1 00:00:00,980 --> 00:00:06,370 GILBERT STRANG: OK well, here we're at the beginning. 2 00:00:06,370 --> 00:00:12,200 And that I think it's worth thinking about what we know. 3 00:00:12,200 --> 00:00:13,830 Calculus. 4 00:00:13,830 --> 00:00:16,950 Differential equations is the big application of calculus, 5 00:00:16,950 --> 00:00:21,530 so it's kind of interesting to see what part of calculus, what 6 00:00:21,530 --> 00:00:24,360 information and what ideas from calculus, 7 00:00:24,360 --> 00:00:27,190 actually get used in differential equations. 8 00:00:27,190 --> 00:00:31,440 And I'm going to show you what I see, 9 00:00:31,440 --> 00:00:34,450 and it's not everything by any means, 10 00:00:34,450 --> 00:00:41,420 it's some basic ideas, but not all the details you learned. 11 00:00:41,420 --> 00:00:43,460 So I'm not saying forget all those, 12 00:00:43,460 --> 00:00:48,930 but just focus on what matters. 13 00:00:48,930 --> 00:00:49,690 OK. 14 00:00:49,690 --> 00:00:54,500 So the calculus you need is my topic. 15 00:00:54,500 --> 00:00:56,760 And the first thing is, you really do 16 00:00:56,760 --> 00:01:00,790 need to know basic derivatives. 17 00:01:00,790 --> 00:01:04,150 The derivative of x to the n, the derivative 18 00:01:04,150 --> 00:01:06,120 of sine and cosine. 19 00:01:06,120 --> 00:01:11,290 Above all, the derivative of e to the x, which is e to the x. 20 00:01:11,290 --> 00:01:13,870 The derivative of e to the x is e to the x. 21 00:01:13,870 --> 00:01:18,353 That's the wonderful equation that is solved by e to the x. 22 00:01:18,353 --> 00:01:21,200 Dy dt equals y. 23 00:01:21,200 --> 00:01:23,070 We'll have to do more with that. 24 00:01:23,070 --> 00:01:28,240 And then the inverse function related to the exponential 25 00:01:28,240 --> 00:01:29,440 is the logarithm. 26 00:01:29,440 --> 00:01:32,901 With that special derivative of 1/x. 27 00:01:32,901 --> 00:01:33,400 OK. 28 00:01:33,400 --> 00:01:35,060 But you know those. 29 00:01:35,060 --> 00:01:41,440 Secondly, out of those few specific facts, 30 00:01:41,440 --> 00:01:46,220 you can create the derivatives of an enormous array 31 00:01:46,220 --> 00:01:50,020 of functions using the key rules. 32 00:01:50,020 --> 00:01:54,100 The derivative of f plus g is the derivative 33 00:01:54,100 --> 00:01:56,780 of f plus the derivative of g. 34 00:01:56,780 --> 00:01:59,730 Derivative is a linear operation. 35 00:01:59,730 --> 00:02:05,300 The product rule fg prime plus gf prime. 36 00:02:05,300 --> 00:02:06,600 The quotient rule. 37 00:02:06,600 --> 00:02:08,259 Who can remember that? 38 00:02:08,259 --> 00:02:12,090 And above all, the chain rule. 39 00:02:12,090 --> 00:02:15,880 The derivative of this-- of that chain of functions, 40 00:02:15,880 --> 00:02:23,080 that composite function is the derivative of f with respect 41 00:02:23,080 --> 00:02:28,360 to g times the derivative of g with respect to x. 42 00:02:28,360 --> 00:02:32,070 That's really-- that it's chains of functions 43 00:02:32,070 --> 00:02:37,120 that really blow open the functions or we can deal with. 44 00:02:37,120 --> 00:02:38,610 OK. 45 00:02:38,610 --> 00:02:42,890 And then the fundamental theorem. 46 00:02:42,890 --> 00:02:45,440 So the fundamental theorem involves the derivative 47 00:02:45,440 --> 00:02:47,040 and the integral. 48 00:02:47,040 --> 00:02:52,340 And it says that one is the inverse operation to the other. 