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# 6.4 - Math 182 Homework 6.4 Problem 8 Consider f(x = 2x(1...

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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 0 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 0 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. There are two real solutions and two imaginary solutions. The real solutions are c 1 0 . 2198 and c 2 1 . 2068.

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6.4 - Math 182 Homework 6.4 Problem 8 Consider f(x = 2x(1...

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