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Definition/Summary
The mean value theorem states that if a real-valued function [itex]f[/itex] is continuous and differentiable on an open interval [itex](a,b)[/itex], then there is a point [itex]c[/itex] in that interval such that [itex]f'(c) \ =\ (f(b) - f(a))/(b - a)[/itex].
It also applies if the condition of differentiability is relaxed to include the existence of "infinite differentials", at points [itex]x[/itex] such that [itex]lim(h\,\to\,0)\ (f(x+h) - f(x))/h\ =\ \pm\,\infty[/itex].
Cauchy's mean value theorem (or the extended mean value theorem) states that if two real-valued functions [itex]f\text{ and }g[/itex] are continuous and differentiable on an open interval [itex](a,b)[/itex], then there is a point [itex]c[/itex] in that interval such that [itex](f(b) - f(a))g'(c)\ =\ (g(b) - g(a))f'(c))[/itex].
Cauchy's mean value theorem may be used to prove l'Hôpital's rule.
These theorems mean, roughly, that a chord of a smooth curve in a plane is parallel to the tangent of the curve at some intermediate point, with Cauchy's version applying in the general case where both coordinates of the curve are defined by the same parameter (a "length"), but the ordinary version only applying where one coordinate is a function of the other (and so there are no "vertical" chords).
Equations
Mean value theorem:
[tex]\exists\ c\,\in\,(a,b): f'(c)\ =\ (f(b) - f(a))/(b - a)[/tex]
Cauchy's mean value theorem:
[tex]\exists\ c\,\in\,(a,b): (f(b) - f(a))\,g'(c)\ =\ (g(b) - g(a))f'(c))[/tex]
Mean value theorems for integration:
If f is continuous and g is positive and integrable:
[tex]\exists\ c\,\in\,(a,b): \int_a^b f(t)g(t)\ =\ f(c)\,\int_a^b g(t)[/tex]
If f is continuous and g is always 1:
[tex]\exists\ c\,\in\,(a,b): \int_a^b f(t)\ =\ f(c)\,(b\ -\ a)[/tex]
If f is monotonic and g is integrable (Okamura's theorem):
[tex]\exists\ c\,\in\,(a,b): \int_a^b f(t)g(t)\ =\ f(a\,+\,0)\int_a^c g(t)\ \ +\ \ f(b\,-\,0)\int_c^b g(t)[/tex]
Extended explanation
The article in wikipeida is particularly good, with clear diagrams, and the reader is referred to it.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
The mean value theorem states that if a real-valued function [itex]f[/itex] is continuous and differentiable on an open interval [itex](a,b)[/itex], then there is a point [itex]c[/itex] in that interval such that [itex]f'(c) \ =\ (f(b) - f(a))/(b - a)[/itex].
It also applies if the condition of differentiability is relaxed to include the existence of "infinite differentials", at points [itex]x[/itex] such that [itex]lim(h\,\to\,0)\ (f(x+h) - f(x))/h\ =\ \pm\,\infty[/itex].
Cauchy's mean value theorem (or the extended mean value theorem) states that if two real-valued functions [itex]f\text{ and }g[/itex] are continuous and differentiable on an open interval [itex](a,b)[/itex], then there is a point [itex]c[/itex] in that interval such that [itex](f(b) - f(a))g'(c)\ =\ (g(b) - g(a))f'(c))[/itex].
Cauchy's mean value theorem may be used to prove l'Hôpital's rule.
These theorems mean, roughly, that a chord of a smooth curve in a plane is parallel to the tangent of the curve at some intermediate point, with Cauchy's version applying in the general case where both coordinates of the curve are defined by the same parameter (a "length"), but the ordinary version only applying where one coordinate is a function of the other (and so there are no "vertical" chords).
Equations
Mean value theorem:
[tex]\exists\ c\,\in\,(a,b): f'(c)\ =\ (f(b) - f(a))/(b - a)[/tex]
Cauchy's mean value theorem:
[tex]\exists\ c\,\in\,(a,b): (f(b) - f(a))\,g'(c)\ =\ (g(b) - g(a))f'(c))[/tex]
Mean value theorems for integration:
If f is continuous and g is positive and integrable:
[tex]\exists\ c\,\in\,(a,b): \int_a^b f(t)g(t)\ =\ f(c)\,\int_a^b g(t)[/tex]
If f is continuous and g is always 1:
[tex]\exists\ c\,\in\,(a,b): \int_a^b f(t)\ =\ f(c)\,(b\ -\ a)[/tex]
If f is monotonic and g is integrable (Okamura's theorem):
[tex]\exists\ c\,\in\,(a,b): \int_a^b f(t)g(t)\ =\ f(a\,+\,0)\int_a^c g(t)\ \ +\ \ f(b\,-\,0)\int_c^b g(t)[/tex]
Extended explanation
The article in wikipeida is particularly good, with clear diagrams, and the reader is referred to it.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!