[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.37,0:00:04.19,Default,,0000,0000,0000,,What I have here in yellow is the graph of y=f（x）.
Dialogue: 0,0:00:04.19,0:00:07.97,Default,,0000,0000,0000,,That here in this move color I’ve graphed y’s equal
Dialogue: 0,0:00:07.97,0:00:11.51,Default,,0000,0000,0000,,to the derivative of f, is f′(x).And then here
Dialogue: 0,0:00:11.51,0:00:15.84,Default,,0000,0000,0000,,in blue I graphed y is equal to the second derivative of our function.
Dialogue: 0,0:00:15.84,0:00:18.51,Default,,0000,0000,0000,,So this is the derivative of this, of the first
Dialogue: 0,0:00:18.51,0:00:23.20,Default,,0000,0000,0000,,derivative right over there. And we’ve already seen examples of how
Dialogue: 0,0:00:23.20,0:00:27.37,Default,,0000,0000,0000,,can we identify minimum and maximum points. Obviously, if we have a graph in front of us,
Dialogue: 0,0:00:27.37,0:00:30.99,Default,,0000,0000,0000,,it’s not hard for human brain to identify this as a local
Dialogue: 0,0:00:30.99,0:00:34.70,Default,,0000,0000,0000,,maximum point. The function might take on higher values later on. And
Dialogue: 0,0:00:34.70,0:00:38.59,Default,,0000,0000,0000,,to identify this as a local minimum point. The function might take on
Dialogue: 0,0:00:38.59,0:00:43.01,Default,,0000,0000,0000,,lower values later on. But we saw, even if we don’t have a graph in front of
Dialogue: 0,0:00:43.01,0:00:46.00,Default,,0000,0000,0000,,us, if we are able to take the derivative of the function, we might…
Dialogue: 0,0:00:46.00,0:00:49.70,Default,,0000,0000,0000,,or if we are not able to take the derivative of the function. We might be able to identify
Dialogue: 0,0:00:49.70,0:00:54.11,Default,,0000,0000,0000,,these points as maximum or minimum. The way that we did it. Ok… what are the critical
Dialogue: 0,0:00:54.11,0:00:57.51,Default,,0000,0000,0000,,points for this function. Well, critical points over the function where the function’s
Dialogue: 0,0:00:57.51,0:01:01.70,Default,,0000,0000,0000,,derivative is either undefined or zero. This the function’s derivative.
Dialogue: 0,0:01:01.70,0:01:06.25,Default,,0000,0000,0000,,It’s zero here and here. So we would call those critical points.
Dialogue: 0,0:01:06.25,0:01:09.50,Default,,0000,0000,0000,,I don’t see any undefined. Any point was the derivative’s undefined
Dialogue: 0,0:01:09.50,0:01:13.38,Default,,0000,0000,0000,,just yet. So we would call here and
Dialogue: 0,0:01:13.38,0:01:17.00,Default,,0000,0000,0000,,here, critical points. So these are candidate
Dialogue: 0,0:01:17.00,0:01:20.92,Default,,0000,0000,0000,,minimum…these are candidate points which are function might take on a minimum or
Dialogue: 0,0:01:20.92,0:01:24.77,Default,,0000,0000,0000,,maximum value. And the way that we figured out whether it was a minimum or maximum
Dialogue: 0,0:01:24.77,0:01:29.19,Default,,0000,0000,0000,,value is to look at the behavior of the derivative around that point
Dialogue: 0,0:01:29.19,0:01:32.44,Default,,0000,0000,0000,,and over here we saw the derivative is de...or the
Dialogue: 0,0:01:32.44,0:01:36.24,Default,,0000,0000,0000,,derivative is positive.The derivative is positive
Dialogue: 0,0:01:36.24,0:01:38.93,Default,,0000,0000,0000,,as we approach that point
Dialogue: 0,0:01:38.93,0:01:44.10,Default,,0000,0000,0000,,and then it becomes negative. It goes from being positive
Dialogue: 0,0:01:44.10,0:01:47.78,Default,,0000,0000,0000,,to negative as we cross that point which means that the function]
Dialogue: 0,0:01:47.78,0:01:52.17,Default,,0000,0000,0000,,was increasing. If the derivative is positive that means the function was increasing
Dialogue: 0,0:01:52.17,0:01:56.11,Default,,0000,0000,0000,,as we approach that point and then decreasing as we leave that point.
