Taylor Polynomials

The relevant information is listed in the following table


Using the formula above we obtain as the second degree Taylor polynomial:


Taylor Polynomials

No reason to only compute second degree Taylor polynomials! If we want to find for example the fourth degree Taylor polynomial for a function f(x) with a given center tex2html_wrap_inline209 , we will insist that the polynomial and f(x) have the same value and the same first four derivatives at tex2html_wrap_inline209 .

A calculation similar to the previous one will yield the formula:


Some more notation.

(1) Usually Taylor polynomials are denoted by tex2html_wrap_inline301 , where n indicates its degree.

(2) The factorial sign comes in handy: Recall that tex2html_wrap_inline305 . For instance tex2html_wrap_inline307 Mathematicians usually say that 0!=1.

(3) Recall that one writes tex2html_wrap_inline311 for the nth derivative of the functionf. tex2html_wrap_inline317 just means f(x).

With these notations, we can then write the nth term of a Taylor polynomial as


Thus we obtain the general formula for the nth Taylor polynomial of a function f(x) with center tex2html_wrap_inline209 :


An alternative way of writing this--for the not easily Mathematese-intimidated-- is provided by the summation notation:


High time to try it yourself!

Find the 5th degree Taylor polynomial for the function tex2html_wrap_inline331 with center tex2html_wrap_inline333 .

Click here for the answer, or to continue.

Helmut Knaust
Sun Jul 7 22:08:09 MDT 1996

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