WEBVTT 00:00:00.000 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.610 Commons license. 00:00:03.610 --> 00:00:05.990 Your support will help MIT OpenCourseWare 00:00:05.990 --> 00:00:10.190 continue to offer high quality educational resources for free. 00:00:10.190 --> 00:00:12.530 To make a donation, or to view additional materials 00:00:12.530 --> 00:00:15.880 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:15.880 --> 00:00:22.040 at ocw.mit.edu. 00:00:22.040 --> 00:00:25.380 PROFESSOR: OK, we're ready to start the eleventh lecture. 00:00:25.380 --> 00:00:29.390 We're still in the middle of sketching. 00:00:29.390 --> 00:00:32.910 And, indeed, one of the reasons why we did not 00:00:32.910 --> 00:00:37.040 talk about hyperbolic functions is 00:00:37.040 --> 00:00:39.540 that we're running just a little bit behind. 00:00:39.540 --> 00:00:41.720 And we'll catch up a tiny bit today. 00:00:41.720 --> 00:00:46.000 And I hope all the way on Tuesday of next week. 00:00:46.000 --> 00:00:58.640 So let me pick up where we left off, with sketching. 00:00:58.640 --> 00:01:05.220 So this is a continuation. 00:01:05.220 --> 00:01:07.020 I want to give you one more example 00:01:07.020 --> 00:01:08.390 of how to sketch things. 00:01:08.390 --> 00:01:10.380 And then we'll go through it systematically. 00:01:10.380 --> 00:01:14.830 So the second example that we did as one example last time, 00:01:14.830 --> 00:01:18.060 is this. 00:01:18.060 --> 00:01:21.970 The function is (x+1)/(x+2). 00:01:21.970 --> 00:01:25.780 And I'm going to save you the time right now. 00:01:25.780 --> 00:01:28.629 This is very typical of me, especially 00:01:28.629 --> 00:01:30.170 if you're in a hurry on an exam, I'll 00:01:30.170 --> 00:01:32.920 just tell you what the derivative is. 00:01:32.920 --> 00:01:34.490 So in this case, it's 1 / (x+2)^2. 00:01:37.760 --> 00:01:42.327 Now, the reason why I'm bringing this example up, 00:01:42.327 --> 00:01:44.660 even though it'll turn out to be a relatively simple one 00:01:44.660 --> 00:01:49.150 to sketch, is that it's easy to fall 00:01:49.150 --> 00:01:53.890 into a black hole with this problem. 00:01:53.890 --> 00:01:57.470 So let me just show you. 00:01:57.470 --> 00:01:59.600 This is not equal to 0. 00:01:59.600 --> 00:02:01.180 It's never equal to 0. 00:02:01.180 --> 00:02:08.940 So that means there are no critical points. 00:02:08.940 --> 00:02:13.710 At this point, students, many students 00:02:13.710 --> 00:02:16.250 who have been trained like monkeys 00:02:16.250 --> 00:02:18.710 to do exactly what they've been told, 00:02:18.710 --> 00:02:21.170 suddenly freeze and give up. 00:02:21.170 --> 00:02:23.970 Because there's nothing to do. 00:02:23.970 --> 00:02:28.000 So this is the one thing that I have to train out of you. 00:02:28.000 --> 00:02:32.260 You can't just give up at this point. 00:02:32.260 --> 00:02:35.060 So what would you suggest? 00:02:35.060 --> 00:02:38.100 Can anybody get us out of this jam? 00:02:38.100 --> 00:02:38.780 Yeah. 00:02:38.780 --> 00:02:47.410 STUDENT: [INAUDIBLE] 00:02:47.410 --> 00:02:48.150 PROFESSOR: Right. 00:02:48.150 --> 00:02:52.810 So the suggestion was to find the x-values 00:02:52.810 --> 00:02:55.800 where f(x) is undefined. 00:02:55.800 --> 00:03:00.680 In fact, so now that's a fairly sophisticated way 00:03:00.680 --> 00:03:03.090 of putting the point that I want to make, 00:03:03.090 --> 00:03:05.380 which is that what we want to do is go back 00:03:05.380 --> 00:03:07.770 to our precalculus skills. 00:03:07.770 --> 00:03:11.150 And just plot points. 00:03:11.150 --> 00:03:13.500 So instead, you go back to precalculus 00:03:13.500 --> 00:03:16.934 and you just plot some points. 00:03:16.934 --> 00:03:18.350 It's a perfectly reasonable thing. 00:03:18.350 --> 00:03:21.970 Now, it turns out that the most important point to plot 00:03:21.970 --> 00:03:24.730 is the one that's not there. 00:03:24.730 --> 00:03:29.300 Namely, the value of x = -2. 00:03:29.300 --> 00:03:31.780 Which is just what was suggested. 00:03:31.780 --> 00:03:38.710 Namely, we plot the points where the function is not defined. 00:03:38.710 --> 00:03:41.220 So how do we do that? 00:03:41.220 --> 00:03:43.670 Well, you have to think about it for a second 00:03:43.670 --> 00:03:46.470 and I'll introduce some new notation when I do it. 00:03:46.470 --> 00:03:49.290 If I evaluate 2 at this place, actually I can't do it. 00:03:49.290 --> 00:03:51.250 I have to do it from the left and the right. 00:03:51.250 --> 00:03:57.840 So if I plug in -2 on the positive side, from the right, 00:03:57.840 --> 00:04:03.500 that's going to be equal to -2 + 1 divided by -2, 00:04:03.500 --> 00:04:06.590 a little bit more than -2, plus 2. 00:04:06.590 --> 00:04:11.110 Which is -1 divided by - now, this denominator is -2, 00:04:11.110 --> 00:04:12.730 a little more than that, plus 2. 00:04:12.730 --> 00:04:20.200 So it's a little more than 0. 00:04:20.200 --> 00:04:24.720 And that is, well we'll fill that in in a second. 00:04:24.720 --> 00:04:25.620 Everybody's puzzled. 00:04:25.620 --> 00:04:26.120 Yes. 00:04:26.120 --> 00:04:30.670 STUDENT: [INAUDIBLE] 00:04:30.670 --> 00:04:36.105 PROFESSOR: No, that's the function. 00:04:36.105 --> 00:04:37.980 I'm plotting points, I'm not differentiating. 00:04:37.980 --> 00:04:38.250 I've already differentiated it. 00:04:38.250 --> 00:04:39.630 I've already got something that's a little puzzling. 00:04:39.630 --> 00:04:41.250 Now I'm focusing on the weird spot. 00:04:41.250 --> 00:04:42.166 Yes, another question. 00:04:42.166 --> 00:04:47.652 STUDENT: Wouldn't it be a little less than 0? 00:04:47.652 --> 00:04:49.610 PROFESSOR: Wouldn't it be a little less than 0? 00:04:49.610 --> 00:04:52.800 OK, that's a very good point and this is a matter of notation 00:04:52.800 --> 00:04:53.310 here. 00:04:53.310 --> 00:04:55.690 And a matter of parentheses. 00:04:55.690 --> 00:04:57.430 So wouldn't this be a little less than 2. 00:04:57.430 --> 00:05:01.900 Well, if the parentheses were this way; that is, 2+ , 00:05:01.900 --> 00:05:06.790 with a minus after I did the 2+ , then it would be less. 00:05:06.790 --> 00:05:09.830 But it's this way. 00:05:09.830 --> 00:05:10.670 OK. 00:05:10.670 --> 00:05:14.420 So the notation is, you have a number 00:05:14.420 --> 00:05:17.190 and you take the plus part of it. 00:05:17.190 --> 00:05:21.550 That's the part which is a little bit bigger than it. 00:05:21.550 --> 00:05:26.840 And so this is what I mean. 00:05:26.840 --> 00:05:29.870 And if you like, here I can put in those parentheses too. 00:05:29.870 --> 00:05:31.270 Yeah, another question. 