1 00:00:00,540 --> 00:00:01,500 Welcome back. 2 00:00:01,500 --> 00:00:05,370 Welcome back for week four of Modeling Complex Systems. 3 00:00:05,370 --> 00:00:07,530 This week, we're gonna be looking 4 00:00:07,530 --> 00:00:09,840 at continuous dynamical systems, 5 00:00:09,840 --> 00:00:12,690 so instead of using discreet rules 6 00:00:12,690 --> 00:00:16,410 that we update for every hour of every day 7 00:00:16,410 --> 00:00:17,760 with some time window, 8 00:00:17,760 --> 00:00:19,740 we're gonna decrease those time window, 9 00:00:19,740 --> 00:00:22,080 so that we have infinitely many small steps 10 00:00:22,080 --> 00:00:24,300 and effectively continuous time. 11 00:00:24,300 --> 00:00:26,280 That's gonna open the entire toolbox 12 00:00:26,280 --> 00:00:28,590 of calculus and differential equation 13 00:00:28,590 --> 00:00:30,720 to help us study these systems. 14 00:00:30,720 --> 00:00:32,970 So this is the bonus video, 15 00:00:32,970 --> 00:00:36,120 that I originally referred to during week three 16 00:00:36,120 --> 00:00:38,760 that was unfortunately recorded without sound, 17 00:00:38,760 --> 00:00:41,100 so you're getting it in week four. 18 00:00:41,100 --> 00:00:43,290 We're gonna be looking at some mathematical tools 19 00:00:43,290 --> 00:00:47,883 that help us take those limits of very small steps. 20 00:00:49,440 --> 00:00:52,200 The first tool we're gonna be looking at are Taylor series, 21 00:00:52,200 --> 00:00:54,510 and here, I wanna highlight 22 00:00:54,510 --> 00:00:58,890 like three important part of the name of this tool. 23 00:00:58,890 --> 00:01:00,690 First of all, that it's a series, 24 00:01:00,690 --> 00:01:02,550 so what that means is that 25 00:01:02,550 --> 00:01:05,100 formally, it's gonna be a power series. 26 00:01:05,100 --> 00:01:06,030 You just have to remember 27 00:01:06,030 --> 00:01:08,670 that we're looking at polynomials essentially, 28 00:01:08,670 --> 00:01:10,800 where we're gonna use the structure of a polynomial. 29 00:01:10,800 --> 00:01:12,330 And this is a trick we're gonna use again 30 00:01:12,330 --> 00:01:13,740 in a different context, 31 00:01:13,740 --> 00:01:15,990 we're gonna use the structure of a polynomial, 32 00:01:15,990 --> 00:01:18,930 where it's easy to take derivatives and all that 33 00:01:18,930 --> 00:01:21,840 to store important pieces of information. 34 00:01:21,840 --> 00:01:23,640 And here, we're gonna store information 35 00:01:23,640 --> 00:01:25,890 about a key function, f of X, 36 00:01:25,890 --> 00:01:30,180 which might simply be too complicated to study efficiently. 37 00:01:30,180 --> 00:01:31,650 So if you have a function, 38 00:01:31,650 --> 00:01:32,610 you wanna take a limit 39 00:01:32,610 --> 00:01:34,560 and its behavior is sort of unknown, 40 00:01:34,560 --> 00:01:35,910 it's a weird function, 41 00:01:35,910 --> 00:01:38,130 then Taylor series can help you do that. 42 00:01:38,130 --> 00:01:40,140 They're gonna help you summarize this function 43 00:01:40,140 --> 00:01:41,583 in a much easier way. 44 00:01:42,780 --> 00:01:44,070 And the last key part is that 45 00:01:44,070 --> 00:01:46,410 it's really valid around one given point, 46 00:01:46,410 --> 00:01:49,320 so if we wanna take a limit of something going to zero, 47 00:01:49,320 --> 00:01:51,690 for example, well that's gonna be convenient, 48 00:01:51,690 --> 00:01:55,020 'cause we know that for power series for polynomial, 49 00:01:55,020 --> 00:01:57,420 if all coefficients go to zero, 50 00:01:57,420 --> 00:01:58,620 well, that's easy to deal with. 51 00:01:58,620 --> 00:02:02,940 We only care about the smaller order terms. 