# 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 , we will insist that the polynomial and f(x) have the same value and the same first four derivatives at .

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

#### Some more notation.

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

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

(3) Recall that one writes for the nth derivative of the functionf. 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 :

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 with center .