00:05:31.270 --> 00:05:34.730 STUDENT: [INAUDIBLE] 00:05:34.730 --> 00:05:37.253 PROFESSOR: Why doesn't the top one have a plus? 00:05:37.253 --> 00:05:39.378 The only reason why the top one doesn't have a plus 00:05:39.378 --> 00:05:42.860 is that I don't need it to evaluate this. 00:05:42.860 --> 00:05:45.410 And when I take the limit, I can just plug in the value. 00:05:45.410 --> 00:05:48.250 Whereas here, I'm still uncertain. 00:05:48.250 --> 00:05:49.430 Because it's going to be 0. 00:05:49.430 --> 00:05:51.740 And I want to know which side of 0 it's on. 00:05:51.740 --> 00:05:55.190 Whether it's on the positive side or the negative side. 00:05:55.190 --> 00:05:58.310 So this one, I could have written here a parentheses 2+, 00:05:58.310 --> 00:06:01.410 but then it would have just simplified to -1. 00:06:01.410 --> 00:06:04.410 In the limit. 00:06:04.410 --> 00:06:07.190 So now, I've got a negative number divided 00:06:07.190 --> 00:06:10.024 by a tiny positive number. 00:06:10.024 --> 00:06:11.940 And so, somebody want to tell me what that is? 00:06:11.940 --> 00:06:16.750 Negative infinity. 00:06:16.750 --> 00:06:21.460 So, we just evaluated this function from one side. 00:06:21.460 --> 00:06:24.830 And if you follow through the other side, 00:06:24.830 --> 00:06:30.700 so this one here, you get something very similar, 00:06:30.700 --> 00:06:34.120 except that this should be-- whoops, what did I do wrong? 00:06:34.120 --> 00:06:37.690 I meant this. 00:06:37.690 --> 00:06:40.500 I wanted -2, the same base point, 00:06:40.500 --> 00:06:43.750 but I want to go from the left. 00:06:43.750 --> 00:06:47.780 So that's going to be -2 + 1, same numerator. 00:06:47.780 --> 00:06:51.840 And then this -2 on the left, plus 2, 00:06:51.840 --> 00:06:57.580 and that's going to come out to be -1 / 0-, -, 00:06:57.580 --> 00:07:00.670 which is plus infinity. 00:07:00.670 --> 00:07:10.520 Or just plain infinity, we don't have to put the plus sign in. 00:07:10.520 --> 00:07:13.090 So this is the first part of the problem. 00:07:13.090 --> 00:07:16.190 And the second piece, to get ourselves started, 00:07:16.190 --> 00:07:18.260 you could evaluate this function at any point. 00:07:18.260 --> 00:07:21.256 This is just the most interesting point, alright? 00:07:21.256 --> 00:07:22.880 This is just the most interesting place 00:07:22.880 --> 00:07:24.936 to evaluate it. 00:07:24.936 --> 00:07:26.560 Now, the next thing that I'd like to do 00:07:26.560 --> 00:07:32.130 is to pay attention to the ends. 00:07:32.130 --> 00:07:34.290 And I haven't really said what the ends are. 00:07:34.290 --> 00:07:37.399 So the ends are just all the way to the left and all the way 00:07:37.399 --> 00:07:37.940 to the right. 00:07:37.940 --> 00:07:42.180 So that means x going to plus or minus infinity. 00:07:42.180 --> 00:07:44.430 So that's the second thing I want to pay attention to. 00:07:44.430 --> 00:07:49.450 Again, this is a little bit like a video screen here. 00:07:49.450 --> 00:07:52.400 And we're about to discover something that's really 00:07:52.400 --> 00:07:55.654 off the screen, in both cases. 00:07:55.654 --> 00:07:58.070 We're taking care of what's happening way to the left, way 00:07:58.070 --> 00:07:59.360 to the right, here. 00:07:59.360 --> 00:08:01.170 And up above, we just took care what 00:08:01.170 --> 00:08:05.280 happens way up and way down. 00:08:05.280 --> 00:08:11.550 So on these ends, I need to do some more analysis. 00:08:11.550 --> 00:08:15.480 Which is related to a precalculus skill 00:08:15.480 --> 00:08:18.310 which is evaluating limits. 00:08:18.310 --> 00:08:21.070 And here, the way to do it is to divide 00:08:21.070 --> 00:08:23.010 by x the numerator and denominator. 00:08:23.010 --> 00:08:27.360 Write it as (1 + 1/x) / (1 + 2/x). 00:08:27.360 --> 00:08:29.650 And then you can see what happens as x 00:08:29.650 --> 00:08:30.980 goes to plus or minus infinity. 00:08:30.980 --> 00:08:33.590 It just goes to 1. 00:08:33.590 --> 00:08:37.690 So, no matter whether x is positive or negative. 00:08:37.690 --> 00:08:42.810 When it gets huge, these two extra numbers here go to 0. 00:08:42.810 --> 00:08:44.550 And so, this tends to 1. 00:08:44.550 --> 00:08:47.560 So if you like, you could abbreviate this 00:08:47.560 --> 00:08:52.830 as f plus or minus infinity is equal to 1. 00:08:52.830 --> 00:08:54.610 So now, I get to draw this. 00:08:54.610 --> 00:08:56.900 And we draw this using asymptotes. 00:08:56.900 --> 00:09:01.800 So there's a level which is y = 1. 00:09:01.800 --> 00:09:06.790 And then there's another line to draw. 00:09:06.790 --> 00:09:11.180 Which is x = -2. 00:09:15.190 --> 00:09:18.520 And now, what information do I have so far? 00:09:18.520 --> 00:09:20.740 Well, the information that I have so far 00:09:20.740 --> 00:09:26.530 is that when we're coming in from the right, that's to -2, 00:09:26.530 --> 00:09:28.570 it plunges down to minus infinity. 00:09:28.570 --> 00:09:33.170 So that's down like this. 00:09:33.170 --> 00:09:39.780 And I also know that it goes up to infinity on the other side 00:09:39.780 --> 00:09:41.850 of the asymptote. 00:09:41.850 --> 00:09:48.130 And over here, I know it's going out to the level 1. 00:09:48.130 --> 00:09:53.410 And here it's also going to the level 1. 00:09:53.410 --> 00:09:57.160 Now, there's an issue. 00:09:57.160 --> 00:09:59.490 I can almost finish this graph now. 00:09:59.490 --> 00:10:01.460 I almost have enough information to finish it. 00:10:01.460 --> 00:10:03.530 But there's one thing which is making 00:10:03.530 --> 00:10:06.780 me hesitate a little bit. 00:10:06.780 --> 00:10:10.160 And that is, I don't know, for instance, over here, 00:10:10.160 --> 00:10:14.230 whether it's going to maybe dip below and come back up. 00:10:14.230 --> 00:10:16.930 Or not. 00:10:16.930 --> 00:10:20.140 So what does it do here? 00:10:20.140 --> 00:10:24.750 Can anybody see? 00:10:24.750 --> 00:10:25.250 Yeah. 00:10:25.250 --> 00:10:29.900 STUDENT: [INAUDIBLE] 00:10:29.900 --> 00:10:32.220 PROFESSOR: It can't dip below because there 00:10:32.220 --> 00:10:33.178 are no critical points. 00:10:33.178 --> 00:10:34.540 What a precisely correct answer. 00:10:34.540 --> 00:10:36.730 So that's exactly right. 00:10:36.730 --> 00:10:43.390 The point here is that because f' is not 0, 00:10:43.390 --> 00:10:45.020 it can't double back on itself. 00:10:45.020 --> 00:10:49.790 Because there can't be any of these horizontal tangents. 00:10:49.790 --> 00:11:00.560 It can't double back, so it can't backtrack. 00:11:00.560 --> 00:11:07.190 So sorry, if f' is not 0, f can't backtrack. 00:11:07.190 --> 00:11:09.410 And so that means that it doesn't look like this. 00:11:09.410 --> 00:11:14.320 It just goes like this. 00:11:14.320 --> 00:11:15.690 So that's basically it. 00:11:15.690 --> 00:11:17.630 And it's practically the end of the problem. 