6.4 - Math 182 Homework 6.4 Problem 8 Consider f ( x ) = 2...

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Unformatted text preview: Math 182 Homework 6.4 Problem 8 Consider f ( x ) = 2 x (1+ x 2 ) 2 : (a) Find the average value of f ( x ) on [0 , 2] Since the length of the interval is 2 this is 1 2 integraltext 2 2 x (1+ x 2 ) 2 dx . To do this we let u = 1 + x 2 so that du = 2 xdx . (Notice the 2 xdx in the numerator which we did not cancel). u (0) = 1 and u (2) = 5 so we have 1 2 integraldisplay 5 1 du u 2 =- 1 2 u- 1 vextendsingle vextendsingle vextendsingle vextendsingle u =5 u =1 = 1 2 u- 1 vextendsingle vextendsingle vextendsingle vextendsingle u =1 u =5 = 1 2 parenleftbigg 1- 1 5 parenrightbigg = 1 2 4 5 = 2 5 (b) Find an approximate value for all real numbers c with c 2 so that f ( c ) equals the average value determined in part ( a ) c must satisfy the following equation: 2 c (1 + c 2 ) 2 = 2 5 5 c = (1 + c 2 ) 2 c 4 + 2 c 2- 5 c + 1 = 0 Unfortunately, this cannot be solved exactly although various numerical schemes can get great approximations....
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This note was uploaded on 09/21/2009 for the course MATH 182 taught by Professor Keppelmann during the Spring '08 term at Nevada.

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6.4 - Math 182 Homework 6.4 Problem 8 Consider f ( x ) = 2...

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