1 00:00:00 --> 00:00:00 2 00:00:00 --> 00:00:02,33 The following content is provided under a Creative 3 00:00:02,33 --> 00:00:03,61 Commons license. 4 00:00:03,61 --> 00:00:06,73 Your support will help MIT OpenCourseWare continue to 5 00:00:06,73 --> 00:00:10,19 offer high quality educational resources for free. 6 00:00:10,19 --> 00:00:12,93 To make a donation, or to view additional materials from 7 00:00:12,93 --> 00:00:15,88 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15,88 --> 00:00:22,04 at ocw.mit.edu. 9 00:00:22,04 --> 00:00:25,38 PROFESSOR: OK, we're ready to start the eleventh lecture. 10 00:00:25,38 --> 00:00:29,39 We're still in the middle of sketching. 11 00:00:29,39 --> 00:00:33,93 And, indeed, one of the reasons why we did not talk about 12 00:00:33,93 --> 00:00:38,44 hyperbolic functions is this that we're running just 13 00:00:38,44 --> 00:00:39,54 a little bit behind. 14 00:00:39,54 --> 00:00:41,72 And we'll catch up a tiny bit today. 15 00:00:41,72 --> 00:00:46 And I hope all the way on Tuesday of next week. 16 00:00:46 --> 00:00:58,64 So let me pick up where we left off, with sketching. 17 00:00:58,64 --> 00:01:05,22 So this is a continuation. 18 00:01:05,22 --> 00:01:07,62 I want to give you one more example of how 19 00:01:07,62 --> 00:01:08,39 to sketch things. 20 00:01:08,39 --> 00:01:10,38 And then we'll go through it systematically. 21 00:01:10,38 --> 00:01:13,45 So the second example that we did as one example 22 00:01:13,45 --> 00:01:18,06 last time, is this. 23 00:01:18,06 --> 00:01:19,76 The function is x + 24 00:01:19,76 --> 00:01:21,97 + 1 / x + + 2. 25 00:01:21,97 --> 00:01:25,78 And I'm going to save you the time right now. 26 00:01:25,78 --> 00:01:29,4 This is very typical of me, especially if you're in a hurry 27 00:01:29,4 --> 00:01:32,92 on an exam, I'll just tell you what the derivative is. 28 00:01:32,92 --> 00:01:37,76 So in this case, it's 1 / (x + 2)^2. 29 00:01:37,76 --> 00:01:42,65 Now, the reason why I'm bringing this example up, even 30 00:01:42,65 --> 00:01:44,72 though it'll turn out to be a relatively simple one to 31 00:01:44,72 --> 00:01:51,69 sketch, is that it's easy to fall into a black hole 32 00:01:51,69 --> 00:01:53,89 with this problem. 33 00:01:53,89 --> 00:01:57,47 So let me just show you. 34 00:01:57,47 --> 00:01:59,6 This is not equal to 0. 35 00:01:59,6 --> 00:02:01,18 It's never equal to 0. 36 00:02:01,18 --> 00:02:08,94 So that means there are no critical points. 37 00:02:08,94 --> 00:02:15,46 At this point, students, many students who have been trained 38 00:02:15,46 --> 00:02:19,23 like monkeys to do exactly what they've been told, suddenly 39 00:02:19,23 --> 00:02:21,17 freeze and give up. 40 00:02:21,17 --> 00:02:23,97 Because there's nothing to do. 41 00:02:23,97 --> 00:02:28 So this is the one thing that I have to train out of you. 42 00:02:28 --> 00:02:32,26 You can't just give up at this point. 43 00:02:32,26 --> 00:02:35,06 So what would you suggest? 44 00:02:35,06 --> 00:02:38,1 Can anybody get us out of this jam? 45 00:02:38,1 --> 00:02:38,78 Yeah. 46 00:02:38,78 --> 00:02:47,41 STUDENT: [INAUDIBLE] 47 00:02:47,41 --> 00:02:48,15 PROFESSOR: Right. 48 00:02:48,15 --> 00:02:53,93 So the suggestion was to find the x values where 49 00:02:53,93 --> 00:02:55,8 f (x) is undefined. 50 00:02:55,8 --> 00:03:01,09 In fact, so now that's a fairly sophisticated way of putting 51 00:03:01,09 --> 00:03:04,69 the point that I want to make, which is that what we want to 52 00:03:04,69 --> 00:03:07,77 do is go back to our precalculus skills. 53 00:03:07,77 --> 00:03:11,15 And just plot points. 54 00:03:11,15 --> 00:03:13,84 So instead, you go back to precalculus and you 55 00:03:13,84 --> 00:03:17,22 just plot some points. 56 00:03:17,22 --> 00:03:18,35 It's a perfectly reasonable thing. 57 00:03:18,35 --> 00:03:22,18 Now, it turns out that the most important point to plot is 58 00:03:22,18 --> 00:03:24,73 the one that's not there. 59 00:03:24,73 --> 00:03:29,3 Namely, the value of x = - 2. 60 00:03:29,3 --> 00:03:31,78 Which is just what was suggested. 61 00:03:31,78 --> 00:03:35,47 Namely, we plot the points where the function 62 00:03:35,47 --> 00:03:38,71 is not defined. 63 00:03:38,71 --> 00:03:41,22 So how do we do that? 64 00:03:41,22 --> 00:03:43,77 Well, you have to think about it for a second and 65 00:03:43,77 --> 00:03:46,47 I'll introduce some new notation when I do it. 66 00:03:46,47 --> 00:03:49,29 If I evaluate 2 at this place, actually I can't do it. 67 00:03:49,29 --> 00:03:51,25 I have to do it from the left and the right. 68 00:03:51,25 --> 00:03:56,61 So if I plug in - 2 on the positive side, from the 69 00:03:56,61 --> 00:04:01,15 right, that's going to be equal to - 2 + 70 00:04:01,15 --> 00:04:05,3 1 / - 2, a little bit more than - 2, 71 00:04:05,3 --> 00:04:06,59 +2. 72 00:04:06,59 --> 00:04:11,11 Which is - 1 divided by - now, this denominator is - 2, 73 00:04:11,11 --> 00:04:12,5 a little more than that, 74 00:04:12,5 --> 00:04:12,73 +2. 75 00:04:12,73 --> 00:04:20,2 So it's a little more than 0. 76 00:04:20,2 --> 00:04:24,72 And that is, well we'll fill that in in a second. 77 00:04:24,72 --> 00:04:25,62 Everybody's puzzled. 78 00:04:25,62 --> 00:04:26,04 Yes. 79 00:04:26,04 --> 00:04:30,67 STUDENT: [INAUDIBLE] 80 00:04:30,67 --> 00:04:37,66 PROFESSOR: No, that's the function. 81 00:04:37,66 --> 00:04:37,98 I'm plotting points, I'm not differentiating. 82 00:04:37,98 --> 00:04:38,25 I've already differentiated it. 83 00:04:38,25 --> 00:04:39,63 I've already got something that's a little puzzling. 84 00:04:39,63 --> 00:04:41,25 Now I'm focusing on the weird spot. 85 00:04:41,25 --> 00:04:41,59 Yes, another question. 86 00:04:41,59 --> 00:04:49,11 STUDENT: Wouldn't it be a little less than 0? 87 00:04:49,11 --> 00:04:49,61 PROFESSOR: Wouldn't it be a little less than 0? 88 00:04:49,61 --> 00:04:50,74 OK, that's a very good point and this is a 89 00:04:50,74 --> 00:04:53,31 matter of notation here. 90 00:04:53,31 --> 00:04:55,69 And a matter of parentheses. 91 00:04:55,69 --> 00:04:57,43 So wouldn't this be a little less than 2. 92 00:04:57,43 --> 00:05:01,45 Well, if the parentheses were this way; that is, 2+ 93 00:05:01,45 --> 00:05:04,595 , with a - after I did the 2+ 94 00:05:04,595 --> 00:05:06,79 , then it would be less. 95 00:05:06,79 --> 00:05:09,83 But it's this way. 96 00:05:09,83 --> 00:05:10,67 OK. 97 00:05:10,67 --> 00:05:16,77 So the notation is, you have a number and you take the 98 00:05:16,77 --> 00:05:17,19 part of it. 99 00:05:17,19 --> 00:05:21,55 That's the part which is a little bit bigger than it. 100 00:05:21,55 --> 00:05:26,84 And so this is what I mean. 101 00:05:26,84 --> 00:05:29,87 And if you like, here I can put in those parentheses too. 102 00:05:29,87 --> 00:05:31,27 Yeah, another question. 