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Especially when math people talk about the beauty or "elegance" of an equation, and I have no idea what they mean.

I'm ashamed to admit I took an entire course in calculus and had no idea why it was invented or what the point of it was.

The main point of maths is to turn it into alien squiggles.

Well, not alien, maths is a foreign language, and if you lose the sense of what it means, you'll not be able to do good maths.

But you should always remember that maths is about patterns, not about the thing you're trying to do maths on, and the patterns have a life of their own.

That's what calculus is all about, it's based on limits. dy/dx is short hand for saying take dx towards zero and dy with it, what do you get when you divide them.. in the limit?

Well, if as you say and I say, limits are required to do the job, then they're required to do the job.I don't think it's possible to do calculus without limits, and having some understanding of them; I think everyone that has ever done calculus gets shown a right-angle triangle of sides dy and dx and is asked to imagine what happens as you make that triangle ever-so-small. That's just exactly the same as taking the limit.

The people who worked out calculus did it to solve specific problems and they knew exactly what those problems were. Why shouldn't everyone else be allowed to start off at the same point where there are clear problems there to be solved and where they can see a mathematical process generate answers.

It should be possible to program a computer to solve these problems without having to type alien squiggles in at all - all of it must be possible using nothing more than a standard CPU instruction set.

Computers can do that, but those are very definitely alien squiggles it's manipulating, you type them in in that form, and you get them back in that form.

Limits are answers where you find that the answer gets closer and closer to something, without ever quite getting there, (if it actually got there the maths usually explodes, but that doesn't matter!!) But not every equation tends to a limit, some don't converge to a limit at all, and some give you a different answer depending on which side of the limit you start from.

For example:y = (x+1)/xy tends to 1/x as x tends to 0, but when you get to 0, it explodes! If you don't believe me, try plugging in a small x and seeing what y is, you'll find it's very close, and gets closer the smaller x is. The reason is, is because when x is small x+1 in the numerator is basically 1, but x in the denominator is still significant to the answer.

So for example in your gravity simulation dx/dt is the speed of a body; if you set dt very small, how fast is the body moving? dx/dt! It's just distance (dx) over time (dt), speed!

Quote from: David Cooper on 02/02/2012 23:05:19It should be possible to program a computer to solve these problems without having to type alien squiggles in at all - all of it must be possible using nothing more than a standard CPU instruction set.Both yes, and no. Mainly no.Normal calculus takes an equation, and changes the equation 'differentiating' or 'integrating' to give new equations. Computers can do that, but those are very definitely alien squiggles it's manipulating, you type them in in that form, and you get them back in that form.

The other way to do it is not to use true calculus, but to use small deltas. So you divide the problem up into small chunks, but you do not take it down to the infinitesimal limit. If the distances between the chunks are small enough, this approximates to using calculus. That's done rather a lot, it's called 'finite element analysis' and similar names.

QuoteSo for example in your gravity simulation dx/dt is the speed of a body; if you set dt very small, how fast is the body moving? dx/dt! It's just distance (dx) over time (dt), speed!It would be helpful if this stuff was taught by retaining words in the early stages so that you can remember what things are meant to be. Why not just write dist/time?

And why does it suddenly go Greek with delta replacing d? The trouble I have with calculus is may be mainly down to a teacher introducing symbols and never restating what they're supposed to mean.

Quote from: wolfekeeper on 03/02/2012 08:30:55Normal calculus takes an equation, and changes the equation 'differentiating' or 'integrating' to give new equations. Computers can do that, but those are very definitely alien squiggles it's manipulating, you type them in in that form, and you get them back in that form.If it can be done at all, it must be possible for a computer to do the job with a set of very simple instructions. It has to be able to turn the squiggle version of the problem into something it can handle through those simple instructions, and that simply requires someone to program it to decode the squiggles.

Normal calculus takes an equation, and changes the equation 'differentiating' or 'integrating' to give new equations. Computers can do that, but those are very definitely alien squiggles it's manipulating, you type them in in that form, and you get them back in that form.

QuoteThe other way to do it is not to use true calculus, but to use small deltas. So you divide the problem up into small chunks, but you do not take it down to the infinitesimal limit. If the distances between the chunks are small enough, this approximates to using calculus. That's done rather a lot, it's called 'finite element analysis' and similar names.If a human can work out exact answers using calculus, why would a machine not solve problems the same way? If a human can do it, it ought to be dead easy for a machine. Maybe it's faster to do it with approximations and that might matter in something like graphics for computer games where precise answers may be unnecessary.

Quote from: wolfekeeper on 03/02/2012 00:38:01Limits are answers where you find that the answer gets closer and closer to something, without ever quite getting there, (if it actually got there the maths usually explodes, but that doesn't matter!!) But not every equation tends to a limit, some don't converge to a limit at all, and some give you a different answer depending on which side of the limit you start from.I thought the whole purpose of calculus was the it could get to an exact answer at some point rather than just getting a good approximation. For example, if you're trying to calculate the slope of a curve through a point, you can calculate the slope of a line and make that line shorter and shorter to get a closer and closer approximation of the slope at the point, but you can't do anything useful with a line of zero length. I was led to believe that calculus would somehow enable you to get an exact value for the slope at a point, but I never saw any evidence that this was actually possible during hundreds of hours of watching a teacher fail to explain how calculus was meant to be done.

