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Part I In 1687 Sir Isaac Newton published Philosophiae Naturalis Principia Mathematica, which, among other things, layed out the mathematics of calculus. It’s given us an innumerable number of advances in science and technology, it’s perhaps the most important field of mathematics for practical application (minus simple arithmetic I suppose). However, calculus is not just math, it’s a way of looking at the world, a new way and one that is immensely powerful. There’s no need to do any math to appreciate this world-view, and that’s what I’ll be doing in part I here, calculus sans math, not an equation to be found. Of course the application of calculus in a practical manner requires some math, and so for those interested part II will explain how to turn the ideas of calculus into equations (this isn’t a math class here, just what the idea of calculus is as written in numbers).
To understand the calculus world view, let’s take an example and look at it from a few different perspectives. Let’s say you pick up a paper clip, hold it at arms length, and then let it fall to the ground. From a normal perspective what you’ll notice is that after you drop this paper clip, it’ll end up on the floor (rather a silly place for a paper clip but this isn’t a philosophy essay … quite ). You might notice how fast it appears to move from your hand to the ground, and if you were well equipped (and had pretty good reflexes) you might even be able to time the fall.
This is a typical, and perfectly valid way of looking at the world, first cause goes to final effect. You let go, gravity pulls the paperclip, the paper clip lands on the floor. Well let’s take one step towards a calculus view and ask ourselves: “what was the paper clip doing in between cause and effect?” If you answered “in the air, falling towards the floor” very good, we’re now one step closer (if you didn’t, I’ll pretend I didn’t hear your answer and you can just stick that one in).
All right, let’s take the crucial step into the realm of calculus by asking: “what was the paper clip doing right after you dropped it?”, or alternately, “what was it doing right before it hit the ground?”. The crucial thing about these questions (as you might imagine from the italics) is the limit in timing. And how you evaluate these questions is how you understand calculus.
Let’s answer the first question (just as good as the second question), about what the paper clip was doing right after you let it go. The first thing to answer is, “where was it?” Well, we decided that it was right after you dropped it, so let’s put it a millimeter down from your fingers, or perhaps a “when was it?” analysis would lead you to the answer of a millisecond after your finger let it go. Well both have a problem in that their not quite close enough, right after means there’s nothing between the event and the right after, and there’s a full millimeter, or one millisecond between your fingers and that paperclip, so clearly that wont cut it. So where should we put it?
The way to answer that question isn’t to keep coming up with smaller and smaller numbers until you get there, it would be exactly like counting to infinity, you’ll never make it! Instead you just say “infinity”, or in this case “infinitely close to my fingers”.
Good! Now we know exactly where the paperclip is, infinitely close to your fingers, but not quite on them. Of course you may be asking yourself what this has to do with calculus, and my answer, if you can look at the world as a series of infinitely close things, you’re looking at it through the lens of calculus, you’re there already!
What happens right after you drop the paper clip? Hopefully your thinking “it’s right below my fingers, but not quite touching them”; now think how that’s different from “on the floor”. Of course both are correct, they’re just different ways of looking at the world, but the first way, the calculus way let’s us answer some questions that we wouldn’t be able to answer with the second perspective.
So let’s finishing answering the first question posed, what is it doing when it leaves your fingers? Well it’s moving down a little bit, and since there’s gravity, it’s beginning to move downwards faster (it’s accelerating). Good! It’s moving downwards, and increasing its rate of downward motion. OK, what’s it doing right after that?
Uh-oh, I’m sure we can answer this question, but how many more are there left to go? If we move the paper clip downward at the rate of an infinitesimally small amount every question we’re going to be answering a lot of them! That doesn’t sound like a very powerful way of doing anything except maybe getting a sore throat and you can do that at a ball game.
All right, if we don’t want to sit around all day answering “right after that?” questions, let’s try and answer all of them at once! The paperclip will be somewhere in the air, moving downward at some speed (increasing that speed as it goes) and eventually it’ll hit the ground. There, that was a lot easier than answering an infinite number of questions, but now how do we distinguish that from our original world view of the paper clip hitting the floor after dropping it?
