1 00:00:00,110 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:03,156 Commons license. 3 00:00:03,156 --> 00:00:05,440 Your support will help MIT OpenCourseWare 4 00:00:05,440 --> 00:00:09,960 continue to offer high quality educational resources for free. 5 00:00:09,960 --> 00:00:12,630 To make a donation or to view additional materials 6 00:00:12,630 --> 00:00:15,910 from hundreds of MIT courses visit MIT OpenCourseWare 7 00:00:15,910 --> 00:00:22,600 at ocw.mit.edu. 8 00:00:22,600 --> 00:00:27,490 Okay so I'd like to begin the second lecture by reminding you 9 00:00:27,490 --> 00:00:30,770 what we did last time. 10 00:00:30,770 --> 00:00:50,520 So last time, we defined the derivative 11 00:00:50,520 --> 00:01:04,260 as the slope of a tangent line. 12 00:01:04,260 --> 00:01:07,080 So that was our geometric point of view 13 00:01:07,080 --> 00:01:10,470 and we also did a couple of computations. 14 00:01:10,470 --> 00:01:18,290 We worked out that the derivative of 1 / x was -1 / 15 00:01:18,290 --> 00:01:20,100 x^2. 16 00:01:20,100 --> 00:01:24,870 And we also computed the derivative of x to the nth 17 00:01:24,870 --> 00:01:32,097 power for n = 1, 2, etc., and that turned out to be x, 18 00:01:32,097 --> 00:01:32,930 I'm sorry, nx^(n-1). 19 00:01:36,970 --> 00:01:46,180 So that's what we did last time, and today I 20 00:01:46,180 --> 00:01:51,580 want to finish up with other points of view 21 00:01:51,580 --> 00:01:53,430 on what a derivative is. 22 00:01:53,430 --> 00:01:56,250 So this is extremely important, it's 23 00:01:56,250 --> 00:01:58,750 almost the most important thing I'll be saying in the class. 24 00:01:58,750 --> 00:02:01,310 But you'll have to think about it again when you start over 25 00:02:01,310 --> 00:02:04,600 and start using calculus in the real world. 26 00:02:04,600 --> 00:02:14,030 So again we're talking about what is a derivative 27 00:02:14,030 --> 00:02:19,660 and this is just a continuation of last time. 28 00:02:19,660 --> 00:02:23,260 So, as I said last time, we talked about geometric 29 00:02:23,260 --> 00:02:28,130 interpretations, and today what we're gonna talk about 30 00:02:28,130 --> 00:02:34,310 is rate of change as an interpretation 31 00:02:34,310 --> 00:02:40,000 of the derivative. 32 00:02:40,000 --> 00:02:46,260 So remember we drew graphs of functions, y = f(x) 33 00:02:46,260 --> 00:02:53,190 and we kept track of the change in x and here the change in y, 34 00:02:53,190 --> 00:02:56,140 let's say. 35 00:02:56,140 --> 00:03:01,872 And then from this new point of view a rate of change, 36 00:03:01,872 --> 00:03:04,080 keeping track of the rate of change of x and the rate 37 00:03:04,080 --> 00:03:07,030 of change of y, it's the relative rate of change 38 00:03:07,030 --> 00:03:12,590 we're interested in, and that's delta y / delta x and that 39 00:03:12,590 --> 00:03:16,010 has another interpretation. 40 00:03:16,010 --> 00:03:21,650 This is the average change. 41 00:03:21,650 --> 00:03:26,880 Usually we would think of that, if x were measuring time and so 42 00:03:26,880 --> 00:03:31,350 the average and that's when this becomes a rate, 43 00:03:31,350 --> 00:03:35,830 and the average is over the time interval delta x. 44 00:03:35,830 --> 00:03:42,670 And then the limiting value is denoted dy/dx 45 00:03:42,670 --> 00:03:47,580 and so this one is the average rate of change 46 00:03:47,580 --> 00:03:59,860 and this one is the instantaneous rate. 47 00:03:59,860 --> 00:04:01,412 Okay, so that's the point of view 48 00:04:01,412 --> 00:04:03,120 that I'd like to discuss now and give you 49 00:04:03,120 --> 00:04:06,200 just a couple of examples. 50 00:04:06,200 --> 00:04:12,980 So, let's see. 51 00:04:12,980 --> 00:04:19,620 Well, first of all, maybe some examples from physics here. 52 00:04:19,620 --> 00:04:26,450 So q is usually the name for a charge, 53 00:04:26,450 --> 00:04:33,660 and then dq/dt is what's known as current. 54 00:04:33,660 --> 00:04:38,600 So that's one physical example. 55 00:04:38,600 --> 00:04:45,200 A second example, which is probably the most tangible one, 56 00:04:45,200 --> 00:04:51,500 is we could denote the letter s by distance 57 00:04:51,500 --> 00:04:58,520 and then the rate of change is what we call speed. 58 00:04:58,520 --> 00:05:02,110 So those are the two typical examples 59 00:05:02,110 --> 00:05:06,550 and I just want to illustrate the second example 60 00:05:06,550 --> 00:05:08,900 in a little bit more detail because I think 61 00:05:08,900 --> 00:05:12,690 it's important to have some visceral sense of this notion 62 00:05:12,690 --> 00:05:16,320 of instantaneous speed. 63 00:05:16,320 --> 00:05:22,570 And I get to use the example of this very building to do that. 64 00:05:22,570 --> 00:05:25,860 Probably you know, or maybe you don't, 65 00:05:25,860 --> 00:05:29,690 that on Halloween there's an event that 66 00:05:29,690 --> 00:05:33,042 takes place in this building or really 67 00:05:33,042 --> 00:05:34,500 from the top of this building which 68 00:05:34,500 --> 00:05:37,000 is called the pumpkin drop. 69 00:05:37,000 --> 00:05:44,450 So let's illustrates this idea of rate of change 70 00:05:44,450 --> 00:05:49,040 with the pumpkin drop. 71 00:05:49,040 --> 00:05:53,800 So what happens is, this building-- well 72 00:05:53,800 --> 00:06:01,070 let's see here's the building, and here's the dot, that's 73 00:06:01,070 --> 00:06:04,890 the beautiful grass out on this side of the building, 74 00:06:04,890 --> 00:06:09,430 and then there's some people up here 75 00:06:09,430 --> 00:06:12,310 and very small objects, well they're 76 00:06:12,310 --> 00:06:15,240 not that small when you're close to them, that 77 00:06:15,240 --> 00:06:19,000 get dumped over the side there. 78 00:06:19,000 --> 00:06:21,510 And they fall down. 79 00:06:21,510 --> 00:06:24,170 You know everything at MIT or a lot of things at MIT 80 00:06:24,170 --> 00:06:28,430 are physics experiments. 81 00:06:28,430 --> 00:06:29,440 That's the pumpkin drop. 82 00:06:29,440 --> 00:06:32,555 So roughly speaking, the building 83 00:06:32,555 --> 00:06:36,360 is about 300 feet high, we're down here 84 00:06:36,360 --> 00:06:39,570 on the first usable floor. 85 00:06:39,570 --> 00:06:44,130 And so we're going to use instead of 300 feet, 86 00:06:44,130 --> 00:06:46,330 just for convenience purposes we'll 87 00:06:46,330 --> 00:06:55,410 use 80 meters because that makes the numbers come out simply. 88 00:06:55,410 --> 00:07:04,380 So we have the height which starts out 89 00:07:04,380 --> 00:07:09,590 at 80 meters at time 0 and then the acceleration due to gravity 90 00:07:09,590 --> 00:07:13,330 gives you this formula for h, this is the height. 91 00:07:13,330 --> 00:07:21,760 So at time t = 0, we're up at the top, h is 80 meters, 92 00:07:21,760 --> 00:07:24,580 the units here are meters. 93 00:07:24,580 --> 00:07:32,200 And at time t = 4 you notice, 5 * 4^2 is 80. 