My mathematical speed bump

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Nexuapex
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My mathematical speed bump

Postby Nexuapex » Thu Sep 06, 2007 3:42 am UTC

So, I'm a computer programmer by nature. And I'm taking Calculus I this semester, and was just reminded of something I've run up against every time I've worked with math in the past.

We're on limits now and, while I can handle the concepts just fine (especially given how many times they're repeated in a standard class period), there are a huge number of problems that I have a huge amount of trouble with.

Maybe it's just that I've been exposed to computer programming far more than math, but it bewilders me how many rules there are for math, and how every rule seems to exist to give birth to more complicated rules. I can't keep track of all of the properties of fractions/square roots/trig functions/limits/etc.

Basically, my huge mathematical toolbox to which is constantly added things like “lim ϴ→0 (sin ϴ)/ϴ = 1â€
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Postby jestingrabbit » Thu Sep 06, 2007 4:02 am UTC

My advice is don't constantly add things. Get used to working out how facts are derived from other facts.

For instance

lim ϴ→0 (sin ϴ)/ϴ = 1

is directly implied by L'hopital's rule. Remember L'hopital's rule, not that particular limit.

Or, a favourite of mine, if you know that e^(ix)=cos(x) + i*sin(x), then you can derive all the angle sum formulas for cos and sin really easily, because

cos(A+B) + i*sin(A+B)
= e^(i(A+B))
= e^(iA)e^(iB)
= (cosA+isinA)(cosB+isinB)
= (cosAcosB-sinAsinB) + i*(cosAsinB+sinAcosB)

so you then have

cos(A+B) = (cosAcosB-sinAsinB).

So you never have to remember the trig angle sum identities, because you can derive them really easily if you remember Euler's formula.

Don't remember a lot of stuff, remember a small kernel of facts that allow you to derive the rest.

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Postby BlochWave » Thu Sep 06, 2007 4:04 am UTC

Well you're learning math theory in that class >_> What you're having trouble with is retaining the theory that you understand just fine in class long enough to use it on a problem, or you understand it just fine but simply haven't practiced enough to KNOW it, both are common symptoms

But also of concern

I can't keep track of all of the properties of fractions/square roots/trig functions/limits/etc.


Properties as in...you can't handle the basics involving them fractions, roots, or trig functions? Limits are excusable because that's sorta what you're learning, but the other three, that's putting a ship to sail before putting in the bottom

If you mean you're having trouble with things you're learning in the class, well, practice more. The fact that you say you understand the theory of the properties ok but can't remember the properties in a problem makes me smirt ^_^ If you're doing a problem and you forget something, REASON IT OUT, don't go to the book. If it takes you an hour to figure it out, you'll never forget it, whereas taking 10 seconds to flip through the book or notes will have you right back there again 10 minutes later. Assume that on any exam you will get nervous, poop yourself, and forget everything, so make sure that in a pinch you could deduce any necessary formula, property, or factoid you need(within reason)

Limits are a small but IMPORTANT portion of that class. You won't be explicitly evaluating any weird limits once you start doing derivatives and integrals, but you'll need to know how to do them to UNDERSTAND derivatives and integrals, and as a computer progammer who will one day be writing code that solves these things numerically, you need a pretty solid understanding

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Postby DrStalker » Thu Sep 06, 2007 4:11 am UTC

I have similar trouble myself; the concepts of limits and integration are wonderful and fun, but the tedium of memorizing which goes along with being able to do it for all cases is not-fun. The result is I remembered enough to pass my maths tests at uni, and then forgot all the special cases. :-)


A cheat-sheet that you made yourself is a huge help, I found. Not that I've had any reason to do complex limits or integrations for a few years, since it doesn't exactly come up a lot in day-to-day life.

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Postby Nexuapex » Thu Sep 06, 2007 4:11 am UTC

jestingrabbit wrote:My advice is don't constantly add things. Get used to working out how facts are derived from other facts.

That's kind of at odds with my five-mornings-a-week class with the problems for a concept due two class periods from when it's explained. :(

jestingrabbit wrote:For instance

lim ϴ→0 (sin ϴ)/ϴ = 1

is directly implied by L'hopital's rule. Remember L'hopital's rule, not that particular limit.

