10.3_and_4-solutions

10.3_and_4-solutions - MthSc 108-Learning Activity 10.3...

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MthSc 108-Learning Activity 10.3 & 10.4 Taylor Series Directions: Do not use a calculator or any other technology unless otherwise specified. To receive full credit you must provide 1) Proper and complete notation!! 2) Legible, organized, and logical (relevant) justification for your answer!! 3) Each response should be presented in a complete sentence, either mathematical or in words as appropriate!! Show all work on a separate sheet of paper. Construct new series from old series using substitution, multiplication, division, integration, or differentiation. Note: See page 609 in your text book for frequently used Taylor series. All series should be expressed in closed form, i.e. expressed in series notation with the nth term. 1. 1. Recall in the activity for Section 10.1 we found Taylor polynomials for the function 2 ( ) x f x e = n ( ) ( ) n f x ( ) (0) n f ( ) ( ) ! 0 n f n 0 2 ( ) x f x e = 2(0) (0) 1 f e = = 1 1 0! = 1 2 '( ) 2 x f x e = 2(0) '(0) 2 2 f e = = 2 2 1! = 2 2 ''( ) 4 x f x e = 2(0) ''(0) 4 4 f e = = 4 2 2! = 3 2 '''( ) 8 x f x e = 2(0) '''(0) 8 8 f e = = 8 4 3! 3 = 4 (4) 2 ( ) 16 x f x e = (4) 2(0) (0) 16 16 f e = = 16 2 4! 3 = Therefore, 2 0 1 2 2 3 2 3 4 3 4 ( ) 1 ( ) 1 2 4 ( ) 1 2 2 4 2 ( ) 1 2 2 ( ) 1 2 3 3 3 2 x T x x T x x x x x x x T T T x x x x x = = + = + + = + + + = + + + + Determine the general (nth term) and give the Taylor (Maclaurin) series for the function at a = 0 and determine the interval of convergence of the series
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10.3_and_4-solutions - MthSc 108-Learning Activity 10.3...

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