Taylor's Theorem - Motivation

Motivation

If a real-valued function f is differentiable at the point a then it has a linear approximation at the point a. This means that there exists a function h₁ such that

Here

is the linear approximation of f at the point a. The graph of y = P₁(x) is the tangent line to the graph of f at x = a. The error in the approximation is

Note that this goes to zero a little bit faster than x − a as x tends to a.

If we wanted a better approximation to f, we might instead try a quadratic polynomial instead of a linear function. Instead of just matching one derivative of f at a, we can match two derivatives, thus producing a polynomial that has the same slope and concavity as f at a. The quadratic polynomial in question is

Taylor's theorem ensures that the quadratic approximation is, in a sufficiently small neighborhood of the point a, a better approximation than the linear approximation. Specifically,

Here the error in the approximation is

which, given the limiting behavior of h₂, goes to zero faster than (x − a)2 as x tends to a.

Similarly, we get still better approximations to f if we use polynomials of higher degree, since then we can match even more derivatives with f at the selected base point. In general, the error in approximating a function by a polynomial of degree k will go to zero a little bit faster than (x − a)k as x tends to a.

This result is of asymptotic nature: it only tells us that the error R_k in an approximation by a k-th order Taylor polynomial P_k tends to zero faster than any nonzero k-th degree polynomial as x → a. It does not tell us how large the error is in any concrete neighborhood of the center of expansion, but for this purpose there are explicit formulae for the remainder term (given below) which are valid under some additional regularity assumptions on f. These enhanced versions of Taylor's theorem typically lead to uniform estimates for the approximation error in a small neighborhood of the center of expansion, but the estimates do not necessarily hold for neighborhoods which are too large, even if the function f is analytic. In that situation one may have to select several Taylor polynomials with different centers of expansion to have reliable Taylor-approximations of the original function (see animation on the right.)

It is also possible that increasing the degree of the approximating polynomial does not increase the quality of approximation at all even if the function f to be approximated is infinitely many times differentiable. An example of this behavior is given below, and it is related to the fact that unlike analytic functions, more general functions are not (locally) determined by the values of their derivatives at a single point.