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Calculus: Graphs of Polynomial Functions


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About this Lesson

  • Type: Video Tutorial
  • Length: 10:14
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 110 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Final Exam Test Prep and Review (45 lessons, $64.35)
Calculus: Curve Sketching (20 lessons, $25.74)
Calculus: Graphing Using the Derivative (4 lessons, $5.94)

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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Curve Sketching
Graphing Using the Derivative
Graphs of Polynomial Functions Page [1 of 3]
So now, let's put these ideas together and actually do a few examples where you actually see finding the critical points, increasing, decreasing regions and concavity to actually sketch very accurate pictures of more elaborate functions. So, the first one I thought we would look at is the following - f of x equals x to the one fourth minus ten x cubed plus five. And the meta question here is to sketch a very accurate drawing of this. But within that meta question, here's what we have to do.
First, we have to locate all of the critical points. Second, we have to find out, using the critical points, the regions where the function is increasing and where the function is decreasing. That, by the first derivative test, will actually enable us to figure out which of those critical points are max, which are mins and which are neither. Then, once we're armed with that then we take a look at the second derivative, which will then give us information about concavity. We can find the points of inflection, where the curvature changes, and then draw a little sign chart and see where we're concave up, where we're concave down and then we can sketch a picture. For each of these problems, we're going to do many, many steps. And in fact, I want to give you an opportunity to try these before you actually watch them with me. So to begin with, why don't you try to locate all the critical points for this function f of x? Give it a shot.
All right. Well, let's see how we made out. The first thing I need to do is take the derivative, because remember the critical points are those places where the derivative either equals zero or the derivative is undefined but the function actually exists there. So we take the derivative - we see four x cubed minus thirty x squared. And what I'd like to do is first set equal to zero. So if we set equal to zero, well, what do we have? Well, you'll notice that I can actually factor out the common factor of two x squared. Well, actually let's do that. Factoring things is always a great idea in these problems, by the way.
So here I'm left with a two x minus fifteen and I'm setting that derivative equal to zero. I want to find out where those slopes potentially are horizontal, where those tangents are zero. Okay, now if I have a product of two things that yield zero, I know that either this is zero or that's zero. Well, if this term is zero that automatically means that x has to equal zero. So there's a critical point right there. And the other possibility is that this equals zero. And if I solve this, I would see that x would have to be fifteen over two or seven point five. So, therefore, I see I have two critical points.
What about that other variety, that other flavor of critical points, namely, where the derivative is undefined? Well, the derivative actually is just some cubic polynomial; it's always defined. If you give me any value of x, I can always cube it. I can always square. I can always define. So, in fact, these are the only two critical points. So these are the critical points.
Okay, let's make a little sign chart now to see if any of these are max, mins or neither. So I make a little sign chart for the first derivative. I wonder what that would look like. Well, what I like to do is I like to actually label these points here. So I put in zero and I put in seven point five. Why don't you right now try to fill in the sign chart with the appropriate signs? And then determine where the function is increasing, where it's decreasing and then using that, tell me if either of these are max, mins or neither. Give it a shot.
Okay. Well, let's see how we made out here. What I just do is I just pick a point, one representative point in each of these regions and see what the function is doing. So here, for example, you might want to pick, say negative one, that's to the left of zero. And when you plug in negative one in here, what do we see? Well, negative one cubed is negative one. So I see minus four, minus thirty, that's very negative. So, in fact, this region here is going to be a negative region. Here the derivative is zero. Now, what happens here? Well, between zero and seven point five, I could pick something like say one. Plug in one in here and I see four minus thirty, well, that's still negative. So I have a negative region here. Here the derivative equals zero, we already saw that.
And what about here? Well, here you could pick something really large; you could pick something like say ten. Ten cubed is a one thousand, so this is four thousand minus only three thousand, so that's positive. So, in fact, this is the sign chart. Using the sign chart for the derivative, what do we see? Well, we see that the function must be falling here because the derivative is negative. It has negative slopes. So the function's falling and the function's also falling here. Here, however, the function is on the rise. So this sign chart actually provides the following information - I now see that the function is decreasing up to zero and then decreasing past zero up to seven point five but then it increases. So what happens to zero? A max or a min? Well, it's neither, because I am decreasing and then I'm decreasing more. That's not a peak and it's not a valley. What about here? Well, that plainly is a valley. We come down and then we go up. So, in fact, that's a min. So this is a min, and this is neither. Well, that tells us all about increasing, decreasing of the function and we also got to create this one minimum, no max, and zero is neither.
