1 00:00:00 --> 00:00:00,11 2 00:00:00,11 --> 00:00:02,49 The following content is provided under a Creative 3 00:00:02,49 --> 00:00:02,74 Commons license. 4 00:00:02,74 --> 00:00:06,57 Your support will help MIT OpenCourseWare continue to 5 00:00:06,57 --> 00:00:09,96 offer high quality educational resources for free. 6 00:00:09,96 --> 00:00:13,17 To make a donation or to view additional materials from 7 00:00:13,17 --> 00:00:15,91 hundreds of MIT courses visit MIT OpenCourseWare 8 00:00:15,91 --> 00:00:22,6 at ocw.mit.edu. 9 00:00:22,6 --> 00:00:27,38 Okay so I'd like to begin the second lecture by reminding 10 00:00:27,38 --> 00:00:30,77 you what we did last time. 11 00:00:30,77 --> 00:00:54,55 So last time, we defined the derivative as the slope 12 00:00:54,55 --> 00:01:04,26 of a tangent line. 13 00:01:04,26 --> 00:01:08,4 So that was our geometric point of view and we also did a 14 00:01:08,4 --> 00:01:10,47 couple of computations. 15 00:01:10,47 --> 00:01:14,5 We worked out that the derivative of 1 / 16 00:01:14,5 --> 00:01:20,1 x was -1 / x^2. 17 00:01:20,1 --> 00:01:27,1 And we also computed the derivative of x ^ nth power for 18 00:01:27,1 --> 00:01:32,31 n = 1, 2, etc., and that turned out to be x, I'm 19 00:01:32,31 --> 00:01:32,93 sorry, nx^(n-1). 20 00:01:32,93 --> 00:01:36,97 21 00:01:36,97 --> 00:01:46,53 So that's what we did last time, and today I want to 22 00:01:46,53 --> 00:01:53,43 finish up with other points of view on what a derivative is. 23 00:01:53,43 --> 00:01:57,35 So this is extremely important, it's almost the most important 24 00:01:57,35 --> 00:01:58,75 thing I'll be saying in the class. 25 00:01:58,75 --> 00:02:01,31 But you'll have to think about it again when you start over 26 00:02:01,31 --> 00:02:04,6 and start using calculus in the real world. 27 00:02:04,6 --> 00:02:14,71 So again we're talking about what is a derivative and this 28 00:02:14,71 --> 00:02:19,66 is just a continuation of last time. 29 00:02:19,66 --> 00:02:23,26 So, as I said last time, we talked about geometric 30 00:02:23,26 --> 00:02:28,81 interpretations, and today what we're gonna talk about is rate 31 00:02:28,81 --> 00:02:40 of change as an interpretation of the derivative. 32 00:02:40 --> 00:02:47,13 So remember we drew graphs of functions, y = f(x) and we 33 00:02:47,13 --> 00:02:51,855 kept track of the change in x and here the change 34 00:02:51,855 --> 00:02:56,14 in y, let's say. 35 00:02:56,14 --> 00:03:02,18 And then from this new point of view a rate of change, keeping 36 00:03:02,18 --> 00:03:05,65 track of the rate of change of x and the rate of change of y, 37 00:03:05,65 --> 00:03:08,71 it's the relative rate of change we're interested in, and 38 00:03:08,71 --> 00:03:14,7 that's delta y / delta x and that has another 39 00:03:14,7 --> 00:03:16,01 interpretation. 40 00:03:16,01 --> 00:03:21,65 This is the average change. 41 00:03:21,65 --> 00:03:26,88 Usually we would think of that, if x were measuring time and so 42 00:03:26,88 --> 00:03:32,63 the average and that's when this becomes a rate, and 43 00:03:32,63 --> 00:03:35,83 the average is over the time interval delta x. 44 00:03:35,83 --> 00:03:45,47 And then the limiting value is denoted dy/dx and so this one 45 00:03:45,47 --> 00:03:48,76 is the average rate of change and this one is the 46 00:03:48,76 --> 00:03:59,86 instantaneous rate. 47 00:03:59,86 --> 00:04:02,67 Okay, so that's the point of view that I'd like to discuss 48 00:04:02,67 --> 00:04:06,2 now and give you just a couple of examples. 49 00:04:06,2 --> 00:04:12,98 So, let's see. 50 00:04:12,98 --> 00:04:19,62 Well, first of all, maybe some examples from physics here. 51 00:04:19,62 --> 00:04:31,34 So q is usually the name for a charge, and then dq/dt is 52 00:04:31,34 --> 00:04:33,66 what's known as current. 53 00:04:33,66 --> 00:04:38,6 So that's one physical example. 54 00:04:38,6 --> 00:04:45,2 A second example, which is probably the most tangible one, 55 00:04:45,2 --> 00:04:51,74 is we could denote know the letter s by distance and 56 00:04:51,74 --> 00:04:58,52 then the rate of change is that what we call speed. 57 00:04:58,52 --> 00:05:03,85 So those are the two typical examples and I just want to 58 00:05:03,85 --> 00:05:08,35 illustrate the second example in a little bit more detail 59 00:05:08,35 --> 00:05:10,56 because I think it's important to have some visceral 60 00:05:10,56 --> 00:05:16,32 sense of this notion of instantaneous speed. 61 00:05:16,32 --> 00:05:22,57 And I get to use the example of this very building to do that. 62 00:05:22,57 --> 00:05:28,72 Probably you know, or maybe you don't, that on Halloween 63 00:05:28,72 --> 00:05:32,65 there's an event that takes place in this building or 64 00:05:32,65 --> 00:05:34,84 really from the top of this building which is called 65 00:05:34,84 --> 00:05:37 the pumpkin drop. 66 00:05:37 --> 00:05:44,45 So let's illustrates this idea of rate of change 67 00:05:44,45 --> 00:05:49,04 with the pumpkin drop. 68 00:05:49,04 --> 00:05:54,5 So what happens is this building, well let's see here's 69 00:05:54,5 --> 00:06:03,15 the building, and here's the dot, that's the beautiful grass 70 00:06:03,15 --> 00:06:07,47 out on this side of the building, and then there's some 71 00:06:07,47 --> 00:06:12,6 people up here and very small objects, well they're not that 72 00:06:12,6 --> 00:06:16,59 small when you're close to them, that get dumped 73 00:06:16,59 --> 00:06:19 over the side there. 74 00:06:19 --> 00:06:21,51 And they fall down. 75 00:06:21,51 --> 00:06:24,17 You know everything at MIT or a lot of things at MIT 76 00:06:24,17 --> 00:06:28,43 are physics experiments. 77 00:06:28,43 --> 00:06:29,44 That's the pumpkin drop. 78 00:06:29,44 --> 00:06:33,7 So roughly speaking, the building is about 300 feet 79 00:06:33,7 --> 00:06:39,57 high, we're down here on the first usable floor. 80 00:06:39,57 --> 00:06:44,61 And so we're going to use instead of 300 feet, just for 81 00:06:44,61 --> 00:06:50,22 convenience purposes we'll use 80 meters because that makes 82 00:06:50,22 --> 00:06:55,41 the numbers come out simply. 83 00:06:55,41 --> 00:07:05,52 So we have the height which starts out at 80 meters at time 84 00:07:05,52 --> 00:07:10,3 0 and then the acceleration due to gravity gives you this 85 00:07:10,3 --> 00:07:13,33 formula for h, this is the height. 86 00:07:13,33 --> 00:07:21,76 So at time t = 0, we're up at the top, h is 80 meters, 87 00:07:21,76 --> 00:07:24,58 the units here are meters. 88 00:07:24,58 --> 00:07:32,2 And at time t = 4 you notice, (5 * 4^2) is 80. 