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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.
Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if x is far from b. That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor ...
Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case = states that = + ′ () + where is the remainder term. The linear approximation is obtained by dropping the remainder: () + ′ ().
Mihăilescu's theorem (number theory) Milliken–Taylor theorem (Ramsey theory) Milliken's tree theorem (Ramsey theory) Milman–Pettis theorem (Banach space) Min-max theorem (functional analysis) Minimax theorem (game theory) Minkowski's theorem (geometry of numbers) Minkowski's second theorem (geometry of numbers) Minkowski–Hlawka theorem ...
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.
Taylor columns were first observed by William Thomson, Lord Kelvin, in 1868. [1] [2] Taylor columns were featured in lecture demonstrations by Kelvin in 1881 [3] and by John Perry in 1890. [4] The phenomenon is explained via the Taylor–Proudman theorem, and it has been investigated by Taylor, [5] Grace, [6] Stewartson, [7] and Maxworthy [8 ...
For example, Michelle Cosgrove's benefits will be cut nearly in half — reduced by $557, to $601. Cosgrove spent the first half of her career as a paralegal, contributing to Social Security ...
Demonstration of this result is fairly straightforward under the assumption that () is differentiable near the neighborhood of and ′ is continuous at with ′ ().To begin, we use the mean value theorem (i.e.: the first order approximation of a Taylor series using Taylor's theorem):