Taylor's theorem: how a few derivatives let a polynomial impersonate a function
Charles Babbage's difference engine computed sines, cosines and logarithms by working with just the first 7 terms of their Taylor series. The trick rests on Taylor's theorem: near a chosen point, a well-behaved function can be mimicked by a polynomial that matches its value and its first few derivatives.
Brook Taylor stated a version in 1715, though James Gregory had mentioned an earlier form in 1671. The first-order Taylor polynomial is the familiar linear approximation; the second-order one, known as the quadratic approximation, also matches the function's second derivative at the base point. For a smooth function, the Taylor polynomial of order k is simply the Taylor series cut off at that order.
The theorem guarantees that the error of a degree-k approximation shrinks faster than the k-th power of the distance from the base point as that distance goes to zero. So, close enough to the point, the quadratic beats the linear version, and higher degrees match ever more derivatives. But the result is asymptotic: on its own it says nothing about how big the error is at any particular distance.
Explicit remainder formulas, valid under extra regularity assumptions, fill that gap. They let you estimate the error for a given degree on an interval, find the smallest degree that meets an error tolerance, or find the widest interval where a given polynomial stays accurate enough. Even for analytic functions, such estimates can break down on intervals that are too large, so several polynomials with different centres may be needed.
Some functions resist altogether: even infinitely differentiable ones can refuse to improve as the degree rises, meaning they are not analytic at that point and are not determined locally by their derivatives. Still, the theorem is taught in introductory calculus, yields simple arithmetic formulas for exponential and trigonometric functions, opens the study of analytic functions, and extends to functions of several variables and to vector-valued functions.
Source: Taylor's theorem
