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In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function , the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.
To find a second-order approximation for the covariance of functions of two random variables (with the same function applied to both), one can proceed as follows.
For the second-order approximations of the third central moment as well as for the derivation of all higher-order approximations see Appendix D of Ref. [3] Taking into account the quadratic terms of the Taylor series and the third moments of the input variables is referred to as second-order third-moment method. [4]
For example, the exponential function is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may equally well define an analytic function by its Taylor series. Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in ...
In fact, for a smooth enough function, we have the similar Taylor expansion (+) = | | ()! + (,), where the last term (the remainder) depends on the exact version of Taylor's formula.
Intuitively, this means that we can express the jet of a curve through p in terms of its Taylor series in local coordinates on M. Examples in local coordinates: As indicated previously, the 1-jet of a curve through p is a tangent vector. A tangent vector at p is a first-order differential operator acting on smooth real-valued functions at p. In ...
NEW YORK (Reuters) -The judge overseeing Donald Trump's criminal hush money case has put off ruling on whether the president-elect's conviction should be thrown out on immunity grounds, enabling ...
In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem.Roughly speaking, the theorem asserts that if A is a closed subset of a Euclidean space, then it is possible to extend a given function of A in such a way as to have prescribed derivatives at the points of A.