The function is .

(a)

Find the Taylor polynomial upto degree .

Definition of Taylor series:

If a function  has derivatives of all orders at  then the series

 is called Taylor series for  at .

First find the successive derivatives of .

Apply derivative on each side with respect to .

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The series is centered at .

Find the values of the function at .

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Observe the values:

Taylor polynomials of odd degree will not exist for because .

Taylor polynomia is

Taylor polynomial of degree is .

The series is centered at .

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Taylor polynomial of degree is .

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Taylor polynomial of degree is .

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Taylor polynomial of degree is .

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Graph the polynomials , , and of function .

(b) Evaluate and these polynomials at and .

Construct the table for , , , and at and .

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(c)

As increases, is a good approxmation to on a larger and larger interval.

(a) , , and .

Graph of the polynomials , , and of function .

(b)

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(c) As increases, is a good approxmation to on a larger and larger interval.