Dialogue: 0,0:01:56.11,0:01:59.34,Default,,0000,0000,0000,,Which is a pretty good way to think about this… Being a maximum point,
Dialogue: 0,0:01:59.34,0:02:03.50,Default,,0000,0000,0000,,for increasing as we approach and decreasing as we leave it. Then this is definitely going
Dialogue: 0,0:02:03.50,0:02:07.24,Default,,0000,0000,0000,,to be a maximum point. Similarly,
Dialogue: 0,0:02:07.24,0:02:11.77,Default,,0000,0000,0000,,right over here, we see that the function is negative or the derivative
Dialogue: 0,0:02:11.77,0:02:15.34,Default,,0000,0000,0000,,is negative as we approach the point which means that the
Dialogue: 0,0:02:15.34,0:02:18.44,Default,,0000,0000,0000,,function is decreasing. And we see the derivative is
Dialogue: 0,0:02:18.44,0:02:23.40,Default,,0000,0000,0000,,positive as we exit that point. We go for having a negative derivative to a positive
Dialogue: 0,0:02:23.40,0:02:26.71,Default,,0000,0000,0000,,derivative which means the function goes from decreasing to
Dialogue: 0,0:02:26.71,0:02:30.50,Default,,0000,0000,0000,,increasing right around that point, which is a pretty good indication.
Dialogue: 0,0:02:30.50,0:02:34.44,Default,,0000,0000,0000,,Or that is an indication, that this critical point is a point at which the
Dialogue: 0,0:02:34.44,0:02:38.17,Default,,0000,0000,0000,,function takes on a minimum…a minimum value.
Dialogue: 0,0:02:38.17,0:02:41.71,Default,,0000,0000,0000,,What I want do now is to extend things by using
Dialogue: 0,0:02:41.71,0:02:46.65,Default,,0000,0000,0000,,the ideal of concavity… con-ca[ei]-vity.
Dialogue: 0,0:02:46.65,0:02:49.77,Default,,0000,0000,0000,,And I know I’m mispronouncing it, maybe it’s conca[æ]vity,
Dialogue: 0,0:02:49.77,0:02:53.44,Default,,0000,0000,0000,,but new thinking about concavity. Start to look at the second
Dialogue: 0,0:02:53.44,0:02:57.41,Default,,0000,0000,0000,,derivative, it rather than kind of seeing just as transition. To think about
Dialogue: 0,0:02:57.41,0:03:01.40,Default,,0000,0000,0000,,whether this is a minimum or maximum point. So
Dialogue: 0,0:03:01.40,0:03:05.32,Default,,0000,0000,0000,,let’s think about what’s happening in this first region. This kind of …this part of
Dialogue: 0,0:03:05.32,0:03:09.11,Default,,0000,0000,0000,,the curve up here where is it looks like an arc where it’s
Dialogue: 0,0:03:09.11,0:03:12.77,Default,,0000,0000,0000,,opening downward. Where it looks kinda like an “A” without the crossbeam or upside
Dialogue: 0,0:03:12.77,0:03:16.90,Default,,0000,0000,0000,,down “U” and then we’ll think about what’s happening in this kind of upward
Dialogue: 0,0:03:16.90,0:03:20.84,Default,,0000,0000,0000,,opening “U”, part of the curve. So over this first
Dialogue: 0,0:03:20.84,0:03:24.90,Default,,0000,0000,0000,,interval right over here, if we start we get this slope is very…is
Dialogue: 0,0:03:24.90,0:03:28.40,Default,,0000,0000,0000,,very ( actually I’ll do it in the same color, exactly the same color that
Dialogue: 0,0:03:28.40,0:03:31.50,Default,,0000,0000,0000,,I used for the actual derivative) the slope is very positive
Dialogue: 0,0:03:31.50,0:03:35.71,Default,,0000,0000,0000,,..slope is very positive. Then it becomes less positive...becomes
Dialogue: 0,0:03:35.71,0:03:39.65,Default,,0000,0000,0000,,less positive…then it becomes even less positive…becomes even less
Dialogue: 0,0:03:39.65,0:03:43.50,Default,,0000,0000,0000,,positive…and eventually gets to zero…eventually gets to zero. Then it keeps
Dialogue: 0,0:03:43.50,0:03:47.32,Default,,0000,0000,0000,,decreasing. Now becomes slightly negative…slightly negative. Then
