1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,610 Commons license. 3 00:00:03,610 --> 00:00:05,990 Your support will help MIT OpenCourseWare 4 00:00:05,990 --> 00:00:10,190 continue to offer high quality educational resources for free. 5 00:00:10,190 --> 00:00:12,530 To make a donation, or to view additional materials 6 00:00:12,530 --> 00:00:15,880 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:15,880 --> 00:00:22,040 at ocw.mit.edu. 8 00:00:22,040 --> 00:00:25,380 PROFESSOR: OK, we're ready to start the eleventh lecture. 9 00:00:25,380 --> 00:00:29,390 We're still in the middle of sketching. 10 00:00:29,390 --> 00:00:32,910 And, indeed, one of the reasons why we did not 11 00:00:32,910 --> 00:00:37,040 talk about hyperbolic functions is 12 00:00:37,040 --> 00:00:39,540 that we're running just a little bit behind. 13 00:00:39,540 --> 00:00:41,720 And we'll catch up a tiny bit today. 14 00:00:41,720 --> 00:00:46,000 And I hope all the way on Tuesday of next week. 15 00:00:46,000 --> 00:00:58,640 So let me pick up where we left off, with sketching. 16 00:00:58,640 --> 00:01:05,220 So this is a continuation. 17 00:01:05,220 --> 00:01:07,020 I want to give you one more example 18 00:01:07,020 --> 00:01:08,390 of how to sketch things. 19 00:01:08,390 --> 00:01:10,380 And then we'll go through it systematically. 20 00:01:10,380 --> 00:01:14,830 So the second example that we did as one example last time, 21 00:01:14,830 --> 00:01:18,060 is this. 22 00:01:18,060 --> 00:01:21,970 The function is (x+1)/(x+2). 23 00:01:21,970 --> 00:01:25,780 And I'm going to save you the time right now. 24 00:01:25,780 --> 00:01:28,629 This is very typical of me, especially 25 00:01:28,629 --> 00:01:30,170 if you're in a hurry on an exam, I'll 26 00:01:30,170 --> 00:01:32,920 just tell you what the derivative is. 27 00:01:32,920 --> 00:01:34,490 So in this case, it's 1 / (x+2)^2. 28 00:01:37,760 --> 00:01:42,327 Now, the reason why I'm bringing this example up, 29 00:01:42,327 --> 00:01:44,660 even though it'll turn out to be a relatively simple one 30 00:01:44,660 --> 00:01:49,150 to sketch, is that it's easy to fall 31 00:01:49,150 --> 00:01:53,890 into a black hole with this problem. 32 00:01:53,890 --> 00:01:57,470 So let me just show you. 33 00:01:57,470 --> 00:01:59,600 This is not equal to 0. 34 00:01:59,600 --> 00:02:01,180 It's never equal to 0. 35 00:02:01,180 --> 00:02:08,940 So that means there are no critical points. 36 00:02:08,940 --> 00:02:13,710 At this point, students, many students 37 00:02:13,710 --> 00:02:16,250 who have been trained like monkeys 38 00:02:16,250 --> 00:02:18,710 to do exactly what they've been told, 39 00:02:18,710 --> 00:02:21,170 suddenly freeze and give up. 40 00:02:21,170 --> 00:02:23,970 Because there's nothing to do. 41 00:02:23,970 --> 00:02:28,000 So this is the one thing that I have to train out of you. 42 00:02:28,000 --> 00:02:32,260 You can't just give up at this point. 43 00:02:32,260 --> 00:02:35,060 So what would you suggest? 44 00:02:35,060 --> 00:02:38,100 Can anybody get us out of this jam? 45 00:02:38,100 --> 00:02:38,780 Yeah. 46 00:02:38,780 --> 00:02:47,410 STUDENT: [INAUDIBLE] 47 00:02:47,410 --> 00:02:48,150 PROFESSOR: Right. 48 00:02:48,150 --> 00:02:52,810 So the suggestion was to find the x-values 49 00:02:52,810 --> 00:02:55,800 where f(x) is undefined. 50 00:02:55,800 --> 00:03:00,680 In fact, so now that's a fairly sophisticated way 51 00:03:00,680 --> 00:03:03,090 of putting the point that I want to make, 52 00:03:03,090 --> 00:03:05,380 which is that what we want to do is go back 53 00:03:05,380 --> 00:03:07,770 to our precalculus skills. 54 00:03:07,770 --> 00:03:11,150 And just plot points. 55 00:03:11,150 --> 00:03:13,500 So instead, you go back to precalculus 56 00:03:13,500 --> 00:03:16,934 and you just plot some points. 57 00:03:16,934 --> 00:03:18,350 It's a perfectly reasonable thing. 58 00:03:18,350 --> 00:03:21,970 Now, it turns out that the most important point to plot 59 00:03:21,970 --> 00:03:24,730 is the one that's not there. 60 00:03:24,730 --> 00:03:29,300 Namely, the value of x = -2. 61 00:03:29,300 --> 00:03:31,780 Which is just what was suggested. 62 00:03:31,780 --> 00:03:38,710 Namely, we plot the points where the function is not defined. 63 00:03:38,710 --> 00:03:41,220 So how do we do that? 64 00:03:41,220 --> 00:03:43,670 Well, you have to think about it for a second 65 00:03:43,670 --> 00:03:46,470 and I'll introduce some new notation when I do it. 66 00:03:46,470 --> 00:03:49,290 If I evaluate 2 at this place, actually I can't do it. 67 00:03:49,290 --> 00:03:51,250 I have to do it from the left and the right. 68 00:03:51,250 --> 00:03:57,840 So if I plug in -2 on the positive side, from the right, 69 00:03:57,840 --> 00:04:03,500 that's going to be equal to -2 + 1 divided by -2, 70 00:04:03,500 --> 00:04:06,590 a little bit more than -2, plus 2. 71 00:04:06,590 --> 00:04:11,110 Which is -1 divided by - now, this denominator is -2, 72 00:04:11,110 --> 00:04:12,730 a little more than that, plus 2. 73 00:04:12,730 --> 00:04:20,200 So it's a little more than 0. 74 00:04:20,200 --> 00:04:24,720 And that is, well we'll fill that in in a second. 75 00:04:24,720 --> 00:04:25,620 Everybody's puzzled. 76 00:04:25,620 --> 00:04:26,120 Yes. 77 00:04:26,120 --> 00:04:30,670 STUDENT: [INAUDIBLE] 78 00:04:30,670 --> 00:04:36,105 PROFESSOR: No, that's the function. 79 00:04:36,105 --> 00:04:37,980 I'm plotting points, I'm not differentiating. 80 00:04:37,980 --> 00:04:38,250 I've already differentiated it. 81 00:04:38,250 --> 00:04:39,630 I've already got something that's a little puzzling. 82 00:04:39,630 --> 00:04:41,250 Now I'm focusing on the weird spot. 83 00:04:41,250 --> 00:04:42,166 Yes, another question. 84 00:04:42,166 --> 00:04:47,652 STUDENT: Wouldn't it be a little less than 0? 85 00:04:47,652 --> 00:04:49,610 PROFESSOR: Wouldn't it be a little less than 0? 86 00:04:49,610 --> 00:04:52,800 OK, that's a very good point and this is a matter of notation 87 00:04:52,800 --> 00:04:53,310 here. 88 00:04:53,310 --> 00:04:55,690 And a matter of parentheses. 89 00:04:55,690 --> 00:04:57,430 So wouldn't this be a little less than 2. 90 00:04:57,430 --> 00:05:01,900 Well, if the parentheses were this way; that is, 2+ , 91 00:05:01,900 --> 00:05:06,790 with a minus after I did the 2+ , then it would be less. 92 00:05:06,790 --> 00:05:09,830 But it's this way. 93 00:05:09,830 --> 00:05:10,670 OK. 94 00:05:10,670 --> 00:05:14,420 So the notation is, you have a number 95 00:05:14,420 --> 00:05:17,190 and you take the plus part of it. 96 00:05:17,190 --> 00:05:21,550 That's the part which is a little bit bigger than it. 97 00:05:21,550 --> 00:05:26,840 And so this is what I mean. 98 00:05:26,840 --> 00:05:29,870 And if you like, here I can put in those parentheses too. 99 00:05:29,870 --> 00:05:31,270 Yeah, another question. 100 00:05:31,270 --> 00:05:34,730 STUDENT: [INAUDIBLE] 101 00:05:34,730 --> 00:05:37,253 PROFESSOR: Why doesn't the top one have a plus? 102 00:05:37,253 --> 00:05:39,378 The only reason why the top one doesn't have a plus 103 00:05:39,378 --> 00:05:42,860 is that I don't need it to evaluate this. 104 00:05:42,860 --> 00:05:45,410 And when I take the limit, I can just plug in the value. 