52 00:02:02,940 --> 00:02:04,290 So that's really the key part. 53 00:02:04,290 --> 00:02:06,660 So to summarize here, 54 00:02:06,660 --> 00:02:08,660 we're gonna be looking for a polynomial, 55 00:02:13,020 --> 00:02:17,193 such that the value of a function, 56 00:02:18,840 --> 00:02:23,840 which I'm gonna write f of X0 and all derivatives, 57 00:02:29,610 --> 00:02:32,043 which here, I'm gonna write f superscript n. 58 00:02:32,910 --> 00:02:35,460 So f(1) would be the first derivative, 59 00:02:35,460 --> 00:02:38,460 the slope of that function at X0. 60 00:02:38,460 --> 00:02:42,310 F(2) would be the curvature of that function at X0 61 00:02:45,210 --> 00:02:50,210 are respected or conserved or preserved 62 00:02:50,940 --> 00:02:54,750 at x = X0, okay. 63 00:02:54,750 --> 00:02:57,490 The insight really is that 64 00:02:58,650 --> 00:03:03,650 we're looking at a function, f of X not f of X0, f of X, 65 00:03:05,460 --> 00:03:07,560 that might be very complicated. 66 00:03:07,560 --> 00:03:10,083 Let me just draw whatever, but it's a function, 67 00:03:12,300 --> 00:03:14,820 sometimes very steep, sometimes not at all. 68 00:03:14,820 --> 00:03:18,510 It's a weird function that might be hard to deal with, 69 00:03:18,510 --> 00:03:20,100 and what we're saying is that, 70 00:03:20,100 --> 00:03:24,183 if I look at a particular value, X0, 71 00:03:26,160 --> 00:03:28,800 if I know where I am and I know the slope 72 00:03:28,800 --> 00:03:31,710 and the curvature and all the derivatives, 73 00:03:31,710 --> 00:03:33,990 I can get a really good local picture 74 00:03:33,990 --> 00:03:36,180 of that function, right, even just the slope. 75 00:03:36,180 --> 00:03:37,770 If you know where you are at X0 76 00:03:37,770 --> 00:03:39,840 and you know in which direction you're going 77 00:03:39,840 --> 00:03:42,510 and how steep you're going, 78 00:03:42,510 --> 00:03:44,490 then it's easy to project your position, 79 00:03:44,490 --> 00:03:47,520 like do a small step in delta X times the slope, 80 00:03:47,520 --> 00:03:51,183 and you get your delta Y kinda thing, right. 81 00:03:52,290 --> 00:03:54,300 So, that's the key insight, 82 00:03:54,300 --> 00:03:56,010 that's what we're looking for here, 83 00:03:56,010 --> 00:03:58,500 and often when people present Taylor series, 84 00:03:58,500 --> 00:04:00,510 they're only gonna give you a formula. 85 00:04:00,510 --> 00:04:01,710 It's almost like a recipe, 86 00:04:01,710 --> 00:04:04,020 like apply this and you're gonna get a polynomial 87 00:04:04,020 --> 00:04:08,130 that reflects your function but is easier to deal with. 88 00:04:08,130 --> 00:04:10,770 The truth is it's super easy. 89 00:04:10,770 --> 00:04:12,660 Looking it up on Wikipedia is easy, 90 00:04:12,660 --> 00:04:16,500 but deriving the Taylor series is almost just as easy. 91 00:04:16,500 --> 00:04:17,800 So we're looking for 92 00:04:18,990 --> 00:04:21,240 an approximation of f of X, 93 00:04:21,240 --> 00:04:23,403 so I'm gonna use this approximation here, 94 00:04:24,960 --> 00:04:27,210 such that if I plug X of X0, 95 00:04:27,210 --> 00:04:30,450 at least I want to to know that I get the right value. 96 00:04:30,450 --> 00:04:32,700 So I'm gonna start with f of X0. 97 00:04:32,700 --> 00:04:36,930 If I plug X0 here, I get the right value. 98 00:04:36,930 --> 00:04:40,800 The problem is that right now, if I plug any other value, 99 00:04:40,800 --> 00:04:42,628 I still get f of X0 100 00:04:42,628 --> 00:04:44,693 'cause I don't have any derivatives right now. 101 00:04:45,690 --> 00:04:48,450 So as I just said, one thing you can do, 102 00:04:48,450 --> 00:04:51,633 lemme go back, back. 103 00:04:53,370 --> 00:04:57,450 One thing you can do is just apply your derivatives, 104 00:04:57,450 --> 00:04:59,640 the first derivative at X0, 105 00:04:59,640 --> 00:05:04,290 so that's the slope of my function, times a change in X, 106 00:05:04,290 --> 00:05:06,570 a delta X, to get a delta Y, right. 