Well now we’ve arrived at our destination, the difference is that in the calculus view the paper clip didn’t just go from your fingers to the floor, it went through an infinite number of tiny steps from your fingers to the floor, and in each step it’s position and velocity were slightly different.
That’s it! If you get that, you get calculus, pretty simple huh? What this means as a perspective on the world is instead of seeing beginning and endings, you can see an infinite number of middles. It may take you thirty minutes to drive to work, but those thirty minutes are filled with an infinite number of tiny steps in which your car is always at different points going at different speeds, the getting to work is the sum of all of those infinite steps.
This is basically the end of Part I, but I’ll speak a little about the way this is used in math without actually using math, a segue perhaps. There’s two things you do with calculus, derivatives and integrals. Derivatives take something (an equation, though if you’re looking at the world through calculus, you can take anything) and then ask what it’s doing at a very specific time and give you the answer. For instance a derivative of the paperclip falling at precisely 1.0000000… seconds after you dropped it (assuming your incredibly tall and it hasn’t hit the floor yet) might tell you that it’s falling at 9.81 meters per second right then. Of course it wont fall 9.81 meters in one second, because this is just the speed it’s falling at right at that point. That’s the key to calculus, you have a series of points and all of them are different from each other, even if only slightly. You let the paperclip keep falling and it’s going to travel a lot more than 9.81 meters in the next second because the velocity wont stay at 9.81 meters.
The other operation, so to speak, is integration, which is taking all those tiny little instants and putting them all together to figure out what’s going on now. The integration of the speed of the paper clip from the time you release it to the time it hit’s the floor will give you the floor. It takes each infinitesimal period of time and multiplies it by the velocity of the paperclip at that time. Kind like if you drive three roads into work, one where you go about 60 for 10 minutes, one where you go 45 for ten minutes, and one where you go about 25 for 10 minutes, then you’d just multiply your speed by the time you spent going that speed and you’d end up with the distance to your office from home. But now I’m using more math than I planned so it’s time to end this.
Part II
Math, Yay! Well despite my enthusiasm I’ll try and keep this as concept oriented as I can with as little math as possible, but since this is explaining math, obviously there will be some.
The most important thing to understand to get calculus as a field of mathematics is something called a “limit”. Limit means in math pretty much what it means in everyday context. If your kid was running widely towards the living room that you had just mopped, you might tell your kid to stop before he got to the living room. You just places a limit on him. He was going towards the living room, but he’ll never quite get there, though he’ll probably get pretty close right? Sneak his toe up to the divider just to try and get you to react? Well while he sits in time-out for trying to get a rise out of you, let’s talk about how this applies to math.
“The limit as x goes to 0 from the right” is a phrase your likely to hear from a mathematician, but what does it mean? Well basically the same thing as what you told your child (though you can make the numbers obey you much easier than the kid). The number represented by ‘x’ will get really, really close to zero without ever reaching it. Just like our paper clip right before it hit the ground, not quite there but as close as it can get without actually being there, and infinitesimal distance away.
The phrase “from the right” means from which side of the number line x is coming from, right being positive and left being negative. Sometimes, when it’s not important, this part is left out, but with 0 it’s very often important since just to the right of 0 (which our original statement would create) means a positive number, and just to the left would yield a negative number, both very, very small, but a sign difference can be big.
OK, so the concept of the limit is that it’s almost to the number, but not quite, so let’s see an example and figure out what exactly it means.
