Approximating Functions with Polynomials

Taylor Polynomial

Let f be a function with f', f'',... and

defined at a. the nth-order Taylor polynomial for f with its center at a, denoted

has the property that it matches f in value, slope, and all derivatives up to the nth derivative at a; that is,

,...., and

The nth-order Taylor polynomial centered at a is

More completely,

, where the coefficients are

, for k=0, 1, 2, ..... , n

Let

be the Taylor polynomial of order n for f. The remainder in using

to approximate f at the point x

Let f have continuous derivatives up to

on an open interval I containing a. For all x in I.

Where

is the nth order Taylor polynomial for f centered at a and the remainder is

for some point c between x and a.

Let n be a fixed positive integer. Suppose there exists a number M such that

, for all c between a and x inclusive. The remainder in the nth-order Taylor polynomial for f centered at a satisfies