A Taylor

polynomial is a

calculus tool that allows one to evaluate hard to
compute

functions by hand, without the aid of a

calculator, assuming one
can do a bit of

math in one's head.

Functions like *f(x)* = cos x, e^{x}, or log x are difficult to compute without the aid of some computation
device. However,
18th century mathematicians couldn't just wait around for someone to invent
the graphing calculator. The English mathematician Brook Taylor (1685-1731)
published a book in 1715 in which (along with defining the field of
differential calculus and inventing integration by parts) contained the
series know as the Taylor expansion. The work went largely unrecognized until
Joseph Lagrange pointed out that the series was the foundation of
differential calculus in 1772. The general form of the Taylor expansion will
yield the Taylor polynomial.

The Taylor polynomial will mimic the function *f(x)* at some point
*x = a* - hence, it will be nearly equal to *f(x)* at points
*x* near *a*. More specifically, we define a linear polynomial

*p*_{}1*(x) = f(a)*

p'_{}1*(x) = f'(a)*

Thus, the polynomial is uniquely given by

*p*_{}1*(x) = f(a) + (x-a)f'(a)*

This first order polynomial defines a tangent line to the function
*f(x)*, and for values near *a*, will evaluate very near to
*f(a)*. This is one of the fundamental ideas of linear calculus: that a
function can be __locally__ approximated by a line to the function at some
defined point. This result can be improved upon, however. The first derivative
of *f(x)* simply gives us a line (the only line, in fact) with the same
slope as *f(x)* at the point *(a, f(a))*; the second derivative will
give us a parabola with the same slope as *f(x)* at *a*, and the
same tonic tendencies (think acceleration) of *f(x)* at *a*.

So, to continue the process of constructing the polynomial, consider a
quadratic polynomial *p*_{2}(x) that approximate *f(x)* at
*x = a*. Specifically,

*p*_{}2*(x) = f(a)*

p'_{}2*(x) = f'(a)*

p''_{}2*(x) = f''(a)*

These are statisfied by

*p*_{}2*(x) = f(a) +(x-a)f'(a) +
*½*(x-a)*^{}2*f''(a)*

This process can be continued to mimic the behavior of *f(x)* at *x =
a* by taking further derivatives of *f(x)* - the higher the degree of
the polynomial (i.e. more derivatives taken of *f(x)*), the closer the
approximation will become. Let *p*_{n}(x) be a polynomial of
degree *n*, and require it to satisfy

*p*^{(j)}_{n}(a) = f^{(j)}(a), j = 0, 1, ...,
*n*

where *f*^{(j)}(x) is the order *j* derivative
of *f(x)*. Then

* *n
p_{n} =
Σ
((x-a)^{j} / j!) f^{(j)}(a)
j = 0

Take e^{x} as an example, at *a* = 0

*f*^{(j)}(x) = e^{x}, f^{(j)}(0) = 1, for all
*j* >= 0

Thus

*p*_{n}(x) = 1+ *x* + (*x*^{2} / 2!) + ... +
(*x*^{n} / n!)

n
=
Σ
(x^{j} / j!)
j = 0

Note: The limit of a Taylor polynomial of degree *n* as *n* goes to
infinity is known as a Taylor series - see that node for more info, and a
number of useful Taylor series formulae.
And many thanks to Palpz for suggesting the link

Sources:
- Dr. Ken Atkinson, University of Iowa
- Stetson University, at
http://www.stetson.edu/~efriedma/periodictable/html/Tl.html