49 00:02:52,340 --> 00:03:03,600 The derivative of the integral of a function is this. 50 00:03:03,600 --> 00:03:09,050 Here is y and the integral goes from 0 51 00:03:09,050 --> 00:03:13,400 to x I don't care what that dummy variable is. 52 00:03:13,400 --> 00:03:17,030 I can-- I'll change that dummy variable to t. 53 00:03:17,030 --> 00:03:17,890 Whatever. 54 00:03:17,890 --> 00:03:20,830 I don't care. 55 00:03:20,830 --> 00:03:23,790 [? ET ?] to show the dummy variable. 56 00:03:23,790 --> 00:03:27,740 The x is the limit of integration. 57 00:03:27,740 --> 00:03:31,780 I won't discuss that fundamental theorem, 58 00:03:31,780 --> 00:03:35,870 but it certainly is fundamental and I'll use it. 59 00:03:35,870 --> 00:03:37,190 Maybe that's better. 60 00:03:37,190 --> 00:03:39,800 I'll use the fundamental theorem right away. 61 00:03:39,800 --> 00:03:42,520 So-- but remember what it says. 62 00:03:42,520 --> 00:03:46,110 It says that if you take a function, you integrate it, 63 00:03:46,110 --> 00:03:50,100 you take the derivative, you get the function back again. 64 00:03:50,100 --> 00:03:54,260 OK can I apply that to a really-- 65 00:03:54,260 --> 00:04:02,140 I see this as a key example in differential equations. 66 00:04:02,140 --> 00:04:04,980 And let me show you the function I have in mind. 67 00:04:04,980 --> 00:04:08,570 The function I have in mind, I'll call it y, 68 00:04:08,570 --> 00:04:13,670 is the interval from 0 to t. 69 00:04:13,670 --> 00:04:17,630 So it's a function of t then, time, It's 70 00:04:17,630 --> 00:04:23,210 the integral of this, e to the t minus s. 71 00:04:23,210 --> 00:04:24,015 Some function. 72 00:04:28,620 --> 00:04:35,650 That's a remarkable formula for the solution 73 00:04:35,650 --> 00:04:38,110 to a basic differential equation. 74 00:04:38,110 --> 00:04:43,770 So with this, that solves the equation dy 75 00:04:43,770 --> 00:04:52,100 dt equals y plus q of t. 76 00:04:52,100 --> 00:04:54,680 So when I see that equation and we'll see it again 77 00:04:54,680 --> 00:04:58,430 and we'll derive this formula, but now I 78 00:04:58,430 --> 00:05:02,870 want to just use the fundamental theorem of calculus 79 00:05:02,870 --> 00:05:05,300 to check the formula. 80 00:05:05,300 --> 00:05:09,570 What as we created-- as we derive the formula-- well 81 00:05:09,570 --> 00:05:15,010 it won't be wrong because our derivation will be good. 82 00:05:15,010 --> 00:05:18,270 But also, it would be nice, I just 83 00:05:18,270 --> 00:05:21,880 think if you plug that in, to that differential equation 84 00:05:21,880 --> 00:05:23,360 it's solved. 85 00:05:23,360 --> 00:05:25,710 OK so I want to take the derivative of that. 86 00:05:25,710 --> 00:05:26,860 That's my job. 87 00:05:26,860 --> 00:05:31,260 And that's why I do it here because it uses all the rules. 88 00:05:31,260 --> 00:05:34,400 OK to take that derivative, I notice 89 00:05:34,400 --> 00:05:38,990 the t is appearing there in the usual place, 90 00:05:38,990 --> 00:05:40,830 and it's also inside the integral. 91 00:05:40,830 --> 00:05:43,310 But this is a simple function. 92 00:05:43,310 --> 00:05:47,140 I can take e to the t-- I'm going to take e 93 00:05:47,140 --> 00:05:51,810 to the t out of the-- outside the integral. 