94 00:07:32,200 --> 00:07:34,030 I picked these numbers conveniently so 95 00:07:34,030 --> 00:07:38,320 that we're down at the bottom. 96 00:07:38,320 --> 00:07:43,620 Okay, so this notion of average change here, 97 00:07:43,620 --> 00:07:51,380 so the average change, or the average speed here, 98 00:07:51,380 --> 00:07:55,636 maybe we'll call it the average speed, 99 00:07:55,636 --> 00:08:02,910 since that's-- over this time that it takes for the pumpkin 100 00:08:02,910 --> 00:08:07,385 to drop is going to be the change in h divided 101 00:08:07,385 --> 00:08:10,170 by the change in t. 102 00:08:10,170 --> 00:08:18,350 Which starts out at, what does it start out as? 103 00:08:18,350 --> 00:08:21,870 It starts out as 80, right? 104 00:08:21,870 --> 00:08:23,930 And it ends at 0. 105 00:08:23,930 --> 00:08:26,520 So actually we have to do it backwards. 106 00:08:26,520 --> 00:08:32,110 We have to take 0 - 80 because the first value is 107 00:08:32,110 --> 00:08:35,500 the final position and the second value 108 00:08:35,500 --> 00:08:37,390 is the initial position. 109 00:08:37,390 --> 00:08:41,470 And that's divided by 4 - 0; times 4 seconds 110 00:08:41,470 --> 00:08:43,680 minus times 0 seconds. 111 00:08:43,680 --> 00:08:49,200 And so that of course is -20 meters per second. 112 00:08:49,200 --> 00:08:56,860 So the average speed of this guy is 20 meters a second. 113 00:08:56,860 --> 00:09:00,970 Now, so why did I pick this example? 114 00:09:00,970 --> 00:09:04,480 Because, of course, the average, although interesting, 115 00:09:04,480 --> 00:09:06,540 is not really what anybody cares about who 116 00:09:06,540 --> 00:09:08,670 actually goes to the event. 117 00:09:08,670 --> 00:09:11,460 All we really care about is the instantaneous speed 118 00:09:11,460 --> 00:09:19,005 when it hits the pavement and so that's can 119 00:09:19,005 --> 00:09:23,610 be calculated at the bottom. 120 00:09:23,610 --> 00:09:25,330 So what's the instantaneous speed? 121 00:09:25,330 --> 00:09:29,530 That's the derivative, or maybe to be 122 00:09:29,530 --> 00:09:31,720 consistent with the notation I've been using so far, 123 00:09:31,720 --> 00:09:35,950 that's d/dt of h. 124 00:09:35,950 --> 00:09:37,580 All right? 125 00:09:37,580 --> 00:09:39,090 So that's d/dt of h. 126 00:09:39,090 --> 00:09:42,020 Now remember we have formulas for these things. 127 00:09:42,020 --> 00:09:43,850 We can differentiate this function now. 128 00:09:43,850 --> 00:09:47,930 We did that yesterday. 129 00:09:47,930 --> 00:09:51,350 So we're gonna take the rate of change and if you take a look 130 00:09:51,350 --> 00:09:56,790 at it, it's just the rate of change of 80 is 0, 131 00:09:56,790 --> 00:10:02,860 minus the rate change for this -5t^2, that's minus 10t. 132 00:10:02,860 --> 00:10:08,750 So that's using the fact that d/dt of 80 is equal to 0 133 00:10:08,750 --> 00:10:12,350 and d/dt of t^2 is equal to 2t. 134 00:10:12,350 --> 00:10:14,420 The special case... 135 00:10:14,420 --> 00:10:17,160 Well I'm cheating here, but there's 136 00:10:17,160 --> 00:10:18,500 a special case that's obvious. 137 00:10:18,500 --> 00:10:19,850 I didn't throw it in over here. 138 00:10:19,850 --> 00:10:23,750 The case n = 2 is that second case there. 139 00:10:23,750 --> 00:10:30,380 But the case n = 0 also works. 140 00:10:30,380 --> 00:10:31,534 Because that's constants. 141 00:10:31,534 --> 00:10:32,950 The derivative of a constant is 0. 142 00:10:32,950 --> 00:10:36,781 And then the factor n there's 0 and that's consistent. 143 00:10:36,781 --> 00:10:38,780 And actually if you look at the formula above it 144 00:10:38,780 --> 00:10:44,090 you'll see that it's the case of n = -1. 145 00:10:44,090 --> 00:10:49,820 So we'll get a larger pattern soon enough with the powers. 146 00:10:49,820 --> 00:10:50,450 Okay anyway. 147 00:10:50,450 --> 00:10:53,770 Back over here we have our rate of change 148 00:10:53,770 --> 00:10:55,380 and this is what it is. 149 00:10:55,380 --> 00:10:59,350 And at the bottom, at that point of impact, 150 00:10:59,350 --> 00:11:04,700 we have t = 4 and so h', which is the derivative, 151 00:11:04,700 --> 00:11:12,860 is equal to -40 meters per second. 152 00:11:12,860 --> 00:11:16,540 So twice as fast as the average speed here, 153 00:11:16,540 --> 00:11:22,900 and if you need to convert that, that's about 90 miles an hour. 154 00:11:22,900 --> 00:11:29,450 Which is why the police are there at midnight on Halloween 155 00:11:29,450 --> 00:11:33,310 to make sure you're all safe and also why when you come 156 00:11:33,310 --> 00:11:37,330 you have to be prepared to clean up afterwards. 157 00:11:37,330 --> 00:11:40,260 So anyway that's what happens, it's 90 miles an hour. 158 00:11:40,260 --> 00:11:42,320 It's actually the buildings a little taller, 159 00:11:42,320 --> 00:11:45,010 there's air resistance and I'm sure you 160 00:11:45,010 --> 00:11:50,350 can do a much more thorough study of this example. 161 00:11:50,350 --> 00:11:54,300 All right so now I want to give you a couple of more examples 162 00:11:54,300 --> 00:11:58,700 because time and these kinds of parameters and variables 163 00:11:58,700 --> 00:12:02,570 are not the only ones that are important for calculus. 164 00:12:02,570 --> 00:12:05,610 If it were only this kind of physics that was involved, 165 00:12:05,610 --> 00:12:09,710 then this would be a much more specialized subject than it is. 166 00:12:09,710 --> 00:12:13,340 And so I want to give you a couple of examples that don't 167 00:12:13,340 --> 00:12:16,570 involve time as a variable. 168 00:12:16,570 --> 00:12:20,175 So the third example I'll give here 169 00:12:20,175 --> 00:12:27,260 is-- The letter T often denotes temperature, 170 00:12:27,260 --> 00:12:35,660 and then dT/dx would be what is known as the temperature 171 00:12:35,660 --> 00:12:38,830 gradient. 172 00:12:38,830 --> 00:12:43,310 Which we really care about a lot when 173 00:12:43,310 --> 00:12:45,820 we're predicting the weather because it's that temperature 174 00:12:45,820 --> 00:12:52,140 difference that causes air flows and causes things to change. 175 00:12:52,140 --> 00:12:54,410 And then there's another theme which 176 00:12:54,410 --> 00:13:01,110 is throughout the sciences and engineering which 177 00:13:01,110 --> 00:13:07,700 I'm going to talk about under the heading of sensitivity 178 00:13:07,700 --> 00:13:15,600 of measurements. 179 00:13:15,600 --> 00:13:18,470 So let me explain this. 180 00:13:18,470 --> 00:13:21,530 I don't want to belabor it because I just 181 00:13:21,530 --> 00:13:23,810 am doing this in order to introduce you 182 00:13:23,810 --> 00:13:26,380 to the ideas on your problem set which 183 00:13:26,380 --> 00:13:29,350 are the first case of this. 184 00:13:29,350 --> 00:13:33,290 So on problem set one you have an example 185 00:13:33,290 --> 00:13:37,360 which is based on a simplified model of GPS, 186 00:13:37,360 --> 00:13:39,550 sort of the Flat Earth Model. 