Extended mean value theorem - Revision history

A WASSERMANN at 11:38, 4 February 2019

2019-02-04T11:38:07Z

← Older revision		Revision as of 11:38, 4 February 2019
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	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>

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	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

A WASSERMANN at 09:28, 30 January 2019

2019-01-30T09:28:32Z

← Older revision		Revision as of 09:28, 30 January 2019
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	and apply Rolle's theorem.		and apply Rolle's theorem.

	'''Visualization'''		'''Visualization:'''
	The extended mean value theorem says that given the curve		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>		:<math> C: [a,b]\to\mathbb{R}, \quad t \mapsto (f(t), g(t)) </math>

A WASSERMANN at 16:50, 29 January 2019

2019-01-29T16:50:06Z

← Older revision		Revision as of 16:50, 29 January 2019
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	// 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 the following 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());

A WASSERMANN at 16:49, 29 January 2019

2019-01-29T16:49:50Z

← Older revision		Revision as of 16:49, 29 January 2019
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	// 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());
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	// 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());

A WASSERMANN at 16:48, 29 January 2019

2019-01-29T16:48:32Z

← Older revision		Revision as of 16:48, 29 January 2019
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	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
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	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

A WASSERMANN at 16:47, 29 January 2019

2019-01-29T16:47:09Z

A WASSERMANN at 16:45, 29 January 2019

2019-01-29T16:45:31Z

← Older revision		Revision as of 16:45, 29 January 2019
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	</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 point <math>\xi</math> on the curve where the curve's
			tangent 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">

A WASSERMANN at 16:37, 29 January 2019

2019-01-29T16:37:44Z

← Older revision		Revision as of 16:37, 29 January 2019
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	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: 'ξ', size: 4, fixed:true, color: 'blue'});

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

A WASSERMANN at 16:37, 29 January 2019

2019-01-29T16:37:08Z

← Older revision		Revision as of 16:37, 29 January 2019
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	The ''extended mean value theorem'' (also called ''Cauchy's mean value theorem'') is:		The ''extended mean value theorem'' (also called ''Cauchy's mean value theorem'') is usually formulated as:

	Let		Let
	:<math> f, g: [a,b] \to \mathbb{R}</math>		:<math> f, g: [a,b] \to \mathbb{R}</math>

A WASSERMANN at 16:36, 29 January 2019

2019-01-29T16:36:14Z

← Older revision		Revision as of 16:36, 29 January 2019
Line 25:		Line 25:
	The proof is similar, just use the function		The proof is similar, just use the function
	:<math>		:<math>
	h(x) = f(x)\cdot(G(b)-g(a)) - (g(x)-g(a))\cdot(f(b)-f(a))		h(x) = f(x)\cdot(g(b)-g(a)) - (g(x)-g(a))\cdot(f(b)-f(a))
	</math>		</math>
	and apply Rolle's theorem.		and apply Rolle's theorem.

← Older revision		Revision as of 16:47, 29 January 2019
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	:<math> C: [a,b]\to\mathbb{R}, \quad t \mapsto (f(t), g(t)) </math>		:<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>,		with the above prerequisites for <math>f</math> and <math>g</math>,
	there exists a ~~point~~ <math>\xi</math> on the curve ~~where~~ the ~~curve's~~		there exists a <math>\xi</math> such that the tangent to the curve in the point <math>C(\xi)</math>
	~~tangent~~ is parallel to the secant through <math>C(a)</math> and <math>C(b)</math>.		is parallel to the secant through <math>C(a)</math> and <math>C(b)</math>.


Line 67:		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: 'ξ', size: 4, fixed:true, color: 'blue'});		graph], {name: 'C(ξ)', size: 4, fixed:true, color: 'blue'});

	board.create('tangent', [r], {strokeColor:'#ff0000'});		board.create('tangent', [r], {strokeColor:'#ff0000'});
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	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(ξ)', size: 4, fixed:true, color: 'blue'});

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