Discontinuous derivative - Revision history

A WASSERMANN at 21:01, 23 March 2021

2021-03-23T21:01:44Z

← Older revision		Revision as of 21:01, 23 March 2021
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	:<math> f: \mathbb{R} \to \mathbb{R}, x \mapsto		:<math> f: \mathbb{R} \to \mathbb{R}, x \mapsto
	\begin{cases}		\begin{cases}
	x^2\sin(1/x),& x\neq 0\\		x^2\sin(1/x),& x\neq 0,\\
	0,& x=0		0,& x=0\,.
	\end{cases}~~\,.~~		\end{cases}
	</math>		</math>
	<math>f</math> is a continous and differentiable function.		<math>f</math> is a continous and differentiable function.

A WASSERMANN at 21:01, 23 March 2021

2021-03-23T21:01:22Z

← Older revision		Revision as of 21:01, 23 March 2021
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	f': \mathbb{R} \to \mathbb{R}, x \mapsto		f': \mathbb{R} \to \mathbb{R}, x \mapsto
	\begin{cases}		\begin{cases}
	2x\sin(1/x) - \cos(1/x), &x \neq 0\\		2x\sin(1/x) - \cos(1/x), &x \neq 0,\\
	0,& x=0		0,& x=0\,.
	\end{cases}~~\,.~~		\end{cases}
	</math>		</math>
	We observe that <math>f'(0) = 0</math>, but <math>\lim_{x\to0}f'(x)</math> does not exist.		We observe that <math>f'(0) = 0</math>, but <math>\lim_{x\to0}f'(x)</math> does not exist.

A WASSERMANN at 20:59, 23 March 2021

2021-03-23T20:59:58Z

← Older revision		Revision as of 20:59, 23 March 2021
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	f': \mathbb{R} \to \mathbb{R}, x \mapsto		f': \mathbb{R} \to \mathbb{R}, x \mapsto
	\begin{cases}		\begin{cases}
	2\sin(1/x) - \cos(1/x), &x \neq 0\\		2x\sin(1/x) - \cos(1/x), &x \neq 0\\
	0,& x=0		0,& x=0
	\end{cases}\,.		\end{cases}\,.
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	var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});		var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});

	var g = board.create('functiongraph', ["2*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});		var g = board.create('functiongraph', ["2xsin(1/x) - cos(1/x)"], {strokeColor: 'red'});
	var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});		var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});
	</jsxgraph>		</jsxgraph>

A WASSERMANN at 20:56, 23 March 2021

2021-03-23T20:56:38Z

← Older revision		Revision as of 20:56, 23 March 2021
Line 6:		Line 6:
	\end{cases}\,.		\end{cases}\,.
	</math>		</math>
	<math>f</math> is a continous and differentiable.		<math>f</math> is a continous and differentiable function.
	The derivative of <math>f</math> is the function (red curve)		The derivative of <math>f</math> is the function (red curve)
	:<math>		:<math>

A WASSERMANN at 09:33, 13 February 2019

2019-02-13T09:33:31Z

← Older revision		Revision as of 09:33, 13 February 2019
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	g: \mathbb{R} \to \mathbb{R}, x \mapsto		g: \mathbb{R} \to \mathbb{R}, x \mapsto
	\begin{cases}		\begin{cases}
	x^2(1-x)^2\sin(1/(\pi x(1-x)),& 0<x< 1\\		x^2(1-x)^2\sin(1/(\pi x(1-x)),& 0 < x < 1\\
	0,& \mbox{otherwise}		0,& \mbox{otherwise}
	\end{cases}\,.		\end{cases}\,.

