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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.

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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

.

.

.

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 .