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Author Topic: Maths thread.  (Read 53346 times)

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pacovf

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Re: Maths thread.
« Reply #825 on: May 09, 2017, 06:38:18 pm »
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Extra credit question I got lately: Find a good (unbinned) Maximum Likelihood Estimator Goodness of Fit.

Too late to get the extra credit, but I would be curious if anyone has any thoughts on this kind of thing.
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Titandrake

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Re: Maths thread.
« Reply #826 on: May 13, 2017, 03:26:46 am »
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For a somewhat ridiculous extra credit question (I don't think the teacher realized what he was doing), I need to solve this integral:

∫(cos(θ) + 2sin(θ))*sqrt((cos(θ)+2sin(θ))^2-4.75) dθ

It's used for a definite integral that can be calculated numerically (and gives the correct answer), but I'm curious if there's any way to figure out the indefinite integral.  No online calculators have been able to find the answer.  It's probably not an elementary function, but I want to see if anybody can figure it out.

Isn't the square root not always defined? I plotted cos(θ)+2sin(θ) and it definitely crosses 0.
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sudgy

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Re: Maths thread.
« Reply #827 on: May 13, 2017, 05:14:11 pm »
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For a somewhat ridiculous extra credit question (I don't think the teacher realized what he was doing), I need to solve this integral:

∫(cos(θ) + 2sin(θ))*sqrt((cos(θ)+2sin(θ))^2-4.75) dθ

It's used for a definite integral that can be calculated numerically (and gives the correct answer), but I'm curious if there's any way to figure out the indefinite integral.  No online calculators have been able to find the answer.  It's probably not an elementary function, but I want to see if anybody can figure it out.

Isn't the square root not always defined? I plotted cos(θ)+2sin(θ) and it definitely crosses 0.

You would have to plot (cos(θ)+2sin(θ))^2-4.75, but yeah, it does cross 0.  Like I said, the context was a definite integral that was within the domain of the function.
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   Quote from: sudgy on June 31, 2011, 11:47:46 pm

silverspawn

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Re: Maths thread.
« Reply #828 on: May 26, 2017, 04:25:00 pm »
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Is there any meta advice which mathy people here would give about how to increase efficiency in solving problems? Anything you've learned over time? E.g. I think I spend too little time thinking broadly and too much following the first approach I've found. Stuff like that.

pacovf

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Re: Maths thread.
« Reply #829 on: May 26, 2017, 06:08:49 pm »
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I don't know about "real" problems, but as far as homework goes, it's just a bunch of doing stuff until you start recognizing the patterns. Homework thinking is affected by the knowledge that you are expected to be able to do it.

It's sometimes useful to start at the answer, as in "if there was a solution, then to be able to find it I would likely have to do X", and then look into X.
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Cuzz

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Re: Maths thread.
« Reply #830 on: May 26, 2017, 06:11:18 pm »
+4

I don't know about "real" problems, but as far as homework goes, it's just a bunch of doing stuff until you start recognizing the patterns. Homework thinking is affected by the knowledge that you are expected to be able to do it.

It's sometimes useful to start at the answer, as in "if there was a solution, then to be able to find it I would likely have to do X", and then look into X.

This is the drastic and sometimes frightening distinction between doing math problems and doing math research.
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Re: Maths thread.
« Reply #831 on: May 26, 2017, 08:42:29 pm »
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Is there any meta advice which mathy people here would give about how to increase efficiency in solving problems? Anything you've learned over time? E.g. I think I spend too little time thinking broadly and too much following the first approach I've found. Stuff like that.

My thoughts:

It is important to reflect after either solving a difficult problem or looking at the solution. Figure out the motivation for the solution, and try to find a framework to view the problem from that makes the solution obvious or natural. When you do lots of problems, you can begin to see when similar frameworks can cause solutions to fall out of problems that might look very different at first.

Also, just do lots of problems.

Most of my experience comes from math olympiad, here are some links from people on solving math olympiad problems (in a broad sense):
https://hcmop.wordpress.com/2012/03/23/how-to-approach-an-olympiad-problem-by-ho-jun-wei/
https://usamo.wordpress.com/ (couldn't find a specific post that really addressed what you are talking about (although it probably exists) but maybe you will find some things interesting.
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pacovf

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Re: Maths thread.
« Reply #832 on: May 27, 2017, 12:28:30 am »
+1

I don't know about "real" problems, but as far as homework goes, it's just a bunch of doing stuff until you start recognizing the patterns. Homework thinking is affected by the knowledge that you are expected to be able to do it.

It's sometimes useful to start at the answer, as in "if there was a solution, then to be able to find it I would likely have to do X", and then look into X.

This is the drastic and sometimes frightening distinction between doing math problems and doing math research.

On the other hand, one advantage of math research, compared to homework, is that there's probably only, like, three people in the whole world that understand what you do, so people can't judge you for not being able to solve all questions :p
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ConMan

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Re: Maths thread.
« Reply #833 on: May 28, 2017, 07:08:37 pm »
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When doing maths research, there are two techniques that often come into play:

1. Generalise the problem into something that might be easier to solve, then show that it holds for the specific case you're looking into.
2. Solve a specific instance of the problem, then try to generalise it to the thing you're interested in.
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