Extended mean value theorem: Difference between revisions
A WASSERMANN (talk | contribs) No edit summary |
A WASSERMANN (talk | contribs) No edit summary |
||
(45 intermediate revisions by the same user not shown) | |||
Line 1: | Line 1: | ||
The ''extended mean value theorem'' (also called ''Cauchy's mean value theorem'') is usually formulated as: | |||
Let | |||
:<math> f, g: [a,b] \to \mathbb{R}</math> | |||
be continuous functions that are differentiable on the open interval <math>(a,b)</math>. | |||
If <math>g'(x)\neq 0</math> for all <math>x\in(a,b)</math>, | |||
then there exists a value <math>\xi \in (a,b)</math> such that | |||
:<math> | |||
\frac{f'(\xi)}{g'(\xi)} = \frac{f(b)-f(a)}{g(b)-g(a)}. | |||
</math> | |||
'''Remark:''' | |||
It seems to be easier to state the extended mean value theorem in the following form: | |||
Let | |||
:<math> f, g: [a,b] \to \mathbb{R}</math> | |||
be continuous functions that are differentiable on the open interval <math>(a,b)</math>. | |||
Then there exists a value <math>\xi \in (a,b)</math> such that | |||
:<math> | |||
f'(\xi)\cdot (g(b)-g(a)) = g'(\xi) \cdot (f(b)-f(a)). | |||
</math> | |||
This second formulation avoids the need that | |||
<math>g'(x)\neq 0</math> for all <math>x\in(a,b)</math> and is therefore much easier to | |||
handle numerically. | |||
The proof is similar, just use the function | |||
:<math> | |||
h(x) = f(x)\cdot(g(b)-g(a)) - (g(x)-g(a))\cdot(f(b)-f(a)) | |||
</math> | |||
and apply Rolle's theorem. | |||
'''Visualization:''' | |||
The extended mean value theorem says that given the curve | |||
:<math> C: [a,b]\to\mathbb{R}, \quad t \mapsto (f(t), g(t)) </math> | |||
with the above prerequisites for <math>f</math> and <math>g</math>, | |||
there exists a <math>\xi</math> such that the tangent to the curve in the point <math>C(\xi)</math> | |||
is parallel to the secant through <math>C(a)</math> and <math>C(b)</math>. | |||
<jsxgraph width="600" height="400" box="box"> | <jsxgraph width="600" height="400" box="box"> | ||
var board = JXG.JSXGraph.initBoard('box', {boundingbox: [-5, 10, 7, -6], axis:true}); | var board = JXG.JSXGraph.initBoard('box', {boundingbox: [-5, 10, 7, -6], axis:true}); | ||
var p = []; | var p = []; | ||
p[0] = board.create('point', [0, -2], {size:2, name: 'C(a)'}); | |||
p[0] = board.create('point', [ | p[1] = board.create('point', [-1.5, 5], {size:2, name: ''}); | ||
p[1] = board.create('point', [-1.5, 5], {size:2}); | p[2] = board.create('point', [1, 4], {size:2, name: ''}); | ||
p[2] = board.create('point', [1,4], {size:2}); | p[3] = board.create('point', [3, 3], {size:2, name: 'C(b)'}); | ||
p[3] = board.create('point', [3, | |||
// Curve | // Curve | ||
var fg = JXG.Math.Numerics.Neville(p); | var fg = JXG.Math.Numerics.Neville(p); | ||
var graph = board.create('curve', fg, {strokeWidth:3, strokeOpacity:0.5}); | var graph = board.create('curve', fg, {strokeWidth:3, strokeOpacity:0.5}); | ||
// Secant | // Secant | ||
Line 20: | Line 57: | ||
var dg = JXG.Math.Numerics.D(fg[1]); | var dg = JXG.Math.Numerics.D(fg[1]); | ||
// Usually, the extended mean value theorem is formulated as | |||
// df(t) / dg(t) == (p[3].X() - p[0].X()) / (p[3].Y() - p[0].Y()) | |||
// We can avoid division by zero with the following formulation: | |||
var quot = function(t) { | var quot = function(t) { | ||
return df(t) * (p[3].Y() - p[0].Y()) - dg(t) * (p[3].X() - p[0].X()); | |||
}; | }; | ||
var r = board.create('glider', [ | var r = board.create('glider', [ | ||
function() { return fg[0](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); }, | |||
function() { return fg[1](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); }, | |||
graph], {name: 'C(ξ)', size: 4, fixed:true, color: 'blue'}); | |||
board.create('tangent', [r], {strokeColor:'#ff0000'}); | board.create('tangent', [r], {strokeColor:'#ff0000'}); | ||
</jsxgraph> | </jsxgraph> | ||
===The underlying JavaScript code=== | ===The underlying JavaScript code=== | ||
<source lang="javascript"> | <source lang="javascript"> | ||
var board = JXG.JSXGraph.initBoard('box', {boundingbox: [-5, 10, 7, -6], axis:true}); | |||
var p = []; | |||
p[0] = board.create('point', [0, -2], {size:2, name: 'C(a)'}); | |||
p[1] = board.create('point', [-1.5, 5], {size:2, name: ''}); | |||
p[2] = board.create('point', [1, 4], {size:2, name: ''}); | |||