103 00:05:31,27 --> 00:05:34,73 STUDENT: [INAUDIBLE] 104 00:05:34,73 --> 00:05:39,19 PROFESSOR: Why doesn't the top one have a plus? 105 00:05:39,19 --> 00:05:39,47 The only reason why the top one doesn't have a plus is that I 106 00:05:39,47 --> 00:05:42,86 don't need it to evaluate this. 107 00:05:42,86 --> 00:05:45,41 And when I take the limit, I can just plug in the value. 108 00:05:45,41 --> 00:05:48,25 Whereas here, I'm still uncertain. 109 00:05:48,25 --> 00:05:49,43 Because it's going to be 0. 110 00:05:49,43 --> 00:05:51,74 And I want to know which side of 0 it's on. 111 00:05:51,74 --> 00:05:55,19 Whether it's on the positive side or the negative side. 112 00:05:55,19 --> 00:05:58,02 So this one, I could have written here a parentheses 113 00:05:58,02 --> 00:06:01,41 2+, but then it would have just simplified to - 1. 114 00:06:01,41 --> 00:06:04,41 In the limit. 115 00:06:04,41 --> 00:06:07,5 So now, I've got a negative number divided by a 116 00:06:07,5 --> 00:06:10,1 tiny positive number. 117 00:06:10,1 --> 00:06:11,94 And so, somebody want to tell me what that is? 118 00:06:11,94 --> 00:06:16,75 Negative infinity. 119 00:06:16,75 --> 00:06:21,46 So, we just evaluated this function from one side. 120 00:06:21,46 --> 00:06:29,19 And if you follow through the other side, so this one here, 121 00:06:29,19 --> 00:06:31,8 you get something very similar, except that this should be -- 122 00:06:31,8 --> 00:06:34,12 whoops, what did I do wrong? 123 00:06:34,12 --> 00:06:37,69 I meant this. 124 00:06:37,69 --> 00:06:40,87 I want it -2 the same base point, but I want to 125 00:06:40,87 --> 00:06:43,75 go from the left. 126 00:06:43,75 --> 00:06:47,78 So that's going to be - 2 + 1, same numerator. 127 00:06:47,78 --> 00:06:52,79 And then this - 2 on the left + 2, and that's going to come out 128 00:06:52,79 --> 00:07:00,67 to be - 1 / 0 -, which is plus infinity. 129 00:07:00,67 --> 00:07:02,348 Or just plain infinity, we don't have to 130 00:07:02,348 --> 00:07:10,52 put the plus sign. 131 00:07:10,52 --> 00:07:13,09 So this is the first part of the problem. 132 00:07:13,09 --> 00:07:16,63 And the second piece, to get ourselves started, you could 133 00:07:16,63 --> 00:07:18,26 evaluate this function at any point. 134 00:07:18,26 --> 00:07:21,34 This is just the most interesting point, alright? 135 00:07:21,34 --> 00:07:22,88 This is just the most interesting place 136 00:07:22,88 --> 00:07:25,27 to evaluate it. 137 00:07:25,27 --> 00:07:27,7 Now, the next thing that I'd like to do is to pay 138 00:07:27,7 --> 00:07:32,13 attention to the ends. 139 00:07:32,13 --> 00:07:34,29 And I haven't really said what the ends are. 140 00:07:34,29 --> 00:07:37,22 So the ends are just all the way to the left and all 141 00:07:37,22 --> 00:07:37,94 the way to the right. 142 00:07:37,94 --> 00:07:42,67 So that means x going to plus or minus infinity. 143 00:07:42,67 --> 00:07:44,43 So that's the second thing I want to pay attention to. 144 00:07:44,43 --> 00:07:49,45 Again, this is a little bit like a video screen here. 145 00:07:49,45 --> 00:07:52,72 And we're about to discover something that's really off 146 00:07:52,72 --> 00:07:55,75 the screen, in both cases. 147 00:07:55,75 --> 00:07:57,95 We're taking care of what's happening way to the left, 148 00:07:57,95 --> 00:07:59,36 way to the right, here. 149 00:07:59,36 --> 00:08:01,82 And up above, we just took care what happens 150 00:08:01,82 --> 00:08:05,28 way up and way down. 151 00:08:05,28 --> 00:08:11,55 So on these ends, I need to do some more analysis. 152 00:08:11,55 --> 00:08:15,65 Which is related to a precalculus skill which 153 00:08:15,65 --> 00:08:18,31 is evaluating limits. 154 00:08:18,31 --> 00:08:21,82 And here, the way to do it is to divide by x the 155 00:08:21,82 --> 00:08:23,01 numerator and denominator. 156 00:08:23,01 --> 00:08:27,36 Write it as (1 + 1 / x)/ (1 + 2 /x). 157 00:08:27,36 --> 00:08:29,95 And then you can see what happens as x goes to 158 00:08:29,95 --> 00:08:30,98 plus or minus infinity. 159 00:08:30,98 --> 00:08:33,59 It just goes to 1. 160 00:08:33,59 --> 00:08:37,69 So, no matter whether x is positive or negative. 161 00:08:37,69 --> 00:08:42,81 When it gets huge, these two extra numbers here go to 0. 162 00:08:42,81 --> 00:08:44,55 And so, this tends to 1. 163 00:08:44,55 --> 00:08:47,79 So if you like, you could abbreviate this as f 164 00:08:47,79 --> 00:08:52,83 (+ or - infinity) = 1. 165 00:08:52,83 --> 00:08:54,61 So now, I get to draw this. 166 00:08:54,61 --> 00:08:56,9 And we draw this using asymptotes. 167 00:08:56,9 --> 00:09:01,8 So there's a level which is y = 1. 168 00:09:01,8 --> 00:09:06,79 And then there's another line to draw. 169 00:09:06,79 --> 00:09:15,19 Which is x = - 2. 170 00:09:15,19 --> 00:09:18,52 And now, what information do I have so far? 171 00:09:18,52 --> 00:09:21,26 Well, the information that I have so far is that when we're 172 00:09:21,26 --> 00:09:27,55 coming in from the right, that's to - 2, it plunges 173 00:09:27,55 --> 00:09:28,57 down to minus infinity. 174 00:09:28,57 --> 00:09:33,17 So that's down like this. 175 00:09:33,17 --> 00:09:38,81 And I also know that it goes up to infinity on the other 176 00:09:38,81 --> 00:09:41,85 side of the asymptote. 177 00:09:41,85 --> 00:09:48,13 And over here, I know it's going out to the level 1. 178 00:09:48,13 --> 00:09:53,41 And here it's also going to the level 1. 179 00:09:53,41 --> 00:09:57,16 Now, there's an issue. 180 00:09:57,16 --> 00:09:59,49 I can almost finish this graph now. 181 00:09:59,49 --> 00:10:01,46 I almost have enough information to finish it. 182 00:10:01,46 --> 00:10:04,45 But there's one thing which is making me hesitate 183 00:10:04,45 --> 00:10:06,78 a little bit. 184 00:10:06,78 --> 00:10:10,6 And that is, I don't know, for instance, over here, whether 185 00:10:10,6 --> 00:10:14,23 it's going to maybe dip below and come back up. 186 00:10:14,23 --> 00:10:16,93 Or not. 187 00:10:16,93 --> 00:10:20,14 So what does it do here? 188 00:10:20,14 --> 00:10:25,04 Can anybody see? 189 00:10:25,04 --> 00:10:25,25 Yeah. 190 00:10:25,25 --> 00:10:29,9 STUDENT: [INAUDIBLE] 191 00:10:29,9 --> 00:10:32,26 PROFESSOR: It can't dip below because there are 192 00:10:32,26 --> 00:10:32,38 no critical points. 193 00:10:32,38 --> 00:10:34,54 What a precisely correct answer. 194 00:10:34,54 --> 00:10:36,73 So that's exactly right. 195 00:10:36,73 --> 00:10:44,02 The point here is that because f' is not 0, it can't 196 00:10:44,02 --> 00:10:45,02 double back on itself. 197 00:10:45,02 --> 00:10:49,79 Because there can't be any of these horizontal tangents. 198 00:10:49,79 --> 00:11:00,56 It can't double back, so it can't backtrack. 199 00:11:00,56 --> 00:11:07,19 So sorry, if f' is not 0, f can't backtrack. 200 00:11:07,19 --> 00:11:09,41 And so that means that it doesn't look like this. 201 00:11:09,41 --> 00:11:14,32 It just goes like this. 202 00:11:14,32 --> 00:11:15,69 So that's basically it. 203 00:11:15,69 --> 00:11:17,63 And it's practically the end of the problem. 