The function is and , , .

(a)

Find the Taylor polynomial with degree  at the number .

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 .

Find the values of the above functions at .

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The series is centered at .

Taylor series centered at .

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(b)

The Taylors inequality is where  .

Here , and .

Substitute in

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Find the fourth derivative.

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Substitute in .

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The Taylors accuracy inequality is .

(c)

The value is .

Here .

Substitute and .

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Graph :

Graph the function .

Observe the graph:

The functions for small value of in the interval .

(a) .

(b) .

(c)

Graph of the function is

The functions for small value of in the interval .