Hmm, so... derivatives (which I haven't gotten to yet) make it easier to compute the limit of (sin ϴ)/ϴ than the massive proof through trig identities that my book has?

jestingrabbit wrote:Or, a favourite of mine, if you know that e^(ix)=cos(x) + i*sin(x), then you can derive all the angle sum formulas for cos and sin really easily, because

cos(A+B) + i*sin(A+B)
= e^(i(A+B))
= e^(iA)e^(iB)
= (cosA+isinA)(cosB+isinB)
= (cosAcosB-sinAsinB) + i*(cosAsinB+sinAcosB)

so you then have

cos(A+B) = (cosAcosB-sinAsinB).

So you never have to remember the trig angle sum identities, because you can derive them really easily if you remember Euler's formula.

Wee, something I'll need to stare at for twenty minutes to make sense of.

I need to find some time to devote to this...

jestingrabbit wrote:Don't remember a lot of stuff, remember a small kernel of facts that allow you to derive the rest.

I would love to do this.
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Postby Nexuapex » Thu Sep 06, 2007 4:18 am UTC

BlochWave wrote:Well you're learning math theory in that class >_> What you're having trouble with is retaining the theory that you understand just fine in class long enough to use it on a problem, or you understand it just fine but simply haven't practiced enough to KNOW it, both are common symptoms

And I think that happened a great deal during my early math education.

BlochWave wrote:But also of concern

I can't keep track of all of the properties of fractions/square roots/trig functions/limits/etc.


Properties as in...you can't handle the basics involving them fractions, roots, or trig functions? Limits are excusable because that's sorta what you're learning, but the other three, that's putting a ship to sail before putting in the bottom

And that's what I'm concerned about.

BlochWave wrote:If you mean you're having trouble with things you're learning in the class, well, practice more. The fact that you say you understand the theory of the properties ok but can't remember the properties in a problem makes me smirt ^_^ If you're doing a problem and you forget something, REASON IT OUT, don't go to the book.

It's the reasoning it out that I find tricky. Ex.: in the context of “lim ϴ→0 (sin ϴ)/ϴ = 1â€
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Postby antonfire » Thu Sep 06, 2007 4:33 am UTC

Nexuapex wrote:Hmm, so... derivatives (which I haven't gotten to yet) make it easier to compute the limit of (sin ϴ)/ϴ than the massive proof through trig identities that my book has?

Yes. However, the fact that that limit is 1 is necessary to prove that the derivative of sin ϴ is cos ϴ, so the massive proof through trig identities is necessary. But the way everyone remembers it is L'Hopital's rule.
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Postby Nexuapex » Thu Sep 06, 2007 4:39 am UTC

DrStalker wrote:A cheat-sheet that you made yourself is a huge help, I found. Not that I've had any reason to do complex limits or integrations for a few years, since it doesn't exactly come up a lot in day-to-day life.

I think that'd be a good idea...

Can anyone recommend a good online resource with a logical progress starting at algebra and working its way up? Just so I can make sure I haven't missed anything.

antonfire wrote:However, the fact that that limit is 1 is necessary to prove that the derivative of sin ϴ is cos ϴ, so the massive proof through trig identities is necessary. But the way everyone remembers it is L'Hopital's rule.

That's cool.
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Postby BlochWave » Thu Sep 06, 2007 4:40 am UTC

Have you learned l'hopital's rule yet? I suppose not, but that's the only way I know to solve it except for good 'ol intuition(I haven't had calc 1 in about 5 years >_>)

for very small x, sin(x) is approximately equal to x(you can look at the graph of sin(x), graph it and look between like -.1 and .1, and graph y=x on top of it, almost the same thing)

so AT x=0, there IS a discontinuity, a nutty one, 0/0, but go just a little bit in either direction and you basically have x/x(because for small x sin(x) ~ x)the closer you get to 0, the closer sin(x) gets to x, so as you're approaching 0 from either direction, you're approaching x/x, or just 1

In your other example, it's the exact same thing, sin(kh) looks just like sin(h) (don't get confused because they changed the variable from x)except the line that it approximates for very small h will have a slope of k instead of 1, so for small h you get h/(hk)=1/k <--thar's your answer

Nary an obscure property in sight, and once you learn l'hopital's rule it's even easier(differentiate top and bottom, you haven't learned how to do it but you get 1/cos(x). Take the limit of THAT, and it's easily 1/1)

Edit: Well I guess I did know how to solve it..kinda, that wasn't an analytic method, that was eyeballing

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Postby recurve boy » Thu Sep 06, 2007 4:52 am UTC

Nexuapex wrote:
jestingrabbit wrote:For instance

lim ϴ→0 (sin ϴ)/ϴ = 1

is directly implied by L'hopital's rule. Remember L'hopital's rule, not that particular limit.

