L28 Taylor and MaClaurin II

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Lecture 28: Taylor and Maclaurin Se- ries (II) ex. Find the Maclaurin series and Taylor Series centered at a = 2 for p ( x ) = x 3 ± 2 x 2 + x ± 3 : 1

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ex. Find the Maclaurin Series for f ( x ) = e 5 x . ex. Find the Maclaurin Series for f ( x ) = xe x . 2
ex. Find the Taylor Series expansions for f ( x ) = sin x centered at a = ± 4 , then calculate sin 48 ± 3

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Find the sum of the series: ex. 1 X 0 ( ± 1) n ± 2 n 6 2 n (2 n )! ex. 1 X 0 ( ± 1) n x 4 n n ! 4
NYTI: : 1. Find the Taylor Series for f ( x ) = 1 x centered at a = 1. Write your answer in sum- mation notation. What is the radius of convergence? 2. Find the Maclaurin Series for g ( x ) = sin( x 2 ) : Use the ±rst 2 non-zero terms of this series to approximate the integral Z 1 0 sin( x 2 ) dx: 3. Find the Maclaurin series for f ( x ) = x cos x . Write your answer in summation notation. 4. Find the MacLaurin series for f ( x ) = ln( x +1) using the de±nition of a Maclaurin series.

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Unformatted text preview: Express your answer in summation notation. 5. Find the sum of the series 1 ln 2 + ( ln 2) 2 2! ( ln 2) 3 3! + 5 ex. Evaluate the indenite integrals as an in-nite series: Z e x 1 x dx; Z arctan( x 2 ) dx . 6 ex. Compute the 10th derivative of f ( x ) = arctan x 2 3 , at x = 0. (hint: Use the MacLaurin series for f ( x ) to nd f 10 (0).) 7 NYTI: Suppose the MacLaurin Series of f ( x ) = x 5 e x 3 about x = 0 is x 5 + x 8 + x 11 2! + x 14 3! + x 17 4! + Find the following: d dx x 5 e x 3 j x =0 = d 11 dx 11 x 5 e x 3 j x =0 = Tonights homework Practice problems in todays lecture. 8...
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