00:11:17.630 --> 00:11:19.500 Goes like this. 00:11:19.500 --> 00:11:21.870 Now you can decorate your thing, right? 00:11:21.870 --> 00:11:24.660 You may notice that maybe it crosses here, the axes, you can 00:11:24.660 --> 00:11:26.790 actually evaluate these places. 00:11:26.790 --> 00:11:27.500 And so forth. 00:11:27.500 --> 00:11:31.130 We're looking right now for qualitative behavior. 00:11:31.130 --> 00:11:34.280 In fact, you can see where these places hit. 00:11:34.280 --> 00:11:36.660 And it's actually a little higher up than I drew. 00:11:36.660 --> 00:11:40.140 Maybe I'll draw it accurately. 00:11:40.140 --> 00:11:44.970 As we'll see in a second. 00:11:44.970 --> 00:11:47.710 So that's what happens to this function. 00:11:47.710 --> 00:11:51.860 Now, let's just take a look in a little bit more detail, 00:11:51.860 --> 00:11:56.470 by double checking. 00:11:56.470 --> 00:11:58.470 So we're just going to double check what happens 00:11:58.470 --> 00:12:01.280 to the sign of the derivative. 00:12:01.280 --> 00:12:03.460 And in the meantime, I'm going to explain to you 00:12:03.460 --> 00:12:05.510 what the derivative is and also talk 00:12:05.510 --> 00:12:07.045 about the second derivative. 00:12:07.045 --> 00:12:11.820 So first of all, the trick for evaluating the derivative 00:12:11.820 --> 00:12:13.570 is an algebraic one. 00:12:13.570 --> 00:12:16.080 I mean, obviously you can do this by the quotient rule. 00:12:16.080 --> 00:12:24.250 But I just point out that this is the same thing as this. 00:12:24.250 --> 00:12:27.330 And now it has, whoops, that should be a 2 00:12:27.330 --> 00:12:28.690 in the denominator. 00:12:28.690 --> 00:12:33.050 And so, now this has the form 1 - 1/(x+2). 00:12:35.950 --> 00:12:39.600 So this makes it easy to see what the derivative is. 00:12:39.600 --> 00:12:42.730 Because the derivative of a constant is 0, right? 00:12:42.730 --> 00:12:49.300 So this is, derivative, is just going to be, switch the sign. 00:12:49.300 --> 00:12:53.410 This is what I wrote before. 00:12:53.410 --> 00:12:55.230 And that explains it. 00:12:55.230 --> 00:12:57.610 But incidentally, it also shows you 00:12:57.610 --> 00:13:04.890 that that this is a hyperbola. 00:13:04.890 --> 00:13:09.620 These are just two curves of a hyperbola. 00:13:09.620 --> 00:13:12.240 So now, let's check the sign. 00:13:12.240 --> 00:13:14.566 It's already totally obvious to us 00:13:14.566 --> 00:13:15.940 that this is just a double check. 00:13:15.940 --> 00:13:18.710 We didn't actually even have to pay any attention to this. 00:13:18.710 --> 00:13:19.790 It had better be true. 00:13:19.790 --> 00:13:22.220 This is just going to check our arithmetic. 00:13:22.220 --> 00:13:24.570 Namely, it's increasing here. 00:13:24.570 --> 00:13:26.610 It's increasing there. 00:13:26.610 --> 00:13:27.830 That's got to be true. 00:13:27.830 --> 00:13:30.970 And, sure enough, this is positive, 00:13:30.970 --> 00:13:32.930 as you can see it's 1 over a square. 00:13:32.930 --> 00:13:34.070 So it is increasing. 00:13:34.070 --> 00:13:35.630 So we checked it. 00:13:35.630 --> 00:13:39.100 But now, there's one more thing that I want to just 00:13:39.100 --> 00:13:40.720 have you watch out about. 00:13:40.720 --> 00:13:46.980 So this means that f is increasing. 00:13:46.980 --> 00:13:51.830 On the interval minus infinity < x < -2. 00:13:51.830 --> 00:13:56.700 And also from -2 all the way out to infinity. 00:13:56.700 --> 00:14:02.260 So I just want to warn you, you cannot say, 00:14:02.260 --> 00:14:10.370 don't say f is increasing on (minus infinity, infinity), 00:14:10.370 --> 00:14:12.020 or all x. 00:14:12.020 --> 00:14:14.826 OK, this is just not true. 00:14:14.826 --> 00:14:16.700 I've written it on the board, but it's wrong. 00:14:16.700 --> 00:14:18.660 I'd better get rid of it. 00:14:18.660 --> 00:14:19.200 There it is. 00:14:19.200 --> 00:14:20.970 Get rid of it. 00:14:20.970 --> 00:14:24.555 And the reason is, so first of all it's totally obvious. 00:14:24.555 --> 00:14:25.390 It's going up here. 00:14:25.390 --> 00:14:28.700 But then it went zooming back down there. 00:14:28.700 --> 00:14:35.860 And here this was true, but only if x is not -2. 00:14:35.860 --> 00:14:37.400 So there's a break. 00:14:37.400 --> 00:14:39.280 And you've got to pay attention to the break. 00:14:39.280 --> 00:14:51.060 So basically, the moral here is that if you ignore this place, 00:14:51.060 --> 00:14:54.390 it's like ignoring Mount Everest, or the Grand Canyon. 00:14:54.390 --> 00:14:56.640 You're ignoring the most important feature 00:14:56.640 --> 00:14:58.174 of this function here. 00:14:58.174 --> 00:14:59.590 If you're going to be figuring out 00:14:59.590 --> 00:15:01.980 where things are going up and down, which is basically 00:15:01.980 --> 00:15:04.460 all we're doing, you'd better pay attention 00:15:04.460 --> 00:15:07.330 to these kinds of places. 00:15:07.330 --> 00:15:09.330 So don't ignore them. 00:15:09.330 --> 00:15:13.140 So that's the first remark. 00:15:13.140 --> 00:15:17.340 And now there's just a little bit of decoration as well. 00:15:17.340 --> 00:15:19.990 Which is the role of the second derivative. 00:15:19.990 --> 00:15:21.990 So we've written down the first derivative here. 00:15:21.990 --> 00:15:33.490 The second derivative is now -2 / (x + 2)^3, right? 00:15:33.490 --> 00:15:36.300 So I get that from differentiating this formula up 00:15:36.300 --> 00:15:39.490 here for the first derivative. 00:15:39.490 --> 00:15:43.970 And now, of course, that's also, only works for x not equal 00:15:43.970 --> 00:15:47.680 to -2. 00:15:47.680 --> 00:15:54.820 And now, we can see that this is going to be negative, let's 00:15:54.820 --> 00:15:56.650 see, where is it negative? 00:15:56.650 --> 00:15:58.540 When this is a positive quantity, 00:15:58.540 --> 00:16:04.590 so when -2 < x < infinity, it's negative. 00:16:04.590 --> 00:16:07.770 And this is where this thing is concave. 00:16:07.770 --> 00:16:08.670 Let's see. 00:16:08.670 --> 00:16:12.650 Did I say that right? 00:16:12.650 --> 00:16:13.330 Negative, right? 00:16:13.330 --> 00:16:19.180 This is concave down. 00:16:19.180 --> 00:16:19.930 Right. 00:16:19.930 --> 00:16:22.570 And similarly, if I look at this expression, 00:16:22.570 --> 00:16:27.830 the numerator is always negative but the denominator becomes 00:16:27.830 --> 00:16:31.130 negative as well when x < -2. 00:16:31.130 --> 00:16:33.710 So this becomes positive. 00:16:33.710 --> 00:16:36.690 So this case, it was negative over positive. 00:16:36.690 --> 00:16:40.950 In this case it was negative divided by negative. 00:16:40.950 --> 00:16:46.300 So here, this is in the range minus infinity < x < -2 And 00:16:46.300 --> 00:16:52.240 here it's concave up. 00:16:52.240 --> 00:16:54.962 Now, again, this is just consistent with what 00:16:54.962 --> 00:16:55.920 we're already guessing. 