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University of Texas M 408D Rusin HW 05

# University of Texas M 408D Rusin HW 05 - byrne(cmb3744 –...

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Unformatted text preview: byrne (cmb3744) – 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 ( t ) = 1 t- 2 . 1. f ( t ) =- ∞ summationdisplay n =0 1 2 n +1 t n correct 2. f ( t ) = ∞ summationdisplay n =0 1 2 n +1 t n 3. f ( t ) = ∞ summationdisplay n =0 (- 1) n − 1 2 n +1 t n 4. f ( t ) = ∞ summationdisplay n =0 (- 1) n 2 n t n 5. f ( t ) =- ∞ summationdisplay n =0 2 n t n Explanation: We know that 1 1- x = 1 + x + x 2 + . . . = ∞ summationdisplay n = 0 x n . On the other hand, 1 t- 2 =- 1 2 parenleftBig 1 1- ( t/ 2) parenrightBig . Thus f ( t ) =- 1 2 ∞ summationdisplay n = 0 parenleftbigg t 2 parenrightbigg n =- 1 2 ∞ summationdisplay n = 0 1 2 n t n . Consequently, f ( t ) =- ∞ summationdisplay n = 0 1 2 n +1 t n with | t | < 2. 002 10.0 points Find a power series representation for the function f ( x ) = x 2 tan − 1 x on (- 1 , 1). 1. f ( x ) = ∞ summationdisplay n =0 1 n + 1 x n +3 2. f ( x ) = ∞ summationdisplay n =0 (- 1) n ( n + 1)! x n +3 3. f ( x ) = ∞ summationdisplay n =0 1 2 n + 1 x 2 n +3 4. f ( x ) = ∞ summationdisplay n =0 (- 1) n (2 n + 1)! x 2 n +3 5. f ( x ) = ∞ summationdisplay n =0 (- 1) n n + 1 x n +3 6. f ( x ) = ∞ summationdisplay n =0 (- 1) n 2 n + 1 x 2 n +3 correct 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 . byrne (cmb3744) – HW 05 – rusin – (55565) 2 Consequently, on (- 1 , 1) f ( x ) = ∞ summationdisplay n =0 (- 1) n 2 n + 1 x 2 n +3 . 003 10.0 points Suppose P ( x ) = 7- 5( x- 1) + 6( x- 1) 2- 7( x- 1) 3 + 6( 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) ≈ 6 . 6536 2. f (1 . 1) ≈ 6 . 8536 3. f (1 . 1) ≈ 6 . 4536 4. f (1 . 1) ≈ 6 . 7536 5. f (1 . 1) ≈ 6 . 5536 correct Explanation: Since p 4 ( x ) is an approximation for f ( x ) we see that f (1 . 1) ≈ 7- 5 10 + 6 10 2- 7 10 3 + 6 10 4 . Consequently, f (1 . 1) ≈ 6 . 5536 . 004 10.0 points Find the Taylor series representation for f centered at x = 1 when f ( x ) = 4 + 5 x- 3 x 2 ....
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University of Texas M 408D Rusin HW 05 - byrne(cmb3744 –...

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