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College Algebra: Linear Function Models

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About this Lesson

  • Type: Video Tutorial
  • Length: 6:17
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 67 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Relations and Functions (57 lessons, $74.25)
College Algebra: Applications of Linear Concepts (4 lessons, $4.95)

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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You know, some of the real power of mathematics is that it allows us to analytically analyze real-world events and real-world situations. In fact, I want to share one with you now, so you can see the power of just using the straight-line things that we talked about, to at least understand and to appreciate better a situation that's actually going on. So, I want to now take a look at the situation with respect to the AIDS epidemic, which is really quite astounding.
If you look here at this graph, you can see that the number - and these are in thousands of recorded AIDS cases from 1986 up through 1994. This is from US News & World Reports from a few years ago. In 1986, there were a reported number of cases of 11,900, and then all the way to 1994, you can see, eight years later, there was a staggering 62,500. That's way up here. Now, if you look at this incredible trend, you might wonder how many AIDS cases should we expect in the future, and how should that dictate policy decisions, and so forth? Mathematics can actually be a powerful tool used to understand that.
Let's just do something very, very simple here to try to capture this. One thing that seems evident by this graph, if you look at the data points, is that if I just put a line down, it seems to approximate, reasonably accurately, the trend in the number of these cases. Just for a very simple point of view, let's find the equation of the line that passes through that first point, that first data point we had, and that last data point. I'm not saying that's the best fit. Maybe, in fact, the best fit of the line might be something like that. Just for simplicity, let's find the equation of the line that passes through those two points and then use that line to try to predict how many cases we would expect to see, for example, in the year 2000.
Well, to do this problem, what we want to do is find the equation of the line. So, what do we know? Well, we have two points. We have the point - these are quite large numbers. Let's call this, by the way, zero - zero, meaning the first year of this experiment. This is year one, year two, year three, year four, year five, year six, year seven, year eight - this is eight years later, eight years after the first time we measured this. So, in fact, I would graph that first point as 0, because it's the start of this counting and the number of AIDS cases back in 1986, where they're 11,900.
Then, the second data point we have is from 1994, which was eight years later, so I have 8 in the x direction, and I have 62,500. We had that many cases then. The question that we now want to find is what is the equation of the line that contains those two points? So, I'm going to use the slope intercept form. You'll notice that, actually, I first need the slopes. Let me compute the slope. The slope is going to be the rise over run. So, I take the change in the y values, divided by the change in the x values. That would be 62,500 - 11,900, all divided by - that's the change in the y, divided by the change in x, which is actually pretty easy. It's 8 - 0, which is just 8. So, what does that equal? Well, that equals - if you subtract these two things out, I think we see 50,600, all divided by 8. In fact, if you divide that out, it's just 6,325. That's the slope. So, the slope = 6,325.
Now, what's the y intercept? Well, actually, that's where it crosses the y axis, and we're given that. That's this point right here. So, in fact, we see that y equals the slope, which is 6325x, plus the y intercept, which is 11,900. So, I see that as the equation for the line that passes through those two points. That line, we see, reasonably accurately - maybe not exactly emulates the data that we have here. So, given that, let's see if we can guess, through this line, how many AIDS cases there were in 2000.
What I would do is plug in 2000 here for x. Actually, plugging in 2000 for x wouldn't be quite right, because remember, we're counting our values as how many years they are from 1986. We have - '94, for example, was 8. So, 1995 would be 9, 1996 would be 10, 1997 would be 11, 1998 would be 12, and so forth. So, what would we have here? How many years away is 2000 from '86? It would be 14 years. So, in fact, I want to plug in 14 for x. If I plug in 14 for x, I would see that the y value would be 6,325 x 14 + 11,900. What is that number? That would be 6325 x 14, and then I add to that, 11,900. So, the estimate that we're coming up with is 100,400 cases, and that's just a staggering amount.
So, in fact, mathematics allows us to model, even in a very simple way, phenomena that are going on around us, and it actually gives us the power to make predictions and to make estimates as so how many of something is going on or how much is happening. This, I think, in turn, will maybe help dictate public policy and help to actually one day find a cure for this or, in other applications, to resolve those issues.
So, this is a simple application, where a straight line allows us to make a prediction, with reasonable accuracy.
Relations and Functions
Linear Functions- Applications
Constructing Linear Function Models of a Set of Data Page [1 of 2]

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