I just don’t get it. Are they like discovering the ideas and formulas or what?
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I assume the word you wanted was “create”.
Inventing a new form of math usually means starting with some new idea for how to think about things, seeing if that idea produces interesting consequences, and then trying to convert that idea into fully mathematical language.
In the case of calculus, we can follow some steps to “invent” it.
So. Let’s imagine that you’re in a car that is initially moving at 5 meters per second, and smoothly accelerates over 10 seconds to 25 meters per second. (That is, it has a speed given by v(t) = 5 + 2t.) We want to know how far you actually move in those 10 seconds.
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Without a formula or some existing math, this isn’t an obvious answer. We know how fast you’re moving at any given time, but the distance you move is your speed times the amount of time you’re at that speed. In this case, you’re only “at” each speed for an instant as you speed up, so that idea won’t work, at least without modification.
Okay, well, let’s try a different approach. Instead of modeling the way you move continuously, let’s imagine you’re moving at a constant speed for each second, and use that to get an approximate answer. That way, we can compute distances for each second. So we approximate that you’re moving at v(0) = 5 m/s for the first second, v(1) = 7 m/s for the second second, and so on. Since you’re traveling at each speed for 1 second in this approximation, the distance you travel is 5 m/s times 1 second = 5 meters in the first second, 7 meters in the second second, and so on. That gets us an approximation of 5 + 7 + 9 + … + 23 (we don’t get a 25 here because we only reach that speed at the end), or 140 meters.
Now, we know that during each second, we’re traveling faster than we were at the beginning of that second. So the approximation we just did was always underestimating our speed, so that 140 meters is a lower bound. We know we went *at least* 140 meters. That’s a nice thing to know! But it’s not a complete answer: **exactly** how far do we move in those 10 seconds?
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Well. The error in our estimate comes from imagining that our speed is constant for a whole second. What if we didn’t do that? What if we just imagined our speed was constant for half a second, instead? That is, we travel at v(0) = 5 m/s for the first half second, v(0.5) = 6 m/s for the second half second, v(1) = 7 m/s for the third half second, and so on. That should be closer to the real answer, because the approximate speed we’re using is tracking more closely with our real speed. That gets us 2.5 meters of travel in the first half second, 3 in the second half second, 3.5 in the third, and so on, for a total of 145 meters.
For the same reason as before, we know this 145 meters is a lower bound. We must be going more than 145 meters. But we’re still not exact.
Okay, what if we made those time windows really, *really* short – only, say, (1/100) of a second? Then we travel at v(0) = 5 m/s for the first hundredth of a second, v(0.01) = 5.02 m/s for the second hundredth of a second, and so on. We have a lot more adding up to do this time (because we now have 1,000 time windows to add up), but if we do add it up, we now get 149.9 meters.
Hmm, okay, what about 1/100000 of a second? Now we have to add up a million little steps, but we get 149.999 meters.
We might, at this point, suspect that the true answer is probably 150 meters. After all, we know it has to be more than 149.999, and we know that 149.999 is probably really close.
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We don’t know that 150 is the correct answer. But we’ve developed a technique that suggests that it might be. Can we make that technique better?
Well, one thing we can do is get an *upper* bound on the true distance using the same technique, by using the speed at the *end* of each window, instead of the beginning. That is, if we’re using one-second-long time windows, we could assume we’re going 7 m/s for the first window rather than 5 m/s. That way, we’re always estimating a *faster* speed than the real one.
It turns out that if we do this, and use really short time windows, we end up with 150.001. So we know our true distance is between 149.999 and 150.001. We should **really** suspect it’s probably exactly 150 at this point, particularly since the window between the lower and upper bounds is shrinking the shorter we make our time steps.
Once we put this idea into formal mathematical language, we can show that both the lower and upper bounds get as close to 150 as you could ever want, and the only number that fits between them is exactly 150 – the true answer. But that formal mathematical language turns out to generalize to a lot of other situations, and it’s that idea that leads you to start really developing calculus.
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