204 00:11:17,63 --> 00:11:19,5 Goes like this. 205 00:11:19,5 --> 00:11:21,87 Now you can decorate your thing, right? 206 00:11:21,87 --> 00:11:24,66 You may notice that maybe it crosses here, the axes, you can 207 00:11:24,66 --> 00:11:26,79 actually evaluate these places. 208 00:11:26,79 --> 00:11:27,5 And so forth. 209 00:11:27,5 --> 00:11:31,13 We're looking right now for qualitative behavior. 210 00:11:31,13 --> 00:11:34,28 In fact, you can see where these places hit. 211 00:11:34,28 --> 00:11:36,66 And it's actually a little higher up than I drew. 212 00:11:36,66 --> 00:11:40,14 Maybe I'll draw it accurately. 213 00:11:40,14 --> 00:11:44,97 As we'll see in a second. 214 00:11:44,97 --> 00:11:47,71 So that's what happens to this function. 215 00:11:47,71 --> 00:11:51,04 Now, let's just take a look in a little bit more 216 00:11:51,04 --> 00:11:56,84 detail, by double checking. 217 00:11:56,84 --> 00:11:58,64 So we're just going to double check what happens to the 218 00:11:58,64 --> 00:12:01,28 sign of the derivative. 219 00:12:01,28 --> 00:12:03,83 And in the meantime, I'm going to explain to you what the 220 00:12:03,83 --> 00:12:07,045 derivative is and also talk about the second derivative. 221 00:12:07,045 --> 00:12:11,82 So first of all, the trick for evaluating the derivative 222 00:12:11,82 --> 00:12:13,57 is an algebraic one. 223 00:12:13,57 --> 00:12:16,08 I mean, obviously you can do this by the quotient rule. 224 00:12:16,08 --> 00:12:24,25 But I just point out that this is the same thing as this. 225 00:12:24,25 --> 00:12:27,33 And now it has, whoops, that should be a 2 226 00:12:27,33 --> 00:12:28,69 in the denominator. 227 00:12:28,69 --> 00:12:35,95 And so, now this has the form 1 - (1 / x + 2). 228 00:12:35,95 --> 00:12:39,6 So this makes it easy to see what the derivative is. 229 00:12:39,6 --> 00:12:42,73 Because the derivative of a constant is 0, right? 230 00:12:42,73 --> 00:12:49,3 So this is, derivative, is just going to be, switch the sign. 231 00:12:49,3 --> 00:12:53,41 This is what I wrote before. 232 00:12:53,41 --> 00:12:55,23 And that explains it. 233 00:12:55,23 --> 00:12:58,14 But incidentally, it also shows you that that 234 00:12:58,14 --> 00:13:04,89 this is a hyperbola. 235 00:13:04,89 --> 00:13:09,62 These are just two curves of a hyperbola. 236 00:13:09,62 --> 00:13:12,24 So now, let's check the sign. 237 00:13:12,24 --> 00:13:14,9 It's already totally obvious to us that this 238 00:13:14,9 --> 00:13:15,94 is just a double check. 239 00:13:15,94 --> 00:13:18,71 We didn't actually even have to pay any attention to this. 240 00:13:18,71 --> 00:13:19,79 It had better be true. 241 00:13:19,79 --> 00:13:22,22 This is just going to check our arithmetic. 242 00:13:22,22 --> 00:13:24,57 Namely, it's increasing here. 243 00:13:24,57 --> 00:13:26,61 It's increasing there. 244 00:13:26,61 --> 00:13:27,83 That's got to be true. 245 00:13:27,83 --> 00:13:31,47 And, sure enough, this is positive, as you can see 246 00:13:31,47 --> 00:13:32,93 it's 1 over a square. 247 00:13:32,93 --> 00:13:34,07 So it is increasing. 248 00:13:34,07 --> 00:13:35,63 So we checked it. 249 00:13:35,63 --> 00:13:39,37 But now, there's one more thing that I want to just have 250 00:13:39,37 --> 00:13:40,72 you watch out about. 251 00:13:40,72 --> 00:13:46,98 So this means that f is increasing. 252 00:13:46,98 --> 00:13:51,83 On the interval minus infinity < x < - 2. 253 00:13:51,83 --> 00:13:56,7 And also from - 2 all the way out to infinity. 254 00:13:56,7 --> 00:14:03,36 So I just want to warn you, you cannot say, don't say f is 255 00:14:03,36 --> 00:14:12,02 increasing on minus infinity < infinity, for all x. 256 00:14:12,02 --> 00:14:15,09 OK, this is just not true. 257 00:14:15,09 --> 00:14:16,7 I've written it on the board, but it's wrong. 258 00:14:16,7 --> 00:14:18,66 I'd better get rid of it. 259 00:14:18,66 --> 00:14:19,2 There it is. 260 00:14:19,2 --> 00:14:20,97 Get rid of it. 261 00:14:20,97 --> 00:14:24,555 And the reason is, so first of all it's totally obvious. 262 00:14:24,555 --> 00:14:25,39 It's going up here. 263 00:14:25,39 --> 00:14:28,7 But then it went zooming back down there. 264 00:14:28,7 --> 00:14:35,86 And here this was true, but only if x is not - 2. 265 00:14:35,86 --> 00:14:37,4 So there's a break. 266 00:14:37,4 --> 00:14:39,28 And you've got to pay attention to the break. 267 00:14:39,28 --> 00:14:51,06 So basically, the moral here is that if you ignore this place, 268 00:14:51,06 --> 00:14:54,39 it's like ignoring Mount Everest, or the Grand Canyon. 269 00:14:54,39 --> 00:14:56,77 You're ignoring the most important feature of 270 00:14:56,77 --> 00:14:58,3 this function here. 271 00:14:58,3 --> 00:15:00,51 If you're going to be figuring out where things are going up 272 00:15:00,51 --> 00:15:03,71 and down, which is basically all we're doing, you'd better 273 00:15:03,71 --> 00:15:07,33 pay attention to these kinds of places. 274 00:15:07,33 --> 00:15:09,33 So don't ignore them. 275 00:15:09,33 --> 00:15:13,14 So that's the first remark. 276 00:15:13,14 --> 00:15:17,34 And now there's just a little bit of decoration as well. 277 00:15:17,34 --> 00:15:19,99 Which is the role of the second derivative. 278 00:15:19,99 --> 00:15:21,99 So we've written down the first derivative here. 279 00:15:21,99 --> 00:15:33,49 The second derivative is now - 2 / (x + 2)^3, right? 280 00:15:33,49 --> 00:15:36,3 So I get that from differentiating this formula up 281 00:15:36,3 --> 00:15:39,49 here for the first derivative. 282 00:15:39,49 --> 00:15:43,475 And now, of course, that's also, only works for 283 00:15:43,475 --> 00:15:47,68 x not equal to - 2. 284 00:15:47,68 --> 00:15:54,82 And now, we can see that this is going to be negative, let's 285 00:15:54,82 --> 00:15:56,65 see, where is it negative? 286 00:15:56,65 --> 00:16:01,66 When this is a positive quantity, so when -2 287 00:16:01,66 --> 00:16:03,405 < x < infinity. 288 00:16:03,405 --> 00:16:04,59 It's negative. 289 00:16:04,59 --> 00:16:07,77 And this is where this thing is concave. 290 00:16:07,77 --> 00:16:08,67 Let's see. 291 00:16:08,67 --> 00:16:12,65 Did I say that right? 292 00:16:12,65 --> 00:16:13,33 Negative, right? 293 00:16:13,33 --> 00:16:19,18 This is concave down. 294 00:16:19,18 --> 00:16:19,93 Right. 295 00:16:19,93 --> 00:16:23,4 And similarly, if I look at this expression, the numerator 296 00:16:23,4 --> 00:16:28,18 is always negative but the denominator becomes negative 297 00:16:28,18 --> 00:16:31,13 as well when x < -2. 298 00:16:31,13 --> 00:16:33,71 So this becomes positive. 299 00:16:33,71 --> 00:16:36,69 So this case, it was negative over positive. 300 00:16:36,69 --> 00:16:40,95 In this case it was negative divided by negative. 301 00:16:40,95 --> 00:16:46,14 So here, this is in the range - infinity < x < -2. 