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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.
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 ...
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: () + ′ ().
Theorem — For any function f(x) continuous on an interval [a,b] there exists a table of nodes for which the sequence of interpolating polynomials () converges to f(x) uniformly on [a,b]. Proof It is clear that the sequence of polynomials of best approximation p n ∗ ( x ) {\displaystyle p_{n}^{*}(x)} converges to f ( x ) uniformly (due to ...
In mathematics, the Regiomontanus's angle maximization problem, is a famous optimization problem [1] posed by the 15th-century German mathematician Johannes Müller [2] (also known as Regiomontanus). The problem is as follows: The two dots at eye level are possible locations of the viewer's eye. A painting hangs from a wall.
Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function = at any = based on the value and slope of the function at =, given that () is differentiable on [,] (or [,]) and that is close to .
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.