187 00:13:39,550 --> 00:13:42,400 And in that situation, well, if the Earth is flat 188 00:13:42,400 --> 00:13:45,490 it's just a horizontal line like this. 189 00:13:45,490 --> 00:13:54,020 And then you have a satellite, which is over here, preferably 190 00:13:54,020 --> 00:14:02,440 above the earth, and the satellite or the system 191 00:14:02,440 --> 00:14:05,810 knows exactly where the point directly below the satellite 192 00:14:05,810 --> 00:14:06,480 is. 193 00:14:06,480 --> 00:14:12,170 So this point is treated as known. 194 00:14:12,170 --> 00:14:22,160 And I'm sitting here with my little GPS device 195 00:14:22,160 --> 00:14:26,440 and I want to know where I am. 196 00:14:26,440 --> 00:14:28,880 And the way I locate where I am is 197 00:14:28,880 --> 00:14:34,750 I communicate with this satellite by radio signals 198 00:14:34,750 --> 00:14:38,750 and I can measure this distance here which is called h. 199 00:14:38,750 --> 00:14:45,060 And then system will compute this horizontal distance which 200 00:14:45,060 --> 00:14:53,360 is L. So in other words what is measured, 201 00:14:53,360 --> 00:15:03,020 so h measured by radios, radio waves and a clock, 202 00:15:03,020 --> 00:15:04,910 or various clocks. 203 00:15:04,910 --> 00:15:13,560 And then L is deduced from h. 204 00:15:13,560 --> 00:15:16,730 And what's critical in all of these systems 205 00:15:16,730 --> 00:15:20,320 is that you don't know h exactly. 206 00:15:20,320 --> 00:15:26,330 There's an error in h which will denote delta h. 207 00:15:26,330 --> 00:15:31,040 There's some degree of uncertainty. 208 00:15:31,040 --> 00:15:35,550 The main uncertainty in GPS is from the ionosphere. 209 00:15:35,550 --> 00:15:38,280 But there are lots of corrections 210 00:15:38,280 --> 00:15:41,340 that are made of all kinds. 211 00:15:41,340 --> 00:15:43,570 And also if you're inside a building 212 00:15:43,570 --> 00:15:44,790 it's a problem to measure it. 213 00:15:44,790 --> 00:15:47,970 But it's an extremely important issue, 214 00:15:47,970 --> 00:15:49,730 as I'll explain in a second. 215 00:15:49,730 --> 00:15:54,040 So the idea is we then get at delta 216 00:15:54,040 --> 00:16:04,246 L is estimated by considering this ratio delta L/delta 217 00:16:04,246 --> 00:16:07,060 h which is going to be approximately 218 00:16:07,060 --> 00:16:13,910 the same as the derivative of L with respect to h. 219 00:16:13,910 --> 00:16:17,815 So this is the thing that's easy because of course it's 220 00:16:17,815 --> 00:16:18,940 calculus. 221 00:16:18,940 --> 00:16:21,960 Calculus is the easy part and that 222 00:16:21,960 --> 00:16:25,330 allows us to deduce something about the real world that's 223 00:16:25,330 --> 00:16:28,600 close by over here. 224 00:16:28,600 --> 00:16:31,980 So the reason why you should care about this quite a bit 225 00:16:31,980 --> 00:16:34,870 is that it's used all the time to land airplanes. 226 00:16:34,870 --> 00:16:36,680 So you really do care that they actually 227 00:16:36,680 --> 00:16:42,386 know to within a few feet or even closer where your plane is 228 00:16:42,386 --> 00:16:48,150 and how high up it is and so forth. 229 00:16:48,150 --> 00:16:48,650 All right. 230 00:16:48,650 --> 00:16:50,850 So that's it for the general introduction 231 00:16:50,850 --> 00:16:52,010 of what a derivative is. 232 00:16:52,010 --> 00:16:53,670 I'm sure you'll be getting used to this 233 00:16:53,670 --> 00:16:56,560 in a lot of different contexts throughout the course. 234 00:16:56,560 --> 00:17:04,510 And now we have to get back down to some rigorous details. 235 00:17:04,510 --> 00:17:09,540 Okay, everybody happy with what we've got so far? 236 00:17:09,540 --> 00:17:10,040 Yeah? 237 00:17:10,040 --> 00:17:13,400 Student: How did you get the equation for height? 238 00:17:13,400 --> 00:17:14,980 Professor: Ah good question. 239 00:17:14,980 --> 00:17:18,560 The question was how did I get this equation for height? 240 00:17:18,560 --> 00:17:24,970 I just made it up because it's the formula from physics 241 00:17:24,970 --> 00:17:29,592 that you will learn when you take 8.01 and, in fact, 242 00:17:29,592 --> 00:17:35,060 it has to do with the fact that this is the speed if you 243 00:17:35,060 --> 00:17:37,340 differentiate another time you get 244 00:17:37,340 --> 00:17:40,550 acceleration and acceleration due to gravity 245 00:17:40,550 --> 00:17:42,330 is 10 meters per second. 246 00:17:42,330 --> 00:17:44,450 Which happens to be the second derivative of this. 247 00:17:44,450 --> 00:17:47,710 But anyway I just pulled it out of a hat from your physics 248 00:17:47,710 --> 00:17:48,350 class. 249 00:17:48,350 --> 00:17:55,510 So you can just say see 8.01 . 250 00:17:55,510 --> 00:18:02,840 All right, other questions? 251 00:18:02,840 --> 00:18:04,970 All right, so let's go on now. 252 00:18:04,970 --> 00:18:09,340 Now I have to be a little bit more systematic about limits. 253 00:18:09,340 --> 00:18:20,130 So let's do that now. 254 00:18:20,130 --> 00:18:30,370 So now what I'd like to talk about is limits and continuity. 255 00:18:30,370 --> 00:18:34,200 And this is a warm up for deriving 256 00:18:34,200 --> 00:18:37,900 all the rest of the formulas, all the rest of the formulas 257 00:18:37,900 --> 00:18:40,430 that I'm going to need to differentiate 258 00:18:40,430 --> 00:18:41,600 every function you know. 259 00:18:41,600 --> 00:18:44,120 Remember, that's our goal and we only have about a week 260 00:18:44,120 --> 00:18:47,510 left so we'd better get started. 261 00:18:47,510 --> 00:18:58,980 So first of all there is what I will call easy limits. 262 00:18:58,980 --> 00:19:00,650 So what's an easy limit? 263 00:19:00,650 --> 00:19:07,285 An easy limit is something like the limit as x goes to 4 of x 264 00:19:07,285 --> 00:19:11,570 plus 3 over x^2 + 1. 265 00:19:11,570 --> 00:19:16,240 And with this kind of limit all I have to do to evaluate it is 266 00:19:16,240 --> 00:19:23,770 to plug in x = 4 because, so what I get here is 4 + 3 267 00:19:23,770 --> 00:19:27,900 divided by 4^2 + 1. 268 00:19:27,900 --> 00:19:31,560 And that's just 7 / 17. 269 00:19:31,560 --> 00:19:33,720 And that's the end of it. 270 00:19:33,720 --> 00:19:38,510 So those are the easy limits. 271 00:19:38,510 --> 00:19:42,669 The second kind of limit - well so this isn't the only 272 00:19:42,669 --> 00:19:44,960 second kind of limit but I just want to point this out, 273 00:19:44,960 --> 00:19:55,680 it's very important - is that: derivatives are are always 274 00:19:55,680 --> 00:19:59,370 harder than this. 275 00:19:59,370 --> 00:20:03,230 You can't get away with nothing here. 276 00:20:03,230 --> 00:20:05,090 So, why is that? 277 00:20:05,090 --> 00:20:07,620 Well, when you take a derivative, 278 00:20:07,620 --> 00:20:13,160 you're taking the limit as x goes to x_0 of f(x), 279 00:20:13,160 --> 00:20:24,520 well we'll write it all out in all its glory. 280 00:20:24,520 --> 00:20:28,790 Here's the formula for the derivative. 