Hmm, so... derivatives (which I haven't gotten to yet) make it easier to compute the limit of (sin ϴ)/ϴ than the massive proof through trig identities that my book has?


Also, there can be multiple ways to get a particular result. Some are easier than others. Once you understand things better, you'll get a feel of what to apply when.

So in this case it is useful to know L'Hopital's rule because the first thing you would do is check lim x-> 0 of sin(x) and x.

It's nothing but practice and for me, mulling over results to get a deeper understanding of what is happening. I think that if you are resorting to cheat sheets you are doing things wrong - unless it is a trig identity cheat sheet!

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Postby Nexuapex » Thu Sep 06, 2007 4:56 am UTC

BlochWave wrote:...but go just a little bit in either direction and you basically have x/x(because for small x sin(x) ~ x)the closer you get to 0, the closer sin(x) gets to x, so as you're approaching 0 from either direction, you're approaching x/x, or just 1

In your other example, it's the exact same thing, sin(kh) looks just like sin(h) (don't get confused because they changed the variable from x)except the line that it approximates for very small h will have a slope of k instead of 1, so for small h you get h/(hk)=1/k <--thar's your answer

Edit: Well I guess I did know how to solve it..kinda, that wasn't an analytic method, that was eyeballing

It makes sense to eyeball it that way, but I still can't see how I would put down my solution on paper, if I need to show my work.

recurve boy wrote:It's nothing but practice and for me, mulling over results to get a deeper understanding of what is happening. I think that if you are resorting to cheat sheets you are doing things wrong - unless it is a trig identity cheat sheet!

Well, I'd only really want simple identities on the sheet...

I can often figure out what's happening and why, the only problems arise when I don't have a foothold on a problem.
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Postby Torn Apart By Dingos » Thu Sep 06, 2007 10:34 am UTC

Try not to use any "facts" to solve problems. Just start from the definitions and apply common sense (no theorems). You'll be amazed at how easy a lot of problems are if you do this. Theorems and useful identities will come to you naturally after a while. There's a famous quote that "the more math you learn, the less you need to remember", and I agree with it.

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Postby Nexuapex » Thu Sep 06, 2007 12:47 pm UTC

Torn Apart By Dingos wrote:Try not to use any "facts" to solve problems. Just start from the definitions and apply common sense (no theorems). You'll be amazed at how easy a lot of problems are if you do this. Theorems and useful identities will come to you naturally after a while. There's a famous quote that "the more math you learn, the less you need to remember", and I agree with it.

I'm definitely hoping for that.

But it's not just the theorems... it's also that I often find myself with a problem and knowing the destination, but not how to get there. Obviously that's a nebulous problem... ah well.
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Postby Quan » Thu Sep 06, 2007 2:27 pm UTC

Everything eventually just sort of mixes together really easy. I think most people at your stage in understanding mathematics have the same problem, so many rules and some of them seem almost arbitrary, but then they go on to imply other things.

I got taught mathematics in a different order, but I do remember having to put some effort in to remembering certain rules that didn't seem worth the time or the effort. I think the best you can do is grind through it and try and find a method of remember that best suits you. Once you get a little further on, things become a lot simpler, powerful tools exists to solve simple or complicated problems, applying definitions to a problem to become 2nd nature.
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Postby Kalak_z » Thu Sep 06, 2007 2:45 pm UTC

[quote="Nexuapex"]It's the reasoning it out that I find tricky. Ex.: in the context of “lim ϴ→0 (sin ϴ)/ϴ = 1â€

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Postby Nexuapex » Thu Sep 06, 2007 3:41 pm UTC

Kalak_z wrote:Here's another suggestion: make what you're given look like what you know. You know lim ϴ→0 (sin ϴ)/ϴ, so find a way to make h/(sin kh) look like it.

That is my general method of solving things. The problem arises when I don't know how.

