12-267/Homework Assignment 3 - Revision history

Drorbn at 13:49, 25 October 2012

2012-10-25T13:49:15Z

← Older revision		Revision as of 09:49, 25 October 2012
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	# <math>y\mapsto\int_a^b\frac{y'^2}{x^3}dx</math>.		# <math>y\mapsto\int_a^b\frac{y'^2}{x^3}dx</math>.
	# <math>y\mapsto\int_a^b(y^2+y'^2+2ye^x)dx</math>.		# <math>y\mapsto\int_a^b(y^2+y'^2+2ye^x)dx</math>.
	# <span style="color: red">Postponed!</span> <math>y\mapsto\int_0^1(y'^2+x^2)dx</math> subject to <math>\int_0^1y^2dx=2</math> and <math>y(0)=0</math> and <math>y(1)=1</math>.		# <span style="color: red">Postponed!</span> <math>y\mapsto\int_0^1(y'^2+x^2)dx</math> subject to <math>\int_0^1y^2dx=2</math> and <math>y(0)=0</math> and <math>y(1)=0</math>.

	'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.		'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.

Twine: Added solution to 1

2012-10-24T22:11:30Z

Added solution to 1

← Older revision		Revision as of 18:11, 24 October 2012
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	{{Template:12-267:Dror/Students Divider}}		{{Template:12-267:Dror/Students Divider}}

			'''Solution to Task 1.''' --[[User:Twine\|Twine]] 18:11, 24 October 2012 (EDT)

			Let <math>\sum_{n=1}^\infty c_n = S</math>. We have that <math>\lim_{n \rightarrow \infty} c_n = 0</math>.

			Let <math>\epsilon > 0</math>.

			<math>\exists N \in \mathbb{Z}^+</math> such that <math>\sum_{n=1}^N c_n > S - \epsilon</math>.

			Choose such an N. Then <math>\sum_{n=N}^{\infty} c_n < \epsilon</math>.

			Then <math>\forall x \in X, \forall m > N, n>N</math>, we have <math>\|\Phi_n(x) - \Phi_m(x)\| < \epsilon</math>.

			As this result is independent of our choice of x, we have by the cauchy criterion that <math>\Phi_n</math> is uniformly convergent and <math>\lim_{n \rightarrow \infty} \Phi_n(x) = \Phi(x)</math> exists.

	Solutions to HW3: [[User:Mathstudent\|Mathstudent]]		Solutions to HW3: [[User:Mathstudent\|Mathstudent]]

Twine: Added student divider

2012-10-24T21:05:56Z

Added student divider

← Older revision		Revision as of 17:05, 24 October 2012
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	'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.		'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.

			{{Template:12-267:Dror/Students Divider}}

	Solutions to HW3: [[User:Mathstudent\|Mathstudent]]		Solutions to HW3: [[User:Mathstudent\|Mathstudent]]

Twine: Changed format of Mathstudent's solutions to a gallery

2012-10-24T21:01:39Z

Changed format of Mathstudent's solutions to a gallery

← Older revision		Revision as of 17:01, 24 October 2012
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	'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.		'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.

	~~[http://drorbn.net/index.php?title=Image:012.jpg Solution~~ to HW3~~, page 1]~~ [[User:Mathstudent\|Mathstudent]]		Solutions to HW3: [[User:Mathstudent\|Mathstudent]]
			<gallery>
	[http://drorbn.net/index.php?title=Image:013.jpg Solution to HW3, page 2] [[User:Mathstudent\|Mathstudent]]
			Image:012.jpg\|page 1
	[http://drorbn.net/index.php?title=Image:014.jpg Solution to HW3, page 3] [[User:Mathstudent\|Mathstudent]]
			Image:013.jpg\|page 2
	[http://drorbn.net/index.php?title=Image:015.jpg Solution to HW3, page 4] [[User:Mathstudent\|Mathstudent]]
			Image:014.jpg\|page 3
			Image:015.jpg\|page 4
			</gallery>

Mathstudent at 23:29, 23 October 2012

2012-10-23T23:29:26Z

← Older revision		Revision as of 19:29, 23 October 2012
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	'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.		'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.

	[http://drorbn.net/index.php?title=Image:~~0012~~.jpg Solution to HW3, page 1] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:012.jpg Solution to HW3, page 1] [[User:Mathstudent\|Mathstudent]]
	[http://drorbn.net/index.php?title=Image:~~0013~~.jpg Solution to HW3, page 2] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:013.jpg Solution to HW3, page 2] [[User:Mathstudent\|Mathstudent]]
	[http://drorbn.net/index.php?title=Image:~~0014~~.jpg Solution to HW3, page 3] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:014.jpg Solution to HW3, page 3] [[User:Mathstudent\|Mathstudent]]
	[http://drorbn.net/index.php?title=Image:~~0015~~.jpg Solution to HW3, page 4] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:015.jpg Solution to HW3, page 4] [[User:Mathstudent\|Mathstudent]]

Mathstudent at 23:28, 23 October 2012

2012-10-23T23:28:55Z

← Older revision		Revision as of 19:28, 23 October 2012
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	'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.		'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.

