1 00:00:01,099 --> 00:00:03,182 The following content is provided under a Creative 2 00:00:03,182 --> 00:00:03,848 Commons license. 3 00:00:03,848 --> 00:00:05,740 Your support will help MIT OpenCourseWare 4 00:00:05,740 --> 00:00:08,850 continue to offer high quality educational resources for free. 5 00:00:08,850 --> 00:00:11,330 To make a donation or to view additional materials 6 00:00:11,330 --> 00:00:13,580 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:13,580 --> 00:00:21,780 at ocw.mit.edu. 8 00:00:21,780 --> 00:00:24,510 PROFESSOR: Today we're going to keep 9 00:00:24,510 --> 00:00:30,660 on going with related rates. 10 00:00:30,660 --> 00:00:32,710 And you may recall that last time we 11 00:00:32,710 --> 00:00:39,000 were in the middle of a problem with this geometry. 12 00:00:39,000 --> 00:00:41,260 There was a right triangle. 13 00:00:41,260 --> 00:00:43,190 There was a road. 14 00:00:43,190 --> 00:00:47,000 Which was going this way, from right to left. 15 00:00:47,000 --> 00:00:53,830 And the police were up here, monitoring the situation. 16 00:00:53,830 --> 00:00:55,990 30 feet from the road. 17 00:00:55,990 --> 00:00:59,990 And you're here. 18 00:00:59,990 --> 00:01:04,214 And you're heading this way. 19 00:01:04,214 --> 00:01:06,130 Maybe it's a two lane highway, but anyway it's 20 00:01:06,130 --> 00:01:07,680 only going this direction. 21 00:01:07,680 --> 00:01:11,670 And this distance was 50 feet. 22 00:01:11,670 --> 00:01:17,070 So, because you're moving, this distance is varying 23 00:01:17,070 --> 00:01:19,120 and so we gave it a letter. 24 00:01:19,120 --> 00:01:22,460 And, similarly, your distance to the foot of the perpendicular 25 00:01:22,460 --> 00:01:25,050 with the road is also varying. 26 00:01:25,050 --> 00:01:28,410 At this instant it's 40, because this is a 3, 27 00:01:28,410 --> 00:01:31,570 4, 5 right triangle. 28 00:01:31,570 --> 00:01:35,830 So this was the situation that we were in last time. 29 00:01:35,830 --> 00:01:38,090 And we're going to pick up where we left off. 30 00:01:38,090 --> 00:01:48,210 The question is, are you speeding 31 00:01:48,210 --> 00:01:54,170 if the rate of change of D with respect to t 32 00:01:54,170 --> 00:01:56,510 is 80 feet per second. 33 00:01:56,510 --> 00:01:59,290 Now, technically that would be -80, 34 00:01:59,290 --> 00:02:03,030 because you're going towards the policemen. 35 00:02:03,030 --> 00:02:10,550 Alright, so D is shrinking at a rate of -80 feet per second. 36 00:02:10,550 --> 00:02:15,620 And I remind you that 95 feet per second 37 00:02:15,620 --> 00:02:17,660 is approximately the speed limit. 38 00:02:17,660 --> 00:02:21,370 Which is 65 miles per hour. 39 00:02:21,370 --> 00:02:24,850 So, again, this is where we were last time. 40 00:02:24,850 --> 00:02:30,150 And, got a little question mark there. 41 00:02:30,150 --> 00:02:34,140 And so let's solve this problem. 42 00:02:34,140 --> 00:02:37,480 So, this is the setup. 43 00:02:37,480 --> 00:02:38,880 There's a right triangle. 44 00:02:38,880 --> 00:02:42,290 So there's a relationship between these lengths. 45 00:02:42,290 --> 00:02:49,800 And the relationship is that x^2 + 30^2 = D^2. 46 00:02:49,800 --> 00:02:53,730 So that's the first relationship that we have. 47 00:02:53,730 --> 00:02:55,540 And the second relationship that we have, 48 00:02:55,540 --> 00:02:57,380 we've already written down. 49 00:02:57,380 --> 00:03:01,540 Which is dx/dt - oops, sorry. 50 00:03:01,540 --> 00:03:05,540 dD/dt = -80. 51 00:03:08,620 --> 00:03:14,920 Now, the idea here is relatively straightforward. 52 00:03:14,920 --> 00:03:17,790 We just want to use differentiation. 53 00:03:17,790 --> 00:03:25,810 Now, you could solve for x. 54 00:03:25,810 --> 00:03:28,960 Alright, x is the square root of D^2 - 30^2. 55 00:03:31,580 --> 00:03:32,820 That's one possibility. 56 00:03:32,820 --> 00:03:36,060 But this is basically a waste of time. 57 00:03:36,060 --> 00:03:37,810 It's a waste of your time. 58 00:03:37,810 --> 00:03:44,340 So it's easier, or easiest, to follow 59 00:03:44,340 --> 00:03:46,350 this method of implicit differentiation, 60 00:03:46,350 --> 00:03:49,080 which I want to encourage you to get used to. 61 00:03:49,080 --> 00:03:51,490 Namely, we just differentiate this equation 62 00:03:51,490 --> 00:03:53,460 with respect to time. 63 00:03:53,460 --> 00:03:56,480 Now, when you do that, you have to remember that you are not 64 00:03:56,480 --> 00:03:59,120 allowed to plug in a constant. 65 00:03:59,120 --> 00:04:00,857 Namely 40, for t. 66 00:04:00,857 --> 00:04:02,440 You have to keep in mind what's really 67 00:04:02,440 --> 00:04:04,565 going on in this problem which is that x is moving, 68 00:04:04,565 --> 00:04:05,670 it's changing. 69 00:04:05,670 --> 00:04:07,250 And D is also changing. 70 00:04:07,250 --> 00:04:10,090 So you have to differentiate first 71 00:04:10,090 --> 00:04:12,030 before you plug in the values. 72 00:04:12,030 --> 00:04:15,160 So the easiest thing is to use, in this case, 73 00:04:15,160 --> 00:04:20,130 implicit differentiation. 74 00:04:20,130 --> 00:04:33,000 And if I do that, I get 2x dx/dt is equal to 2D dD/dt. 75 00:04:33,000 --> 00:04:34,910 No more DDT left. 76 00:04:34,910 --> 00:04:36,640 We hope. 77 00:04:36,640 --> 00:04:39,160 Except in this blackboard. 78 00:04:39,160 --> 00:04:40,690 So there's our situation. 79 00:04:40,690 --> 00:04:50,670 Now, if I just plug in, now I can plug in values. 80 00:04:50,670 --> 00:04:52,820 So this is after taking the derivative. 81 00:04:52,820 --> 00:04:54,820 And, indeed, we have here 2 times 82 00:04:54,820 --> 00:04:57,500 the value for x which is 40 at this instant. 83 00:04:57,500 --> 00:05:01,070 And then we have dx/dt. 84 00:05:01,070 --> 00:05:07,510 And that's equal to 2 times D, which is 50. 85 00:05:07,510 --> 00:05:12,900 And then dD/dt is -80. 86 00:05:12,900 --> 00:05:20,130 So the 80's cancel and we see that dx/dt = -100 feet 87 00:05:20,130 --> 00:05:21,660 per second. 88 00:05:21,660 --> 00:05:26,985 And so the answer to the question is yes. 89 00:05:26,985 --> 00:05:28,360 Although you probably wouldn't be 90 00:05:28,360 --> 00:05:32,540 pulled over for this much of a violation. 91 00:05:32,540 --> 00:05:36,240 So that's-- right, it's more than 65 miles an hour, 92 00:05:36,240 --> 00:05:41,190 by a little bit. 93 00:05:41,190 --> 00:05:43,380 So that's the end of this question. 94 00:05:43,380 --> 00:05:46,460 And usually in these rate of change or related rates 95 00:05:46,460 --> 00:05:49,100 questions, this is considered to be the answer to the question, 96 00:05:49,100 --> 00:05:53,530 if you like. 97 00:05:53,530 --> 00:05:55,600 So that's one example. 98 00:05:55,600 --> 00:05:57,890 I'm going to give one more example 99 00:05:57,890 --> 00:06:01,285 before we go on to some other applications 100 00:06:01,285 --> 00:06:04,560 of implicit differentiation. 101 00:06:04,560 --> 00:06:17,250 So my second example is going to be, you have a conical tank. 102 00:06:17,250 --> 00:06:26,640 With top of radius 4 feet, let's say. 103 00:06:26,640 --> 00:06:31,770 And depth 10 feet. 104 00:06:31,770 --> 00:06:34,710 So that's our situation. 105 00:06:34,710 --> 00:06:37,730 And then it's being filled with water. 106 00:06:37,730 --> 00:06:42,830 So, is being filled with water. 107 00:06:42,830 --> 00:06:49,900 At 2 cubic feet per minute. 108 00:06:49,900 --> 00:06:52,980 So there is our situation. 109 00:06:52,980 --> 00:07:05,750 And then the question is, how fast is the water rising 110 00:07:05,750 --> 00:07:15,040 when it is at depth 5 feet? 111 00:07:15,040 --> 00:07:19,020 So if this thing is half-full in the sense-- 112 00:07:19,020 --> 00:07:21,770 well not half-full in terms of total volume, 113 00:07:21,770 --> 00:07:25,640 but half-full in terms of height. 114 00:07:25,640 --> 00:07:36,540 What's the speed at which the water is rising. 