Looking at your solution, I can puzzle out the rules (you can multiply in a value in the numerator of a fraction as long as you multiply the fraction by the reciprocal of that value, and you can take a fraction's reciprocal and put it under 1 without changing the value) and they make sense, I just wouldn't've arrived at them myself. I haven't done enough problems in which said rules are used to be able to apply them whenever they're needed.

And it gets worse when I look at a solution and can't understand the rules they use.
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Postby Pesto » Thu Sep 06, 2007 7:08 pm UTC

Try to build new concepts on old concepts. Don't learn in a vacuum.

Have you ever seen the following identity?

sin^2(x) + cos^2(x) = 1

I didn't have to memorize that equation, because I learned why it works, not simply that it works. It works because of two basic mathematical concepts: the unit circle and the Pythagorean Theorem.

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Notice that sin(), cos() and the unit radius form a right triangle. That means that these values can be substituted in for a, b and c in the Pythagorean Theorem.

a^2 + b^2 = c^2

becomes

sin^2(x) + cos^2(x) = 1

In other words, learn how all the different rules and formulas are connected to each other. This way you won't be "adding more tools" to you toolbox. You'll simply enhance the tools that are already in there.

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Postby BlochWave » Thu Sep 06, 2007 11:43 pm UTC

Looking at your solution, I can puzzle out the rules (you can multiply in a value in the numerator of a fraction as long as you multiply the fraction by the reciprocal of that value, and you can take a fraction's reciprocal and put it under 1 without changing the value)


That's part of your problem if you try to put things in your head like THAT, my goodness

A slightly more succint way to say it would be "You can always multiply by 1"

that other thing, so very natural to me now that I can't tell you HOW I remember it any more than how I remember to breathe, is just arithmetic. 1/(x/y)=1/1 [divided by] x/y, and you remember what to do when dividing fractions? it becomes 1/1*y/x, you should always remember that, that's like the DEFINITION of dividing(multiplying by the reciprocal)and subtraction is just addition of a negative number, that's why PEMDAS can safely be PEMA or PEDS or whatever

Truth be told I was gonna say "in problems like those when you know one thing and have something similar, use math to make it look like the one thing" but since I couldn't back it up with actions I didn't mention it, but other poster showed you how. Just remember, you're not doing anything there you haven't done since the 9th grade(except the limit at the end which is the easy part)

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Postby zenten » Fri Sep 07, 2007 12:10 am UTC

I think of the numbers/letters jumping around and whatnot, with sound effects. That way I become so entertained that I can actually stay focused enough to remember.

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Postby eattre » Fri Sep 07, 2007 7:57 am UTC

Nexuapex wrote:It's the reasoning it out that I find tricky. Ex.: in the context of “lim ϴ→0 (sin ϴ)/ϴ = 1” I find the problem “lim h→0- h/(sin kh)” with a given constant k and, for the life of me, I can't figure out how the properties I know are supposed to apply to this.

I'm guessing here that k is nonzero (if k=0 then the expression is positive infinity for positive h). In this case, can you think of a change of variable that puts k outside of the expression for a limit you already know?

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Postby Nexuapex » Fri Sep 07, 2007 12:36 pm UTC

BlochWave wrote:That's part of your problem if you try to put things in your head like THAT, my goodness

Well, I don't, really, it just took me a minute to see what was going on. You can always multiply by one, but there are so many different ways of doing so.

eattre wrote:I'm guessing here that k is nonzero (if k=0 then the expression is positive infinity for positive h). In this case, can you think of a change of variable that puts k outside of the expression for a limit you already know?

Yeah, k is given and non-zero. And I don't know what you mean.
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Postby Herman » Fri Sep 07, 2007 12:57 pm UTC

cos(A+B) + i*sin(A+B)
= e^(i(A+B))
= e^(iA)e^(iB)
= (cosA+isinA)(cosB+isinB)
= (cosAcosB-sinAsinB) + i*(cosAsinB+sinAcosB)

so you then have

cos(A+B) = (cosAcosB-sinAsinB).


Well, as long as you know that sin(A+B) = cosAsinB + sinAcosB. So you still have to memorize one of the sum angle identities.

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Postby Token » Fri Sep 07, 2007 1:03 pm UTC

Herman wrote:
cos(A+B) + i*sin(A+B)
= e^(i(A+B))
= e^(iA)e^(iB)
= (cosA+isinA)(cosB+isinB)
= (cosAcosB-sinAsinB) + i*(cosAsinB+sinAcosB)

so you then have

cos(A+B) = (cosAcosB-sinAsinB).


