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| PAGE: 798 | SET: Exercises | PROBLEM: 1 |
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 
.
.
.
.
.
.
.
The series is centered at 
.
Find the values of the function at 
.
.
.
.
.
.
.
.
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 .
.
Taylor polynomial of degree 
is 
.
.
Taylor polynomial of degree 
is 
.
.
Taylor polynomial of degree is 
.
.
Graph the polynomials 
, 
, 
and 
of function 
.
.gif)
(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.
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