1 00:00:24,206 --> 00:00:25,800 PROFESSOR: Hi. 2 00:00:25,800 --> 00:00:31,460 Well, this is exponential day, the day for the function that 3 00:00:31,460 --> 00:00:33,850 only calculus could create, y is e to the x. 4 00:00:36,450 --> 00:00:41,700 And it couldn't have come from algebra because, however we 5 00:00:41,700 --> 00:00:47,620 approach e to the x, there's some limiting step. 6 00:00:47,620 --> 00:00:49,190 Something goes to 0. 7 00:00:49,190 --> 00:00:51,400 Something goes to infinity. 8 00:00:51,400 --> 00:00:55,170 I've got different ways to reach e to the x, but all of 9 00:00:55,170 --> 00:00:58,810 them involve that limiting process, which we haven't 10 00:00:58,810 --> 00:01:01,160 discussed in full. 11 00:01:01,160 --> 00:01:06,100 Let me come back at a later time to the whole theory, 12 00:01:06,100 --> 00:01:11,630 discussion of limits and just go forward here with this 13 00:01:11,630 --> 00:01:14,360 highly important function. 14 00:01:14,360 --> 00:01:18,930 And I'd like to start with its most important 15 00:01:18,930 --> 00:01:22,320 property, which is-- 16 00:01:22,320 --> 00:01:27,350 so it has this remarkable property that its slope is 17 00:01:27,350 --> 00:01:28,600 equal to itself. 18 00:01:33,500 --> 00:01:37,250 That's what is special about e to the x. 19 00:01:41,110 --> 00:01:43,550 The slope is equal to the function. 20 00:01:43,550 --> 00:01:51,486 Now, I have to admit that if we had a function like that, y 21 00:01:51,486 --> 00:01:55,950 equals e to the x, then 2e to the x, x would work just as 22 00:01:55,950 --> 00:01:58,450 well, or 10e to the x. 23 00:01:58,450 --> 00:01:59,720 Those would-- 24 00:01:59,720 --> 00:02:04,600 the factor 2 or the factor 10 would be in y and it would 25 00:02:04,600 --> 00:02:09,600 also be in the slope and it would cancel and-- 26 00:02:09,600 --> 00:02:14,310 this is a differential equation, our first 27 00:02:14,310 --> 00:02:16,360 differential equation. 28 00:02:16,360 --> 00:02:21,060 A differential equation is an equation that involves, as 29 00:02:21,060 --> 00:02:23,910 this one does, the function and the slope. 30 00:02:23,910 --> 00:02:24,720 It connects them. 31 00:02:24,720 --> 00:02:29,040 And that's the fantastic description of nature, is by 32 00:02:29,040 --> 00:02:30,580 differential equations. 33 00:02:30,580 --> 00:02:34,180 So it's great to see this one early and it's the most 34 00:02:34,180 --> 00:02:36,170 important one. 35 00:02:36,170 --> 00:02:38,210 When you get this one, you've got a whole 36 00:02:38,210 --> 00:02:40,510 lot of others solved. 37 00:02:40,510 --> 00:02:41,210 OK. 38 00:02:41,210 --> 00:02:46,080 But I needed to give it a starting point so that the 39 00:02:46,080 --> 00:02:50,220 solution would be e to the x and not 10e to the x. 40 00:02:50,220 --> 00:02:51,960 So where should I start it? 41 00:02:51,960 --> 00:02:58,700 Well if I want it to be an e to the x, then when x is 0, e 42 00:02:58,700 --> 00:03:05,490 to the 0 power, some number to the 0 power, is always 1. 43 00:03:05,490 --> 00:03:13,980 So let me start this y equals 1 at x equals 0. 44 00:03:13,980 --> 00:03:17,390 Differential equations, you have to tell where they begin. 45 00:03:17,390 --> 00:03:19,800 So that's our starting point. 46 00:03:19,800 --> 00:03:22,070 And do you see what this means? 47 00:03:22,070 --> 00:03:27,720 This means that it starts at 1. 48 00:03:27,720 --> 00:03:31,760 And what's it's slope at the starting point? 49 00:03:31,760 --> 00:03:33,200 The slope is also 1. 50 00:03:33,200 --> 00:03:35,720 So it's climbing. 51 00:03:35,720 --> 00:03:38,680 As it climbs-- 52 00:03:38,680 --> 00:03:42,330 so y gets larger because it's got a positive slope. 53 00:03:42,330 --> 00:03:46,940 As y gets larger, the slope gets larger. 54 00:03:46,940 --> 00:03:49,080 So it climbs faster. 55 00:03:49,080 --> 00:03:54,560 And then it's gone higher, y is bigger, the slope is equal, 56 00:03:54,560 --> 00:03:57,350 so the slope is also bigger, so it climbs even faster. 57 00:03:57,350 --> 00:03:58,600 It just takes off. 58 00:04:02,150 --> 00:04:05,590 It climbs much faster than x to the 100th power. 59 00:04:05,590 --> 00:04:09,150 You might think x to the 100th, that's 60 00:04:09,150 --> 00:04:10,760 climbing pretty well. 61 00:04:10,760 --> 00:04:14,115 2 to the 100th, 10 to the 100th, but now way. 62 00:04:14,115 --> 00:04:20,203 It doesn't come close to keeping up with y 63 00:04:20,203 --> 00:04:21,453 equals e to the x. 64 00:04:23,550 --> 00:04:25,390 OK. 65 00:04:25,390 --> 00:04:27,300 I've got several things to do. 66 00:04:27,300 --> 00:04:31,530 And one more thing I have to do, this is a key property, 67 00:04:31,530 --> 00:04:37,310 but there's another key property that is true for any 68 00:04:37,310 --> 00:04:40,740 2 to the x, 3 to the x, e to the x. 69 00:04:40,740 --> 00:04:44,370 And that key property is also to show-- 70 00:04:44,370 --> 00:04:52,713 I have to show this, that my function, e to the x, times e 71 00:04:52,713 --> 00:04:58,892 to the possibly a different x is equal to-- 72 00:04:58,892 --> 00:05:02,020 do you know what we want here? 73 00:05:02,020 --> 00:05:03,830 This has got to come out of the 74 00:05:03,830 --> 00:05:06,930 construction, out of this property. 75 00:05:06,930 --> 00:05:08,260 It's got to come-- 76 00:05:08,260 --> 00:05:12,950 but we want this to deserve, to be called some number to 77 00:05:12,950 --> 00:05:14,360 the x power. 