Most mathematical functions (exp, sin, cos, etc) cannot be in general evaluated exactly. By hand, and on a computer as well, we usually calculate some approximations to these functions. The approximate functions need to be easy to evaluate, and have to provide values which are close enough to the values of the original function. One of the most convenient ways to compute functions is to approximate them by polynomials.
A Taylor polynomial of degree is
constructed to mimic the behavior of near a point .
If is a degree Taylor polynomial,
We have to choose the coefficients such that for . For this, it is natural to require that the value of the polynomial (and the values of its first derivatives) coincides with the value of the function (and the values of function's first derivatives, respectively) for . This means that , and since we have determined . This one condition is clearly not sufficient to determine all coefficients of the polynomial. An extra condition, that forces more resemblance between and when is . Since , we have determined the second coefficient also. If this is enough; if not, we continue imposing conditions on higher derivates of , until we have conditions for the coefficients:
To fix the ideas, let , and . Then
For , we add the third condition
which reads , or .
Thus the quadratic approximation of near is
In the general case, if is easy to see that
In order to be able to write this formula, we need of course to assume that is at least times continuously differentiable; we will actually assume for the purpose of writing an error estimate also.
( has derivatives of any order)
we can increase indefinitely; the approximations
get better and better, and, in the limit, they will coincide
with . In the limit the summation becomes
infinite, and we obtain the Taylor series expansion
Question: to define we have to evaluate , together with several of its derivatives. Aren't we better off just evaluating , and forgetting about this whole approximation stuff? The answer is that, indeed, we have to evaluate at once. Then , once constructed, will be a cheap approximation for thousands of future calls. Moreover, and its derivatives are evaluated at one point , while is a good approximation of for an entire interval .