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Calculus: Defining the Hyperbolic Functions


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About this Lesson

  • Type: Video Tutorial
  • Length: 8:38
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 92 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Inverse and Hyperbolic Functions (14 lessons, $19.80)
Calculus: The Hyperbolic Functions (3 lessons, $4.95)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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Elementary Functions and Their Inverses
The Hyperbolic Functions
Defining the Hyperbolic Functions Page [1 of 2]
Oh, hey, so I'm trying to get into the theater and it's a long line. That's not good. Anyway, so while you're standing there in line waiting for the movie to begin or, if you have a lot of money and you go to the bank and you're standing in line, or if you fly a lot and you're standing in line - basically, if you're standing in line, people want you to stay in line. Have you noticed that it's demanded of you? And they way they get you to do that is they put up these poles, and then they have some sort of chain there. Now, in a bank, it's very fancy velvet, because they're taking your money and they're using it to buy the velvet. But, in general, it's just this thing, it's like a rope or a chain hanging from two poles. And did you ever wonder what that is? In fact, is there math here? Is there math in the movie theater? Well, the answer, my friends, is, yes, there is. And, in fact, right now what I want to do is show you what this curve looks like.
This curve actually looks like this. And let me show you that this curve, this graph right here, whatever that is, matches up perfectly with this. Look at that, it's perfect. Not quite a parabola, it sort of looks like a parabola, but it's really not. It's sort of a wider mouth, looks like a big mouth. And it fits just perfectly there. So the question is what kind of math thing will actually generate this? This thing, by they way, is called a catinary curve. And you see it all the time. Next time you're at a movie theater, next time you're at a bank, next time you're waiting in line to go on an airplane, take a look at this and there's math right there. In fact, if you go for hikes and stuff and you see those big wires hanging where they have electrically wires and they're dangling, those are actually catinary curves, too, but usually you block those things out. When you're at a movie theater and you're waiting there and you have nothing to say to your date, bring this up and it'll be great. So let me show you where this comes from. In fact, first of all, let me just try to break down the set here, which, by the way, is quite extensive, as you can imagine. So we don't need that anymore and we don't need the chain anymore.
Okay, where are these curves coming from? Well, in fact, now it's time, folks for hyperbolic functions to go Hollywood. So what's a hyperbolic function? Well, a hyperbolic function is actually very much what I just showed you. In fact, what I showed you was an example of a hyperbolic function. And these functions actually have names that look sort of triggy. For example, let me introduce to you the first one, which is hyperbolic cosine. That's the name of it, here's how we denote it: cosh x. So it's not cosine, it's hyperbolic cosine. So what's the definition of it? Well, the definition of it is actually a whole bunch of stuff with e's in it. The definition of a hyperbolic cosine is actually . Why cosine? it looks like it should be called hyperbolic e or something. So where's the cosine coming from? I'll leave that for now, but stay tuned.
Now, what does this curve look like? Well, it turns out that this curve has a picture that looks just like this. That is the graph of this function. Now, how can you see that or make some sense out of it? Well, if you think back to the definition, what you actually see is I'm adding two functions together. I'm adding to and I'm getting this result. So if you graph just and then and sort of add those two graphs up, we should get this picture. And, in fact, that's exactly what happens, and let me show you a picture of that, just so we can really see what's going on here. Now, we know what the e^x function looks like. The e^x function is sort of this blue curve right here. And is just the same thing, it's just been a little bit squashed, but there it is. And then the is the same thing, just reflected over to the y-axis. So, in fact, it goes this way. And now, what happens if you add the values of those blue curves? For example, suppose we take a point right here, so one blue curve has value , the other blue curve has value . When you take and add it to , you get the red curve. I'm adding the two curves together, in order to make up the hyperbolic cosine. Notice, for example, here, to get to that red point, what I do is I take this little blue height and add it to this other blue height. If you take that height and then stick on that little extra bit, I hit the red curve. So, in fact, if you add these two curves, you get this very pretty curve. And it turns out, the amazing is it turns out that, in fact, that's the exact curve of a hanging rope between two things. So there it is. That's hyperbolic cosine. And you can't do that with a lot of functions; just pick up a chain and say, "Hyperbolic cosine." But you can say it with hyperbolic cosine.
Now, if you've got hyperbolic cosine, I bet you're thinking, "Well, I want a hyperbolic tangent. I want a hyperbolic cosecant. What about those functions?" Well, it's okay. If you have those desires, we can help you out, because all you've got to do is define them in the following way. So for hyperbolic sine, that's defined to be . And you're saying, "Wait a minute, that sounds like hyperbolic cosine." No, no, no, because hyperbolic cosine is a plus rather than a minus. And if you stick in that minus and actually do the adding of the functions, this is the curve of hyperbolic sine.
Now what about the other functions? Well, we've got as many functions as you have time for. For example, hyperbolic tangent we'll just define it to be hyperbolic sine/hyperbolic cosine. That's sort of natural, because remember that good old-fashioned tangent is actually sine/cosine. And when you plug that in with the definitions of sine and cosine? You see . So that's the definition for hyperbolic tangent. What's the graph look like? It looks like this. Pretty funky, you like? Yeah, I thought you'd dig it. See, it's pretty good.
Now, what about hyperbolic secant? Well, here's hyperbolic secant. We define it to be 1/hyperbolic cosine. And that's . So that's the hyperbolic secant. What does the graph of that look like? It looks like this. It's got a big spike; it goes up to 1 and then comes down really fast. So it's sort of very, very timid, and then has a burst of intuition, and then it immediately drops. That's the graph of that, so that's pretty exciting.
What about a hyperbolic cosecant? That's defined to be 1/hyperbolic sine. So you can see sort of these relationships, because good old-fashioned cosecant is defined to be 1/good old-fashioned sine. So new fancy hyperbolic cosecant is defined to be 1/hyperbolic sine, and that's . You have to make sure that x doesn't equal 0. Otherwise, this would be undefined. So the graph of that actually is not defined when x = 0, and then you can see this asymptote right there when x = 0. So there's the picture of that. It looks pretty if you like that kind of thing.
What about hyperbolic cotangent? Not surprising now, it's defined to be 1/hyperbolic tangent, which would be . Again, x can't be 0, otherwise you'll have something undefined. Again, the graph of this, you can see these big spike things here. So there's the graph.
So those are sort of new functions and they really actually do appear in nature, as we've seen with the chain and so forth and so on.
So there are two questions that I think are worth asking. There are not a lot of questions, by the way, that are worth asking with respect to these functions, because they're using sort of advanced mathematics, but, for our purposes, I think there are two questions that are worth asking. One is, why the trig names? Now, we've seen some of that already, how we define hyperbolic tangent to be hyperbolic sine/hyperbolic cosine and so forth. So we've seen some indication, but really why are we calling all these e-things trig names? That's a good question. And then the other question I think is worth asking is where is the hyperbolic part of it? What's hyperbolic here? We haven't seen any hyperbolic stuff. So I think it's appropriate to end this Hollywood lecture with a cliffhanger. So stay tuned, those questions will be answered. See you at the next lecture, or see you inside. Let's go to the movies. See ya.

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