115 00:07:36,540 --> 00:07:39,970 So, how do we set up problems like this? 116 00:07:39,970 --> 00:07:42,240 Well, we talked about this last time. 117 00:07:42,240 --> 00:07:51,780 The first step is to set up a diagram and variables. 118 00:07:51,780 --> 00:07:54,450 And I'm just going to draw the picture. 119 00:07:54,450 --> 00:07:58,320 And I'm actually going to draw the picture twice. 120 00:07:58,320 --> 00:08:01,590 So here's the conical tank. 121 00:08:01,590 --> 00:08:05,040 And we have this radius, which is 4. 122 00:08:05,040 --> 00:08:07,970 And we have this height, which is 10. 123 00:08:07,970 --> 00:08:10,950 So that's just to allow me to think about this problem. 124 00:08:10,950 --> 00:08:18,270 Now, it turns out because we have a varying depth 125 00:08:18,270 --> 00:08:21,730 and so on, and this is just the top. 126 00:08:21,730 --> 00:08:24,530 That I'd better depict also the level at which the water 127 00:08:24,530 --> 00:08:26,700 actually is at present. 128 00:08:26,700 --> 00:08:30,710 And furthermore, it's better to do this schematically, 129 00:08:30,710 --> 00:08:31,830 as you'll see. 130 00:08:31,830 --> 00:08:38,200 So the key point here is to draw this triangle here. 131 00:08:38,200 --> 00:08:45,230 Which shows me the 10 and shows me the 4, over here. 132 00:08:45,230 --> 00:08:48,290 And then imagine that I'm filling it partway. 133 00:08:48,290 --> 00:08:52,400 So maybe we'll put that water level in in another color here. 134 00:08:52,400 --> 00:08:54,370 So here's the water level. 135 00:08:54,370 --> 00:08:56,130 And the water level, I'm going to depict 136 00:08:56,130 --> 00:08:59,659 that horizontal distance here, as r. 137 00:08:59,659 --> 00:09:00,950 That's going to be my variable. 138 00:09:00,950 --> 00:09:04,150 That's the radius of the top of the water. 139 00:09:04,150 --> 00:09:06,359 So I'm taking a cross-section of this, 140 00:09:06,359 --> 00:09:08,150 because that geometrically the only thing I 141 00:09:08,150 --> 00:09:10,230 have to keep track of. 142 00:09:10,230 --> 00:09:11,820 At least initially. 143 00:09:11,820 --> 00:09:14,630 So this is our water level. 144 00:09:14,630 --> 00:09:18,540 And it's really rotated around. 145 00:09:18,540 --> 00:09:23,130 But I'm depicting just this one half-slice of the picture. 146 00:09:23,130 --> 00:09:26,300 And then similarly, I have the height. 147 00:09:26,300 --> 00:09:30,390 Which is this dimension there. 148 00:09:30,390 --> 00:09:32,570 Or, if you like, the depth of the water. 149 00:09:32,570 --> 00:09:42,140 So the water has filled us up, up to this point here. 150 00:09:42,140 --> 00:09:46,050 So I set it up this way so that it's clear 151 00:09:46,050 --> 00:09:48,830 that we have two triangles here, and that one 152 00:09:48,830 --> 00:09:51,210 piece of information we can get from the geometry 153 00:09:51,210 --> 00:09:53,880 is the similar triangles information. 154 00:09:53,880 --> 00:09:59,870 Namely, that r / h = 4 / 10. 155 00:09:59,870 --> 00:10:09,760 So this is by far the most typical geometric fact 156 00:10:09,760 --> 00:10:13,450 that you'll have to glean from a picture. 157 00:10:13,450 --> 00:10:16,150 So that's one piece of information 158 00:10:16,150 --> 00:10:18,920 that we get from this picture. 159 00:10:18,920 --> 00:10:20,420 Now, the second thing we have to do 160 00:10:20,420 --> 00:10:25,622 is set up formulas for the volume of water, 161 00:10:25,622 --> 00:10:27,330 and then figure out what's going on here. 162 00:10:27,330 --> 00:10:33,460 So the volume of water is-- of a cone. 163 00:10:33,460 --> 00:10:36,950 So again, you have to know something about geometry 164 00:10:36,950 --> 00:10:38,540 to do many of these problems. 165 00:10:38,540 --> 00:10:41,190 So you have to know that the volume of a cone 166 00:10:41,190 --> 00:10:44,500 is 1/3 base times height. 167 00:10:44,500 --> 00:10:45,900 Now, this one is upside down. 168 00:10:45,900 --> 00:10:49,130 The base is on the top and it's going down 169 00:10:49,130 --> 00:10:50,720 but it works the same way. 170 00:10:50,720 --> 00:10:52,270 That doesn't affect volume. 171 00:10:52,270 --> 00:10:57,450 So it's 1/3, and the base is pi r^2, that's the base, 172 00:10:57,450 --> 00:11:01,890 and times h, which is the height. 173 00:11:01,890 --> 00:11:06,290 So this is the setup for this problem. 174 00:11:06,290 --> 00:11:11,480 And now, having our relationship, 175 00:11:11,480 --> 00:11:14,580 we have one relationship left that we have to remember. 176 00:11:14,580 --> 00:11:17,630 Because we have one more piece of information in this problem. 177 00:11:17,630 --> 00:11:20,990 Namely, how fast the volume is changing. 178 00:11:20,990 --> 00:11:22,960 It's going at 2 cubic feet per minute. 179 00:11:22,960 --> 00:11:29,740 It's increasing, so that means that dV/dt = 2. 180 00:11:29,740 --> 00:11:33,970 So I've now gotten rid of all the words 181 00:11:33,970 --> 00:11:38,010 and I have only formulas left. 182 00:11:38,010 --> 00:11:43,574 I started here with the words, I drew some pictures, 183 00:11:43,574 --> 00:11:44,740 and I derived some formulas. 184 00:11:44,740 --> 00:11:46,240 Actually, there's one thing missing. 185 00:11:46,240 --> 00:11:49,930 What's missing? 186 00:11:49,930 --> 00:11:52,570 Yeah. 187 00:11:52,570 --> 00:11:56,949 STUDENT: [INAUDIBLE] 188 00:11:56,949 --> 00:11:57,740 PROFESSOR: Exactly. 189 00:11:57,740 --> 00:11:58,656 What you want to find. 190 00:11:58,656 --> 00:12:00,690 What I left out is the question. 191 00:12:00,690 --> 00:12:15,110 So the question is, what is dh(dt when h = 5? 192 00:12:15,110 --> 00:12:20,486 So that's the question here. 193 00:12:20,486 --> 00:12:22,360 Now, we've got the whole problem and we never 194 00:12:22,360 --> 00:12:25,510 have to look at it again if you like. 195 00:12:25,510 --> 00:12:32,610 We just have to pay attention to this piece here. 196 00:12:32,610 --> 00:12:34,780 So let's carry it out. 197 00:12:34,780 --> 00:12:36,870 So what happens here. 198 00:12:36,870 --> 00:12:39,190 So look, you could do this by implicit differentiation. 199 00:12:39,190 --> 00:12:42,830 But it's so easy to express r as a function of h 200 00:12:42,830 --> 00:12:44,560 that that seems kind of foolish. 201 00:12:44,560 --> 00:12:48,900 So let's write r as 2/5 h. 202 00:12:48,900 --> 00:12:51,280 That's coming from this first equation here. 203 00:12:51,280 --> 00:12:53,800 And then let's substitute that in. 204 00:12:53,800 --> 00:12:58,800 That means that V = 1/3 pi (2/5 h)^2 h. 205 00:13:04,810 --> 00:13:06,630 And now I have to differentiate that. 206 00:13:06,630 --> 00:13:08,850 So now I will use implicit differentiation. 207 00:13:08,850 --> 00:13:12,030 It's very foolish at this point to take cube roots 208 00:13:12,030 --> 00:13:13,320 to solve for h. 209 00:13:13,320 --> 00:13:16,080 You just get yourself into a bunch of junk. 210 00:13:16,080 --> 00:13:18,180 So there is a stage at which we're still using 211 00:13:18,180 --> 00:13:19,490 implicit differentiation here. 212 00:13:19,490 --> 00:13:24,772 I'm not going to try to solve for h as a function of V. 213 00:13:24,772 --> 00:13:26,480 Instead I'm just differentiating straight 214 00:13:26,480 --> 00:13:28,230 out from this formula, which is relatively 215 00:13:28,230 --> 00:13:29,530 easy to differentiate. 216 00:13:29,530 --> 00:13:33,795 So this is dV/dt, which of course is 2, 217 00:13:33,795 --> 00:13:37,440 is equal to, and if I differentiate it I just get 218 00:13:37,440 --> 00:13:42,390 this constant, pi/3, this other constant, (2/5)^2. 219 00:13:42,390 --> 00:13:45,200 And then I have to differentiate h^3. 220 00:13:45,200 --> 00:13:47,290 (h^2)h. 221 00:13:47,290 --> 00:13:50,050 So that's 3h^2 dh/dt. 222 00:13:53,140 --> 00:13:59,760 That's the chain rule. 223 00:13:59,760 --> 00:14:02,080 So now let's plug in all of our numbers. 224 00:14:02,080 --> 00:14:04,810 Again, the other theme is, you don't 225 00:14:04,810 --> 00:14:08,300 plug in numbers, fixed numbers, until everything 226 00:14:08,300 --> 00:14:09,600 has stopped moving. 