mean value theorem

Noun

 * 1)  Any of various theorems that saliently concern mean values.
 * 2) * 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.
 * 1)  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.
 * 2) * 1990, A. Neumaier, Interval Methods for Systems of Equations,, 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 $$\check x$$.
 * 1) * 1990, A. Neumaier, Interval Methods for Systems of Equations,, 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 $$\check x$$.

Usage notes

 * 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

 * Lagrange mean value theorem, mean value theorem for derivatives

Translations

 * Finnish: differentiaalilaskennan väliarvolause, väliarvolause
 * German: Mittelwertsatz
 * Hebrew: משפט הערך הממוצע, משפט לגראנז'
 * Italian: teorema di Lagrange, teorema del valor medio, teorema dell'incremento finito
 * Tagalog: hunain ng tamtaming halga