107 00:05:06,570 --> 00:05:08,340 That's what the derivative means. 108 00:05:08,340 --> 00:05:12,990 And now, if I plug anything other than X0, 109 00:05:12,990 --> 00:05:15,300 I'm using this first order slope 110 00:05:15,300 --> 00:05:17,850 to sort of project or extrapolate from my position, 111 00:05:17,850 --> 00:05:19,440 so that's good. 112 00:05:19,440 --> 00:05:21,840 It's also good because, if I take the derivative, 113 00:05:21,840 --> 00:05:25,230 this is just a constant, so this goes away. 114 00:05:25,230 --> 00:05:27,240 The derivative of this term is one. 115 00:05:27,240 --> 00:05:30,960 So if I take the derivative of my polynomial here, 116 00:05:30,960 --> 00:05:33,780 I know that I'm gonna get the right value here, f prime, 117 00:05:33,780 --> 00:05:35,043 so we're doin' great. 118 00:05:37,830 --> 00:05:39,900 If we take the second derivative though, 119 00:05:39,900 --> 00:05:41,070 right now, we're at zero, 120 00:05:41,070 --> 00:05:43,530 'cause we don't have any second order term, 121 00:05:43,530 --> 00:05:46,083 so we wanna introduce a second order term, 122 00:05:48,960 --> 00:05:53,343 oops, which is also gonna be evaluated at X0. 123 00:06:00,210 --> 00:06:02,040 And now, this one is interesting. 124 00:06:02,040 --> 00:06:04,620 Here, I've made it a second order term. 125 00:06:04,620 --> 00:06:07,740 I respect the structure of my polynomial, 126 00:06:07,740 --> 00:06:11,793 so I got this little square here, second power. 127 00:06:12,630 --> 00:06:16,500 But if I take the derivative of f of X, 128 00:06:16,500 --> 00:06:18,390 of my power series, what happens? 129 00:06:18,390 --> 00:06:20,190 The first term goes away, it's a constant. 130 00:06:20,190 --> 00:06:24,930 The second term gives me my f(1) for the first derivative. 131 00:06:24,930 --> 00:06:27,453 I take a second derivative, it goes away. 132 00:06:29,280 --> 00:06:31,350 This term that I just wrote though, 133 00:06:31,350 --> 00:06:33,270 if I take two derivatives, 134 00:06:33,270 --> 00:06:35,550 the two is gonna fall in the front, right, 135 00:06:35,550 --> 00:06:40,290 so f prime prime or f(2) of x is gonna give me 136 00:06:40,290 --> 00:06:43,763 two f(2) of X0, and that's it. 137 00:06:47,190 --> 00:06:49,683 So, I'm getting twice the derivative now, 138 00:06:51,060 --> 00:06:54,873 so really, I can just add 1/2 to control for that. 139 00:06:56,285 --> 00:06:57,840 And now I could add all derivatives, 140 00:06:57,840 --> 00:07:00,685 but if I have the term that's like an X(n), 141 00:07:00,685 --> 00:07:01,980 then if I take n derivatives, 142 00:07:01,980 --> 00:07:05,400 I'm gonna get n times n minus one times n minus two 143 00:07:05,400 --> 00:07:07,830 for all the derivatives of these powers. 144 00:07:07,830 --> 00:07:10,230 So the general form of this is really 145 00:07:10,230 --> 00:07:15,230 just that you include the nth derivative with this form, 146 00:07:18,990 --> 00:07:22,710 which you divide by n factorial, 147 00:07:22,710 --> 00:07:25,112 to control for the fact that your derivatives 148 00:07:25,112 --> 00:07:27,870 are adding factors in front 149 00:07:27,870 --> 00:07:30,630 of the true slope that you want. 150 00:07:30,630 --> 00:07:31,470 And that's it. 151 00:07:31,470 --> 00:07:36,470 The sum of all f(n) of X0 times x minus X0 to the n, 152 00:07:36,900 --> 00:07:40,050 normalize with this n factorial, that's our power series. 153 00:07:40,050 --> 00:07:43,110 It's just a way to store the value and all the derivatives 154 00:07:43,110 --> 00:07:44,850 in a way that makes sense, 155 00:07:44,850 --> 00:07:47,130 and that if you derive your approximation, 156 00:07:47,130 --> 00:07:48,930 you're gonna get the right function. 