227 00:14:09,600 --> 00:14:12,790 At this point, we've already calculated our rates of change. 228 00:14:12,790 --> 00:14:14,140 So now I can put in the numbers. 229 00:14:14,140 --> 00:14:23,790 So, 2 is equal to pi/3 (2/5)^2 times 3, and then h was 5, 230 00:14:23,790 --> 00:14:26,010 so this is 5^2. 231 00:14:26,010 --> 00:14:27,870 And then I have dh/dt. 232 00:14:27,870 --> 00:14:31,840 There's only one unknown thing left in this problem, which 233 00:14:31,840 --> 00:14:32,560 is dh/dt. 234 00:14:32,560 --> 00:14:33,990 Everything else is a number. 235 00:14:33,990 --> 00:14:35,650 And if you do all the cancellations, 236 00:14:35,650 --> 00:14:38,210 you see that this cancels this. 237 00:14:38,210 --> 00:14:41,149 One of the 2's cancels - well, this cancels this. 238 00:14:41,149 --> 00:14:42,690 And then one of the 2's cancels that. 239 00:14:42,690 --> 00:14:49,600 So all told what we have here is that dh/dt = 1/2 pi. 240 00:14:54,080 --> 00:14:58,700 And so that happens to be feet per second. 241 00:14:58,700 --> 00:15:04,597 This is the whole story. 242 00:15:04,597 --> 00:15:05,680 Questions, way back there. 243 00:15:05,680 --> 00:15:09,297 STUDENT: [INAUDIBLE] 244 00:15:09,297 --> 00:15:11,630 PROFESSOR: Where did I get-- the question is where did I 245 00:15:11,630 --> 00:15:14,590 get r = 2/5 h from. 246 00:15:14,590 --> 00:15:17,680 The answer was, it came from similar triangles, 247 00:15:17,680 --> 00:15:18,964 which is over here. 248 00:15:18,964 --> 00:15:20,380 I did this similar triangle thing. 249 00:15:20,380 --> 00:15:23,530 And I got this relationship here. 250 00:15:23,530 --> 00:15:29,620 r/h = 4/10, but then I canceled it, got 2/5 251 00:15:29,620 --> 00:15:31,724 and brought the h over. 252 00:15:31,724 --> 00:15:32,890 Another question, over here. 253 00:15:32,890 --> 00:15:45,470 STUDENT: [INAUDIBLE] 254 00:15:45,470 --> 00:15:53,100 PROFESSOR: The question was the following. 255 00:15:53,100 --> 00:15:56,815 Suppose you're at this stage, can you write from here dV/dh 256 00:15:56,815 --> 00:16:01,390 - so, suppose you're here - and work out what this is. 257 00:16:01,390 --> 00:16:05,030 It's going to end up being some constant times h^2. 258 00:16:05,030 --> 00:16:14,990 And then also use dV/dt = dV/dh dh/dt. 259 00:16:14,990 --> 00:16:17,520 Which the chain rule. 260 00:16:17,520 --> 00:16:20,830 And the answer is yes. 261 00:16:20,830 --> 00:16:26,310 We can do that, and indeed that is what my next sentence is. 262 00:16:26,310 --> 00:16:27,750 That's exactly what I'm saying. 263 00:16:27,750 --> 00:16:32,780 So when I said this-- sorry, when you said this, I did that. 264 00:16:32,780 --> 00:16:34,080 That's exactly what I did. 265 00:16:34,080 --> 00:16:41,110 This chunk is exactly dV/dh. 266 00:16:41,110 --> 00:16:43,390 That's just what I'm doing. 267 00:16:43,390 --> 00:16:44,730 OK. 268 00:16:44,730 --> 00:16:48,750 So, definitely, that's what I had in mind. 269 00:16:48,750 --> 00:16:49,760 Yeah, another question. 270 00:16:49,760 --> 00:16:55,224 STUDENT: [INAUDIBLE] 271 00:16:55,224 --> 00:16:57,390 PROFESSOR: You're asking me whether my arithmetic is 272 00:16:57,390 --> 00:16:58,040 right or not? 273 00:16:58,040 --> 00:17:03,890 STUDENT: [INAUDIBLE] 274 00:17:03,890 --> 00:17:06,850 PROFESSOR: Pi - per second. 275 00:17:06,850 --> 00:17:07,350 Oh. 276 00:17:07,350 --> 00:17:09,480 This should - no, OK, right. 277 00:17:09,480 --> 00:17:11,474 I guess it's per minute. 278 00:17:11,474 --> 00:17:12,890 Since the other one is per minute. 279 00:17:12,890 --> 00:17:14,400 Thank you. 280 00:17:14,400 --> 00:17:16,756 Yes. 281 00:17:16,756 --> 00:17:17,880 Was there another question? 282 00:17:17,880 --> 00:17:21,810 Probably also fixing my seconds to minutes. 283 00:17:21,810 --> 00:17:23,150 Way back there. 284 00:17:23,150 --> 00:17:27,490 STUDENT: I don't understand, why did you have to do all that. 285 00:17:27,490 --> 00:17:30,930 Isn't the speed 2 cubic feet per minute? 286 00:17:30,930 --> 00:17:33,000 PROFESSOR: The speed at which you're 287 00:17:33,000 --> 00:17:35,920 filling it is 2 cubic feet, but the water level is rising 288 00:17:35,920 --> 00:17:38,750 at a different rate, depending on whether you're low 289 00:17:38,750 --> 00:17:40,350 down or high up. 290 00:17:40,350 --> 00:17:44,220 It depends on how wide the pond, the surface, is. 291 00:17:44,220 --> 00:17:45,310 So in fact it's not. 292 00:17:45,310 --> 00:17:48,950 In fact, the answer wasn't 2 cubic-- it wasn't. 293 00:17:48,950 --> 00:17:49,900 There's a rate there. 294 00:17:49,900 --> 00:17:52,090 That is, that's how much volume is being added. 295 00:17:52,090 --> 00:17:54,650 But then there's another number that we're keeping track of, 296 00:17:54,650 --> 00:17:55,780 which is the height. 297 00:17:55,780 --> 00:17:58,790 Or, if you like, the depth of the water. 298 00:17:58,790 --> 00:17:59,792 OK. 299 00:17:59,792 --> 00:18:01,750 So this is the whole point about related rates. 300 00:18:01,750 --> 00:18:03,670 Is you have one variable, which is V, 301 00:18:03,670 --> 00:18:04,882 which is volume of something. 302 00:18:04,882 --> 00:18:06,340 You have another variable, which is 303 00:18:06,340 --> 00:18:10,675 h, which is the height of the cone of water there. 304 00:18:10,675 --> 00:18:13,300 And you're keeping track of one variable in terms of the other. 305 00:18:13,300 --> 00:18:15,990 And the relationship will always be a chain rule 306 00:18:15,990 --> 00:18:16,866 type of relationship. 307 00:18:16,866 --> 00:18:18,573 So, therefore, you'll be able to-- if you 308 00:18:18,573 --> 00:18:20,810 know one you'll be able to figure out the other. 309 00:18:20,810 --> 00:18:22,864 Provided you get all of the related rates. 310 00:18:22,864 --> 00:18:24,530 These are what are called related rates. 311 00:18:24,530 --> 00:18:28,050 This is a rate of something with respect to something, etc. etc. 312 00:18:28,050 --> 00:18:30,540 So it's really all about the chain rule. 313 00:18:30,540 --> 00:18:38,540 And just fitting it to word problems. 314 00:18:38,540 --> 00:18:41,900 So there's a couple of examples. 315 00:18:41,900 --> 00:18:44,430 And I'll let you work out some more. 316 00:18:44,430 --> 00:18:50,100 So now, the next thing that I want to do 317 00:18:50,100 --> 00:18:54,080 is to give you one more max-min problem. 318 00:18:54,080 --> 00:18:58,670 Which has to do with this device, which 319 00:18:58,670 --> 00:19:00,400 I brought with me. 320 00:19:00,400 --> 00:19:02,480 So this is a ring. 321 00:19:02,480 --> 00:19:04,150 Happens to be a napkin ring, and this 322 00:19:04,150 --> 00:19:09,450 is some parachute string, which I use when I go backpacking. 323 00:19:09,450 --> 00:19:15,382 And the question is if you have a-- you 324 00:19:15,382 --> 00:19:17,090 think of this is a weight, it's flexible. 325 00:19:17,090 --> 00:19:19,040 It's allowed to move along here. 326 00:19:19,040 --> 00:19:23,480 And the question is, if I fix these two ends where my two 327 00:19:23,480 --> 00:19:25,290 hands are, where my right hand is here 328 00:19:25,290 --> 00:19:28,410 and where my left hand is over there. 329 00:19:28,410 --> 00:19:32,500 And the question is, where does this ring settle down. 330 00:19:32,500 --> 00:19:35,130 Now, it's obvious, or should be maybe obvious, 331 00:19:35,130 --> 00:19:38,020 is that if my two hands are at equal heights, 332 00:19:38,020 --> 00:19:41,232 it should settle in the middle. 333 00:19:41,232 --> 00:19:42,940 The question that we're trying to resolve 334 00:19:42,940 --> 00:19:47,290 is what if one hand is a little higher than the other. 335 00:19:47,290 --> 00:19:49,040 What happens, or if the other way. 336 00:19:49,040 --> 00:19:52,370 Where does it settle down? 337 00:19:52,370 --> 00:19:55,210 So in order to depict this problem properly, 338 00:19:55,210 --> 00:19:57,480 I need two volunteers to help me out. 339 00:19:57,480 --> 00:20:00,400 Can I have some help? 340 00:20:00,400 --> 00:20:03,370 OK. 341 00:20:03,370 --> 00:20:06,320 So I need one of you to hold the right side, and one of you 342 00:20:06,320 --> 00:20:08,460 to hold the left side. 343 00:20:08,460 --> 00:20:09,260 OK. 344 00:20:09,260 --> 00:20:11,180 And just hold it against the blackboard. 