Quote from: David Cooper on 03/02/2012 22:06:30It would be helpful if this stuff was taught by retaining words in the early stages so that you can remember what things are meant to be. Why not just write dist/time?Because if you do dist/time, that's actually the average speed.

It would be helpful if this stuff was taught by retaining words in the early stages so that you can remember what things are meant to be. Why not just write dist/time?

What you're trying to calculate is the speed at a particular instant in time. For example, a car that is accelerating may travel 100 metres in 10 seconds, but it wasn't going at 10m/s the whole way. At the beginning it was stationary, at the end it was going at (say) 20 m/s.How would you work out the speed at the end of the course, if it wasn't accelerating completely uniformly?

QuoteAnd why does it suddenly go Greek with delta replacing d? The trouble I have with calculus is may be mainly down to a teacher introducing symbols and never restating what they're supposed to mean.Delta means it's not a limit, it's a real number you could type into your calculator, like 0.0001, it's just a small number.

No, it doesn't decode them, it treats them as squiggles and messes around with the squiggles. It doesn't have a clue what the squiggles really mean.

You take the limit of the gradient as the line length tends to zero, and you get the answer; that's the gradient at that point.The answer is exact, the limit of something (when it's soluble) is an EXACT answer, it's not approximate.The approximation gets better and better as the line length gets shorter, and the limit is precise.1+1/2+1/4+... the limit of that is 2. It's exactly 2. Likewise the limit of the gradient is the actual gradient at that point (by definition- they define it that way!!)

How do we know it's right? That the gradient really is that? Probably, mathematicians don't, really. But they find that the answers always make sense, they're always self consistent, so it doesn't matter. But really it's pretty obvious that it ought to work.

Quote from: wolfekeeper on 03/02/2012 22:20:00Quote from: David Cooper on 03/02/2012 22:06:30It would be helpful if this stuff was taught by retaining words in the early stages so that you can remember what things are meant to be. Why not just write dist/time?Because if you do dist/time, that's actually the average speed.Then so is dx/dy. If the d has some function other than being a constant, wouldn't it be better to have a special symbol for whatever it's supposed to be to avoid confusion? What does the d actually mean?

QuoteWhat you're trying to calculate is the speed at a particular instant in time. For example, a car that is accelerating may travel 100 metres in 10 seconds, but it wasn't going at 10m/s the whole way. At the beginning it was stationary, at the end it was going at (say) 20 m/s.How would you work out the speed at the end of the course, if it wasn't accelerating completely uniformly?I probably couldn't, but I can imagine that there might be a way to do it if you had a formula to describe the acceleration.

x = 0.5 t^3 + 3 t^2what speed is it going at, at time t=10?

Yes, Ok, if I give you the acceleration, but what if I give you an equation of its position:x = 0.5 t^3 + 3 t^2what speed is it going at, at time t=10?

dx/dt = 1.5 t^2 + 6 t

Um. Well delta x/delta t is the average speed over a time delta t. So for example delta t might be .1 second and the distance travelled might be 5 metres so the average speed would be 50 m/s.

If the delta is made very small, then you're averaging the speed over a very small time, then the average speed tends to the limit of the actual, instantaneous speed.

In other words for the exact speed, rather than just the close average, you want dx/dt which is the limit of delta-x/delta t.

So dx/dt is the speed. Geezer already calculated that (correctly!) for you.So all you have to do is plug t=10 into Geezer's equation, and you have the exact speed at that moment.

I poked around a bit and came up with this ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN This lady has gone to some trouble to explain an example in practical terms. (I confess I didn't sit through the entire lecture.)

Say we start with a constant Acceleration=10gSo, now if I set my distance to be the diameter of the moon (3,475 km = 3,475,000 m) and solve for T, I get:So, t=266.3 seconds.Plugging that back into the velocity equation, one getsv=10gt = 10*9.8*266.3 = 26097 m/s = 26 km/s

Anyway, you can use calculus to change from distance to velocity to acceleration, and visa versa.

Derivatives gives one slopes (or instantaneous rate of change) of a curve.Integrals gives one the area under a curve.

Certainly it can be applied to many other things such as population models.

It's a startlingly simple process which I spent months doing in school without ever being allowed to know why.

I'm having trouble following that, partly because I can't tell whether letters are constants or units, and I don't know what the dt is doing.

Try this:A population of bacteria squares every second. Let's call the population p, and the time t...Differentiation allows you to determine just how steep a slope is at a particular point on the slope.

if f(t) is a distance function with respect to time t.

Then Velocity(t) is a function of the rate of change of the distance at any time t. I.E. If you are driving from LA to New York. If Velocity = 0, that means you are parked, and distance is not changing. If the Velocity is > 0, then you are going forward.If the Velocity is < 0, then you are going backward.