94 00:05:51,810 --> 00:05:54,410 e to the t. 95 00:05:54,410 --> 00:06:01,410 So I have a function t times another function of t. 96 00:06:01,410 --> 00:06:03,610 I'm going to use the product rule 97 00:06:03,610 --> 00:06:07,730 and show that the derivative of that product 98 00:06:07,730 --> 00:06:13,310 is one term will be y and the other term will be q. 99 00:06:13,310 --> 00:06:18,065 Can I just apply the product rule to this function 100 00:06:18,065 --> 00:06:21,660 that I've pulled out of a hat, but you'll see it again. 101 00:06:21,660 --> 00:06:25,350 OK so it's a product of this times this. 102 00:06:25,350 --> 00:06:33,480 So the derivative dy dt is-- the product rule 103 00:06:33,480 --> 00:06:36,400 says take the derivative of [INAUDIBLE] that 104 00:06:36,400 --> 00:06:38,350 is e to the [INAUDIBLE]. 105 00:06:42,315 --> 00:06:50,916 Plus, the first thing times the derivative of the second. 106 00:06:50,916 --> 00:06:53,110 Now I'm using the product rule. 107 00:06:53,110 --> 00:06:56,820 It just-- you have to notice that e to the t came twice 108 00:06:56,820 --> 00:07:02,270 because it is there and its derivative is the same. 109 00:07:02,270 --> 00:07:08,290 OK now, what's the derivative of that? 110 00:07:08,290 --> 00:07:10,180 Fundamental theorem of calculus. 111 00:07:10,180 --> 00:07:13,910 We've integrated something, I want to take its derivative, 112 00:07:13,910 --> 00:07:15,520 so I get that something. 113 00:07:15,520 --> 00:07:20,770 I get e to the minus tq of t. 114 00:07:20,770 --> 00:07:23,230 That's the fundamental theorem. 115 00:07:23,230 --> 00:07:25,240 Are you good with that? 116 00:07:25,240 --> 00:07:28,160 So let's just look and see what we have. 117 00:07:28,160 --> 00:07:33,720 First term was exactly y. 118 00:07:33,720 --> 00:07:35,580 Exactly what is above because when 119 00:07:35,580 --> 00:07:38,660 I took the derivative of the first guy, 120 00:07:38,660 --> 00:07:42,690 the f it didn't change it, so I still have y. 121 00:07:42,690 --> 00:07:45,010 What have I-- what do I have here? 122 00:07:45,010 --> 00:07:49,650 E to the t times e to the minus t is one. 123 00:07:49,650 --> 00:07:52,500 So e to the t cancels e to the minus t 124 00:07:52,500 --> 00:07:55,930 and I'm left with q of t Just what I want. 125 00:07:55,930 --> 00:07:57,880 So the two terms from the product rule 126 00:07:57,880 --> 00:08:01,460 are the two terms in the differential equation. 127 00:08:01,460 --> 00:08:05,940 I just think as you saw the fundamental theorem was needed 128 00:08:05,940 --> 00:08:08,930 right there to find the derivative of what's 129 00:08:08,930 --> 00:08:12,610 in that box, is what's in those parentheses. 130 00:08:12,610 --> 00:08:15,820 I just like that the use of the fundamental theorem. 131 00:08:15,820 --> 00:08:21,860 OK one more topic of calculus we need. 132 00:08:21,860 --> 00:08:24,250 And here we go. 133 00:08:24,250 --> 00:08:31,570 So it involves the tangent line to the graph. 134 00:08:31,570 --> 00:08:34,789 This tangent to the graph. 135 00:08:34,789 --> 00:08:44,140 So it's a straight line and what we need is y of t plus delta t. 136 00:08:44,140 --> 00:08:47,410 That's taking any function, maybe 137 00:08:47,410 --> 00:08:49,600 you'd rather I just called the function f. 138 00:08:52,600 --> 00:08:56,910 A function at a point a little beyond t, 139 00:08:56,910 --> 00:09:01,860 is approximately the function at t 140 00:09:01,860 --> 00:09:06,260 plus the correction because it-- plus a delta f, right? 