345 00:20:11,180 --> 00:20:12,240 We're going to trace. 346 00:20:12,240 --> 00:20:15,335 So stick it about here, in the middle somewhere. 347 00:20:15,335 --> 00:20:17,210 And now we want to make sure that this one is 348 00:20:17,210 --> 00:20:18,960 a little higher, all right? 349 00:20:18,960 --> 00:20:22,430 So we'll have to-- yeah, let's get a little higher up. 350 00:20:22,430 --> 00:20:25,730 That's probably good enough. 351 00:20:25,730 --> 00:20:29,770 So the experiment has been done. 352 00:20:29,770 --> 00:20:31,100 We now see where this thing is. 353 00:20:31,100 --> 00:20:33,810 But, so now hold on tight. 354 00:20:33,810 --> 00:20:35,960 This thing stretches. 355 00:20:35,960 --> 00:20:38,050 So we want to get it stretched so that we 356 00:20:38,050 --> 00:20:40,667 can see what it is properly. 357 00:20:40,667 --> 00:20:43,000 So this thing isn't heavy enough for this demonstration. 358 00:20:43,000 --> 00:20:46,480 I should've had a ten-ton brick attached there. 359 00:20:46,480 --> 00:20:49,830 But if I did that, than I would tax my, 360 00:20:49,830 --> 00:20:52,310 right, I would tax your abilities to-- right, 361 00:20:52,310 --> 00:20:55,400 so we're going to try to trace out what the possibilities are 362 00:20:55,400 --> 00:21:03,810 here. 363 00:21:03,810 --> 00:21:06,560 So this is, roughly speaking, where this thing-- 364 00:21:06,560 --> 00:21:07,960 and so now where does it settle. 365 00:21:07,960 --> 00:21:10,550 Well, it settles about here. 366 00:21:10,550 --> 00:21:12,630 So we're going to put that as X marks the spot. 367 00:21:12,630 --> 00:21:15,900 OK, thank you very much. 368 00:21:15,900 --> 00:21:20,960 Got to remember where those dots-- OK, all right. 369 00:21:20,960 --> 00:21:24,040 Sorry, I forgot to mark the spots. 370 00:21:24,040 --> 00:21:27,410 OK, so here's the situation. 371 00:21:27,410 --> 00:21:29,860 We have a problem. 372 00:21:29,860 --> 00:21:33,170 And we've hung a string. 373 00:21:33,170 --> 00:21:39,630 And it went down like this and then it went like that. 374 00:21:39,630 --> 00:21:43,710 And we discovered where it settled. 375 00:21:43,710 --> 00:21:44,750 Physically. 376 00:21:44,750 --> 00:21:47,820 So we want to explain this mathematically, and see what's 377 00:21:47,820 --> 00:21:49,280 going on with this problem. 378 00:21:49,280 --> 00:21:50,940 So this is a real-life problem. 379 00:21:50,940 --> 00:21:52,850 It honestly is the problem you have to solve 380 00:21:52,850 --> 00:21:54,100 if you want to build a bridge. 381 00:21:54,100 --> 00:21:55,570 Like, a suspension bridge. 382 00:21:55,570 --> 00:21:57,010 This, among many problems. 383 00:21:57,010 --> 00:21:59,350 It's a very serious and important problem. 384 00:21:59,350 --> 00:22:02,940 And this is the simplest one of this type. 385 00:22:02,940 --> 00:22:04,990 So we've got our shape here. 386 00:22:04,990 --> 00:22:08,255 This should be a straight line, maybe not quite as angled 387 00:22:08,255 --> 00:22:10,920 as that. 388 00:22:10,920 --> 00:22:13,030 The first, we've already drawn the diagram 389 00:22:13,030 --> 00:22:14,670 and we've more or less visualized 390 00:22:14,670 --> 00:22:16,010 what's going on here. 391 00:22:16,010 --> 00:22:21,460 But the first step after the diagram is to give letters. 392 00:22:21,460 --> 00:22:24,630 Is to label things appropriately. 393 00:22:24,630 --> 00:22:29,170 And I do not expect you to be able to do this, at this stage. 394 00:22:29,170 --> 00:22:31,390 Because it requires a lot of experience. 395 00:22:31,390 --> 00:22:33,040 But I'm going to do it for you. 396 00:22:33,040 --> 00:22:34,630 We're going to just do that. 397 00:22:34,630 --> 00:22:37,300 So the simplest thing to do is to use 398 00:22:37,300 --> 00:22:39,800 the coordinates of the plane. 399 00:22:39,800 --> 00:22:43,750 And if you do that, it's also easiest to use the origin. 400 00:22:43,750 --> 00:22:46,590 My favorite number is 0 and it should be yours, too. 401 00:22:46,590 --> 00:22:51,420 So we're going to make this point be (0, 0). 402 00:22:51,420 --> 00:22:54,550 Now, there's another fixed point in this problem. 403 00:22:54,550 --> 00:22:57,089 And it's this point over here. 404 00:22:57,089 --> 00:22:58,880 And we don't know what its coordinates are. 405 00:22:58,880 --> 00:23:00,850 So we're just going to give them letters, a and b. 406 00:23:00,850 --> 00:23:02,225 But those letters are going to be 407 00:23:02,225 --> 00:23:06,110 fixed numbers in this problem. 408 00:23:06,110 --> 00:23:08,840 And we want to solve it for all possible a's and b's. 409 00:23:08,840 --> 00:23:10,470 Now, the interesting thing, remember, 410 00:23:10,470 --> 00:23:12,630 is what happens when b Is not 0. 411 00:23:12,630 --> 00:23:15,825 If b = 0, we already have a clue that the point 412 00:23:15,825 --> 00:23:16,950 should be the center point. 413 00:23:16,950 --> 00:23:19,190 It should be exactly that X, the middle point, 414 00:23:19,190 --> 00:23:22,730 which I'm going to label in a second, is halfway in between. 415 00:23:22,730 --> 00:23:25,930 So now the variable point that I'm going to use is down here. 416 00:23:25,930 --> 00:23:30,220 I'm going to call this point (x, y). 417 00:23:30,220 --> 00:23:31,600 So here's my setup. 418 00:23:31,600 --> 00:23:37,800 I've now given labels to all the things on the diagram so far. 419 00:23:37,800 --> 00:23:42,260 Most of the things on the diagram. 420 00:23:42,260 --> 00:23:47,700 So now, what else do I have to do? 421 00:23:47,700 --> 00:23:55,920 Well, I have to explain to you that this is a minimization 422 00:23:55,920 --> 00:23:56,850 problem. 423 00:23:56,850 --> 00:23:59,010 What happens, actually, physically 424 00:23:59,010 --> 00:24:03,050 is that the weight settles to the lowest point. 425 00:24:03,050 --> 00:24:05,820 That's the thing that has the lowest potential energy. 426 00:24:05,820 --> 00:24:09,370 So we're minimizing a function. 427 00:24:09,370 --> 00:24:13,260 And it's this curve here. 428 00:24:13,260 --> 00:24:17,000 The constraint is that we're restricted to this curve. 429 00:24:17,000 --> 00:24:18,660 So this is a constraint curve. 430 00:24:18,660 --> 00:24:24,590 And we want the lowest point of this curve. 431 00:24:24,590 --> 00:24:29,040 So now, we need a little bit more language in order 432 00:24:29,040 --> 00:24:31,180 to describe what it is that we've got. 433 00:24:31,180 --> 00:24:36,500 And the constraint curve, we got it in a particular way. 434 00:24:36,500 --> 00:24:39,320 Namely, we strung some string from here to there. 435 00:24:39,320 --> 00:24:41,090 And what happens at all these points 436 00:24:41,090 --> 00:24:45,920 is that the total length of the string is the same. 437 00:24:45,920 --> 00:24:48,530 So one way of expressing the constraint 438 00:24:48,530 --> 00:24:52,270 is that the length of the string is constant. 439 00:24:52,270 --> 00:24:54,520 And so in order to figure out what the constraint is, 440 00:24:54,520 --> 00:24:58,120 what this curve is, I have to describe that analytically. 441 00:24:58,120 --> 00:25:01,830 And I'm going to do that by drawing in some helping lines. 442 00:25:01,830 --> 00:25:05,400 Namely, some right triangles to figure out what this length is. 443 00:25:05,400 --> 00:25:07,290 And what the other length is. 444 00:25:07,290 --> 00:25:11,410 So this length is pretty easy if I draw a right triangle here. 445 00:25:11,410 --> 00:25:14,410 Because we go over x and we go down y. 446 00:25:14,410 --> 00:25:19,150 So this length is the square root of x^2 + y^2. 447 00:25:19,150 --> 00:25:22,420 That's the Pythagorean theorem. 448 00:25:22,420 --> 00:25:26,200 Similarly, over here, I'm going to get another length. 