281 00:20:28,790 --> 00:20:39,110 Now notice that if you plug in x = x:0, always gives 0 / 0. 282 00:20:39,110 --> 00:20:42,080 So it just basically never works. 283 00:20:42,080 --> 00:20:50,940 So we always are going to need some cancellation 284 00:20:50,940 --> 00:21:05,960 to make sense out of the limit. 285 00:21:05,960 --> 00:21:12,570 Now in order to make things a little easier for myself 286 00:21:12,570 --> 00:21:15,700 to explain what's going on with limits 287 00:21:15,700 --> 00:21:18,660 I need to introduce just one more piece of notation. 288 00:21:18,660 --> 00:21:20,490 What I'm gonna introduce here is what's 289 00:21:20,490 --> 00:21:23,380 known as a left-hand and a right limit. 290 00:21:23,380 --> 00:21:29,500 If I take the limit as x tends to x_0 with a plus sign here 291 00:21:29,500 --> 00:21:42,280 of some function, this is what's known as the right-hand limit. 292 00:21:42,280 --> 00:21:44,870 And I can display it visually. 293 00:21:44,870 --> 00:21:45,950 So what does this mean? 294 00:21:45,950 --> 00:21:47,530 It means practically the same thing 295 00:21:47,530 --> 00:21:51,160 as x tends to x_0 except there is one more restriction which 296 00:21:51,160 --> 00:21:53,630 has to do with this plus sign, which is we're going 297 00:21:53,630 --> 00:21:55,370 from the plus side of x_0. 298 00:21:55,370 --> 00:21:58,710 That means x is bigger than x_0. 299 00:21:58,710 --> 00:22:01,770 And I say right-hand, so there should be a hyphen here, 300 00:22:01,770 --> 00:22:06,600 right-hand limit because on the number line, 301 00:22:06,600 --> 00:22:14,580 if x_0 is over here the x is to the right. 302 00:22:14,580 --> 00:22:15,080 All right? 303 00:22:15,080 --> 00:22:16,750 So that's the right-hand limit. 304 00:22:16,750 --> 00:22:19,550 And then this being the left side of the board, 305 00:22:19,550 --> 00:22:22,716 I'll put on the right side of the board the left limit, 306 00:22:22,716 --> 00:22:24,560 just to make things confusing. 307 00:22:24,560 --> 00:22:30,520 So that one has the minus sign here. 308 00:22:30,520 --> 00:22:33,940 I'm just a little dyslexic and I hope you're not. 309 00:22:33,940 --> 00:22:38,200 So I may have gotten that wrong. 310 00:22:38,200 --> 00:22:41,510 So this is the left-hand limit, and I'll draw it. 311 00:22:41,510 --> 00:22:45,705 So of course that just means x goes to x_0 but x is 312 00:22:45,705 --> 00:22:48,260 to the left of x_0 . 313 00:22:48,260 --> 00:22:52,290 And again, on the number line, here's the x_0 314 00:22:52,290 --> 00:22:56,570 and the x is on the other side of it. 315 00:22:56,570 --> 00:22:58,970 Okay, so those two notations are going 316 00:22:58,970 --> 00:23:01,830 to help us to clarify a bunch of things. 317 00:23:01,830 --> 00:23:04,580 It's much more convenient to have 318 00:23:04,580 --> 00:23:08,520 this extra bit of description of limits 319 00:23:08,520 --> 00:23:15,880 than to just consider limits from both sides. 320 00:23:15,880 --> 00:23:25,980 Okay so I want to give an example of this. 321 00:23:25,980 --> 00:23:29,310 And also an example of how you're going to 322 00:23:29,310 --> 00:23:32,110 think about these sorts of problems. 323 00:23:32,110 --> 00:23:38,020 So I'll take a function which has two different definitions. 324 00:23:38,020 --> 00:23:47,570 Say it's x + 1, when x > 0 and -x + 2, when x < 0. 325 00:23:47,570 --> 00:23:51,280 So maybe put commas there. 326 00:23:51,280 --> 00:23:58,540 So when x > 0, it's x + 1. 327 00:23:58,540 --> 00:24:01,030 Now I can draw a picture of this. 328 00:24:01,030 --> 00:24:02,955 It's gonna be kind of a little small 329 00:24:02,955 --> 00:24:04,830 because I'm gonna try to fit it down in here, 330 00:24:04,830 --> 00:24:07,670 but maybe I'll put the axis down below. 331 00:24:07,670 --> 00:24:13,990 So at height 1, I have to the right something of slope 332 00:24:13,990 --> 00:24:16,890 1 so it goes up like this. 333 00:24:16,890 --> 00:24:18,240 All right? 334 00:24:18,240 --> 00:24:26,500 And then to the left of 0 I have something which has slope -1, 335 00:24:26,500 --> 00:24:30,720 but it hits the axis at 2 so it's up here. 336 00:24:30,720 --> 00:24:34,175 So I had this sort of strange antenna figure here, 337 00:24:34,175 --> 00:24:35,150 which is my graph. 338 00:24:35,150 --> 00:24:43,710 Maybe I should draw these in another color to depict that. 339 00:24:43,710 --> 00:24:47,780 And then if I calculate these two limits here, 340 00:24:47,780 --> 00:24:54,670 what I see is that the limit as x 341 00:24:54,670 --> 00:25:00,860 goes to 0 from above of f(x), that's the same as the limit 342 00:25:00,860 --> 00:25:07,990 as x goes to 0 of the formula here, x + 1. 343 00:25:07,990 --> 00:25:10,430 Which turns out to be 1. 344 00:25:10,430 --> 00:25:15,360 And if I take the limit, so that's the left-hand limit. 345 00:25:15,360 --> 00:25:20,700 Sorry, I told you I was dyslexic. 346 00:25:20,700 --> 00:25:23,320 This is the right, so it's that right-hand. 347 00:25:23,320 --> 00:25:25,080 Here we go. 348 00:25:25,080 --> 00:25:31,530 So now I'm going from the left, and it's f(x) again, 349 00:25:31,530 --> 00:25:35,180 but now because I'm on that side the thing I need to plug 350 00:25:35,180 --> 00:25:43,540 is the other formula, -x + 2, and that's gonna give us 2. 351 00:25:43,540 --> 00:25:48,310 Now, notice that the left and right limits, and this 352 00:25:48,310 --> 00:25:51,470 is one little tiny subtlety and it's almost the only thing 353 00:25:51,470 --> 00:25:53,770 that I need you to really pay attention to a little bit 354 00:25:53,770 --> 00:26:06,210 right now, is that this, we did not need x = 0 value. 355 00:26:06,210 --> 00:26:11,860 In fact I never even told you what f(0) was here. 356 00:26:11,860 --> 00:26:14,650 If we stick it in we could stick it in. 357 00:26:14,650 --> 00:26:20,050 Okay let's say we stick it in on this side. 358 00:26:20,050 --> 00:26:22,970 Let's make it be that it's on this side. 359 00:26:22,970 --> 00:26:32,860 So that means that this point is in and this point is out. 360 00:26:32,860 --> 00:26:37,680 So that's a typical notation: this little open circle 361 00:26:37,680 --> 00:26:41,530 and this closed dot for when you include the. 362 00:26:41,530 --> 00:26:44,830 So in that case the value of f(x) 363 00:26:44,830 --> 00:26:48,360 happens to be the same as its right-hand limit, 364 00:26:48,360 --> 00:26:56,530 namely the value is 1 here and not 2. 365 00:26:56,530 --> 00:27:01,140 Okay, so that's the first kind of example. 366 00:27:01,140 --> 00:27:06,610 Questions? 367 00:27:06,610 --> 00:27:13,420 Okay, so now our next job is to introduce 368 00:27:13,420 --> 00:27:17,270 the definition of continuity. 369 00:27:17,270 --> 00:27:20,080 So that was the other topic here. 370 00:27:20,080 --> 00:27:23,490 So we're going to define. 371 00:27:23,490 --> 00:27:39,515 So f is continuous at x_0 means that the limit of f(x) as x 372 00:27:39,515 --> 00:27:44,440 tends to x_0 is equal to f(x_0) . 