105 00:05:45,410 --> 00:05:48,250 Whereas here, I'm still uncertain. 106 00:05:48,250 --> 00:05:49,430 Because it's going to be 0. 107 00:05:49,430 --> 00:05:51,740 And I want to know which side of 0 it's on. 108 00:05:51,740 --> 00:05:55,190 Whether it's on the positive side or the negative side. 109 00:05:55,190 --> 00:05:58,310 So this one, I could have written here a parentheses 2+, 110 00:05:58,310 --> 00:06:01,410 but then it would have just simplified to -1. 111 00:06:01,410 --> 00:06:04,410 In the limit. 112 00:06:04,410 --> 00:06:07,190 So now, I've got a negative number divided 113 00:06:07,190 --> 00:06:10,024 by a tiny positive number. 114 00:06:10,024 --> 00:06:11,940 And so, somebody want to tell me what that is? 115 00:06:11,940 --> 00:06:16,750 Negative infinity. 116 00:06:16,750 --> 00:06:21,460 So, we just evaluated this function from one side. 117 00:06:21,460 --> 00:06:24,830 And if you follow through the other side, 118 00:06:24,830 --> 00:06:30,700 so this one here, you get something very similar, 119 00:06:30,700 --> 00:06:34,120 except that this should be-- whoops, what did I do wrong? 120 00:06:34,120 --> 00:06:37,690 I meant this. 121 00:06:37,690 --> 00:06:40,500 I wanted -2, the same base point, 122 00:06:40,500 --> 00:06:43,750 but I want to go from the left. 123 00:06:43,750 --> 00:06:47,780 So that's going to be -2 + 1, same numerator. 124 00:06:47,780 --> 00:06:51,840 And then this -2 on the left, plus 2, 125 00:06:51,840 --> 00:06:57,580 and that's going to come out to be -1 / 0-, -, 126 00:06:57,580 --> 00:07:00,670 which is plus infinity. 127 00:07:00,670 --> 00:07:10,520 Or just plain infinity, we don't have to put the plus sign in. 128 00:07:10,520 --> 00:07:13,090 So this is the first part of the problem. 129 00:07:13,090 --> 00:07:16,190 And the second piece, to get ourselves started, 130 00:07:16,190 --> 00:07:18,260 you could evaluate this function at any point. 131 00:07:18,260 --> 00:07:21,256 This is just the most interesting point, alright? 132 00:07:21,256 --> 00:07:22,880 This is just the most interesting place 133 00:07:22,880 --> 00:07:24,936 to evaluate it. 134 00:07:24,936 --> 00:07:26,560 Now, the next thing that I'd like to do 135 00:07:26,560 --> 00:07:32,130 is to pay attention to the ends. 136 00:07:32,130 --> 00:07:34,290 And I haven't really said what the ends are. 137 00:07:34,290 --> 00:07:37,399 So the ends are just all the way to the left and all the way 138 00:07:37,399 --> 00:07:37,940 to the right. 139 00:07:37,940 --> 00:07:42,180 So that means x going to plus or minus infinity. 140 00:07:42,180 --> 00:07:44,430 So that's the second thing I want to pay attention to. 141 00:07:44,430 --> 00:07:49,450 Again, this is a little bit like a video screen here. 142 00:07:49,450 --> 00:07:52,400 And we're about to discover something that's really 143 00:07:52,400 --> 00:07:55,654 off the screen, in both cases. 144 00:07:55,654 --> 00:07:58,070 We're taking care of what's happening way to the left, way 145 00:07:58,070 --> 00:07:59,360 to the right, here. 146 00:07:59,360 --> 00:08:01,170 And up above, we just took care what 147 00:08:01,170 --> 00:08:05,280 happens way up and way down. 148 00:08:05,280 --> 00:08:11,550 So on these ends, I need to do some more analysis. 149 00:08:11,550 --> 00:08:15,480 Which is related to a precalculus skill 150 00:08:15,480 --> 00:08:18,310 which is evaluating limits. 151 00:08:18,310 --> 00:08:21,070 And here, the way to do it is to divide 152 00:08:21,070 --> 00:08:23,010 by x the numerator and denominator. 153 00:08:23,010 --> 00:08:27,360 Write it as (1 + 1/x) / (1 + 2/x). 154 00:08:27,360 --> 00:08:29,650 And then you can see what happens as x 155 00:08:29,650 --> 00:08:30,980 goes to plus or minus infinity. 156 00:08:30,980 --> 00:08:33,590 It just goes to 1. 157 00:08:33,590 --> 00:08:37,690 So, no matter whether x is positive or negative. 158 00:08:37,690 --> 00:08:42,810 When it gets huge, these two extra numbers here go to 0. 159 00:08:42,810 --> 00:08:44,550 And so, this tends to 1. 160 00:08:44,550 --> 00:08:47,560 So if you like, you could abbreviate this 161 00:08:47,560 --> 00:08:52,830 as f plus or minus infinity is equal to 1. 162 00:08:52,830 --> 00:08:54,610 So now, I get to draw this. 163 00:08:54,610 --> 00:08:56,900 And we draw this using asymptotes. 164 00:08:56,900 --> 00:09:01,800 So there's a level which is y = 1. 165 00:09:01,800 --> 00:09:06,790 And then there's another line to draw. 166 00:09:06,790 --> 00:09:11,180 Which is x = -2. 167 00:09:15,190 --> 00:09:18,520 And now, what information do I have so far? 168 00:09:18,520 --> 00:09:20,740 Well, the information that I have so far 169 00:09:20,740 --> 00:09:26,530 is that when we're coming in from the right, that's to -2, 170 00:09:26,530 --> 00:09:28,570 it plunges down to minus infinity. 171 00:09:28,570 --> 00:09:33,170 So that's down like this. 172 00:09:33,170 --> 00:09:39,780 And I also know that it goes up to infinity on the other side 173 00:09:39,780 --> 00:09:41,850 of the asymptote. 174 00:09:41,850 --> 00:09:48,130 And over here, I know it's going out to the level 1. 175 00:09:48,130 --> 00:09:53,410 And here it's also going to the level 1. 176 00:09:53,410 --> 00:09:57,160 Now, there's an issue. 177 00:09:57,160 --> 00:09:59,490 I can almost finish this graph now. 178 00:09:59,490 --> 00:10:01,460 I almost have enough information to finish it. 179 00:10:01,460 --> 00:10:03,530 But there's one thing which is making 180 00:10:03,530 --> 00:10:06,780 me hesitate a little bit. 181 00:10:06,780 --> 00:10:10,160 And that is, I don't know, for instance, over here, 182 00:10:10,160 --> 00:10:14,230 whether it's going to maybe dip below and come back up. 183 00:10:14,230 --> 00:10:16,930 Or not. 184 00:10:16,930 --> 00:10:20,140 So what does it do here? 185 00:10:20,140 --> 00:10:24,750 Can anybody see? 186 00:10:24,750 --> 00:10:25,250 Yeah. 187 00:10:25,250 --> 00:10:29,900 STUDENT: [INAUDIBLE] 188 00:10:29,900 --> 00:10:32,220 PROFESSOR: It can't dip below because there 189 00:10:32,220 --> 00:10:33,178 are no critical points. 190 00:10:33,178 --> 00:10:34,540 What a precisely correct answer. 191 00:10:34,540 --> 00:10:36,730 So that's exactly right. 192 00:10:36,730 --> 00:10:43,390 The point here is that because f' is not 0, 193 00:10:43,390 --> 00:10:45,020 it can't double back on itself. 194 00:10:45,020 --> 00:10:49,790 Because there can't be any of these horizontal tangents. 195 00:10:49,790 --> 00:11:00,560 It can't double back, so it can't backtrack. 196 00:11:00,560 --> 00:11:07,190 So sorry, if f' is not 0, f can't backtrack. 197 00:11:07,190 --> 00:11:09,410 And so that means that it doesn't look like this. 198 00:11:09,410 --> 00:11:14,320 It just goes like this. 199 00:11:14,320 --> 00:11:15,690 So that's basically it. 200 00:11:15,690 --> 00:11:17,630 And it's practically the end of the problem. 201 00:11:17,630 --> 00:11:19,500 Goes like this. 202 00:11:19,500 --> 00:11:21,870 Now you can decorate your thing, right? 