449 00:25:26,200 --> 00:25:28,360 Which is a little bit of a mess. 450 00:25:28,360 --> 00:25:30,670 It's the vertical. 451 00:25:30,670 --> 00:25:35,290 So I'm just going to label one half of it, so that you see. 452 00:25:35,290 --> 00:25:38,600 So this horizontal distance is x. 453 00:25:38,600 --> 00:25:41,180 And this horizontal distance from this top 454 00:25:41,180 --> 00:25:44,000 point with this right angle, over there. 455 00:25:44,000 --> 00:25:47,660 It starts at x and ends at a. 456 00:25:47,660 --> 00:25:50,820 The right-most point is a in the x-coordinate. 457 00:25:50,820 --> 00:25:56,150 So the whole distance is a - x. 458 00:25:56,150 --> 00:26:00,700 So that's this leg of this right triangle. 459 00:26:00,700 --> 00:26:05,770 And, similarly, the vertical distance will be b - y. 460 00:26:05,770 --> 00:26:08,300 And so, the formula here, which is a little complicated 461 00:26:08,300 --> 00:26:13,770 for this length, is the square root of (a-x)^2 + (b-y)^2. 462 00:26:17,160 --> 00:26:20,502 So here are the two formulas that 463 00:26:20,502 --> 00:26:22,460 are going to allow me to set up my problem now. 464 00:26:22,460 --> 00:26:25,240 So, my goal is to set it up the way I did here, just 465 00:26:25,240 --> 00:26:26,940 with formulas. 466 00:26:26,940 --> 00:26:40,590 And not with diagrams and not with names. 467 00:26:40,590 --> 00:26:42,910 So here's what I'd like to do. 468 00:26:42,910 --> 00:26:46,840 I claim that what's constrained, if I'm along that curve, 469 00:26:46,840 --> 00:26:48,910 is that the total length is constant. 470 00:26:48,910 --> 00:26:50,730 So that's this statement here. 471 00:26:50,730 --> 00:26:55,830 The square root of x^2 + y^2 plus this other square root. 472 00:26:55,830 --> 00:27:02,360 These are the two lengths of string. 473 00:27:02,360 --> 00:27:08,210 Is equal to some number, L, which is constant. 474 00:27:08,210 --> 00:27:15,790 And this, as I said, is what I'm calling my constraint. 475 00:27:15,790 --> 00:27:16,290 Yeah. 476 00:27:16,290 --> 00:27:20,480 STUDENT: [INAUDIBLE] 477 00:27:20,480 --> 00:27:24,750 PROFESSOR: So the question is, shouldn't it be b+y. 478 00:27:24,750 --> 00:27:28,210 No, and the reason is that y is a negative number. 479 00:27:28,210 --> 00:27:30,300 It's below 0. 480 00:27:30,300 --> 00:27:39,210 So it's actually the sum, -y is a positive number. 481 00:27:39,210 --> 00:27:43,840 All right, so here's the formula. 482 00:27:43,840 --> 00:27:53,000 And then, we want to find the minimum of something. 483 00:27:53,000 --> 00:27:56,250 So what is it that we're finding the minimum of? 484 00:27:56,250 --> 00:27:58,370 This is actually the hardest part of the problem, 485 00:27:58,370 --> 00:27:59,030 conceptually. 486 00:27:59,030 --> 00:28:02,650 I tried to prepare it, but it's very hard to figure this out. 487 00:28:02,650 --> 00:28:07,540 We're finding the least what? 488 00:28:07,540 --> 00:28:10,740 It's y. 489 00:28:10,740 --> 00:28:11,830 We got a name for that. 490 00:28:11,830 --> 00:28:17,220 So we want to find the lowest y. 491 00:28:17,220 --> 00:28:19,690 Now, the reason why it seems a little weird 492 00:28:19,690 --> 00:28:21,770 is you might think of y as just being a variable. 493 00:28:21,770 --> 00:28:25,180 But really, y is a function. 494 00:28:25,180 --> 00:28:29,920 It's really y = y(x) is defined implicitly 495 00:28:29,920 --> 00:28:34,930 by the constraint equation. 496 00:28:34,930 --> 00:28:36,420 That's what that curve is. 497 00:28:36,420 --> 00:28:40,250 And notice the bottom point is exactly 498 00:28:40,250 --> 00:28:46,160 the place where the tangent line will be horizontal. 499 00:28:46,160 --> 00:28:48,180 Which is just what we want. 500 00:28:48,180 --> 00:29:03,600 So from the diagram, the bottom point is where y' = 0. 501 00:29:03,600 --> 00:29:15,195 So this is the critical point. 502 00:29:15,195 --> 00:29:25,720 Yeah? 503 00:29:25,720 --> 00:29:27,540 STUDENT: [INAUDIBLE] 504 00:29:27,540 --> 00:29:28,440 PROFESSOR: Exactly. 505 00:29:28,440 --> 00:29:31,780 So I'm deriving for you-- so the question 506 00:29:31,780 --> 00:29:35,590 is, could I have just tried to find y' = 0 to begin with. 507 00:29:35,590 --> 00:29:37,320 The answer is yes, absolutely. 508 00:29:37,320 --> 00:29:39,070 And in fact I'm leading in that direction. 509 00:29:39,070 --> 00:29:41,820 I'm just showing you, so I'm trying 510 00:29:41,820 --> 00:29:43,930 to make the following, very subtle, point. 511 00:29:43,930 --> 00:29:48,500 Which is in maximum-minimum problems, 512 00:29:48,500 --> 00:29:51,130 we always have to keep track of two things. 513 00:29:51,130 --> 00:29:54,924 Often the interesting point is the critical point. 514 00:29:54,924 --> 00:29:56,840 And that indeed turns out to be the case here. 515 00:29:56,840 --> 00:29:59,840 But we always have to check the ends. 516 00:29:59,840 --> 00:30:02,125 And so there are several ways of checking the ends. 517 00:30:02,125 --> 00:30:03,666 One is, we did this physical problem. 518 00:30:03,666 --> 00:30:05,410 We can see that it's coming up here. 519 00:30:05,410 --> 00:30:06,910 We can see that it's coming up here. 520 00:30:06,910 --> 00:30:10,480 Therefore the bottom must be at this critical point. 521 00:30:10,480 --> 00:30:14,140 So that's OK, so that's one way of checking it. 522 00:30:14,140 --> 00:30:17,872 Another way of checking it is the reasoning that I just gave. 523 00:30:17,872 --> 00:30:19,330 But it's really the same reasoning. 524 00:30:19,330 --> 00:30:20,650 I'm pointing to this thing and I'm 525 00:30:20,650 --> 00:30:23,010 showing you that the bottom is somewhere in the middle. 526 00:30:23,010 --> 00:30:26,470 So, therefore, it is a place of horizontal tangency. 527 00:30:26,470 --> 00:30:28,970 That's the reasoning that I'm using. 528 00:30:28,970 --> 00:30:30,510 So, again, this is to avoid having 529 00:30:30,510 --> 00:30:33,775 to evaluate a limit of an end or to use 530 00:30:33,775 --> 00:30:35,150 the second derivative test, which 531 00:30:35,150 --> 00:30:42,310 is a total catastrophe in this case. 532 00:30:42,310 --> 00:30:45,470 OK, now. 533 00:30:45,470 --> 00:30:47,530 There's one other thing that you might 534 00:30:47,530 --> 00:30:51,081 know about this if you've seen this geometric construction 535 00:30:51,081 --> 00:30:51,580 before. 536 00:30:51,580 --> 00:30:54,300 With a string and chalk. 537 00:30:54,300 --> 00:30:57,917 Which is that this curve is an eclipse. 538 00:30:57,917 --> 00:30:59,750 It turns out, this is a piece of an eclipse. 539 00:30:59,750 --> 00:31:00,870 It's a huge ellipse. 540 00:31:00,870 --> 00:31:03,580 These two points turn out to be the so-called foci 541 00:31:03,580 --> 00:31:05,500 of the ellipse. 542 00:31:05,500 --> 00:31:09,530 However, that geometric fact is totally useless for solving 543 00:31:09,530 --> 00:31:10,860 this problem. 544 00:31:10,860 --> 00:31:12,390 Completely useless. 545 00:31:12,390 --> 00:31:14,850 If you actually write out the formulas for the ellipse, 546 00:31:14,850 --> 00:31:17,130 you'll discover that you have a much harder problem 547 00:31:17,130 --> 00:31:18,090 on your hands. 548 00:31:18,090 --> 00:31:20,150 And it will take you twice or ten times 549 00:31:20,150 --> 00:31:23,720 as long, so, it's true that it's an ellipse, 550 00:31:23,720 --> 00:31:25,670 but it doesn't help. 551 00:31:25,670 --> 00:31:29,510 OK, so what we're going to do instead is much simpler. 552 00:31:29,510 --> 00:31:32,390 We're going to leave this expression alone 553 00:31:32,390 --> 00:31:35,480 and we're just going to differentiate implicitly. 554 00:31:35,480 --> 00:31:42,430 So again, we use implicit differentiation 555 00:31:42,430 --> 00:31:49,430 on the constraint equation. 556 00:31:49,430 --> 00:31:52,380 So that's the equation which is directly above me there, 557 00:31:52,380 --> 00:31:54,310 at the top. 558 00:31:54,310 --> 00:31:57,470 And I have to differentiate it with respect to x. 559 00:31:57,470 --> 00:31:59,190 So that's a little ugly, but we've 560 00:31:59,190 --> 00:32:01,080 done this a few times before. 