00:16:55.920 --> 00:16:57.628 Of course we already know it in this case 00:16:57.628 --> 00:16:59.680 if we know that this is a hyperbola. 00:16:59.680 --> 00:17:01.600 That it's going to be concave down 00:17:01.600 --> 00:17:04.270 to the right of the vertical line, dotted vertical line. 00:17:04.270 --> 00:17:07.770 And concave up to the left. 00:17:07.770 --> 00:17:10.640 So what extra piece of information 00:17:10.640 --> 00:17:16.100 is it that this is giving us? 00:17:16.100 --> 00:17:17.800 Did I say this backwards? 00:17:17.800 --> 00:17:18.520 No. 00:17:18.520 --> 00:17:19.222 That's OK. 00:17:19.222 --> 00:17:21.430 So what extra piece of information is this giving us? 00:17:21.430 --> 00:17:23.400 It looks like it's giving us hardly anything. 00:17:23.400 --> 00:17:25.680 And it really is giving us hardly anything. 00:17:25.680 --> 00:17:28.840 But it is giving us something that's a little aesthetic. 00:17:28.840 --> 00:17:34.280 It's ruling out the possibility of a wiggle. 00:17:34.280 --> 00:17:37.590 There isn't anything like that in the curve. 00:17:37.590 --> 00:17:39.790 It can't shift from curving this way 00:17:39.790 --> 00:17:41.590 to curving that way to curving this way. 00:17:41.590 --> 00:17:42.860 That doesn't happen. 00:17:42.860 --> 00:17:59.070 So these properties say there's no wiggle in the graph of that. 00:17:59.070 --> 00:17:59.790 Alright. 00:17:59.790 --> 00:18:01.390 So. 00:18:01.390 --> 00:18:01.890 Question. 00:18:01.890 --> 00:18:05.570 STUDENT: Do we define the increasing and decreasing based 00:18:05.570 --> 00:18:09.250 purely on the derivative, or the sort 00:18:09.250 --> 00:18:13.390 of more general definition of picking any two points 00:18:13.390 --> 00:18:14.310 and seeing. 00:18:14.310 --> 00:18:16.610 Because sometimes there can be an inconsistency 00:18:16.610 --> 00:18:20.300 between the two definitions. 00:18:20.300 --> 00:18:26.110 PROFESSOR: OK, so the question is, in this course, 00:18:26.110 --> 00:18:29.290 are we going to define positive derivative as being 00:18:29.290 --> 00:18:31.360 the same thing as increasing. 00:18:31.360 --> 00:18:33.030 And the answer is no. 00:18:33.030 --> 00:18:36.210 We'll try to use these terms separately. 00:18:36.210 --> 00:18:40.330 What's always true is that if f' is positive, 00:18:40.330 --> 00:18:42.530 then f is increasing. 00:18:42.530 --> 00:18:45.060 But the reverse is not necessarily true. 00:18:45.060 --> 00:18:47.190 It could be very flat, the derivative can be 0 00:18:47.190 --> 00:18:50.050 and still the function can be increasing. 00:18:50.050 --> 00:18:53.730 OK, the derivative can be 0 at a few places. 00:18:53.730 --> 00:18:59.300 For instance, like some cubics. 00:18:59.300 --> 00:19:05.370 Other questions? 00:19:05.370 --> 00:19:09.860 So that's as much as I need to say in general. 00:19:09.860 --> 00:19:11.390 I mean, in a specific case. 00:19:11.390 --> 00:19:13.240 But I want to get you a general scheme 00:19:13.240 --> 00:19:16.500 and I want to go through a more complicated example that 00:19:16.500 --> 00:19:22.750 gets all the features of this kind of thing. 00:19:22.750 --> 00:19:34.710 So let's talk about a general strategy for sketching. 00:19:34.710 --> 00:19:40.300 So the first part of this strategy, if you like, 00:19:40.300 --> 00:19:40.800 let's see. 00:19:40.800 --> 00:19:42.430 I have it all plotted out here. 00:19:42.430 --> 00:19:45.930 So I'm going to make sure I get it exactly the way 00:19:45.930 --> 00:19:47.780 I wanted you to see. 00:19:47.780 --> 00:19:51.440 So I have, it's plotting. 00:19:51.440 --> 00:19:52.720 The plot thickens. 00:19:52.720 --> 00:19:54.040 Here we go. 00:19:54.040 --> 00:19:57.710 So plot, what is it that you should plot first? 00:19:57.710 --> 00:20:01.070 Before you even think about derivatives, 00:20:01.070 --> 00:20:08.250 you should plot discontinuities. 00:20:08.250 --> 00:20:18.490 Especially the infinite ones. 00:20:18.490 --> 00:20:20.210 That's the first thing you should do. 00:20:20.210 --> 00:20:27.160 And then, you should plot end points, for ends. 00:20:27.160 --> 00:20:30.580 For x going to plus or minus infinity 00:20:30.580 --> 00:20:35.640 if there don't happen to be any finite ends to the problem. 00:20:35.640 --> 00:20:44.600 And the third thing you can do is plot any easy points. 00:20:44.600 --> 00:20:49.000 This is optional. 00:20:49.000 --> 00:20:50.710 At your discretion. 00:20:50.710 --> 00:20:53.810 You might, for instance, on this example, 00:20:53.810 --> 00:20:59.550 plot the places where the graph crosses the axis. 00:20:59.550 --> 00:21:04.540 If you want to. 00:21:04.540 --> 00:21:05.810 So that's the first part. 00:21:05.810 --> 00:21:08.190 And again, this is all precalculus. 00:21:08.190 --> 00:21:18.050 So now, in the second part we're going to solve this equation 00:21:18.050 --> 00:21:29.360 and we're going to plot the critical points and values. 00:21:29.360 --> 00:21:32.810 In the problem which we just discussed, there weren't any. 00:21:32.810 --> 00:21:38.640 So this part was empty. 00:21:38.640 --> 00:21:49.685 So the third step is to decide whether f', sorry, whether, 00:21:49.685 --> 00:22:01.120 f' is positive or negative on each interval. 00:22:01.120 --> 00:22:17.210 Between critical points, discontinuities. 00:22:17.210 --> 00:22:22.560 The direction of the sign, in this case it doesn't change. 00:22:22.560 --> 00:22:24.900 It goes up here and it also goes up here. 00:22:24.900 --> 00:22:27.770 But it could go up here and then come back down. 00:22:27.770 --> 00:22:31.460 So the direction can change at every critical point. 00:22:31.460 --> 00:22:33.850 It can change at every discontinuity. 00:22:33.850 --> 00:22:35.350 And you don't know. 00:22:35.350 --> 00:22:39.310 However, this particular step has 00:22:39.310 --> 00:22:47.230 to be consistent with 1 and 2, with steps 1 and 2. 00:22:47.230 --> 00:22:51.800 In fact, it will never, if you can 00:22:51.800 --> 00:22:56.970 succeed in doing steps 1 and 2, you'll never need step 3. 00:22:56.970 --> 00:23:02.190 All it's doing is double-checking. 00:23:02.190 --> 00:23:05.770 So if you made an arithmetic mistake somewhere, 00:23:05.770 --> 00:23:09.160 you'll be able to see it. 00:23:09.160 --> 00:23:10.870 So that's maybe the most important thing. 00:23:10.870 --> 00:23:13.050 And it's actually the most frustrating thing for me 00:23:13.050 --> 00:23:17.700 when I see people working on problems, is they start step 3, 00:23:17.700 --> 00:23:21.140 they get it wrong, and then they start trying to draw the graph 00:23:21.140 --> 00:23:22.480 and it doesn't work. 00:23:22.480 --> 00:23:23.620 Because it's inconsistent. 