Dialogue: 0,0:03:47.32,0:03:51.24,Default,,0000,0000,0000,,it becomes even more negative…becomes even more negative…and
Dialogue: 0,0:03:51.24,0:03:55.59,Default,,0000,0000,0000,,then it stops decreasing right around. It looks like it stops decreasing right
Dialogue: 0,0:03:55.59,0:03:59.81,Default,,0000,0000,0000,,around there. So the slope stops decreasing right around there. You see that in the red ,
Dialogue: 0,0:03:59.81,0:04:03.15,Default,,0000,0000,0000,,the slope is decreasing…decreasing…decreasing……until that point and
Dialogue: 0,0:04:03.15,0:04:06.40,Default,,0000,0000,0000,,then it starts to increase. So this entire section,
Dialogue: 0,0:04:06.40,0:04:09.00,Default,,0000,0000,0000,,this entire section right over here…
Dialogue: 0,0:04:09.00,0:04:14.64,Default,,0000,0000,0000,,the slope is decreasing. “Slope…
Dialogue: 0,0:04:14.64,0:04:19.06,Default,,0000,0000,0000,,slope is decreasing” and
Dialogue: 0,0:04:19.06,0:04:21.85,Default,,0000,0000,0000,,you see it right over here when we take the derivative, the deri…ative
Dialogue: 0,0:04:21.85,0:04:27.15,Default,,0000,0000,0000,,right over here… the entire, over this entire interval is decreasing.
Dialogue: 0,0:04:27.15,0:04:30.73,Default,,0000,0000,0000,,And we also see that when we take the second derivative. If the derivative is
Dialogue: 0,0:04:30.73,0:04:34.57,Default,,0000,0000,0000,,decreasing that means that the second, the derivative of the derivative is
Dialogue: 0,0:04:34.57,0:04:38.17,Default,,0000,0000,0000,,negative and we see that is indeed the case
Dialogue: 0,0:04:38.17,0:04:42.48,Default,,0000,0000,0000,,over this entire interval. The second derivative, the second
Dialogue: 0,0:04:42.48,0:04:46.31,Default,,0000,0000,0000,,derivative is indeed negative. Now what
Dialogue: 0,0:04:46.31,0:04:50.11,Default,,0000,0000,0000,,happens as we start to transition to this upward opening ”U” part of the curve.
Dialogue: 0,0:04:50.11,0:04:54.31,Default,,0000,0000,0000,,Well here the derivative is reasonably negative,
Dialogue: 0,0:04:54.31,0:04:58.05,Default,,0000,0000,0000,,it’s reasonably negative right there. But then it starts gets…it’s
Dialogue: 0,0:04:58.05,0:05:00.90,Default,,0000,0000,0000,,still negative but it becomes less negative and less negative
Dialogue: 0,0:05:00.90,0:05:06.31,Default,,0000,0000,0000,,…then it becomes zero,
Dialogue: 0,0:05:06.31,0:05:09.47,Default,,0000,0000,0000,,it becomes zero right over here. And then it becomes more and
Dialogue: 0,0:05:09.47,0:05:13.90,Default,,0000,0000,0000,,more and more positive, and you see that right over here. So over this
Dialogue: 0,0:05:13.90,0:05:18.17,Default,,0000,0000,0000,,entire interval, the slope or the derivative is increasing.
Dialogue: 0,0:05:18.17,0:05:21.90,Default,,0000,0000,0000,,So the slope...slope is
Dialogue: 0,0:05:21.90,0:05:25.90,Default,,0000,0000,0000,,is increasing…the slope is increasing.And you
Dialogue: 0,0:05:25.90,0:05:28.84,Default,,0000,0000,0000,,see this over here, over there the slope is zero. The slope of the derivative is
Dialogue: 0,0:05:28.84,0:05:32.57,Default,,0000,0000,0000,,zero, the slope of the derivative self isn’t changing right this moment and then
Dialogue: 0,0:05:32.57,0:05:37.37,Default,,0000,0000,0000,,…and then you see that the slope is increasing.
Dialogue: 0,0:05:37.37,0:05:40.73,Default,,0000,0000,0000,,And once again we can visualize that on the second derivative, the derivative of the derivative.
Dialogue: 0,0:05:40.73,0:05:44.72,Default,,0000,0000,0000,,If the derivative is increasing that means the derivative of that must be positive.
