1 00:00:00,012 --> 00:00:05,577 [MUSIC] Earlier, we saw how the sign, the S I G N, of the derivative, encoded 2 00:00:05,577 --> 00:00:09,297 whether the function was increasing or decreasing. 3 00:00:09,297 --> 00:00:14,127 Thinking back to the graph, here I've just drawn some random graph. 4 00:00:14,127 --> 00:00:19,697 What is the derivative encoding? Well, here at this point a, the slope of this 5 00:00:19,697 --> 00:00:25,512 tangent of negative, the derivative is negative, and yeah, the function's going 6 00:00:25,512 --> 00:00:28,219 down here. At this point B, the slope of this 7 00:00:28,219 --> 00:00:31,854 tangent line's positive, and the function's increasing through here. 8 00:00:31,854 --> 00:00:35,371 All right, the derivative's negative here, and it's positive here. 9 00:00:35,371 --> 00:00:38,091 The function's decreasing here and increasing here. 10 00:00:38,091 --> 00:00:40,363 So that's what the derivative is measuring. 11 00:00:40,363 --> 00:00:45,167 What is the sign of the second derivative really encoding? Maybe we don't have such 12 00:00:45,167 --> 00:00:48,412 a good word for it, so we'll just make up a new word. 13 00:00:48,412 --> 00:00:53,082 The sign, the sign of the second derivative, the sign of the derivative of 14 00:00:53,082 --> 00:00:58,176 the derivative, measures concavity. The word's concavity and here's the two 15 00:00:58,176 --> 00:01:01,807 possibilities. Concave up where the second derivative 16 00:01:01,807 --> 00:01:06,487 was positive and concave down where the second derivative is negative. 17 00:01:06,487 --> 00:01:11,778 And I've drawn sort of cartoony pictures of what the graphs look like in these two 18 00:01:11,778 --> 00:01:14,729 cases. Now note, it's not just increasing or 19 00:01:14,729 --> 00:01:19,707 decreasing but this concavity is recording sort of the shape Of the graph 20 00:01:19,707 --> 00:01:23,012 in some sense. Positive second derivative makes it look 21 00:01:23,012 --> 00:01:27,522 like this, negative second derivative makes the graph look like this and I'm 22 00:01:27,522 --> 00:01:31,137 just labeling these two things, concave up and concave down. 23 00:01:31,137 --> 00:01:35,542 And this makes sense, if we think of the second derivative as measuring the 24 00:01:35,542 --> 00:01:39,557 change, in the derivative. So let's think back to this graph again, 25 00:01:39,557 --> 00:01:42,097 here's this graph of some random function. 26 00:01:42,097 --> 00:01:44,732 Look at this part of the graph right here. 27 00:01:44,732 --> 00:01:50,040 That looks like the concave up shape from before where the second derivative was 28 00:01:50,040 --> 00:01:53,373 positive. So we might think the second derivative 29 00:01:53,373 --> 00:01:57,126 is positive here. That would mean that the derivative is 30 00:01:57,126 --> 00:02:00,632 increasing. What that really means is that the slope 31 00:02:00,632 --> 00:02:04,252 of a tangent line through this region is increasing, 32 00:02:04,252 --> 00:02:09,667 and that's exactly what's happening. The slope is negative here. 33 00:02:09,667 --> 00:02:16,441 And as I move this tangent line over, the slope of that tangent line is increasing. 34 00:02:16,441 --> 00:02:23,047 The second derivative is positive here. You can tell yourself the same story for 35 00:02:23,047 --> 00:02:27,346 concave down. So look over here in our sample graph. 36 00:02:27,346 --> 00:02:32,045 That part of the graph. Looks like this concave down picture, 37 00:02:32,045 --> 00:02:37,600 where the second derivative is negative. Now if the second derivative is negative, 38 00:02:37,600 --> 00:02:42,958 that means the derivative is decreasing. And yeah, the slope of the tangent line 39 00:02:42,958 --> 00:02:47,600 through this region is going down, right? The slope starts off pretty 40 00:02:47,600 --> 00:02:52,465 positive over here and as I move this tangent line over the slope is zero, and 41 00:02:52,465 --> 00:02:58,046 now getting more and more negative. So in this part of the graph, the 2nd 42 00:02:58,046 --> 00:03:03,505 derivative is negative. What happens inbetween? Where does the 43 00:03:03,505 --> 00:03:09,189 regime change take place? So over here, the second derivative is negative. 44 00:03:09,189 --> 00:03:12,417 Over here, the second derivative is positive. 45 00:03:12,417 --> 00:03:17,949 There's a point in between, maybe it's right here, and at that point, the second 46 00:03:17,949 --> 00:03:23,071 derivative is equal to zero and on one side, it's concave down and on the other 47 00:03:23,071 --> 00:03:27,175 side, it's concave up. A point where the concavity actually 48 00:03:27,175 --> 00:03:32,441 changes is called an inflection point. All right it's concave down over here and 49 00:03:32,441 --> 00:03:37,586 it's concave up over here and the place where the change is taking place, we're 50 00:03:37,586 --> 00:03:40,779 just going to call those points inflection points. 51 00:03:40,779 --> 00:03:45,509 It's not that the terminology itself is so important but we want words to 52 00:03:45,509 --> 00:03:50,442 describe the qualitative phenomenon that we're seeing in these graphs. 53 00:03:50,442 --> 00:03:53,052 Inflection points are something you can really feel. 54 00:03:53,052 --> 00:03:55,603 I mean, if you're driving in a car, you're braking. 55 00:03:55,603 --> 00:03:58,238 Right? That means the second derivative is negative. 56 00:03:58,238 --> 00:04:01,316 You're slowing down. And then, suddenly, you step on the gas. 57 00:04:01,316 --> 00:04:04,782 Now you're accelerating, and your second derivative is positive. 58 00:04:04,782 --> 00:04:09,367 What happened, right? Something big happened. 59 00:04:09,367 --> 00:04:15,771 [MUSIC] You're changing regimes from concave down to concave up. 60 00:04:15,771 --> 00:04:19,786 And you want to denote that change somehow. 61 00:04:19,786 --> 00:04:24,589 We're going to call that change an inflection point. 62 00:04:24,589 --> 00:04:25,365 [MUSIC]