00:23:23.620 --> 00:23:25.880 And the reason is some arithmetic error 00:23:25.880 --> 00:23:27.630 with the derivative or something like that 00:23:27.630 --> 00:23:29.880 or some other misinterpretation. 00:23:29.880 --> 00:23:31.930 And then there's a total mess. 00:23:31.930 --> 00:23:34.170 If you start with these two steps, 00:23:34.170 --> 00:23:36.454 then you're going to know when you get to this step 00:23:36.454 --> 00:23:37.620 that you're making mistakes. 00:23:37.620 --> 00:23:41.860 People don't generally make as many mistakes in the first two 00:23:41.860 --> 00:23:42.360 steps. 00:23:42.360 --> 00:23:45.220 Anyway, in fact you can skip this step if you want. 00:23:45.220 --> 00:23:49.470 But that's at risk of not double-checking your work. 00:23:49.470 --> 00:23:51.280 So what's the fourth step? 00:23:51.280 --> 00:23:59.400 Well, we take a look at whether f'' is positive or negative. 00:23:59.400 --> 00:24:02.300 And so we're deciding on things like whether it's concave 00:24:02.300 --> 00:24:07.640 up or down. 00:24:07.640 --> 00:24:15.120 And we have these points, f''(x) = 0, 00:24:15.120 --> 00:24:24.570 which are called inflection points. 00:24:24.570 --> 00:24:31.550 And the last step is just to combine everything. 00:24:31.550 --> 00:24:35.710 So this is this the scheme, the general scheme. 00:24:35.710 --> 00:24:58.850 And let's just carry it out in a particular case. 00:24:58.850 --> 00:25:02.280 So here's the function that I'm going to use as an example. 00:25:02.280 --> 00:25:08.250 I'll use f(x) = x / ln x. 00:25:08.250 --> 00:25:11.460 And because the logarithm-- yeah, question. 00:25:11.460 --> 00:25:11.960 Yeah. 00:25:11.960 --> 00:25:18.660 STUDENT: [INAUDIBLE] 00:25:18.660 --> 00:25:21.350 PROFESSOR: The question is, is this optional. 00:25:21.350 --> 00:25:25.570 So that's a good question. 00:25:25.570 --> 00:25:26.390 Is this optional. 00:25:26.390 --> 00:25:31.516 STUDENT: [INAUDIBLE] 00:25:31.516 --> 00:25:37.590 PROFESSOR: OK, the question is is this optional, 00:25:37.590 --> 00:25:38.660 this kind of question. 00:25:38.660 --> 00:25:48.620 And the answer is, it's more than just-- 00:25:48.620 --> 00:25:51.700 so, in many instances, I'm not going to ask you to. 00:25:51.700 --> 00:25:55.170 I strongly recommend that if I don't ask you to do it, 00:25:55.170 --> 00:25:57.050 that you not try. 00:25:57.050 --> 00:26:01.050 Because it's usually awful to find the second derivative. 00:26:01.050 --> 00:26:02.940 Any time you can get away without computing 00:26:02.940 --> 00:26:06.330 a second derivative, you're better off. 00:26:06.330 --> 00:26:07.834 So in many, many instances. 00:26:07.834 --> 00:26:09.500 On the other hand, if I ask you to do it 00:26:09.500 --> 00:26:13.060 it's because I want you to have the work to do it. 00:26:13.060 --> 00:26:16.470 But basically, if nobody forces you to, 00:26:16.470 --> 00:26:22.130 I would say never do step 4 here. 00:26:22.130 --> 00:26:26.750 Other questions. 00:26:26.750 --> 00:26:27.610 All right. 00:26:27.610 --> 00:26:29.150 So we're going to force ourselves 00:26:29.150 --> 00:26:31.810 to do step 4, however, in this instance. 00:26:31.810 --> 00:26:35.010 But maybe this will be one of the few times. 00:26:35.010 --> 00:26:39.140 So here we go, just for illustrative purposes. 00:26:39.140 --> 00:26:43.240 OK, now. 00:26:43.240 --> 00:26:46.140 So here's the function that I want to discuss. 00:26:46.140 --> 00:26:49.120 And the range has to be x positive, 00:26:49.120 --> 00:26:55.500 because the logarithm is not defined for negative values. 00:26:55.500 --> 00:26:57.660 So the first thing that I'm going to do 00:26:57.660 --> 00:27:02.560 is, I'd like to follow the scheme here. 00:27:02.560 --> 00:27:04.770 Because if I don't follow the scheme, 00:27:04.770 --> 00:27:06.490 I'm going to get a little mixed up. 00:27:06.490 --> 00:27:13.980 So the first part is to find the singularities. 00:27:13.980 --> 00:27:17.060 That is, the places where f is infinite. 00:27:17.060 --> 00:27:20.720 And that's when the logarithm, the denominator, vanishes. 00:27:20.720 --> 00:27:25.630 So that's f(1+), if you like. 00:27:25.630 --> 00:27:32.000 So that's 1 / ln(1+), which is 1 / 0, 00:27:32.000 --> 00:27:34.280 with a little bit of positiveness to it. 00:27:34.280 --> 00:27:37.270 Which is infinity. 00:27:37.270 --> 00:27:39.830 And second, we do it the other way. 00:27:39.830 --> 00:27:46.850 And not surprisingly, this comes out to be negative infinity. 00:27:46.850 --> 00:27:51.980 Now, the next thing I want to do is the ends. 00:27:51.980 --> 00:27:56.980 So I call these the ends. 00:27:56.980 --> 00:28:01.380 And there are two of them. 00:28:01.380 --> 00:28:06.250 One of them is f(0) from the right. 00:28:06.250 --> 00:28:08.300 f(0+). 00:28:08.300 --> 00:28:21.120 So that is 0+ / ln(0+), which is 0+ divided by, well, 00:28:21.120 --> 00:28:25.360 ln(0+) is actually minus infinity. 00:28:25.360 --> 00:28:27.180 That's what happens to the logarithm, goes 00:28:27.180 --> 00:28:28.170 to minus infinity. 00:28:28.170 --> 00:28:31.160 So this is 0 over infinity, which is definitely 0, 00:28:31.160 --> 00:28:37.100 there's no problem about what happens to this. 00:28:37.100 --> 00:28:42.910 The other side, so this is the end, this is the first end. 00:28:42.910 --> 00:28:44.920 The range is this. 00:28:44.920 --> 00:28:48.019 And I just did the left endpoint. 00:28:48.019 --> 00:28:49.810 And so now I have to do the right endpoint, 00:28:49.810 --> 00:28:51.870 I have to let x go to infinity. 00:28:51.870 --> 00:28:53.591 So if I let x go to infinity, I'm 00:28:53.591 --> 00:28:55.090 just going to have to think about it 00:28:55.090 --> 00:28:57.690 a little bit by plugging in a very large number. 00:28:57.690 --> 00:29:01.960 I'll plug in 10^10, to see what happens. 00:29:01.960 --> 00:29:07.890 So if I plug in 10^10 into x / ln x, I get 10^10 / ln(10^10). 00:29:11.730 --> 00:29:17.590 Which is 10^10 / (10 ln(10)). 00:29:17.590 --> 00:29:20.110 So the denominator, this number here, 00:29:20.110 --> 00:29:23.180 is about 2 point something. 00:29:23.180 --> 00:29:25.130 2.3 or so. 00:29:25.130 --> 00:29:27.470 So this is maybe 230 in the denominator, 00:29:27.470 --> 00:29:31.900 and this is a number with ten 0's after it. 00:29:31.900 --> 00:29:33.530 So it's very, very large. 00:29:33.530 --> 00:29:35.300 I claim it's big. 00:29:35.300 --> 00:29:38.080 And that gives us the clue that what's happening 00:29:38.080 --> 00:29:40.540 is that this thing is infinite. 00:29:40.540 --> 00:29:42.150 So, in other words, our conclusion 00:29:42.150 --> 00:29:52.120 is that f of infinity is infinity. 00:29:52.120 --> 00:29:59.650 So what do we have so far for our function? 