89 00:07:32,2 --> 00:07:34,24 I picked these numbers conveniently so that we're 90 00:07:34,24 --> 00:07:38,32 down at the bottom. 91 00:07:38,32 --> 00:07:45,77 Okay, so this notion of average change here, so the average 92 00:07:45,77 --> 00:07:53,08 change, or the average speed here, maybe we'll call it the 93 00:07:53,08 --> 00:08:02,29 average speed since that's, over this time that it takes 94 00:08:02,29 --> 00:08:06,48 for the pumpkin to drop is going to be the change 95 00:08:06,48 --> 00:08:10,17 in h / the change in t. 96 00:08:10,17 --> 00:08:18,35 Which starts out at, what does it start out as? 97 00:08:18,35 --> 00:08:21,87 It starts out as 80, right? 98 00:08:21,87 --> 00:08:23,93 And it ends at 0. 99 00:08:23,93 --> 00:08:26,52 So actually we have to do it backwards. 100 00:08:26,52 --> 00:08:33,29 We have to take 0 - 80 because the first value is the final 101 00:08:33,29 --> 00:08:37,39 position and the second value is the initial position. 102 00:08:37,39 --> 00:08:41,47 And that's divided by 4 - 0; times 4 seconds 103 00:08:41,47 --> 00:08:43,68 minus times 0 seconds. 104 00:08:43,68 --> 00:08:49,2 And so that of course is -20 meters per second. 105 00:08:49,2 --> 00:08:56,86 So the average speed of this guy is 20 meters a second. 106 00:08:56,86 --> 00:09:00,97 Now, so why did I pick this example? 107 00:09:00,97 --> 00:09:04,48 Because, of course, the average, although interesting, 108 00:09:04,48 --> 00:09:06,86 is not really what anybody cares about who actually 109 00:09:06,86 --> 00:09:08,67 goes to the event. 110 00:09:08,67 --> 00:09:12,37 All we really care about is the instantaneous speed when it 111 00:09:12,37 --> 00:09:22,95 hits the pavement and so that's can be calculated 112 00:09:22,95 --> 00:09:23,61 at the bottom. 113 00:09:23,61 --> 00:09:25,33 So what's the instantaneous speed? 114 00:09:25,33 --> 00:09:30,36 That's the derivative, or maybe to be consistent with the 115 00:09:30,36 --> 00:09:35,95 notation I've been using so far, that's d/dt of h. 116 00:09:35,95 --> 00:09:37,58 All right? 117 00:09:37,58 --> 00:09:39,09 So that's d/dt of h. 118 00:09:39,09 --> 00:09:42,02 Now remember we have formulas for these things. 119 00:09:42,02 --> 00:09:43,85 We can differentiate this function now. 120 00:09:43,85 --> 00:09:47,93 We did that yesterday. 121 00:09:47,93 --> 00:09:51,35 So we're gonna take the rate of change and if you take a look 122 00:09:51,35 --> 00:09:57,25 at it, it's just the rate of change of 80 is 0, minus 123 00:09:57,25 --> 00:10:02,86 the rate change for this -5t^2, that's minus 10t. 124 00:10:02,86 --> 00:10:09 So that's using the fact that d/dt of 80 is equal to 0 and 125 00:10:09 --> 00:10:12,35 d/dt of t^2 is equal to 2t. 126 00:10:12,35 --> 00:10:14,42 The special case... 127 00:10:14,42 --> 00:10:17,69 Well I'm cheating here, but there's a special 128 00:10:17,69 --> 00:10:18,5 case that's obvious. 129 00:10:18,5 --> 00:10:19,85 I didn't throw it in over here. 130 00:10:19,85 --> 00:10:23,75 The case n = 2 is that second case there. 131 00:10:23,75 --> 00:10:30,38 But the case n = 0 also works. 132 00:10:30,38 --> 00:10:31,62 Because that's constants. 133 00:10:31,62 --> 00:10:32,95 The derivative of a constant is 0. 134 00:10:32,95 --> 00:10:36,96 And then the factor n there's 0 and that's consistent. 135 00:10:36,96 --> 00:10:39,14 And actually if you look at the formula above it you'll see 136 00:10:39,14 --> 00:10:44,09 that it's the case of n = -1. 137 00:10:44,09 --> 00:10:49,82 So we'll get a larger pattern soon enough with the powers. 138 00:10:49,82 --> 00:10:50,45 Okay anyway. 139 00:10:50,45 --> 00:10:54,09 Back over here we have our rate of change and 140 00:10:54,09 --> 00:10:55,38 this is what it is. 141 00:10:55,38 --> 00:11:00,01 And at the bottom, at that point of impact, we have t 142 00:11:00,01 --> 00:11:07,75 = 4 and so h' which is the derivative is equal to 143 00:11:07,75 --> 00:11:12,86 -40 meters per second. 144 00:11:12,86 --> 00:11:19,07 So twice as fast as the average speed here, and if you need 145 00:11:19,07 --> 00:11:22,9 to convert that, that's about 90 miles an hour. 146 00:11:22,9 --> 00:11:29,45 Which is why the police are there at midnight on Halloween 147 00:11:29,45 --> 00:11:33,41 to make sure you're all safe and also why when you come you 148 00:11:33,41 --> 00:11:37,33 have to be prepared to clean up afterwards. 149 00:11:37,33 --> 00:11:40,26 So anyway that's what happens, it's 90 miles an hour. 150 00:11:40,26 --> 00:11:42,63 It's actually the buildings a little taller, there's air 151 00:11:42,63 --> 00:11:46,88 resistance and I'm sure you can do a much more thorough 152 00:11:46,88 --> 00:11:50,35 study of this example. 153 00:11:50,35 --> 00:11:54,3 All right so now I want to give you a couple of more examples 154 00:11:54,3 --> 00:11:58,81 because time and these kinds of parameters and variables are 155 00:11:58,81 --> 00:12:02,57 not the only ones that are important for calculus. 156 00:12:02,57 --> 00:12:06,3 If it were only this kind of physics that was involved, then 157 00:12:06,3 --> 00:12:08,54 this would be a much more specialized subject than it. 158 00:12:08,54 --> 00:12:13,34 Is And so I want to give you a couple of examples that don't 159 00:12:13,34 --> 00:12:16,57 involved time as a variable. 160 00:12:16,57 --> 00:12:21,72 So the third example I'll give here is. 161 00:12:21,72 --> 00:12:31,27 The letter t often denotes temperature, and then dt/dx 162 00:12:31,27 --> 00:12:38,83 would be what is known as the temperature gradient. 163 00:12:38,83 --> 00:12:43,94 Which we really care about a lot when we're predicting the 164 00:12:43,94 --> 00:12:46,29 weather because it's that temperature difference 165 00:12:46,29 --> 00:12:52,14 that causes air flows and causes things to change. 166 00:12:52,14 --> 00:12:59,16 And then there's another theme which is throughout the 167 00:12:59,16 --> 00:13:02,76 sciences and engineering which I'm going to talk about under 168 00:13:02,76 --> 00:13:15,6 the heading of sensitivity of measurements. 169 00:13:15,6 --> 00:13:18,47 So let me explain this. 170 00:13:18,47 --> 00:13:22,26 I don't want to belabor it because I just am doing this in 171 00:13:22,26 --> 00:13:26,38 order to introduce you to the ideas on your problem set which 172 00:13:26,38 --> 00:13:29,35 are the first case of this. 173 00:13:29,35 --> 00:13:35,26 So on problem set one you have an example which is based on a 174 00:13:35,26 --> 00:13:39,55 simplified model of GPS, sort of the Flat Earth Model. 