203 00:11:21,870 --> 00:11:24,660 You may notice that maybe it crosses here, the axes, you can 204 00:11:24,660 --> 00:11:26,790 actually evaluate these places. 205 00:11:26,790 --> 00:11:27,500 And so forth. 206 00:11:27,500 --> 00:11:31,130 We're looking right now for qualitative behavior. 207 00:11:31,130 --> 00:11:34,280 In fact, you can see where these places hit. 208 00:11:34,280 --> 00:11:36,660 And it's actually a little higher up than I drew. 209 00:11:36,660 --> 00:11:40,140 Maybe I'll draw it accurately. 210 00:11:40,140 --> 00:11:44,970 As we'll see in a second. 211 00:11:44,970 --> 00:11:47,710 So that's what happens to this function. 212 00:11:47,710 --> 00:11:51,860 Now, let's just take a look in a little bit more detail, 213 00:11:51,860 --> 00:11:56,470 by double checking. 214 00:11:56,470 --> 00:11:58,470 So we're just going to double check what happens 215 00:11:58,470 --> 00:12:01,280 to the sign of the derivative. 216 00:12:01,280 --> 00:12:03,460 And in the meantime, I'm going to explain to you 217 00:12:03,460 --> 00:12:05,510 what the derivative is and also talk 218 00:12:05,510 --> 00:12:07,045 about the second derivative. 219 00:12:07,045 --> 00:12:11,820 So first of all, the trick for evaluating the derivative 220 00:12:11,820 --> 00:12:13,570 is an algebraic one. 221 00:12:13,570 --> 00:12:16,080 I mean, obviously you can do this by the quotient rule. 222 00:12:16,080 --> 00:12:24,250 But I just point out that this is the same thing as this. 223 00:12:24,250 --> 00:12:27,330 And now it has, whoops, that should be a 2 224 00:12:27,330 --> 00:12:28,690 in the denominator. 225 00:12:28,690 --> 00:12:33,050 And so, now this has the form 1 - 1/(x+2). 226 00:12:35,950 --> 00:12:39,600 So this makes it easy to see what the derivative is. 227 00:12:39,600 --> 00:12:42,730 Because the derivative of a constant is 0, right? 228 00:12:42,730 --> 00:12:49,300 So this is, derivative, is just going to be, switch the sign. 229 00:12:49,300 --> 00:12:53,410 This is what I wrote before. 230 00:12:53,410 --> 00:12:55,230 And that explains it. 231 00:12:55,230 --> 00:12:57,610 But incidentally, it also shows you 232 00:12:57,610 --> 00:13:04,890 that that this is a hyperbola. 233 00:13:04,890 --> 00:13:09,620 These are just two curves of a hyperbola. 234 00:13:09,620 --> 00:13:12,240 So now, let's check the sign. 235 00:13:12,240 --> 00:13:14,566 It's already totally obvious to us 236 00:13:14,566 --> 00:13:15,940 that this is just a double check. 237 00:13:15,940 --> 00:13:18,710 We didn't actually even have to pay any attention to this. 238 00:13:18,710 --> 00:13:19,790 It had better be true. 239 00:13:19,790 --> 00:13:22,220 This is just going to check our arithmetic. 240 00:13:22,220 --> 00:13:24,570 Namely, it's increasing here. 241 00:13:24,570 --> 00:13:26,610 It's increasing there. 242 00:13:26,610 --> 00:13:27,830 That's got to be true. 243 00:13:27,830 --> 00:13:30,970 And, sure enough, this is positive, 244 00:13:30,970 --> 00:13:32,930 as you can see it's 1 over a square. 245 00:13:32,930 --> 00:13:34,070 So it is increasing. 246 00:13:34,070 --> 00:13:35,630 So we checked it. 247 00:13:35,630 --> 00:13:39,100 But now, there's one more thing that I want to just 248 00:13:39,100 --> 00:13:40,720 have you watch out about. 249 00:13:40,720 --> 00:13:46,980 So this means that f is increasing. 250 00:13:46,980 --> 00:13:51,830 On the interval minus infinity < x < -2. 251 00:13:51,830 --> 00:13:56,700 And also from -2 all the way out to infinity. 252 00:13:56,700 --> 00:14:02,260 So I just want to warn you, you cannot say, 253 00:14:02,260 --> 00:14:10,370 don't say f is increasing on (minus infinity, infinity), 254 00:14:10,370 --> 00:14:12,020 or all x. 255 00:14:12,020 --> 00:14:14,826 OK, this is just not true. 256 00:14:14,826 --> 00:14:16,700 I've written it on the board, but it's wrong. 257 00:14:16,700 --> 00:14:18,660 I'd better get rid of it. 258 00:14:18,660 --> 00:14:19,200 There it is. 259 00:14:19,200 --> 00:14:20,970 Get rid of it. 260 00:14:20,970 --> 00:14:24,555 And the reason is, so first of all it's totally obvious. 261 00:14:24,555 --> 00:14:25,390 It's going up here. 262 00:14:25,390 --> 00:14:28,700 But then it went zooming back down there. 263 00:14:28,700 --> 00:14:35,860 And here this was true, but only if x is not -2. 264 00:14:35,860 --> 00:14:37,400 So there's a break. 265 00:14:37,400 --> 00:14:39,280 And you've got to pay attention to the break. 266 00:14:39,280 --> 00:14:51,060 So basically, the moral here is that if you ignore this place, 267 00:14:51,060 --> 00:14:54,390 it's like ignoring Mount Everest, or the Grand Canyon. 268 00:14:54,390 --> 00:14:56,640 You're ignoring the most important feature 269 00:14:56,640 --> 00:14:58,174 of this function here. 270 00:14:58,174 --> 00:14:59,590 If you're going to be figuring out 271 00:14:59,590 --> 00:15:01,980 where things are going up and down, which is basically 272 00:15:01,980 --> 00:15:04,460 all we're doing, you'd better pay attention 273 00:15:04,460 --> 00:15:07,330 to these kinds of places. 274 00:15:07,330 --> 00:15:09,330 So don't ignore them. 275 00:15:09,330 --> 00:15:13,140 So that's the first remark. 276 00:15:13,140 --> 00:15:17,340 And now there's just a little bit of decoration as well. 277 00:15:17,340 --> 00:15:19,990 Which is the role of the second derivative. 278 00:15:19,990 --> 00:15:21,990 So we've written down the first derivative here. 279 00:15:21,990 --> 00:15:33,490 The second derivative is now -2 / (x + 2)^3, right? 280 00:15:33,490 --> 00:15:36,300 So I get that from differentiating this formula up 281 00:15:36,300 --> 00:15:39,490 here for the first derivative. 282 00:15:39,490 --> 00:15:43,970 And now, of course, that's also, only works for x not equal 283 00:15:43,970 --> 00:15:47,680 to -2. 284 00:15:47,680 --> 00:15:54,820 And now, we can see that this is going to be negative, let's 285 00:15:54,820 --> 00:15:56,650 see, where is it negative? 286 00:15:56,650 --> 00:15:58,540 When this is a positive quantity, 287 00:15:58,540 --> 00:16:04,590 so when -2 < x < infinity, it's negative. 288 00:16:04,590 --> 00:16:07,770 And this is where this thing is concave. 289 00:16:07,770 --> 00:16:08,670 Let's see. 290 00:16:08,670 --> 00:16:12,650 Did I say that right? 291 00:16:12,650 --> 00:16:13,330 Negative, right? 292 00:16:13,330 --> 00:16:19,180 This is concave down. 293 00:16:19,180 --> 00:16:19,930 Right. 294 00:16:19,930 --> 00:16:22,570 And similarly, if I look at this expression, 295 00:16:22,570 --> 00:16:27,830 the numerator is always negative but the denominator becomes 296 00:16:27,830 --> 00:16:31,130 negative as well when x < -2. 297 00:16:31,130 --> 00:16:33,710 So this becomes positive. 298 00:16:33,710 --> 00:16:36,690 So this case, it was negative over positive. 299 00:16:36,690 --> 00:16:40,950 In this case it was negative divided by negative. 300 00:16:40,950 --> 00:16:46,300 So here, this is in the range minus infinity < x < -2 And 301 00:16:46,300 --> 00:16:52,240 here it's concave up. 302 00:16:52,240 --> 00:16:54,962 Now, again, this is just consistent with what 303 00:16:54,962 --> 00:16:55,920 we're already guessing. 