157 00:07:50,190 --> 00:07:52,203 Here's an example of why that's useful. 158 00:07:53,520 --> 00:07:56,940 So, we dealt a lot with small probabilities, 159 00:07:56,940 --> 00:08:01,940 and I've played a little loose with probabilities, 160 00:08:06,420 --> 00:08:10,313 with spelling probability, okay. 161 00:08:12,780 --> 00:08:14,830 I need more coffee, it's early right now. 162 00:08:16,980 --> 00:08:19,470 I've played a little loose with some of those terms. 163 00:08:19,470 --> 00:08:22,410 Sometimes I would write, for example, 164 00:08:22,410 --> 00:08:24,453 in the case of an SIS model, 165 00:08:25,592 --> 00:08:26,767 I would do one step and I would say, 166 00:08:26,767 --> 00:08:30,937 "Well, to know who's susceptible at time t plus delta t 167 00:08:30,937 --> 00:08:35,187 "or t plus one steps, I take who's susceptible at time t," 168 00:08:37,080 --> 00:08:38,580 and then I've had multiple version, 169 00:08:38,580 --> 00:08:43,580 but one of them was those that are still susceptible 170 00:08:43,650 --> 00:08:45,390 are those that don't get infected. 171 00:08:45,390 --> 00:08:47,280 So a contact doesn't infect 172 00:08:47,280 --> 00:08:50,640 with probability one minus beta transmission rate 173 00:08:50,640 --> 00:08:53,343 times delta t, the step that we're considering, 174 00:08:54,240 --> 00:08:59,190 and I have I of t susceptible individuals. 175 00:08:59,190 --> 00:09:01,560 And let's say it's a SI model, no recovery, 176 00:09:01,560 --> 00:09:04,770 so this is fine, this is a complete model. 177 00:09:04,770 --> 00:09:07,050 Now if delta, if beta times delta t, 178 00:09:07,050 --> 00:09:10,230 the probability of transition is really small, 179 00:09:10,230 --> 00:09:11,890 really what I can say is that 180 00:09:14,160 --> 00:09:18,153 this is my one minus x to I of t, let's say, 181 00:09:19,380 --> 00:09:20,917 and I can say like, 182 00:09:20,917 --> 00:09:24,757 "Well lemme do a Taylor series around X equals zero, 183 00:09:24,757 --> 00:09:27,217 "so that I'm gonna know what's the behavior 184 00:09:27,217 --> 00:09:30,687 "for very small probabilities, for very small X." 185 00:09:31,590 --> 00:09:35,403 And so well, if you evaluate that X equals zero, 186 00:09:36,420 --> 00:09:38,973 we're gonna get, I'm just gonna write f of X, 187 00:09:41,190 --> 00:09:42,240 I'm just gonna get one. 188 00:09:42,240 --> 00:09:44,970 If I evaluate it at X0 equal to 2(0), 189 00:09:44,970 --> 00:09:47,850 'cause I have one to some power, that's always one. 190 00:09:47,850 --> 00:09:49,893 Now I take my first derivative, 191 00:09:50,790 --> 00:09:55,670 and my first derivative is simply gonna be X times I of t. 192 00:09:57,900 --> 00:09:59,760 And then usually, we would write, 193 00:09:59,760 --> 00:10:04,200 we got some other terms in X square, and that's a big O, 194 00:10:04,200 --> 00:10:07,050 meaning we have terms of order X square, 195 00:10:07,050 --> 00:10:08,970 but because X is already really small, 196 00:10:08,970 --> 00:10:11,310 X square is even smaller, 197 00:10:11,310 --> 00:10:14,670 potentially so small that we ignore it, right. 198 00:10:14,670 --> 00:10:16,170 So that's what we're gonna do. 199 00:10:17,220 --> 00:10:19,650 And what this means is that this equation 200 00:10:19,650 --> 00:10:21,160 could have been written 201 00:10:23,880 --> 00:10:28,880 simply as S of t times one minus beta delta t I, 202 00:10:32,700 --> 00:10:34,953 I'm running out of space here, I apologize. 203 00:10:37,710 --> 00:10:40,410 Let's just go back here. 204 00:10:40,410 --> 00:10:45,410 S of t times one minus beta delta t times I of t. 205 00:10:48,330 --> 00:10:52,290 And sometimes I use this form because it's easier, right, 206 00:10:52,290 --> 00:10:54,930 and the average is respected. 