Okay, now let's take a look at the curvature. So, what do we do for curvature? Well, we want to look at the second derivative. And so we take a look at the second derivative and what do we get? Well, the second derivative is the derivative of the first derivative, so that's twelve x squared minus sixty x. So, the first thing I'll do is actually figure out when that second derivative equals zero. Those are finding possible points of inflection, places where the concavity might switch from concave up, concave down, concave down to concave up. So, let's set that equal to zero, just like before. You'll notice that this is very parallel to what we were doing before. So, in fact, let me right now have you try to find the points of inflection for this. Give it a shot right now.
Okay, well if you set this equal to zero, I see I can factor out a common factor of twelve x. And when I do that I see twelve x times x minus five. And if I set that equal to zero, I see either x is zero - that sounds familiar, doesn't it? We already saw that x equals zero produces a place where the derivative is zero. But now we see the second derivative is also zero there. And the other possibility is that x equals five. So these are two candidates for points of inflection. To see if they really are, what I do is I sort of parallel this theory. I make a little sign chart now for the second derivative. And I just mark down these points, so I'll mark down zero and five and I want to fill in with the sign of the second derivative. Why don't you try that right now and determine a couple of things - figure out where this curve is concave up, where the curve is concave down and see if either of these points are points of inflections. Give it a shot.
Well, let's see, if I pick a point to the left of zero, say one, and go back to the second derivative, if I plug in one for x, I see negative. So there's a negative region here. Here I see the second derivative equals zero. Between zero and five, I could pick one. X equals one. Plug that in here. If you put in x equals one; you'd still see negative region. And then what happens here after five? Well, after five what I see is if I pick something like ten, if I put in ten here, what would I see? Well, if I put in ten, this would give me six hundred. This would give me one hundred, so this would give me times twelve would be twelve hundred, so this actually becomes positive. Now let's recap and make sure this is all okay.
So, what if I plugged in a negative one? If I plugged in a negative one, actually look what happens if you plug in a negative one. This actually is a plus sixty and this is still positive. So, in fact, these should really be positive here. So you put in a negative one, negative one actually makes this positive. So, be careful. Well, I see positive, negative, positive, it's zero here. So I see both of these points are points of inflection - so these are points of inflections. Great!
And what about the concavity? Well, since this is positive in this region, I see this is concave up so I sort of put a little happy face here. Here negative region, concave down and here I see positive region, concave up. So that's all the data and using that we can actually figure out what the graph of this looks like. One thing we might want to have are the values of the functions at all these points. So you, in fact, you would know to just plug into the original function, which gives you height to determine what the value of the function is in each of these points. So, for example, f of zero - now I go back to the original function to get the height, where the dot is. And I see immediately that's five. The next point I'll need is f of seven point five. And if you plug in seven point five here and work that out, I think you'll see minus one thousand forty-nine point six eight and it goes on for a while. And then if you plug in this five, which is a point of inflection, and evaluate I think you'll see minus six hundred twenty.
Now, using all this information and these two charts see if you can sketch an accurate picture of the graph. Try it right now. Okay, well, let's see if we can do this together. We have all the information we need right over there and so let's take a look at what we have. Okay, well we know that at zero, the function is zero. Sorry, at zero we know that the function is five - so let me put in a point right here zero five. At seven point five, right around here, we know the function is minus one thousand forty nine, so it's way down. So I'm not going to be able to draw this perfectly to scale, I'm going to put the point right down here. So this is minus one thousand forty-nine and this is nearly seven point nine. And then at five, which is the other point that we're going to need, I'll put the five right here - where are we? We're at minus six hundred and something, so I'll put a point right in here.
And what do I know? I know that the function is decreasing and the function decreases all the way to here and then it increases. That's what that charts says, that first chart tells me. So I'm going to get some sort of vee effect here. That's the minimum, as we already saw.
Okay, but now what about the curvature? Well, what we saw was that it's concave up, so it's going to be sitting up like this and then concave down and then concave up. And so we put all that information together, decreasing, increasing, concave up, then concave down, then concave up. It would look something like this. The graph would look something like this. I have a pretty version of that so you can really see all of the detail. There's a really pretty version of this and you can see how the function is concave up and then it levels off and then it's concave down for a while and then it becomes concave up again and only now does it start to increase. The derivative now is positive and so you get a sketch like this. Take a look at this. Give this a try yourself and I'll be back with another example in a second. See you there.

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