mean value theorem

1. calculusThe 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.

   for some ζ on the line segment between ̃x and ̌x.

   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(̃x)=f(̌x)+f'(ζ)(̃x-̌x)

   2 more examples

   In what follows, we will use the mean value theorem, another one of Lagrange's many contributions to numerical analysis.

   2003, Sylvain Raynes, Ann Rutledge, The Analysis of Structured Securities, Oxford University Press, page 397

   The main existence theorems in calculus are the Intermediate Value Theorem, the Extreme Value Theorem, Rolle's Theorem, and the Mean Value Theorem.

   2007, Denise Szecsei, Calculus, The Career Press, page 10

   Synonyms: Lagrange's mean value theorem, mean value theorem for derivatives

plural mean value theorems

1. mathematicsAny of various theorems that saliently concern mean values.

   Several mean value theorems in the theory of elasticity have appeared in the recent literature[…].

   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

   4 more examples

   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.

   However, Anton switches, unannounced, to another conceptualization in justifying the Fundamental Theorem - he bases it on the mean value theorem for integrals.

   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

   We prove mean value theorems in the context of integrals which are analogous to the ones studied in Chapter 5.

   2013, Elimhan Mahmudov, Single Variable Differential and Integral Calculus: Mathematical Analysis, Springer, page 259

   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.

   2013, Peter D. Lax, Maria Shea Terrell, Calculus With Applications, Springer, page 171