302 00:16:46,14 --> 00:16:52,24 And here it's concave up. 303 00:16:52,24 --> 00:16:55,15 Now, again, this is just consistent with what 304 00:16:55,15 --> 00:16:55,92 we're already guessing. 305 00:16:55,92 --> 00:16:57,77 Of course we already know it in this case if we know 306 00:16:57,77 --> 00:16:59,68 that this is a hyperbola. 307 00:16:59,68 --> 00:17:02,31 That it's going to be concave down to the right of the 308 00:17:02,31 --> 00:17:04,27 vertical line, dotted vertical line. 309 00:17:04,27 --> 00:17:07,77 And concave up to the left. 310 00:17:07,77 --> 00:17:11,91 So what extra piece of information is it that 311 00:17:11,91 --> 00:17:16,1 this is giving us? 312 00:17:16,1 --> 00:17:17,8 Did I say this backwards? 313 00:17:17,8 --> 00:17:18,52 No. 314 00:17:18,52 --> 00:17:19,57 That's OK. 315 00:17:19,57 --> 00:17:21,43 So what extra piece of information is this giving us? 316 00:17:21,43 --> 00:17:23,4 It looks like it's giving us hardly anything. 317 00:17:23,4 --> 00:17:25,68 And it really is giving us hardly anything. 318 00:17:25,68 --> 00:17:28,84 But it is giving us something that's a little aesthetic. 319 00:17:28,84 --> 00:17:34,28 It's ruling out the possibility of a wiggle. 320 00:17:34,28 --> 00:17:37,59 There isn't anything like that in the curve. 321 00:17:37,59 --> 00:17:40,286 It can't shift from curving this way to curving that 322 00:17:40,286 --> 00:17:41,59 way to curving this way. 323 00:17:41,59 --> 00:17:42,86 That doesn't happen. 324 00:17:42,86 --> 00:17:59,07 So these properties say there's no wiggle in the graph of that. 325 00:17:59,07 --> 00:17:59,79 Alright. 326 00:17:59,79 --> 00:18:01,41 So. 327 00:18:01,41 --> 00:18:01,89 Question. 328 00:18:01,89 --> 00:18:05,57 STUDENT: Do we define the increasing and decreasing base 329 00:18:05,57 --> 00:18:10,63 purely on the derivative, or the sort of more general 330 00:18:10,63 --> 00:18:14,31 definition of picking any two points and seeing. 331 00:18:14,31 --> 00:18:17,07 Because sometimes there can be an inconsistency between 332 00:18:17,07 --> 00:18:20,3 the two definitions. 333 00:18:20,3 --> 00:18:26,3 PROFESSOR: OK, so the question is, in this course are we 334 00:18:26,3 --> 00:18:29,37 going to define positive derivative as being the 335 00:18:29,37 --> 00:18:31,36 same thing as increasing. 336 00:18:31,36 --> 00:18:33,03 And the answer is no. 337 00:18:33,03 --> 00:18:36,21 We'll try to use these terms separately. 338 00:18:36,21 --> 00:18:40,33 What's always true is that if f' is positive, 339 00:18:40,33 --> 00:18:42,53 then f is increasing. 340 00:18:42,53 --> 00:18:45,06 But the reverse is not necessarily true. 341 00:18:45,06 --> 00:18:47,79 It could be very flat, the derivative can be 0 and still 342 00:18:47,79 --> 00:18:50,05 the function can be increasing. 343 00:18:50,05 --> 00:18:53,73 OK, the derivative can be 0 at a few places. 344 00:18:53,73 --> 00:18:59,3 For instance, like some cubics. 345 00:18:59,3 --> 00:19:05,37 Other questions? 346 00:19:05,37 --> 00:19:09,86 So that's as much as I need to say in general. 347 00:19:09,86 --> 00:19:11,39 I mean, in a specific case. 348 00:19:11,39 --> 00:19:14,6 But I want to get you a general scheme and I want to go through 349 00:19:14,6 --> 00:19:20,32 a more complicated example that gets all the features 350 00:19:20,32 --> 00:19:22,75 of this kind of thing. 351 00:19:22,75 --> 00:19:34,71 So let's talk about a general strategy for sketching. 352 00:19:34,71 --> 00:19:38,6 So the first part of this strategy, if 353 00:19:38,6 --> 00:19:40,8 you like, let's see. 354 00:19:40,8 --> 00:19:42,43 I have it all plotted out here. 355 00:19:42,43 --> 00:19:45,93 So I'm going to make sure I get it exactly the way 356 00:19:45,93 --> 00:19:47,78 I wanted you to see. 357 00:19:47,78 --> 00:19:51,44 So I have, its plotting. 358 00:19:51,44 --> 00:19:52,72 The plot thickens. 359 00:19:52,72 --> 00:19:54,04 Here we go. 360 00:19:54,04 --> 00:19:57,71 So plot, what is it that you should plot first? 361 00:19:57,71 --> 00:20:01,47 Before you even think about derivatives, you should 362 00:20:01,47 --> 00:20:08,25 plot discontinuities. 363 00:20:08,25 --> 00:20:18,49 Especially the infinite ones. 364 00:20:18,49 --> 00:20:20,21 That's the first thing you should do. 365 00:20:20,21 --> 00:20:27,16 And then, you should plot end points, for ends. 366 00:20:27,16 --> 00:20:31,44 For x going to plus or minus infinity if there don't 367 00:20:31,44 --> 00:20:35,64 happen to be any finite ends to the problem. 368 00:20:35,64 --> 00:20:44,6 And the third thing you can do is plot any easy points. 369 00:20:44,6 --> 00:20:49 This is optional. 370 00:20:49 --> 00:20:50,71 At your discretion. 371 00:20:50,71 --> 00:20:54,43 You might, for instance, on this example, plot the 372 00:20:54,43 --> 00:20:59,55 places where the graph crosses the axis. 373 00:20:59,55 --> 00:21:04,54 If you want to. 374 00:21:04,54 --> 00:21:05,81 So that's the first part. 375 00:21:05,81 --> 00:21:08,19 And again, this is all precalculus. 376 00:21:08,19 --> 00:21:17,23 So now, in the second part we're going to solve this 377 00:21:17,23 --> 00:21:22,82 equation and we're going to plot the critical 378 00:21:22,82 --> 00:21:29,36 points and values. 379 00:21:29,36 --> 00:21:32,81 In the problem which we just discussed, there weren't any. 380 00:21:32,81 --> 00:21:38,64 So this part was empty. 381 00:21:38,64 --> 00:21:50,2 So the third step is to decide whether f', sorry, whether, f' 382 00:21:50,2 --> 00:22:01,12 is positive or negative on each interval. 383 00:22:01,12 --> 00:22:17,21 Between critical points, discontinuities. 384 00:22:17,21 --> 00:22:22,56 The direction of the sign, in this case it doesn't change. 385 00:22:22,56 --> 00:22:24,9 It goes up here and it also goes up here. 386 00:22:24,9 --> 00:22:27,77 But it could go up here and then come back down. 387 00:22:27,77 --> 00:22:31,46 So the direction can change at every critical point. 388 00:22:31,46 --> 00:22:33,85 It can change at every discontinuity. 389 00:22:33,85 --> 00:22:35,35 And you don't know. 390 00:22:35,35 --> 00:22:44,52 However, this particular step has to be consistent with 1 391 00:22:44,52 --> 00:22:47,23 and 2, with steps 1 and 2. 392 00:22:47,23 --> 00:22:53,205 In fact, it will never, if you can succeed in doing steps 1 393 00:22:53,205 --> 00:22:56,97 and 2, you'll never need step 3. 394 00:22:56,97 --> 00:23:02,19 All it's doing is double-checking. 395 00:23:02,19 --> 00:23:05,99 So if you made an arithmetic mistake somewhere, you'll 396 00:23:05,99 --> 00:23:09,16 be able to see it. 397 00:23:09,16 --> 00:23:10,87 So that's maybe the most important thing. 398 00:23:10,87 --> 00:23:13,3 And it's actually the most frustrating thing for me when I 399 00:23:13,3 --> 00:23:17,98 see people working on problems, is they start step 3, they get 400 00:23:17,98 --> 00:23:21,14 it wrong, and then they start trying to draw the graph 401 00:23:21,14 --> 00:23:22,48 and it doesn't work. 402 00:23:22,48 --> 00:23:23,62 Because it's inconsistent. 