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Calculus: The Cycloid

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About this Lesson

  • Type: Video Tutorial
  • Length: 12:31
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 134 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Parametric Equations, Polar Coordinates (18 lessons, $27.72)
Calculus: Understanding Parametric Equations (3 lessons, $5.94)

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 http://www.thinkwell.com/student/product/calculus. 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.

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Parametric Equations and Polar Coordinates
Calculus and Parametric Equations
The Cycloid Page [1 of 3]
Bicycles are great. They get you from here to there, or actually from here to anywhere, in fact, and they're great. Why are they called bicycles? Well, the bi part is because there are two - there are two of them. And why are they called cycles? Well, let's think about that for a second. If you think about the wheel of a bicycle - there are two of them; here's a sample of one. If you were to bicycle in place - in fact, people do this, right? If you go to a gym, it's a beautiful day outside - beautiful - and you'll see a hundred people inside on a bicycle spinning and literally spinning their wheels going nowhere. They could be doing that outside, and it's nice, and go somewhere, but no, they stay inside. Now if you do that, there's nothing really exciting going on. You just literally see this - this is going around in circles.
But happily if you actually use a bicycle the way it was intended to be used, in fact what you actually do is you have movement. Once you have movement, one has to wonder what's actually happening to the wheel, as it's moving in space. In fact, here I've marked this little area here. If I just keep the bike in position and turn it, that thing is actually sweeping out a circle - not very exciting. Of course, when you're at the gym using a stationary bicycle, it's not very exciting; but what happens when you use a bicycle outside? When you use a bicycle outside, you start to move; and notice that as you move, that thing actually is moving in space. What's the path of that?
Well, in fact, that's a neat question. Let's take a look at an actual sample here of someone moving on a bicycle. What I've done here is I've actually marked one point just like I did on the bicycle wheel. I marked one point here. Now let's watch, and the question is: What is the path of that point? So watch that point as it traverses through space. Let's see what happens. If we let it go - well, I won't let it go, but if you just watch it - now the point is sort of going up and it's going to the left. Now it's coming down but still going to the left. Then it sort of stops here and then comes up again, and then it starts to go down again, and then it starts to go up again. So it seems to be going up and down. It's like life - it has its ups and downs. Let's do it again. Here we go. So it's going up, then down, then up, then down. So it's in place, it's sweeping out a circle, but I'm also now moving in this direction. So it's sweeping out something that's sort of going like this.
Well, what does that path look like? That path actually is given by this curve. Maybe that doesn't seem to make a lot of sense. Like how come it's not round? It should be like a circle maybe or something. Well, I'm going to prove to you that, in fact, this is the right answer.
Here is the bicycle wheel that made this path. This bicycle wheel actually has, you can see, I'll put it down here...and here's the dot, I'll start it here. If we start here, then it touches right here. Let's follow that dot as the bicycle wheel turns and moves this way. Now watch this really carefully, this is really cool. So here we go. I'm trying to do this with my limited amount of bodies in the way. Watch this now. Watch the dot. The question is: Does the dot hug the curve? So far, yes; so far, yes. And look at it, even if it goes up here, it's hugging the curve beautifully. It hugs the curve just perfectly up to that top point there, and look, now it starts to come down. The point is just hugging that curve. Isn't that amazing? Now, there's that sharp corner there. Let's see if that really hugs the curve past the sharp corner. Here we go...here we go...sharp corner, sharp corner, hit. Now what happens next? Just watch it...keep watching, keep watching, keep watching...and then bang! It goes up. It really does follow that curve. Amazing, absolutely amazing!
How can you actually figure out this path? That is a hard question, but if we use parametric equations, we can actually produce this path parametrically. That is to say, I want to give you an x-location and a y-location for every point on this curve. So as it moves through time, if I say time is this, it will produce that green point. So I want to parametrically describe that green point moving through space just like that. Now I can do it really fast now that you've seen it. It goes down, and comes up, and comes out there. Okay, so that's what I want to do, I want to parametrically model that.
How am I going to model that? What I have to do is for every single point, I have to be able to produce the x-value and the y-value for that. This is really, really tricky. In fact, by the way, this is called a cycloid. The path of a bicycle wheel is a cycloid. So what I want to do now is get parametric equations to describe this very complicated-looking curve.
How do we proceed? The way I'm going to proceed is to model this. So let me model this now. Let's pretend that the bicycle wheel - in fact, we made a large bicycle wheel here so you can really see it - but let's pretend that the bicycle wheel just had radius 1. This actually has a radius 2, but we wanted to make it big so you can actually see it. But, in fact, let's just pretend - it's the same path no matter what the radius is, you'll just get a little smaller curve - let's pretend that in fact this is a one here.