00:29:59.650 --> 00:30:03.300 We're just trying to build the scaffolding of the function. 00:30:03.300 --> 00:30:07.270 And we're doing it by taking the most important points. 00:30:07.270 --> 00:30:09.147 And from a mathematician's point of view, 00:30:09.147 --> 00:30:10.980 the most important points are the ones which 00:30:10.980 --> 00:30:13.490 are sort of infinitely obvious. 00:30:13.490 --> 00:30:15.190 For the ends of the problem. 00:30:15.190 --> 00:30:19.730 So that's where we're heading. 00:30:19.730 --> 00:30:22.600 We have a vertical asymptote, which is at x = 1. 00:30:22.600 --> 00:30:29.150 So this gives us x = 1. 00:30:29.150 --> 00:30:34.270 And we have a value which is that it's 0 here. 00:30:34.270 --> 00:30:38.640 And we also know that when we come in from the-- sorry, 00:30:38.640 --> 00:30:42.330 so we come in from the left, that's 00:30:42.330 --> 00:30:46.060 f, the one from the left, we get negative infinity. 00:30:46.060 --> 00:30:47.550 So it's diving down. 00:30:47.550 --> 00:30:52.460 It's going down like this. 00:30:52.460 --> 00:30:55.925 And, furthermore, on the other side we know it's climbing up. 00:30:55.925 --> 00:30:58.560 So it's going up like this. 00:30:58.560 --> 00:31:00.270 Just start a little higher. 00:31:00.270 --> 00:31:00.770 Right, so. 00:31:00.770 --> 00:31:02.520 So far, this is what we know. 00:31:02.520 --> 00:31:05.810 Oh, and there's one other thing that we know. 00:31:05.810 --> 00:31:12.420 When we go to plus infinity, it's going back up. 00:31:12.420 --> 00:31:15.150 So, so far we have this. 00:31:15.150 --> 00:31:17.750 Now, already it should be pretty obvious what's 00:31:17.750 --> 00:31:19.619 going to happen to this function. 00:31:19.619 --> 00:31:21.160 So there shouldn't be many surprises. 00:31:21.160 --> 00:31:23.160 It's going to come down like this. 00:31:23.160 --> 00:31:27.090 Go like this, it's going to turn around and go back up. 00:31:27.090 --> 00:31:29.290 That's what we expect. 00:31:29.290 --> 00:31:33.600 So we don't know that yet, but we're pretty sure. 00:31:33.600 --> 00:31:36.800 So at this point, we can start looking at the critical points. 00:31:36.800 --> 00:31:41.994 We can do our step 2 here - we need a little bit more room 00:31:41.994 --> 00:31:45.490 here - and see what's happening with this function. 00:31:45.490 --> 00:31:49.220 So I have to differentiate it. 00:31:49.220 --> 00:31:52.070 And it's, this is the quotient rule. 00:31:52.070 --> 00:31:54.880 So remember the function is up here, x / ln x. 00:31:54.880 --> 00:31:59.400 So I have a (ln x)^2 in the denominator. 00:31:59.400 --> 00:32:02.430 And I get here the derivative of x is 1, so we get 1 * 00:32:02.430 --> 00:32:07.174 ln x minus x times the derivative of ln 00:32:07.174 --> 00:32:07.840 x, which is 1/x. 00:32:10.560 --> 00:32:16.850 So all told, that's (ln x - 1) / (ln x)^2. 00:32:20.160 --> 00:32:27.770 So here's our derivative. 00:32:27.770 --> 00:32:35.080 And now, if I set this equal to 0, at least in the numerator, 00:32:35.080 --> 00:32:40.970 the numerator is 0 when x = e. 00:32:40.970 --> 00:32:43.490 The log of e is 1. 00:32:43.490 --> 00:32:46.290 So here's our critical point. 00:32:46.290 --> 00:32:51.320 And we have a critical value, which is f(e). 00:32:51.320 --> 00:32:55.610 And that's going to be e / ln e. 00:32:55.610 --> 00:32:57.240 Which is e, again. 00:32:57.240 --> 00:32:59.090 Because ln e = 1. 00:32:59.090 --> 00:33:01.890 So now I can also plot the critical point, 00:33:01.890 --> 00:33:03.110 which is down here. 00:33:03.110 --> 00:33:07.680 And there's only one of them, and it's at (e, e). 00:33:07.680 --> 00:33:09.530 That's kind of not to scale here, 00:33:09.530 --> 00:33:12.520 because my blackboard isn't quite tall enough. 00:33:12.520 --> 00:33:15.360 It should be over here and then, it's slope 1. 00:33:15.360 --> 00:33:17.140 But I dipped it down. 00:33:17.140 --> 00:33:18.730 So this is not to scale, and indeed 00:33:18.730 --> 00:33:20.490 that's one of the things that we're not 00:33:20.490 --> 00:33:22.680 going to attempt to do with these pictures, 00:33:22.680 --> 00:33:24.680 is to make them to scale. 00:33:24.680 --> 00:33:29.560 So the scale's a little squashed. 00:33:29.560 --> 00:33:32.710 So, so far I have this critical point. 00:33:32.710 --> 00:33:36.180 And, in fact, I'm going to label it with a C. 00:33:36.180 --> 00:33:38.030 Whenever I have a critical point I'll just 00:33:38.030 --> 00:33:41.490 make sure that I remember that that's what it is. 00:33:41.490 --> 00:33:45.120 And since there's only one, the rest of this picture 00:33:45.120 --> 00:33:49.900 is now correct. 00:33:49.900 --> 00:33:54.880 That's the same mechanism that we used for the hyperbola. 00:33:54.880 --> 00:33:56.977 Namely, we know there's only one place where 00:33:56.977 --> 00:33:57.810 the derivative is 0. 00:33:57.810 --> 00:33:59.980 So that means there no more horizontals, 00:33:59.980 --> 00:34:01.950 so there's no more backtracking. 00:34:01.950 --> 00:34:03.330 It has to come down to here. 00:34:03.330 --> 00:34:03.970 Get to there. 00:34:03.970 --> 00:34:06.060 This is the only place it can turn around. 00:34:06.060 --> 00:34:06.834 Goes back up. 00:34:06.834 --> 00:34:09.000 It has to start here and it has to go down to there. 00:34:09.000 --> 00:34:10.600 It can't go above 0. 00:34:10.600 --> 00:34:13.600 Do not pass go, do not get positive. 00:34:13.600 --> 00:34:20.230 It has to head down here. 00:34:20.230 --> 00:34:21.690 So that's great. 00:34:21.690 --> 00:34:25.080 That means that this picture is almost completely correct now. 00:34:25.080 --> 00:34:27.520 And the rest is more or less decoration. 00:34:27.520 --> 00:34:30.250 We're pretty much done with the way it looks, 00:34:30.250 --> 00:34:34.570 at least schematically. 00:34:34.570 --> 00:34:37.700 However, I am going to punish you, because I warned you. 00:34:37.700 --> 00:34:40.120 We are going to go over here and do this step 4 00:34:40.120 --> 00:34:44.216 and fix up the concavity. 00:34:44.216 --> 00:34:45.840 And we're also going to do a little bit 00:34:45.840 --> 00:35:00.770 of that double-checking. 00:35:00.770 --> 00:35:04.960 So now, let's again-- just, I want 00:35:04.960 --> 00:35:10.660 to emphasize-- We're going to do a double-check. 00:35:10.660 --> 00:35:12.240 This is part 3. 00:35:12.240 --> 00:35:16.870 But in advance, I already have, based on this picture 00:35:16.870 --> 00:35:19.000 I already know what has to be true. 00:35:19.000 --> 00:35:35.450 That f is decreasing on 0 to 1. f is also decreasing on 1 to e. 00:35:35.450 --> 00:35:45.490 And f is increasing on e to infinity. 