403 00:23:23,62 --> 00:23:26,28 And the reason is some arithmetic error with the 404 00:23:26,28 --> 00:23:28,3 derivative or something like that or some other 405 00:23:28,3 --> 00:23:29,88 misinterpretation. 406 00:23:29,88 --> 00:23:31,93 And then there's a total mess. 407 00:23:31,93 --> 00:23:35,04 If you start with these two steps, then you're going to 408 00:23:35,04 --> 00:23:37,62 know when you get to this step that you're making mistakes. 409 00:23:37,62 --> 00:23:40,31 People don't generally make as many mistakes 410 00:23:40,31 --> 00:23:42,36 in the first two steps. 411 00:23:42,36 --> 00:23:45,22 Anyway, in fact you can skip this step if you want. 412 00:23:45,22 --> 00:23:49,47 But that's at risk of not double-checking your work. 413 00:23:49,47 --> 00:23:51,28 So what's the fourth step? 414 00:23:51,28 --> 00:23:59,4 Well, we take a look at whether f'' is positive or negative. 415 00:23:59,4 --> 00:24:01,89 And so we're deciding on things like whether it's 416 00:24:01,89 --> 00:24:07,64 concave up or down. 417 00:24:07,64 --> 00:24:15,49 And we have these points, f'' ( x ) = 0, which are 418 00:24:15,49 --> 00:24:24,57 called inflection points. 419 00:24:24,57 --> 00:24:31,55 And the last step is just to combine everything. 420 00:24:31,55 --> 00:24:35,71 So this is this the scheme, the general scheme. 421 00:24:35,71 --> 00:24:58,85 And let's just carry it out in a particular case. 422 00:24:58,85 --> 00:25:02,28 So here's the function that I'm going to use as an example. 423 00:25:02,28 --> 00:25:08,25 I'll use f ( x ) = x / ln x. 424 00:25:08,25 --> 00:25:11,64 And because the logarithm - yeah, question. 425 00:25:11,64 --> 00:25:11,96 Yeah. 426 00:25:11,96 --> 00:25:18,66 STUDENT: [INAUDIBLE] 427 00:25:18,66 --> 00:25:21,35 PROFESSOR: The question is, is this optional. 428 00:25:21,35 --> 00:25:25,57 So that's a good question. 429 00:25:25,57 --> 00:25:26,39 Is this optional. 430 00:25:26,39 --> 00:25:31,516 STUDENT: [INAUDIBLE] 431 00:25:31,516 --> 00:25:37,59 PROFESSOR: OK, the question is is this optional; 432 00:25:37,59 --> 00:25:38,66 this kind of question. 433 00:25:38,66 --> 00:25:48,94 And the answer is, it's more than just -- so, in 434 00:25:48,94 --> 00:25:51,7 many instances, I'm not going to ask you to. 435 00:25:51,7 --> 00:25:54,78 I strongly recommend that if I don't ask you to do 436 00:25:54,78 --> 00:25:57,05 it, that you not try. 437 00:25:57,05 --> 00:26:01,05 Because it's usually awful to find the second derivative. 438 00:26:01,05 --> 00:26:03,29 Any time you can get away without computing a second 439 00:26:03,29 --> 00:26:06,33 derivative, you're better off. 440 00:26:06,33 --> 00:26:07,92 So in many, many instances. 441 00:26:07,92 --> 00:26:10,72 On the other hand, if I ask you to do it it's because I want 442 00:26:10,72 --> 00:26:13,06 you to have the, work to do it. 443 00:26:13,06 --> 00:26:16,72 But basically, if nobody forces you to, I would 444 00:26:16,72 --> 00:26:22,13 say never do step 4. 445 00:26:22,13 --> 00:26:26,75 Other questions. 446 00:26:26,75 --> 00:26:27,61 Alright. 447 00:26:27,61 --> 00:26:29,95 So we're going to force ourselves to do step 4, 448 00:26:29,95 --> 00:26:31,81 however, in this instance. 449 00:26:31,81 --> 00:26:35,01 But maybe this will be one of the few times. 450 00:26:35,01 --> 00:26:39,14 So here we go, just for illustrative purposes. 451 00:26:39,14 --> 00:26:43,24 OK, now. 452 00:26:43,24 --> 00:26:46,14 So here's the function that I want to discuss. 453 00:26:46,14 --> 00:26:50,01 And the range has to be x positive, because the 454 00:26:50,01 --> 00:26:55,5 logarithm is not defined for negative values. 455 00:26:55,5 --> 00:26:58,85 So the first thing that I'm going to do is, I'd like to 456 00:26:58,85 --> 00:27:02,56 follow the scheme here. 457 00:27:02,56 --> 00:27:05,14 Because if I don't follow the scheme, I'm going to 458 00:27:05,14 --> 00:27:06,49 get a little mixed up. 459 00:27:06,49 --> 00:27:13,98 So the first part is to find the singularities. 460 00:27:13,98 --> 00:27:17,06 That is, the places where f is infinite. 461 00:27:17,06 --> 00:27:20,72 And that's when the logarithm, the denominator, vanishes. 462 00:27:20,72 --> 00:27:25,63 So that's f ( 1 +), if you like. 463 00:27:25,63 --> 00:27:32,37 So that's 1 / ln 1 +, which is 1 / 0, with a little 464 00:27:32,37 --> 00:27:34,28 bit of positiveness to it. 465 00:27:34,28 --> 00:27:37,27 Which is infinity. 466 00:27:37,27 --> 00:27:39,83 And second, we do it the other way. 467 00:27:39,83 --> 00:27:42,84 And not surprisingly, this comes out to be 468 00:27:42,84 --> 00:27:46,85 negative infinity. 469 00:27:46,85 --> 00:27:51,98 Now, the next thing I want to do is the ends. 470 00:27:51,98 --> 00:27:56,98 So I call these the ends. 471 00:27:56,98 --> 00:28:01,38 And there are two of them. 472 00:28:01,38 --> 00:28:08,3 One of them is f ( 0 ) from the right. f ( 0+). 473 00:28:08,3 --> 00:28:21,49 So that is 0 + / ln 0 +, which is 0 plus divided by, well, ln 474 00:28:21,49 --> 00:28:25,36 0 + is actually minus infinity. 475 00:28:25,36 --> 00:28:27,18 That's what happens to the logarithm, goes 476 00:28:27,18 --> 00:28:28,17 to minus infinity. 477 00:28:28,17 --> 00:28:31,25 So this is 0 over infinity, which is definitely 0, there's 478 00:28:31,25 --> 00:28:37,1 no problem. about what happens. 479 00:28:37,1 --> 00:28:42,91 The other side, so this is the end, this is the first end. 480 00:28:42,91 --> 00:28:44,92 The range is this. 481 00:28:44,92 --> 00:28:48,48 And I just did the left endpoint. 482 00:28:48,48 --> 00:28:50,63 And so now I have to do the right endpoint, I have 483 00:28:50,63 --> 00:28:51,87 to let x go to infinity. 484 00:28:51,87 --> 00:28:54,67 So if I let x go to infinity, I'm just going to have to think 485 00:28:54,67 --> 00:28:56,52 about it a little bit by plugging in a very 486 00:28:56,52 --> 00:28:57,69 large number. 487 00:28:57,69 --> 00:29:01,96 I'll plug in 10 ^ 10, just to see what happens. 488 00:29:01,96 --> 00:29:11,73 So if I plug in 10 ^ 10 into x ln x, I get 10 ^ 10 / ln 10^10. 489 00:29:11,73 --> 00:29:17,59 Which is 10 ^ 10 / 10 ( log 10). 490 00:29:17,59 --> 00:29:23,18 So the denominator, this number here, is about 2.something. 491 00:29:23,18 --> 00:29:25,13 2.3 or something. 492 00:29:25,13 --> 00:29:28,31 So this is maybe 230 in the denominator, and this is a 493 00:29:28,31 --> 00:29:31,9 number with ten 0's after it. 494 00:29:31,9 --> 00:29:33,53 So it's very, very large. 495 00:29:33,53 --> 00:29:35,3 I claim it's big. 496 00:29:35,3 --> 00:29:38,73 And that gives us the clue that what's happening is 497 00:29:38,73 --> 00:29:40,54 that this thing is infinite. 498 00:29:40,54 --> 00:29:43,766 So, in other words, our conclusion is that f of 499 00:29:43,766 --> 00:29:52,12 infinity is infinity. 500 00:29:52,12 --> 00:29:59,65 So what do we have so far for our function? 