The Taylor Series

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The Taylor Series Problems 3

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Topics The Taylor Series Problems 3

Problem : Compute the Taylor polynomial at 0 of degree 3 for f (x) = sin(e^x).

The first few derivatives are

f'(x)	=	cos(e^x)e^x
f''(x)	=	cos(e^x)e^x - sin(e^x)e^2x
f'''(x)	=	f''(x) - [sin(e^x)2e^2x + cos(e^x)e^3x]

Substituting 0 for x yields f (0) = sin(1), f'(0) = cos(1), f''(0) = cos(1) - sin(1), f'''(0) = - 3 sin(1), so the desired Taylor polynomial is:

sin(1) + cos(1)x +

+

x³

Problem : Approximate log(0.8) using the Taylor polynomial of degree 3 at x = 1 for log(x).

The desired Taylor polynomial is

(x - 1) -

+

Substituting 0.8 for x, we have log(0.8)

-0.222

.

Problem : Find the Taylor series for sin(x) + cos(x).

Adding term-by-term, we have

1 + x -

-

+

+ ^...

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