# mean value theorem
## English
### Noun

mean value theorem (plural mean value theorems)
* (mathematics) Any of various theorems that saliently concern mean values.
** 1964, J. H. Bramble, L. E. Payne, Some Mean Value Theorems in Electrostatics: Journal of the Society for Industrial and Applied Mathematics, volume 12, page 105:Several mean value theorems in the theory of elasticity have appeared in the recent literature […] .
** 1984 [Nauka, Moscow], Sergey Ermakov, V. V. Nekrutkin (authors and translators), A. S. Sipin (author), Random Processes for Classical Equations of Mathematical Physics, [1984, С. М. Ермаков, В. В. Некруткин, А. С. Сипин, Случайные процессы для решения классических уравнений математической физики], 1989, Kluwer, Softcover Reprint, page xiii,
    For parabolic equations (Section 5.1) and for the exterior Dirichlet problem (Section 5.2), it is possible to apply the well known mean value theorems.
** 1994, Patrick W. Thompson, “Images of Rate and Operational Understanding of the Fundamental Theorem of Calculus”, in Paul Cobb, editor, Learning Mathematics, Kluwer, page 167:However, Anton switches, unannounced, to another conceptualization in justifying the Fundamental Theorem - he bases it on the mean value theorem for integrals.
** 2013, Elimhan Mahmudov, Single Variable Differential and Integral Calculus: Mathematical Analysis, Springer, page 259:We prove mean value theorems in the context of integrals which are analogous to the ones studied in Chapter 5.
** 2013, Peter D. Lax, Maria Shea Terrell, Calculus With Applications, Springer, page 171:The mean value theorem for derivatives provides an important link between the derivative of f on an interval and the behavior of f over the interval.
* (calculus, uncountable) The theorem that for any real-valued function that is differentiable on an interval, there is a point in that interval where the derivative of the curve equals the slope of the straight line between the graphed function values at the interval's end points.
** 1990, A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, page 51,
    In order to get a true bound for the range we may replace the Taylor series in (2) by the mean value theorem, which tells us that
       $f({\tilde {x}})=f({\check {x}})+f'(\zeta )({\tilde {x}}-{\check {x}})$
    for some $\zeta$ on the line segment between ${\tilde {x}}$ and ${\check {x}}$.
** 2003, Sylvain Raynes, Ann Rutledge, The Analysis of Structured Securities, Oxford University Press, page 397:In what follows, we will use the mean value theorem, another one of Lagrange's many contributions to numerical analysis.
** 2007, Denise Szecsei, Calculus, The Career Press, page 10:The main existence theorems in calculus are the Intermediate Value Theorem, the Extreme Value Theorem, Rolle's Theorem, and the Mean Value Theorem.

#### Usage notes

* (theorem that a point exists where the derivative equals the average slope): In mathematical terms, if $f:\mathbb {R} \rightarrow \mathbb {R}$ is continuous on $[a,b]$ and differentiable on $(a,b)$ (where $a<b$) then $\exists c\in (a,b):f'(c)={\frac {f(b)-f(a)}{b-a}}$. (Note that since nothing is assumed about the function outside the interval, it cannot, strictly speaking, be said to be differentiable at the end points. However, the continuity condition means that it is right differentiable at $a$ and left differentiable at $b$.)

#### Synonyms

* (theorem that a point exists where the derivative equals the overall slope): Lagrange mean value theorem, mean value theorem for derivatives

#### Derived terms

* Cauchy's mean value theorem
* extended mean value theorem
* Gauss's mean value theorem

#### Translations
### See also

* intermediate value theorem