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''Lagrange finite-increments formula''
 
''Lagrange finite-increments formula''
  
A formula expressing the increment of a function in terms of the value of its derivative at an intermediate point. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f0403001.png" /> is continuous on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f0403002.png" /> on the real axis and is differentiable at the interior points of it, then
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<span id="Fig1">
 
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[[File:Finite-increments-formula-1.gif| right| frame| Figure 1. Given the [[chord]] of the graph of the function $f$ with end points $(a,f(a))$, $(b,f(b))$, then there exists a point $\xi$, $a<\xi<b$, such that the [[Tangent line|tangent]] to the graph of the function at the point $(\xi,f(\xi))$ is parallel to the chord.]]
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f0403003.png" /></td> </tr></table>
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</span>
  
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A formula expressing the increment of a function in terms of the value of its derivative at an intermediate point. If a function $f$ is continuous on an interval $[a,b]$ on the real axis and is differentiable at the interior points of it, then
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\begin{equation}
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f(b)-f(a)=f'(\xi)(b-a),\quad a<\xi<b.
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\end{equation}
 
The finite-increments formula can also be written in the form
 
The finite-increments formula can also be written in the form
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\begin{equation}
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f(x+\Delta x)-f(x)=f'(x+\theta\Delta x)\Delta x,\quad 0<\theta<1.
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\end{equation}
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The geometric meaning of the finite-increments formula is illustrated in [[#Fig1|Figure&nbsp;1]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f0403004.png" /></td> </tr></table>
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The finite-increments formula can be generalized to functions of several variables: If a function $f$ is differentiable at each point of a [[convex domain]] $G$ in an $n$-dimensional [[Euclidean space]], then there exists for each pair of points $x=(x_1,\dots,x_n)\in G$, $x+\Delta x=(x_1+\Delta x_1,\dots,x_n+\Delta x_n)\in G$ a point $\xi=(\xi_1,\ldots,\xi_n)$ lying on the segment joining $x$ and $x+\Delta x$ and such that
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\begin{equation}
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f(x+\Delta x)-f(x)=\sum_{i=1}^n\dfrac{\partial f(\xi)}{\partial x_i}\Delta x_i,\quad \xi_i=x_i+\theta\Delta x_i,\quad 0<\theta<1,\quad i=1,\ldots,n.
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\end{equation}
  
The geometric meaning of the finite-increments formula is: Given the chord of the graph of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f0403005.png" /> with end points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f0403006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f0403007.png" />, then there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f0403008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f0403009.png" />, such that the tangent to the graph of the function at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030010.png" /> is parallel to the chord (see Fig.).
 
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f040300a.gif" />
 
  
Figure: f040300a
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====Comments====
  
The finite-increments formula can be generalized to functions of several variables: If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030011.png" /> is differentiable at each point of a convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030012.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030013.png" />-dimensional Euclidean space, then there exists for each pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030015.png" /> a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030016.png" /> lying on the segment joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030018.png" /> and such that
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This formula is usually called the mean-value theorem (for derivatives). It is a statement for real-valued functions only; consider, e.g., $f(x)=e^{ix}$.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030019.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030020.png" /></td> </tr></table>
 
 
 
 
 
 
 
====Comments====
 
This formula is usually called the mean-value theorem (for derivatives). It is a statement for real-valued functions only; consider, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040300/f04030021.png" />.
 

Latest revision as of 08:41, 28 April 2016


Lagrange finite-increments formula

Figure 1. Given the chord of the graph of the function $f$ with end points $(a,f(a))$, $(b,f(b))$, then there exists a point $\xi$, $a<\xi<b$, such that the tangent to the graph of the function at the point $(\xi,f(\xi))$ is parallel to the chord.

A formula expressing the increment of a function in terms of the value of its derivative at an intermediate point. If a function $f$ is continuous on an interval $[a,b]$ on the real axis and is differentiable at the interior points of it, then \begin{equation} f(b)-f(a)=f'(\xi)(b-a),\quad a<\xi<b. \end{equation} The finite-increments formula can also be written in the form \begin{equation} f(x+\Delta x)-f(x)=f'(x+\theta\Delta x)\Delta x,\quad 0<\theta<1. \end{equation} The geometric meaning of the finite-increments formula is illustrated in Figure 1.

The finite-increments formula can be generalized to functions of several variables: If a function $f$ is differentiable at each point of a convex domain $G$ in an $n$-dimensional Euclidean space, then there exists for each pair of points $x=(x_1,\dots,x_n)\in G$, $x+\Delta x=(x_1+\Delta x_1,\dots,x_n+\Delta x_n)\in G$ a point $\xi=(\xi_1,\ldots,\xi_n)$ lying on the segment joining $x$ and $x+\Delta x$ and such that \begin{equation} f(x+\Delta x)-f(x)=\sum_{i=1}^n\dfrac{\partial f(\xi)}{\partial x_i}\Delta x_i,\quad \xi_i=x_i+\theta\Delta x_i,\quad 0<\theta<1,\quad i=1,\ldots,n. \end{equation}


Comments

This formula is usually called the mean-value theorem (for derivatives). It is a statement for real-valued functions only; consider, e.g., $f(x)=e^{ix}$.

How to Cite This Entry:
Finite-increments formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Finite-increments_formula&oldid=12197
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article