Dialogue: 0,0:05:44.72,0:05:48.00,Default,,0000,0000,0000,,And it is indeed the case that the derivative is
Dialogue: 0,0:05:48.00,0:05:51.48,Default,,0000,0000,0000,,positive. And we have a word for this downward
Dialogue: 0,0:05:51.48,0:05:56.40,Default,,0000,0000,0000,,opening “U” and this upward opening “U”. we call this
Dialogue: 0,0:05:56.40,0:05:58.31,Default,,0000,0000,0000,,“ concave downwards”
Dialogue: 0,0:05:58.31,0:06:03.73,Default,,0000,0000,0000,,(let me make this clear)…
Dialogue: 0,0:06:03.73,0:06:07.100,Default,,0000,0000,0000,,concave downwards.
Dialogue: 0,0:06:07.100,0:06:11.71,Default,,0000,0000,0000,,And we call this “ concave upwards”… concave
Dialogue: 0,0:06:11.71,0:06:15.31,Default,,0000,0000,0000,,upwards. So let’s review how we can identify concave
Dialogue: 0,0:06:15.31,0:06:19.90,Default,,0000,0000,0000,,downwards intervals and upwards intervals. So we are talking about concave
Dialogue: 0,0:06:19.90,0:06:24.77,Default,,0000,0000,0000,,downwards…”concave downwards”.
Dialogue: 0,0:06:24.77,0:06:27.85,Default,,0000,0000,0000,,We see several things,
Dialogue: 0,0:06:27.85,0:06:32.77,Default,,0000,0000,0000,,we see that the slope is decreasing, the slope is
Dialogue: 0,0:06:32.77,0:06:37.17,Default,,0000,0000,0000,,is decreasing.“The slope
Dialogue: 0,0:06:37.17,0:06:40.100,Default,,0000,0000,0000,,is decreasing” which is another way of saying,
Dialogue: 0,0:06:40.100,0:06:42.65,Default,,0000,0000,0000,,which is another way of saying that f’(x)
Dialogue: 0,0:06:42.65,0:06:50.90,Default,,0000,0000,0000,,is decreasing.
Dialogue: 0,0:06:50.90,0:06:54.25,Default,,0000,0000,0000,,decreasing. Which is another way of saying that the
Dialogue: 0,0:06:54.25,0:06:57.57,Default,,0000,0000,0000,,second derivative must be negative. If the first derivative is decreasing, the second
Dialogue: 0,0:06:57.57,0:07:01.25,Default,,0000,0000,0000,,the second derivative must be negative. Which is another way
Dialogue: 0,0:07:01.25,0:07:04.90,Default,,0000,0000,0000,,of saying that the second derivative of that interval must
Dialogue: 0,0:07:04.90,0:07:09.31,Default,,0000,0000,0000,,be… must be negative. So if you have
Dialogue: 0,0:07:09.31,0:07:12.77,Default,,0000,0000,0000,,negative second derivative, then you are in a concave
Dialogue: 0,0:07:12.77,0:07:16.77,Default,,0000,0000,0000,,downward interval. Similarly…similarly
Dialogue: 0,0:07:16.77,0:07:20.39,Default,,0000,0000,0000,,(I have trouble saying that word), let’s think about concave upwards,
Dialogue: 0,0:07:20.39,0:07:25.24,Default,,0000,0000,0000,,where you have an upward opening “U”. Concave upwards.