78 00:05:14,360 --> 00:05:19,450 If we take some number x times multiplied by that same number 79 00:05:19,450 --> 00:05:22,190 capital x times, then we've got that 80 00:05:22,190 --> 00:05:24,100 number how many times? 81 00:05:24,100 --> 00:05:25,860 x plus capital x. 82 00:05:28,920 --> 00:05:32,220 So that's a key property to be proved. 83 00:05:32,220 --> 00:05:35,150 So what will I do? 84 00:05:35,150 --> 00:05:40,050 Let me summarize in advance, outline in advance. 85 00:05:40,050 --> 00:05:47,660 I'm going to construct this function from its property. 86 00:05:47,660 --> 00:05:54,030 Then I'm going check that it's got this property, that 87 00:05:54,030 --> 00:05:56,690 important equality there. 88 00:05:56,690 --> 00:05:58,160 Then, of course, I'll graph it. 89 00:05:58,160 --> 00:06:02,830 I'll figure out what e is, and I'll say something about cases 90 00:06:02,830 --> 00:06:04,770 where this comes up. 91 00:06:04,770 --> 00:06:07,380 I could even say something right away about, where does 92 00:06:07,380 --> 00:06:14,620 this happen that growth is equal or proportional to the 93 00:06:14,620 --> 00:06:15,870 function itself? 94 00:06:18,540 --> 00:06:24,010 It happens with interest, with money in a bank. 95 00:06:24,010 --> 00:06:29,260 When you get interest, the interest is proportional, of 96 00:06:29,260 --> 00:06:32,050 course, the amount there. 97 00:06:32,050 --> 00:06:35,680 And if they add that interest in, if you don't take it out 98 00:06:35,680 --> 00:06:39,440 and spend it but you compound it, put it in there, then you 99 00:06:39,440 --> 00:06:41,170 have more money. 100 00:06:41,170 --> 00:06:44,990 When they compute the interest again, it's computed on that 101 00:06:44,990 --> 00:06:50,470 larger amount and is more interest than the first time. 102 00:06:50,470 --> 00:06:52,820 And so it goes. 103 00:06:52,820 --> 00:07:00,400 So money in the bank is a case of exponential growth. 104 00:07:00,400 --> 00:07:05,820 A hedge fund grows faster than our bank account does, but all 105 00:07:05,820 --> 00:07:07,410 following e to the x. 106 00:07:07,410 --> 00:07:11,400 If you just hang on long enough, you're way up there. 107 00:07:11,400 --> 00:07:12,770 OK. 108 00:07:12,770 --> 00:07:16,120 So here's my job. 109 00:07:16,120 --> 00:07:19,120 Follow this rule and start at y equals 1. 110 00:07:19,120 --> 00:07:22,990 So can I just do it this way? 111 00:07:22,990 --> 00:07:26,610 Here is my function, y of x, that I want to construct. 112 00:07:26,610 --> 00:07:28,410 I want to build that function. 113 00:07:28,410 --> 00:07:31,120 And I know that it starts at 1. 114 00:07:31,120 --> 00:07:34,760 But it's going to have some more things. 115 00:07:34,760 --> 00:07:38,200 Now, this has to equal dy dx. 116 00:07:41,430 --> 00:07:43,720 These have to be the same. 117 00:07:43,720 --> 00:07:45,870 That's my rule. 118 00:07:45,870 --> 00:07:52,780 So dy dx is going to start with a 1. 119 00:07:52,780 --> 00:08:00,370 But now I can't stop because if the derivative is a 1, I 120 00:08:00,370 --> 00:08:01,150 better put-- 121 00:08:01,150 --> 00:08:06,260 I have to put an x up here so that its 122 00:08:06,260 --> 00:08:08,280 derivative will be 1, right? 123 00:08:08,280 --> 00:08:11,680 Its slope will be 1. 124 00:08:11,680 --> 00:08:16,340 That's that steadily climbing x whose slope is 1. 125 00:08:16,340 --> 00:08:21,650 But now, these are supposed to be equal again. 126 00:08:21,650 --> 00:08:26,100 So I have to put this x also here. 127 00:08:26,100 --> 00:08:31,380 But now, I've got to add something more on the top so 128 00:08:31,380 --> 00:08:34,669 that the slope will be 1 plus x. 129 00:08:34,669 --> 00:08:36,220 The slope of the x was 1. 130 00:08:36,220 --> 00:08:41,460 What do I need here to give the slope to be x? 131 00:08:41,460 --> 00:08:47,660 Remember, x squared had the slope 2x, so I need half of x 132 00:08:47,660 --> 00:08:49,710 squared so that I'll have 1x. 133 00:08:49,710 --> 00:08:54,310 So I need a half of x squared. 134 00:08:54,310 --> 00:08:55,730 Good. 135 00:08:55,730 --> 00:08:57,880 The slope of that is this. 136 00:08:57,880 --> 00:09:01,740 But I'm also trying to get the 2 to be equal. 137 00:09:01,740 --> 00:09:02,960 So I better-- 138 00:09:02,960 --> 00:09:03,950 I have no choice. 139 00:09:03,950 --> 00:09:07,070 I have to put in the 1/2 x squared there. 140 00:09:07,070 --> 00:09:09,980 You see, I'm never going to catch up. 141 00:09:09,980 --> 00:09:13,840 Or only if I go forever. 142 00:09:13,840 --> 00:09:14,560 That's the point. 143 00:09:14,560 --> 00:09:16,750 I'll have to go forever. 144 00:09:16,750 --> 00:09:18,640 And what will the next one be? 145 00:09:18,640 --> 00:09:19,640 Oh yeah. 146 00:09:19,640 --> 00:09:22,620 If you see the next one, then we can see the pattern. 147 00:09:27,030 --> 00:09:27,930 Now what am I doing? 148 00:09:27,930 --> 00:09:33,870 This one has to have this slope. 149 00:09:33,870 --> 00:09:37,180 I'm fixing the top line now. 150 00:09:37,180 --> 00:09:40,660 If I'm aiming for a slope of x squared, then I need some 151 00:09:40,660 --> 00:09:43,010 number of x cubes. 152 00:09:43,010 --> 00:09:45,000 So how many x cubes do I need? 153 00:09:47,700 --> 00:09:51,420 Well, I need to know, what's the slope of x cube? 154 00:09:51,420 --> 00:09:56,360 The rule for powers of x, x to the n, is n 155 00:09:56,360 --> 00:09:58,330 times one smaller power. 156 00:09:58,330 --> 00:10:05,040 The slope of x cube is 3 times x squared. 