304 00:16:55,920 --> 00:16:57,628 Of course we already know it in this case 305 00:16:57,628 --> 00:16:59,680 if we know that this is a hyperbola. 306 00:16:59,680 --> 00:17:01,600 That it's going to be concave down 307 00:17:01,600 --> 00:17:04,270 to the right of the vertical line, dotted vertical line. 308 00:17:04,270 --> 00:17:07,770 And concave up to the left. 309 00:17:07,770 --> 00:17:10,640 So what extra piece of information 310 00:17:10,640 --> 00:17:16,100 is it that this is giving us? 311 00:17:16,100 --> 00:17:17,800 Did I say this backwards? 312 00:17:17,800 --> 00:17:18,520 No. 313 00:17:18,520 --> 00:17:19,222 That's OK. 314 00:17:19,222 --> 00:17:21,430 So what extra piece of information is this giving us? 315 00:17:21,430 --> 00:17:23,400 It looks like it's giving us hardly anything. 316 00:17:23,400 --> 00:17:25,680 And it really is giving us hardly anything. 317 00:17:25,680 --> 00:17:28,840 But it is giving us something that's a little aesthetic. 318 00:17:28,840 --> 00:17:34,280 It's ruling out the possibility of a wiggle. 319 00:17:34,280 --> 00:17:37,590 There isn't anything like that in the curve. 320 00:17:37,590 --> 00:17:39,790 It can't shift from curving this way 321 00:17:39,790 --> 00:17:41,590 to curving that way to curving this way. 322 00:17:41,590 --> 00:17:42,860 That doesn't happen. 323 00:17:42,860 --> 00:17:59,070 So these properties say there's no wiggle in the graph of that. 324 00:17:59,070 --> 00:17:59,790 Alright. 325 00:17:59,790 --> 00:18:01,390 So. 326 00:18:01,390 --> 00:18:01,890 Question. 327 00:18:01,890 --> 00:18:05,570 STUDENT: Do we define the increasing and decreasing based 328 00:18:05,570 --> 00:18:09,250 purely on the derivative, or the sort 329 00:18:09,250 --> 00:18:13,390 of more general definition of picking any two points 330 00:18:13,390 --> 00:18:14,310 and seeing. 331 00:18:14,310 --> 00:18:16,610 Because sometimes there can be an inconsistency 332 00:18:16,610 --> 00:18:20,300 between the two definitions. 333 00:18:20,300 --> 00:18:26,110 PROFESSOR: OK, so the question is, in this course, 334 00:18:26,110 --> 00:18:29,290 are we going to define positive derivative as being 335 00:18:29,290 --> 00:18:31,360 the same thing as increasing. 336 00:18:31,360 --> 00:18:33,030 And the answer is no. 337 00:18:33,030 --> 00:18:36,210 We'll try to use these terms separately. 338 00:18:36,210 --> 00:18:40,330 What's always true is that if f' is positive, 339 00:18:40,330 --> 00:18:42,530 then f is increasing. 340 00:18:42,530 --> 00:18:45,060 But the reverse is not necessarily true. 341 00:18:45,060 --> 00:18:47,190 It could be very flat, the derivative can be 0 342 00:18:47,190 --> 00:18:50,050 and still the function can be increasing. 343 00:18:50,050 --> 00:18:53,730 OK, the derivative can be 0 at a few places. 344 00:18:53,730 --> 00:18:59,300 For instance, like some cubics. 345 00:18:59,300 --> 00:19:05,370 Other questions? 346 00:19:05,370 --> 00:19:09,860 So that's as much as I need to say in general. 347 00:19:09,860 --> 00:19:11,390 I mean, in a specific case. 348 00:19:11,390 --> 00:19:13,240 But I want to get you a general scheme 349 00:19:13,240 --> 00:19:16,500 and I want to go through a more complicated example that 350 00:19:16,500 --> 00:19:22,750 gets all the features of this kind of thing. 351 00:19:22,750 --> 00:19:34,710 So let's talk about a general strategy for sketching. 352 00:19:34,710 --> 00:19:40,300 So the first part of this strategy, if you like, 353 00:19:40,300 --> 00:19:40,800 let's see. 354 00:19:40,800 --> 00:19:42,430 I have it all plotted out here. 355 00:19:42,430 --> 00:19:45,930 So I'm going to make sure I get it exactly the way 356 00:19:45,930 --> 00:19:47,780 I wanted you to see. 357 00:19:47,780 --> 00:19:51,440 So I have, it's plotting. 358 00:19:51,440 --> 00:19:52,720 The plot thickens. 359 00:19:52,720 --> 00:19:54,040 Here we go. 360 00:19:54,040 --> 00:19:57,710 So plot, what is it that you should plot first? 361 00:19:57,710 --> 00:20:01,070 Before you even think about derivatives, 362 00:20:01,070 --> 00:20:08,250 you should plot discontinuities. 363 00:20:08,250 --> 00:20:18,490 Especially the infinite ones. 364 00:20:18,490 --> 00:20:20,210 That's the first thing you should do. 365 00:20:20,210 --> 00:20:27,160 And then, you should plot end points, for ends. 366 00:20:27,160 --> 00:20:30,580 For x going to plus or minus infinity 367 00:20:30,580 --> 00:20:35,640 if there don't happen to be any finite ends to the problem. 368 00:20:35,640 --> 00:20:44,600 And the third thing you can do is plot any easy points. 369 00:20:44,600 --> 00:20:49,000 This is optional. 370 00:20:49,000 --> 00:20:50,710 At your discretion. 371 00:20:50,710 --> 00:20:53,810 You might, for instance, on this example, 372 00:20:53,810 --> 00:20:59,550 plot the places where the graph crosses the axis. 373 00:20:59,550 --> 00:21:04,540 If you want to. 374 00:21:04,540 --> 00:21:05,810 So that's the first part. 375 00:21:05,810 --> 00:21:08,190 And again, this is all precalculus. 376 00:21:08,190 --> 00:21:18,050 So now, in the second part we're going to solve this equation 377 00:21:18,050 --> 00:21:29,360 and we're going to plot the critical points and values. 378 00:21:29,360 --> 00:21:32,810 In the problem which we just discussed, there weren't any. 379 00:21:32,810 --> 00:21:38,640 So this part was empty. 380 00:21:38,640 --> 00:21:49,685 So the third step is to decide whether f', sorry, whether, 381 00:21:49,685 --> 00:22:01,120 f' is positive or negative on each interval. 382 00:22:01,120 --> 00:22:17,210 Between critical points, discontinuities. 383 00:22:17,210 --> 00:22:22,560 The direction of the sign, in this case it doesn't change. 384 00:22:22,560 --> 00:22:24,900 It goes up here and it also goes up here. 385 00:22:24,900 --> 00:22:27,770 But it could go up here and then come back down. 386 00:22:27,770 --> 00:22:31,460 So the direction can change at every critical point. 387 00:22:31,460 --> 00:22:33,850 It can change at every discontinuity. 388 00:22:33,850 --> 00:22:35,350 And you don't know. 389 00:22:35,350 --> 00:22:39,310 However, this particular step has 390 00:22:39,310 --> 00:22:47,230 to be consistent with 1 and 2, with steps 1 and 2. 391 00:22:47,230 --> 00:22:51,800 In fact, it will never, if you can 392 00:22:51,800 --> 00:22:56,970 succeed in doing steps 1 and 2, you'll never need step 3. 393 00:22:56,970 --> 00:23:02,190 All it's doing is double-checking. 394 00:23:02,190 --> 00:23:05,770 So if you made an arithmetic mistake somewhere, 395 00:23:05,770 --> 00:23:09,160 you'll be able to see it. 396 00:23:09,160 --> 00:23:10,870 So that's maybe the most important thing. 397 00:23:10,870 --> 00:23:13,050 And it's actually the most frustrating thing for me 398 00:23:13,050 --> 00:23:17,700 when I see people working on problems, is they start step 3, 399 00:23:17,700 --> 00:23:21,140 they get it wrong, and then they start trying to draw the graph 400 00:23:21,140 --> 00:23:22,480 and it doesn't work. 401 00:23:22,480 --> 00:23:23,620 Because it's inconsistent. 