141 00:09:06,260 --> 00:09:07,870 A delta f. 142 00:09:07,870 --> 00:09:10,210 And what's the delta f approximately? 143 00:09:10,210 --> 00:09:20,920 It's approximately delta t times the derivative at t. 144 00:09:20,920 --> 00:09:25,720 That-- there's a lot of symbols on that line, 145 00:09:25,720 --> 00:09:31,390 but it expresses the most basic fact of differential calculus. 146 00:09:31,390 --> 00:09:35,590 If I put that f of t on this side with a minus sign, 147 00:09:35,590 --> 00:09:38,230 then I have delta f. 148 00:09:38,230 --> 00:09:44,240 If I divide by that delta t, then the same rule 149 00:09:44,240 --> 00:09:49,280 is saying that this is approximately df dt. 150 00:09:49,280 --> 00:09:51,380 That's a fundamental idea of calculus, 151 00:09:51,380 --> 00:09:55,070 that the derivative is quite close. 152 00:09:55,070 --> 00:09:58,460 At the point t-- the derivative at the point t 153 00:09:58,460 --> 00:10:01,930 is close to delta f divided by delta t. 154 00:10:01,930 --> 00:10:05,100 It changes over a short time interval. 155 00:10:05,100 --> 00:10:10,270 OK so that's the tangent line because it starts with that's 156 00:10:10,270 --> 00:10:12,350 the constant term. 157 00:10:12,350 --> 00:10:16,710 It's a function of delta t and that's the slope. 158 00:10:16,710 --> 00:10:18,436 Just draw a picture. 159 00:10:18,436 --> 00:10:20,940 So I'm drawing a picture here. 160 00:10:20,940 --> 00:10:24,680 So let me draw a graph of-- oh there's 161 00:10:24,680 --> 00:10:27,090 the graph of e to the t. 162 00:10:27,090 --> 00:10:29,122 So it starts up with slope 1. 163 00:10:29,122 --> 00:10:30,580 Let me give it a little slope here. 164 00:10:33,380 --> 00:10:36,250 OK the tangent line, and of course it 165 00:10:36,250 --> 00:10:39,380 comes down here Not below. 166 00:10:39,380 --> 00:10:42,630 So the tangent line is that line. 167 00:10:47,070 --> 00:10:48,510 That's the tangent line. 168 00:10:48,510 --> 00:10:51,080 That's this approximation to f. 169 00:10:51,080 --> 00:10:55,560 And you see as I-- here is t equals 0 let's say. 170 00:10:55,560 --> 00:10:58,400 And here's t equal delta t. 171 00:10:58,400 --> 00:11:00,210 And you see if I take a big step, 172 00:11:00,210 --> 00:11:03,070 my line is far from the curve. 173 00:11:03,070 --> 00:11:06,080 And we want to get closer. 174 00:11:06,080 --> 00:11:09,330 So the way to get closer is we have 175 00:11:09,330 --> 00:11:11,130 to take into account the bending. 176 00:11:11,130 --> 00:11:12,660 The curve is bending. 177 00:11:12,660 --> 00:11:17,060 What derivative tells us about bending? 178 00:11:17,060 --> 00:11:24,571 That is delta t squared times the second derivative. 179 00:11:27,320 --> 00:11:27,820 One half. 180 00:11:27,820 --> 00:11:31,450 It turns out a one half shows in there. 181 00:11:31,450 --> 00:11:35,760 So this is the term that changes the tangent line, 182 00:11:35,760 --> 00:11:38,820 to a tangent parabola. 183 00:11:38,820 --> 00:11:41,300 It notices the bending at that point. 184 00:11:41,300 --> 00:11:43,620 The second derivative at that point. 185 00:11:43,620 --> 00:11:45,210 So it curves up. 186 00:11:45,210 --> 00:11:49,400 It doesn't follow it perfectly, but as well-- much better 187 00:11:49,400 --> 00:11:51,100 than the other. 188 00:11:51,100 --> 00:11:53,735 So this is the line. 189 00:11:53,735 --> 00:11:54,610 Here is the parabola. 190 00:11:57,264 --> 00:11:58,305 And here is the function. 