501 00:29:59,65 --> 00:30:03,3 We're just trying to build the scaffolding of the function. 502 00:30:03,3 --> 00:30:07,27 And we're doing it by taking the most important points. 503 00:30:07,27 --> 00:30:09,52 And from a mathematician's point of view, the most 504 00:30:09,52 --> 00:30:11,38 important points are the ones which are sort of 505 00:30:11,38 --> 00:30:13,49 infinitely obvious. 506 00:30:13,49 --> 00:30:15,19 For the ends of the problem. 507 00:30:15,19 --> 00:30:19,73 So that's where we're heading. 508 00:30:19,73 --> 00:30:22,6 We have a vertical asymptote, which is at x = 1. 509 00:30:22,6 --> 00:30:29,15 So this gives us x = 1. 510 00:30:29,15 --> 00:30:34,27 And we have a value which is that it's 0 here. 511 00:30:34,27 --> 00:30:38,97 And we also know that when we come in from the - sorry, so we 512 00:30:38,97 --> 00:30:44,91 come in from the left, that's f, the one from the left, 513 00:30:44,91 --> 00:30:46,06 we get negative infinity. 514 00:30:46,06 --> 00:30:47,55 So it's diving down. 515 00:30:47,55 --> 00:30:52,46 It's going down like this. 516 00:30:52,46 --> 00:30:55,925 And, furthermore, on the other side we know it's climbing up. 517 00:30:55,925 --> 00:30:58,56 So it's going up like this. 518 00:30:58,56 --> 00:31:00,46 Just start a little higher. 519 00:31:00,46 --> 00:31:00,77 Right, so. 520 00:31:00,77 --> 00:31:02,52 So far, this is what we know. 521 00:31:02,52 --> 00:31:05,81 Oh, and there's one other thing that we know. 522 00:31:05,81 --> 00:31:12,42 When we go to plus infinity, it's going back up. 523 00:31:12,42 --> 00:31:15,15 So, so far we have this. 524 00:31:15,15 --> 00:31:17,85 Now, already it should be pretty obvious what's going 525 00:31:17,85 --> 00:31:19,76 to happen to this function. 526 00:31:19,76 --> 00:31:21,16 So there shouldn't be many surprises. 527 00:31:21,16 --> 00:31:23,16 It's going to come down like this. 528 00:31:23,16 --> 00:31:27,09 Go like this, it's going to turn around and go back up. 529 00:31:27,09 --> 00:31:29,29 That's what we expect. 530 00:31:29,29 --> 00:31:33,6 So we don't know that yet, but we're pretty sure. 531 00:31:33,6 --> 00:31:36,8 So at this point, we can start looking at the critical points. 532 00:31:36,8 --> 00:31:41,995 We can do our step 2 here -- we need a little bit more room 533 00:31:41,995 --> 00:31:45,49 here -- and see what's happening with this function. 534 00:31:45,49 --> 00:31:49,22 So I have to differentiate it. 535 00:31:49,22 --> 00:31:52,07 And it's, this is the quotient rule. 536 00:31:52,07 --> 00:31:54,88 So remember the function is up here, x / ln x. 537 00:31:54,88 --> 00:31:59,4 So I have a ln x^2 in the denominator. 538 00:31:59,4 --> 00:32:04,19 And I get here the derivative of x is 1, so we get 1 ( ln x) 539 00:32:04,19 --> 00:32:10,56 - x ( the derivative of ln x, which is 1 /x). 540 00:32:10,56 --> 00:32:20,16 So all told, that's (ln x - 1) / ln x^2. 541 00:32:20,16 --> 00:32:27,77 So here's our derivative. 542 00:32:27,77 --> 00:32:35,08 And now, if I set this equal to 0, at least in the numerator, 543 00:32:35,08 --> 00:32:40,97 the numerator is 0 when x = e. 544 00:32:40,97 --> 00:32:43,49 The ln e = 1. 545 00:32:43,49 --> 00:32:46,29 So here's our critical point. 546 00:32:46,29 --> 00:32:51,32 And we have a critical value, which is f(e). 547 00:32:51,32 --> 00:32:55,61 And that's going to be e / ln e. 548 00:32:55,61 --> 00:32:57,24 Which is e, again. 549 00:32:57,24 --> 00:32:59,09 Because ln e = 1. 550 00:32:59,09 --> 00:33:01,89 So now I can also plot the critical point, 551 00:33:01,89 --> 00:33:03,11 which is down here. 552 00:33:03,11 --> 00:33:07,68 And there's only one of them, and it's at (e e). 553 00:33:07,68 --> 00:33:10,8 That's kind of not to scale here, because my blackboard 554 00:33:10,8 --> 00:33:12,52 isn't quite tall enough. 555 00:33:12,52 --> 00:33:15,36 It should be over here and then, it's slope 1. 556 00:33:15,36 --> 00:33:17,14 But I dipped it down. 557 00:33:17,14 --> 00:33:19,93 So this is not to scale, and indeed that's one of the things 558 00:33:19,93 --> 00:33:22,68 that we're not going to attempt to do with these pictures, 559 00:33:22,68 --> 00:33:24,68 is to make them to scale. 560 00:33:24,68 --> 00:33:29,56 So the scale's a little squashed. 561 00:33:29,56 --> 00:33:32,71 So, so far I have this critical point. 562 00:33:32,71 --> 00:33:36,18 And, in fact, I'm going to label it with a c. 563 00:33:36,18 --> 00:33:38,46 Whenever I have a critical point I'll just make sure 564 00:33:38,46 --> 00:33:41,49 that I remember that that's what it is. 565 00:33:41,49 --> 00:33:44,79 And since there's only one, the rest of this 566 00:33:44,79 --> 00:33:49,9 picture is now correct. 567 00:33:49,9 --> 00:33:54,88 That's the same mechanism that we used for the hyperbola. 568 00:33:54,88 --> 00:33:57,14 Namely, we know there's only one place where 569 00:33:57,14 --> 00:33:57,81 the derivative is 0. 570 00:33:57,81 --> 00:34:00,34 So that means there no more horizontals, so there's 571 00:34:00,34 --> 00:34:01,95 no more backtracking. 572 00:34:01,95 --> 00:34:03,33 It has to come down to here. 573 00:34:03,33 --> 00:34:03,97 Get to there. 574 00:34:03,97 --> 00:34:06,06 This is the only place it can turn around. 575 00:34:06,06 --> 00:34:07,17 Goes back up. 576 00:34:07,17 --> 00:34:09 It has to start here and it has to go down to there. 577 00:34:09 --> 00:34:10,6 It can't go above 0. 578 00:34:10,6 --> 00:34:13,6 Do not pass go, do not get positive. 579 00:34:13,6 --> 00:34:20,23 It has to head down here. 580 00:34:20,23 --> 00:34:21,69 So that's great. 581 00:34:21,69 --> 00:34:25,08 That means that this picture is almost completely correct now. 582 00:34:25,08 --> 00:34:27,52 And the rest is more or less decoration. 583 00:34:27,52 --> 00:34:30,25 We're pretty much done with the way it looks, 584 00:34:30,25 --> 00:34:34,57 at least schematically. 585 00:34:34,57 --> 00:34:37,7 However, I am going to punish you, because I warned you. 586 00:34:37,7 --> 00:34:40,72 We are going to go over here and do this step 4 and 587 00:34:40,72 --> 00:34:44,28 fix up the concavity. 588 00:34:44,28 --> 00:34:45,93 And we're also going to do a little bit of 589 00:34:45,93 --> 00:35:00,77 that double-checking. 590 00:35:00,77 --> 00:35:06,13 So now, let's again, just, I want to emphasize. 591 00:35:06,13 --> 00:35:10,66 We're going to do a double-check. 592 00:35:10,66 --> 00:35:12,24 This is part 3. 593 00:35:12,24 --> 00:35:17,4 But in advance, I already have, based on this picture I already 594 00:35:17,4 --> 00:35:19 know what has to be true. 595 00:35:19 --> 00:35:35,45 That f is decreasing on 0 to 1. f is also decreasing on 1 to e. 596 00:35:35,45 --> 00:35:45,49 And f is increasing on e to infinity. 