			[http://drorbn.net/index.php?title=Image:0012.jpg Solution to HW3, page 1] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:0013.jpg Solution to HW3, page 2] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:0014.jpg Solution to HW3, page 3] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:0015.jpg Solution to HW3, page 4] [[User:Mathstudent\|Mathstudent]]

Eyjaffe at 04:22, 7 October 2012

2012-10-07T04:22:26Z

← Older revision		Revision as of 00:22, 7 October 2012
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	'''Task 1.''' Let <math>\phi_n\colon X\to{\mathbb R}</math> be a sequence of functions defined on some set <math>X</math>, and suppose that some sequence <math>c_n</math> of non-negative reals is given such that for every <math>x\in X</math>, <math>\|\phi_n(x)-\phi_{n+1}(x)\|\leq c_n</math>. Suppose also that <math>\sum_{n=1}^\infty c_n</math> is finite. Prove that the sequence <math>\phi_n</math> is uniformly convergent.		'''Task 1.''' Let <math>\phi_n\colon X\to{\mathbb R}</math> be a sequence of functions defined on some set <math>X</math>, and suppose that some sequence <math>c_n</math> of non-negative reals is given such that for every <math>x\in X</math>, <math>\|\phi_n(x)-\phi_{n+1}(x)\|\leq c_n</math>. Suppose also that <math>\sum_{n=1}^\infty c_n</math> is finite. Prove that the sequence <math>\phi_n</math> is uniformly convergent.

	'''Task 2.''' Find the ~~extremas~~ of the following functionals:		'''Task 2.''' Find the extrema of the following functionals:
	# <math>y\mapsto \int_0^1y'dx</math> subject to <math>y(0)=0</math> and <math>y(1)=1</math>.		# <math>y\mapsto \int_0^1y'dx</math> subject to <math>y(0)=0</math> and <math>y(1)=1</math>.
	# <math>y\mapsto \int_0^1yy'dx</math> subject to <math>y(0)=0</math> and <math>y(1)=1</math>.		# <math>y\mapsto \int_0^1yy'dx</math> subject to <math>y(0)=0</math> and <math>y(1)=1</math>.

Drorbn at 12:15, 5 October 2012

2012-10-05T12:15:54Z

← Older revision		Revision as of 08:15, 5 October 2012
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	# <math>y\mapsto\int_a^b\frac{y'^2}{x^3}dx</math>.		# <math>y\mapsto\int_a^b\frac{y'^2}{x^3}dx</math>.
	# <math>y\mapsto\int_a^b(y^2+y'^2+2ye^x)dx</math>.		# <math>y\mapsto\int_a^b(y^2+y'^2+2ye^x)dx</math>.
	# <math>y\mapsto\int_0^1(y'^2+x^2)dx</math> subject to <math>\int_0^1y^2dx=2</math> and <math>y(0)=0</math> and <math>y(1)=1</math>.		# <span style="color: red">Postponed!</span> <math>y\mapsto\int_0^1(y'^2+x^2)dx</math> subject to <math>\int_0^1y^2dx=2</math> and <math>y(0)=0</math> and <math>y(1)=1</math>.

	'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.		'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.

Twine: minor grammar edit

2012-10-04T21:01:53Z

minor grammar edit

← Older revision		Revision as of 17:01, 4 October 2012
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	{{12-267/Navigation}}		{{12-267/Navigation}}

	This assignment is due at the tutorial on Tuesday October 9. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if ~~the~~ your readers ~~will be~~ ~~having~~ hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorial on Tuesday October 9. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.

	'''Task 0.''' Identify yourself in the [[12-267/Class Photo\|Class Photo]]!		'''Task 0.''' Identify yourself in the [[12-267/Class Photo\|Class Photo]]!

Drorbn at 11:04, 2 October 2012

2012-10-02T11:04:46Z

← Older revision		Revision as of 07:04, 2 October 2012
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	{{12-267/Navigation}}		{{12-267/Navigation}}
	{{In Preparation}}

	This assignment is due at the tutorial on Tuesday October 9. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the your readers will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorial on Tuesday October 9. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the your readers will be having hard time deciphering your work they will give up and assume it is wrong.

← Older revision		Revision as of 17:01, 24 October 2012
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	'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.		'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.

	~~[http://drorbn.net/index.php?title=Image:012.jpg Solution~~ to HW3~~, page 1]~~ [[User:Mathstudent\|Mathstudent]]		Solutions to HW3: [[User:Mathstudent\|Mathstudent]]
			<gallery>
	[http://drorbn.net/index.php?title=Image:013.jpg Solution to HW3, page 2] [[User:Mathstudent\|Mathstudent]]
			Image:012.jpg\|page 1
	[http://drorbn.net/index.php?title=Image:014.jpg Solution to HW3, page 3] [[User:Mathstudent\|Mathstudent]]
			Image:013.jpg\|page 2
	[http://drorbn.net/index.php?title=Image:015.jpg Solution to HW3, page 4] [[User:Mathstudent\|Mathstudent]]
			Image:014.jpg\|page 3
			Image:015.jpg\|page 4
			</gallery>

← Older revision		Revision as of 19:29, 23 October 2012
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	'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.		'''Task 3.''' A roach <math>R</math> I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying <math>R</math>'s life, help it find the fastest path from one point in the upper half plane <math>\{y>0\}</math> to another point in the upper half plane, assuming there is only one wall around, built along the <math>x</math>-axis <math>y=0</math>.

	[http://drorbn.net/index.php?title=Image:~~0012~~.jpg Solution to HW3, page 1] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:012.jpg Solution to HW3, page 1] [[User:Mathstudent\|Mathstudent]]
	[http://drorbn.net/index.php?title=Image:~~0013~~.jpg Solution to HW3, page 2] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:013.jpg Solution to HW3, page 2] [[User:Mathstudent\|Mathstudent]]
	[http://drorbn.net/index.php?title=Image:~~0014~~.jpg Solution to HW3, page 3] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:014.jpg Solution to HW3, page 3] [[User:Mathstudent\|Mathstudent]]
	[http://drorbn.net/index.php?title=Image:~~0015~~.jpg Solution to HW3, page 4] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:015.jpg Solution to HW3, page 4] [[User:Mathstudent\|Mathstudent]]