Difference between revisions of "Extended mean value theorem"

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Line 7: Line 7:
 
then there exists a value <math>\xi \in (a,b)</math> such that
 
then there exists a value <math>\xi \in (a,b)</math> such that
 
:<math>
 
:<math>
\frac{f'(\xi)}{g'(\xi)} = \frac{f(b)-f(a)}{g(b)-g(b)}.
+
\frac{f'(\xi)}{g'(\xi)} = \frac{f(b)-f(a)}{g(b)-g(a)}.
 
</math>
 
</math>
  
Line 18: Line 18:
 
Then there exists a value <math>\xi \in (a,b)</math> such that
 
Then there exists a value <math>\xi \in (a,b)</math> such that
 
:<math>
 
:<math>
f'(\xi)\cdot (g(b)-g(a))  = g'(\xi) \cdot (f(b)-f(b)).
+
f'(\xi)\cdot (g(b)-g(a))  = g'(\xi) \cdot (f(b)-f(a)).
 
</math>
 
</math>
 
This second formulation avoids the need that  
 
This second formulation avoids the need that  
Line 29: Line 29:
 
</math>
 
</math>
 
and apply Rolle's theorem.
 
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">
Line 34: Line 42:
 
var p = [];
 
var p = [];
  
p[0] = board.create('point', [0, -2], {size:2});
+
p[0] = board.create('point', [0, -2], {size:2, name: 'C(a)'});
p[1] = board.create('point', [-1.5, 5], {size:2});
+
p[1] = board.create('point', [-1.5, 5], {size:2, name: ''});
p[2] = board.create('point', [1, 4], {size:2});
+
p[2] = board.create('point', [1, 4], {size:2, name: ''});
p[3] = board.create('point', [3, 3], {size:2});
+
p[3] = board.create('point', [3, 3], {size:2, name: 'C(b)'});
  
 
// Curve
 
// Curve
Line 51: Line 59:
 
// Usually, the extended mean value theorem is formulated as
 
// 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())
 
// 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:
+
// 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());
 
     return df(t) * (p[3].Y() - p[0].Y()) - dg(t) * (p[3].X() - p[0].X());
Line 59: Line 67:
 
             function() { return fg[0](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
 
             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)); },
 
             function() { return fg[1](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
             graph], {name: '&xi;', size: 4, fixed:true, color: 'blue'});
+
             graph], {name: 'C(&xi;)', size: 4, fixed:true, color: 'blue'});
  
 
board.create('tangent', [r], {strokeColor:'#ff0000'});
 
board.create('tangent', [r], {strokeColor:'#ff0000'});
Line 70: Line 78:
 
var p = [];
 
var p = [];
  
p[0] = board.create('point', [0, -2], {size:2});
+
p[0] = board.create('point', [0, -2], {size:2, name: 'C(a)'});
p[1] = board.create('point', [-1.5, 5], {size:2});
+
p[1] = board.create('point', [-1.5, 5], {size:2, name: ''});
p[2] = board.create('point', [1, 4], {size:2});
+
p[2] = board.create('point', [1, 4], {size:2, name: ''});
p[3] = board.create('point', [3, 3], {size:2});
+
p[3] = board.create('point', [3, 3], {size:2, name: 'C(b)'});
  
 
// Curve
 
// Curve
Line 87: Line 95:
 
// Usually, the extended mean value theorem is formulated as
 
// 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())
 
// 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:
+
// 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());
 
     return df(t) * (p[3].Y() - p[0].Y()) - dg(t) * (p[3].X() - p[0].X());
Line 95: Line 103:
 
             function() { return fg[0](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
 
             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)); },
 
             function() { return fg[1](JXG.Math.Numerics.root(quot, (fg[3]() + fg[2]) * 0.5)); },
             graph], {name: '', size: 4, fixed:true, color: 'blue'});
+
             graph], {name: 'C(&xi;)', size: 4, fixed:true, color: 'blue'});
  
 
board.create('tangent', [r], {strokeColor:'#ff0000'});
 
board.create('tangent', [r], {strokeColor:'#ff0000'});

Latest revision as of 12:38, 4 February 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(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].


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(&xi;)', size: 4, fixed:true, color: 'blue'});

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