175 00:13:39,55 --> 00:13:42,76 And in that situation, well, if the Earth is flat it's just 176 00:13:42,76 --> 00:13:45,49 a horizontal line like this. 177 00:13:45,49 --> 00:13:54,02 And then you have a satellite, which is over here, preferably 178 00:13:54,02 --> 00:14:03,06 above the earth, and the satellite or the system knows 179 00:14:03,06 --> 00:14:05,2 exactly where the point directly below the 180 00:14:05,2 --> 00:14:06,48 satellite is. 181 00:14:06,48 --> 00:14:12,17 So this point is treated as known. 182 00:14:12,17 --> 00:14:24,43 And I'm sitting here with my little GPS device and I 183 00:14:24,43 --> 00:14:26,44 want to know where I am. 184 00:14:26,44 --> 00:14:30,06 And the way I locate where I am is I communicate with this 185 00:14:30,06 --> 00:14:36,74 satellite by radio signals and I can measure this distance 186 00:14:36,74 --> 00:14:38,75 here which is called h. 187 00:14:38,75 --> 00:14:42,84 And then system will computer this horizontal 188 00:14:42,84 --> 00:14:47,07 distance which is L. 189 00:14:47,07 --> 00:14:58,91 So in other words what is measured, so h measured by 190 00:14:58,91 --> 00:15:04,91 radios, radio waves and a clock, or various clocks. 191 00:15:04,91 --> 00:15:13,56 And then L is deduced from h. 192 00:15:13,56 --> 00:15:17,63 And what's critical in all of these systems is that you 193 00:15:17,63 --> 00:15:20,32 don't know h exactly. 194 00:15:20,32 --> 00:15:26,33 There's an error in h which will denote delta h. 195 00:15:26,33 --> 00:15:31,04 There's some degree of uncertainty. 196 00:15:31,04 --> 00:15:35,55 The main uncertainty in GPS is from the ionosphere. 197 00:15:35,55 --> 00:15:38,45 But there are lots of corrections that are 198 00:15:38,45 --> 00:15:41,34 made of all kinds. 199 00:15:41,34 --> 00:15:43,65 And also if you're inside a building it's a 200 00:15:43,65 --> 00:15:44,79 problem to measure it. 201 00:15:44,79 --> 00:15:48,22 But it's an extremely important issue, as I'll 202 00:15:48,22 --> 00:15:49,73 explain in a second. 203 00:15:49,73 --> 00:16:01,44 So the idea is we then get at delta L is estimated by 204 00:16:01,44 --> 00:16:06 considering this ratio delta L/delta h which is going to be 205 00:16:06 --> 00:16:11,01 approximately the same as the derivative of L 206 00:16:11,01 --> 00:16:13,91 with respect to h. 207 00:16:13,91 --> 00:16:16,32 So this is the thing that's easy because of 208 00:16:16,32 --> 00:16:18,94 course it's calculus. 209 00:16:18,94 --> 00:16:22,85 Calculus is the easy part and that allows us to deduce 210 00:16:22,85 --> 00:16:28,6 something about the real world that's close by over here. 211 00:16:28,6 --> 00:16:31,98 So the reason why you should care about this quite a bit 212 00:16:31,98 --> 00:16:34,87 is that it's used all the time to land airplanes. 213 00:16:34,87 --> 00:16:38,22 So you really do care that they actually know to within a few 214 00:16:38,22 --> 00:16:42,77 feet or even closer where your plane is and how high 215 00:16:42,77 --> 00:16:48,57 up it is and so forth. 216 00:16:48,57 --> 00:16:48,65 All right. 217 00:16:48,65 --> 00:16:51,15 So that's it for the general introduction of what 218 00:16:51,15 --> 00:16:52,01 a derivative is. 219 00:16:52,01 --> 00:16:54,26 I'm sure you'll be getting used to this in a lot of different 220 00:16:54,26 --> 00:16:56,56 contexts throughout the course. 221 00:16:56,56 --> 00:17:04,51 And now we have to get back down to some rigorous details. 222 00:17:04,51 --> 00:17:09,72 Ok, everybody happy with what we've got so far? 223 00:17:09,72 --> 00:17:10,04 Yeah? 224 00:17:10,04 --> 00:17:13,4 Student: How did you get the equation for height? 225 00:17:13,4 --> 00:17:14,98 Professor: Ah good question question. 226 00:17:14,98 --> 00:17:18,56 The question was how did I get this equation for height? 227 00:17:18,56 --> 00:17:25,38 I just made it up because it's the formula from physics that 228 00:17:25,38 --> 00:17:30,06 you will learn when you take 8.01 and, in fact, it has to do 229 00:17:30,06 --> 00:17:35,87 with the fact that this is the speed if you differentiate 230 00:17:35,87 --> 00:17:39,86 another time you get acceleration and acceleration 231 00:17:39,86 --> 00:17:42,33 due to gravity is 10 meters per second. 232 00:17:42,33 --> 00:17:44,45 Which happens to be the second derivative of this. 233 00:17:44,45 --> 00:17:46,96 But anyway I just pulled it out of a hat from 234 00:17:46,96 --> 00:17:48,35 your physics class. 235 00:17:48,35 --> 00:17:55,51 So you can just say see 8.01 . 236 00:17:55,51 --> 00:18:02,84 All right, other questions? 237 00:18:02,84 --> 00:18:04,97 All right, so let's go on now. 238 00:18:04,97 --> 00:18:09,34 Now I have to be a little bit more systematic about limits. 239 00:18:09,34 --> 00:18:20,13 So let's do that now. 240 00:18:20,13 --> 00:18:30,37 So now what I'd like to talk about is limits and continuity. 241 00:18:30,37 --> 00:18:34,96 And this is a warm up for deriving all the rest of the 242 00:18:34,96 --> 00:18:38,19 formulas, all the rest of the formulas that I'm going 243 00:18:38,19 --> 00:18:41,6 to need to differentiate every function you know. 244 00:18:41,6 --> 00:18:44,99 Remember, that's our goal and we only have about a week left 245 00:18:44,99 --> 00:18:47,51 so we'd better get started. 246 00:18:47,51 --> 00:18:58,98 So first of all there is what I will call easy limits. 247 00:18:58,98 --> 00:19:00,65 So what's an easy limit? 248 00:19:00,65 --> 00:19:07,55 An easy limit is something like the limit as x goes to 4 of (x 249 00:19:07,55 --> 00:19:09,81 3 / x^2 250 00:19:09,81 --> 00:19:11,57 1). 251 00:19:11,57 --> 00:19:16,24 And with this kind of limit all I have to do to evaluate it is 252 00:19:16,24 --> 00:19:21,48 to plug in x = 4 because, so what I get here 253 00:19:21,48 --> 00:19:24,26 is 4 + 3 / (4^2 254 00:19:24,26 --> 00:19:27,9 1). 255 00:19:27,9 --> 00:19:31,56 And that's just 7 / 17. 256 00:19:31,56 --> 00:19:33,72 And that's the end of it. 257 00:19:33,72 --> 00:19:38,51 So those are the easy limits. 258 00:19:38,51 --> 00:19:43,07 The second kind of limit, well so this isn't the only second 259 00:19:43,07 --> 00:19:45,29 kind of limit but I just want to point this out, it's very 260 00:19:45,29 --> 00:19:55,68 important is that: derivatives are are always 261 00:19:55,68 --> 00:19:59,37 harder than this. 262 00:19:59,37 --> 00:20:03,23 You can't get away with nothing here. 263 00:20:03,23 --> 00:20:05,09 So, why is that? 264 00:20:05,09 --> 00:20:08,2 Well, when you take a derivative, you're taking the 265 00:20:08,2 --> 00:20:20,27 limit as x goes to x0 of f(x), well we'll write it all 266 00:20:20,27 --> 00:20:24,52 out in all its glory. 267 00:20:24,52 --> 00:20:28,79 Here's the formula for the derivative. 268 00:20:28,79 --> 00:20:39,11 Now notice that if you plug in x = x0, always gives 0 / 0. 