Difference between revisions of "Extended mean value theorem"

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The ''extended mean value theorem'' (also called ''Cauchy's mean value theorem'') is:
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The ''extended mean value theorem'' (also called ''Cauchy's mean value theorem'') is usually formulated as:
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Let
 
Let
 
:<math> f, g: [a,b] \to \mathbb{R}</math>
 
:<math> f, g: [a,b] \to \mathbb{R}</math>

Revision as of 18:37, 29 January 2019

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(b)}. [/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(b)). [/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.

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});
p[1] = board.create('point', [-1.5, 5], {size:2});
p[2] = board.create('point', [1, 4], {size:2});
p[3] = board.create('point', [3, 3], {size:2});

// 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 that 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: '', size: 4, fixed:true, color: 'blue'});

board.create('tangent', [r], {strokeColor:'#ff0000'});