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Calculus: Concavity and Inflection Points


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About this Lesson

  • Type: Video Tutorial
  • Length: 13:12
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 142 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: Concavity (2 lessons, $4.95)

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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Recent Reviews

Thank you
~ Stephen32

At first i wasnt sure where he was going with this but at the end he really ties it together so that the second time i watched it i was understanding. Thanks bunches man this was worth every penny.

Thank you
~ Stephen32

At first i wasnt sure where he was going with this but at the end he really ties it together so that the second time i watched it i was understanding. Thanks bunches man this was worth every penny.

Curve Sketching
Concavity and Inflection Points Page [1 of 3]
Well, now we've come to the moment that we've been waiting for since our initial discussion about graphing--in particular, that childhood experience we had with the parabola where we saw that it was curved up by connecting those points and getting that beautiful parabolic curve. But now, I threw a little wrench into the fire, because I said, "Well, maybe it's sort of nice and smooth for awhile, but then off, like, at a billion or a trillion or a gillion or whatever, all of a sudden it starts to weave, maybe, back and forth, and the curvature actually begins to be a little more complicated."
How do we know for sure that it's curved beautifully up like that? The answer is up to this point, we don't. But now, we're going to tackle--I think--the intriguing issue of curvature.
Let's take a look at curvature and see how curvature can come. Well, there are two types of curvatures. There's this kind of curvature that's sort of curved upward. Now, I'm talking now about curvature, not the notion of increasing, decreasing. So notice that this function is actually decreasing and then increasing, but the curvature I consider to be the same. This curvature I consider to be called curved upward. Sometimes we refer to curvature as concavity, which is a very exotic math term--concavity. And I would say this is concave up--well, curved up.
And the reason is, is because it's sort of like a cup that's sitting up. In fact, if you were our sound person, who needs a very large cup of coffee to keep him awake when he listens to my lectures, then it would be as though the cup is sitting up--just like that. So, this is concave up
The other kind of curvature we could possibly have is one that looks like this. Now, here again, notice that we have both increasing possibilities and decreasing possibilities. But it's the curvature I want you to look at--the sort of bending down, as it is. It's always sort of pointing downward, in some sense--curving downward. This is called downward curvature or concave down. And that would be if we took the cup upside down and see that curvature of this sort.
So, concave down is any part of the cup that's like this. So, understand that this part is concave down and decreasing. This part, though, is also concave down, even though the function would be increasing here. Any portion of this is called concave down. Whereas, this kind of thing, I would call concave up, even if it's this point here where I see it's actually decreasing, but it's concave up or this area where it's increasing, concave up. So, independent to the notion of increasing or decreasing, we now see the notion of curvature, whether it's saying it's curving upward, concave up, or curving downward, concave down.
So, the question that I want us to now answer, and which we're empowered to answer, is to think to ourselves, "How can we determine whether a curve is concave down or concave up?" All right, well to do this, not surprisingly, we have to take a look at the calculus. So, let's think about what's going on here. Now, this here is concave up. Okay. Now, I'm trying to figure out what that means.
Now, notice if I look at the slopes of tangent lines here, in this region, you'll see that the tangent line is negatively sloped, and then it moves from being negatively sloped to being zero sloped. It's horizontal, and in fact, in this case, you can see that's a minimum. And then if I keep going, you'll notice that it becomes positively sloped.
Okay, so that actually allows me to determine when the function is increasing and decreasing. In particular, if the slope is negative, I know it's decreasing. If the slope is positive, I know it's increasing. I want to now figure out how do I explain this kind of behavior, this curvature. And so, what I want to do is notice something interesting.
Let's now forget the actual sign of the derivative but ask ourselves as we move along this curve, what is the derivative doing? Is the derivative increasing or is the derivative decreasing? Now this is actually a new idea. Usually, we look at the derivative and say, "If it's negative, then I know the function is decreasing. If the derivative is positive, I know the function is increasing."
But now I want us to ask what is the derivative doing as we move along here? Well, let's see. Here, it's very, very negative. And what is it doing as we move closer and closer to that low point? Well, it's still negative, but notice it's less negative. This is actually a bigger slope than this. Because this is a negative number like maybe negative fifty, and here, this is maybe, like, negative ten. And then we get to zero, which is bigger than every single negative number.
And then what happens after that? Well after that, we now see positive numbers, and those positive numbers get larger and larger and larger--steeper and steeper. So, what happens to the derivative during this journey? Well, the derivative starts off very, very small--very negative--moves to zero, and then becomes quite positive. The derivative is increasing. The derivative is getting bigger and bigger. It starts off very negative, and then moves to very positive. So, the derivative is increasing in this region.