269 00:20:39,11 --> 00:20:42,08 So it just basically never works. 270 00:20:42,08 --> 00:20:51,22 So we always are going to need some cancellation to make 271 00:20:51,22 --> 00:21:05,96 sense out of the limit. 272 00:21:05,96 --> 00:21:12,99 Now in order to make things a little easier for myself to 273 00:21:12,99 --> 00:21:16,85 explain what's going on with limits I need to introduce just 274 00:21:16,85 --> 00:21:18,66 one more piece of notation. 275 00:21:18,66 --> 00:21:21,3 What I'm gonna introduce here is what's known as a 276 00:21:21,3 --> 00:21:23,38 left-hand and a right limit. 277 00:21:23,38 --> 00:21:27,295 If I take the limit as x tends to x0 with a 278 00:21:27,295 --> 00:21:32,63 sign here of some function, this is what's known as 279 00:21:32,63 --> 00:21:42,28 the right hand limit. 280 00:21:42,28 --> 00:21:44,87 And I can display it visually. 281 00:21:44,87 --> 00:21:45,95 So what does this mean? 282 00:21:45,95 --> 00:21:49,15 It means practically the same thing as x tends to x0 except 283 00:21:49,15 --> 00:21:52,08 there is one more restriction which has to do with this 284 00:21:52,08 --> 00:21:55,37 sign, which is we're going from the plus side of x0. 285 00:21:55,37 --> 00:21:58,71 That means x is bigger than x0. 286 00:21:58,71 --> 00:22:01,77 And I say right-hand, so there should be a hyphen here, 287 00:22:01,77 --> 00:22:07,64 right-hand limit because on the number line, if x zero is over 288 00:22:07,64 --> 00:22:14,86 here the x is to the right. 289 00:22:14,86 --> 00:22:15,08 All right? 290 00:22:15,08 --> 00:22:16,75 So that's the right-hand limit. 291 00:22:16,75 --> 00:22:20,13 And then this being the left side of the board, I'll put on 292 00:22:20,13 --> 00:22:23,454 the right side of the board the left limit, just to 293 00:22:23,454 --> 00:22:24,56 make things confusing. 294 00:22:24,56 --> 00:22:30,52 So that one has the minus sign here. 295 00:22:30,52 --> 00:22:33,94 I'm just a little dyslexic and I hope you're not. 296 00:22:33,94 --> 00:22:38,2 So I may have gotten that wrong. 297 00:22:38,2 --> 00:22:41,51 So this is the left-hand limit, and I'll draw it. 298 00:22:41,51 --> 00:22:45,705 So of course that just means x goes to x0 but x is 299 00:22:45,705 --> 00:22:48,26 to the left of x0 . 300 00:22:48,26 --> 00:22:54,27 And again, on the number line, here's the x0 and the x is 301 00:22:54,27 --> 00:22:56,57 on the other side of it. 302 00:22:56,57 --> 00:22:59,66 Okay, so those two notations are going to help us to 303 00:22:59,66 --> 00:23:01,83 clarify a bunch of things. 304 00:23:01,83 --> 00:23:06,22 It's much more convenient to have this extra bit of 305 00:23:06,22 --> 00:23:10,37 description of limits than to just consider limits 306 00:23:10,37 --> 00:23:15,88 from both sides. 307 00:23:15,88 --> 00:23:25,98 Okay so I want to give an example of this. 308 00:23:25,98 --> 00:23:30,45 And also an example of how you're going to think about 309 00:23:30,45 --> 00:23:32,11 these sorts of problems. 310 00:23:32,11 --> 00:23:38,02 So I'll take a function which has two different definitions. 311 00:23:38,02 --> 00:23:44,085 Say it's x + 1, when x > 0 and -x 312 00:23:44,085 --> 00:23:47,57 2, when x < 0. 313 00:23:47,57 --> 00:23:51,28 So maybe put commas there. 314 00:23:51,28 --> 00:23:58,54 So when x > 0, it's x + 1. 315 00:23:58,54 --> 00:24:01,03 Now I can draw a picture of this. 316 00:24:01,03 --> 00:24:04,33 It's gonna be kind of little small because I'm gonna try to 317 00:24:04,33 --> 00:24:07,67 fit it down in here, but maybe I'll put the axis down below. 318 00:24:07,67 --> 00:24:14,5 So at height 1, I have to the right something of slope 1 319 00:24:14,5 --> 00:24:16,89 so it goes up like this. 320 00:24:16,89 --> 00:24:18,24 All right? 321 00:24:18,24 --> 00:24:24,12 And then to the left of 0 I have something which has slope 322 00:24:24,12 --> 00:24:30,72 -1, but it hits the axis at 2 so it's up here. 323 00:24:30,72 --> 00:24:33,94 So I had this sort of strange antennae figure 324 00:24:33,94 --> 00:24:35,15 here is my graph. 325 00:24:35,15 --> 00:24:43,71 Maybe I should draw these in another color to depict that. 326 00:24:43,71 --> 00:24:50,72 And then if I calculate these two limits here, what I see is 327 00:24:50,72 --> 00:25:00,04 that the limit as x goes to 0 from above of f(x), that's the 328 00:25:00,04 --> 00:25:07,99 same as the limit as x goes to 0 of the formula here, x + 1. 329 00:25:07,99 --> 00:25:10,43 Which turns out to be 1. 330 00:25:10,43 --> 00:25:15,36 And if I take the limit, so that's the left-hand limit. 331 00:25:15,36 --> 00:25:20,7 Sorry, I told you I was dyslexic. 332 00:25:20,7 --> 00:25:23,32 This is the right, so it's that right-hand. 333 00:25:23,32 --> 00:25:25,08 Here we go. 334 00:25:25,08 --> 00:25:32,2 So now I'm going from the left, and it's f(x) again, but now 335 00:25:32,2 --> 00:25:35,67 because I'm on that side the thing I need to plug is the 336 00:25:35,67 --> 00:25:43,54 other formula, -x + 2, and that's gonna give us 2. 337 00:25:43,54 --> 00:25:48,77 Now, notice that the left and right limits, and this is one 338 00:25:48,77 --> 00:25:51,65 little tiny subtlety and it's almost the only thing that I 339 00:25:51,65 --> 00:25:54,02 need you to really pay attention to a little bit right 340 00:25:54,02 --> 00:26:06,21 now, is that this, we did not need x = 0 value. 341 00:26:06,21 --> 00:26:11,86 In fact I never even told you what f(0) was here. 342 00:26:11,86 --> 00:26:14,65 If we stick it in we could stick it in. 343 00:26:14,65 --> 00:26:20,05 Okay let's say we stick it in on this side. 344 00:26:20,05 --> 00:26:22,97 Let's make it be that it's on this side. 345 00:26:22,97 --> 00:26:32,86 So that means that this point is in and this point is out. 346 00:26:32,86 --> 00:26:37,68 So that's a typical notation: this little open circle 347 00:26:37,68 --> 00:26:41,53 and this closed dot for when you include the. 348 00:26:41,53 --> 00:26:46,36 So in that case the value of f(x) happens to be the same as 349 00:26:46,36 --> 00:26:56,53 its right hand limit, namely the value is 1 here and not 2. 350 00:26:56,53 --> 00:27:01,14 Okay, so that's the first kind of example. 351 00:27:01,14 --> 00:27:06,61 Questions? 352 00:27:06,61 --> 00:27:13,5 Okay, so now our next job is to introduce the 353 00:27:13,5 --> 00:27:17,27 definition of continuity. 354 00:27:17,27 --> 00:27:20,08 So that was the other topic here. 355 00:27:20,08 --> 00:27:23,49 So we're going to define. 356 00:27:23,49 --> 00:27:39,3 So f is continuous at x0 means that the limit of f(x) as 357 00:27:39,3 --> 00:27:44,44 x tends to x0 = f(x0) . 