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Calculus: Parabolas


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About this Lesson

  • Type: Video Tutorial
  • Length: 21:30
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 232 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: The Basics (8 lessons, $11.88)
Calculus: Pre-Calculus Review (4 lessons, $9.90)

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 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.

About this Author

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The Basics
Precalculus Review
Parabolas Page [1 of 1]
Lines - well, we've got them down. Great. But you know, lines - beautiful, straight, right, sort of the essence of beauty and straightness, but you know lines only go so far. What about more exotic looking curves? Well, you could certainly make an exotic curve by just, you know, distorting the line. Look at that. Look how subtle that is or this - or even more dramatic. Look at that. You can make all sorts of things, all sorts of curves, very subtle and very dramatic.
Now of course, remember the line sort of this extremely simple example of a curve, you just need two pieces of information to really nail that thing down. Right? Either two points, or a slope and a point. Just two pieces of information and you can nail down the entire curve, which goes on forever. Notice the simplicity and the beauty of that when you think about more exotic curves, right. I just can't give you two pieces of information for you to completely understand the many mysteries and nuances of this very pretty curve, right. It's much deeper than that. In fact, an interesting question to ask is how would you understand this curve. It feels very smooth and very sort of nice to the touch but how would you understand it, turns out, actually very subtle. But the next most natural and the next most, in some sense basic object is a curve and of course, that's the parabola.
So we'll leave this very exotic one and see if we can think about and move our thinking and in fact, move our pictures to our parabola. You already know what a parabola is and I'm going to make one for you right now, live. This is a live parabola happening as we speak, and you know, parabolas you think, you know, are sort of easy to make, but no. But in fact, there is an absolutely beautiful, I would say, near perfect parabola, the parabola. Okay, and then what makes the parabola, so parabola. Well, let me write down the sort of basic equations for the parabola. Parabolas actually come in two flavors; come from two families.
The first family of parabolas - the f of x equals x squared. By the way, notice how I'm trying to get in the habit of using f of x equals x squared instead of the more traditional, perhaps, y equals x squared. I want to get us thinking about these things as functions. I want to use this notation where you plug in the x and you spit out the y value. So this one - let me show you what this looks like, let me draw my axes here. This is the very standard, ever popular, much beloved parabola that looks like this. It's, it's symmetric along the y axis, passes right through zero, zero, right at that low point there and has this beautiful symmetry on both sides of the y axis. That is the standard vanilla f of x equals x squared. Now its, its nemesis - and in fact, in fact the way I think about this by the way, let me stick a little thing here. I think about this, I classify this kind of a parabola as a happy face parabola, and there you go. Look at that. Happy, happy parabola.
Now it's nemesis is f of x equals minus x squared. That is the exact same picture but reflected down. So it's a parabola that is very symmetric to the y-axis but now is downward, still passes the origin. You think of this as a sad faced parabola. Happy, sad. In fact, to really emphasize that, I should do it in red -- very sad, very sad. Okay, now - well, now there are variations of course on this theme. Let me actually put down some possibilities here. One way to modify these kind of vanilla parabolas, very, very readily, in fact, would be to - my parabola is bending a little out shape - sorry about that, folks. Don't you hate when that happens? How many times have you had like people over and all of sudden realized, your parabola's out of whack. You know, very embarrassing especially on those sort of intimate moments, but I digress. There we go, we're back.
One thing you could do is just put a coefficient, put a number in there and multiply through. For example, we could put like a three and then an x squared. Well, since that number is positive, it's going to be one of the happy face parabolas. Let's move this off to the side for a second. Well, the question is what is that three going to do? Well, that three is either going to squash the parabola in more or it's going to sort of flatten it out. It's going to make it sort of come out and be a little flatter. What do you think the three's going to do? Is the three going to squash it in or is the three going to flatten it out slightly? I'll give you a second just to think about that. We're not going to stop, but just see what your guess is. Your guess, and please guess - and either it's going to be shrink in or flatten out. And if you don't know, just guess.