561 00:32:01,080 --> 00:32:02,940 When you differentiate a square root, 562 00:32:02,940 --> 00:32:08,230 the square root goes into the denominator. 563 00:32:08,230 --> 00:32:11,500 And there's a factor of 1/2, so there's a 2x which cancels. 564 00:32:11,500 --> 00:32:12,750 So I claim it's this. 565 00:32:12,750 --> 00:32:17,180 Now, because y depends on x, there's also a y y' here. 566 00:32:17,180 --> 00:32:22,170 So technically speaking, it's twice this with a half. 567 00:32:22,170 --> 00:32:24,310 2/2 times that. 568 00:32:24,310 --> 00:32:30,790 So that's the differentiation of the first piece of this guy. 569 00:32:30,790 --> 00:32:32,360 Now I'm going to do this second one, 570 00:32:32,360 --> 00:32:34,060 and it's also the chain rule. 571 00:32:34,060 --> 00:32:39,150 But you're just going to have to let me do it for you. 572 00:32:39,150 --> 00:32:41,690 Because it's just a little bit too long for you 573 00:32:41,690 --> 00:32:43,847 to pay attention to. 574 00:32:43,847 --> 00:32:45,430 It turns out there's a minus sign that 575 00:32:45,430 --> 00:32:49,710 comes out, because there's a - x and a - y there. 576 00:32:49,710 --> 00:32:51,940 And then the numerator, the denominator 577 00:32:51,940 --> 00:32:55,100 is the same massive square root. 578 00:32:55,100 --> 00:32:59,920 So it's (a - (a-x)^2, (b-y)^2. 579 00:32:59,920 --> 00:33:05,670 And the numerator is a - x, which is 580 00:33:05,670 --> 00:33:07,750 what replaces the x over here. 581 00:33:07,750 --> 00:33:13,660 And then another term, which is (b-y)y'. 582 00:33:13,660 --> 00:33:17,140 I claim that that's analogous to what I did in the first term. 583 00:33:17,140 --> 00:33:20,086 And you'll just have to check this on your own. 584 00:33:20,086 --> 00:33:21,460 Because I did it too fast for you 585 00:33:21,460 --> 00:33:23,607 to be able to check yourself. 586 00:33:23,607 --> 00:33:26,190 Now, that's going to be equal to what, on the right-hand side? 587 00:33:26,190 --> 00:33:29,510 What's the derivative of L with respect to x? 588 00:33:29,510 --> 00:33:30,540 It's 0. 589 00:33:30,540 --> 00:33:32,300 It's not changing in the problem. 590 00:33:32,300 --> 00:33:37,630 Although my parachute stuff stretches. 591 00:33:37,630 --> 00:33:40,700 We tried to stretch it to its fullest extent. 592 00:33:40,700 --> 00:33:45,100 So that we kept it fixed, that was the goal here. 593 00:33:45,100 --> 00:33:50,380 So now, this looks like a total mess. 594 00:33:50,380 --> 00:33:52,940 But, it's not. 595 00:33:52,940 --> 00:33:54,440 And let me show you why. 596 00:33:54,440 --> 00:33:57,110 It simplifies a great deal. 597 00:33:57,110 --> 00:34:01,140 And let me show you exactly how. 598 00:34:01,140 --> 00:34:03,400 So, first of all, the whole point 599 00:34:03,400 --> 00:34:06,860 is we're looking for the place where y' = 0. 600 00:34:06,860 --> 00:34:14,080 So that means that these terms go away. y' = 0. 601 00:34:14,080 --> 00:34:16,690 So they're gone. 602 00:34:16,690 --> 00:34:21,070 And now what we have is the following equation. 603 00:34:21,070 --> 00:34:27,810 It's x divided by square root of x^2 + y^2 is equal to, if I put 604 00:34:27,810 --> 00:34:32,450 it on the other side the minus sign is changed to a plus sign, 605 00:34:32,450 --> 00:34:37,470 a - x divided by this other massive object, 606 00:34:37,470 --> 00:34:39,020 (a-x)^2 + (b-y)^2. 607 00:34:43,300 --> 00:34:45,510 So this is what it simplifies to. 608 00:34:45,510 --> 00:34:51,020 Now again, that also might look complicated to you. 609 00:34:51,020 --> 00:34:55,890 But I claim that this is something, 610 00:34:55,890 --> 00:34:59,620 this is a kind of equilibrium equation 611 00:34:59,620 --> 00:35:01,980 that can be interpreted geometrically, 612 00:35:01,980 --> 00:35:05,360 in a way that is very meaningful and important. 613 00:35:05,360 --> 00:35:09,330 So first of all, let me observe for you that this x is 614 00:35:09,330 --> 00:35:10,730 something on our picture. 615 00:35:10,730 --> 00:35:14,160 And the square root of x^2 + y^2 is something on our picture. 616 00:35:14,160 --> 00:35:18,870 Namely, if I go over to the picture, here was x 617 00:35:18,870 --> 00:35:20,760 and this was a right triangle. 618 00:35:20,760 --> 00:35:25,700 And this hypotenuse was square root of x^2 + y^2. 619 00:35:25,700 --> 00:35:34,200 So, if I call this angle alpha, then this is the sine of alpha. 620 00:35:34,200 --> 00:35:34,700 Right? 621 00:35:34,700 --> 00:35:36,950 It's a right triangle, that's the opposite leg. 622 00:35:36,950 --> 00:35:44,940 So this guy is the sine of alpha. 623 00:35:44,940 --> 00:35:48,510 Similarly, the other side has an interpretation 624 00:35:48,510 --> 00:35:50,800 for the other right triangle. 625 00:35:50,800 --> 00:35:56,150 If this angle is beta, then the opposite side is a-x, 626 00:35:56,150 --> 00:36:00,850 and the hypotenuse is what was in the denominator over there. 627 00:36:00,850 --> 00:36:10,130 So this side is sine of beta. 628 00:36:10,130 --> 00:36:14,080 And so what this condition is telling us 629 00:36:14,080 --> 00:36:19,230 is that alpha = beta. 630 00:36:19,230 --> 00:36:22,620 Which is the hidden symmetry in the problem. 631 00:36:22,620 --> 00:36:24,940 I don't know if you can actually see it 632 00:36:24,940 --> 00:36:28,490 when I show you this thing. 633 00:36:28,490 --> 00:36:32,920 But, no matter how I tilt it, actually the two angles 634 00:36:32,920 --> 00:36:39,330 from the horizontal are the same. 635 00:36:39,330 --> 00:36:40,920 In the middle it's kind of obvious 636 00:36:40,920 --> 00:36:42,540 that that should be the case. 637 00:36:42,540 --> 00:36:44,360 But on the sides it's not obvious 638 00:36:44,360 --> 00:36:46,820 that that's what's happening. 639 00:36:46,820 --> 00:36:49,882 Now, this has even-- so that's a symmetry, if you like, 640 00:36:49,882 --> 00:36:50,590 of the situation. 641 00:36:50,590 --> 00:36:52,590 These two angles are equal. 642 00:36:52,590 --> 00:36:55,440 But there's something more to be said. 643 00:36:55,440 --> 00:36:58,290 If you do a force diagram for this, what you'll discover 644 00:36:58,290 --> 00:37:03,680 is that the tension on the two lines is the same. 645 00:37:03,680 --> 00:37:05,690 Which means that when you build something 646 00:37:05,690 --> 00:37:11,350 which is hanging like this, it will involve the least stress. 647 00:37:11,350 --> 00:37:13,260 If you hang something very heavy, 648 00:37:13,260 --> 00:37:15,790 and one side carries twice as much load as the other, 649 00:37:15,790 --> 00:37:18,060 then you have twice as much chance of its falling 650 00:37:18,060 --> 00:37:19,260 and breaking. 651 00:37:19,260 --> 00:37:22,530 If they each hold the same strength, 652 00:37:22,530 --> 00:37:26,320 then you've distributed the load in a much more balanced way. 653 00:37:26,320 --> 00:37:28,660 So this is a kind of a balance condition, 654 00:37:28,660 --> 00:37:31,740 and it's very typical of minimization problems. 655 00:37:31,740 --> 00:37:34,040 And fortunately, there are nice solutions 656 00:37:34,040 --> 00:37:37,020 which distribute the weight reasonably well. 657 00:37:37,020 --> 00:37:41,290 That's certainly the principle of suspension bridges. 658 00:37:41,290 --> 00:37:42,820 So yeah, one more question. 659 00:37:42,820 --> 00:37:44,760 STUDENT: [INAUDIBLE] 660 00:37:44,760 --> 00:37:48,767 PROFESSOR: OK, so the question where 661 00:37:48,767 --> 00:37:50,600 it hangs, that is, what the formula for x is 662 00:37:50,600 --> 00:37:53,030 and what the formula for y is. 663 00:37:53,030 --> 00:37:55,900 And other things, like the equation for the ellipse 664 00:37:55,900 --> 00:37:58,310 and lots of other stuff like that. 665 00:37:58,310 --> 00:38:01,950 Those are things that I will answer for you in a set 666 00:38:01,950 --> 00:38:03,700 of notes which I will hand out. 667 00:38:03,700 --> 00:38:06,310 And they're just a mess. 668 00:38:06,310 --> 00:38:08,400 You see they're, just as in that other problem 669 00:38:08,400 --> 00:38:11,604 that we did last time, there's some illuminating things you 670 00:38:11,604 --> 00:38:13,020 can say about the problem and then 671 00:38:13,020 --> 00:38:15,890 there's some messy formulas. 