So if this is a one here, what's going to happen? First of all, if the radius is one, let's just look at some facts here. So here are some facts about a circle. If the radius were to be one, then what is the perimeter - or also known as circumference - of this? Well, the circumference of this would be 2pr - in this case, r is one. So, in fact, once around would mean 2p radians. What would that mean? What that would mean is that if this thing were just going around once like this and hits here, that once-around would chart out a length of 2p...of 2pr.
So really if you think about it, since I have the radius being one, then as I go, the distance I've traveled is precisely the angle around that I've gone. If I go halfway, for example to here, that would be p. If I go all the way around, that would be 2p. So in fact in this direction, I could think of this as sweeping out the angle - the angle that I'm located at.
So that's sort of helpful. But then how do I get this point - sort of that arbitrary point that's sort of hanging in space? Let's draw a model of that and see if we can figure out exactly how that would look. Here's a model of that. What I'm going to do is I'm going to draw some axes. Of course, you see that. And I'm only going to try to draw the circle in it. In fact, I could put the circle in here, but I don't want to ruin it because it's such a pretty circle. Let me put in my own special circle - special in that I'm making it handmade, and you never know...when you start drawing a circle, you never know how well it's going to come out, but that's not bad. It may not be the world's best, but it's certainly not bad. Let's pretend that the dot is located right here. So it would look like this. So there it is in actuality, and I'm modeling it here. Now if the dot is right there, what I want to do is as this thing turns, the dot's going to move, and I want to tell you the x-location of the dot and the y-location of the dot at any point in time.
So what I want to do here is I want to think about that as (x,y) - that's a point in space, (x,y) - and I want to try to figure out a formula for x and y in terms of time as I sweep this thing out. What I'm going to do is I'm going to draw some extra things in here to help me out. Let's remember that the radius of this circle is one. That means that this length right here is one. So that's one. Unfortunately, that dot is not always going to be at one, it's going to be going up and down, up and down, and who knows.
This is a right triangle. As I sweep this thing out, I know that this is actually going to be q, or the angle that I've traveled, which could correspond to this as a reference angle of q. Let me put in q right here. Remember that we said that as I sweep this thing out that determines what degrees I am. If go halfway around, for example, then I'm at p. If I go all the way, I'm at 2 p. So this sweeps out sort of the angle that I've traversed. So that can be represented here with this little reference angle.
Now, let's think about this. Since I know that the radius is one, then this hypotenuse is also one, because remember this is a circle of radius 1. So now we're almost done, believe it or not. All I've got to do is figure out how to describe those points in terms of this q, in terms of how far I've traveled if you think of q as time to take you away, and I'm going out as I travel to the right.
How can I do that? Well, let's see. What is this length right here? If this angle is q and a little right triangle, and this is hypotenuse(1), then this would be the opposite. So it seems to me that this is actually sinq. Let me write that in. So this is sinq. Let's just check that because sinq is opposite over hypotenuse. That would be . What would this be? This would be cosq.
Now let's see if we can figure out the x-location. What's this x-location? That value right there looks like this entire thing, which we know is q, and then I have to back up this much. You see, that distance right there is this entire journey minus that. So that would be q, and then I subtract off sinq. So, in fact, this length from here to here is nothing more than q minus sinq. So the x-direction would be this q minus sinq. What about this height, what about the y-direction? Well, the y-direction would be one, and then I've got to subtract off cosq. So to get that height, take this one and then subtract off the cosq.
If you do that, these give the parametric equations for this movement - and the bicycle stills go on - for this general movement. And that general movement is given in terms of q, and you can think of q as time, for example, or you can think of q as representing the angle that the dot is sort of located on as it goes on its journey making one complete cycle or continuing, and so forth.
So, in fact, these two parametric equations describe curves that look like this - that look like these cycloid curves. And that's really amazing that a sort of complicated curve can be modeled using parametric equations, and the trick is to sort of draw this picture and then really analyze where the point is located in space. Notice that both x and y are both moving, so a really tricky example but sort of famous example, and an example that you see all of the time. If you ever watch someone biking by you, just imagine picking a point on that tire and watching where that point goes. It's going to sweep out something that can be represented like this.
I'll see you at the next lecture, and we'll talk more about parametric equations.

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