Well, as long as you know that sin(A+B) = cosAsinB + sinAcosB. So you still have to memorize one of the sum angle identities.

No you don't. You deduce cos(A+B) from the real part and sin(A+B) from the imaginary part.

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Postby Zohar » Fri Sep 07, 2007 4:14 pm UTC

The best advice I can give you is to keep practicing. I remember how difficult it was to do some things in math three years ago. If I go back to them now they're much easier. A lot of it has to do with a greater understanding of math. But also, I practiced *a lot*.

And something a physics professor said to our class in our first year - "I'm not smarter than you, I just have more experience". So don't get discouraged, everyone has trouble with things at first. I think it took me at least a month just to understand what a limit was...

Lastly, while things don't get easier, you get used to them with time.
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Postby Herman » Fri Sep 07, 2007 4:36 pm UTC

Token wrote:
Herman wrote:
cos(A+B) + i*sin(A+B)
= e^(i(A+B))
= e^(iA)e^(iB)
= (cosA+isinA)(cosB+isinB)
= (cosAcosB-sinAsinB) + i*(cosAsinB+sinAcosB)

so you then have

cos(A+B) = (cosAcosB-sinAsinB).


Well, as long as you know that sin(A+B) = cosAsinB + sinAcosB. So you still have to memorize one of the sum angle identities.

No you don't. You deduce cos(A+B) from the real part and sin(A+B) from the imaginary part.


Oh yeah. Hehe. Thanks.

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Postby aguacate » Fri Sep 07, 2007 5:05 pm UTC

Nothing beats locking yourself in your room (preferrably on a saturday) and opening your book and working out all the problems until you figure it out. When you do figure something out and learn something new endorphins are released in your brain which provides motivation to learn more and work harder. The key is to start figuring out something, then you'll get better and better.

Also pay attention to your physiological state. Eat good fruits and vegetables and do so damn exercise damnit.
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Postby Yakk » Fri Sep 07, 2007 10:06 pm UTC

So mathematics is not a huge collection of facts.

Mathematics is a web of facts.

Each fact depends on other facts being true, and proves that other facts are true.

If a fact you know is true depends on something you are not certain of being true, then the thing you are not certain of is true!

And if a fact you know is true proves that something you are not certain of is true, then the thing you are not certain of is true.

..

The limit as theta goes to zero of sin theta / theta being 1 is a fact.

Now, suppose you cannot quite remember it -- you remember it is some constant, but not which one it is.

You can make an educated guess by plugging in really small values of theta. Hmm, it gets close to 1.

You can also try doing the trig. gymnastics. Or you could learn facts that depend on the sin theta/theta fact and let you get back to it (d sin / dx = cos x does it, as noted). There are lots of approaches.

Understanding the web, and how to move around in it, and enough facts to reconstruct it, is the art of mathematics.

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Postby GBog » Fri Sep 07, 2007 11:02 pm UTC

Herman wrote:
Token wrote:
Herman wrote:
cos(A+B) + i*sin(A+B)
= e^(i(A+B))
= e^(iA)e^(iB)
= (cosA+isinA)(cosB+isinB)
= (cosAcosB-sinAsinB) + i*(cosAsinB+sinAcosB)

so you then have

cos(A+B) = (cosAcosB-sinAsinB).


Well, as long as you know that sin(A+B) = cosAsinB + sinAcosB. So you still have to memorize one of the sum angle identities.

No you don't. You deduce cos(A+B) from the real part and sin(A+B) from the imaginary part.


Oh yeah. Hehe. Thanks.


Euler's formula. It's the only trigonometric identity you'll ever need. :wink:

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Postby Amicitia » Sat Sep 08, 2007 1:45 am UTC

Yakk wrote:Understanding the web, and how to move around in it, and enough facts to reconstruct it, is the art of mathematics.

Don't you mean the art of the Internet?

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Postby Pathway » Sat Sep 08, 2007 4:08 am UTC

Nexuapex wrote:
jestingrabbit wrote:For instance

lim ϴ→0 (sin ϴ)/ϴ = 1

is directly implied by L'hopital's rule. Remember L'hopital's rule, not that particular limit.

Hmm, so... derivatives (which I haven't gotten to yet) make it easier to compute the limit of (sin ϴ)/ϴ than the massive proof through trig identities that my book has?