Well, it turns out that multiplying by three, if you think about that for a second, what does it do to the little function machine. You put an x and instead of getting a value that used to be here, it's now going to be three times as big. So instead of being here, it's going to be three times as big. So at this point, that has to be way up there, so I've got to actually shrink this down. You've got to be pretty strong by the way, to bend these little parabolas, and it looks like that. So if you put in - moral--if you put in a positive number that's greater than one, the bigger than one it is, than the more tighter it's going to be. If I put a number that's less than one but still positive like a half, that's actually going to bow it out. And if you multiply it by like a fourth, it bows it out even more and the more that you - the smaller the number you put in, the more flat it gets, the more flat it gets. And similarly, if I put a negative three in there, then we see the same kind of phenomena but now with the sad face parabola. It would crunch in for the three; you can crunch this, crunch, crunch, crunch. But if you multiply it by let's say negative of four, you would come out and be like an upside bowl. So this is how the parabola works out, of course, then there are other variations on the theme.
You could add a number here. Like I could add the number plus one, what would that do? Well, the number plus one - here's my generic parabola. I multiply it by three that squeezes it in a little bit, right, squeezes it in. And then what does the plus one do? Well, it just takes every y value and now I add one to it. So this moves the whole thing up one unit, except this is a one right here. I go from the three x squared to the three x squared plus one. I just shift it up one. Similarly if I looked at three x squared minus four, guess what I do. I would start with this one and then I would drag down, one, two, three, four. That shifts it up and down.
Now of course, you could do much more exotic things. For example, you could do three x squared minus two x plus one. Well, now that minus two x, it's not so apparent what effect that's going to have on the picture. But I'll just tell you that it will actually be involved in the sort of the bowing of this, and not only will it be involved in the sort of up and downness, but also it's going to potentially move it in this kind of way, side to side, and so that could actually be up here, down here and so forth. And we'll see more of this, by the way, when we talk about actually graphing these parabolas later in the course, but for now any parabola, it looks like this. This is still positive which means this is still going to be a happy face parabola. Moral of the story.
Now let's try to graph some functions armed with this. And the first thing I wanted to graph here, in fact, I did this for you in advance. You know sort of like those cooking shows. You ever see those cooking shows, you know, you want to make the soufflé. Well, it takes them three seconds. Oh, this is it. Zip, dip, you put in flour and whup, it's out. You know, right. Well, I wrote this out in advance.
Now, the question is can we graph this function and I hope that your reaction would be what my reaction would be, is first of all, ouch. And second of all, why in the world is Professor Burger asking me to graph this function - there's a division here, there's no quadratic stuff. There's a cube thing here. Has he lost his marbles? Well, potentially, you're saying yes. But the thing to remember is if something looks hard, if something looks challenging, don't panic. Look at it and see what you can do.
Now for example in this problem, I'm very threatened. I'm threatened by that denominator and if you remember, maybe you might remember from graphing functions in high school, when the denominator is -there might even be things like asymptotes and all sorts of things, oh goodness, what does all that mean? Well, I don't know, but it's scary. So I'm intimidated, however, if I look at the top, I do notice something and I bet you notice it too. And that is that there is an x in every single thing on the top, so I could actually, if I wanted to, if I was sort of impish and you know what I'm the king of imp. I could actually factor out an x.
Let's do that. If I factor out an x, what am I left with? I'm left with x squared plus one times x all divided by x. Now that actually is pretty interesting. It's interesting for a couple of reasons. First of all, notice the appearance of a x squared. Well, that sounds good because we're talking about you know, parabolas. It would be nice to have a quadratic in there somewhere. It's also interesting because notice that these x's seem to be in position to be canceled. In fact, you know what, I am going to cancel them for you right now, live, using my red cancellation pen. Isn't this a nice pen but I could use it as a pointer. Look at this - this x together with this x can be cancelled. Very effective pointer, but I'm going to use it like this.
