HW05-solutions - zakaria (mmz255) HW05 gilbert (55485) 1...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 10.0 points 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 10.0 points 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 , zakaria (mmz255) HW05 gilbert (55485) 2 or the fact that ln(1- x ) =- integraldisplay x 1 1- s ds =- integraldisplay x braceleftBig summationdisplay n =0 s n bracerightBig ds =-...
View Full Document

This note was uploaded on 06/05/2011 for the course MATH 408 D taught by Professor Gilbert during the Spring '11 term at University of Texas at Austin.

Page1 / 6

HW05-solutions - zakaria (mmz255) HW05 gilbert (55485) 1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online