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College Algebra: An Introduction to Variation

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About this Lesson

  • Type: Video Tutorial
  • Length: 5:56
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 63 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Equations & Inequalities (50 lessons, $65.34)
College Algebra: Variation (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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Thinkwell
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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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Now I thought we would talk about something that's sort of interesting and actually appears a lot, not only in mathematics, but actually beyond. And it's sort of an interesting idea, sort of philosophically. You know, if I tell you that this thing is equal to that thing, then if I up this thing and this ups by the exact same amount, they'd always be equal. For example, if I say something like the water level versus, you know, how much water is there. If I raise the water level, then I'll have more there and so forth and so on. But this actually can be generalized to the idea of two things being proportional. Sometimes they're referred to as variation, but really it's a notion of proportionality. What that means is if two things are directly proportional, what that means is that they might not, in fact, be identical. They may be different, however if one changes by a certain percentage and increases a certain amount, this other thing which is a different value would increase by the same percentage. So even though the values wouldn't be the same, the truth remains that they would either increase or decrease by the same percentage. So for two things to be proportional, it means that if one goes up, then the other one would go on. If one goes down, the other one would go down. But they might not necessarily be the same values. Now I want to illustrate this with a very simple example but I think it would sort of drive home the idea. So let's take a look down here and you'll see all I have here is a circle. Now notice that if I were to put--I'm going to try to do this right now live so I have no idea how this going to work or not. But if I put, for example, two marks here and I now move this thing around a certain fixed amount, let's say right to here; if I stop it, you could see that those lengths are plainly different. I don't know if you can see the green length in the hole or not, but it's really there. I promise. There's the green length. I'm just out shadowing what already exists. There's this purple length. They're plainly different. But notice that I sweeped out just by the same amount. It was just this thing going from here all the way to there. I just sweeped that out. Now what if I were to say instead of the sweeping out from here to here, I'm now going to go just further. If I go just further, what's going to happen? Well, what's going to happen is they're both going to increase but notice they're going to increase by different amounts. The important thing is even though they increase by different amounts, the relative amount that they increase remains the same. So, in fact, now instead of just going from this point to this point, we've increased it up to here and here we've increased it out to here. So you can see I went from here to here. And even though those values are still different, the point is the increase was the same. These two quantities are proportional. For example, if I were to start here and then go to here and now go all the way around, you would see that even though they would be increasing by different lengths, the percentage of increase is always the same. These two things are actually directly proportional. A 50% increase in one would lead to a 50% increase in the other. So those two things are related even though the actual numerical value, those lengths are not the same.
Let me try to illustrate this idea a little further with an actual sort of how this works with nuts and bolts sort of in math. So if two ideas or two values are proportional, I'll say x, in fact, here is a symbol; x ~ y. It sort of almost looks like an equal sign, but they're not equal. They might have different values, but they're proportional. What that means is that they actually are equal up to a multiplicative constant. So these two things are directly proportional meaning that there is some fixed number so that if I take 1 and multiply it by this fixed number, I get the other number. And so therefore, if this thing goes up, notice this quantity would go up as well. And if this quantity goes up, this would go up, even though the values would always differ. But the point is that they differ by a multiplicative constant of another. So if two things are directly proportional, if, in fact, one is equal to a constant; this is some fixed number times the other.
There's another kind of proportionality which also occurs a lot in nature, especially in physics, and that's called inversely proportional. Inversely proportional means that a quantity, x, is proportional to the reciprocal of another quantity; which means that x equals some constant divided by y. Why do you think this is called inversely proportional? Because look what happens. If the y were to go up, what happens to the x? Well, if y goes up, then this whole fraction would actually get smaller since the y is in the denominator. So if y goes up, that forces x to go down. Conversely if y now goes down, that would make this actually get larger because if the denominator of a fraction gets smaller, the whole thing gets larger. So if y goes down, x would go up. So we call these things inversely proportional because they sort of work against each other. If one goes up, the other goes down by the same percentage; and vice versa, if one goes down, the other one goes up. These are called inversely proportional and this is directly proportional.
So if two things are directly proportional, it means that they're equal, but you have to put in a multiplicative constant in front, some value; and similarly if we have two things that are inversely proportional, then, in fact, we have them equal to times some constant here. That's all proportionality means. It may seem a little bit sort of strange and obscure now, but actually this is a very real thing that happens in nature and we'll take a look at a couple of examples of both of these up next and you'll see these things actually in practice.
We'll see you there.
Equations and Inequalities
Variation
Introduction to Variation Page [2 of 2]

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