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About this Lesson
 Type: Video Tutorial
 Length: 8:45
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 95 MB
 Posted: 06/26/2009
This lesson is part of the following series:
Calculus (279 lessons, $198.00)
Calculus: Curve Sketching (20 lessons, $25.74)
Calculus: Introduction (3 lessons, $2.97)
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 UTAustin 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, padic 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
 Thinkwell
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11/13/2008
Founded in 1997, Thinkwell has succeeded in creating "nextgeneration" 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 technologybased textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.
Thinkwell lessons feature a starstudded cast of outstanding university professors: Edward Burger (PreAlgebra through...
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 04/26/2010

He's ALWAYS GREAT!!!! MUST WATCH!! THANK YOU!!!! Hopefully people can truly appreciate how rare it is to have the ability to watch one of the best professors utilize all skills of teaching to produce such power packed lessons that stick in the brain. Every detail is thought of in the video presentation. THANKS AGAIN!
Curve Sketching
Introduction
An Introduction to CurveSketching Page [1 of 1]
As we come to this last application of the derivative, I thought it would be nice, before going further, to take a look back, sit back, and get a sense of where we came from and where we are now.
Now when we first developed the notion of the derivative we saw that there are two different ways of looking at it. On the one hand, we could think about it as representing instantaneous rate of change. So, for example, we are now empowered with the ability to find the instantaneous velocity of things that are moving. However, one thing we discovered along that intellectual journey was, in fact, that we could also use the derivative to find the slopes of tangent lines. And we went off and took, looked at examples of each of these aspects of the derivative.
For example, we looked at some interesting problems involving finding velocities, and finding when things stopped, and when they got their highest when they're just thrown up in the air, and all sorts of velocity acceleration problems. We looked at the notion of looking at the derivative as a, as a slope, and, and looking at tangent lines in order to actually do calculations of computated things. Like we looked at the cubed root of seven point nine and whatnot, so we saw how to use the, the line, the tangent line approximation as a means of, of approximating numbers. And, then, more recently, we took a look at the notion of optimizing things, finding the maximum or minimum that a, that a function could take on by looking at, again, the derivative, and thinking about it as a slope.
And then there's the penultimate thing, the thing we just did, was think about the derivative. Again, as a rate and look at these more elaborate questions where, in fact, one rate is related to another rate, and that, of course, was this business of related rates that we just looked at, is the slope.
So, now I'd like to close this collection of interesting applications, and important applications of the derivative, to returning to again, the tangent line issue. We just looked at the rate issue. And now I want to return and close with the tangent line issue. And, in particular, think about pictures, specifically, graphs of functions.
So, let's think about this for a second. You know, when I was a kid in high school, somewhere along the line I learned about the parabola. Now, I don't want to insult you. Of course, you all know about the parabola too. But let me just recap and tell you about my high school days. So, we looked at the parabola Y equals X squared, and then we wanted to graph it. And here's how we graphed it. Now, maybe some of you actually graphed it the same way. We set up a little table, and here's how to do it. You wrote X in this column and Y in this column, and then you wrote some numbers. You wrote like zero, one, two. Then you made a little graph. You made a little graph like this. Do you remember this? Doesn't this sort of hearken you back to Ms. Fizby's eighth or ninth grade Algebra? And you graphed zero, zero, one, one, two, four, minus one, one, minus two, four, and then you connected those points and you got the graph of the parabola. You were well on your way to graphing. That's how we did it. That's how I did it. And, so we have the parabola.
Okay. Now, there are a couple of things here that are obvious and let me just point them out to you. First of all, I think if you think about it a little teeny bit, you can convince yourself that this very pretty symmetry that we're seeing, sort of almost a mirrored symmetry, that if you just took this side and reflected it, it would land perfectly on this side. That actually makes a lot of sense, because when you square something, the square is blind to the positiveness or the negativeness of the number, right? The square function is sort of an equal opportunity lender. It will just take any number, positive or negative, and report the same value independent of its sign. If you put an A here or minus A, it spits back the same value. So, it is a mirror image of itself. It is symmetric around this Yaxis.
So, that's actually not surprising. Okay. It's also not surprising, I think, that as you migrate off with X values that are getting bigger, and bigger, and bigger, this graph should get bigger, and bigger, and bigger, because if you take bigger numbers and square them you're going to get bigger numbers. So, to me, the fact that this goes up is not a big shock. The fact that this side is coming down this way is not a big surprise for the same reason. I'm looking at negative numbers that are actually quite large after you square them, and as you move this way, they get smaller and smaller.
Okay, well, not a big surprise. So, in fact, this method seems to do quite a nice job. Why am I even wasting your time right now, and wasting the information superhighway with telling you all this? Well, because there's one thing that I don't think is obvious and, in fact, there's one thing that has yet to be established in our own minds. And that's the following fundamental question.
Why couldn't the sine function do this? Starts at zero. We know it has to pass through those points, because we just verified that. So, why couldn't you do this? Start to go like this and then maybe wiggle a little bit. And then the same reflected thing here. We know it has to be reflected. Why couldn't this be the sine function? Notice that this has all the properties I mentioned. It's completely reflective. It increases, comes down, then goes up, passes through all my points. But somehow it starts to wiggle a little bit. Maybe it starts to wiggle way out, far away. How do I know that that's not the sine function?
Well, the answer is this table will not be able to prove whether this is the sine function or whether the green is really the sine function. Because all I'm looking at is just a discrete collection of points, just a few points. Well, there are a lot of curves that pass through those points. How do I know that this um, not the sine function, I'm sorry. How do I know that the quadratic function is really the one that goes through here?
Well, the answer is we don't know that right now. But that seems like an interesting and important question. Namely, how do we know, with certainty, what a graph of a function looks like. Now, we've seen applications, and the advantage, and the importance of knowing what a graph looks like, and in many of the applications we've already seen. But now the question comes down to, how do we actually know what a graph looks like, in some sense, with certainty. And how do we know that the parabola doesn't look like this, but instead looks like this pretty green thing, which we were told. Up to this point, it's just been blind faith.
I think this is an opportunity for you to think back and say, "Gee, I always knew this was a parabola, but no one ever explained to me why it sort of curves like that." And that subtle curvature could be something like this.
Well, in what we're about to do, we're going to take a long, careful look at how can we determine what the graph of a function looks like and how can we identify the difference between this kind of thing and the green alleged parabola. Now, we've already seen a little bit of this. In particular, we've already seen the notion of increasing and decreasing. So, we know that a function is decreasing, which means that the derivative of a tangent is negative. So, the function is falling as you read it from left to right, precisely when the derivative is negative. And we know that the function is increasing when the derivative is positive. We also know that where the derivative is zero, we have a leveling off effect, which may be a max, may be a min, may be neither. So, we can actually use that information to get at least a rough, a rough sense of what a picture looks like for a graph.
What we'll do first is take a look at that. Consider some examples. Get a sense of how to find where functions are increasing or decreasing. And then we'll take a look at the more interesting idea of curvature. And how do I know that this wavy orange curve is not a parabola, but instead, it's that beautiful, more familiar green curve, that's the parabola.
So, first take a look at maxima, minima, increasing, decreasing, and then move the more exotic, but no more difficult, issue of curvature. Stay with us.
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He's ALWAYS GREAT!!!! MUST WATCH!! THANK YOU!!!! Hopefully people can truly appreciate how rare it is to have the ability to watch one of the best professors utilize all skills of teaching to produce such power packed lessons that stick in the brain. Every detail is thought of in the video presentation. THANKS AGAIN!