157 00:10:05,040 --> 00:10:11,540 So I had better divide by that 3 so that the 3 cancels the 3. 158 00:10:11,540 --> 00:10:16,120 Now the slope of that, the 3x squared, the threes would 159 00:10:16,120 --> 00:10:17,740 cancel and I would get x squared. 160 00:10:17,740 --> 00:10:20,080 But I'm looking for 1/2 x squared. 161 00:10:20,080 --> 00:10:24,300 I need also a 2. 162 00:10:24,300 --> 00:10:28,410 Do you see that it's 1/6 of x cube that's 163 00:10:28,410 --> 00:10:29,740 going to do the job. 164 00:10:29,740 --> 00:10:33,520 1/6 of x cubed because the slope-- 165 00:10:33,520 --> 00:10:37,600 the 3 cancels the 3 and I wanted to end up with a 2. 166 00:10:37,600 --> 00:10:40,700 And now, do you know what's coming? 167 00:10:40,700 --> 00:10:42,040 These are supposed to be equal. 168 00:10:42,040 --> 00:10:45,140 I have to have this 1/6 x cubed down here too. 169 00:10:48,290 --> 00:10:49,775 And I never get to stop. 170 00:10:54,170 --> 00:10:58,570 We have to see, OK, what is a typical-- 171 00:10:58,570 --> 00:11:03,260 after I've done this, say, n times, I'd like to have some 172 00:11:03,260 --> 00:11:10,660 idea of what is it when I get up to x to some nth power, 173 00:11:10,660 --> 00:11:14,860 then it's multiplied by some fraction and I'm looking to 174 00:11:14,860 --> 00:11:17,840 see, what is that fraction? 175 00:11:17,840 --> 00:11:19,950 What is that fraction? 176 00:11:19,950 --> 00:11:24,050 And then, of course, they'll all show up down there again. 177 00:11:24,050 --> 00:11:30,780 Well, if you see this pattern, this was 3 times 2-- you could 178 00:11:30,780 --> 00:11:33,380 say 3 times 2 times 1. 179 00:11:33,380 --> 00:11:35,250 This one was 2 times 1. 180 00:11:35,250 --> 00:11:36,940 This one was just 1. 181 00:11:36,940 --> 00:11:40,200 It's n factorial. 182 00:11:40,200 --> 00:11:42,420 n factorial is what I need. 183 00:11:42,420 --> 00:11:46,620 I need n times n minus 1. 184 00:11:46,620 --> 00:11:50,100 I need all these numbers all the way and I'll throw in the 185 00:11:50,100 --> 00:11:51,420 1 at the end. 186 00:11:51,420 --> 00:11:58,580 And I have to put the mathematicians take it away 187 00:11:58,580 --> 00:12:01,870 symbol, the little three dots that mean 188 00:12:01,870 --> 00:12:04,230 don't stop, keep going. 189 00:12:04,230 --> 00:12:07,090 But do you see that this will be OK? 190 00:12:07,090 --> 00:12:12,670 This is called n factorial, x to the nth over n factorial, 191 00:12:12,670 --> 00:12:15,830 because when I take the slope of x to the nth, 192 00:12:15,830 --> 00:12:17,220 an n will come down. 193 00:12:17,220 --> 00:12:19,810 Cancel that n. 194 00:12:19,810 --> 00:12:23,180 x, I'll have one lower power. 195 00:12:23,180 --> 00:12:26,720 You see, when I take the slope of this, I'll have the n will 196 00:12:26,720 --> 00:12:27,960 cancel the n. 197 00:12:27,960 --> 00:12:33,900 So I'll still have these other guys down below. 198 00:12:33,900 --> 00:12:36,990 And I'll have x to the n minus 1. 199 00:12:36,990 --> 00:12:39,470 And that will be x to the n minus 1 200 00:12:39,470 --> 00:12:40,960 over n minus 1 factorial. 201 00:12:40,960 --> 00:12:42,750 That will be the previous one. 202 00:12:42,750 --> 00:12:50,860 But now I have to add in the x to the nth over n factorial 203 00:12:50,860 --> 00:12:54,850 because y and dy to the x have to be the same, so I have to 204 00:12:54,850 --> 00:12:55,610 keep going. 205 00:12:55,610 --> 00:12:56,640 OK. 206 00:12:56,640 --> 00:13:03,010 So you might say, well, you're going to blow up. 207 00:13:03,010 --> 00:13:06,590 Not personally, the series. 208 00:13:06,590 --> 00:13:09,830 But what saves you? 209 00:13:09,830 --> 00:13:15,690 What saves you is the fact that these n factorials, those 210 00:13:15,690 --> 00:13:19,840 fractions, that n factorial gets to be really large really 211 00:13:19,840 --> 00:13:25,650 fast, faster than this x to nth could grow. 212 00:13:25,650 --> 00:13:31,370 So altogether, these terms, x to the nth over n factorial, 213 00:13:31,370 --> 00:13:36,020 they get extremely, extremely small. 214 00:13:36,020 --> 00:13:42,790 And then this series of things, it comes to a limit. 215 00:13:42,790 --> 00:13:46,270 It doesn't keep going, getting bigger, and bigger, and bigger 216 00:13:46,270 --> 00:13:48,620 as I had more terms, because what I'm adding is 217 00:13:48,620 --> 00:13:50,935 so small, so small. 218 00:13:53,810 --> 00:13:59,200 And that's the point where we have to discuss limits later. 219 00:13:59,200 --> 00:13:59,980 OK. 220 00:13:59,980 --> 00:14:04,600 So that's my construction. 221 00:14:04,600 --> 00:14:05,850 Construction complete. 222 00:14:09,050 --> 00:14:11,380 The exponential function e to the x-- 223 00:14:11,380 --> 00:14:13,390 this is e to the x-- 224 00:14:13,390 --> 00:14:19,470 is being defined by 1 plus x plus 1/2 x squared plus 6 x 225 00:14:19,470 --> 00:14:21,350 cubed, and so on. 226 00:14:21,350 --> 00:14:22,600 OK. 227 00:14:24,430 --> 00:14:26,480 I've got a function. 228 00:14:26,480 --> 00:14:30,510 Now, its property. 229 00:14:30,510 --> 00:14:34,950 And the key property is this one. 230 00:14:34,950 --> 00:14:37,780 Can I move to the next board? 231 00:14:37,780 --> 00:14:40,580 So the next step is, check-- 232 00:14:40,580 --> 00:14:43,370 well, I've asked you. 233 00:14:43,370 --> 00:14:44,620 I've got e to the x. 234 00:14:46,562 --> 00:14:48,715 And let me write again what it is. 235 00:14:48,715 --> 00:14:57,810 1 plus x plus 1/2 x squared plus 1/6 x cubed and so on. 236 00:14:57,810 --> 00:15:01,880 And then I've got e to the any other power, or even the same 237 00:15:01,880 --> 00:15:03,990 power, 1 plus-- 238 00:15:03,990 --> 00:15:08,970 I'll just use capital x four this power. 