672 00:38:15,890 --> 00:38:19,222 You know, you want to try to pick out the simple things 673 00:38:19,222 --> 00:38:19,930 that you can say. 674 00:38:19,930 --> 00:38:21,740 In fact, that's a property of math, 675 00:38:21,740 --> 00:38:24,500 you want to focus on the more comprehensible things. 676 00:38:24,500 --> 00:38:26,770 On the other hand, it can be done. 677 00:38:26,770 --> 00:38:30,290 It just takes a little bit of computation. 678 00:38:30,290 --> 00:38:33,630 So I didn't answer the question of what the lowest y was. 679 00:38:33,630 --> 00:38:39,020 But I'll do that for you. 680 00:38:39,020 --> 00:38:42,284 Maybe I'll just mention one more thing about this problem. 681 00:38:42,284 --> 00:38:44,700 This is a very amusing problem from a completely different 682 00:38:44,700 --> 00:38:45,890 point of view. 683 00:38:45,890 --> 00:38:50,280 If you sort of roll the ellipse around, 684 00:38:50,280 --> 00:38:55,580 you get the same phenomenon from each place here. 685 00:38:55,580 --> 00:38:58,170 So it doesn't matter where a and 0 are. 686 00:38:58,170 --> 00:38:59,490 You'll get the same phenomenon. 687 00:38:59,490 --> 00:39:02,040 That is, the tangent line. 688 00:39:02,040 --> 00:39:06,030 So, this angle and that angle will be equal. 689 00:39:06,030 --> 00:39:09,934 So you can also read that as being the angle over here 690 00:39:09,934 --> 00:39:11,350 and the angle over here are equal. 691 00:39:11,350 --> 00:39:15,730 That is, beta-- that is the complementary angles 692 00:39:15,730 --> 00:39:17,290 are also equal. 693 00:39:17,290 --> 00:39:21,170 And if you interpret this as a ray of light, 694 00:39:21,170 --> 00:39:23,540 and this as a mirror, then this would 695 00:39:23,540 --> 00:39:27,040 be saying that if you start at one focus, every ray of light 696 00:39:27,040 --> 00:39:29,310 will bounce and go to the other focus. 697 00:39:29,310 --> 00:39:32,690 So that's a property that an ellipse has. 698 00:39:32,690 --> 00:39:36,460 More precisely, a property that this kind of curve has. 699 00:39:36,460 --> 00:39:39,080 And in fact, a few years ago there 700 00:39:39,080 --> 00:39:43,590 was this great piece of art at something 701 00:39:43,590 --> 00:39:46,330 called the DeCordova Museum, which I recommended very highly 702 00:39:46,330 --> 00:39:52,460 to you go sometime to visit in your four years here. 703 00:39:52,460 --> 00:39:59,000 There was a collection of miniature golf holes. 704 00:39:59,000 --> 00:40:02,070 So they had a bunch of mini golf pieces of art. 705 00:40:02,070 --> 00:40:04,320 And every one was completely different from the other. 706 00:40:04,320 --> 00:40:07,910 And one of them was called hole in one. 707 00:40:07,910 --> 00:40:11,110 And the tee was at one focus of the ellipse. 708 00:40:11,110 --> 00:40:15,390 And the hole was at the other focus of the ellipse. 709 00:40:15,390 --> 00:40:18,660 So, no matter how you hit the golf ball, 710 00:40:18,660 --> 00:40:21,790 it always goes into the hole. 711 00:40:21,790 --> 00:40:24,595 No matter where it bounces, it just, one bounce 712 00:40:24,595 --> 00:40:26,820 and it's in the hole. 713 00:40:26,820 --> 00:40:30,550 So that's actually a consequence of the computation 714 00:40:30,550 --> 00:40:40,260 that we just did. 715 00:40:40,260 --> 00:40:42,170 Time to go on. 716 00:40:42,170 --> 00:41:06,150 We're going to now talk about something else. 717 00:41:06,150 --> 00:41:10,990 So our next topic is Newton's method. 718 00:41:10,990 --> 00:41:24,020 Which is one of the greatest applications of calculus. 719 00:41:24,020 --> 00:41:27,610 And I'm going to describe it here for you. 720 00:41:27,610 --> 00:41:34,870 And we'll illustrate it on an example, 721 00:41:34,870 --> 00:41:39,970 which is solving the equation x^2 = 5. 722 00:41:39,970 --> 00:41:42,000 We're going to find the square root of 5. 723 00:41:42,000 --> 00:41:45,520 Now, you can actually solve any equation this way. 724 00:41:45,520 --> 00:41:46,910 Any equation that you understand, 725 00:41:46,910 --> 00:41:49,822 you can solve this way, essentially. 726 00:41:49,822 --> 00:41:52,280 So even though I'm doing it for the square root of 5, which 727 00:41:52,280 --> 00:41:56,990 is something that you can figure out on your calculator, 728 00:41:56,990 --> 00:41:59,000 in fact this is really at the heart 729 00:41:59,000 --> 00:42:02,210 of many of the ways in which calculators work. 730 00:42:02,210 --> 00:42:04,880 So, the first thing is to make this problem a little bit more 731 00:42:04,880 --> 00:42:06,420 abstract. 732 00:42:06,420 --> 00:42:13,830 We're going to set f(x) = x^2 - 5. 733 00:42:13,830 --> 00:42:19,840 And then we're going to solve f(x) = 0. 734 00:42:19,840 --> 00:42:24,120 So this is the sort of standard form for solving such a-- 735 00:42:24,120 --> 00:42:27,360 So you take some either complicated or simple function 736 00:42:27,360 --> 00:42:29,490 of x, linear functions are boring, 737 00:42:29,490 --> 00:42:32,550 quadratic functions are already interesting. 738 00:42:32,550 --> 00:42:36,272 And cubic functions, as I've said a few times, 739 00:42:36,272 --> 00:42:37,980 you don't even have formulas for solving. 740 00:42:37,980 --> 00:42:39,688 So this would be the only method you have 741 00:42:39,688 --> 00:42:43,750 for solving them numerically. 742 00:42:43,750 --> 00:42:45,810 So here's how it works. 743 00:42:45,810 --> 00:42:50,010 So the idea, I'll plot this function. 744 00:42:50,010 --> 00:42:55,810 Here's the function, it's a parabola, y = x^2 - 5. 745 00:42:55,810 --> 00:43:01,090 It dips below 0, sorry, it should be centered, but anyway. 746 00:43:01,090 --> 00:43:06,530 And now the idea here is to start with a guess. 747 00:43:06,530 --> 00:43:11,380 And square root of 5 is pretty close to the square root of 4, 748 00:43:11,380 --> 00:43:12,360 which is 2. 749 00:43:12,360 --> 00:43:17,030 So my first guess is going to be 2, here. 750 00:43:17,030 --> 00:43:30,510 So start with initial guess. 751 00:43:30,510 --> 00:43:32,320 So that's our first guess. 752 00:43:32,320 --> 00:43:35,860 And now, what we're going to do is 753 00:43:35,860 --> 00:43:40,080 we're going to pretend that the function is linear. 754 00:43:40,080 --> 00:43:41,300 That's all we're going to do. 755 00:43:41,300 --> 00:43:44,530 And then if the function were linear, 756 00:43:44,530 --> 00:43:47,432 we're going to try to find where the 0 is. 757 00:43:47,432 --> 00:43:49,265 So if the function is linear, what we'll use 758 00:43:49,265 --> 00:43:54,990 is we'll plot the point where 2 is on, that is, 759 00:43:54,990 --> 00:43:59,040 the point (2, f(2)), and then we're 760 00:43:59,040 --> 00:44:06,660 going to draw the tangent line here. 761 00:44:06,660 --> 00:44:14,560 And this is going to be our new guess. 762 00:44:14,560 --> 00:44:22,420 x equals x, which I'll call x_1. 763 00:44:22,420 --> 00:44:26,610 So the idea here is that this point may be somewhat far 764 00:44:26,610 --> 00:44:31,250 from where it crosses, but this point will be a little closer. 765 00:44:31,250 --> 00:44:33,830 And now we're going to do this over and over again. 766 00:44:33,830 --> 00:44:38,340 And see how fast it gets to the place we're aiming for. 767 00:44:38,340 --> 00:44:42,020 So we have to work out what the formulas are. 768 00:44:42,020 --> 00:44:53,700 And that's the strategy. 769 00:44:53,700 --> 00:45:01,360 So now, the first step here is, we have our guess, 770 00:45:01,360 --> 00:45:06,020 we have our tangent line. 771 00:45:06,020 --> 00:45:13,540 Which has the formula y - y_0 = m(x - x_0). 