207 00:10:54,930 --> 00:10:57,390 But the important part here is that 208 00:10:57,390 --> 00:10:58,890 there is an approximation, 209 00:10:58,890 --> 00:11:00,060 so let me just highlight it here. 210 00:11:00,060 --> 00:11:02,490 This is approximately equal 211 00:11:02,490 --> 00:11:05,160 to the simulations we would do with discrete time steps 212 00:11:05,160 --> 00:11:10,160 and discrete number of infectious neighbors, okay. 213 00:11:14,040 --> 00:11:15,570 There was one that was even harder 214 00:11:15,570 --> 00:11:17,340 that showed up in a video we had. 215 00:11:17,340 --> 00:11:18,780 This, I just copy and pasted it. 216 00:11:18,780 --> 00:11:21,060 We had a cumulative distribution function 217 00:11:21,060 --> 00:11:23,160 for our Poisson process, right, 218 00:11:23,160 --> 00:11:26,850 and Poisson processes, like formally, 219 00:11:26,850 --> 00:11:29,610 and that was part of the quiz in week three. 220 00:11:29,610 --> 00:11:30,900 They're defined by a rate, 221 00:11:30,900 --> 00:11:34,200 so by the expected time until the next event. 222 00:11:34,200 --> 00:11:36,300 In fact, that's all they're defined by, 223 00:11:36,300 --> 00:11:37,590 so they don't have any memory. 224 00:11:37,590 --> 00:11:40,530 If all that matters is the expected rate 225 00:11:40,530 --> 00:11:43,380 at which the next event occurs or the expected time, 226 00:11:43,380 --> 00:11:45,390 this is, time and rate are just the same thing, 227 00:11:45,390 --> 00:11:47,730 one over the other until the next event, 228 00:11:47,730 --> 00:11:49,500 then it's a Poisson processes. 229 00:11:49,500 --> 00:11:52,683 But it is a continuous time process. 230 00:11:53,850 --> 00:11:56,880 And we were doing this cumulative distribution function 231 00:11:56,880 --> 00:11:58,140 and then showing how, 232 00:11:58,140 --> 00:11:59,610 which we expect is an exponential 233 00:11:59,610 --> 00:12:03,810 or Poisson distribution of recovery time 234 00:12:03,810 --> 00:12:05,910 or of me leaving a building. 235 00:12:05,910 --> 00:12:08,250 And we took this derivative already, 236 00:12:08,250 --> 00:12:11,490 so when time steps get really small, right. 237 00:12:11,490 --> 00:12:14,403 We took this limit, not this derivative, this limit. 238 00:12:17,100 --> 00:12:19,470 Well, so you could play with Taylor series, 239 00:12:19,470 --> 00:12:22,050 doing something similar here. 240 00:12:22,050 --> 00:12:25,920 The important part is that the delta t occurs twice, right, 241 00:12:25,920 --> 00:12:27,600 so it is a complicated function. 242 00:12:27,600 --> 00:12:28,823 And what's interesting here is that 243 00:12:28,823 --> 00:12:33,823 there's one term that goes to one, and that's the term here, 244 00:12:34,050 --> 00:12:36,360 but then the power goes to infinity. 245 00:12:36,360 --> 00:12:38,370 So as time gets really small, 246 00:12:38,370 --> 00:12:40,650 the probability of transition 247 00:12:40,650 --> 00:12:42,900 during every time step is effectively zero. 248 00:12:42,900 --> 00:12:45,390 So nothing happens, the argument goes to one, 249 00:12:45,390 --> 00:12:48,290 but the number of times steps that we do becomes infinite. 250 00:12:49,350 --> 00:12:50,730 And there's a second tool 251 00:12:50,730 --> 00:12:54,033 that's really useful for dealing with that, 252 00:12:56,760 --> 00:12:57,873 and that's the, 253 00:12:59,841 --> 00:13:02,010 I apologize, I have a hard time saying this in English. 