373 00:27:44,440 --> 00:27:47,090 Right? 374 00:27:47,090 --> 00:27:51,750 So the reason why I spend all this time paying 375 00:27:51,750 --> 00:27:54,540 attention to the left and the right and so on and so forth 376 00:27:54,540 --> 00:27:57,340 and focusing is that I want you to pay attention for one moment 377 00:27:57,340 --> 00:28:01,820 to what the content of this definition is. 378 00:28:01,820 --> 00:28:12,640 What it's saying is the following: continuous at x_0 379 00:28:12,640 --> 00:28:15,450 has various ingredients here. 380 00:28:15,450 --> 00:28:24,540 So the first one is that this limit exists. 381 00:28:24,540 --> 00:28:27,080 And what that means is that there's 382 00:28:27,080 --> 00:28:35,150 an honest limiting value both from the left and right. 383 00:28:35,150 --> 00:28:39,250 And they also have to be the same. 384 00:28:39,250 --> 00:28:41,980 All right, so that's what's going on here. 385 00:28:41,980 --> 00:28:50,380 And the second property is that f(x_0) is defined. 386 00:28:50,380 --> 00:28:52,100 So I can't be in one of these situations 387 00:28:52,100 --> 00:28:54,770 where I haven't even specified what 388 00:28:54,770 --> 00:29:05,220 f(x_0) is and they're equal. 389 00:29:05,220 --> 00:29:09,190 Okay, so that's the situation. 390 00:29:09,190 --> 00:29:13,310 Now again let me emphasize a tricky part 391 00:29:13,310 --> 00:29:15,560 of the definition of a limit. 392 00:29:15,560 --> 00:29:20,320 This side, the left-hand side is completely independent, 393 00:29:20,320 --> 00:29:23,790 is evaluated by a procedure which does not 394 00:29:23,790 --> 00:29:25,070 involve the right-hand side. 395 00:29:25,070 --> 00:29:26,900 These are separate things. 396 00:29:26,900 --> 00:29:34,310 This one is, to evaluate it, you always avoid the limit point. 397 00:29:34,310 --> 00:29:37,670 So that's if you like a paradox, because it's exactly 398 00:29:37,670 --> 00:29:41,290 the question: is it true that if you plug in x_0 399 00:29:41,290 --> 00:29:44,300 you get the same answer as if you move in the limit? 400 00:29:44,300 --> 00:29:46,270 That's the issue that we're considering here. 401 00:29:46,270 --> 00:29:48,270 We have to make that distinction in order 402 00:29:48,270 --> 00:29:50,880 to say that these are two, otherwise 403 00:29:50,880 --> 00:29:55,270 this is just tautological. 404 00:29:55,270 --> 00:29:56,630 It doesn't have any meaning. 405 00:29:56,630 --> 00:29:58,046 But in fact it does have a meaning 406 00:29:58,046 --> 00:30:00,620 because one thing is evaluated separately with reference 407 00:30:00,620 --> 00:30:03,850 to all the other points and the other 408 00:30:03,850 --> 00:30:06,870 is evaluated right at the point in question. 409 00:30:06,870 --> 00:30:11,090 And indeed what these things are, 410 00:30:11,090 --> 00:30:17,896 are exactly the easy limits. 411 00:30:17,896 --> 00:30:19,770 That's exactly what we're talking about here. 412 00:30:19,770 --> 00:30:24,150 They're the ones you can evaluate this way. 413 00:30:24,150 --> 00:30:25,640 So we have to make the distinction. 414 00:30:25,640 --> 00:30:27,685 And these other ones are gonna be the ones which 415 00:30:27,685 --> 00:30:29,670 we can't evaluate that way. 416 00:30:29,670 --> 00:30:31,540 So these are the nice ones and that's 417 00:30:31,540 --> 00:30:33,830 why we care about them, why we have a whole definition 418 00:30:33,830 --> 00:30:36,470 associated with them. 419 00:30:36,470 --> 00:30:38,700 All right? 420 00:30:38,700 --> 00:30:40,400 So now what's next? 421 00:30:40,400 --> 00:30:48,910 Well, I need to give you a a little tour, very brief tour, 422 00:30:48,910 --> 00:30:54,090 of the zoo of what are known as discontinuous functions. 423 00:30:54,090 --> 00:30:57,430 So sort of everything else that's not continuous. 424 00:30:57,430 --> 00:31:04,550 So, the first example here, let me just write it down here. 425 00:31:04,550 --> 00:31:13,670 It's jump discontinuities. 426 00:31:13,670 --> 00:31:15,300 So what would a jump discontinuity be? 427 00:31:15,300 --> 00:31:18,730 Well we've actually already seen it. 428 00:31:18,730 --> 00:31:21,790 The jump discontinuity is the example 429 00:31:21,790 --> 00:31:23,230 that we had right there. 430 00:31:23,230 --> 00:31:32,490 This is when the limit from the left and right 431 00:31:32,490 --> 00:31:42,180 exist, but are not equal. 432 00:31:42,180 --> 00:31:50,940 Okay, so that's as in the example. 433 00:31:50,940 --> 00:31:51,440 Right? 434 00:31:51,440 --> 00:31:53,680 In this example, the two limits, one of them 435 00:31:53,680 --> 00:31:57,890 was 1 and of them was 2. 436 00:31:57,890 --> 00:32:02,150 So that's a jump discontinuity. 437 00:32:02,150 --> 00:32:09,310 And this kind of issue, of whether something 438 00:32:09,310 --> 00:32:14,940 is continuous or not, may seem a little bit technical 439 00:32:14,940 --> 00:32:26,120 but it is true that people have worried about it a lot. 440 00:32:26,120 --> 00:32:28,820 Bob Merton, who was a professor at MIT when 441 00:32:28,820 --> 00:32:33,410 he did his work for the Nobel prize in economics, 442 00:32:33,410 --> 00:32:36,180 was interested in this very issue 443 00:32:36,180 --> 00:32:39,320 of whether stock prices of various kinds 444 00:32:39,320 --> 00:32:42,540 are continuous from the left or right in a certain model. 445 00:32:42,540 --> 00:32:44,580 And that was a very serious issue 446 00:32:44,580 --> 00:32:49,150 in developing the model that priced things 447 00:32:49,150 --> 00:32:51,840 that our hedge funds use all the time now. 448 00:32:51,840 --> 00:32:57,630 So left and right can really mean something very different. 449 00:32:57,630 --> 00:33:01,507 In this case left is the past and right is the future 450 00:33:01,507 --> 00:33:03,340 and it makes a big difference whether things 451 00:33:03,340 --> 00:33:06,840 are continuous from the left or continuous from the right. 452 00:33:06,840 --> 00:33:09,120 Right, is it true that the point is here, 453 00:33:09,120 --> 00:33:11,720 here, somewhere in the middle, somewhere else. 454 00:33:11,720 --> 00:33:13,480 That's a serious issue. 455 00:33:13,480 --> 00:33:18,210 So the next example that I want to give you 456 00:33:18,210 --> 00:33:22,720 is a little bit more subtle. 457 00:33:22,720 --> 00:33:32,140 It's what's known as a removable discontinuity. 458 00:33:32,140 --> 00:33:43,010 And so what this means is that the limit from left and right 459 00:33:43,010 --> 00:33:46,190 are equal. 460 00:33:46,190 --> 00:33:47,980 So a picture of that would be, you 461 00:33:47,980 --> 00:33:50,480 have a function which is coming along like this 462 00:33:50,480 --> 00:33:52,820 and there's a hole maybe where, who knows 463 00:33:52,820 --> 00:33:56,270 either the function is undefined or maybe it's defined up here, 464 00:33:56,270 --> 00:33:58,751 and then it just continues on. 465 00:33:58,751 --> 00:33:59,250 All right? 