239 00:15:08,970 --> 00:15:15,240 1/6 of capital x cubed plus so on. 240 00:15:15,240 --> 00:15:18,620 And I want to multiply those and see what I get. 241 00:15:21,200 --> 00:15:21,580 OK. 242 00:15:21,580 --> 00:15:23,840 I apologize. 243 00:15:23,840 --> 00:15:29,360 Here I ask you to believe in this infinite series, and 244 00:15:29,360 --> 00:15:34,560 yeah, a little dodgy, but it works. 245 00:15:34,560 --> 00:15:38,730 And now I ask you to multiply two of the things. 246 00:15:38,730 --> 00:15:41,380 You might say, OK, you're asking a lot here. 247 00:15:41,380 --> 00:15:45,030 But just hang on. 248 00:15:45,030 --> 00:15:47,430 Let's multiply these. 249 00:15:47,430 --> 00:15:52,850 e to the x times e to the capital x, because that's what 250 00:15:52,850 --> 00:15:55,560 I'm interested in knowing. 251 00:15:55,560 --> 00:15:56,810 OK. 252 00:15:59,760 --> 00:16:01,470 Just do all the multiplications. 253 00:16:05,590 --> 00:16:07,750 And we'll see what we get. 254 00:16:07,750 --> 00:16:09,350 OK, so 1 times 1 is 1. 255 00:16:09,350 --> 00:16:10,580 No problem. 256 00:16:10,580 --> 00:16:13,000 1 times x is the x. 257 00:16:13,000 --> 00:16:16,210 1 times this x is the big x. 258 00:16:16,210 --> 00:16:17,490 Now can I keep going? 259 00:16:17,490 --> 00:16:22,640 All right, well, 1 times 1/2 x squared is-- 260 00:16:22,640 --> 00:16:27,620 and now I have x times a big x. 261 00:16:27,620 --> 00:16:30,870 And now I have a 1 times 1/2 big x squared. 262 00:16:34,050 --> 00:16:36,570 And more, of course. 263 00:16:36,570 --> 00:16:42,370 Notice the way I'm doing is like I'm keeping all the 264 00:16:42,370 --> 00:16:45,600 things that have two x's together. 265 00:16:45,600 --> 00:16:48,020 And then I would keep all the things that have three x's 266 00:16:48,020 --> 00:16:50,160 together, and so on. 267 00:16:50,160 --> 00:16:54,140 Now what is it that I'm hoping? 268 00:16:54,140 --> 00:17:04,890 I'm hoping that this is the same as the series for x plus 269 00:17:04,890 --> 00:17:08,240 capital x, OK? 270 00:17:08,240 --> 00:17:09,750 What's that? 271 00:17:09,750 --> 00:17:12,099 That's my exponential series. 272 00:17:12,099 --> 00:17:16,030 And every time, I have to put in x plus capital x. 273 00:17:16,030 --> 00:17:19,180 In other words, of course, it starts with 1. 274 00:17:19,180 --> 00:17:22,210 Then it has the x plus capital x. 275 00:17:22,210 --> 00:17:27,380 And then it has the 1/2 of x plus capital x squared. 276 00:17:27,380 --> 00:17:28,630 And it keeps going. 277 00:17:32,030 --> 00:17:35,170 And I just wanted you to say, yes. 278 00:17:35,170 --> 00:17:40,920 I guess I hope you say yes when I ask, is this big 279 00:17:40,920 --> 00:17:44,080 multiplication the same as this one? 280 00:17:44,080 --> 00:17:45,760 Well, I think it is. 281 00:17:45,760 --> 00:17:48,530 Let's just start to check, anyway. 282 00:17:48,530 --> 00:17:50,020 The ones are good. 283 00:17:50,020 --> 00:17:50,890 The x and the x-- 284 00:17:50,890 --> 00:17:56,680 I'm really just putting parentheses around all the-- 285 00:17:56,680 --> 00:17:59,440 now I'm going to put parentheses around all the 286 00:17:59,440 --> 00:18:05,360 second degree terms and say, is that the same as that? 287 00:18:05,360 --> 00:18:06,330 Yeah. 288 00:18:06,330 --> 00:18:09,080 This is the critical point here. 289 00:18:09,080 --> 00:18:13,690 Do we, at least, start out correctly? 290 00:18:13,690 --> 00:18:16,730 So we have to remember, how do you do-- but, of course, you 291 00:18:16,730 --> 00:18:22,000 do remember how to multiply x plus capital x by itself. 292 00:18:22,000 --> 00:18:24,790 You just do the multiplications. 293 00:18:24,790 --> 00:18:29,990 x, when I multiply that by itself, I get x squared. 294 00:18:29,990 --> 00:18:32,060 With 1/2, I get that. 295 00:18:32,060 --> 00:18:33,900 And then, you remember? 296 00:18:33,900 --> 00:18:37,360 How many x times x's do I get? 297 00:18:37,360 --> 00:18:41,380 Little x times big x, there'd be two of those. 298 00:18:41,380 --> 00:18:44,700 But then the 1/2 factor leaves me with 1, and 299 00:18:44,700 --> 00:18:47,030 that's what I want. 300 00:18:47,030 --> 00:18:51,940 And then, finally, this guy by himself squared is the 1/2 301 00:18:51,940 --> 00:18:54,520 capital x squared that I also want. 302 00:18:54,520 --> 00:18:55,770 So far, so good. 303 00:18:59,430 --> 00:19:03,530 Do you want to see the cubed terms? 304 00:19:07,150 --> 00:19:12,550 Well, I'd rather you did it, but I should at least show 305 00:19:12,550 --> 00:19:16,260 that I'm willing to try. 306 00:19:16,260 --> 00:19:19,230 So what do I mean by the cubed terms? 307 00:19:19,230 --> 00:19:24,800 I mean that here, I want to get-- the next one should be 308 00:19:24,800 --> 00:19:29,360 1/6 of x plus x cubed. 309 00:19:29,360 --> 00:19:35,650 And from the multiplication, I get some separate pieces. 310 00:19:35,650 --> 00:19:38,750 I get 1 times-- 311 00:19:38,750 --> 00:19:43,690 when I do that multiplication, I get 1/6 x cubed. 312 00:19:43,690 --> 00:19:49,750 And then I maybe get some 1/2 x squared times x. 313 00:19:49,750 --> 00:19:56,040 You see why I would rather you did this. 314 00:19:56,040 --> 00:19:58,210 But I'll finish this little line. 315 00:19:58,210 --> 00:20:04,500 There's also an x times 1/2 x squared. 