Difference between revisions of "Mean Value Theorem"
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− | The | + | The Mean Value Theorem (often abbreviated MVT) is considered one of the most important results in [[real analysis]]. An elegant proof of the [[Fundamental Theorem of Calculus]] can be given using Mean Value Theorem. |
− | <cmath>f(c)=\ | + | == Statement == |
+ | Let <math>f:[a,b]\rightarrow\mathbb R</math> be differentiable function on the [[interval]] <math>(a,b)</math>. The '''Mean Value Theorem''' states there exists a point <math>c\in(a,b)</math> such that <cmath>f'(c)=\frac{f(b)-f(a)}{b-a}.</cmath> | ||
− | + | == Proof == | |
− | + | Note that <math>\frac{f(b)-f(a)}{b-a}</math> is the slope of the [[secant]] [[line]] which passes through the graph of <math>f</math> at points <math>(a,f(a))</math> and <math>(b,f(b))</math>. Let <math>g(x)=f(x)-\frac{f(b)-f(a)}{b-a}(x-a)</math>. As <math>g</math> is continuous on <math>[a,b]</math> since it is the linear combinatoin of the continuous functions <math>f(x)</math> and <math>(x-a)</math>, and that <cmath>g'(x)=f'(x)-\frac{f(b)-f(a)}{b-a}.</cmath> Since <math>g</math> is differentiable on <math>(a,b)</math> and <math>g(a)=g(b)=f(a)</math>, by [[Rolle's Theorem]], there exists some <math>c\in(a,b)</math> such that <math>g'(c)=0</math>. Thus <cmath>f'(c)=g'(c)+\frac{f(b)-f(a)}{b-a}=\frac{f(b)-f(a)}{b-a}</cmath> as desired. <math>\fbox{}</math> | |
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Latest revision as of 18:19, 18 September 2020
The Mean Value Theorem (often abbreviated MVT) is considered one of the most important results in real analysis. An elegant proof of the Fundamental Theorem of Calculus can be given using Mean Value Theorem.
Statement
Let be differentiable function on the interval
. The Mean Value Theorem states there exists a point
such that
Proof
Note that is the slope of the secant line which passes through the graph of
at points
and
. Let
. As
is continuous on
since it is the linear combinatoin of the continuous functions
and
, and that
Since
is differentiable on
and
, by Rolle's Theorem, there exists some
such that
. Thus
as desired.
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