358 00:27:44,44 --> 00:27:47,09 Right? 359 00:27:47,09 --> 00:27:52,36 So the reason why I spend all this time paying attention to 360 00:27:52,36 --> 00:27:55,04 the left and the right and so on and so forth and focusing 361 00:27:55,04 --> 00:27:57,76 is that I want you to pay attention for one moment to 362 00:27:57,76 --> 00:28:01,82 what the content of this definition is. 363 00:28:01,82 --> 00:28:12,64 What it's saying is the following: continuous at x0 364 00:28:12,64 --> 00:28:15,45 has various ingredients here. 365 00:28:15,45 --> 00:28:24,54 So the first one is that this limit exists. 366 00:28:24,54 --> 00:28:28,23 And what that means is that there's an honest limiting 367 00:28:28,23 --> 00:28:35,15 value both from the left and right. 368 00:28:35,15 --> 00:28:39,25 And they also have to be the same. 369 00:28:39,25 --> 00:28:41,98 All right, so that's what's going on here. 370 00:28:41,98 --> 00:28:50,38 And the second property is that f(x0) is defined. 371 00:28:50,38 --> 00:28:53,23 So I can't be in one of these situations where I haven't 372 00:28:53,23 --> 00:29:05,22 even specified what f(x0) is and they're equal. 373 00:29:05,22 --> 00:29:09,19 Okay, so that's the situation. 374 00:29:09,19 --> 00:29:13,56 Now again let me emphasize a tricky part of the 375 00:29:13,56 --> 00:29:15,56 definition of a limit. 376 00:29:15,56 --> 00:29:20,71 This side, the left-hand side is completely independent, is 377 00:29:20,71 --> 00:29:24,38 evaluated by a procedure which does not involve the 378 00:29:24,38 --> 00:29:25,07 right-hand side. 379 00:29:25,07 --> 00:29:26,9 These are separate things. 380 00:29:26,9 --> 00:29:32,68 This one is, to evaluate it, you always avoid 381 00:29:32,68 --> 00:29:34,31 the limit point. 382 00:29:34,31 --> 00:29:37,67 So that's if you like a paradox, because it's exactly 383 00:29:37,67 --> 00:29:41,68 the question: is it true that if you plug in x0 you get the 384 00:29:41,68 --> 00:29:44,3 same answer as if you move in the limit? 385 00:29:44,3 --> 00:29:46,27 That's the issue that we're considering here. 386 00:29:46,27 --> 00:29:49,01 We have to make that distinction in order to say 387 00:29:49,01 --> 00:29:55,27 that these are two, otherwise this is just tautalogical. 388 00:29:55,27 --> 00:29:56,63 It doesn't have any meaning. 389 00:29:56,63 --> 00:29:58,64 But in fact it does have a meaning because one thing is 390 00:29:58,64 --> 00:30:01,25 evaluated separately with reference to all the other 391 00:30:01,25 --> 00:30:05,85 points and the other is evaluated right at the 392 00:30:05,85 --> 00:30:06,87 point in question. 393 00:30:06,87 --> 00:30:12,5 And indeed what these things are, are exactly 394 00:30:12,5 --> 00:30:18,16 the easy limits. 395 00:30:18,16 --> 00:30:19,77 That's exactly what we're talking about here. 396 00:30:19,77 --> 00:30:24,15 They're the ones you can evaluate this way. 397 00:30:24,15 --> 00:30:25,64 So we have to make the distinction. 398 00:30:25,64 --> 00:30:27,77 And these other ones are gonna be the ones which we 399 00:30:27,77 --> 00:30:29,67 can't evaluate that way. 400 00:30:29,67 --> 00:30:32,47 So these are the nice ones and that's why we care about them 401 00:30:32,47 --> 00:30:36,47 why we have a whole definitions associated with them. 402 00:30:36,47 --> 00:30:38,7 All right? 403 00:30:38,7 --> 00:30:40,4 So now what's next? 404 00:30:40,4 --> 00:30:48,91 Well, I need to give you a a little tour, very brief tour, 405 00:30:48,91 --> 00:30:54,09 of the zoo of what are known as discontinuous functions. 406 00:30:54,09 --> 00:30:57,43 So sort of everything else that's not continuous. 407 00:30:57,43 --> 00:31:04,55 So, the first example here, let me just write it down here. 408 00:31:04,55 --> 00:31:13,67 It's jump discontinuities. 409 00:31:13,67 --> 00:31:15,3 So what would a jump discontinuity be? 410 00:31:15,3 --> 00:31:18,73 Well we've actually already seen it. 411 00:31:18,73 --> 00:31:21,89 The jump discontinuity is the example that 412 00:31:21,89 --> 00:31:23,23 we had right there. 413 00:31:23,23 --> 00:31:35,26 This is when the limit from the left and right exist, 414 00:31:35,26 --> 00:31:42,18 but are not equal. 415 00:31:42,18 --> 00:31:51,22 Okay, so that's as in the example. 416 00:31:51,22 --> 00:31:51,44 Right? 417 00:31:51,44 --> 00:31:53,82 In this example, the two limits, one of them was 418 00:31:53,82 --> 00:31:57,89 1 and of them was 2. 419 00:31:57,89 --> 00:32:02,15 So that's a jump discontinuity. 420 00:32:02,15 --> 00:32:09,87 And this kind of issue, of whether something is continuous 421 00:32:09,87 --> 00:32:21,31 or not, may seem a little bit technical but it is true that 422 00:32:21,31 --> 00:32:26,12 people have worried about it a lot. 423 00:32:26,12 --> 00:32:30,08 Bob Merton, who was a professor at MIT when he did his work for 424 00:32:30,08 --> 00:32:35,27 the Nobel prize in economics, was interested in this very 425 00:32:35,27 --> 00:32:39,86 issue of whether stock prices of various kinds are continuous 426 00:32:39,86 --> 00:32:42,54 from the left or right in a certain model. 427 00:32:42,54 --> 00:32:46,53 And that was a very serious issue in developing the model 428 00:32:46,53 --> 00:32:50,43 that priced things that our hedge funds use 429 00:32:50,43 --> 00:32:51,84 all the time now. 430 00:32:51,84 --> 00:32:57,63 So left and right can really mean something very different. 431 00:32:57,63 --> 00:33:01,96 In this case left is the past and right is the future and it 432 00:33:01,96 --> 00:33:04,29 makes a big difference whether things are continuous from the 433 00:33:04,29 --> 00:33:06,84 left or continuous from the right. 434 00:33:06,84 --> 00:33:09,37 Right, is it true that the point is here, here, somewhere 435 00:33:09,37 --> 00:33:11,72 in the middle, somewhere else. 436 00:33:11,72 --> 00:33:13,48 That's a serious issue. 437 00:33:13,48 --> 00:33:18,45 So the next example that I want to give you is a 438 00:33:18,45 --> 00:33:22,72 little bit more subtle. 439 00:33:22,72 --> 00:33:32,14 It's what's known as a removable discontinuity. 440 00:33:32,14 --> 00:33:39,65 And so what this means is that the limit from left 441 00:33:39,65 --> 00:33:46,19 and right are equal. 442 00:33:46,19 --> 00:33:48,78 So a picture of that would be, you have a function which is 443 00:33:48,78 --> 00:33:52,5 coming along like this and there's a hole maybe where, who 444 00:33:52,5 --> 00:33:55,05 knows either the function is undefined or maybe it's defined 445 00:33:55,05 --> 00:33:58,97 up here, and then it just continues on. 446 00:33:58,97 --> 00:33:59,25 All right? 