254 00:13:02,010 --> 00:13:06,210 So it's the L'Hospital or the Hospital Rule. 255 00:13:11,370 --> 00:13:13,320 Okay, and this is gonna be our example, 256 00:13:13,320 --> 00:13:15,603 this CDF that we dealt with. 257 00:13:16,740 --> 00:13:20,583 Fun fact, a bonus trivia fact as part of this bonus video, 258 00:13:21,600 --> 00:13:25,807 you will also see this written as Hopital Rule. 259 00:13:31,170 --> 00:13:33,510 Basically, in Old French, 260 00:13:33,510 --> 00:13:35,970 we would add a lot of os and es, 261 00:13:35,970 --> 00:13:37,380 that are now, in Modern French, 262 00:13:37,380 --> 00:13:38,970 written with this circumflex 263 00:13:38,970 --> 00:13:41,640 or little hat on the vowel. 264 00:13:41,640 --> 00:13:43,803 So, it's the same in my name. 265 00:13:46,200 --> 00:13:50,340 My last name, Dufresne, means from the Ash Tree, 266 00:13:50,340 --> 00:13:53,520 but in Modern French, we would write so from, Du, 267 00:13:53,520 --> 00:13:57,033 and then Frene, which is Ash Tree. 268 00:13:58,230 --> 00:13:59,553 Okay, fun fact, 269 00:14:01,200 --> 00:14:02,940 but let me go back to Hospitals Rule. 270 00:14:02,940 --> 00:14:05,640 You can Google it with both spelling, that's my point. 271 00:14:08,100 --> 00:14:09,870 So this works really well, 272 00:14:09,870 --> 00:14:12,900 when you have this idea of two terms 273 00:14:12,900 --> 00:14:14,460 that sort of fight each other, right. 274 00:14:14,460 --> 00:14:17,310 When you take the limit, they fight each other. 275 00:14:17,310 --> 00:14:19,320 And the rule is best written 276 00:14:19,320 --> 00:14:21,090 when you're dealing with ratios. 277 00:14:21,090 --> 00:14:22,290 So when you wanna apply this rule, 278 00:14:22,290 --> 00:14:25,050 the first thing to do is to write it as a ratio, 279 00:14:25,050 --> 00:14:27,720 and here, we're just gonna use 280 00:14:27,720 --> 00:14:29,853 a little dumb trick to do that. 281 00:14:32,940 --> 00:14:36,640 We're gonna take the exponential 282 00:14:39,847 --> 00:14:41,200 of the natural logarithm 283 00:14:44,010 --> 00:14:45,450 of our expression. 284 00:14:45,450 --> 00:14:48,210 So taking the exponential of the natural logarithm 285 00:14:48,210 --> 00:14:50,673 means doing nothing, right, 286 00:14:52,290 --> 00:14:55,830 but here, that means that the limit of an exponential, 287 00:14:55,830 --> 00:14:56,663 what you care about 288 00:14:56,663 --> 00:14:59,250 is what the argument of the exponential is going to. 289 00:14:59,250 --> 00:15:04,143 So we can say we're taking the exponential, the limit, 290 00:15:05,010 --> 00:15:08,640 of a logarithm, then logarithm of something to some power, 291 00:15:08,640 --> 00:15:09,990 you can take the power out, 292 00:15:09,990 --> 00:15:12,900 and it applies a multiplicative factor. 293 00:15:12,900 --> 00:15:15,753 So now we get L over delta t, 294 00:15:18,150 --> 00:15:22,323 and then the logarithm, one minus L delta t. 295 00:15:25,080 --> 00:15:30,080 And now we have what we want, so as delta t goes to zero, 296 00:15:31,440 --> 00:15:34,410 the denominator here is delta t goes to zero, 297 00:15:34,410 --> 00:15:36,570 so that makes the whole thing blow up. 298 00:15:36,570 --> 00:15:38,700 But we also get the logarithm, 299 00:15:38,700 --> 00:15:43,700 which goes to one as L times delta t goes to zero, 300 00:15:43,920 --> 00:15:45,960 and logarithm of one goes to zero. 301 00:15:45,960 --> 00:15:49,320 So we have one term, well, we have two terms going to zero, 302 00:15:49,320 --> 00:15:51,090 the numerator and the denominator, 303 00:15:51,090 --> 00:15:52,410 so they're sort of fighting. 