316 00:20:04,500 --> 00:20:08,780 So that's 1/2 of x times the big x squared. 317 00:20:08,780 --> 00:20:12,775 And then there is the 1 times the 1/6 x cubed. 318 00:20:19,270 --> 00:20:23,560 So those are the four pieces that come, third degree, when 319 00:20:23,560 --> 00:20:25,360 I do the big multiplication. 320 00:20:25,360 --> 00:20:27,570 And they have to match the third degree 321 00:20:27,570 --> 00:20:30,820 term in the last line. 322 00:20:30,820 --> 00:20:32,270 And they do match. 323 00:20:35,200 --> 00:20:40,530 Do you remember the right words to say now? 324 00:20:40,530 --> 00:20:42,970 Binomial theorem. 325 00:20:42,970 --> 00:20:48,680 The binomial theorem tells you how to take the nth power all 326 00:20:48,680 --> 00:20:54,180 a sum like x plus capital x to the nth power. 327 00:20:54,180 --> 00:20:58,200 It tells you all the many pieces you get. 328 00:20:58,200 --> 00:21:02,960 And those many pieces are exactly the pieces that we get 329 00:21:02,960 --> 00:21:10,490 directly by multiplying that line by that line. 330 00:21:10,490 --> 00:21:15,920 So the binomial theorem, at long last, pays off and 331 00:21:15,920 --> 00:21:19,780 confirms our great property here. 332 00:21:19,780 --> 00:21:22,020 So this is a big deal. 333 00:21:25,800 --> 00:21:27,050 OK. 334 00:21:30,560 --> 00:21:33,880 So let me now come back here, having checked that. 335 00:21:36,760 --> 00:21:41,950 I wanted to say something about this series, 1 plus x 336 00:21:41,950 --> 00:21:45,430 plus 1/2 x squared, where the typical term is x to the nth 337 00:21:45,430 --> 00:21:46,900 over n factorial. 338 00:21:46,900 --> 00:21:53,740 This is the, I would say, the second most important infinite 339 00:21:53,740 --> 00:21:58,510 series in mathematics, the exponential series. 340 00:21:58,510 --> 00:22:06,010 And it's the way I wanted to construct e to the x by 341 00:22:06,010 --> 00:22:11,460 matching term by term and seeing that these n 342 00:22:11,460 --> 00:22:13,260 factorials show up. 343 00:22:13,260 --> 00:22:17,900 You might want to know, what's the most important series? 344 00:22:17,900 --> 00:22:20,220 Reasonable question. 345 00:22:20,220 --> 00:22:25,380 For me, the most important series would be the one 346 00:22:25,380 --> 00:22:30,060 looking like this, except it doesn't have the fractions. 347 00:22:30,060 --> 00:22:33,230 For me, the most important series would be the one-- 348 00:22:33,230 --> 00:22:35,410 I'll slip it up here-- 349 00:22:35,410 --> 00:22:43,500 1 plus x plus x squared, without the 1/2, plus x cubed, 350 00:22:43,500 --> 00:22:49,570 without the 1/6, plus so on, plus x to the n without this n 351 00:22:49,570 --> 00:22:52,445 factorial that's making it so small. 352 00:22:55,310 --> 00:23:00,200 Can you see this 1 plus x plus x squared plus x cubed 353 00:23:00,200 --> 00:23:02,640 plus x to the n? 354 00:23:02,640 --> 00:23:06,560 That, I think it's called the geometric series. 355 00:23:06,560 --> 00:23:08,620 Powers of x. 356 00:23:08,620 --> 00:23:14,260 Now, it's simpler because it doesn't have these fractions. 357 00:23:14,260 --> 00:23:19,320 But it's riskier because those fractions were making the 358 00:23:19,320 --> 00:23:22,420 exponential series succeed. 359 00:23:22,420 --> 00:23:26,160 Whereas here, with the geometric series, well, look 360 00:23:26,160 --> 00:23:29,330 what happens when x is 1. 361 00:23:29,330 --> 00:23:35,550 When x is 1, we have 1 plus 1 plus 1 plus 1 plus 1 forever. 362 00:23:35,550 --> 00:23:36,270 All ones. 363 00:23:36,270 --> 00:23:38,710 It blows up. 364 00:23:38,710 --> 00:23:40,960 And when x is bigger than 1, that series 365 00:23:40,960 --> 00:23:44,130 blows up even faster. 366 00:23:44,130 --> 00:23:47,350 So in this series, the geometric series, this most 367 00:23:47,350 --> 00:23:54,260 important one, does succeed but only when x is below 1. 368 00:23:54,260 --> 00:23:58,050 x equal 1 is the cutoff and it fails after that. 369 00:23:58,050 --> 00:24:03,760 There is no cutoff for the exponential series because of 370 00:24:03,760 --> 00:24:07,940 dividing by these bigger and bigger numbers. 371 00:24:07,940 --> 00:24:09,910 This works for all x. 372 00:24:09,910 --> 00:24:12,290 OK, so those are the two series. 373 00:24:12,290 --> 00:24:13,540 OK. 374 00:24:15,410 --> 00:24:21,470 So let me ask you, what happens if I put x equal 1 in 375 00:24:21,470 --> 00:24:25,550 the exponential series? 376 00:24:25,550 --> 00:24:31,630 That gives me e to the first power, which is e. 377 00:24:31,630 --> 00:24:35,770 So finally, you may say, it's rather late in the day. 378 00:24:35,770 --> 00:24:41,100 i'm going to figure out what e is from this series. 379 00:24:41,100 --> 00:24:47,770 Put in set x equal 1 and you learn that e to the first 380 00:24:47,770 --> 00:24:51,770 power, which is e, is-- 381 00:24:51,770 --> 00:24:53,060 can I just put it in? 382 00:24:53,060 --> 00:25:02,180 1 plus x is 1 plus 1/2 of 1 squared plus 1/6 of 1 cubed. 383 00:25:02,180 --> 00:25:06,470 What's the next term in this? 384 00:25:06,470 --> 00:25:09,160 So these are numbers now, and I'm getting a number. 385 00:25:09,160 --> 00:25:14,290 I'm getting this incredible number e, named after Euler. 386 00:25:14,290 --> 00:25:16,360 Euler was a fantastic mathematician. 