447 00:33:59,25 --> 00:34:01,21 So the two limits are the same. 448 00:34:01,21 --> 00:34:05,01 And then of course the function in begging to be redefined 449 00:34:05,01 --> 00:34:07,37 so that we remove that hole. 450 00:34:07,37 --> 00:34:14,47 And that's why it's called a removable discontinuity. 451 00:34:14,47 --> 00:34:18,14 Now let me give you an example of this, or actually 452 00:34:18,14 --> 00:34:22,46 a couple of examples. 453 00:34:22,46 --> 00:34:28,47 So these are quite important examples which you will be 454 00:34:28,47 --> 00:34:34,02 working with in a few minutes. 455 00:34:34,02 --> 00:34:41,79 So the first is the function g(x), which is sin x / x, and 456 00:34:41,79 --> 00:34:50,52 the second will be the function h(x), which is 1 - cos x / x. 457 00:34:50,52 --> 00:35:00,29 So we have a problem at g(0) , g(0) is undefined. 458 00:35:00,29 --> 00:35:03,97 On the other hand it turns out this function has what's called 459 00:35:03,97 --> 00:35:05,71 a removable singularity. 460 00:35:05,71 --> 00:35:14,63 Namely the limit as x goes to 0 of sin x / x does exist. 461 00:35:14,63 --> 00:35:17,05 In fact it's equal to 1. 462 00:35:17,05 --> 00:35:19,96 So that's a very important limit that we will work out 463 00:35:19,96 --> 00:35:22,145 either at the end of this lecture or the beginning 464 00:35:22,145 --> 00:35:23,42 of next lecture. 465 00:35:23,42 --> 00:35:35,37 And similarly, the limit of 1 - cos x / x, as x goes to 0 is 0. 466 00:35:35,37 --> 00:35:38,44 Maybe I'll put that a little farther away 467 00:35:38,44 --> 00:35:40,36 so you can read it. 468 00:35:40,36 --> 00:35:45,43 Okay, so these are very useful facts that we're going 469 00:35:45,43 --> 00:35:47,8 to need later on. 470 00:35:47,8 --> 00:35:51,09 And what they say is that these things have removable 471 00:35:51,09 --> 00:36:04,6 singularities, sorry removable discontinuity at x = 0. 472 00:36:04,6 --> 00:36:13,03 All right so as I say, we'll get to that in a few minutes. 473 00:36:13,03 --> 00:36:16,59 Okay so are there any questions before I move on? 474 00:36:16,59 --> 00:36:16,9 Yeah? 475 00:36:16,9 --> 00:36:30,63 Student: [INAUDIBLE] 476 00:36:30,63 --> 00:36:38,3 Professor: The question is: why is this true? 477 00:36:38,3 --> 00:36:40,3 Is that what your question is? 478 00:36:40,3 --> 00:36:44,963 The answer is it's very, very unobvious, I haven't shown it 479 00:36:44,963 --> 00:36:49,67 to you yet, and if you were not surprised by it then that 480 00:36:49,67 --> 00:36:51,56 would be very strange indeed. 481 00:36:51,56 --> 00:36:53,39 So we haven't done it yet. 482 00:36:53,39 --> 00:36:55,99 You have to stay tuned until we do. 483 00:36:55,99 --> 00:36:57,21 Okay? 484 00:36:57,21 --> 00:36:59,25 We haven't shown it yet. 485 00:36:59,25 --> 00:37:02,49 And actually even this other statement, which maybe seems 486 00:37:02,49 --> 00:37:05,76 stranger still, is also not yet explained. 487 00:37:05,76 --> 00:37:09,33 Okay, so we are going to get there, as I said, either at 488 00:37:09,33 --> 00:37:15,41 the end of this lecture or at the beginning of next. 489 00:37:15,41 --> 00:37:22,56 Other questions? 490 00:37:22,56 --> 00:37:29,04 All right, so let me just continue my tour of the 491 00:37:29,04 --> 00:37:34 zoo of discontinuities. 492 00:37:34 --> 00:37:38,24 And, I guess, I want to illustrate something with the 493 00:37:38,24 --> 00:37:42,37 convenience of right and left hand limits so I'll save this 494 00:37:42,37 --> 00:37:52,18 board about right and left-hand limits. 495 00:37:52,18 --> 00:37:55,445 So a third type of discontinuity is what's known 496 00:37:55,445 --> 00:38:07,32 as an infinite discontinuity. 497 00:38:07,32 --> 00:38:11,95 And we've already encountered one of these. 498 00:38:11,95 --> 00:38:14,45 I'm going to draw them over here. 499 00:38:14,45 --> 00:38:19,37 Remember the function y is 1 / x. 500 00:38:19,37 --> 00:38:22,45 That's this function here. 501 00:38:22,45 --> 00:38:26,68 But now I'd like to draw also the other branch of the 502 00:38:26,68 --> 00:38:31,14 hyperbola down here and allow myself to consider 503 00:38:31,14 --> 00:38:32,32 negative values of x. 504 00:38:32,32 --> 00:38:35,91 So here's the graph of 1 / x. 505 00:38:35,91 --> 00:38:42,82 And the convenience here of distinguishing the left and the 506 00:38:42,82 --> 00:38:46,62 right hand limits is very important because here I 507 00:38:46,62 --> 00:38:49,39 can write down that the limit as x goes to 0 508 00:38:49,39 --> 00:38:51,8 of 1 / x. 509 00:38:51,8 --> 00:38:57,3 Well that's coming from the right and it's going up. 510 00:38:57,3 --> 00:39:00,58 So this limit is infinity. 511 00:39:00,58 --> 00:39:07,12 Whereas, the limit in the other direction, from the left, 512 00:39:07,12 --> 00:39:10,63 that one is going down. 513 00:39:10,63 --> 00:39:16,51 And so it's quite different, it's minus infinity. 514 00:39:16,51 --> 00:39:19,98 Now some people say that these limits are undefined but 515 00:39:19,98 --> 00:39:22,94 actually they're going in some very definite direction. 516 00:39:22,94 --> 00:39:26,12 So you should, whenever possible, specify what 517 00:39:26,12 --> 00:39:26,64 these limits are. 518 00:39:26,64 --> 00:39:30,97 On the other hand, the statement that the limit as 519 00:39:30,97 --> 00:39:37,25 x goes to 0 of 1 / x is infinity is simply wrong. 520 00:39:37,25 --> 00:39:40,34 Okay, it's not that people don't write this. 521 00:39:40,34 --> 00:39:41,68 It's just that it's wrong. 522 00:39:41,68 --> 00:39:43,47 It's not that they don't write it down. 523 00:39:43,47 --> 00:39:45 In fact you'll probably see it. 524 00:39:45 --> 00:39:47,04 It's because people are just thinking of the 525 00:39:47,04 --> 00:39:48,79 right hand branch. 526 00:39:48,79 --> 00:39:51,22 It's not that they're making a mistake necessarily, 527 00:39:51,22 --> 00:39:53,19 but anyway, it's sloppy. 528 00:39:53,19 --> 00:39:56,2 And there's some sloppiness that we'll endure and others 529 00:39:56,2 --> 00:39:57,08 that we'll try to avoid. 530 00:39:57,08 --> 00:40:00,12 So here, you want to say this, and it does make a difference. 531 00:40:00,12 --> 00:40:04,99 You know, plus infinity is an infinite number of dollars 532 00:40:04,99 --> 00:40:07,45 and minus infinity is and infinite amount of debt. 533 00:40:07,45 --> 00:40:08,98 They're actually different. 534 00:40:08,98 --> 00:40:09,89 They're not the same. 535 00:40:09,89 --> 00:40:13,79 So, you know, this is sloppy and this is actually 536 00:40:13,79 --> 00:40:15,54 more correct. 537 00:40:15,54 --> 00:40:18,38 Okay, so now in addition, I just want to point 538 00:40:18,38 --> 00:40:21,35 out one more thing. 