Now, now remember one thing though - what can't we look at? What's the one thing we can't look at? What over what? We talked about this awhile back if you were with us, zero over zero - can't divide by zero. So I can cancel these things only on one condition, you have to make a promise with me. You have to make a covenant with me right now. You have to promise me that if I cancel these away, what can x never equal? Can't equal zero. So I'm going to cancel these things away right now, live, right now, live on the web. But there's a promise with that and the promise is that x is not going to equal zero. You promise? Of course you do, because if x was zero then I couldn't cancel zero over zero. So let's suppose that x is not zero. Okay, I mean, if that's the way you have to play the game, then fine. Okay. Well, what am I left with, then I see that f of x equals x squared plus one, but don't forget, we have to remember that little rule. Oh, by the way, look at this, look at this. You can pull the pen out this way and you can write with there. That's actually a pen down there, but it's very fine. You would never see if I wrote there. I can pull it out this way. There's five different pens in this thing. It's ridiculous. I'm going to use the fat one so you can see it.
So f of x equals x squared plus one as long as x is not zero. Well, that's a parabola. We can graph that. So that's a good old-fashioned happy face parabola. You already saw this in fact; this can be shifted up one. There is one little thing that we have to remember. You have to remember the promise, the promise that x doesn't equal zero so we can't have anything when x equals zero.
So in fact the graph of this parabola, I think you might find interesting. Let me do it in blue, just like this. This parabola tends to be - and it's going to look like this. A good old-fashioned generic parabola with what condition? The condition that we don't have anything at zero. And right here, there's a hole. There's a hole at x equals zero because we made a promise that x cannot be zero. Look at this. This exotic looking function that had a denominator and had a cubics thing here, all these complicated thing. What is the graph of that look like? It's actually just a parabola with a hole. Notice that for me to get that hole in a parabola, I had to divide and do all this stuff. I think this is really sort of neat. That the graph of this is actually a very happy looking thing but with one interesting condition and that creates this nice smooth thing with the one point removed, that's just one point missing. Isn't that neat? So in fact, these are sort of punctured parabolas in a way, punctured parabola. So just because a thing looks really exotic, we shouldn't panic. We should look at it, look for a pattern. Always in life, search for a pattern if you don't know how to proceed. In this case, I just know there's a common factor of x, factored it out. Now here, we're had to be a little careful, made a little promise and moved on and got a very interesting graph of this function. Neat, let's show one more.
This one, again done in advance, is to graph this very complicated-looking thing. Look at that. That is f of x equals - again I'm f of x instead of y - equals three x squared minus two x minus one all divided by x minus one. Whew! Well, this one is really scary. Of course, here there is, there is that quadratic looking thing here. So maybe it's tough, but then what do you do with all this stuff? Well, what do you do?
Well, I would look for a pattern and see if there's anything I can do. Well, I see a quadratic here. Maybe I could factor that. Let's see if that actually can be factored. A little practice in factoring here. How would that go? Well, I don't know how you factor but I draw these little parentheses there. If nothing else, it feels like I made progress, little parentheses. Okay, now I need to break up a three x squared, so my guess is going to be to three x and x, might not work but I don't know, I'm going to try. This is so important, by the way. That you know, you know what - in fact, let me just say one thing here. Do you know why sometimes people are frozen when they study mathematics, like a deer in headlights? They go - ooh. It's because they can't see how the whole thing is going to end and they don't want to just try something unless they're absolutely certain of the path they're going to take. You know what this is like? It's like the soccer player that receives the ball and then instead of kicking the ball, holds the ball, stops right there and refuses to kick the ball until that player is absolutely sure of how the ball is going to find it's way inevitably into the goal. Well, you can't do that. What would happen? Some other - some team's person from the other side of the team would grab the ball and he'd kick it, dada, dada, da and the person would still be standing there paralyzed.
