points where the function has a relative    extrema.          1.   f ( x, y ) = 2 x 2 + 3xy + 2 y 2 − 5 x + 5 y                     2.   z = − y 4 + 4 xy − 2 x 2 + 1  Example:  Section 9.3 #33:  Suppose that the profit (in   hundreds of dollars) of a certain firm is  approximated by the function  P ( x, y ) = 1500 + 36 x − 1.5 x 2 + 120 y − 2 y 2 ,  where  x  is the cost of a unit of labor and  y  is the cost  of a unit of goods.  Find the values of  x  and  y  that  will maximize the profit for the firm.    Clicker Check­out:  Choose any letter to check out now.      Tomorrow in recitation: Quiz on Sections 6.1, 6.2, 6.3, 9.1    Next time – Section 7.1        ...
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## This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

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