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In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the kth-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k 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. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.

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rdf:type	Thing
rdfs:label	Taylor's theorem (en)
rdfs:comment	In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the kth-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k 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. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. (en)
sameAs	Taylor's theorem Taylor's theorem
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Subject	Approximations Articles containing proofs Theorems in calculus Theorems in real analysis
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has abstract	In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the kth-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order k 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. 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, although 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. It gives simple arithmetic formulas to accurately compute values of many transcendental functions such as the exponential function and trigonometric functions.It is the starting point of the study of analytic functions, and is fundamental in various areas of mathematics, as well as in numerical analysis and mathematical physics. Taylor's theorem also generalizes to multivariate and vector valued functions. (en)
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is differentFrom of	Taylor rule
is Wikipage redirect of	Taylor's Theorem Taylors theorem Lagrange error bound Lagrange form of the remainder Lagrange remainder Lagrange remainder theorem Quadratic approximation Cauchy's estimate Proof of Taylor's theorem Taylor's formula Taylor's inequality Taylor approximation Taylor theorem
is Link from a Wikipage to another Wikipage of	List of theorems List of things named after Jacques Hadamard List of things named after Joseph-Louis Lagrange Numerical modeling (geology) 1671 in science 1715 in science 1 − 2 + 3 − 4 + ⋯ Adaptive step size Analytic continuation Augustin-Louis Cauchy Barnes G-function Binomial approximation Bloch's theorem (complex variables) Brook Taylor Calculus on Euclidean space Cauchy momentum equation Central differencing scheme Central limit theorem Frenet–Serret formulas Generalized filtering Geometric series Glossary of calculus Joseph-Louis Lagrange Differential calculus Differential geometry of surfaces Differential of a function Divided differences Jet (mathematics) Céa's lemma Delta method Linear approximation Linearization List of Italian inventions and discoveries List of multivariable calculus topics List of real analysis topics Concave function Convex function Convolution Cotangent bundle Hoeffding's lemma List of calculus topics Newton's method Equicontinuity Finite difference Finite difference method Flat function Function of a real variable Function of several real variables Fundamental theorem of calculus Gateaux derivative Gauss–Newton algorithm Indian mathematics Lagrange error bound Lagrange form of the remainder Lagrange remainder

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