387 00:25:16,360 --> 00:25:21,110 I think he wrote more important papers then any 388 00:25:21,110 --> 00:25:22,360 mathematician in history. 389 00:25:25,450 --> 00:25:31,620 So he was allowed to name this number after himself, e. 390 00:25:31,620 --> 00:25:34,890 E-U-L-E-R, his name is spelled. 391 00:25:34,890 --> 00:25:36,180 OK, what's the next term? 392 00:25:38,930 --> 00:25:40,910 This is 3 factorial, right? 393 00:25:40,910 --> 00:25:41,990 3 times 2 times 1. 394 00:25:41,990 --> 00:25:43,920 The next term will be 4 factorial. 395 00:25:43,920 --> 00:25:45,570 I'll multiply that by 4. 396 00:25:45,570 --> 00:25:47,140 It'll be 1/24. 397 00:25:47,140 --> 00:25:48,750 And then times 5. 398 00:25:48,750 --> 00:25:51,090 1/120, and so on. 399 00:25:51,090 --> 00:25:54,310 They're getting small. 400 00:25:54,310 --> 00:25:58,090 What can I tell you about this number? 401 00:25:58,090 --> 00:25:59,355 It will be a definite number. 402 00:26:02,300 --> 00:26:04,230 And is more than-- 403 00:26:04,230 --> 00:26:06,840 well, it's certainly more than 2 1/2, because I 404 00:26:06,840 --> 00:26:08,250 start with 2 1/2 here. 405 00:26:08,250 --> 00:26:09,720 And then I add these. 406 00:26:09,720 --> 00:26:12,510 Well, I could even throw in 1/6. 407 00:26:12,510 --> 00:26:16,350 That's more than 2 2/3, would that be? 408 00:26:16,350 --> 00:26:19,090 If I quit here, I'd have 2 2/3. 409 00:26:19,090 --> 00:26:21,460 And then I get a little more. 410 00:26:21,460 --> 00:26:22,740 It's easy to show. 411 00:26:22,740 --> 00:26:25,790 No way you would reach as far as 3. 412 00:26:25,790 --> 00:26:30,450 These later terms are dropping too fast. And actually, the 413 00:26:30,450 --> 00:26:32,590 number turns out to be-- 414 00:26:32,590 --> 00:26:34,850 so it's 2 point something. 415 00:26:34,850 --> 00:26:36,360 2 point-- 416 00:26:36,360 --> 00:26:39,820 let's see, a little more than 2 2/3, so it's around 2.7. 417 00:26:43,690 --> 00:26:47,290 But it's it's not exactly 2.7. 418 00:26:47,290 --> 00:26:50,530 In fact, it's not exactly any fraction 419 00:26:50,530 --> 00:26:53,690 or any finite decimal. 420 00:26:53,690 --> 00:26:54,980 It goes on and on. 421 00:26:54,980 --> 00:26:59,800 1, 8, 2, 8, something. 422 00:26:59,800 --> 00:27:04,280 I think there are more eights than you'd expect right here 423 00:27:04,280 --> 00:27:10,370 at the beginning, but then, in the long run, not. 424 00:27:10,370 --> 00:27:13,010 So that's the number, e. 425 00:27:13,010 --> 00:27:14,025 OK. 426 00:27:14,025 --> 00:27:16,330 Oh, so now we know e. 427 00:27:16,330 --> 00:27:17,480 We know e to the x. 428 00:27:17,480 --> 00:27:18,340 We know e. 429 00:27:18,340 --> 00:27:19,810 We know this thing. 430 00:27:19,810 --> 00:27:21,750 I should draw a graph, right? 431 00:27:21,750 --> 00:27:23,080 That's the other thing you do with a 432 00:27:23,080 --> 00:27:25,860 function is draw a graph. 433 00:27:25,860 --> 00:27:26,610 OK. 434 00:27:26,610 --> 00:27:29,740 So here's a graph. 435 00:27:29,740 --> 00:27:31,410 This is x. 436 00:27:31,410 --> 00:27:36,040 Let me put in x equals 0 here and x equal 1 here. 437 00:27:36,040 --> 00:27:39,460 And this is going to be a graph of e to the x. 438 00:27:39,460 --> 00:27:44,820 And at x equals 0, what is it? 439 00:27:44,820 --> 00:27:46,060 We started with that. 440 00:27:46,060 --> 00:27:46,950 It should be-- 441 00:27:46,950 --> 00:27:48,090 so this is y. 442 00:27:48,090 --> 00:27:50,060 I'm graphing y. 443 00:27:50,060 --> 00:27:53,430 And it starts at 1. 444 00:27:53,430 --> 00:27:54,210 That's what we said. 445 00:27:54,210 --> 00:27:58,280 At x equals 0, I've started at 1 with a slope of 1. 446 00:27:58,280 --> 00:28:00,920 So I have a slope of 1, but the slope, the slope, the 447 00:28:00,920 --> 00:28:02,260 slope is climbing up. 448 00:28:02,260 --> 00:28:05,530 And it reaches here. 449 00:28:05,530 --> 00:28:06,890 That height is what-- 450 00:28:09,490 --> 00:28:10,830 e. 451 00:28:10,830 --> 00:28:12,080 That height is e. 452 00:28:14,410 --> 00:28:18,140 Because when we said x equal 1 here, we got e. 453 00:28:18,140 --> 00:28:20,410 So it's climbing, climbing, climbing. 454 00:28:20,410 --> 00:28:23,130 And now what about on the other side? 455 00:28:23,130 --> 00:28:28,470 That had a slope of 1, so it was more like that. 456 00:28:28,470 --> 00:28:32,890 Now what about when x is negative? 457 00:28:32,890 --> 00:28:38,770 When x is negative, this is a highly useful fact. 458 00:28:38,770 --> 00:28:44,010 Suppose I want to think about e to the minus x. 459 00:28:44,010 --> 00:28:47,240 Well now, let me just take capital x to be 460 00:28:47,240 --> 00:28:48,420 minus little x. 461 00:28:48,420 --> 00:28:52,410 So I get e to the x times e to the minus x. 462 00:28:52,410 --> 00:28:54,220 What is that? 463 00:28:54,220 --> 00:28:57,110 What does that equal if I multiply e to the x times e to 464 00:28:57,110 --> 00:29:00,090 the minus x? 465 00:29:00,090 --> 00:29:03,310 As usual, I'm supposed to add these. 466 00:29:03,310 --> 00:29:07,100 I get 0, so I get e to the 0, which is 1. 467 00:29:07,100 --> 00:29:13,030 In other words, e to the minus x is 1 over e to the x, which 468 00:29:13,030 --> 00:29:15,670 we fully expected. 469 00:29:15,670 --> 00:29:22,430 So that at x equal minus 1 here, I'm down to 1 over e, 470 00:29:22,430 --> 00:29:24,770 1/3, approximately. 471 00:29:24,770 --> 00:29:28,120 So it's going down. 472 00:29:28,120 --> 00:29:32,100 In this way, it's decaying very fast. It almost touches 473 00:29:32,100 --> 00:29:34,480 that line, but never quite. 