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.
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