466 00:33:59,250 --> 00:34:01,210 So the two limits are the same. 467 00:34:01,210 --> 00:34:05,010 And then of course the function is begging to be redefined 468 00:34:05,010 --> 00:34:07,370 so that we remove that hole. 469 00:34:07,370 --> 00:34:14,470 And that's why it's called a removable discontinuity. 470 00:34:14,470 --> 00:34:17,710 Now let me give you an example of this, 471 00:34:17,710 --> 00:34:22,460 or actually a couple of examples. 472 00:34:22,460 --> 00:34:28,130 So these are quite important examples 473 00:34:28,130 --> 00:34:34,020 which you will be working with in a few minutes. 474 00:34:34,020 --> 00:34:41,660 So the first is the function g(x), which is sin x / x, 475 00:34:41,660 --> 00:34:45,260 and the second will be the function h(x), which is 1 - 476 00:34:45,260 --> 00:34:50,520 cos x over x. 477 00:34:50,520 --> 00:35:00,290 So we have a problem at g(0), g(0) is undefined. 478 00:35:00,290 --> 00:35:03,760 On the other hand it turns out this function has what's 479 00:35:03,760 --> 00:35:05,710 called a removable singularity. 480 00:35:05,710 --> 00:35:14,630 Namely the limit as x goes to 0 of sin x / x does exist. 481 00:35:14,630 --> 00:35:17,050 In fact it's equal to 1. 482 00:35:17,050 --> 00:35:20,430 So that's a very important limit that we will work out either 483 00:35:20,430 --> 00:35:23,420 at the end of this lecture or the beginning of next lecture. 484 00:35:23,420 --> 00:35:30,940 And similarly, the limit of 1 - cos x 485 00:35:30,940 --> 00:35:35,370 divided by x, as x goes to 0, is 0. 486 00:35:35,370 --> 00:35:38,051 Maybe I'll put that a little farther 487 00:35:38,051 --> 00:35:40,360 away so you can read it. 488 00:35:40,360 --> 00:35:44,940 Okay, so these are very useful facts 489 00:35:44,940 --> 00:35:47,800 that we're going to need later on. 490 00:35:47,800 --> 00:35:50,460 And what they say is that these things have 491 00:35:50,460 --> 00:35:58,520 removable singularities, sorry removable discontinuity at x 492 00:35:58,520 --> 00:36:04,600 = 0. 493 00:36:04,600 --> 00:36:13,030 All right so as I say, we'll get to that in a few minutes. 494 00:36:13,030 --> 00:36:16,400 Okay so are there any questions before I move on? 495 00:36:16,400 --> 00:36:16,900 Yeah? 496 00:36:16,900 --> 00:36:30,630 Student: [INAUDIBLE] 497 00:36:30,630 --> 00:36:38,300 Professor: The question is: why is this true? 498 00:36:38,300 --> 00:36:40,300 Is that what your question is? 499 00:36:40,300 --> 00:36:44,070 The answer is it's very, very unobvious, 500 00:36:44,070 --> 00:36:48,360 I haven't shown it to you yet, and if you were not 501 00:36:48,360 --> 00:36:51,560 surprised by it then that would be very strange indeed. 502 00:36:51,560 --> 00:36:53,390 So we haven't done it yet. 503 00:36:53,390 --> 00:36:55,990 You have to stay tuned until we do. 504 00:36:55,990 --> 00:36:57,210 Okay? 505 00:36:57,210 --> 00:36:59,250 We haven't shown it yet. 506 00:36:59,250 --> 00:37:01,320 And actually even this other statement, 507 00:37:01,320 --> 00:37:03,600 which maybe seems stranger still, 508 00:37:03,600 --> 00:37:05,760 is also not yet explained. 509 00:37:05,760 --> 00:37:08,865 Okay, so we are going to get there, as I said, 510 00:37:08,865 --> 00:37:10,240 either at the end of this lecture 511 00:37:10,240 --> 00:37:15,410 or at the beginning of next. 512 00:37:15,410 --> 00:37:22,560 Other questions? 513 00:37:22,560 --> 00:37:28,180 All right, so let me just continue my tour 514 00:37:28,180 --> 00:37:34,000 of the zoo of discontinuities. 515 00:37:34,000 --> 00:37:37,050 And, I guess, I want to illustrate something 516 00:37:37,050 --> 00:37:41,440 with the convenience of right and left hand limits 517 00:37:41,440 --> 00:37:52,180 so I'll save this board about right and left-hand limits. 518 00:37:52,180 --> 00:37:54,970 So a third type of discontinuity is 519 00:37:54,970 --> 00:38:07,320 what's known as an infinite discontinuity. 520 00:38:07,320 --> 00:38:11,950 And we've already encountered one of these. 521 00:38:11,950 --> 00:38:14,450 I'm going to draw them over here. 522 00:38:14,450 --> 00:38:19,370 Remember the function y is 1 / x. 523 00:38:19,370 --> 00:38:22,450 That's this function here. 524 00:38:22,450 --> 00:38:25,500 But now I'd like to draw also the other branch 525 00:38:25,500 --> 00:38:31,140 of the hyperbola down here and allow myself to consider 526 00:38:31,140 --> 00:38:32,320 negative values of x. 527 00:38:32,320 --> 00:38:35,910 So here's the graph of 1 / x. 528 00:38:35,910 --> 00:38:42,640 And the convenience here of distinguishing the left 529 00:38:42,640 --> 00:38:46,620 and the right hand limits is very important because here I 530 00:38:46,620 --> 00:38:51,800 can write down that the limit as x goes to 0+ of 1 / x. 531 00:38:51,800 --> 00:38:57,300 Well that's coming from the right and it's going up. 532 00:38:57,300 --> 00:39:00,580 So this limit is infinity. 533 00:39:00,580 --> 00:39:05,380 Whereas, the limit in the other direction, 534 00:39:05,380 --> 00:39:10,630 from the left, that one is going down. 535 00:39:10,630 --> 00:39:16,510 And so it's quite different, it's minus infinity. 536 00:39:16,510 --> 00:39:19,860 Now some people say that these limits are undefined 537 00:39:19,860 --> 00:39:22,940 but actually they're going in some very definite direction. 538 00:39:22,940 --> 00:39:24,950 So you should, whenever possible, 539 00:39:24,950 --> 00:39:26,640 specify what these limits are. 540 00:39:26,640 --> 00:39:30,860 On the other hand, the statement that the limit 541 00:39:30,860 --> 00:39:37,250 as x goes to 0 of 1 / x is infinity is simply wrong. 542 00:39:37,250 --> 00:39:40,340 Okay, it's not that people don't write this. 543 00:39:40,340 --> 00:39:41,680 It's just that it's wrong. 544 00:39:41,680 --> 00:39:43,470 It's not that they don't write it down. 545 00:39:43,470 --> 00:39:45,000 In fact you'll probably see it. 546 00:39:45,000 --> 00:39:48,055 It's because people are just thinking of the right hand 547 00:39:48,055 --> 00:39:48,790 branch. 548 00:39:48,790 --> 00:39:51,220 It's not that they're making a mistake necessarily, 549 00:39:51,220 --> 00:39:53,116 but anyway, it's sloppy. 550 00:39:53,116 --> 00:39:54,990 And there's some sloppiness that we'll endure 551 00:39:54,990 --> 00:39:57,080 and others that we'll try to avoid. 552 00:39:57,080 --> 00:40:00,120 So here, you want to say this, and it does make a difference. 553 00:40:00,120 --> 00:40:04,990 You know, plus infinity is an infinite number of dollars 554 00:40:04,990 --> 00:40:07,450 and minus infinity is and infinite amount of debt. 555 00:40:07,450 --> 00:40:08,980 They're actually different. 556 00:40:08,980 --> 00:40:09,890 They're not the same. 557 00:40:09,890 --> 00:40:15,540 So, you know, this is sloppy and this is actually more correct. 558 00:40:15,540 --> 00:40:17,885 Okay, so now in addition, I just want 559 00:40:17,885 --> 00:40:21,350 to point out one more thing. 