Exactly.

Calculus was invented to make things easier, not to make them harder. It's like learning to write in Hindu-Arabic numerals (0, 1, 2, 3, 4, 5, ...) instead of Roman numerals (I, II, III, IV, V, ...).

It might be a pain at first--but it helps in the end.

zenten wrote:I think of the numbers/letters jumping around and whatnot, with sound effects. That way I become so entertained that I can actually stay focused enough to remember.


I think of the same thing--except it's not really a sound effect, it's some sort of mental trigger that isn't explainable by external sensory imagery. It's not a conscious thing, though--so don't try to think about it too much.

It's kind of like feeling a rhythm--if a drummer is tapping out a pattern, you know what comes next. Well, you don't really (just like in math), because it's possible that what comes next is a variation on or expansion of the theme, but you have a sense for what's allowed to come next, depending on the style of music and other factors. Don't take that metaphorically.
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Postby danpilon54 » Tue Sep 11, 2007 1:47 am UTC

didnt really have time to read every reply, but I saw sin(t)/t (too lazy to figure out how to type theta).

An easy way to do this is to notice that as you get closer and closer to t=0, sin(t) becomes more and more like t. This can be seen with a taylor polynomial estimation of sine or just the fact that if you zoom in on a graphing calculator it just looks linear. If you are arbitrarily close to 0 then, sin(t) can be replaced by t, making the function just 1. Lim t->0 1 =1

done :P

Magitek
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Postby Magitek » Tue Sep 11, 2007 3:04 am UTC

As a fellow programmer, I've found that a big difference between programming and math is that when you're learning to code, you have a compiler that is able to give you meaningful feedback as to what might possibly be wrong and where. No such analog exists in math unless someone is watching your work and tells you where you messed up.

Even computer math packages such as Moodle, Blackboard, and Webwork only give correct/wrong feedback.

I'd really like to develop a system where you could work out problems on a computer line by line and have the computer check the validity of those "moves" and find any logical errors I've made and point out where in a problem I've gone wrong.

zenten
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Postby zenten » Tue Sep 11, 2007 11:22 am UTC

Magitek wrote:I'd really like to develop a system where you could work out problems on a computer line by line and have the computer check the validity of those "moves" and find any logical errors I've made and point out where in a problem I've gone wrong.


Problem is its not possible, at least not for every valid "move".

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Yakk
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Postby Yakk » Tue Sep 11, 2007 1:13 pm UTC

zenten wrote:
Magitek wrote:I'd really like to develop a system where you could work out problems on a computer line by line and have the computer check the validity of those "moves" and find any logical errors I've made and point out where in a problem I've gone wrong.


Problem is its not possible, at least not for every valid "move".


It is possible[1], but we do invalid moves and call them valid all of the time. And making all of the moves really valid is really really tedious.

[1] Or at least we think it is.

jtniehof
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Re: My mathematical speed bump

Postby jtniehof » Tue Sep 11, 2007 2:46 pm UTC

Nexuapex wrote:Or maybe I just need to print out a bunch of cheat sheets and bind them in a hardcover book.

Or pay someone to do it for you

taemyr
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Postby taemyr » Thu Sep 13, 2007 1:01 pm UTC

Yakk wrote:
zenten wrote:
Magitek wrote:It is possible[1], but we do invalid moves and call them valid all of the time. And making all of the moves really valid is really really tedious.

[1] Or at least we think it is.


Even if it is possible it would tend to be rather useless. Because you often have several valid moves that you can make, but not all of them lead to an answer. I am fairly sure that the question of which moves is helpful is undecidable.

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Postby Tac-Tics » Fri Sep 14, 2007 2:36 pm UTC

Mathematical laws, identities, and theorems are just like a complicated API in programming. You should know the key players (sin^2(x) + cos^2(x) = 1 and e^(xi) = cos x + i*sin x), but you shouldn't try to engulf ALL of them. Just use your texts as reference (or memorize the ones on the tests). Math is supposed to be an exercise in problem solving (just like computer science!).

Route memorization is really good for professors because it's easy to quiz and test you on facts. However, it's the concepts that are what are really important. Doing something useful in math or CS isn't about individual theorems or algorithms, but combining creating new theorems and algorithms out of your knowledge of existing ones.


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