402 00:23:23,620 --> 00:23:25,880 And the reason is some arithmetic error 403 00:23:25,880 --> 00:23:27,630 with the derivative or something like that 404 00:23:27,630 --> 00:23:29,880 or some other misinterpretation. 405 00:23:29,880 --> 00:23:31,930 And then there's a total mess. 406 00:23:31,930 --> 00:23:34,170 If you start with these two steps, 407 00:23:34,170 --> 00:23:36,454 then you're going to know when you get to this step 408 00:23:36,454 --> 00:23:37,620 that you're making mistakes. 409 00:23:37,620 --> 00:23:41,860 People don't generally make as many mistakes in the first two 410 00:23:41,860 --> 00:23:42,360 steps. 411 00:23:42,360 --> 00:23:45,220 Anyway, in fact you can skip this step if you want. 412 00:23:45,220 --> 00:23:49,470 But that's at risk of not double-checking your work. 413 00:23:49,470 --> 00:23:51,280 So what's the fourth step? 414 00:23:51,280 --> 00:23:59,400 Well, we take a look at whether f'' is positive or negative. 415 00:23:59,400 --> 00:24:02,300 And so we're deciding on things like whether it's concave 416 00:24:02,300 --> 00:24:07,640 up or down. 417 00:24:07,640 --> 00:24:15,120 And we have these points, f''(x) = 0, 418 00:24:15,120 --> 00:24:24,570 which are called inflection points. 419 00:24:24,570 --> 00:24:31,550 And the last step is just to combine everything. 420 00:24:31,550 --> 00:24:35,710 So this is this the scheme, the general scheme. 421 00:24:35,710 --> 00:24:58,850 And let's just carry it out in a particular case. 422 00:24:58,850 --> 00:25:02,280 So here's the function that I'm going to use as an example. 423 00:25:02,280 --> 00:25:08,250 I'll use f(x) = x / ln x. 424 00:25:08,250 --> 00:25:11,460 And because the logarithm-- yeah, question. 425 00:25:11,460 --> 00:25:11,960 Yeah. 426 00:25:11,960 --> 00:25:18,660 STUDENT: [INAUDIBLE] 427 00:25:18,660 --> 00:25:21,350 PROFESSOR: The question is, is this optional. 428 00:25:21,350 --> 00:25:25,570 So that's a good question. 429 00:25:25,570 --> 00:25:26,390 Is this optional. 430 00:25:26,390 --> 00:25:31,516 STUDENT: [INAUDIBLE] 431 00:25:31,516 --> 00:25:37,590 PROFESSOR: OK, the question is is this optional, 432 00:25:37,590 --> 00:25:38,660 this kind of question. 433 00:25:38,660 --> 00:25:48,620 And the answer is, it's more than just-- 434 00:25:48,620 --> 00:25:51,700 so, in many instances, I'm not going to ask you to. 435 00:25:51,700 --> 00:25:55,170 I strongly recommend that if I don't ask you to do it, 436 00:25:55,170 --> 00:25:57,050 that you not try. 437 00:25:57,050 --> 00:26:01,050 Because it's usually awful to find the second derivative. 438 00:26:01,050 --> 00:26:02,940 Any time you can get away without computing 439 00:26:02,940 --> 00:26:06,330 a second derivative, you're better off. 440 00:26:06,330 --> 00:26:07,834 So in many, many instances. 441 00:26:07,834 --> 00:26:09,500 On the other hand, if I ask you to do it 442 00:26:09,500 --> 00:26:13,060 it's because I want you to have the work to do it. 443 00:26:13,060 --> 00:26:16,470 But basically, if nobody forces you to, 444 00:26:16,470 --> 00:26:22,130 I would say never do step 4 here. 445 00:26:22,130 --> 00:26:26,750 Other questions. 446 00:26:26,750 --> 00:26:27,610 All right. 447 00:26:27,610 --> 00:26:29,150 So we're going to force ourselves 448 00:26:29,150 --> 00:26:31,810 to do step 4, however, in this instance. 449 00:26:31,810 --> 00:26:35,010 But maybe this will be one of the few times. 450 00:26:35,010 --> 00:26:39,140 So here we go, just for illustrative purposes. 451 00:26:39,140 --> 00:26:43,240 OK, now. 452 00:26:43,240 --> 00:26:46,140 So here's the function that I want to discuss. 453 00:26:46,140 --> 00:26:49,120 And the range has to be x positive, 454 00:26:49,120 --> 00:26:55,500 because the logarithm is not defined for negative values. 455 00:26:55,500 --> 00:26:57,660 So the first thing that I'm going to do 456 00:26:57,660 --> 00:27:02,560 is, I'd like to follow the scheme here. 457 00:27:02,560 --> 00:27:04,770 Because if I don't follow the scheme, 458 00:27:04,770 --> 00:27:06,490 I'm going to get a little mixed up. 459 00:27:06,490 --> 00:27:13,980 So the first part is to find the singularities. 460 00:27:13,980 --> 00:27:17,060 That is, the places where f is infinite. 461 00:27:17,060 --> 00:27:20,720 And that's when the logarithm, the denominator, vanishes. 462 00:27:20,720 --> 00:27:25,630 So that's f(1+), if you like. 463 00:27:25,630 --> 00:27:32,000 So that's 1 / ln(1+), which is 1 / 0, 464 00:27:32,000 --> 00:27:34,280 with a little bit of positiveness to it. 465 00:27:34,280 --> 00:27:37,270 Which is infinity. 466 00:27:37,270 --> 00:27:39,830 And second, we do it the other way. 467 00:27:39,830 --> 00:27:46,850 And not surprisingly, this comes out to be negative infinity. 468 00:27:46,850 --> 00:27:51,980 Now, the next thing I want to do is the ends. 469 00:27:51,980 --> 00:27:56,980 So I call these the ends. 470 00:27:56,980 --> 00:28:01,380 And there are two of them. 471 00:28:01,380 --> 00:28:06,250 One of them is f(0) from the right. 472 00:28:06,250 --> 00:28:08,300 f(0+). 473 00:28:08,300 --> 00:28:21,120 So that is 0+ / ln(0+), which is 0+ divided by, well, 474 00:28:21,120 --> 00:28:25,360 ln(0+) is actually minus infinity. 475 00:28:25,360 --> 00:28:27,180 That's what happens to the logarithm, goes 476 00:28:27,180 --> 00:28:28,170 to minus infinity. 477 00:28:28,170 --> 00:28:31,160 So this is 0 over infinity, which is definitely 0, 478 00:28:31,160 --> 00:28:37,100 there's no problem about what happens to this. 479 00:28:37,100 --> 00:28:42,910 The other side, so this is the end, this is the first end. 480 00:28:42,910 --> 00:28:44,920 The range is this. 481 00:28:44,920 --> 00:28:48,019 And I just did the left endpoint. 482 00:28:48,019 --> 00:28:49,810 And so now I have to do the right endpoint, 483 00:28:49,810 --> 00:28:51,870 I have to let x go to infinity. 484 00:28:51,870 --> 00:28:53,591 So if I let x go to infinity, I'm 485 00:28:53,591 --> 00:28:55,090 just going to have to think about it 486 00:28:55,090 --> 00:28:57,690 a little bit by plugging in a very large number. 487 00:28:57,690 --> 00:29:01,960 I'll plug in 10^10, to see what happens. 488 00:29:01,960 --> 00:29:07,890 So if I plug in 10^10 into x / ln x, I get 10^10 / ln(10^10). 489 00:29:11,730 --> 00:29:17,590 Which is 10^10 / (10 ln(10)). 490 00:29:17,590 --> 00:29:20,110 So the denominator, this number here, 491 00:29:20,110 --> 00:29:23,180 is about 2 point something. 492 00:29:23,180 --> 00:29:25,130 2.3 or so. 493 00:29:25,130 --> 00:29:27,470 So this is maybe 230 in the denominator, 494 00:29:27,470 --> 00:29:31,900 and this is a number with ten 0's after it. 495 00:29:31,900 --> 00:29:33,530 So it's very, very large. 496 00:29:33,530 --> 00:29:35,300 I claim it's big. 497 00:29:35,300 --> 00:29:38,080 And that gives us the clue that what's happening 498 00:29:38,080 --> 00:29:40,540 is that this thing is infinite. 499 00:29:40,540 --> 00:29:42,150 So, in other words, our conclusion 500 00:29:42,150 --> 00:29:52,120 is that f of infinity is infinity. 501 00:29:52,120 --> 00:29:59,650 So what do we have so far for our function? 502 00:29:59,650 --> 00:30:03,300 We're just trying to build the scaffolding of the function. 