The mean value theorem states that for a function continuous on a closed interval [a, b] and differentiable on the open interval (a, b), there exists some point c in (a, b) where the function's instantaneous rate of change equals its average rate of change over the whole interval. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor a function f continuous on [a, b] and differentiable on (a, b), there exists some c in (a, b) such that f prime of c equals f(b) minus f(a), divided by b minus a. 1 Classification
Statement Form Connections
Associated With
Derivative, Concepts The theorem states a differentiable function's average rate of change equals its derivative at some interior point.
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1. Mean Value Theorem (Wikipedia)
Wikimedia Foundationopening paragraph
the average rate of change of such a function over an interval is equal to the instantaneous rate of change at some point in the interval
History
The mean value theorem in its modern form was stated and proved by Augustin Louis Cauchy in 1823.
History section
The mean value theorem in its modern form was stated and proved by Augustin Louis Cauchy in 1823.
lead section
the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions
Lead section, statement-form reference
In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating that the average rate of change of such a function over an interval is equal to the instantaneous rate of change at some point in the interval.
Associated With: Derivative, Statement section
Let f : [a, b] to R be a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).
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