Mean value theorem (divided differences)
- Mean value theorem (divided differences)
-
In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.[1] It states that for any n + 1 points x0, ..., xn in the domain of an n-times differentiable function ƒ there exists an interior point
where the nth derivative of ƒ equals n ! times the nth divided difference at these points:
![f[x_0,\dots,x_n] = \frac{f^{(n)}(\xi)}{n!}.](f/51fa88c963aaa5856a79dece88a846b2.png)
For n = 1, that is two function points, one obtains the simple mean value theorem.
Applications
The theorem can be used to generalise the Stolarsky mean to more than two variables.
References
Wikimedia Foundation.
2010.
Look at other dictionaries:
Mean value theorem — For the theorem in harmonic function theory, see Harmonic function#Mean value property. Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus Derivative Change of variables … Wikipedia
Divided differences — In mathematics divided differences is a recursive division process. The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form. Contents 1 Definition 2 Notation 3 Example … Wikipedia
Logarithmic mean — In mathematics, the logarithmic mean is a function of two numbers which is equal to their difference divided by the logarithm of their quotient. In symbols::egin{matrix}M {mbox{lm(x,y) =lim {(xi,eta) o(x,y)} frac{eta xi}{ln eta ln xi}… … Wikipedia
Stolarsky mean — In mathematics, the Stolarsky mean of two positive real numbers x,y is defined as: :egin{matrix}S p(x,y) = lim {(xi,eta) o(x,y)} left({frac{xi^p eta^p}{p (xi eta) ight)^{1over p 1} = egin{cases}x mbox{if }x=y left({frac{x^p y^p}{p (x y)… … Wikipedia
Identric mean — The Identric mean of two positive real numbers x,y is defined as: :egin{matrix}I(x,y) = frac{1}{e}cdotlim {(xi,eta) o(x,y)}sqrt [xi eta] {frac{xi^xi}{eta^eta = lim {(xi,eta) o(x,y)}expleft(frac{xicdotlnxi etacdotlneta}{xi eta} 1 ight) =… … Wikipedia
List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… … Wikipedia
Difference quotient — The primary vehicle of calculus and other higher mathematics is the function. Its input value is its argument, usually a point ( P ) expressible on a graph. The difference between two points, themselves, is known as their Delta (ΔP), as is the… … Wikipedia
List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra … Wikipedia
Indian mathematics — mdash;which here is the mathematics that emerged in South Asia from ancient times until the end of the 18th century mdash;had its beginnings in the Bronze Age Indus Valley civilization (2600 1900 BCE) and the Iron Age Vedic culture (1500 500 BCE) … Wikipedia
Derivative — This article is an overview of the term as used in calculus. For a less technical overview of the subject, see Differential calculus. For other uses, see Derivative (disambiguation) … Wikipedia