474 00:29:34,480 --> 00:29:35,730 This way, it's climbing. 475 00:29:38,860 --> 00:29:41,600 It's growing, growing really-- 476 00:29:41,600 --> 00:29:43,243 well, it's growing exponentially. 477 00:29:46,650 --> 00:29:49,710 And that's what this graph looks like. 478 00:29:49,710 --> 00:29:55,690 And now I would like to connect back, at the end of 479 00:29:55,690 --> 00:30:01,650 this lecture, to the insurance business-- 480 00:30:01,650 --> 00:30:07,680 sorry, the interest business, the bank compounding interest. 481 00:30:07,680 --> 00:30:15,940 Can I take your time with that important example of the 482 00:30:15,940 --> 00:30:17,130 exponential function? 483 00:30:17,130 --> 00:30:20,430 And we'll see a new way to reach e. 484 00:30:20,430 --> 00:30:22,300 I like this way. 485 00:30:22,300 --> 00:30:26,320 I like the way we did it with the infinite series. 486 00:30:26,320 --> 00:30:29,280 But here's another way. 487 00:30:29,280 --> 00:30:37,540 So suppose you're getting 100% interest. Generous bank. 488 00:30:37,540 --> 00:30:38,870 OK. 489 00:30:38,870 --> 00:30:43,870 And you start with $1 at 100% now. 490 00:30:43,870 --> 00:30:46,696 It's 100%. 491 00:30:46,696 --> 00:30:49,450 And the bank gives you interest at the 492 00:30:49,450 --> 00:30:51,770 end of every year. 493 00:30:51,770 --> 00:30:57,020 So at the end of the first year, you had $1 dollar in the 494 00:30:57,020 --> 00:31:00,890 bank, it adds in 100%. 495 00:31:00,890 --> 00:31:02,780 It adds in another dollar. 496 00:31:02,780 --> 00:31:08,360 So now you've got $2 in the bank after the first year. 497 00:31:08,360 --> 00:31:13,390 At the end of the second year, it gives you 100% of what 498 00:31:13,390 --> 00:31:14,360 you've got in the bank. 499 00:31:14,360 --> 00:31:15,500 So it gives you 2 more. 500 00:31:15,500 --> 00:31:18,030 It give you 4. 501 00:31:18,030 --> 00:31:22,050 At the end of the third year, it gives you an additional 4. 502 00:31:22,050 --> 00:31:24,220 You're up 50 to 8. 503 00:31:24,220 --> 00:31:26,190 And you see what's happening. 504 00:31:26,190 --> 00:31:29,130 It's the powers of two. 505 00:31:29,130 --> 00:31:33,770 Well, that's pretty good growth. 506 00:31:33,770 --> 00:31:37,140 But it's not calculus. 507 00:31:37,140 --> 00:31:41,660 Calculus doesn't do things in steps of a year. 508 00:31:41,660 --> 00:31:44,160 Calculus says cut that step down. 509 00:31:44,160 --> 00:31:48,120 You would want to ask your bank, couldn't you just, like, 510 00:31:48,120 --> 00:31:51,430 figure the interest a little more often and put it in 511 00:31:51,430 --> 00:31:53,890 there-- like, figure it every month? 512 00:31:53,890 --> 00:31:55,550 So what would happen if you figured the 513 00:31:55,550 --> 00:31:58,100 interest every month? 514 00:31:58,100 --> 00:32:01,590 Of course, you wouldn't get 100% interest in a month. 515 00:32:01,590 --> 00:32:06,710 You'd get 100% divided by 12, because we're only talking 516 00:32:06,710 --> 00:32:08,540 about one month. 517 00:32:08,540 --> 00:32:12,900 So if it was months, you start with 1. 518 00:32:12,900 --> 00:32:18,350 You have 1 plus 1/12. 519 00:32:18,350 --> 00:32:21,880 That's what you'd have after a month. 520 00:32:21,880 --> 00:32:24,970 Now, what would you have after 2 months and what would you 521 00:32:24,970 --> 00:32:29,640 have after 12 months? 522 00:32:29,640 --> 00:32:31,935 Well, we're going to follow the rule. 523 00:32:35,670 --> 00:32:39,030 They gave you the 1/12 at the end of January. 524 00:32:39,030 --> 00:32:41,730 So through all of February, you've got 1 525 00:32:41,730 --> 00:32:43,680 plus 1/12 in there. 526 00:32:43,680 --> 00:32:51,420 At the end of February, they take 1/12 of that, add it in. 527 00:32:51,420 --> 00:33:00,400 What you get the next time is 1 plus 1/12 squared. 528 00:33:00,400 --> 00:33:04,860 That's what you have. 529 00:33:04,860 --> 00:33:08,310 Essentially every time, they're going to multiply what 530 00:33:08,310 --> 00:33:10,860 you've got by this number 1 plus 1/12. 531 00:33:10,860 --> 00:33:13,140 1 to give you-- 532 00:33:13,140 --> 00:33:14,320 leave the money in. 533 00:33:14,320 --> 00:33:15,560 You have to leave your money. 534 00:33:15,560 --> 00:33:17,790 I'm sorry. 535 00:33:17,790 --> 00:33:23,060 Plus 1/12 of it for the interest. And then twice, and 536 00:33:23,060 --> 00:33:27,060 after 1 year, it's done this. 537 00:33:27,060 --> 00:33:30,360 You see what happens after 1 year, it's 538 00:33:30,360 --> 00:33:34,540 multiplied 12 times. 539 00:33:34,540 --> 00:33:38,010 1 plus 1/12 to the 12th power. 540 00:33:38,010 --> 00:33:40,720 And that's better than 2, right? 541 00:33:40,720 --> 00:33:45,000 You've got the 2 only when they put the interest in just 542 00:33:45,000 --> 00:33:45,630 once a year. 543 00:33:45,630 --> 00:33:51,380 Now we're speeding up the bank and getting more out of it. 544 00:33:51,380 --> 00:33:55,170 So I don't know exactly what 1 plus 1/12 to the 12th power 545 00:33:55,170 --> 00:33:57,700 is, but I know it's more than 2. 546 00:33:57,700 --> 00:34:01,080 And actually, I'm sure it's not more than 3. 547 00:34:04,160 --> 00:34:09,420 In fact, yeah, I'm claiming that it's not 548 00:34:09,420 --> 00:34:12,699 as much as e, 2.7. 549 00:34:12,699 --> 00:34:18,610 But it was worth doing, to get them to compound every month. 550 00:34:18,610 --> 00:34:21,790 But, of course, you think, okay, I'm on to a good thing. 