539 00:40:21,35 --> 00:40:24,34 Remember, we calculated the derivative, and 540 00:40:24,34 --> 00:40:26,88 that was -1/x^2. 541 00:40:26,88 --> 00:40:31,71 But, I want to draw the graph of that and make a few 542 00:40:31,71 --> 00:40:32,57 comments about it. 543 00:40:32,57 --> 00:40:36,91 So I'm going to draw the graph directly underneath the 544 00:40:36,91 --> 00:40:38,82 graph of the function. 545 00:40:38,82 --> 00:40:41,29 And notice what this graphs is. 546 00:40:41,29 --> 00:40:48,53 It goes like this, it's always negative, and it points down. 547 00:40:48,53 --> 00:40:52,3 So now this may look a little strange, that the derivative 548 00:40:52,3 --> 00:40:57,1 of this thing is this guy, but that's because of 549 00:40:57,1 --> 00:40:58,63 something very important. 550 00:40:58,63 --> 00:41:01,03 And you should always remember this about derivatives. 551 00:41:01,03 --> 00:41:03,13 The derivative function looks nothing like the 552 00:41:03,13 --> 00:41:04,86 function, necessarily. 553 00:41:04,86 --> 00:41:07,78 So you should just forget about that as being an idea. 554 00:41:07,78 --> 00:41:10,33 Some people feel like if one thing goes down, the other 555 00:41:10,33 --> 00:41:11,47 thing has to go down. 556 00:41:11,47 --> 00:41:13,03 Just forget that intuition. 557 00:41:13,03 --> 00:41:14,16 It's wrong. 558 00:41:14,16 --> 00:41:20,17 What we're dealing with here, if you remember, is the slope. 559 00:41:20,17 --> 00:41:24,33 So if you have a slope here, that corresponds to just a 560 00:41:24,33 --> 00:41:29,22 place over here and as the slope gets a little bit less 561 00:41:29,22 --> 00:41:31,92 steep, that's why we're approaching the 562 00:41:31,92 --> 00:41:33,32 horizontal axis. 563 00:41:33,32 --> 00:41:36,48 The number is getting a little smaller as we close in. 564 00:41:36,48 --> 00:41:41,12 Now over here, the slope is also negative. 565 00:41:41,12 --> 00:41:43,32 It is going down and as we get down here it's getting 566 00:41:43,32 --> 00:41:44,58 more and more negative. 567 00:41:44,58 --> 00:41:48,27 As we go here the slope, this function is going up, but 568 00:41:48,27 --> 00:41:50,05 its slope is going down. 569 00:41:50,05 --> 00:41:55,79 All right, so the slope is down on both sides and the notation 570 00:41:55,79 --> 00:42:03,69 that we use for that is well suited to this left 571 00:42:03,69 --> 00:42:09,41 and right business. 572 00:42:09,41 --> 00:42:16,45 Namely, the limit as x goes to 0 of -1 / x^2, that's going to 573 00:42:16,45 --> 00:42:18,14 be equal to minus infinity. 574 00:42:18,14 --> 00:42:21,41 And that applies to x going to 0 575 00:42:21,41 --> 00:42:24,76 and x going to 0-. 576 00:42:24,76 --> 00:42:31,78 So both have this property. 577 00:42:31,78 --> 00:42:34,34 Finally let me just make one last comment about 578 00:42:34,34 --> 00:42:37,66 these two graphs. 579 00:42:37,66 --> 00:42:43 This function here is an odd function and when you take the 580 00:42:43 --> 00:42:45,33 derivative of an odd function you always get an 581 00:42:45,33 --> 00:42:50,74 even function. 582 00:42:50,74 --> 00:42:54,38 That's closely related to the fact that this 1 / x is an odd 583 00:42:54,38 --> 00:43:01,17 power and x^1 is an odd power and x^2 is an even power. 584 00:43:01,17 --> 00:43:05,62 So all of this your intuition should be reinforcing the fact 585 00:43:05,62 --> 00:43:11,07 that these pictures look right. 586 00:43:11,07 --> 00:43:16,74 Okay, now there's one last kind of discontinuity that I want to 587 00:43:16,74 --> 00:43:27,46 mention briefly, which I will call other ugly 588 00:43:27,46 --> 00:43:33,99 discontinuities. 589 00:43:33,99 --> 00:43:39,77 And there are lots and lots of them. 590 00:43:39,77 --> 00:43:44,42 So one example would be the function y = sin 1 591 00:43:44,42 --> 00:43:50,08 / x, as x goes to 0. 592 00:43:50,08 --> 00:43:59,29 And that looks a little bit like this. 593 00:43:59,29 --> 00:44:00,33 Back and forth and back and forth. 594 00:44:00,33 --> 00:44:06,17 It oscillates infinitely often as we tend to 0. 595 00:44:06,17 --> 00:44:19,26 There's no left or right limit in this case. 596 00:44:19,26 --> 00:44:25,33 So there is a very large quantity of things like that. 597 00:44:25,33 --> 00:44:29,35 Fortunately we're not gonna deal with them in this course. 598 00:44:29,35 --> 00:44:32,62 A lot of times in real life there are things that oscillate 599 00:44:32,62 --> 00:44:35,94 as time goes to infinity, but we're not going to worry 600 00:44:35,94 --> 00:44:40,18 about that right now. 601 00:44:40,18 --> 00:44:49,4 Okay, so that's our final mention of a discontinuity, and 602 00:44:49,4 --> 00:44:56,38 now I need to do just one more piece of groundwork for 603 00:44:56,38 --> 00:44:59,36 our formulas next time. 604 00:44:59,36 --> 00:45:09,13 Namely, I want to check for you one basic fact, 605 00:45:09,13 --> 00:45:10,28 one limiting tool. 606 00:45:10,28 --> 00:45:12,96 So this is going to be a theorem. 607 00:45:12,96 --> 00:45:17,72 Fortunately it's a very short theorem and has 608 00:45:17,72 --> 00:45:19,58 a very short proof. 609 00:45:19,58 --> 00:45:22,09 So the theorem goes under the name differentiable 610 00:45:22,09 --> 00:45:28,21 implies continuous. 611 00:45:28,21 --> 00:45:33,89 And what it says is the following: it says that if f is 612 00:45:33,89 --> 00:45:41,88 differentiable, in other words its the derivative exists at 613 00:45:41,88 --> 00:45:59,84 x0, then f is continuous at x0. 614 00:45:59,84 --> 00:46:02,47 So, we're gonna need this is as a tool, it's a key step in the 615 00:46:02,47 --> 00:46:05,75 product and quotient rules. 616 00:46:05,75 --> 00:46:12,38 So I'd like to prove it right now for you. 617 00:46:12,38 --> 00:46:16,27 So here is the proof. 618 00:46:16,27 --> 00:46:20,43 Fortunately the proof is just one line. 619 00:46:20,43 --> 00:46:24,12 So first of all, I want to write in just the right 620 00:46:24,12 --> 00:46:27,41 way what it is that we have to check. 621 00:46:27,41 --> 00:46:31,64 So what we have to check is that the limit, as x goes 622 00:46:31,64 --> 00:46:41,39 to x0 of f(x) - f(x0) = 0. 623 00:46:41,39 --> 00:46:42,68 So this is what we want to know. 624 00:46:42,68 --> 00:46:45,9 We don't know it yet, but we're trying to check 625 00:46:45,9 --> 00:46:47,65 whether this is true or not. 626 00:46:47,65 --> 00:46:50,735 So that's the same as the statement that the function is 627 00:46:50,735 --> 00:46:54,66 continuous because the limit of f(x) is supposed to be f(x0) 628 00:46:54,66 --> 00:46:59,69 and so this difference should have limit 0. 