597 00:35:45,49 --> 00:35:49,88 So, already, because we've plot a bunch of points and we know 598 00:35:49,88 --> 00:35:51,47 that there aren't any places where the derivative 599 00:35:51,47 --> 00:35:55,5 vanishes, we already know it goes down, down, up. 600 00:35:55,5 --> 00:35:56,97 That's what it's got to do. 601 00:35:56,97 --> 00:35:58,98 Now, we'll just make sure that we didn't make any 602 00:35:58,98 --> 00:36:00,72 arithmetic mistakes, now. 603 00:36:00,72 --> 00:36:03,59 By actually computing the derivative, or staring 604 00:36:03,59 --> 00:36:04,65 at it, anyway. 605 00:36:04,65 --> 00:36:10,35 And making sure that it's correct. 606 00:36:10,35 --> 00:36:17,6 So first of all, we just take a look at the numerator. 607 00:36:17,6 --> 00:36:26,655 So f,' remember, was (ln x - 1) / ln x^2, the quantity squared. 608 00:36:26,655 --> 00:36:28,48 So the denominator is positive. 609 00:36:28,48 --> 00:36:32,65 So let's just take a look at the three ranges. 610 00:36:32,65 --> 00:36:37,33 So we have 0 < x < 1. 611 00:36:37,33 --> 00:36:40,57 And on that range, the logarithm is negative, so 612 00:36:40,57 --> 00:36:45,23 this is negative divided by positive, which is negative. 613 00:36:45,23 --> 00:36:47,22 That's decreasing, that's good. 614 00:36:47,22 --> 00:36:50,16 And in fact, that also works on the next range. 615 00:36:50,16 --> 00:36:55,37 1 l< x < e, it's negative divided by positive. 616 00:36:55,37 --> 00:36:57,72 And the only reason why we skipped 1, again, is that 617 00:36:57,72 --> 00:36:58,433 it's undefined there. 618 00:36:58,433 --> 00:37:01,04 And there's something dramatic happening there. 619 00:37:01,04 --> 00:37:05,42 And then, at the last range, when x is bigger than e, that 620 00:37:05,42 --> 00:37:07,86 means the logarithm is already bigger than 1. 621 00:37:07,86 --> 00:37:09,923 So the numerator is now positive, and the denominator's 622 00:37:09,923 --> 00:37:13,65 still positive, so it's increasing. 623 00:37:13,65 --> 00:37:22,91 So we've just double-checked something that we already knew. 624 00:37:22,91 --> 00:37:26,94 Alright, so that's pretty much all there is to 625 00:37:26,94 --> 00:37:29,2 say about step 3. 626 00:37:29,2 --> 00:37:33,98 So this is checking the positivity and negativity. 627 00:37:33,98 --> 00:37:35,66 And now, step 4. 628 00:37:35,66 --> 00:37:38,48 There is one small point which I want to make before we go on. 629 00:37:38,48 --> 00:37:42,9 Which is that sometimes, you can't evaluate the 630 00:37:42,9 --> 00:37:45,445 function or its derivative particularly well. 631 00:37:45,445 --> 00:37:48,62 So sometimes you can't plot the points very well. 632 00:37:48,62 --> 00:37:50,41 And if you can't plot the points very well, then you 633 00:37:50,41 --> 00:37:53,16 might have to do 3 first, to figure out what's 634 00:37:53,16 --> 00:37:55,29 going on a little bit. 635 00:37:55,29 --> 00:37:59,15 You might have to skip. 636 00:37:59,15 --> 00:38:02,4 So now we're going to go on the second derivative. 637 00:38:02,4 --> 00:38:07,34 But first, I want to use an algebraic trick to 638 00:38:07,34 --> 00:38:08,47 rearrange the terms. 639 00:38:08,47 --> 00:38:10,82 And I want to notice one more little point. 640 00:38:10,82 --> 00:38:16,32 Which I, as I say, this is decoration for the graph. 641 00:38:16,32 --> 00:38:18,29 So I want to rewrite the formula. 642 00:38:18,29 --> 00:38:22,32 Maybe I'll do it right over here. 643 00:38:22,32 --> 00:38:31,44 Another way of writing this is (1 / ln x) - (1 / (ln x)^2). 644 00:38:31,44 --> 00:38:35,24 So that's another way of writing the derivative. 645 00:38:35,24 --> 00:38:38,64 And that allows me to notice something that 646 00:38:38,64 --> 00:38:40,83 I missed, before. 647 00:38:40,83 --> 00:38:46,88 When I solved the equation ln x - 1 - this is equal to 0 648 00:38:46,88 --> 00:38:48,59 here, this equation here. 649 00:38:48,59 --> 00:38:51,46 I missed a possibility. 650 00:38:51,46 --> 00:38:54,02 I missed the possibility that the denominator 651 00:38:54,02 --> 00:38:58,69 could be infinity. 652 00:38:58,69 --> 00:39:02,11 So actually, if the denominator's infinity, as 653 00:39:02,11 --> 00:39:05,6 you can see from the other expression there, it 654 00:39:05,6 --> 00:39:09 actually is true that the derivative is 0. 655 00:39:09 --> 00:39:16,71 So also when x = 0 +, the slope is going to be 0. 656 00:39:16,71 --> 00:39:19,05 Let me just emphasize that again. 657 00:39:19,05 --> 00:39:23,5 If you evaluate using this other formula over here, 658 00:39:23,5 --> 00:39:31,54 this is (1 / ln 0+) - (1 / (ln 0+)^2). 659 00:39:31,54 --> 00:39:37,31 That's 1 / - infinity - 1 / infinity, if you like, squared. 660 00:39:37,31 --> 00:39:40,63 Anyway, it's 0. 661 00:39:40,63 --> 00:39:42,33 So this is 0. 662 00:39:42,33 --> 00:39:43,41 The slope is 0 there. 663 00:39:43,41 --> 00:39:46,64 That is a little piece of decoration on our graph. 664 00:39:46,64 --> 00:39:50,92 It's telling us, going back to our graph here, it's telling us 665 00:39:50,92 --> 00:39:53,56 this is coming in with slope horizontal. 666 00:39:53,56 --> 00:39:57,53 So we're starting out this way. 667 00:39:57,53 --> 00:40:01,013 That's just a little start here to the graph. 668 00:40:01,013 --> 00:40:02,62 It's a horizontal slope. 669 00:40:02,62 --> 00:40:07,94 So there really were two places where the slope was horizontal. 670 00:40:07,94 --> 00:40:13,55 Now, with the help of this second formula I can also 671 00:40:13,55 --> 00:40:17,05 differentiate a second time. 672 00:40:17,05 --> 00:40:19,69 So it's a little bit easier to do that if I differentiate 1 / 673 00:40:19,69 --> 00:40:34,27 ln, that's -( ln x) ^ - 2 ( 1 / x) + 2 (ln x) ^ -3 (1/x). 674 00:40:34,27 --> 00:40:42,15 And that, if I put it over a common denominator, is x ln 675 00:40:42,15 --> 00:40:48,69 x^3 times, let's see here. 676 00:40:48,69 --> 00:40:55,34 I guess I'll have to take the 2 - ln x. 677 00:40:55,34 --> 00:40:57,88 So I've now rewritten the formula for the second 678 00:40:57,88 --> 00:41:03,45 derivative as a ratio. 679 00:41:03,45 --> 00:41:09,43 Now, to decide the sign, you see there are two places 680 00:41:09,43 --> 00:41:11,91 where the sign flips. 681 00:41:11,91 --> 00:41:16,37 The numerator crosses when the logarithm is 2, that's 682 00:41:16,37 --> 00:41:18,62 going to be when x = e ^2. 683 00:41:18,62 --> 00:41:24,27 And the denominator flips when x = 1, that's when the log 684 00:41:24,27 --> 00:41:28,02 flips from positive to negative. 685 00:41:28,02 --> 00:41:32,28 So we have a couple of ranges here. 686 00:41:32,28 --> 00:41:37,58 So, first of all, we have the range from 0 to 1. 687 00:41:37,58 --> 00:41:42,27 And then we have the range from 1 to e^2. 688 00:41:42,27 --> 00:41:45,995 And then we have the range from e ^2 all the 689 00:41:45,995 --> 00:41:49,78 way out to infinity. 690 00:41:49,78 --> 00:41:58,64 So between 0 and 1, the numerator is, well this is a 691 00:41:58,64 --> 00:42:00,695 negative number in this, so minus a negative number 692 00:42:00,695 --> 00:42:04,65 is positive, so the numerator is positive. 