551 00:34:21,790 --> 00:34:22,600 Every day. 552 00:34:22,600 --> 00:34:24,120 Why not? 553 00:34:24,120 --> 00:34:26,770 So what would every day be? 554 00:34:26,770 --> 00:34:31,830 1 plus 1/365. 555 00:34:31,830 --> 00:34:34,620 That's the interest you would get for just that day. 556 00:34:34,620 --> 00:34:39,810 But then they would compound it 365 times. 557 00:34:39,810 --> 00:34:44,090 So that would be a little more than this because they're 558 00:34:44,090 --> 00:34:47,600 adding the interest in more frequently. 559 00:34:47,600 --> 00:34:51,880 And, in general, I'm going to divide the 560 00:34:51,880 --> 00:34:54,540 year up into n pieces. 561 00:34:54,540 --> 00:34:59,730 In every piece, they multiply my wealth by 1 plus 1 over n. 562 00:34:59,730 --> 00:35:01,685 And they do it n times in a year. 563 00:35:05,020 --> 00:35:11,110 And the beautiful thing is that as n goes to infinity, 564 00:35:11,110 --> 00:35:15,150 and calculus comes in, because we're asking them to compound 565 00:35:15,150 --> 00:35:20,450 interest continuously, not just every month, not every 566 00:35:20,450 --> 00:35:26,100 day, every second even, but all the time. 567 00:35:26,100 --> 00:35:28,850 You don't get an infinite amount out of this. 568 00:35:28,850 --> 00:35:31,430 You get e. 569 00:35:31,430 --> 00:35:35,140 As n gets bigger, that approaches this number e. 570 00:35:35,140 --> 00:35:39,330 That's another way to construct e, as the limit-- 571 00:35:39,330 --> 00:35:45,000 you see, as n gets bigger, it's like 1 to the infinity, 572 00:35:45,000 --> 00:35:48,400 which is kind of meaningless. 573 00:35:48,400 --> 00:35:50,460 I don't want to say that 1 to the-- 574 00:35:50,460 --> 00:35:53,990 I had an email the other day that said, well, 1 to the 575 00:35:53,990 --> 00:35:56,350 infinity is e. 576 00:35:56,350 --> 00:35:57,140 What's happening? 577 00:35:57,140 --> 00:35:58,890 That's not true. 578 00:35:58,890 --> 00:36:04,190 It's this thing that's going to 1, this thing that's going 579 00:36:04,190 --> 00:36:05,610 to infinity. 580 00:36:05,610 --> 00:36:08,780 Then the combination goes to e. 581 00:36:08,780 --> 00:36:09,610 OK. 582 00:36:09,610 --> 00:36:16,030 So that's the application that shows the 583 00:36:16,030 --> 00:36:18,850 number e appearing again. 584 00:36:18,850 --> 00:36:20,100 OK. 585 00:36:23,140 --> 00:36:25,240 You've got the essence of e to the x. 586 00:36:28,280 --> 00:36:31,780 I just would like to say one thing, coming back to the very 587 00:36:31,780 --> 00:36:33,310 beginning here. 588 00:36:33,310 --> 00:36:37,730 The great differential equation, dy dx equal y. 589 00:36:37,730 --> 00:36:39,480 That was beautiful. 590 00:36:39,480 --> 00:36:41,620 Which we've now solved. 591 00:36:41,620 --> 00:36:46,560 Now I want to ask, what if the differential equation was dy 592 00:36:46,560 --> 00:36:51,570 dx is some multiple of y? 593 00:36:51,570 --> 00:36:53,620 How would that come up? 594 00:36:53,620 --> 00:36:57,360 Well, up to now, c was 1. 595 00:36:57,360 --> 00:37:02,840 We were getting 100% interest per year. 596 00:37:02,840 --> 00:37:06,260 But now, if c is sort of the interest rate, the growth 597 00:37:06,260 --> 00:37:10,580 rate, or the decay rate of c is negative, we may be losing 598 00:37:10,580 --> 00:37:15,890 money in this bank. 599 00:37:15,890 --> 00:37:22,090 So can I just tell you what is the solution to this 600 00:37:22,090 --> 00:37:24,750 differential equation? 601 00:37:24,750 --> 00:37:27,960 When I tell you, and we learned about taking 602 00:37:27,960 --> 00:37:30,650 derivatives, you'll see, of course, that's all it is. 603 00:37:30,650 --> 00:37:33,550 It's just the solution to this one. 604 00:37:33,550 --> 00:37:35,800 I'll also start at one. 605 00:37:35,800 --> 00:37:40,010 The solution to that one is y of x is e-- 606 00:37:40,010 --> 00:37:41,710 e is coming in again-- 607 00:37:41,710 --> 00:37:42,960 to the cx. 608 00:37:45,430 --> 00:37:49,790 What I'm doing is like changing the rate at-- 609 00:37:49,790 --> 00:37:52,490 I've made the rate of chance c. 610 00:37:52,490 --> 00:37:56,850 And then that c is going to come up there and in the 611 00:37:56,850 --> 00:38:03,030 derivative, the slope of this guy, that c will come down. 612 00:38:03,030 --> 00:38:07,530 The slope of this will be c e to the cx, which is cy, which 613 00:38:07,530 --> 00:38:09,680 is what that second differential 614 00:38:09,680 --> 00:38:11,540 equation tells us. 615 00:38:11,540 --> 00:38:16,460 So that's just a comment looking ahead, that we've 616 00:38:16,460 --> 00:38:19,290 solved not only the most important differential 617 00:38:19,290 --> 00:38:21,380 equation with the most important function that 618 00:38:21,380 --> 00:38:26,680 calculus creates but a whole collection of related 619 00:38:26,680 --> 00:38:34,390 equations in which the rate can be any fixed number, c. 620 00:38:34,390 --> 00:38:35,050 OK. 621 00:38:35,050 --> 00:38:35,780 Thank you. 622 00:38:35,780 --> 00:38:38,010 FEMALE SPEAKER: This has been a production of MIT 623 00:38:38,010 --> 00:38:40,390 OpenCourseWare and Gilbert Strang. 624 00:38:40,390 --> 00:38:42,670 Funding for this video was provided by the Lord 625 00:38:42,670 --> 00:38:43,890 Foundation. 626 00:38:43,890 --> 00:38:47,020 To help OCW continue to provide free and open access 627 00:38:47,020 --> 00:38:50,090 to MIT courses, please make a donation at 628 00:38:50,090 --> 00:38:51,650 ocw.mit.edu/donate.