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HW05-solutions

# HW05-solutions - zakaria(mmz255 HW05 gilbert(55485 This...

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zakaria (mmz255) – HW05 – gilbert – (55485) 1 This print-out should have 12 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Determine the value of f ( - 2) when f ( x ) = x 4 2 + 2 x 3 4 4 + 3 x 5 4 6 + ... . ( Hint : differentiate the power series represen- tation of ( x 2 - 4 2 ) 1 .) 1. f ( - 2) = - 2 9 correct 2. f ( - 2) = 1 16 3. f ( - 2) = - 2 25 4. f ( - 2) = 2 9 5. f ( - 2) = - 1 16 Explanation: The geometric series 1 4 2 - x = 1 4 2 parenleftBig 1 1 - x/ 4 2 parenrightBig = 1 4 2 parenleftBig 1 + x 4 2 + x 2 4 4 + x 3 4 6 + ... parenrightBig has interval of convergence ( - 16 , 16), and so 1 x - 4 2 = - 1 4 2 - x = - 1 4 2 parenleftBig 1 + x 4 2 + x 2 4 4 + x 3 4 6 + ... parenrightBig on the interval ( - 16 , 16). Thus, if we restrict x to the interval ( - 4 , 4), we can replace x by x 2 in this series. Consequently, 1 x 2 - 4 2 = - 1 4 2 parenleftBig 1 + x 2 4 2 + x 4 4 4 + x 6 4 6 + ... parenrightBig on the interval ( - 4 , 4), On this interval the derivative of the left hand side is the term- by-term derivative of the series on the right hand. Hence 2 x ( x 2 - 4 2 ) 2 = 1 4 2 parenleftBig 2 x 4 2 + 4 x 3 4 4 + 6 x 5 4 6 + ... parenrightBig , and so f ( x ) = 4 2 x ( x 2 - 4 2 ) 2 . As x = - 2 lies in ( - 4 , 4), f ( - 2) = - 2 9 . 002 10.0points Find a power series representation for the function f ( y ) = ln(6 - y ) . 1. f ( y ) = ln 6 + summationdisplay n =1 y n 6 n 2. f ( y ) = summationdisplay n =0 y n n 6 n 3. f ( y ) = ln 6 - summationdisplay n =0 y n 6 n 4. f ( y ) = ln 6 - summationdisplay n =1 y n n 6 n correct 5. f ( y ) = - summationdisplay n =1 y n n 6 n 6. f ( y ) = ln 6 + summationdisplay n =0 y n n 6 n Explanation: We can either use the known power series representation ln(1 - x ) = - summationdisplay n =1 x n n ,

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zakaria (mmz255) – HW05 – gilbert – (55485) 2 or the fact that ln(1 - x ) = - integraldisplay x 0 1 1 - s ds = - integraldisplay x 0 braceleftBig summationdisplay n =0 s n bracerightBig ds = - summationdisplay n =0 integraldisplay x 0 s n ds = - summationdisplay n =1 x n n .
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HW05-solutions - zakaria(mmz255 HW05 gilbert(55485 This...

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