Editing Taylor's theorem (section)

{{short description|Approximation of a function by a truncated power series}}
[[File:Taylorspolynomialexbig.svg|thumb|right|300px|The exponential function <math display="inline">y=e^x</math> (red) and the corresponding Taylor polynomial of degree four (dashed green) around the origin.]]
{{Calculus |Differential}}

In [[calculus]], '''Taylor's theorem''' gives an approximation of a <math display="inline">k</math>-times [[differentiable function]] around a given point by a [[polynomial]] of degree <math display="inline">k</math>, called the <math display="inline">k</math>-th-order '''Taylor polynomial'''. For a [[smooth function]], the Taylor polynomial is the truncation at the order ''<math display="inline">k</math>'' of the [[Taylor series]] of the function. The first-order Taylor polynomial is the [[linear approximation]] of the function, and the second-order Taylor polynomial is often referred to as the '''quadratic approximation'''.<ref>(2013). [http://www.math.ubc.ca/~sujatha/2013/103/week10-12/Linearapp.pdf"Linear and quadratic approximation"] Retrieved December 6, 2018</ref> There are several versions of Taylor's theorem, some giving explicit estimates of 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 1715,<ref>{{cite book|language=la|last=Taylor |first=Brook |title=Methodus Incrementorum Directa et Inversa |url=https://archive.org/details/UFIE003454_TO0324_PNI-2529_000000|trans-title=Direct and Reverse Methods of Incrementation |location=London |date=1715 |at=p. 21–23 (Prop. VII, Thm. 3, Cor. 2)}} Translated into English in {{cite book|first=D. J. |last=Struik|title=A Source Book in Mathematics 1200–1800 |location=Cambridge, Massachusetts |publisher=Harvard University Press |date=1969 |pages= 329–332}}</ref> although an earlier version of the result was already mentioned in [[1671 in science|1671]] by [[James Gregory (astronomer and mathematician)|James Gregory]].<ref>{{harvnb|Kline|1972|pp=442, 464}}.</ref>

Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in [[mathematical analysis]]. It gives simple arithmetic formulas to accurately compute values of many [[transcendental function]]s such as the [[exponential function]] and [[trigonometric function]]s.
It is the starting point of the study of [[analytic function]]s, and is fundamental in various areas of mathematics, as well as in [[numerical analysis]] and [[mathematical physics]]. Taylor's theorem also generalizes to [[multivariate function|multivariate]] and [[vector valued function|vector valued]] functions. It provided the mathematical basis for some landmark early computing machines: [[Charles Babbage]]'s [[Difference Engine]] calculated sines, cosines, logarithms, and other transcendental functions by numerically integrating the first 7 terms of their Taylor series.