Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. For analytic functions the Taylor polynomials at a given point are finite-order truncations of its Taylor series, which completely determines the function in some neighborhood of the point. It can be thought of as the extension of linear approximation to higher order polynomials, and in the case of k equals 2 is often referred to as a quadratic approximation. The exact content of "Taylor's theorem" is not universally agreed upon. Indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1712. Yet an explicit expression of the error was not provided until much later on by Joseph-Louis Lagrange. An earlier version of the result was already mentioned in 1671 by James Gregory. Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. Within pure mathematics it is the starting point of more advanced asymptotic analysis and is commonly used in more applied fields of numerics, as well as in mathematical physics. Taylor's theorem also generalizes to multivariate and vector valued functions f : R n → R m {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}} on any dimensions n and m. This generalization of Taylor's theorem is the basis for the definition of so-called jets, which appear in differential geometry and partial differential equations.


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