What do you have to do? What you have to do in mathematics as what you have to do in life, is you just got to kick the ball a little bit. Move it a foot and then see what happens and then assess where you are and then move forward and that is really, probably the biggest secret in mathematics. Don't be afraid to try something even if you don't know where it's going to end up or where it's going to take you. You might have to actually back up. You might have to - the ball might kicked behind you and you have to get it and go forward. Don't be afraid by that, who cares. The thing to do is to move forward and try something.
So here, I'm trying to factor and I hope that you'll do the same kind of thing. Anyway, so I'm trying three x and x. Is it going to work? I have no idea. Am I afraid? No. Okay. This minus one tells me - this minus sign tells me that I'm going to have to have two signs here, so either plus minus or minus plus, I don't know which one. And the product has to be one, so maybe one and one if we're lucky. I'm going to try that. And now I've got to put in a plus or minus so that my net result, my foil to the inside and outside is negative two. Well, it seems like if I put a negative sign here, that would give me a negative three x and then a plus sign here, that would give me then, plus x. So minus three x and x gives me a minus two x. Hey, look. It actually factored. By not panicking, no problem.
So it turns out the top is actually a very friendly function. It can be factored in this way and you'll notice that x is a common factor on the top. This whole thing is a factor on the top is also a factor on the bottom and so what can I do. I can cancel and again, I use my cancel pen. But there's a promise you have to make me. What's the promise in this case? Is it that x doesn't equal zero? No, it's not that x doesn't equal zero. It's that this thing is not zero. It's that x minus one is not zero, x minus one is not zero, which is a fancy way of saying that x does not equal one. So this is true, I'm allowed to cancel as long as you promise that x will never be one. In that case though, what we get is that the function is just what? Well, it's - all that's left three x plus one, but remember the promise. Remember your promise; x is not equal to one.
So what does this parabola looking thing actually become? It actually looks like a straight line, but it's a straight line with one minor deficiency - it's missing the value in x equals one. So the graph of this is sort of fun. Let me sketch it for you. Look, it's in a y intercept form. Right, it's in slope intercept form. By the way, if someone hands you something in slope intercept form, grab it. It's great. It's just that you don't want to spit answers out in slope intercept form. So here, what do I see. Well, the y intercept is one, so I go up one. So that's the y intercept and the slope is three. So that's actually three over one, so that means I can go one unit over in the x direction and then three units up in the y direction. So one, two, three. It's a very steep looking thing. Okay, but there's a promise you have to make me. The promise is that x cannot equal one, so in fact, here is a hole. Okay, so, so this actually, this interesting quotient is actually just a line with a hole, punctured line. Interesting, neat.
Okay, now the last thing I wanted to tell you about before we break is just how to find distances, how do you find distances between two points. Suppose I give you two points. Let's say this point is x one and this y one and this is x two and this y two, then how would I find the distance? Well, that would be the straight-line distance between those two points, between those two points. It looks just like this. How would you find that distance? Well, let me pause here for a second and I'm going to let you think about this, then we'll come back and see if we can figure out how to do it.
Okay. Well, maybe you remembered the formula for distance, sort of a long complicated formula, but let me actually tell you that I don't want you to memorize that formula. You know, you don't want to memorize things in life. You want to understand them and in fact, I'll tell you something right now. I do not have memorized all the math stuff that a lot of students actually have memorized. Cause you know what, my head is only this big, you know and if I start memorizing all these little puny stupid little things, you know what's going to happen? Something that really is important is going to slip out, like my telephone number or something, right. So you can't do that. You can't just fill your brain with all these little stupid things; it's just not worth it. What you do is understand the things and empower yourself to actually be able to figure them out.
