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1124worksheet12

# 1124worksheet12 - a = 0 for each function(a y = e-x 2 cos...

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Polytechnic Institute of NYU MA 1124/1424 Worksheet 12 Print Name: Signature: ID #: Instructor/Section: Directions: Show all your work for every problem. Problem Possible Points 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100 1

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2 (1) Use a Taylor series to find the Taylor series for f ( x ) at 0. (a) f ( x ) = x 2 cos( x/ 2) (b) f ( x ) = sin 2 x (Hint:Use sin 2 x = 1 2 (1 - cos 2 x ))
3 (2) (a) Expand cos x in powers of x - π (b) Expand sin 1 2 πx in powers of x - 1

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4 (3) (a) Expand x - 1 in powers of x - 1 (b) Expand x 2 + e 3 x in powers of x - 2
5 (4) Use series to evaluate the limit. lim x 0 1 - cos x 1 + x - e x

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6 (5) Find the sum of the series. (a) X n =0 ( - 1) n π 2 n 6 2 n (2 n )! (b) X n =0 x n +1 ( n + 1)!
7 (6) Find the sum of the series. (a) X n =0 x 4 n n ! (b) X n =0 x n 2 n ( n + 1)!

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8 (7) Find the first three nonzero terms in the Taylor series around

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Unformatted text preview: a = 0 for each function. (a) y = e-x 2 cos x (b) y = e x ln(1-x ) 9 (8) Solve exactly for the variable. (a) x-x 2 2 + x 3 3-x 4 4 + ... = 0 . 6 (b) x-x 3 2! + x 5 4!-x 7 6! + ... = . 5 x 10 (9) (a) Expand e x in powers of x-a . (b) Use the expansion to show that e x 1 + x 2 = e x 1 e x 2 11 (10) Let i = √-1. We deﬁne e iθ by substituting iθ in the Taylor series for e x . Use the Taylor series for sin θ and cos θ to show that e iθ = cos θ + i sin θ...
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1124worksheet12 - a = 0 for each function(a y = e-x 2 cos...

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