629 00:46:59,69 --> 00:47:03,97 And now, the way this is proved is just by rewriting 630 00:47:03,97 --> 00:47:09,72 it by multiplying and dividing by (x - x0). 631 00:47:09,72 --> 00:47:18,15 So I'll rewrite the limit as x goes to x0 of (f(x) - f(x0) 632 00:47:18,15 --> 00:47:25,57 / by x - x0) (x - x0). 633 00:47:25,57 --> 00:47:29,23 Okay, so I wrote down the same expression that I had here. 634 00:47:29,23 --> 00:47:31,67 This is just the same limit, but I multiplied and 635 00:47:31,67 --> 00:47:38,07 divided by (x - x0). 636 00:47:38,07 --> 00:47:45,25 And now when I take the limit what happens is the limit of 637 00:47:45,25 --> 00:47:48,83 the first factor is f'(x0). 638 00:47:48,83 --> 00:47:53,94 That's the thing we know exists by our assumption. 639 00:47:53,94 --> 00:48:00,37 And the limit of the second factor is 0 because the 640 00:48:00,37 --> 00:48:06,7 limit as x goes to x0 of (x - x0) is clearly 0 . 641 00:48:06,7 --> 00:48:09,21 So that's it. 642 00:48:09,21 --> 00:48:12,21 The answer is 0, which is what we wanted. 643 00:48:12,21 --> 00:48:14,98 So that's the proof. 644 00:48:14,98 --> 00:48:19,9 Now there's something exceedingly fishy looking about 645 00:48:19,9 --> 00:48:26,37 this proof and let me just point to it before we proceed. 646 00:48:26,37 --> 00:48:33,05 Namely, you're used in limits to setting x equal to 0. 647 00:48:33,05 --> 00:48:35,88 And this looks like we're multiplying, dividing by 0, 648 00:48:35,88 --> 00:48:40,82 exactly the thing which makes all proofs wrong in all kinds 649 00:48:40,82 --> 00:48:43,52 of algebraic situations and so on and so forth. 650 00:48:43,52 --> 00:48:45,78 You've been taught that that never works. 651 00:48:45,78 --> 00:48:47,75 Right? 652 00:48:47,75 --> 00:48:52,71 But somehow these limiting tricks have found a way around 653 00:48:52,71 --> 00:48:55,88 this and let me just make explicit what it is. 654 00:48:55,88 --> 00:49:03,5 In this limit we never are using x = x0. 655 00:49:03,5 --> 00:49:06,11 That's exactly the one value of x that we don't 656 00:49:06,11 --> 00:49:09,12 consider in this limit. 657 00:49:09,12 --> 00:49:11,91 That's how limits are cooked up. 658 00:49:11,91 --> 00:49:15,91 And that's sort of been the themes so far today, is that we 659 00:49:15,91 --> 00:49:18,96 don't have to consider that and so this multiplication and 660 00:49:18,96 --> 00:49:21,45 division by this number is legal. 661 00:49:21,45 --> 00:49:25,2 It may be small, this number, but it's always non-zero. 662 00:49:25,2 --> 00:49:28,21 So this really works, and it's really true, and we just 663 00:49:28,21 --> 00:49:32,56 checked that a differentiable function is continuous. 664 00:49:32,56 --> 00:49:38,93 So I'm gonna have to carry out these limits, which are very 665 00:49:38,93 --> 00:49:42,04 interesting 0 / 0 limits next time. 666 00:49:42,04 --> 00:49:46,03 But let's hang on for one second to see if there any 667 00:49:46,03 --> 00:49:47,96 questions before we stop. 668 00:49:47,96 --> 00:49:48,99 Yeah, there is a question. 669 00:49:48,99 --> 00:49:53,45 Student: [INAUDIBLE] 670 00:49:53,45 --> 00:50:00,97 Professor: Repeat this proof right here? 671 00:50:00,97 --> 00:50:02,83 Just say again. 672 00:50:02,83 --> 00:50:08,23 Student: [INAUDIBLE] 673 00:50:08,23 --> 00:50:13,21 Professor: Okay, so there are two steps to the proof and 674 00:50:13,21 --> 00:50:17,87 the step that you're asking about is the first step. 675 00:50:17,87 --> 00:50:18,58 Right? 676 00:50:18,58 --> 00:50:21,1 And what I'm saying is if you have a number, and you 677 00:50:21,1 --> 00:50:24,64 multiply it by 10 / 10 it's the same number. 678 00:50:24,64 --> 00:50:26,92 If you multiply it by 3 / 3 it's the same number. 679 00:50:26,92 --> 00:50:30,11 2 / 2, 1 / 1, and so on. 680 00:50:30,11 --> 00:50:32,48 So it is okay to change this to this, it's 681 00:50:32,48 --> 00:50:34,4 exactly the same thing. 682 00:50:34,4 --> 00:50:36,35 That's the first step. 683 00:50:36,35 --> 00:50:36,72 Yes? 684 00:50:36,72 --> 00:50:41,56 Student: [INAUDIBLE] 685 00:50:41,56 --> 00:50:45,01 Professor: Shhhh... 686 00:50:45,01 --> 00:50:52,51 The question was how does the proof, how does this line, yeah 687 00:50:52,51 --> 00:50:53,96 where the question mark is. 688 00:50:53,96 --> 00:50:56,5 So what I checked was that this number which is on the left 689 00:50:56,5 --> 00:51:02,58 hand side is equal to this very long complicated number which 690 00:51:02,58 --> 00:51:06,27 is equal to this number which is equal to this number. 691 00:51:06,27 --> 00:51:08,87 And so I've checked that this number is equal to 0 because 692 00:51:08,87 --> 00:51:12,12 the last thing is 0. 693 00:51:12,12 --> 00:51:16,12 This is equal to that is equal to that is equal to 0. 694 00:51:16,12 --> 00:51:17,6 And that's the proof. 695 00:51:17,6 --> 00:51:17,92 Yes? 696 00:51:17,92 --> 00:51:21,91 Student: [INAUDIBLE] 697 00:51:21,91 --> 00:51:30,58 Professor: So that's a different question. 698 00:51:30,58 --> 00:51:36,04 Okay, so the hypothesis of differentiability I use 699 00:51:36,04 --> 00:51:39,42 because this limit is equal to this number. 700 00:51:39,42 --> 00:51:40,52 That that limit exits. 701 00:51:40,52 --> 00:51:44,17 That's how I use the hypothesis of the theorem. 702 00:51:44,17 --> 00:51:47,84 The conclusion of the theorem is the same as this because 703 00:51:47,84 --> 00:51:52,65 being continuous is the same as limit as x goes to 704 00:51:52,65 --> 00:51:56,02 x0 of f(x) = f(x0). 705 00:51:56,02 --> 00:51:57,53 That's the definition of continuity. 706 00:51:57,53 --> 00:52:02,15 And I subtracted f(x0) from both sides to get this 707 00:52:02,15 --> 00:52:02,86 as being the same thing. 708 00:52:02,86 --> 00:52:06,46 So this claim is continuity and it's the same as 709 00:52:06,46 --> 00:52:10,35 this question here. 710 00:52:10,35 --> 00:52:11,18 Last question. 711 00:52:11,18 --> 00:52:16,97 Student: How did you get the 0 [INAUDIBLE] 712 00:52:16,97 --> 00:52:18,52 Professor: How did we get the 0 from this? 713 00:52:18,52 --> 00:52:20,88 So the claim that is being made, so the claim is why 714 00:52:20,88 --> 00:52:24,67 is this tending to that. 715 00:52:24,67 --> 00:52:27,41 So for example, I'm going to have to erase something 716 00:52:27,41 --> 00:52:28,73 to explain that. 717 00:52:28,73 --> 00:52:35,24 So the claim is that the limit as x goes to x0 of x - x0 = 0. 718 00:52:35,24 --> 00:52:37,16 That's what I'm claiming. 719 00:52:37,16 --> 00:52:39,49 Okay, does that answer your question? 720 00:52:39,49 --> 00:52:40,99 Okay. 721 00:52:40,99 --> 00:52:42,42 All right. 722 00:52:42,42 --> 00:52:45,37 Ask me other stuff after lecture. 723 00:52:45,37 --> 00:52:46,18