693 00:42:04,65 --> 00:42:07,505 And the denominator is negative, because the ln 694 00:42:07,505 --> 00:42:09,66 is negative it's taken to the third power. 695 00:42:09,66 --> 00:42:12,372 So this is a negative numbers, so it's positive divided by a 696 00:42:12,372 --> 00:42:15,19 negative number, which is less than 0. 697 00:42:15,19 --> 00:42:18,77 That means it's concave down. 698 00:42:18,77 --> 00:42:26,04 So this is concave down plot. 699 00:42:26,04 --> 00:42:28,01 And that's a good thing, because over here 700 00:42:28,01 --> 00:42:29,12 this is concave down. 701 00:42:29,12 --> 00:42:30,56 So there are no wiggles. 702 00:42:30,56 --> 00:42:34,26 It goes straight down, like this. 703 00:42:34,26 --> 00:42:41,59 And then the other two pieces are f'' is equal to, well 704 00:42:41,59 --> 00:42:43,26 it's going to switch here. 705 00:42:43,26 --> 00:42:44,856 The denominator becomes positive. 706 00:42:44,856 --> 00:42:48,19 So it's positive over positive. 707 00:42:48,19 --> 00:42:56,67 So this is concave up. 708 00:42:56,67 --> 00:42:58,41 And that's going over here. 709 00:42:58,41 --> 00:43:02,715 But notice that it's not the bottom where it turns around, 710 00:43:02,715 --> 00:43:07,74 it's somewhere else. 711 00:43:07,74 --> 00:43:09,93 So there's another transition here. 712 00:43:09,93 --> 00:43:12,09 This is e ^2. 713 00:43:12,09 --> 00:43:15,09 This is e. 714 00:43:15,09 --> 00:43:20,44 So what happens at the end is, again, the sign flips again. 715 00:43:20,44 --> 00:43:23,57 Because the numerator, now, when x > e 716 00:43:23,57 --> 00:43:26,58 ^2, becomes negative. 717 00:43:26,58 --> 00:43:29,965 And this is negative divided by positive, which is negative. 718 00:43:29,965 --> 00:43:35,63 And part is concave down. 719 00:43:35,63 --> 00:43:39,1 And so we didn't quite draw the graph right. 720 00:43:39,1 --> 00:43:41,18 There's an inflection point right here, which 721 00:43:41,18 --> 00:43:45 I'll label with i. 722 00:43:45 --> 00:43:47,34 Makes a turn the other way at that point. 723 00:43:47,34 --> 00:43:49,48 So there was a wiggle. 724 00:43:49,48 --> 00:43:51,21 There's the wiggle. 725 00:43:51,21 --> 00:43:53,71 Still going up, still going to infinity. 726 00:43:53,71 --> 00:43:57,04 But kind of the slope of the mountain, right? 727 00:43:57,04 --> 00:44:01,24 It's going the other way. 728 00:44:01,24 --> 00:44:09,35 This point happens to be (e^2, e ^2 / 2). 729 00:44:09,35 --> 00:44:11,98 So that's as detailed as we'll ever get. 730 00:44:11,98 --> 00:44:16,91 And indeed, the next game is going to be avoid being, is to 731 00:44:16,91 --> 00:44:18,73 avoid being this detailed. 732 00:44:18,73 --> 00:44:21,76 So let me introduce the next subject. 733 00:44:21,76 --> 00:44:48,31 Which is maxima and minima. 734 00:44:48,31 --> 00:45:04,16 OK, now, maxima and minima, maximum and minimum problems 735 00:45:04,16 --> 00:45:06,55 can be described graphically in the following ways. 736 00:45:06,55 --> 00:45:13,15 Suppose you have a function, right, here it is. 737 00:45:13,15 --> 00:45:14,42 OK? 738 00:45:14,42 --> 00:45:24,63 Now, find the maximum. 739 00:45:24,63 --> 00:45:30,74 And find the minimum. 740 00:45:30,74 --> 00:45:31,41 OK. 741 00:45:31,41 --> 00:45:38,88 So this problem is done. 742 00:45:38,88 --> 00:45:50,54 The point being, that it is easy to find max and the 743 00:45:50,54 --> 00:45:58,67 min with the sketch. 744 00:45:58,67 --> 00:46:00,33 It's very easy. 745 00:46:00,33 --> 00:46:05,13 The goal, the problem, is that the sketch is a lot of work. 746 00:46:05,13 --> 00:46:10,18 We just spent 20 minutes sketching something. 747 00:46:10,18 --> 00:46:13,24 We would not like to spend all that time every single 748 00:46:13,24 --> 00:46:14,97 time we want to find a maximum and minimum. 749 00:46:14,97 --> 00:46:19,576 So the goal is to do it with, so our goal is 750 00:46:19,576 --> 00:46:25,17 to use shortcuts. 751 00:46:25,17 --> 00:46:31,52 And, indeed, as I said earlier, we certainly never want to use 752 00:46:31,52 --> 00:46:33,84 the second derivative if we can avoid it. 753 00:46:33,84 --> 00:46:37,27 And we don't want to decorate the graph and do all of these 754 00:46:37,27 --> 00:46:40,62 elaborate, subtle, things which make the graph look nicer and 755 00:46:40,62 --> 00:46:42,64 really, or aesthetically appropriate. 756 00:46:42,64 --> 00:46:45,64 But are totally unnecessary to see whether the 757 00:46:45,64 --> 00:46:54,29 graph is up or down. 758 00:46:54,29 --> 00:46:58,03 So essentially, this whole business is out, which 759 00:46:58,03 --> 00:47:00,03 is a good thing. 760 00:47:00,03 --> 00:47:04,31 And, unfortunately, those early parts are the parts that 761 00:47:04,31 --> 00:47:06,17 people tend to ignore. 762 00:47:06,17 --> 00:47:10,15 Which are typically, often, very important. 763 00:47:10,15 --> 00:47:22,94 So let me first tell you the main point here. 764 00:47:22,94 --> 00:47:32,76 So the key idea. 765 00:47:32,76 --> 00:47:39,65 Key to finding maximum. 766 00:47:39,65 --> 00:47:42,19 So the key point is, we only need to look 767 00:47:42,19 --> 00:48:00,13 at critical points. 768 00:48:00,13 --> 00:48:04,54 Well, that's actually what it seems like to, in 769 00:48:04,54 --> 00:48:05,49 many calculus classes. 770 00:48:05,49 --> 00:48:06,81 But that's not true. 771 00:48:06,81 --> 00:48:15,59 This is not the end of the sentence. 772 00:48:15,59 --> 00:48:35,32 And, end points, and points of discontinuity. 773 00:48:35,32 --> 00:48:37,98 So you must watch out for those. 774 00:48:37,98 --> 00:48:42,37 If you look at the example that I just drew here, which is the 775 00:48:42,37 --> 00:48:48,08 one that I carried out, you can see that there are actually 776 00:48:48,08 --> 00:48:51,25 five extreme points on this picture. 777 00:48:51,25 --> 00:48:52,75 So let's switch. 778 00:48:52,75 --> 00:48:58,05 So we'll take a look. 779 00:48:58,05 --> 00:49:04,84 There are five places where the max or the min might be. 780 00:49:04,84 --> 00:49:08,05 There's this important point. 781 00:49:08,05 --> 00:49:10,71 This is, as I say, the scaffolding of the function. 782 00:49:10,71 --> 00:49:13,65 There's this point, there down at minus infinity. 783 00:49:13,65 --> 00:49:18,37 There's this, there's this, and there's this. 784 00:49:18,37 --> 00:49:23,62 Only one out of five is a critical point. 785 00:49:23,62 --> 00:49:25,51 So there's more that you have to pay attention 786 00:49:25,51 --> 00:49:26,45 to on the function. 787 00:49:26,45 --> 00:49:29,71 And you always have to keep the schema, the picture of 788 00:49:29,71 --> 00:49:31,31 the function, in the back of your head. 789 00:49:31,31 --> 00:49:33,76 Even though this may be the most interesting point, 790 00:49:33,76 --> 00:49:36,52 and the one that you're going to be looking at. 791 00:49:36,52 --> 00:49:40,832 So we'll do a few examples of that next time. 792 00:49:40,832 --> 00:49:41,715