Taylor's theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Taylor%27s_theorem


00:02:24 1 Motivation
00:04:22 2 Taylor's theorem in one real variable
00:12:09 2.1 Statement of the theorem
00:12:20 2.2 Explicit formulas for the remainder
00:18:33 2.3 Estimates for the remainder
00:24:18 2.4 Example
00:27:14 3 Relationship to analyticity
00:28:42 3.1 Taylor expansions of real analytic functions
00:33:40 3.2 Taylor's theorem and convergence of Taylor series
00:33:51 3.3 Taylor's theorem in complex analysis
00:39:11 3.4 Example
00:46:55 4 Generalizations of Taylor's theorem
01:03:24 4.1 Higher-order differentiability
01:05:25 4.2 Taylor's theorem for multivariate functions
01:05:57 4.3 Example in two dimensions
01:06:08 5 Proofs
01:11:38 5.1 Proof for Taylor's theorem in one real variable
01:16:56 5.2 Derivation for the mean value forms of the remainder
01:21:56 5.3 Derivation for the integral form of the remainder
01:22:05 5.4 Derivation for the remainder of multivariate Taylor polynomials
01:26:40 6 See also
01:29:38 7 Footnotes
01:29:42 8 References
01:37:39 9 External links
01:45:50 1, it must hold for every positive integer k.
01:45:57 Derivation for the remainder of multivariate Taylor polynomials
01:46:10 We prove the special case, where f : Rn → R has continuous partial derivatives up to the order k+1 in some closed ball B with center a.  The strategy of the proof is to apply the one-variable case of Taylor's theorem to the restriction of f to the line segment adjoining x and a.  Parametrize the line segment between a and x by u(t)
01:53:06 See also



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SUMMARY
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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 
  
    
      
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    {\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.