00:35:45.490 --> 00:35:49.144 So, already, because we plotted a bunch of points 00:35:49.144 --> 00:35:51.060 and we know that there aren't any places where 00:35:51.060 --> 00:35:52.518 the derivative vanishes, we already 00:35:52.518 --> 00:35:55.500 know it goes down, down, up. 00:35:55.500 --> 00:35:56.970 That's what it's got to do. 00:35:56.970 --> 00:35:59.550 Now, we'll just make sure that we didn't make any arithmetic 00:35:59.550 --> 00:36:00.720 mistakes, now. 00:36:00.720 --> 00:36:02.980 By actually computing the derivative, 00:36:02.980 --> 00:36:04.650 or staring at it, anyway. 00:36:04.650 --> 00:36:10.350 And making sure that it's correct. 00:36:10.350 --> 00:36:17.600 So first of all, we just take a look at the numerator. 00:36:17.600 --> 00:36:26.570 So f', remember, was (ln x - 1) / (ln x)^2. 00:36:26.570 --> 00:36:28.480 So the denominator is positive. 00:36:28.480 --> 00:36:32.650 So let's just take a look at the three ranges. 00:36:32.650 --> 00:36:37.330 So we have 0 < x < 1. 00:36:37.330 --> 00:36:40.310 And on that range, the logarithm is negative, 00:36:40.310 --> 00:36:45.230 so this is negative divided by positive, which is negative. 00:36:45.230 --> 00:36:47.220 That's decreasing, that's good. 00:36:47.220 --> 00:36:50.160 And in fact, that also works on the next range. 00:36:50.160 --> 00:36:55.357 1 < x < e, it's negative divided by positive. 00:36:55.357 --> 00:36:57.190 And the only reason why we skipped 1, again, 00:36:57.190 --> 00:36:58.432 is that it's undefined there. 00:36:58.432 --> 00:37:01.040 And there's something dramatic happening there. 00:37:01.040 --> 00:37:05.240 And then, at the last range, when x is bigger than e, 00:37:05.240 --> 00:37:07.770 that means the logarithm is already bigger than 1. 00:37:07.770 --> 00:37:09.145 So the numerator is now positive, 00:37:09.145 --> 00:37:13.650 and the denominator's still positive, so it's increasing. 00:37:13.650 --> 00:37:22.910 So we've just double-checked something that we already knew. 00:37:22.910 --> 00:37:26.540 Alright, so that's pretty much all 00:37:26.540 --> 00:37:29.200 there is to say about step 3. 00:37:29.200 --> 00:37:33.980 So this is checking the positivity and negativity. 00:37:33.980 --> 00:37:35.660 And now, step 4. 00:37:35.660 --> 00:37:38.480 There is one small point which I want to make before we go on. 00:37:38.480 --> 00:37:42.110 Which is that sometimes, you can't 00:37:42.110 --> 00:37:45.445 evaluate the function or its derivative particularly well. 00:37:45.445 --> 00:37:48.359 So sometimes you can't plot the points very well. 00:37:48.359 --> 00:37:50.150 And if you can't plot the points very well, 00:37:50.150 --> 00:37:52.440 then you might have to do 3 first, 00:37:52.440 --> 00:37:55.290 to figure out what's going on a little bit. 00:37:55.290 --> 00:37:59.150 You might have to skip. 00:37:59.150 --> 00:38:02.400 So now we're going to go on to the second derivative. 00:38:02.400 --> 00:38:07.180 But first, I want to use an algebraic trick 00:38:07.180 --> 00:38:08.470 to rearrange the terms. 00:38:08.470 --> 00:38:10.820 And I want to notice one more little point. 00:38:10.820 --> 00:38:16.320 Which I-- as I say, this is decoration for the graph. 00:38:16.320 --> 00:38:18.290 So I want to rewrite the formula. 00:38:18.290 --> 00:38:22.320 Maybe I'll do it right over here. 00:38:22.320 --> 00:38:27.270 Another way of writing this is 1/(ln x) - 1/(ln x)^2. 00:38:31.440 --> 00:38:35.240 So that's another way of writing the derivative. 00:38:35.240 --> 00:38:38.520 And that allows me to notice something 00:38:38.520 --> 00:38:40.830 that I missed, before. 00:38:40.830 --> 00:38:47.410 When I solved the equation ln x - 1 - this is equal to 0 here, 00:38:47.410 --> 00:38:48.590 this equation here. 00:38:48.590 --> 00:38:51.460 I missed a possibility. 00:38:51.460 --> 00:38:54.020 I missed the possibility that the denominator 00:38:54.020 --> 00:38:58.690 could be infinity. 00:38:58.690 --> 00:39:02.040 So actually, if the denominator's infinity, 00:39:02.040 --> 00:39:05.460 as you can see from the other expression there, 00:39:05.460 --> 00:39:09.000 it actually is true that the derivative is 0. 00:39:09.000 --> 00:39:16.710 So also when x = 0+, the slope is going to be 0. 00:39:16.710 --> 00:39:19.050 Let me just emphasize that again. 00:39:19.050 --> 00:39:23.500 If you evaluate using this other formula over here, 00:39:23.500 --> 00:39:28.410 this is 1/(ln(0+)) - 1/(ln(0+))^2. 00:39:31.540 --> 00:39:36.500 That's 1 over -infinity - minus 1 over infinity, if you like, 00:39:36.500 --> 00:39:37.310 squared. 00:39:37.310 --> 00:39:40.630 Anyway, it's 0. 00:39:40.630 --> 00:39:42.330 So this is 0. 00:39:42.330 --> 00:39:43.410 The slope is 0 there. 00:39:43.410 --> 00:39:46.640 That is a little piece of decoration on our graph. 00:39:46.640 --> 00:39:50.330 It's telling us, going back to our graph here, 00:39:50.330 --> 00:39:53.560 it's telling us this is coming in with slope horizontal. 00:39:53.560 --> 00:39:57.530 So we're starting out this way. 00:39:57.530 --> 00:40:01.013 That's just a little start here to the graph. 00:40:01.013 --> 00:40:02.620 It's a horizontal slope. 00:40:02.620 --> 00:40:07.940 So there really were two places where the slope was horizontal. 00:40:07.940 --> 00:40:11.470 Now, with the help of this second formula 00:40:11.470 --> 00:40:16.972 I can also differentiate a second time. 00:40:16.972 --> 00:40:19.430 So it's a little bit easier to do that if I differentiate 1 00:40:19.430 --> 00:40:28.050 over the log, that's -(ln x)^(-2) 1/x + 2 (ln x)^(-3) 00:40:28.050 --> 00:40:28.550 1/x. 00:40:34.270 --> 00:40:39.090 And that, if I put it over a common denominator, 00:40:39.090 --> 00:40:48.690 is x (ln x)^3 times, let's see here, 00:40:48.690 --> 00:40:55.340 I guess I'll have to take the 2 - ln x. 00:40:55.340 --> 00:40:57.260 So I've now rewritten the formula 00:40:57.260 --> 00:41:03.450 for the second derivative as a ratio. 00:41:03.450 --> 00:41:09.610 Now, to decide the sign, you see there are two places where 00:41:09.610 --> 00:41:11.910 the sign flips. 00:41:11.910 --> 00:41:16.030 The numerator crosses when the logarithm is 2, 00:41:16.030 --> 00:41:18.620 that's going to be when x = e^2. 00:41:18.620 --> 00:41:22.400 And the denominator flips when x = 1, 00:41:22.400 --> 00:41:28.020 that's when the log flips from positive to negative. 00:41:28.020 --> 00:41:32.280 So we have a couple of ranges here. 00:41:32.280 --> 00:41:37.580 So, first of all, we have the range from 0 to 1. 00:41:37.580 --> 00:41:42.270 And then we have the range from 1 to e^2. 00:41:42.270 --> 00:41:46.380 And then we have the range from e^2 all the way out 00:41:46.380 --> 00:41:49.780 to infinity. 00:41:49.780 --> 00:41:57.210 So between 0 and 1, the numerator 00:41:57.210 --> 00:41:59.630 is, well this is a negative number and this, 00:41:59.630 --> 00:42:01.300 so minus a negative number is positive, 00:42:01.300 --> 00:42:04.650 so the numerator is positive. 