503 00:30:03,300 --> 00:30:07,270 And we're doing it by taking the most important points. 504 00:30:07,270 --> 00:30:09,147 And from a mathematician's point of view, 505 00:30:09,147 --> 00:30:10,980 the most important points are the ones which 506 00:30:10,980 --> 00:30:13,490 are sort of infinitely obvious. 507 00:30:13,490 --> 00:30:15,190 For the ends of the problem. 508 00:30:15,190 --> 00:30:19,730 So that's where we're heading. 509 00:30:19,730 --> 00:30:22,600 We have a vertical asymptote, which is at x = 1. 510 00:30:22,600 --> 00:30:29,150 So this gives us x = 1. 511 00:30:29,150 --> 00:30:34,270 And we have a value which is that it's 0 here. 512 00:30:34,270 --> 00:30:38,640 And we also know that when we come in from the-- sorry, 513 00:30:38,640 --> 00:30:42,330 so we come in from the left, that's 514 00:30:42,330 --> 00:30:46,060 f, the one from the left, we get negative infinity. 515 00:30:46,060 --> 00:30:47,550 So it's diving down. 516 00:30:47,550 --> 00:30:52,460 It's going down like this. 517 00:30:52,460 --> 00:30:55,925 And, furthermore, on the other side we know it's climbing up. 518 00:30:55,925 --> 00:30:58,560 So it's going up like this. 519 00:30:58,560 --> 00:31:00,270 Just start a little higher. 520 00:31:00,270 --> 00:31:00,770 Right, so. 521 00:31:00,770 --> 00:31:02,520 So far, this is what we know. 522 00:31:02,520 --> 00:31:05,810 Oh, and there's one other thing that we know. 523 00:31:05,810 --> 00:31:12,420 When we go to plus infinity, it's going back up. 524 00:31:12,420 --> 00:31:15,150 So, so far we have this. 525 00:31:15,150 --> 00:31:17,750 Now, already it should be pretty obvious what's 526 00:31:17,750 --> 00:31:19,619 going to happen to this function. 527 00:31:19,619 --> 00:31:21,160 So there shouldn't be many surprises. 528 00:31:21,160 --> 00:31:23,160 It's going to come down like this. 529 00:31:23,160 --> 00:31:27,090 Go like this, it's going to turn around and go back up. 530 00:31:27,090 --> 00:31:29,290 That's what we expect. 531 00:31:29,290 --> 00:31:33,600 So we don't know that yet, but we're pretty sure. 532 00:31:33,600 --> 00:31:36,800 So at this point, we can start looking at the critical points. 533 00:31:36,800 --> 00:31:41,994 We can do our step 2 here - we need a little bit more room 534 00:31:41,994 --> 00:31:45,490 here - and see what's happening with this function. 535 00:31:45,490 --> 00:31:49,220 So I have to differentiate it. 536 00:31:49,220 --> 00:31:52,070 And it's, this is the quotient rule. 537 00:31:52,070 --> 00:31:54,880 So remember the function is up here, x / ln x. 538 00:31:54,880 --> 00:31:59,400 So I have a (ln x)^2 in the denominator. 539 00:31:59,400 --> 00:32:02,430 And I get here the derivative of x is 1, so we get 1 * 540 00:32:02,430 --> 00:32:07,174 ln x minus x times the derivative of ln 541 00:32:07,174 --> 00:32:07,840 x, which is 1/x. 542 00:32:10,560 --> 00:32:16,850 So all told, that's (ln x - 1) / (ln x)^2. 543 00:32:20,160 --> 00:32:27,770 So here's our derivative. 544 00:32:27,770 --> 00:32:35,080 And now, if I set this equal to 0, at least in the numerator, 545 00:32:35,080 --> 00:32:40,970 the numerator is 0 when x = e. 546 00:32:40,970 --> 00:32:43,490 The log of e is 1. 547 00:32:43,490 --> 00:32:46,290 So here's our critical point. 548 00:32:46,290 --> 00:32:51,320 And we have a critical value, which is f(e). 549 00:32:51,320 --> 00:32:55,610 And that's going to be e / ln e. 550 00:32:55,610 --> 00:32:57,240 Which is e, again. 551 00:32:57,240 --> 00:32:59,090 Because ln e = 1. 552 00:32:59,090 --> 00:33:01,890 So now I can also plot the critical point, 553 00:33:01,890 --> 00:33:03,110 which is down here. 554 00:33:03,110 --> 00:33:07,680 And there's only one of them, and it's at (e, e). 555 00:33:07,680 --> 00:33:09,530 That's kind of not to scale here, 556 00:33:09,530 --> 00:33:12,520 because my blackboard isn't quite tall enough. 557 00:33:12,520 --> 00:33:15,360 It should be over here and then, it's slope 1. 558 00:33:15,360 --> 00:33:17,140 But I dipped it down. 559 00:33:17,140 --> 00:33:18,730 So this is not to scale, and indeed 560 00:33:18,730 --> 00:33:20,490 that's one of the things that we're not 561 00:33:20,490 --> 00:33:22,680 going to attempt to do with these pictures, 562 00:33:22,680 --> 00:33:24,680 is to make them to scale. 563 00:33:24,680 --> 00:33:29,560 So the scale's a little squashed. 564 00:33:29,560 --> 00:33:32,710 So, so far I have this critical point. 565 00:33:32,710 --> 00:33:36,180 And, in fact, I'm going to label it with a C. 566 00:33:36,180 --> 00:33:38,030 Whenever I have a critical point I'll just 567 00:33:38,030 --> 00:33:41,490 make sure that I remember that that's what it is. 568 00:33:41,490 --> 00:33:45,120 And since there's only one, the rest of this picture 569 00:33:45,120 --> 00:33:49,900 is now correct. 570 00:33:49,900 --> 00:33:54,880 That's the same mechanism that we used for the hyperbola. 571 00:33:54,880 --> 00:33:56,977 Namely, we know there's only one place where 572 00:33:56,977 --> 00:33:57,810 the derivative is 0. 573 00:33:57,810 --> 00:33:59,980 So that means there no more horizontals, 574 00:33:59,980 --> 00:34:01,950 so there's no more backtracking. 575 00:34:01,950 --> 00:34:03,330 It has to come down to here. 576 00:34:03,330 --> 00:34:03,970 Get to there. 577 00:34:03,970 --> 00:34:06,060 This is the only place it can turn around. 578 00:34:06,060 --> 00:34:06,834 Goes back up. 579 00:34:06,834 --> 00:34:09,000 It has to start here and it has to go down to there. 580 00:34:09,000 --> 00:34:10,600 It can't go above 0. 581 00:34:10,600 --> 00:34:13,600 Do not pass go, do not get positive. 582 00:34:13,600 --> 00:34:20,230 It has to head down here. 583 00:34:20,230 --> 00:34:21,690 So that's great. 584 00:34:21,690 --> 00:34:25,080 That means that this picture is almost completely correct now. 585 00:34:25,080 --> 00:34:27,520 And the rest is more or less decoration. 586 00:34:27,520 --> 00:34:30,250 We're pretty much done with the way it looks, 587 00:34:30,250 --> 00:34:34,570 at least schematically. 588 00:34:34,570 --> 00:34:37,700 However, I am going to punish you, because I warned you. 589 00:34:37,700 --> 00:34:40,120 We are going to go over here and do this step 4 590 00:34:40,120 --> 00:34:44,216 and fix up the concavity. 591 00:34:44,216 --> 00:34:45,840 And we're also going to do a little bit 592 00:34:45,840 --> 00:35:00,770 of that double-checking. 593 00:35:00,770 --> 00:35:04,960 So now, let's again-- just, I want 594 00:35:04,960 --> 00:35:10,660 to emphasize-- We're going to do a double-check. 595 00:35:10,660 --> 00:35:12,240 This is part 3. 596 00:35:12,240 --> 00:35:16,870 But in advance, I already have, based on this picture 597 00:35:16,870 --> 00:35:19,000 I already know what has to be true. 598 00:35:19,000 --> 00:35:35,450 That f is decreasing on 0 to 1. f is also decreasing on 1 to e. 599 00:35:35,450 --> 00:35:45,490 And f is increasing on e to infinity. 