A WASSERMANN at 09:32, 13 February 2019

2019-02-13T09:32:48Z

← Older revision		Revision as of 09:32, 13 February 2019
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	var board = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox:[-1/2,0.08,1.5,-0.02]});		var board = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox:[-1/2,0.08,1.5,-0.02]});

	var g = board.create('functiongraph', ["((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2)))))"], {strokeColor: 'red'});		var g_der = board.create('functiongraph', ["(0 < x && x < 1) ? ((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2))))) : 0"], {strokeColor: 'red'});
	var f = board.create('functiongraph', ["(0 < x && x < 1) ? x^2(1-x)^2sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2});		var g = board.create('functiongraph', ["(0 < x && x < 1) ? x^2(1-x)^2sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2});
	</jsxgraph>		</jsxgraph>

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	var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});		var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});

	var g = board.create('functiongraph', [""], {strokeColor: 'red'});		var g = board.create('functiongraph', ["2*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});
	var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});		var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});
			</source>

			Second example:
			<source lang="javascript">
			var board = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox:[-1/2,0.08,1.5,-0.02]});

			var g_der = board.create('functiongraph', ["(0 < x && x < 1) ? ((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2))))) : 0"], {strokeColor: 'red'});
			var g = board.create('functiongraph', ["(0 < x && x < 1) ? x^2(1-x)^2sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2});
	</source>		</source>

	[[Category:Examples]]		[[Category:Examples]]
	[[Category:Calculus]]		[[Category:Calculus]]

A WASSERMANN at 09:31, 13 February 2019

2019-02-13T09:31:09Z

← Older revision		Revision as of 09:31, 13 February 2019
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	var board = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox:[-1/2,0.08,1.5,-0.02]});		var board = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox:[-1/2,0.08,1.5,-0.02]});

	//var g = board.create('functiongraph', ["2*~~sin~~(1/x) - ~~cos~~(1/x)"], {strokeColor: 'red'});		var g = board.create('functiongraph', ["((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2)))))"], {strokeColor: 'red'});
	var f = board.create('functiongraph', ["(0 < x && x < 1) ? x^2(1-x)^2sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2});		var f = board.create('functiongraph', ["(0 < x && x < 1) ? x^2(1-x)^2sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2});
	</jsxgraph>		</jsxgraph>
Line 48:		Line 48:
	var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});		var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});

	var g = board.create('functiongraph', ["((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2)))))"], {strokeColor: 'red'});		var g = board.create('functiongraph', [""], {strokeColor: 'red'});
	var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});		var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});
	</source>		</source>

A WASSERMANN at 09:30, 13 February 2019

2019-02-13T09:30:01Z

← Older revision		Revision as of 09:30, 13 February 2019
Line 48:		Line 48:
	var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});		var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});

	var g = board.create('functiongraph', ["2*~~sin~~(1/x) - ~~cos~~(1/x)"], {strokeColor: 'red'});		var g = board.create('functiongraph', ["((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2)))))"], {strokeColor: 'red'});
	var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});		var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});
	</source>		</source>

A WASSERMANN at 09:23, 13 February 2019

2019-02-13T09:23:58Z

← Older revision		Revision as of 09:23, 13 February 2019
Line 39:		Line 39:

	//var g = board.create('functiongraph', ["2*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});		//var g = board.create('functiongraph', ["2*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});
	var f = board.create('functiongraph', ["(0 < x \|\| x < 1) ? x^2(1-x)^2sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2});		var f = board.create('functiongraph', ["(0 < x && x < 1) ? x^2(1-x)^2sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2});
	</jsxgraph>		</jsxgraph>

A WASSERMANN at 09:22, 13 February 2019

2019-02-13T09:22:06Z

← Older revision		Revision as of 09:22, 13 February 2019
Line 39:		Line 39:

	//var g = board.create('functiongraph', ["2*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});		//var g = board.create('functiongraph', ["2*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});
	var f = board.create('functiongraph', ["(x < 1) ? x^2(1-x)^2sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2});		var f = board.create('functiongraph', ["(0 < x \|\| x < 1) ? x^2(1-x)^2sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2});
	</jsxgraph>		</jsxgraph>