Lim x->0+ x
Now if this were a true equation the “x->0” would be written under “Lim” but I can’t really format that here so just imagine. This statement reads “x, as x goes to zero from the right”. The “Lim” obviously stands for “limit”, meaning what’s coming up next is setting up some number getting really close to another number but not quite there. The “x->0+” tells you that the number x is getting really close to zero (it’s “going to” zero), but is not quite there. The plus sign tells you that it’s coming from the plus, or right side. Finally, the “x” says, OK we have the limit, but what are we doing with it? Of course a more interesting equation might be:
Lim x->0+ x*2
Which is “x times two as x goes to zero”. In other words, that last bit is the equation your trying to figure out the answer to, and your substituting “x” with “x->0” to find the answer. So let’s go back to the original equation, what is the result of x when x goes to zero? Well, the answer is zero. Now I’m sure this is very frustrating, we just spend all that time making it so x gets close to zero but isn’t quite there and now I’m saying we’re going to call it zero? Why not just stick in a zero instead of doing all this? Well basically because it isn’t quite zero, but it’s so close to zero we can just call it zero. However, it’s let’s us give actual answers to a question like this:
Lim x->0+ x/x
If we just called x zero then we’d be up a creek wouldn’t we? Because now we have a number divided by zero, and even when it’s zero divided by zero you’ll still get an undefined answer. But luckily, we put a limit in, so since x isn’t quite zero we don’t have to worry about dividing by zero and we can just cancel it out. The answer is 1! Any non-zero number divided by itself is always one, and x isn’t quite zero. So the answer to 0/0 is different than Lim x->0+ x/x, and that’s why wee use limits.
All right, hopefully you get the concept of limits (one number gets really close to another) and are at least willing to wait to judge if their useful or not, so let’s get into heavy duty calculus. Now all I’m going to talk about here is derivatives, but it should be enough to last you.
So a derivative is where something, anything, is going at one point. The paper clip had a derivative that told us how fast it was moving downwards for instance. Well since we’re doing math, let’s talk about an equation, and let’s use the all time most popular equation for calculus, x^2 (x squared).
Where is x^2 going when x is equal to one? Well up, it’s certainly increasing, but how fast? This is where calculus becomes basically the only way to answer the question exactly, we’ll take a derivative!
So, in general, how do you figure out how fast something is changing (moving)? Say a car driving down the road. Well one way to do it is to see how far it goes in a specific time period, and then divide by the length of the time period. For instance if your car went 60 miles in one hour, then you would divide 60 by one hour and discover (not surprisingly) that your car was going 60 miles per hour.
Well that’s all well and good, but we need to know what it’s doing at an exact time, not over the space of an hour. How do we get more exact? Well, we can shorten the time, and discover that the car travels 30 miles in one half hour, and so for that half an hour it’s going 30/(.5), or 60 miles per hour. Of course this is still too long, so we would continue to shorten it until the time we measure is infinitely small and the distance it travels is infinitely small as well. Sound familiar?
Well with an equation like x^2 we can do the same thing. When x is equal to 0 it’s position (y=x^2, position at 0 would be what ‘y’ equals when x is equal to 0) is 0 (0^2 = 0). When x equals 2, then it’s position is 4. So over a period of 2 spaces (think of x as time) the equation has moved 4 periods (think of y as distance). So it’s 4/2 some things per something, or 2 some things per something. Only of course that’s not accurate enough to decide what it’s speed is when x equals one, but it does give us a framework for looking closer.
[2^2 – 0^2] / (2 – 0) = 4 / 2 = 2
That’s what we just did, but now let’s right it in terms of another number ‘n’, which will be the difference between x when it’s at 0 and x when it’s at 2. This is pretty easy, 2-0 = 2. So in this case n is two. If we started at 1.5 and ended at 3.2 then n would be 3.2-1.5, or 1.7. All right, so let’s rewrite this sucker:
X = 0 --- [(X+n)^2 – (X)^2]/n = 2
First we declared the number x to be 0, the original, then instead of just written another number 2, we described 2 as the original x plus the difference. Or in other words, we picked the starting number, and then wrote the second number not as ‘2’, but as the original number, plus the difference between the original number and the second number; that equation could also be written as:
[ (0 + 2)^2 – (0)^2]/2 = 2
or
X = 0 --- [ (X + 2) ^ 2 - X^2] / 2= 2
Well we haven’t made whole lot of progress it seems, just playing around making the equation a little different (not to mention a little more complicated) but now it’s set up perfectly so that we can tackle the problem of the making that interval really, really small with limits! So instead of checking how much this equation is changing between 0 and 2 we can check it between 1 and right after 1, and we’ve entered the realm of calculus. So let’s get to it!