560 00:40:21,350 --> 00:40:24,210 Remember, we calculated the derivative, 561 00:40:24,210 --> 00:40:26,880 and that was -1/x^2. 562 00:40:26,880 --> 00:40:31,196 But, I want to draw the graph of that 563 00:40:31,196 --> 00:40:32,570 and make a few comments about it. 564 00:40:32,570 --> 00:40:34,420 So I'm going to draw the graph directly 565 00:40:34,420 --> 00:40:38,820 underneath the graph of the function. 566 00:40:38,820 --> 00:40:41,290 And notice what this graphs is. 567 00:40:41,290 --> 00:40:48,530 It goes like this, it's always negative, and it points down. 568 00:40:48,530 --> 00:40:51,480 So now this may look a little strange, 569 00:40:51,480 --> 00:40:55,080 that the derivative of this thing is this guy, 570 00:40:55,080 --> 00:40:58,630 but that's because of something very important. 571 00:40:58,630 --> 00:41:01,030 And you should always remember this about derivatives. 572 00:41:01,030 --> 00:41:03,995 The derivative function looks nothing like the function, 573 00:41:03,995 --> 00:41:04,860 necessarily. 574 00:41:04,860 --> 00:41:07,780 So you should just forget about that as being an idea. 575 00:41:07,780 --> 00:41:10,040 Some people feel like if one thing goes down, 576 00:41:10,040 --> 00:41:11,470 the other thing has to go down. 577 00:41:11,470 --> 00:41:13,030 Just forget that intuition. 578 00:41:13,030 --> 00:41:14,160 It's wrong. 579 00:41:14,160 --> 00:41:20,170 What we're dealing with here, if you remember, is the slope. 580 00:41:20,170 --> 00:41:23,870 So if you have a slope here, that corresponds 581 00:41:23,870 --> 00:41:26,960 to just a place over here and as the slope 582 00:41:26,960 --> 00:41:30,190 gets a little bit less steep, that's 583 00:41:30,190 --> 00:41:33,320 why we're approaching the horizontal axis. 584 00:41:33,320 --> 00:41:36,480 The number is getting a little smaller as we close in. 585 00:41:36,480 --> 00:41:41,120 Now over here, the slope is also negative. 586 00:41:41,120 --> 00:41:42,980 It is going down and as we get down here 587 00:41:42,980 --> 00:41:44,580 it's getting more and more negative. 588 00:41:44,580 --> 00:41:48,170 As we go here the slope, this function is going up, 589 00:41:48,170 --> 00:41:50,050 but its slope is going down. 590 00:41:50,050 --> 00:41:55,790 All right, so the slope is down on both sides and the notation 591 00:41:55,790 --> 00:42:03,690 that we use for that is well suited to this left 592 00:42:03,690 --> 00:42:09,410 and right business. 593 00:42:09,410 --> 00:42:16,030 Namely, the limit as x goes to 0 of -1 / x^2, 594 00:42:16,030 --> 00:42:18,140 that's going to be equal to minus infinity. 595 00:42:18,140 --> 00:42:24,760 And that applies to x going to 0+ and x going to 0-. 596 00:42:24,760 --> 00:42:31,780 So both have this property. 597 00:42:31,780 --> 00:42:34,040 Finally let me just make one last comment 598 00:42:34,040 --> 00:42:37,660 about these two graphs. 599 00:42:37,660 --> 00:42:42,220 This function here is an odd function 600 00:42:42,220 --> 00:42:44,620 and when you take the derivative of an odd function 601 00:42:44,620 --> 00:42:50,740 you always get an even function. 602 00:42:50,740 --> 00:42:54,380 That's closely related to the fact that this 1 / x is an odd 603 00:42:54,380 --> 00:43:01,170 power and-- x^1 is an odd power and x^2 is an even power. 604 00:43:01,170 --> 00:43:05,620 So all of this your intuition should be reinforcing the fact 605 00:43:05,620 --> 00:43:11,070 that these pictures look right. 606 00:43:11,070 --> 00:43:16,010 Okay, now there's one last kind of discontinuity 607 00:43:16,010 --> 00:43:20,460 that I want to mention briefly, which I will call 608 00:43:20,460 --> 00:43:33,990 other ugly discontinuities. 609 00:43:33,990 --> 00:43:39,770 And there are lots and lots of them. 610 00:43:39,770 --> 00:43:44,220 So one example would be the function y = sin 611 00:43:44,220 --> 00:43:50,080 1 / x, as x goes to 0. 612 00:43:50,080 --> 00:43:58,914 And that looks a little bit like this. 613 00:43:58,914 --> 00:44:00,330 Back and forth and back and forth. 614 00:44:00,330 --> 00:44:06,170 It oscillates infinitely often as we tend to 0. 615 00:44:06,170 --> 00:44:19,260 There's no left or right limit in this case. 616 00:44:19,260 --> 00:44:25,330 So there is a very large quantity of things like that. 617 00:44:25,330 --> 00:44:29,350 Fortunately we're not gonna deal with them in this course. 618 00:44:29,350 --> 00:44:31,500 A lot of times in real life there 619 00:44:31,500 --> 00:44:34,800 are things that oscillate as time goes to infinity, 620 00:44:34,800 --> 00:44:40,180 but we're not going to worry about that right now. 621 00:44:40,180 --> 00:44:49,090 Okay, so that's our final mention of a discontinuity, 622 00:44:49,090 --> 00:44:54,130 and now I need to do just one more piece of groundwork 623 00:44:54,130 --> 00:44:59,360 for our formulas next time. 624 00:44:59,360 --> 00:45:09,130 Namely, I want to check for you one basic fact, 625 00:45:09,130 --> 00:45:10,280 one limiting tool. 626 00:45:10,280 --> 00:45:12,960 So this is going to be a theorem. 627 00:45:12,960 --> 00:45:17,450 Fortunately it's a very short theorem 628 00:45:17,450 --> 00:45:19,580 and has a very short proof. 629 00:45:19,580 --> 00:45:22,090 So the theorem goes under the name differentiable 630 00:45:22,090 --> 00:45:28,210 implies continuous. 631 00:45:28,210 --> 00:45:30,190 And what it says is the following: 632 00:45:30,190 --> 00:45:35,600 it says that if f is differentiable, in other words 633 00:45:35,600 --> 00:45:45,560 its-- the derivative exists at x_0, then 634 00:45:45,560 --> 00:45:59,245 f is continuous at x_0. 635 00:45:59,245 --> 00:46:00,870 So, we're gonna need this is as a tool, 636 00:46:00,870 --> 00:46:05,750 it's a key step in the product and quotient rules. 637 00:46:05,750 --> 00:46:12,380 So I'd like to prove it right now for you. 638 00:46:12,380 --> 00:46:16,270 So here is the proof. 639 00:46:16,270 --> 00:46:20,430 Fortunately the proof is just one line. 640 00:46:20,430 --> 00:46:24,740 So first of all, I want to write in just the right way what 641 00:46:24,740 --> 00:46:27,410 it is that we have to check. 642 00:46:27,410 --> 00:46:33,540 So what we have to check is that the limit, as x goes to x_0, 643 00:46:33,540 --> 00:46:41,347 of f(x) - f(x_0) is equal to 0. 644 00:46:41,347 --> 00:46:42,680 So this is what we want to know. 645 00:46:42,680 --> 00:46:44,930 We don't know it yet, but we're trying 646 00:46:44,930 --> 00:46:47,650 to check whether this is true or not. 647 00:46:47,650 --> 00:46:49,790 So that's the same as the statement 648 00:46:49,790 --> 00:46:52,180 that the function is continuous because the limit of f(x) 649 00:46:52,180 --> 00:46:56,300 is supposed to be f(x_0) and so this difference should 650 00:46:56,300 --> 00:46:59,690 have limit 0. 651 00:46:59,690 --> 00:47:02,730 And now, the way this is proved is just 652 00:47:02,730 --> 00:47:09,720 by rewriting it by multiplying and dividing by (x - x_0). 