191 00:12:02,320 --> 00:12:04,060 The real one. 192 00:12:04,060 --> 00:12:05,690 OK. 193 00:12:05,690 --> 00:12:10,580 I won't review the theory there that it pulls out that one 194 00:12:10,580 --> 00:12:12,350 half, but you could check it. 195 00:12:12,350 --> 00:12:16,690 Now finally, what if we want to do even better? 196 00:12:16,690 --> 00:12:19,010 Well we need to take into account the third derivative 197 00:12:19,010 --> 00:12:21,480 and then the fourth derivative and so on, 198 00:12:21,480 --> 00:12:24,970 and if we get all those derivatives then, 199 00:12:24,970 --> 00:12:29,075 all of them that means, we will be at the function 200 00:12:29,075 --> 00:12:32,100 because that's a nice function, e to the t. 201 00:12:32,100 --> 00:12:37,110 We can recreate that function from knowing 202 00:12:37,110 --> 00:12:42,565 its height, its slope, its bending 203 00:12:42,565 --> 00:12:44,350 and all the rest of the terms. 204 00:12:44,350 --> 00:12:48,130 So there's a whole lot more-- Infinitely many terms. 205 00:12:48,130 --> 00:12:51,010 That one over two-- the good way to think of one 206 00:12:51,010 --> 00:12:55,830 over two, one half, is one over two factorial, two times one. 207 00:12:55,830 --> 00:12:59,650 Because this is one over n factorial, 208 00:12:59,650 --> 00:13:04,170 times t to the nth, pretty small, 209 00:13:04,170 --> 00:13:07,175 times the nth derivative of the function. 210 00:13:10,170 --> 00:13:13,020 And keep going. 211 00:13:13,020 --> 00:13:19,310 That's called the Taylor series named after Taylor. 212 00:13:19,310 --> 00:13:25,550 Kind of frightening at first. 213 00:13:25,550 --> 00:13:28,910 It's frightening because it's got infinitely many terms. 214 00:13:28,910 --> 00:13:31,731 And the terms are getting a little more comp-- 215 00:13:31,731 --> 00:13:33,740 For most functions, you really don't want 216 00:13:33,740 --> 00:13:35,680 to compute the nth derivative. 217 00:13:35,680 --> 00:13:39,100 For e to the t, I don't mind computing the nth derivative 218 00:13:39,100 --> 00:13:44,790 because it's still e to the t, but usually that's-- this 219 00:13:44,790 --> 00:13:46,950 isn't so practical. 220 00:13:46,950 --> 00:13:48,570 [INAUDIBLE] very practical. 221 00:13:48,570 --> 00:13:51,380 Tangent parabola, quite practical. 222 00:13:51,380 --> 00:13:55,150 Higher order terms, less-- much less practical. 223 00:13:55,150 --> 00:13:59,210 But the formula is beautiful because you 224 00:13:59,210 --> 00:14:02,340 see the pattern, that's really what mathematics 225 00:14:02,340 --> 00:14:04,300 is about patterns, and here you're 226 00:14:04,300 --> 00:14:08,880 seeing the pattern in the higher, higher terms. 227 00:14:08,880 --> 00:14:14,050 They all fit that pattern and when you add up all the terms, 228 00:14:14,050 --> 00:14:18,280 if you have a nice function, then the approximation 229 00:14:18,280 --> 00:14:21,560 becomes perfect and you would have equality. 230 00:14:21,560 --> 00:14:27,800 So to end this lecture, approximate to equal provided 231 00:14:27,800 --> 00:14:30,080 we have a nice function. 232 00:14:30,080 --> 00:14:34,510 And those are the best functions of mathematics and exponential 233 00:14:34,510 --> 00:14:36,010 is of course one of them. 234 00:14:36,010 --> 00:14:39,030 OK that's calculus. 235 00:14:39,030 --> 00:14:40,990 Well, part of calculus. 236 00:14:40,990 --> 00:14:42,790 Thank you.