What does it mean for the derivative to be increasing? Well, for the slopes to be increasing; that means what property must the function that gives me the slopes have? Well, if the slope function is increasing, its derivative must be positive. Because derivative tells me increasing or decreasing.
And so, the function that gives me slopes of tangents must be increasing, therefore it's derivative must be positive, and let's think about that. Well, what is the function that gives me tangent slopes? Well, that is the derivative.
But now, I'm seeing that this function--forget about the fact that it's a derivative for a second--but now, think of it as a function. It's a little machine that spits out slopes. And I see that that machine is doing what as I move along here? That machine is increasing. That function is getting bigger and bigger and bigger, which means what about the derivative of that function? The derivative of that function must be positive, because the function is increasing. Remember what it means for a function to be increasing. It means that derivative is positive. So, the derivative of that function must be positive. So, the derivative of that function is the second derivative, and it must be positive.
Do you see that? The idea is that the rate of the slopes of the tangents are increasing, so the rate, which is a derivative of the tangent, which is a derivative, must be positive. Well, now we've established when we see a curve that's concave up, curves are concave up when the second derivative is positive. So, the curve is concave up--so concave up, when the second derivative is positive.
And I really want you to think about this, because I would love for you to really understand this, not memorize it, but understand it. The derivative right here is the function that gives me the slopes. And if it's concave up, I see those slopes are actually getting larger and larger and larger. So, the slope function is increasing, and for a function to be increasing, it's derivative must be positive.
So, if I take the derivative of this machine, it must be positive. Do you see it? I want you to think about it. It's subtle. But notice the derivative machine gives me slopes. Those slopes are increasing; therefore the derivative of that machine must be positive. When is something concave up? When the second derivative--the derivative of the derivative--is positive.
What about concave down? Well, you could make a really good guess now, and that guess would be perfectly correct. Let's take a look at what's happening to the derivative function. This is a machine that generates slopes. Let's take a look at what happens.
Well, here the slope is positive, then the slope gets smaller and smaller and smaller until it gets to zero. Then beyond that point, it begins to get negative and then gets more negative and more negative and more negative. What's happening to the slopes of these tangent lines? We're going from a very positive number to zero to a very negative number. The slopes are decreasing. So, what about this function? This function is decreasing. What does it mean for that function to be decreasing? Its derivative must be negative. So, in fact, the derivative of the derivative must be negative.
So, to find out when something is curved downward, you want to ask where is the second derivative less than zero. And, again, the idea is that we're seeing the slopes changing. And they're changing from very positive, to zero, to very negative. So, they're decreasing. The slopes are decreasing. That means the derivative of that function is negative, but that function is the derivative. So, the derivative of the derivative--the second derivative--must be negative. That what it means for something to be curving down or concave down.
So, let's recap. To find out where a function is concave up, we take a look at where the second derivative is positive, because then the slopes would actually be increasing. To find out where a function is curved downward like that, we ask when is the second derivative negative, because that would indicate that we go from big slopes to negative slopes--small slopes, so we're curving downward like this.
So, the second derivative gives us information about curvature. The first derivative gives us increasing, decreasing of the function, and the second derivative gives us the more delicate issue of how it's curving. And the last point I want to tell you before I close this introduction is the notion of what happens when the curvature changes? So, what happens when I go from concave down to concave up? This is concave up. The second derivative here is positive because the slopes here are increasing, right?
The slopes go from negative to positive. But there's got to be a point where this curvature stops being curved down and begins to be curved up. Do you see it? See, this is all curved up here. That's easy. This is all curved down here. This is down. This is up. This is down. This is up. This is down--say--oops! There must be a point where that curvature changes--a change in curvature. And that change in curvature is called a point of inflection or an inflection point, and that's where the curvature changes--a change in curvature.
And what do you think has to happen at a point of inflection? You can figure this out now. If in one area, the second derivative is less than zero, and in the next area, the second derivative is greater than zero, what do you think identifies the point of inflection? Well, it must be where the second derivative equals zero.
We passed from the negative second derivatives, and to get to the positive second derivatives, we have to pass through zero. So, this happens where the second derivative equals zero. So, to find out the point of inflection, we take the second derivative and set it equal to zero and find those points.
We'll take a look at a lot of examples, and also talk a little more about this. But, in the meantime, I really want you to think about this notion, this notion of how the slopes are changing. They're getting smaller here, and therefore it's curved down. Second derivative is negative. Here they're getting larger, they're increasing, and therefore, the second derivative is positive.
It's a subtle point, but curvature is really a subtle issue. Think about it. We'll take a look at some examples and practice in a second. See you there.

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