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 .

.

.

.

The series is centered at . 

Taylor series centered at . 

.

. \ \

(b)

The taylors inequality is where  .

Here , and .

Substitute  in

 in \ \

Hence, .

.

.

The value of ,hence the value of should be .

The taylors accuracy inequality is .

(c)

The value is .

Here .

Substitute and .

.

Graph :

Graph the function .

Observe the graph:

The functions for small value of in the interval .

(a) .

(b)

(c)

Graph of the function is

Observe the graph:

The functions for small value of in the interval .