Dialogue: 0,0:07:25.24,0:07:28.64,Default,,0000,0000,0000,,In these intervals, the slope is increasing,
Dialogue: 0,0:07:28.64,0:07:32.71,Default,,0000,0000,0000,,we have negative slope, less negative, less negative…zero, positive, more positive, more
Dialogue: 0,0:07:32.71,0:07:36.26,Default,,0000,0000,0000,,positive…even more positive. So slope...slope
Dialogue: 0,0:07:36.26,0:07:39.64,Default,,0000,0000,0000,,is increasing. "Slope is
Dialogue: 0,0:07:39.64,0:07:43.44,Default,,0000,0000,0000,,increasing which means
Dialogue: 0,0:07:43.44,0:07:48.04,Default,,0000,0000,0000,,that the derivative of the function is
Dialogue: 0,0:07:48.04,0:07:52.04,Default,,0000,0000,0000,,increasing. And you see that right over
Dialogue: 0,0:07:52.04,0:07:55.37,Default,,0000,0000,0000,,here, this derivative is increasing in value,
Dialogue: 0,0:07:55.37,0:07:59.26,Default,,0000,0000,0000,,which means that the second derivative，the second derivative
Dialogue: 0,0:07:59.26,0:08:03.43,Default,,0000,0000,0000,,over the interval where we are concave upwards must be greater than zero,
Dialogue: 0,0:08:03.43,0:08:07.44,Default,,0000,0000,0000,,the second derivative is greater than zero that means the first derivative is increasing,
Dialogue: 0,0:08:07.44,0:08:11.68,Default,,0000,0000,0000,,which means that the slope is increasing. We are in a concave upward,
Dialogue: 0,0:08:11.68,0:08:14.70,Default,,0000,0000,0000,,we are in a concave upward interval. Now,
Dialogue: 0,0:08:14.70,0:08:19.18,Default,,0000,0000,0000,,given all these definitions we’ve just given for concave downwards
Dialogue: 0,0:08:19.18,0:08:23.37,Default,,0000,0000,0000,,and concave upwards interval, can we come out with another way of indentifying whether a critical point
Dialogue: 0,0:08:23.37,0:08:26.77,Default,,0000,0000,0000,,is a minimum point or maximum point.
Dialogue: 0,0:08:26.77,0:08:30.60,Default,,0000,0000,0000,,Well, if you have a maximum point, if you have a critical point where the
Dialogue: 0,0:08:30.60,0:08:34.37,Default,,0000,0000,0000,,function...where the function is concave downwards,
Dialogue: 0,0:08:34.37,0:08:38.37,Default,,0000,0000,0000,,then it going to be a maximum point."Concave downwards". Let’s just be clear here,
Dialogue: 0,0:08:38.37,0:08:42.60,Default,,0000,0000,0000,,means that it’s opening down like this
Dialogue: 0,0:08:42.60,0:08:46.59,Default,,0000,0000,0000,,and we are talking about a critical point. If we’re assuming it’s concave downwards
Dialogue: 0,0:08:46.59,0:08:49.44,Default,,0000,0000,0000,,over here, we’re assuming differentiability over this interval and so the critical point
Dialogue: 0,0:08:49.44,0:08:53.32,Default,,0000,0000,0000,,is gonna be one where the slope is zero, so it’s gonna be that point
Dialogue: 0,0:08:53.32,0:08:57.77,Default,,0000,0000,0000,,right over there. So if you have a concave upwards and you have a point where
Dialogue: 0,0:08:57.77,0:09:01.87,Default,,0000,0000,0000,,f’(a) = 0
Dialogue: 0,0:09:01.87,0:09:04.87,Default,,0000,0000,0000,,then we have a maximum point at a.
Dialogue: 0,0:09:04.87,0:09:14.04,Default,,0000,0000,0000,,And similarly if we are a concave upwards
Dialogue: 0,0:09:14.04,0:09:17.71,Default,,0000,0000,0000,,that means that our function looks something like this and if we
Dialogue: 0,0:09:17.71,0:09:21.93,Default,,0000,0000,0000,,found the point. Obviously a critical point could also be where the function is not
Dialogue: 0,0:09:21.93,0:09:25.68,Default,,0000,0000,0000,,defined. But if we are assuming that our first derivative and second derivative is
Dialogue: 0,0:09:25.68,0:09:29.98,Default,,0000,0000,0000,,defined here then the critical point is going to be one where the first derivative is
Dialogue: 0,0:09:29.98,0:09:33.37,Default,,0000,0000,0000,,going to be zero, so f’(a)
Dialogue: 0,0:09:33.37,0:09:37.70,Default,,0000,0000,0000,,f’(a)= 0.If f’(a)= 0
Dialogue: 0,0:09:37.70,0:09:41.97,Default,,0000,0000,0000,,and if we are concave upwards and the interval around
Dialogue: 0,0:09:41.97,0:09:44.98,Default,,0000,0000,0000,,a, so the second derivative is greater than zero, then it’s pretty
Dialogue: 0,0:09:44.98,0:09:49.18,Default,,0000,0000,0000,,clear you see here that we are dealing with… we are dealing with a minimum,
Dialogue: 0,0:09:49.18,0:09:53.18,Default,,0000,0000,0000,,a minimum point at a .