p[3] = board.create('point', [3, 3], {size:2, name: 'C(b)'}); | |||
// Curve | |||
var fg = JXG.Math.Numerics.Neville(p); | |||
var graph = board.create('curve', fg, {strokeWidth:3, strokeOpacity:0.5}); | |||
// Secant | |||
line = board.create('line', [p[0], p[3]], {strokeColor:'#ff0000', dash:1}); | |||
var df = JXG.Math.Numerics.D(fg[0]); | |||
var dg = JXG.Math.Numerics.D(fg[1]); | |||
// Usually, the extended mean value theorem is formulated as | |||
// df(t) / dg(t) == (p[3].X() - p[0].X()) / (p[3].Y() - p[0].Y()) | |||
// We can avoid division by zero with the following formulation: | |||
var quot = function(t) { | |||
return df(t) * (p[3].Y() - p[0].Y()) - dg(t) * (p[3].X() - p[0].X()); | |||
}; | |||
var r = board.create('glider', [ | |||
function() { return fg[0](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); }, | |||
function() { return fg[1](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); }, | |||
graph], {name: 'C(ξ)', size: 4, fixed:true, color: 'blue'}); | |||
board.create('tangent', [r], {strokeColor:'#ff0000'}); | |||
</source> | </source> | ||
[[Category:Examples]] | [[Category:Examples]] | ||
[[Category:Calculus]] | [[Category:Calculus]] |
Latest revision as of 11:38, 4 February 2019
The extended mean value theorem (also called Cauchy's mean value theorem) is usually formulated as:
Let
- [math]\displaystyle{ f, g: [a,b] \to \mathbb{R} }[/math]
be continuous functions that are differentiable on the open interval [math]\displaystyle{ (a,b) }[/math]. If [math]\displaystyle{ g'(x)\neq 0 }[/math] for all [math]\displaystyle{ x\in(a,b) }[/math], then there exists a value [math]\displaystyle{ \xi \in (a,b) }[/math] such that
- [math]\displaystyle{ \frac{f'(\xi)}{g'(\xi)} = \frac{f(b)-f(a)}{g(b)-g(a)}. }[/math]
Remark: It seems to be easier to state the extended mean value theorem in the following form:
Let
- [math]\displaystyle{ f, g: [a,b] \to \mathbb{R} }[/math]
be continuous functions that are differentiable on the open interval [math]\displaystyle{ (a,b) }[/math]. Then there exists a value [math]\displaystyle{ \xi \in (a,b) }[/math] such that
- [math]\displaystyle{ f'(\xi)\cdot (g(b)-g(a)) = g'(\xi) \cdot (f(b)-f(a)). }[/math]
This second formulation avoids the need that [math]\displaystyle{ g'(x)\neq 0 }[/math] for all [math]\displaystyle{ x\in(a,b) }[/math] and is therefore much easier to handle numerically.
The proof is similar, just use the function
- [math]\displaystyle{ h(x) = f(x)\cdot(g(b)-g(a)) - (g(x)-g(a))\cdot(f(b)-f(a)) }[/math]
and apply Rolle's theorem.
Visualization: The extended mean value theorem says that given the curve
- [math]\displaystyle{ C: [a,b]\to\mathbb{R}, \quad t \mapsto (f(t), g(t)) }[/math]
with the above prerequisites for [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math], there exists a [math]\displaystyle{ \xi }[/math] such that the tangent to the curve in the point [math]\displaystyle{ C(\xi) }[/math] is parallel to the secant through [math]\displaystyle{ C(a) }[/math] and [math]\displaystyle{ C(b) }[/math].
The underlying JavaScript code
var board = JXG.JSXGraph.initBoard('box', {boundingbox: [-5, 10, 7, -6], axis:true});
var p = [];
p[0] = board.create('point', [0, -2], {size:2, name: 'C(a)'});
p[1] = board.create('point', [-1.5, 5], {size:2, name: ''});
p[2] = board.create('point', [1, 4], {size:2, name: ''});
p[3] = board.create('point', [3, 3], {size:2, name: 'C(b)'});
// Curve
var fg = JXG.Math.Numerics.Neville(p);
var graph = board.create('curve', fg, {strokeWidth:3, strokeOpacity:0.5});
// Secant
line = board.create('line', [p[0], p[3]], {strokeColor:'#ff0000', dash:1});
var df = JXG.Math.Numerics.D(fg[0]);
var dg = JXG.Math.Numerics.D(fg[1]);
// Usually, the extended mean value theorem is formulated as
// df(t) / dg(t) == (p[3].X() - p[0].X()) / (p[3].Y() - p[0].Y())
// We can avoid division by zero with the following formulation:
var quot = function(t) {
return df(t) * (p[3].Y() - p[0].Y()) - dg(t) * (p[3].X() - p[0].X());
};
var r = board.create('glider', [
function() { return fg[0](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
function() { return fg[1](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
graph], {name: 'C(ξ)', size: 4, fixed:true, color: 'blue'});
board.create('tangent', [r], {strokeColor:'#ff0000'});