304 00:15:52,410 --> 00:15:54,697 And then what L'Hospital's Rule is saying is, 305 00:15:54,697 --> 00:15:57,577 "Well, if you can look at how fast 306 00:15:57,577 --> 00:16:00,330 "they're both going to zero, you're gonna know who win." 307 00:16:00,330 --> 00:16:04,890 If the tops goes, if the numerator goes to zero faster, 308 00:16:04,890 --> 00:16:06,630 the whole thing is going to zero. 309 00:16:06,630 --> 00:16:08,580 If the denominator goes to zero faster, 310 00:16:08,580 --> 00:16:09,840 the whole thing might blow up, 311 00:16:09,840 --> 00:16:11,790 and then you're not only gonna know where they go 312 00:16:11,790 --> 00:16:13,173 but how fast they're going. 313 00:16:14,430 --> 00:16:15,750 So that's the idea. 314 00:16:15,750 --> 00:16:19,230 So more formally saying what L'Hospital's Rule says is 315 00:16:19,230 --> 00:16:21,000 when you have a ratio like this, 316 00:16:21,000 --> 00:16:23,310 take the derivative of the numerator 317 00:16:23,310 --> 00:16:25,380 and the derivative of the denominator, 318 00:16:25,380 --> 00:16:27,573 and then reevaluate your limit. 319 00:16:35,220 --> 00:16:38,610 So if I do exactly that here, 320 00:16:38,610 --> 00:16:43,610 I'm simply gonna get the exponential of the limit 321 00:16:43,920 --> 00:16:46,293 when delta t goes to zero. 322 00:16:48,990 --> 00:16:50,670 The derivative of my denominator, 323 00:16:50,670 --> 00:16:55,670 which is my first term here, is simply one delta t, 324 00:16:55,770 --> 00:16:58,953 it's derivative as a function of delta t goes to one, 325 00:17:01,740 --> 00:17:06,397 and then, I have minus L over 326 00:17:12,810 --> 00:17:17,810 one minus L delta t here at the bottom. 327 00:17:20,970 --> 00:17:23,103 So then if I apply my limit, 328 00:17:32,280 --> 00:17:34,203 my denominator goes to one, 329 00:17:36,900 --> 00:17:39,783 and I get my derivative of minus LL, 330 00:17:41,460 --> 00:17:43,803 which is what we had in the previous video. 331 00:17:45,150 --> 00:17:46,440 So those two rules, 332 00:17:46,440 --> 00:17:48,639 sometimes they give you the same results, 333 00:17:48,639 --> 00:17:51,240 well, they always give you the same result, 334 00:17:51,240 --> 00:17:52,650 but basically is that, 335 00:17:52,650 --> 00:17:54,630 sometimes one is gonna be easier to deal with. 336 00:17:54,630 --> 00:17:56,880 Here we didn't have to do a lot of derivatives, 337 00:17:56,880 --> 00:17:59,820 we just took the derivatives of logarithm term 338 00:17:59,820 --> 00:18:02,010 and the derivative of a linear term, 339 00:18:02,010 --> 00:18:03,180 so that was easy. 340 00:18:03,180 --> 00:18:04,500 And dealing with a function, 341 00:18:04,500 --> 00:18:07,410 like one minus X to some power over X, 342 00:18:07,410 --> 00:18:11,220 would've been a little harder with Taylor series. 343 00:18:11,220 --> 00:18:13,530 So Taylor series is the more general tool, 344 00:18:13,530 --> 00:18:14,970 but L'Hospital's Rule is really good, 345 00:18:14,970 --> 00:18:16,830 when you notice that there's the structure 346 00:18:16,830 --> 00:18:18,880 of two terms fighting against each other. 347 00:18:19,740 --> 00:18:21,000 So that's it for now. 348 00:18:21,000 --> 00:18:22,650 In the next few videos for this week, 349 00:18:22,650 --> 00:18:24,180 we're gonna be applying some of these rules 350 00:18:24,180 --> 00:18:28,530 to show how discrete models and continuous models 351 00:18:28,530 --> 00:18:32,100 are connected mathematically 352 00:18:32,100 --> 00:18:33,630 in a way that's very quantifiable, 353 00:18:33,630 --> 00:18:37,110 so that we're gonna be able to use discrete simulations 354 00:18:37,110 --> 00:18:40,440 to compare with continuous equations and vice versa, 355 00:18:40,440 --> 00:18:41,610 as long as we know 356 00:18:41,610 --> 00:18:44,580 what approximations we're making mathematically. 357 00:18:44,580 --> 00:18:46,770 So that's why these tools 358 00:18:46,770 --> 00:18:49,020 are gonna be super useful going forward. 359 00:18:49,020 --> 00:18:51,423 So, I'll see you in the next one.