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, ex, 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

p1(x) = f(a)
p'
1(x) = f'(a)

Thus, the polynomial is uniquely given by

p1(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 p2(x) that approximate f(x) at x = a. Specifically,

p2(x) = f(a)
p'
2(x) = f'(a)
p''
2(x) = f''(a)

These are statisfied by

p2(x) = f(a) +(x-a)f'(a) + ½(x-a)2f''(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 pn(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 pn = Σ ((x-a)j / j!) f(j)(a) j = 0

Take ex as an example, at a = 0

f(j)(x) = ex, f(j)(0) = 1, for all j >= 0

Thus

pn(x) = 1+ x + (x2 / 2!) + ... + (xn / n!)

        n
     = 
Σ
  (xj / 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

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