# HW5 - hussain (tsh476) HW 05 rusin (55565) 1 This print-out...

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Unformatted text preview: hussain (tsh476) HW 05 rusin (55565) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. Some Taylor series and some curves for your mathematical pleasure! 001 10.0 points Find a power series representation for the function f ( z ) = 1 z- 4 . 1. f ( z ) = summationdisplay n = 0 (- 1) n 4 n z n 2. f ( z ) =- summationdisplay n = 0 1 4 n +1 z n correct 3. f ( z ) = summationdisplay n = 0 (- 1) n 1 4 n +1 z n 4. f ( z ) = summationdisplay n = 0 1 4 n +1 z n 5. f ( z ) =- summationdisplay n = 0 4 n z n Explanation: We know that 1 1- x = 1 + x + x 2 + . . . = summationdisplay n = 0 x n . On the other hand, 1 z- 4 =- 1 4 parenleftBig 1 1- ( z/ 4) parenrightBig . Thus f ( z ) =- 1 4 summationdisplay n = 0 parenleftBig z 4 parenrightBig n =- 1 4 summationdisplay n =0 1 4 n z n . Consequently, f ( z ) =- summationdisplay n = 0 1 4 n +1 z n with | z | &amp;lt; 4. 002 10.0 points Find a power series representation for the function f ( x ) = x 3 tan 1 x on (- 1 , 1). 1. f ( x ) = summationdisplay n =0 (- 1) n ( n + 1)! x n +4 2. f ( x ) = summationdisplay n =0 (- 1) n n + 1 x n +4 3. f ( x ) = summationdisplay n =0 (- 1) n (2 n + 1)! x 2 n +4 4. f ( x ) = summationdisplay n =0 (- 1) n 2 n + 1 x 2 n +4 correct 5. f ( x ) = summationdisplay n =0 1 2 n + 1 x 2 n +4 6. f ( x ) = summationdisplay n =0 1 n + 1 x n +4 Explanation: The interval of convergence of the geomet- ric series 1 1- x = 1 + x + x 2 + . . . is (- 1 , 1). Thus on (- 1 , 1) 1 1 + x 2 = 1- x 2 + x 4- . . . = summationdisplay n = 0 (- 1) n x 2 n . On the other hand, tan 1 x = integraldisplay x 1 1 + t 2 dt . Thus on (- 1 , 1) tan 1 x = integraldisplay x summationdisplay n = 0 (- 1) n t 2 n dt = summationdisplay n = 0 braceleftBig integraldisplay x (- 1) n t 2 n dt bracerightBig = summationdisplay n = 0 (- 1) n 2 n + 1 x 2 n +1 . hussain (tsh476) HW 05 rusin (55565) 2 Consequently, on (- 1 , 1) f ( x ) = summationdisplay n =0 (- 1) n 2 n + 1 x 2 n +4 . 003 10.0 points Suppose P ( x ) = 4- 5( x- 1) + 6( x- 1) 2- 8( x- 1) 3 + 2( x- 1) 4 is the degree 4 Taylor polynomial centered at x = 1 for a certain function f . Use p 4 to estimate the value of f (1 . 1). 1. f (1 . 1) 3 . 7522 2. f (1 . 1) 3 . 6522 3. f (1 . 1) 3 . 5522 correct 4. f (1 . 1) 3 . 4522 5. f (1 . 1) 3 . 8522 Explanation: Since p 4 ( x ) is an approximation for f ( x ) we see that f (1 . 1) 4- 5 10 + 6 10 2- 8 10 3 + 2 10 4 . Consequently, f (1 . 1) 3 . 5522 . 004 10.0 points Find the Taylor series representation for f centered at x = 2 when f ( x ) = 4 + 3 x- 2 x 2 ....
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## This note was uploaded on 04/03/2011 for the course MATH 408D taught by Professor Chu during the Spring '09 term at University of Texas at Austin.

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HW5 - hussain (tsh476) HW 05 rusin (55565) 1 This print-out...

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