600 00:35:45,490 --> 00:35:49,144 So, already, because we plotted a bunch of points 601 00:35:49,144 --> 00:35:51,060 and we know that there aren't any places where 602 00:35:51,060 --> 00:35:52,518 the derivative vanishes, we already 603 00:35:52,518 --> 00:35:55,500 know it goes down, down, up. 604 00:35:55,500 --> 00:35:56,970 That's what it's got to do. 605 00:35:56,970 --> 00:35:59,550 Now, we'll just make sure that we didn't make any arithmetic 606 00:35:59,550 --> 00:36:00,720 mistakes, now. 607 00:36:00,720 --> 00:36:02,980 By actually computing the derivative, 608 00:36:02,980 --> 00:36:04,650 or staring at it, anyway. 609 00:36:04,650 --> 00:36:10,350 And making sure that it's correct. 610 00:36:10,350 --> 00:36:17,600 So first of all, we just take a look at the numerator. 611 00:36:17,600 --> 00:36:26,570 So f', remember, was (ln x - 1) / (ln x)^2. 612 00:36:26,570 --> 00:36:28,480 So the denominator is positive. 613 00:36:28,480 --> 00:36:32,650 So let's just take a look at the three ranges. 614 00:36:32,650 --> 00:36:37,330 So we have 0 < x < 1. 615 00:36:37,330 --> 00:36:40,310 And on that range, the logarithm is negative, 616 00:36:40,310 --> 00:36:45,230 so this is negative divided by positive, which is negative. 617 00:36:45,230 --> 00:36:47,220 That's decreasing, that's good. 618 00:36:47,220 --> 00:36:50,160 And in fact, that also works on the next range. 619 00:36:50,160 --> 00:36:55,357 1 < x < e, it's negative divided by positive. 620 00:36:55,357 --> 00:36:57,190 And the only reason why we skipped 1, again, 621 00:36:57,190 --> 00:36:58,432 is that it's undefined there. 622 00:36:58,432 --> 00:37:01,040 And there's something dramatic happening there. 623 00:37:01,040 --> 00:37:05,240 And then, at the last range, when x is bigger than e, 624 00:37:05,240 --> 00:37:07,770 that means the logarithm is already bigger than 1. 625 00:37:07,770 --> 00:37:09,145 So the numerator is now positive, 626 00:37:09,145 --> 00:37:13,650 and the denominator's still positive, so it's increasing. 627 00:37:13,650 --> 00:37:22,910 So we've just double-checked something that we already knew. 628 00:37:22,910 --> 00:37:26,540 Alright, so that's pretty much all 629 00:37:26,540 --> 00:37:29,200 there is to say about step 3. 630 00:37:29,200 --> 00:37:33,980 So this is checking the positivity and negativity. 631 00:37:33,980 --> 00:37:35,660 And now, step 4. 632 00:37:35,660 --> 00:37:38,480 There is one small point which I want to make before we go on. 633 00:37:38,480 --> 00:37:42,110 Which is that sometimes, you can't 634 00:37:42,110 --> 00:37:45,445 evaluate the function or its derivative particularly well. 635 00:37:45,445 --> 00:37:48,359 So sometimes you can't plot the points very well. 636 00:37:48,359 --> 00:37:50,150 And if you can't plot the points very well, 637 00:37:50,150 --> 00:37:52,440 then you might have to do 3 first, 638 00:37:52,440 --> 00:37:55,290 to figure out what's going on a little bit. 639 00:37:55,290 --> 00:37:59,150 You might have to skip. 640 00:37:59,150 --> 00:38:02,400 So now we're going to go on to the second derivative. 641 00:38:02,400 --> 00:38:07,180 But first, I want to use an algebraic trick 642 00:38:07,180 --> 00:38:08,470 to rearrange the terms. 643 00:38:08,470 --> 00:38:10,820 And I want to notice one more little point. 644 00:38:10,820 --> 00:38:16,320 Which I-- as I say, this is decoration for the graph. 645 00:38:16,320 --> 00:38:18,290 So I want to rewrite the formula. 646 00:38:18,290 --> 00:38:22,320 Maybe I'll do it right over here. 647 00:38:22,320 --> 00:38:27,270 Another way of writing this is 1/(ln x) - 1/(ln x)^2. 648 00:38:31,440 --> 00:38:35,240 So that's another way of writing the derivative. 649 00:38:35,240 --> 00:38:38,520 And that allows me to notice something 650 00:38:38,520 --> 00:38:40,830 that I missed, before. 651 00:38:40,830 --> 00:38:47,410 When I solved the equation ln x - 1 - this is equal to 0 here, 652 00:38:47,410 --> 00:38:48,590 this equation here. 653 00:38:48,590 --> 00:38:51,460 I missed a possibility. 654 00:38:51,460 --> 00:38:54,020 I missed the possibility that the denominator 655 00:38:54,020 --> 00:38:58,690 could be infinity. 656 00:38:58,690 --> 00:39:02,040 So actually, if the denominator's infinity, 657 00:39:02,040 --> 00:39:05,460 as you can see from the other expression there, 658 00:39:05,460 --> 00:39:09,000 it actually is true that the derivative is 0. 659 00:39:09,000 --> 00:39:16,710 So also when x = 0+, the slope is going to be 0. 660 00:39:16,710 --> 00:39:19,050 Let me just emphasize that again. 661 00:39:19,050 --> 00:39:23,500 If you evaluate using this other formula over here, 662 00:39:23,500 --> 00:39:28,410 this is 1/(ln(0+)) - 1/(ln(0+))^2. 663 00:39:31,540 --> 00:39:36,500 That's 1 over -infinity - minus 1 over infinity, if you like, 664 00:39:36,500 --> 00:39:37,310 squared. 665 00:39:37,310 --> 00:39:40,630 Anyway, it's 0. 666 00:39:40,630 --> 00:39:42,330 So this is 0. 667 00:39:42,330 --> 00:39:43,410 The slope is 0 there. 668 00:39:43,410 --> 00:39:46,640 That is a little piece of decoration on our graph. 669 00:39:46,640 --> 00:39:50,330 It's telling us, going back to our graph here, 670 00:39:50,330 --> 00:39:53,560 it's telling us this is coming in with slope horizontal. 671 00:39:53,560 --> 00:39:57,530 So we're starting out this way. 672 00:39:57,530 --> 00:40:01,013 That's just a little start here to the graph. 673 00:40:01,013 --> 00:40:02,620 It's a horizontal slope. 674 00:40:02,620 --> 00:40:07,940 So there really were two places where the slope was horizontal. 675 00:40:07,940 --> 00:40:11,470 Now, with the help of this second formula 676 00:40:11,470 --> 00:40:16,972 I can also differentiate a second time. 677 00:40:16,972 --> 00:40:19,430 So it's a little bit easier to do that if I differentiate 1 678 00:40:19,430 --> 00:40:28,050 over the log, that's -(ln x)^(-2) 1/x + 2 (ln x)^(-3) 679 00:40:28,050 --> 00:40:28,550 1/x. 680 00:40:34,270 --> 00:40:39,090 And that, if I put it over a common denominator, 681 00:40:39,090 --> 00:40:48,690 is x (ln x)^3 times, let's see here, 682 00:40:48,690 --> 00:40:55,340 I guess I'll have to take the 2 - ln x. 683 00:40:55,340 --> 00:40:57,260 So I've now rewritten the formula 684 00:40:57,260 --> 00:41:03,450 for the second derivative as a ratio. 685 00:41:03,450 --> 00:41:09,610 Now, to decide the sign, you see there are two places where 686 00:41:09,610 --> 00:41:11,910 the sign flips. 687 00:41:11,910 --> 00:41:16,030 The numerator crosses when the logarithm is 2, 688 00:41:16,030 --> 00:41:18,620 that's going to be when x = e^2. 689 00:41:18,620 --> 00:41:22,400 And the denominator flips when x = 1, 690 00:41:22,400 --> 00:41:28,020 that's when the log flips from positive to negative. 691 00:41:28,020 --> 00:41:32,280 So we have a couple of ranges here. 692 00:41:32,280 --> 00:41:37,580 So, first of all, we have the range from 0 to 1. 693 00:41:37,580 --> 00:41:42,270 And then we have the range from 1 to e^2. 694 00:41:42,270 --> 00:41:46,380 And then we have the range from e^2 all the way out 695 00:41:46,380 --> 00:41:49,780 to infinity. 696 00:41:49,780 --> 00:41:57,210 So between 0 and 1, the numerator 697 00:41:57,210 --> 00:41:59,630 is, well this is a negative number and this, 698 00:41:59,630 --> 00:42:01,300 so minus a negative number is positive, 699 00:42:01,300 --> 00:42:04,650 so the numerator is positive. 