Math query

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PriamNevhausten
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Math query

Unread postby PriamNevhausten » Sat Jun 10, 2006 1:37 am

So, I've been wondering for a while, thanks to the mysteries of collegiate employment. What exactly are integrals (and I guess derivatives, by relation), conceptually and computationally speaking? I ask here and not on google because it's been years since I've done upper (read: beyond high-school) mathematics and I don't trust J. Random Website to say things in terms that I can understand. <p><span style="font-size:xx-small;">"It's in the air, in the headlines in the newspapers, in the blurry images on television. It is a secret you have yet to grasp, although the first syllable has been spoken in a dream you cannot quite recall." --Unknown Armies</span></p>Edited by: [url=http://p068.ezboard.com/brpgww60462.showUserPublicProfile?gid=priamnevhausten>PriamNevhausten</A]&nbsp; Image at: 6/10/06 1:37

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Re: Math query

Unread postby E Mouse » Sat Jun 10, 2006 2:15 am

The best explanation is probably the one that my Calc professors have been beating me over the head with for the past few years.

You know how you can graph a curved line on a 2-d peice of paper? Turn it into a proper x-and-y graph. Now find an equation that will always return the right y value for the graphed line if you give it the corresonding x value for that point (this isn't always easy, and only works if the line doesn't 'overlap' with two values for a given x). This is the function that defines that line.

Taking the derivative of a function gives you a seperate function in terms of x and y that can be used, at any point, to determine the rate of change of the original line/graph/function at that same point. Basically, it tells you if the line is going to the upper-right or lower-right at the point you solve it at, and how fast. (Derivatives are always defined going from left to right.) The value of the derivative at a point is called the slope of the original line (at that point). Slopes are only constant for straight lines.

Sometimes you can take the derivative of the derivative itself, and this second(-order) derivative can be used for less useful applications on the original equation, like concavity, but I doubt you care about that.

If Physics was more your strong point, here's an alternate description, and a useful application: the position of an object compared to time can be considered a basic graph. If you find the function for it and take the derivative, you get the function for its velocity. If you take the derivative of that, you get the acceleration formula.

So basically, the location of something at a point of time is the normal equation, before taking any derivatives. The derivative of that equation will give the speed of the object at any valid point. The second derivative of the position (derivative of velocity) gives how much the object is speeding up or slowing down.

Integrals are... basically derivatives in reverse, but more difficult to deal with. The integral of a function that only has points above the x-axis (i.e. positive y) will give the function to find the area under the curve. (If part of the original line is below the x-axis, this part will be considered 'negative area' and actually be SUBTRACTED from the 'area.' It's really weird, so they're kinda useless on their own.)

As for how to mathematically take derivatives and integrals, that's a lengthly thing to explain, since it's different depending on what functions you're playing with. Any triginometric functions screw things up (guess what the derivative of Secant is!), as do any non-basic functions multiplied/divided by one another. If you really want to know, take a night course in it or something. It's mildly interesting.

Of course, I probably screwed this up by being too detailed and taking too long to post, but oh well. <p>


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Idran1701
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Re: Math query

Unread postby Idran1701 » Sat Jun 10, 2006 2:50 am

E-Mus has pretty much got the right of it, though second-order derivatives are more useful than he gives them credit for in an applied sense. (Acceleration is the second-order derivative of position, for example.)

In the most general sense: the derivative of a function describes the rate of change of the function at a given point. The integral of a function describes how to find the area under the curve of a given function.

If you're interested in finding out how to actually _find_ derivatives and integrals of a function, we can go into a bit more detail, of course. <p>

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</p>Edited by: [url=http://p068.ezboard.com/brpgww60462.showUserPublicProfile?gid=idran1701>Idran1701</A] at: 6/10/06 2:52

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Re: Math query

Unread postby PriamNevhausten » Sat Jun 10, 2006 4:07 am

I was mostly confused about integrals, actually. I'd read about derivatives (and actually applied the velocity-acceleration metaphor you mentioned, Mus!) some years ago, but never learned about integrals--and being thrust into advanced physics courses in which integrals and differential equation systems are commonplace and taken for granted did not give me much, contextually, to work with.

So integrals give area, though. Interesting.

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Re: Math query

Unread postby Idran1701 » Sat Jun 10, 2006 6:32 am

They are also equivalent in certain circumstances to an infinite sum, actually, because of one way it works. If you're curious, look into a Riemann Sum. Basically, you approximate the area using identically-wide rectangles that fit under the curve. Then, you do it again with skinnier rectangles. You repeat this, making the rectangles skinnier each time. thus, the limit of the value as the width approaches 0 is the total area of the function. <p>

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Re: Math query

Unread postby Zemyla » Sun Jun 11, 2006 2:35 pm

Simulation is where I use most of my derivatives and integrals. Nothing like setting up a variable stepsize Runge-Kutta for a video game. <p>-----
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Re: Math query

Unread postby BrainWalker » Wed Jun 14, 2006 2:27 pm

The maths and I do not get along. While I sorta get what's goign on with derivatives, at least in theory, I still remain largely mystified by integrals. So you can approximate the area beneath a curve. That's dandy, but what's the point? I have a hard time figuring out why that is such an important value that it would be commonplace in physics lectures, but then again, physics looks like arcane wizardry to me.

So, beyond physics lectures and apparently programming, according to Zem, what are the practical applications of integrals? Architecture, maybe? Engineering? <p><div style="text-align:center">Image</div></p>Edited by: [url=http://p068.ezboard.com/brpgww60462.showUserPublicProfile?gid=brainwalker>BrainWalker</A]&nbsp; Image at: 6/14/06 14:27

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Re: Math query

Unread postby Idran1701 » Wed Jun 14, 2006 3:20 pm

Practical applications? Those are boring, who cares about those? Only thing special about the universe is it's the mathematical model we happen to live in. :D



More seriously: Not just a curve. Using integrals, you can, for one, find the volume of any shape, two or three dimensional.

Other than that, I'm theoretical, not applied, so you'd probably need to talk to Zem for specific applications of integrals. It's important in Fourier series, which is basically a way of approximating any periodic function (or any function that you can pretend is periodic) as an infinite sum of sines and cosines. It's also important for finding the sum of a function, which allows you to evaluate the sums of infinite series. <p>

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Re: Math query

Unread postby Zemyla » Wed Jun 14, 2006 3:56 pm

Fourier series, simulations (every physics engine in videogames is just an evaluation of an integral), some compression schemes use integrals too...

It's just so many applications that I can't think of just one. <p>-----
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PriamNevhausten
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Re: Math query

Unread postby PriamNevhausten » Wed Jun 14, 2006 4:10 pm

I've seen them used for...what was it...diffusion rates from compartment to compartment. Given a certain rate of transfer from A to B and B to C, how much is in C after a certain time has passed? What if these are nonlinear rates? Integrals.

Well, actually, for that I've heard more mention of Laplace transforms, but they also mentioned that integration was a possible, if messier, solution method. <p><span style="font-size:xx-small;">"It's in the air, in the headlines in the newspapers, in the blurry images on television. It is a secret you have yet to grasp, although the first syllable has been spoken in a dream you cannot quite recall." --Unknown Armies</span></p>

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Re: Math query

Unread postby Idran1701 » Wed Jun 14, 2006 4:45 pm

One of the cool parts of math; often, something that looks like it's an entirely new idea, once you do some work with it, turns out to really be just another way of expressing something you've looked at already. <p>

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