Let me show you how I would figure out the distance formula for the distance between these two points. So remember I have these two points. This point, by the way, is the point x one comma y one. That's an x there and this point here is x two comma y two. Look how fat that font is, you know, this is a purple pen but look how fat that font is. Okay, well, if I want to find this distance, I think of one thing and I think to myself, "That is a hard distance to find." You know what, if the distance was like across, if the distance was like, you know, up that would be easy, just subtract the x's and subtract the y's, but this is a skewed thing. Even though it's a straight line, it's sort of a subtle straight line, isn't it? What I would just do then is just do the easier problems, say we'll forget about that one. That's too hard. Instead, let me just look at this and this.
Well, this distance is actually really easy; it's just the differences in the x's. So in fact this distance is pretty straight forward, that's just the change here of the x's, so it would be x two minus x one. That's that length; that was real easy because that line is just parallel to the floor. Now this length is actually just as easy. It's parallel to the wall so to find its length; all I have to do is subtract the y values. So y two minus y one. Okay, well, that doesn't help me really over here until I remember one of the most important results in the world, beyond mathematics - the Pythagorean theorem. Because this, my friends, is a right triangle. So what do I know? This distance squared is going to equal this squared plus that squared, so in fact I can actually solve -I can write that d squared equals - well this length squared. Now I got to square the whole quantity now, so x two minus x one all squared plus that length squared which is the quantity y two minus y one squared. And now I've got to take square roots of both sides. Well, remember you actually have to take plus or minus the square root, but this is a length and happily, when you look at a ruler - have you looked at a ruler. It starts at zero. This doesn't start negative seven. Length always comes in positive numbers, so in fact, I've got take the positive square root and I would see that distance equals the square root of x two minus x one squared plus y two minus y one all squared. It's the square root of that whole big thing and that might have been what you actually guessed but see, I wouldn't even guess it. I always have to reconstruct it; I reconstruct it from the Pythagorean theorem.
So there is the actual formula, but please don't remember it; instead remember this picture, this is much more important than this little red box. So I guess really what I should do to is be completely honest. I should actually make this thing be a box, right. This is the important thing. Making a highlight in the book. Now isn't' that stupid. Can you imagine, by the way, if you were writing a book, right and then you saw someone using your book and saw a couple of words highlighted. You're like, "Hey, buddy, I mean like I spent a lot of time writing those other words; now you're only going to highlight those. I mean that's crazy." So to be fair, I'm highlighting almost everything here. Here's the formula, forget it. Here's the idea, make it your own. Okay.
There's a great proof of the Pythagorean theorem and that actually leads us to finding - let me remind you why we brought that up - how to find distances, shortest distance of straight lines. By the way, are shortest distances always a straight line? Well, maybe yes, maybe no, not going to tell you the answer but in one of our math breaks, which are an opportunity for you to sort of stop calculus for a second and just take a break and in some sense surf math. On one of the math breaks, I'm actually going to talk about that and show you a really, really fun thing, which in fact you might want to just stop right now and take a sneak peek at and take a look at - where you're actually going to see that sometimes maybe the answer isn't as obvious as you think. So we'll have fun with that but before we do that, why don't try some of these sort of fun questions and see if you can get some of them. Okay, now after this by the way, before we leave - before you close, don't go yet.
We finally have, I think a lot of stuff built up. Remember original problem was to find the instantaneous velocity. The problem with that is when you plugged in instantaneous, you got zero over zero. Yuck. Okay, so what did we do? We said okay, we want to sort of inch up to that. How did we do that? First, we built a little bit of language. We talked about functions, f of x and so forth. Then we talked about graphs, special functions like lines and parabolas and what not. Did all that stuff that was all background, folks. Now I'm ready to move forward back on the original problem and the question is how can we figure out instantaneous velocity without plugging in zero over zero, and the idea is to inch up to answer. Just head to the answer and never actually get there. That is the new topic that's going to be up on the griddle next and it's the first real new idea that calculus empowers us with. Okay, try some of these problems and I'll see you in a bit. Bye.

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