700 00:42:04,650 --> 00:42:08,030 And the denominator is negative, because the log is negative 701 00:42:08,030 --> 00:42:09,660 and it's taken to the third power. 702 00:42:09,660 --> 00:42:12,170 So this is a negative number, so it's positive 703 00:42:12,170 --> 00:42:15,190 divided by negative, which is less than 0. 704 00:42:15,190 --> 00:42:18,770 That means it's concave down. 705 00:42:18,770 --> 00:42:26,040 So this is a concave down part. 706 00:42:26,040 --> 00:42:28,230 And that's a good thing, because over here this 707 00:42:28,230 --> 00:42:29,120 was concave down. 708 00:42:29,120 --> 00:42:30,560 So there are no wiggles. 709 00:42:30,560 --> 00:42:34,260 It goes straight down, like this. 710 00:42:34,260 --> 00:42:41,590 And then the other two pieces are f'' is equal to, well 711 00:42:41,590 --> 00:42:43,260 it's going to switch here. 712 00:42:43,260 --> 00:42:44,856 The denominator becomes positive. 713 00:42:44,856 --> 00:42:48,190 So it's positive over positive. 714 00:42:48,190 --> 00:42:56,670 So this is concave up. 715 00:42:56,670 --> 00:42:58,410 And that's going over here. 716 00:42:58,410 --> 00:43:02,715 But notice that it's not the bottom where it turns around, 717 00:43:02,715 --> 00:43:07,740 it's somewhere else. 718 00:43:07,740 --> 00:43:09,930 So there's another transition here. 719 00:43:09,930 --> 00:43:12,090 This is e^2. 720 00:43:12,090 --> 00:43:15,090 This is e. 721 00:43:15,090 --> 00:43:20,440 So what happens at the end is, again, the sign flips again. 722 00:43:20,440 --> 00:43:26,580 Because the numerator, now, when x > e^2, becomes negative. 723 00:43:26,580 --> 00:43:29,965 And this is negative divided by positive, which is negative. 724 00:43:29,965 --> 00:43:35,630 And this is concave down. 725 00:43:35,630 --> 00:43:39,100 And so we didn't quite draw the graph right. 726 00:43:39,100 --> 00:43:40,990 There's an inflection point right here, 727 00:43:40,990 --> 00:43:45,670 which I'll label with an I. And it makes a turn 728 00:43:45,670 --> 00:43:47,340 the other way at that point. 729 00:43:47,340 --> 00:43:49,480 So there was a wiggle. 730 00:43:49,480 --> 00:43:51,210 There's the wiggle. 731 00:43:51,210 --> 00:43:53,710 Still going up, still going to infinity. 732 00:43:53,710 --> 00:43:57,040 But kind of the slope of a mountain, right? 733 00:43:57,040 --> 00:44:01,240 It's going the other way. 734 00:44:01,240 --> 00:44:03,510 This point happens to be (e^2, e^2 / 2). 735 00:44:09,350 --> 00:44:11,980 So that's as detailed as we'll ever get. 736 00:44:11,980 --> 00:44:14,760 And indeed, the next game is going 737 00:44:14,760 --> 00:44:18,730 to be avoid being-- is to avoid being this detailed. 738 00:44:18,730 --> 00:44:21,760 So let me introduce the next subject. 739 00:44:21,760 --> 00:44:48,310 Which is maxima and minima. 740 00:44:48,310 --> 00:45:04,160 OK, now, maxima and minima, maximum and minimum problems 741 00:45:04,160 --> 00:45:06,550 can be described graphically in the following ways. 742 00:45:06,550 --> 00:45:13,150 Suppose you have a function, right, here it is. 743 00:45:13,150 --> 00:45:14,420 OK? 744 00:45:14,420 --> 00:45:24,630 Now, find the maximum. 745 00:45:24,630 --> 00:45:30,740 And find the minimum. 746 00:45:30,740 --> 00:45:31,410 OK. 747 00:45:31,410 --> 00:45:38,880 So this problem is done. 748 00:45:38,880 --> 00:45:51,625 The point being, that it is easy to find max and the min 749 00:45:51,625 --> 00:45:58,670 with the sketch. 750 00:45:58,670 --> 00:46:00,330 It's very easy. 751 00:46:00,330 --> 00:46:05,130 The goal, the problem, is that the sketch is a lot of work. 752 00:46:05,130 --> 00:46:10,180 We just spent 20 minutes sketching something. 753 00:46:10,180 --> 00:46:12,637 We would not like to spend all that time 754 00:46:12,637 --> 00:46:14,970 every single time we want to find a maximum and minimum. 755 00:46:14,970 --> 00:46:19,576 So the goal is to do it with-- so our goal is 756 00:46:19,576 --> 00:46:25,170 to use shortcuts. 757 00:46:25,170 --> 00:46:30,650 And, indeed, as I said earlier, we certainly 758 00:46:30,650 --> 00:46:33,840 never want to use the second derivative if we can avoid it. 759 00:46:33,840 --> 00:46:36,450 And we don't want to decorate the graph 760 00:46:36,450 --> 00:46:38,550 and do all of these elaborate, subtle, 761 00:46:38,550 --> 00:46:40,900 things which make the graph look nicer and really, 762 00:46:40,900 --> 00:46:42,640 or aesthetically appropriate. 763 00:46:42,640 --> 00:46:46,700 But are totally unnecessary to see whether the graph is 764 00:46:46,700 --> 00:46:54,290 up or down. 765 00:46:54,290 --> 00:46:57,030 So essentially, this whole business 766 00:46:57,030 --> 00:47:00,030 is out, which is a good thing. 767 00:47:00,030 --> 00:47:03,640 And, unfortunately, those early parts 768 00:47:03,640 --> 00:47:06,170 are the parts that people tend to ignore. 769 00:47:06,170 --> 00:47:10,150 Which are typically, often, very important. 770 00:47:10,150 --> 00:47:22,940 So let me first tell you the main point here. 771 00:47:22,940 --> 00:47:32,760 So the key idea. 772 00:47:32,760 --> 00:47:39,650 Key to finding maximum. 773 00:47:39,650 --> 00:47:41,850 So the key point is, we only need 774 00:47:41,850 --> 00:48:00,130 to look at critical points. 775 00:48:00,130 --> 00:48:01,980 Well, that's actually what it seems 776 00:48:01,980 --> 00:48:05,490 like in many calculus classes. 777 00:48:05,490 --> 00:48:06,810 But that's not true. 778 00:48:06,810 --> 00:48:15,590 This is not the end of the sentence. 779 00:48:15,590 --> 00:48:35,320 And, end points, and points of discontinuity. 780 00:48:35,320 --> 00:48:37,980 So you must watch out for those. 781 00:48:37,980 --> 00:48:41,580 If you look at the example that I just drew here, 782 00:48:41,580 --> 00:48:43,680 which is the one that I carried out, 783 00:48:43,680 --> 00:48:50,120 you can see that there are actually five extreme points 784 00:48:50,120 --> 00:48:51,250 on this picture. 785 00:48:51,250 --> 00:48:52,750 So let's switch. 786 00:48:52,750 --> 00:48:58,050 So we'll take a look. 787 00:48:58,050 --> 00:49:04,840 There are five places where the max or the min might be. 788 00:49:04,840 --> 00:49:08,050 There's this important point. 789 00:49:08,050 --> 00:49:10,710 This is, as I say, the scaffolding of the function. 790 00:49:10,710 --> 00:49:13,650 There's this point, there down at minus infinity. 791 00:49:13,650 --> 00:49:18,370 There's this, there's this, and there's this. 792 00:49:18,370 --> 00:49:23,620 Only one out of five is a critical point. 793 00:49:23,620 --> 00:49:25,784 So there's more that you have to pay attention to 794 00:49:25,784 --> 00:49:26,450 on the function. 795 00:49:26,450 --> 00:49:28,960 And you always have to keep the schema, 796 00:49:28,960 --> 00:49:31,310 the picture of the function, in the back of your head. 797 00:49:31,310 --> 00:49:33,760 Even though this may be the most interesting point, 798 00:49:33,760 --> 00:49:36,520 and the one that you're going to be looking at. 799 00:49:36,520 --> 00:49:41,180 So we'll do a few examples of that next time.