653 00:47:09,720 --> 00:47:17,381 So I'll rewrite the limit as x goes to x_0 of f(x) - 654 00:47:17,381 --> 00:47:25,570 f(x_0) divided by x - x_0 times x - x_0. 655 00:47:25,570 --> 00:47:29,230 Okay, so I wrote down the same expression that I had here. 656 00:47:29,230 --> 00:47:32,080 This is just the same limit, but I multiplied and divided 657 00:47:32,080 --> 00:47:38,070 by (x - x_0). 658 00:47:38,070 --> 00:47:45,150 And now when I take the limit what happens is the limit 659 00:47:45,150 --> 00:47:48,830 of the first factor is f'(x_0). 660 00:47:48,830 --> 00:47:53,940 That's the thing we know exists by our assumption. 661 00:47:53,940 --> 00:48:00,640 And the limit of the second factor is 0 because the limit 662 00:48:00,640 --> 00:48:06,700 as x goes to x_0 of (x - x_0) is clearly 0 . 663 00:48:06,700 --> 00:48:09,210 So that's it. 664 00:48:09,210 --> 00:48:12,210 The answer is 0, which is what we wanted. 665 00:48:12,210 --> 00:48:14,980 So that's the proof. 666 00:48:14,980 --> 00:48:19,500 Now there's something exceedingly fishy-looking 667 00:48:19,500 --> 00:48:26,370 about this proof and let me just point to it before we proceed. 668 00:48:26,370 --> 00:48:33,050 Namely, you're used in limits to setting x equal to 0. 669 00:48:33,050 --> 00:48:35,880 And this looks like we're multiplying, dividing by 0, 670 00:48:35,880 --> 00:48:38,430 exactly the thing which makes all proofs 671 00:48:38,430 --> 00:48:42,562 wrong in all kinds of algebraic situations 672 00:48:42,562 --> 00:48:43,520 and so on and so forth. 673 00:48:43,520 --> 00:48:45,780 You've been taught that that never works. 674 00:48:45,780 --> 00:48:47,750 Right? 675 00:48:47,750 --> 00:48:51,040 But somehow these limiting tricks 676 00:48:51,040 --> 00:48:54,100 have found a way around this and let me just 677 00:48:54,100 --> 00:48:55,880 make explicit what it is. 678 00:48:55,880 --> 00:49:03,500 In this limit we never are using x = x_0. 679 00:49:03,500 --> 00:49:05,720 That's exactly the one value of x that we 680 00:49:05,720 --> 00:49:09,120 don't consider in this limit. 681 00:49:09,120 --> 00:49:11,910 That's how limits are cooked up. 682 00:49:11,910 --> 00:49:14,840 And that's sort of been the themes so far today, 683 00:49:14,840 --> 00:49:17,100 is that we don't have to consider that 684 00:49:17,100 --> 00:49:19,990 and so this multiplication and division by this number 685 00:49:19,990 --> 00:49:21,450 is legal. 686 00:49:21,450 --> 00:49:25,200 It may be small, this number, but it's always non-zero. 687 00:49:25,200 --> 00:49:27,670 So this really works, and it's really true, 688 00:49:27,670 --> 00:49:31,040 and we just checked that a differentiable function is 689 00:49:31,040 --> 00:49:32,560 continuous. 690 00:49:32,560 --> 00:49:38,580 So I'm gonna have to carry out these limits, which 691 00:49:38,580 --> 00:49:42,040 are very interesting 0 / 0 limits next time. 692 00:49:42,040 --> 00:49:46,512 But let's hang on for one second to see if there any questions 693 00:49:46,512 --> 00:49:47,907 before we stop. 694 00:49:47,907 --> 00:49:48,990 Yeah, there is a question. 695 00:49:48,990 --> 00:50:00,970 Student: [INAUDIBLE] Professor: Repeat this proof right here? 696 00:50:00,970 --> 00:50:02,830 Just say again. 697 00:50:02,830 --> 00:50:08,230 Student: [INAUDIBLE] 698 00:50:08,230 --> 00:50:13,060 Professor: Okay, so there are two steps to the proof 699 00:50:13,060 --> 00:50:17,870 and the step that you're asking about is the first step. 700 00:50:17,870 --> 00:50:18,580 Right? 701 00:50:18,580 --> 00:50:20,890 And what I'm saying is if you have a number, 702 00:50:20,890 --> 00:50:24,640 and you multiply it by 10 / 10 it's the same number. 703 00:50:24,640 --> 00:50:26,920 If you multiply it by 3 / 3 it's the same number. 704 00:50:26,920 --> 00:50:30,110 2 / 2, 1 / 1, and so on. 705 00:50:30,110 --> 00:50:32,385 So it is okay to change this to this, 706 00:50:32,385 --> 00:50:34,400 it's exactly the same thing. 707 00:50:34,400 --> 00:50:36,220 That's the first step. 708 00:50:36,220 --> 00:50:36,720 Yes? 709 00:50:36,720 --> 00:50:41,560 Student: [INAUDIBLE] 710 00:50:41,560 --> 00:50:45,010 Professor: Shhhh... 711 00:50:45,010 --> 00:50:52,100 The question was how does the proof, how does this line, 712 00:50:52,100 --> 00:50:53,960 yeah where the question mark is. 713 00:50:53,960 --> 00:50:55,910 So what I checked was that this number which 714 00:50:55,910 --> 00:50:59,850 is on the left hand side is equal to this very long 715 00:50:59,850 --> 00:51:04,800 complicated number which is equal to this number which 716 00:51:04,800 --> 00:51:06,270 is equal to this number. 717 00:51:06,270 --> 00:51:08,600 And so I've checked that this number is equal to 0 718 00:51:08,600 --> 00:51:12,120 because the last thing is 0. 719 00:51:12,120 --> 00:51:16,120 This is equal to that is equal to that is equal to 0. 720 00:51:16,120 --> 00:51:17,420 And that's the proof. 721 00:51:17,420 --> 00:51:17,920 Yes? 722 00:51:17,920 --> 00:51:21,910 Student: [INAUDIBLE] 723 00:51:21,910 --> 00:51:30,580 Professor: So that's a different question. 724 00:51:30,580 --> 00:51:35,750 Okay, so the hypothesis of differentiability I 725 00:51:35,750 --> 00:51:39,420 use because this limit is equal to this number. 726 00:51:39,420 --> 00:51:40,520 That that limit exits. 727 00:51:40,520 --> 00:51:44,170 That's how I use the hypothesis of the theorem. 728 00:51:44,170 --> 00:51:46,560 The conclusion of the theorem is the same 729 00:51:46,560 --> 00:51:51,300 as this because being continuous is the same as limit 730 00:51:51,300 --> 00:51:56,020 as x goes to x_0 of f(x) is equal to f(x_0). 731 00:51:56,020 --> 00:51:57,530 That's the definition of continuity. 732 00:51:57,530 --> 00:52:00,990 And I subtracted f(x_0) from both sides 733 00:52:00,990 --> 00:52:02,860 to get this as being the same thing. 734 00:52:02,860 --> 00:52:08,100 So this claim is continuity and it's the same as this question 735 00:52:08,100 --> 00:52:10,350 here. 736 00:52:10,350 --> 00:52:11,180 Last question. 737 00:52:11,180 --> 00:52:16,771 Student: How did you get the 0 [INAUDIBLE] 738 00:52:16,771 --> 00:52:18,520 Professor: How did we get the 0 from this? 739 00:52:18,520 --> 00:52:20,450 So the claim that is being made, so the claim 740 00:52:20,450 --> 00:52:24,670 is why is this tending to that. 741 00:52:24,670 --> 00:52:27,410 So for example, I'm going to have to erase something 742 00:52:27,410 --> 00:52:28,730 to explain that. 743 00:52:28,730 --> 00:52:33,900 So the claim is that the limit as x goes to x_0 of x - x_0 744 00:52:33,900 --> 00:52:35,240 is equal to 0. 745 00:52:35,240 --> 00:52:37,160 That's what I'm claiming. 746 00:52:37,160 --> 00:52:39,490 Okay, does that answer your question? 747 00:52:39,490 --> 00:52:40,990 Okay. 748 00:52:40,990 --> 00:52:42,420 All right. 749 00:52:42,420 --> 00:52:45,320 Ask me other stuff after lecture.