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ePosted June 10, 2001 I was over at Kathy's house a few days ago, and she was studying for an exam in some business course she's taking. She ran across something in her textbook that seemed wrong, and she asked me about it. To my delight, not only was it wrong, but it was wrong about math. As if that weren't enough, it was also ironic.
This textbook, whose title and author I fortunately didn't catch, was
trying to tell
So what There are numerous interesting properties of the exponential function. Never mind that it always increases. Lots of functions always increase. This one is special. Picture a graph of e^x. If you don't know what e is, it's one of those special numbers like pi, and it's about 2.7. I'll tell you how to calculate the exact value later. On the far left, at e^-10, it's just a little above the X axis, and it's pointing nearly exactly horizontal. It gets a little steeper as it approaches the Y axis, which it intersects at Y=1. This should be no surprise because at the Y axis, X=0 and e^0=1, just like anything else to the power of 0. On the right hand side of the Y axis, it gets really big really fast. The further you look to the right, the bigger it is, and the faster it grows. At X=10, it's around 22000. At X=20, it's about half a billion. At X=100, well, I don't know a handy abbreviation for 44-digit numbers so I can't tell you. If you followed that, you'll agree that e^x is quite a dramatic function. It grows too fast for normal graph paper, so if you plan to graph it, you would do well to compress the Y axis so you can get a better view. Let's try compressing the Y axis by 22000, essentially graphing e^x/22000, and see what shape we get. Well, starting at the far left with X=-10, we're even closer than before to the X axis, but you wouldn't know the difference by looking. At the Y axis where X=0, we've got e^x=1, but if we divide by 22000, we'll still be pretty darn close to the X axis. On the right at X=10, we'll get up to one square of graph paper. This is more reasonable than last time, where we were off the page by several miles at X=10. So far so good, but if we try X=20, we get 22000. And if X=30, we get half a billion again. So all we achieved by compressing the Y axis by 22000 was to move the problem 10 squares to the right. But the graph is still exactly the same shape. This is one of the handy things about the exponential function: it's exactly the same shape no matter how much you compress the Y axis. Another cool thing about the exponential function, which is only cool if you think calculus is cool -- and in case you're wondering, yes, I think calculus is cool -- is that it's equal to its first derivative. And its second derivative, and its third derivative, and all the rest of them too. A derivative is a function that describes how fast another function changes. Speed describes how fast your position is changing, so speed is the derivative of position. Acceleration describes how fast your speed is changing, so acceleration is the derivative of speed. Acceleration is the derivative of the derivative of position; friends call it the second derivative. Case in point: e^x describes how fast e^x changes. When e^x is 1, it's got a slope of 1. When e^x is 22000, it's got a slope of 22000. This is a pretty good definition of e, because there's only one number that works out that way. e^x is not the only function which is it's own derivative; there is one other important but boring function which is equal to its first, second, third, and all the rest of its derivatives. That function would be zero. The rate at which zero changes is zero.
So ---
I said earlier that I would tell you how to calculate
But if it were futile, I certainly wouldn't be doing it on this show,
now would I? The point here is that F is e^x and we don't know what
You can try that on a calculator. As you add more inverses of
factorials, you'll get closer to the actual value of e. By the time
you add 1/4! you'll have 2.7; if you get to 12 terms, you'll see
something like 2.71828182828, which may lead you to think there's some
kind of pattern. But don't let that thought carry you away because
soon enough you'll get to 2.71828182845905 which will start to look
just as unfathomable as 3.14159265358979. The patterns come and go
but those numbers are definitely irrational. That means that there is
no pair of integers you can divide in order to get either Oh, and the factorial of X? Just multiply X by all the natural numbers below X. 3 factorial is 3 x 2. 4 factorial is 4 x 3 x 2, also known as 24. 1 factorial is 1. 150 factorial is about 260 digits long. Your calculator might even have a button on it that calculates factorials for you; if so, it'll say x with an exclamation point. Like it's really excited. X! And if you want to check your Taylor approximation, any calculator with a factorial button, and for that matter any calculator that deserves to exist at all, has a button labelled e^x. So you can press 1, then press the e^x button, and you'll get as much of 2.71828182845905 as can fit on the screen. |