00:42:04.650 --> 00:42:08.030 And the denominator is negative, because the log is negative 00:42:08.030 --> 00:42:09.660 and it's taken to the third power. 00:42:09.660 --> 00:42:12.170 So this is a negative number, so it's positive 00:42:12.170 --> 00:42:15.190 divided by negative, which is less than 0. 00:42:15.190 --> 00:42:18.770 That means it's concave down. 00:42:18.770 --> 00:42:26.040 So this is a concave down part. 00:42:26.040 --> 00:42:28.230 And that's a good thing, because over here this 00:42:28.230 --> 00:42:29.120 was concave down. 00:42:29.120 --> 00:42:30.560 So there are no wiggles. 00:42:30.560 --> 00:42:34.260 It goes straight down, like this. 00:42:34.260 --> 00:42:41.590 And then the other two pieces are f'' is equal to, well 00:42:41.590 --> 00:42:43.260 it's going to switch here. 00:42:43.260 --> 00:42:44.856 The denominator becomes positive. 00:42:44.856 --> 00:42:48.190 So it's positive over positive. 00:42:48.190 --> 00:42:56.670 So this is concave up. 00:42:56.670 --> 00:42:58.410 And that's going over here. 00:42:58.410 --> 00:43:02.715 But notice that it's not the bottom where it turns around, 00:43:02.715 --> 00:43:07.740 it's somewhere else. 00:43:07.740 --> 00:43:09.930 So there's another transition here. 00:43:09.930 --> 00:43:12.090 This is e^2. 00:43:12.090 --> 00:43:15.090 This is e. 00:43:15.090 --> 00:43:20.440 So what happens at the end is, again, the sign flips again. 00:43:20.440 --> 00:43:26.580 Because the numerator, now, when x > e^2, becomes negative. 00:43:26.580 --> 00:43:29.965 And this is negative divided by positive, which is negative. 00:43:29.965 --> 00:43:35.630 And this is concave down. 00:43:35.630 --> 00:43:39.100 And so we didn't quite draw the graph right. 00:43:39.100 --> 00:43:40.990 There's an inflection point right here, 00:43:40.990 --> 00:43:45.670 which I'll label with an I. And it makes a turn 00:43:45.670 --> 00:43:47.340 the other way at that point. 00:43:47.340 --> 00:43:49.480 So there was a wiggle. 00:43:49.480 --> 00:43:51.210 There's the wiggle. 00:43:51.210 --> 00:43:53.710 Still going up, still going to infinity. 00:43:53.710 --> 00:43:57.040 But kind of the slope of a mountain, right? 00:43:57.040 --> 00:44:01.240 It's going the other way. 00:44:01.240 --> 00:44:03.510 This point happens to be (e^2, e^2 / 2). 00:44:09.350 --> 00:44:11.980 So that's as detailed as we'll ever get. 00:44:11.980 --> 00:44:14.760 And indeed, the next game is going 00:44:14.760 --> 00:44:18.730 to be avoid being-- is to avoid being this detailed. 00:44:18.730 --> 00:44:21.760 So let me introduce the next subject. 00:44:21.760 --> 00:44:48.310 Which is maxima and minima. 00:44:48.310 --> 00:45:04.160 OK, now, maxima and minima, maximum and minimum problems 00:45:04.160 --> 00:45:06.550 can be described graphically in the following ways. 00:45:06.550 --> 00:45:13.150 Suppose you have a function, right, here it is. 00:45:13.150 --> 00:45:14.420 OK? 00:45:14.420 --> 00:45:24.630 Now, find the maximum. 00:45:24.630 --> 00:45:30.740 And find the minimum. 00:45:30.740 --> 00:45:31.410 OK. 00:45:31.410 --> 00:45:38.880 So this problem is done. 00:45:38.880 --> 00:45:51.625 The point being, that it is easy to find max and the min 00:45:51.625 --> 00:45:58.670 with the sketch. 00:45:58.670 --> 00:46:00.330 It's very easy. 00:46:00.330 --> 00:46:05.130 The goal, the problem, is that the sketch is a lot of work. 00:46:05.130 --> 00:46:10.180 We just spent 20 minutes sketching something. 00:46:10.180 --> 00:46:12.637 We would not like to spend all that time 00:46:12.637 --> 00:46:14.970 every single time we want to find a maximum and minimum. 00:46:14.970 --> 00:46:19.576 So the goal is to do it with-- so our goal is 00:46:19.576 --> 00:46:25.170 to use shortcuts. 00:46:25.170 --> 00:46:30.650 And, indeed, as I said earlier, we certainly 00:46:30.650 --> 00:46:33.840 never want to use the second derivative if we can avoid it. 00:46:33.840 --> 00:46:36.450 And we don't want to decorate the graph 00:46:36.450 --> 00:46:38.550 and do all of these elaborate, subtle, 00:46:38.550 --> 00:46:40.900 things which make the graph look nicer and really, 00:46:40.900 --> 00:46:42.640 or aesthetically appropriate. 00:46:42.640 --> 00:46:46.700 But are totally unnecessary to see whether the graph is 00:46:46.700 --> 00:46:54.290 up or down. 00:46:54.290 --> 00:46:57.030 So essentially, this whole business 00:46:57.030 --> 00:47:00.030 is out, which is a good thing. 00:47:00.030 --> 00:47:03.640 And, unfortunately, those early parts 00:47:03.640 --> 00:47:06.170 are the parts that people tend to ignore. 00:47:06.170 --> 00:47:10.150 Which are typically, often, very important. 00:47:10.150 --> 00:47:22.940 So let me first tell you the main point here. 00:47:22.940 --> 00:47:32.760 So the key idea. 00:47:32.760 --> 00:47:39.650 Key to finding maximum. 00:47:39.650 --> 00:47:41.850 So the key point is, we only need 00:47:41.850 --> 00:48:00.130 to look at critical points. 00:48:00.130 --> 00:48:01.980 Well, that's actually what it seems 00:48:01.980 --> 00:48:05.490 like in many calculus classes. 00:48:05.490 --> 00:48:06.810 But that's not true. 00:48:06.810 --> 00:48:15.590 This is not the end of the sentence. 00:48:15.590 --> 00:48:35.320 And, end points, and points of discontinuity. 00:48:35.320 --> 00:48:37.980 So you must watch out for those. 00:48:37.980 --> 00:48:41.580 If you look at the example that I just drew here, 00:48:41.580 --> 00:48:43.680 which is the one that I carried out, 00:48:43.680 --> 00:48:50.120 you can see that there are actually five extreme points 00:48:50.120 --> 00:48:51.250 on this picture. 00:48:51.250 --> 00:48:52.750 So let's switch. 00:48:52.750 --> 00:48:58.050 So we'll take a look. 00:48:58.050 --> 00:49:04.840 There are five places where the max or the min might be. 00:49:04.840 --> 00:49:08.050 There's this important point. 00:49:08.050 --> 00:49:10.710 This is, as I say, the scaffolding of the function. 00:49:10.710 --> 00:49:13.650 There's this point, there down at minus infinity. 00:49:13.650 --> 00:49:18.370 There's this, there's this, and there's this. 00:49:18.370 --> 00:49:23.620 Only one out of five is a critical point. 00:49:23.620 --> 00:49:25.784 So there's more that you have to pay attention to 00:49:25.784 --> 00:49:26.450 on the function. 00:49:26.450 --> 00:49:28.960 And you always have to keep the schema, 00:49:28.960 --> 00:49:31.310 the picture of the function, in the back of your head. 00:49:31.310 --> 00:49:33.760 Even though this may be the most interesting point, 00:49:33.760 --> 00:49:36.520 and the one that you're going to be looking at. 00:49:36.520 --> 00:49:41.180 So we'll do a few examples of that next time.