Anyway, the Velocity(t) is the rate of change of the distance function, or the derivative of the distance function d(f(x))

Acceleration tells you how fast you are changing the velocity function (how much you've pressed the gas pedal down). So, it would be the slope of the velocity function at any given time. Acceleration(t) = d(Velocity(t)) = d(d(f(t))

I suppose I forgot that when one takes derivatives, one looses one's constant factors. So, taking integrals, one should add them back in when doing integrals. I.E. What is the initial distance or velocity, which in some cases can be set to zero.

Anyway, the mathematics is relevant for scientific discussions. However, one can often understand a lot about a subject without needing anything very complex.

Indeed one can, but there's no reason why anyone should be shut out of the maths - it is simplicity obscured by squiggles. What people really need to see to get an easy ride into calculus (or anything else of this kind) is clear rules laid out as to how to go about things, clear descriptions of what things are, and clear demonstrations of the procedures you have to go through to solve problems. If any of the tools aren't available to you or if any part of the language isn't clear to you, you're stuffed - you can appear to be a failure at maths, and yet that is not the case. The real problem is that you haven't been given a checklist of tools so that you can be sure you're working with the complete set, you haven't been given an adequate (or even an inadequate) manual explaining the squiggles which you can go back to consult at any time, and you're pushed into doing everything through squiggles before you're ready. It's no wonder that so many people get stuck here (or somewhere else in maths) and never get any further. The problem isn't with the learners, but with the teaching.

Quote from: wolfekeeper on 04/02/2012 23:29:20Um. Well delta x/delta t is the average speed over a time delta t. So for example delta t might be .1 second and the distance travelled might be 5 metres so the average speed would be 50 m/s.As soon as these deltas come into it, I get confused because they seem to have no purpose - all it's actually saying is that distance divided by time = average speed.

QuoteIf the delta is made very small, then you're averaging the speed over a very small time, then the average speed tends to the limit of the actual, instantaneous speed.Which it appears will always be zero because you're hacking it back to the speed at the start when it wan't moving.

At the risk of hopelessly derailing the thread (not that that ever stopped me) I saw something recently about people seeing numbers in particular colo(u)rs, at least I think it was numbers.

Q. Are some people born good at maths? some people are born bad at maths ... ...sorry, you cannot view external links. To see them, please REGISTER or LOGIN

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Quote from: David Cooper on 06/02/2012 01:05:44As soon as these deltas come into it, I get confused because they seem to have no purpose - all it's actually saying is that distance divided by time = average speed.Nearly, it's the distance CHANGE and the time CHANGE.distance change divided by time change = average speed over that timei.e.delta x / delta t = average speed

As soon as these deltas come into it, I get confused because they seem to have no purpose - all it's actually saying is that distance divided by time = average speed.

QuoteQuoteIf the delta is made very small, then you're averaging the speed over a very small time, then the average speed tends to the limit of the actual, instantaneous speed.Which it appears will always be zero because you're hacking it back to the speed at the start when it wan't moving.Wrong!!!! And it's wrong in an important way.These are DELTAS so it would be delta x (0) over delta t (0) = 0/0 = undefinedSo you can't put it all the way back, because the maths blows up when you set the time delta to exactly zero, a zero sized triangle has no calculable gradient.But by doing the LIMIT, you are effectively setting the time delta an infinitesimally small amount above zero, so it doesn't blow up, and you get a sensible answer. And that's how calculus works.

But it isn't a change in distance - it's a change in location, and the distance represents the amount of change in location.

Either way, I still can't see why the delta is necessary because metres and seconds are primarily units of distance and duration rather than locations.

Quote from: David Cooper on 07/02/2012 23:44:23But it isn't a change in distance - it's a change in location, and the distance represents the amount of change in location.For example at t=3, you might be at x=4, 4 is a distance from the originand at t=3.3 you might be at 4.2, 4.2 is the distance from the originBut delta x is NOT 4 or 4.2, no, it's the change in distance, the distance delta (delta literally just means change) in the distance: 4.2 - 4 = 0.2 = delta x when you make a small step in time.

It's not just any distance, it's specifically a very small distance, as small as possible.

Quote from: wolfekeeper on 08/02/2012 22:10:34It's not just any distance, it's specifically a very small distance, as small as possible.Right! You really could care less what the actual distance is because you are only interested in the ratio of the distances in two axes.Perhaps you could substitute something along the lines of "the ratio of the axes at any point" for the shorthand "dx/dy"

Technically , dy/dx is not a ratio - it is the limit or just a notation.

Mad mathematicians would tell you that the ratio of two infinitessimals can be anything you choose (kind of like the ration of two infinities it is not well defined).

It is a technical point and not that important - I have seen thinking of dy/dx as a ratio described as a "useful crutch that could get you into difficulties later". The reasons are abstruse and arcane - and I couldn't rehearse them off hand (or possibly at all), but I have followed some very heated exchanges in which mathematicians lose their rag trying to explain to the math forum visitor why it should not technically be thought of as ratio