772 00:45:13,540 --> 00:45:16,220 So that's the general form for a tangent line. 773 00:45:16,220 --> 00:45:20,140 And now, I have to tell you what x_1 is. 774 00:45:20,140 --> 00:45:29,970 In terms of this tangent line. x_1 is the x-intercept. 775 00:45:29,970 --> 00:45:32,810 The tangent line, if you look over here at the diagram, 776 00:45:32,810 --> 00:45:36,220 the tangent line is the orange line. 777 00:45:36,220 --> 00:45:41,340 Where that crosses the axis, that's where I want to put x_1. 778 00:45:41,340 --> 00:45:43,530 So that's the x-intercept. 779 00:45:43,530 --> 00:45:46,670 Now, how do you find the x-intercept? 780 00:45:46,670 --> 00:45:49,760 You find it by setting y = 0. 781 00:45:49,760 --> 00:45:52,270 That horizontal line is y = 0. 782 00:45:52,270 --> 00:45:59,790 So I set y = 0, and I get 0 - y_0 = m (x_1 - x_0). 783 00:45:59,790 --> 00:46:02,300 So I changed two things in this equation. 784 00:46:02,300 --> 00:46:04,970 I plugged in 0 here, for y. 785 00:46:04,970 --> 00:46:07,020 And I said that the place where that happens 786 00:46:07,020 --> 00:46:13,050 is going to be where x is x_1. 787 00:46:13,050 --> 00:46:15,530 So now let's solve. 788 00:46:15,530 --> 00:46:19,700 And what we get here is -.y divided by-- sorry, 789 00:46:19,700 --> 00:46:26,390 -y_0 / m = x_1 - x_0. 790 00:46:26,390 --> 00:46:30,800 And now I can get a formula for x_1. 791 00:46:30,800 --> 00:46:46,600 So x_1 = x_0 - y_0 / m. 792 00:46:46,600 --> 00:46:53,140 I now need to tell you what this formula means, 793 00:46:53,140 --> 00:46:56,070 in terms of the function f. 794 00:46:56,070 --> 00:46:59,230 So first of all, x_0 is x_0, whatever x_0 is. 795 00:46:59,230 --> 00:47:06,230 And y_0, I claim, is f(x_0). 796 00:47:06,230 --> 00:47:11,080 And m is the slope at that same place. 797 00:47:11,080 --> 00:47:17,230 So it's f'(x_0). 798 00:47:17,230 --> 00:47:19,650 And this is the whole story. 799 00:47:19,650 --> 00:47:25,440 This is the formula which will enable us to calculate 800 00:47:25,440 --> 00:47:34,520 basically any root. 801 00:47:34,520 --> 00:47:36,072 I'm going to repeat this formula, 802 00:47:36,072 --> 00:47:37,530 so I'm going to tell you again what 803 00:47:37,530 --> 00:47:40,620 Newton's method is, and put a little more 804 00:47:40,620 --> 00:47:43,080 colorful box around it. 805 00:47:43,080 --> 00:47:53,210 So Newton's method involves the following formula. 806 00:47:53,210 --> 00:47:57,470 In order to get the (n+1)st point, 807 00:47:57,470 --> 00:48:00,250 that's our better and better guess, 808 00:48:00,250 --> 00:48:03,530 I'm going to take the nth one and then I'm going to plug 809 00:48:03,530 --> 00:48:04,330 in this formula. 810 00:48:04,330 --> 00:48:09,870 So f(x_n) / f'(x_n). 811 00:48:09,870 --> 00:48:13,630 So this is the basic formula, and if you like, 812 00:48:13,630 --> 00:48:18,820 this is the idea of just repeating what I had before. 813 00:48:18,820 --> 00:48:21,900 Now, we've gone from geometry, from just pictures, 814 00:48:21,900 --> 00:48:23,520 to an honest to goodness formula which 815 00:48:23,520 --> 00:48:25,690 is completely implementable and very easy 816 00:48:25,690 --> 00:48:29,450 to implement in any case. 817 00:48:29,450 --> 00:48:33,540 So let's see how it works in the case that we've got. 818 00:48:33,540 --> 00:48:41,160 Which is x_0 = 2. f(x) = x^2 - 5. 819 00:48:41,160 --> 00:48:44,850 Let's see how to implement this formula. 820 00:48:44,850 --> 00:48:49,920 So first of all, I have to calculate for you, f'(x). 821 00:48:49,920 --> 00:48:55,200 That's 2x. 822 00:48:55,200 --> 00:49:05,420 And so, x_1 is equal to x_0 minus, so f', sorry, 823 00:49:05,420 --> 00:49:08,690 f(x) would be x_0^2 - 5. 824 00:49:08,690 --> 00:49:10,550 That's what's in the numerator. 825 00:49:10,550 --> 00:49:13,220 And in the denominator I have the derivative, 826 00:49:13,220 --> 00:49:18,820 so that's 2 x_0. 827 00:49:18,820 --> 00:49:23,920 And so all told, well, let's work it out in two steps here. 828 00:49:23,920 --> 00:49:27,190 This is -1/2 x_0 for the first term, 829 00:49:27,190 --> 00:49:31,750 and then plus (5/2) / x_0 for the second term. 830 00:49:31,750 --> 00:49:40,450 And these two terms combine, so we have here 1/2 x_0 plus 5/2 831 00:49:40,450 --> 00:49:43,010 with an x_0 in the denominator. 832 00:49:43,010 --> 00:49:53,310 So here's the formula for x_1. in this case. 833 00:49:53,310 --> 00:50:03,620 Now I'd like to show you how well this works. 834 00:50:03,620 --> 00:50:10,790 So first of all, we have x_1 is 1/2 * 2, if I plug in x1, 835 00:50:10,790 --> 00:50:16,630 plus 5/4, which is 9/4. 836 00:50:16,630 --> 00:50:25,620 And x_2, I have 1/2 * 9/4 + + 5/2 * 4/9. 837 00:50:28,550 --> 00:50:30,360 That's the next one. 838 00:50:30,360 --> 00:50:37,610 And if you work this out, it's 161/72. 839 00:50:37,610 --> 00:50:41,302 And then x_3 is kind of long. 840 00:50:41,302 --> 00:50:43,760 But I will just write down what it is, so that you can see. 841 00:50:43,760 --> 00:50:49,270 It's 1/2 * 161/72 plus 5/2 and then 842 00:50:49,270 --> 00:50:50,600 I do the reciprocal of that. 843 00:50:50,600 --> 00:50:56,620 So 72/161. 844 00:50:56,620 --> 00:51:00,770 So let's see how good these are. 845 00:51:00,770 --> 00:51:05,850 I carefully calculated how far off they are. 846 00:51:05,850 --> 00:51:07,510 Somewhere on my notes. 847 00:51:07,510 --> 00:51:11,660 So I'll just take a look and see what I said. 848 00:51:11,660 --> 00:51:13,930 Oh yeah, I did do it. 849 00:51:13,930 --> 00:51:20,460 So, what's the square root of 5 minus-- so here's n, 850 00:51:20,460 --> 00:51:24,420 here's the square root of 5 minus x_n, if you like. 851 00:51:24,420 --> 00:51:26,800 Or the other way around. 852 00:51:26,800 --> 00:51:28,900 The size of this. 853 00:51:28,900 --> 00:51:31,540 You'll have to decide on your homework 854 00:51:31,540 --> 00:51:33,960 whether it comes out positive or negative, to the right 855 00:51:33,960 --> 00:51:35,660 or to the left of the answer. 856 00:51:35,660 --> 00:51:37,460 But let's do this. 857 00:51:37,460 --> 00:51:41,070 So when n = 0, the guess was 2. 858 00:51:41,070 --> 00:51:45,330 And we're off by about 2 * 10^(-1). 859 00:51:45,330 --> 00:51:48,690 And if n = 1, so that's this 9/4, 860 00:51:48,690 --> 00:51:53,120 that's off by about 10^(-2). 861 00:51:53,120 --> 00:51:58,240 And then n = 2, that's this number here, right? 862 00:51:58,240 --> 00:52:03,540 And that's off by about 4 * 10^(-5). 863 00:52:03,540 --> 00:52:05,840 That's already as good an approximation 864 00:52:05,840 --> 00:52:10,450 to the square root of 5 as you'll ever need in your life. 865 00:52:10,450 --> 00:52:17,050 If you do 3, this number here turns out to be accurate 866 00:52:17,050 --> 00:52:20,140 to 10^(-10) or so. 867 00:52:20,140 --> 00:52:23,391 This goes right off to the edge of my calculator, this one 868 00:52:23,391 --> 00:52:23,890 here. 869 00:52:23,890 --> 00:52:26,500 So already, with the third iterate. 870 00:52:26,500 --> 00:52:31,340 you're of way past the accuracy that you need for most things. 871 00:52:31,340 --> 00:52:32,070 Yep, question. 872 00:52:32,070 --> 00:52:32,903 STUDENT: [INAUDIBLE] 873 00:52:32,903 --> 00:52:38,660 PROFESSOR: How come the x_0 disappears? 874 00:52:38,660 --> 00:52:55,360 STUDENT: [INAUDIBLE] PROFESSOR: So, from here to here 875 00:52:55,360 --> 00:52:56,010 [INAUDIBLE] 876 00:52:56,010 --> 00:52:56,380 STUDENT: [INAUDIBLE] 877 00:52:56,380 --> 00:52:58,296 PROFESSOR: Hang on, folks, we have a question. 878 00:52:58,296 --> 00:52:59,390 Let's just answer it. 879 00:52:59,390 --> 00:53:02,870 So here we have an x_0, and here we have -1/2, 880 00:53:02,870 --> 00:53:08,050 there's an x_0^2 and an x which cancel. 881 00:53:08,050 --> 00:53:11,480 And here we have a minus, and a -5/2 x0. 882 00:53:11,480 --> 00:53:16,150 So I combine the 1 - 1/2, I got +1/2, that's all. 883 00:53:16,150 --> 00:53:17,680 OK? 884 00:53:17,680 --> 00:53:18,890 All right, thanks. 885 00:53:18,890 --> 00:53:21,910 We'll have to ask other questions after class.