Lim n -> 0 [ (X + n)^2 - X^2]/n
First question to ask, what happened to the x equaling 1? The answer is, it went away! That equation is for any x, not just 1, if you wrote it for just 1 (which you certainly can do) it would look like this:
Lim n->0 [(1 + n)^2 – 1^2]/n
But let’s stick with the original one, since then we can answer our question for any x, not just when x equals one. All right, we’ve got the equation, we’ve got limits, let’s go at it! From what we know we replace ‘n’ with zero and we get:
[X^2 – X^2] / 0 = ?
Uh-oh, our method has failed! We replaced n to soon, let’s try it again. First, square everything.
Lim n->0 [X^2 – X^2 – 2*X*n – n^2] / n
OK, now we’re getting some where! We can get rid of the X^2s, since they cancel out, and (here’s the tricky part) we can get rid of n^2. Why? Because almost zero times almost zero is much, much smaller than an actual number (2*X) times almost 0, so much smaller it can be ignored completely without any effect on the end result. So now we have:
Lim n->0 [2*X*n]/n = 2*X
And there’s your answer! At any point on the equation X^2, the equation is increasing at a rate of 2*X. So at one, it’s increasing at 2*1, or 2. You’ve just done a calculus problem!
Boy, that was a lot of work, wasn’t it? Is all calculus this bad? No, because people have discovered some tricks. Almost never do you actually take the limit of n going to zero when doing calculus, instead you use short cuts that have been proven. For instance, the derivative of c*x^n is always c*n*x^(n-1). But at the heart of calculus is that limit, and when you understand it, you understand the heart of calculus.
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Thank you, Hobbes. I have always wondered about calculus. I read part I, and I think I get it. But a little ways into part II, I got lost. It may be that it's just too late at night. Or I might just have to read and reread it several times. I like the way you explain things, though. So this is probably my best chance for ever understanding calculus.
I'm quitting *before* my head starts to hurt. Rain
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I like your explanations, Hobbes. One thing that I found useful in understanding calculus was the graphical representations of derivatives and integrals. Unfortunately, it requires knowledge of algebra to make much sense.
Derivatives = slope of the line tangent to the curve at a given point (draw a curve, then mark a point on it. Find the one line that goes through that point [locally] and that point only. The slope of that line is the derivative of the curve at that point.)
Integrals = The area under a curve. Usually this is limited to a specific part of the curve since in many cases the area under the entire curve is infinite or zero.
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I read this and I never knew I used to think about Calculus when I was a kid! When I was in elementary school I remember asking my dad that if a car was driving towards a place but decreasing its speed constantly, so that it was always going x miles per hour and x also equaled the distance from the car to the place, would it never get there, even if it continued moving forever? In other words, the car is going 60 MPH when it's 60 miles away, and 59 MPH when it's 59 miles away, and all the speeds in between. I don't know how to say it in math terms, although I probably said it better here than when I was trying to explain my train of thought to my dad 15 years ago.
Anyway, I rememeber asking him, "Wouldn't the place you're driving to always be an hour away if you did that? How can it always be an hour away if you keep getting closer? How can it be an hour away even if you're one foot away from it?"
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Perhaps this is what you were thinking of mathematically (this is pretending that your description is slightly altered and thus doesn't need a step function)
v (velociity) = 60-x;
Where x is in miles and is the distance traveled. This is actually a more complicated problem than one might think, sine you want time and it's in terms of distance, and perhaps I'll explain how that works after my upcoming 4 hours of class.... grrr.
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In it's similest form, basically means integers, you'll probably be doing basic counting theory, probability, that sort of thing. And then you move into different stuff later on that I've never done, enjoy.
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Thank you, Hobbes... that was really fun and well done. My calculus was all the way back in high school (um, can I express how many years ago as a derivative? ) -- and I still remember some of those lovely moments in class of 'getting it.' Your essay(s) brought it back to me.
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). You might notice how fast it appears to move from your hand to the ground, and if you were well equipped (and had pretty good reflexes) you might even be able to time the fall.![[Smile]](smile.gif)
(if you didn’t, I’ll pretend I didn’t hear your answer and you can just stick that one in).
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