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College Algebra: Intro to Conic Sections

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About this Lesson

  • Type: Video Tutorial
  • Length: 4:32
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 49 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Trigonometry: Full Course (152 lessons, $148.50)
Trigonometry: Conic Sections (12 lessons, $26.73)
Trigonometry: Parabolas (4 lessons, $5.94)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/collegealgebra. The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra 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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Okay, so now I want to start talking about something that's very, very geometric, and turns out will have sort of algebraic ramifications and guises. But really these are objects geometric. And they're objects that, in fact, occur in three space. And this is really sort of a discussion about, well, analytic geometry, and in particular, conic sections. So what is, in fact, a conic section? Well if you think about it, it's exactly what it sounds like. It's actually sections of a cone. So imagine having a cone like this, like sort of an ice cream cone, but inverted. And the question is what kind of sections can you get if you were to cut? Well it turns out you can get all sorts of neat things. For example, if you cut right at the tiptop point, you would get a point. So in fact, a point is a conic section. And that's sort of a fundamental object. If I were to take a cleaver, and to cut it right along there, just right there, just graze it, you'd notice I would have a line. So in fact, that's the way to think about a line. It's just a very special conic section where you just cut, just graze the surface. You get a line. You cut right here, you get a point. What about some more exotic things? Well, for example, you would get a circle. How could you get a circle? Well if you just took this thing, sort of like cooking with Professor Berger, if you just took this thing and make a nice straight cut, you see you would get a circle. And you can see that. I don't know if you can see this or not, but see the boundary of the, still falling, the boundary of that right there, can you see that? I think you can understand, that's a circle. There's a circle. Now what if you actually cut, but instead of cutting flat like this, you sort of cut at an angle. I'll try to do this right now live for you. Let's see what happens. Ooh, not bad, well then you actually get an ellipse. And in fact, this is a perfect ellipse. It's not just sort of an oval. An oval is just sort of a random curvy thing that looks like an ellipse. But this is a genuine, pure ellipse. And in fact, you can take this to be the definition of what an ellipse is. You might say, "Hey, what is an ellipse?" Just whack off part of a cone, and you've got an ellipse, really neat.
All right, but there's something else you can build. What if I cut, and cut parallel to this side right here, right down there? Let's see what happens, shall we? I'm going to try to do this. I hope this works. Well when you do that, what you actually end up with, besides a mess, is a beautiful, perfect parabola. In fact, we've been talking about parabolas throughout this course. And now you can see how to make a parabola. You just literally take a cone, and you can see there is a beautiful curved symmetry. I mean you can see it right there. You just take a cone and flush it out like that. And you see a beautiful parabola. There's a happy face parabola that we saw, and there's a sad face parabola. You can see them all. They're all there. So that's pretty cool.
But in fact, there's more. If you were to take the cone, and I'm sort of losing cone juice here, and just cut straight down, have it sort of sitting right up and cutting straight down, you get another kind of curve that looks like a parabola, but really isn't. In this case, you get something that looks parabolesque, but not really a parabola. And in fact, what you get is a hyperbola wing. That really is part of a hyperbola. And in fact there are two wings of a hyperbola. And to really see them both, what you've got to do is realize that what's really going on here is I have actually two cones, one on top of each other, so when I cut down on one, you see, I actually cut also a piece on the other. So I see a hyperbola wing that's up, and I see a hyperbola wing that's down. They both look like this. So I see sort of this kind of thing, and then also this kind of thing. But it all just comes down to this. So it's really amazing that through the cone, through these conic sections, you can actually find circles right here, you can find hyperbola, you can find parabola, and you can find ellipses. And you can also, of course, find the degenerates of those things, the line and the dot.
So coming up next what we're going to take a look at is how do you describe these things analytically. A lot of them we've already seen. The parabola we know is something x^2, and so forth. But with a geometric eye, how do you actually describe these curves in space, and in the plane. So we'll take a look at the parabola. We'll take a look at the ellipse. And then we'll also take a look at the ever popular, and much beloved hyperbola. I'll see you soon.
Conic Sections
Parabolas
Introduction to Conic Sections Page [1 of 1]

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