1 00:00:01,230 --> 00:00:06,742 In this video, we're going to be looking at how curves behave and 2 00:00:06,742 --> 00:00:09,286 how to find specific points on 3 00:00:09,286 --> 00:00:16,550 those curves. Let's begin by looking at a sketch of a curve. 4 00:00:21,230 --> 00:00:26,906 Now we've got a curve. There's some noticeable things about it. 5 00:00:27,460 --> 00:00:29,048 First of all here. 6 00:00:30,860 --> 00:00:38,290 The curve turns if we draw tangent to the curve. 7 00:00:38,840 --> 00:00:44,972 It's level parallel to the X axis. Here I've exaggerated it a 8 00:00:44,972 --> 00:00:50,593 little bit. The curve is flat and the tangent is again 9 00:00:50,593 --> 00:00:53,148 parallel to the X axis. 10 00:00:53,720 --> 00:01:01,700 Here the curve turns and again a tangent is parallel 11 00:01:01,700 --> 00:01:04,892 to the X axis. 12 00:01:05,780 --> 00:01:12,150 Now, one of the things that we do know is that the gradient of 13 00:01:12,150 --> 00:01:18,975 a curve is given by DY by The X. What's clear is that each of 14 00:01:18,975 --> 00:01:20,340 these points here. 15 00:01:21,310 --> 00:01:23,428 Here and here. 16 00:01:24,030 --> 00:01:29,761 Because the tangent is flat parallel to the X axis, the 17 00:01:29,761 --> 00:01:35,492 gradient is zero and so D why by DX is 0. 18 00:01:36,360 --> 00:01:43,080 Now for points like these where DY by the X is zero, we have 19 00:01:43,080 --> 00:01:45,960 a special name, we call them. 20 00:01:46,920 --> 00:01:50,015 Stationary 21 00:01:50,015 --> 00:01:56,407 points. If you like the curve, sort of 22 00:01:56,407 --> 00:02:02,059 isn't moving, it isn't changing. At that point. The why by DX, 23 00:02:02,059 --> 00:02:05,356 which is the rate of change, is 24 00:02:05,356 --> 00:02:11,920 0. Now two of these I discribed the curve as turning. I said 25 00:02:11,920 --> 00:02:17,080 there the curb was turning and here again the curve is turning 26 00:02:17,080 --> 00:02:22,670 so that's going from coming up to going down and this one is 27 00:02:22,670 --> 00:02:28,260 from going down to coming up. This one isn't this one is very 28 00:02:28,260 --> 00:02:29,980 different so these two. 29 00:02:30,910 --> 00:02:33,740 Are special types of stationary 30 00:02:33,740 --> 00:02:36,670 points. These two. 31 00:02:37,450 --> 00:02:41,056 Are cold turning 32 00:02:41,056 --> 00:02:47,530 points. So at turning points divided 33 00:02:47,530 --> 00:02:50,692 by DX is 34 00:02:50,692 --> 00:02:57,932 0. Thing to realize though, is that if the wide by the X is 35 00:02:57,932 --> 00:03:02,486 zero, that doesn't mean we've got a turning point. We might 36 00:03:02,486 --> 00:03:07,454 have one of these, that's what makes life just a little bit 37 00:03:07,454 --> 00:03:11,180 tricky, but nevertheless thing to remember is that turning 38 00:03:11,180 --> 00:03:16,562 points divided by DX is 0. Now let's have a look at the 39 00:03:16,562 --> 00:03:19,046 nature of this particular turning point. 40 00:03:20,590 --> 00:03:23,836 In the immediate area of it. 41 00:03:24,980 --> 00:03:30,846 The curve has no bigger value. We go up, we reach a value and 42 00:03:30,846 --> 00:03:33,779 then we go down again. So this 43 00:03:33,779 --> 00:03:40,400 point. Is called a local maximum. 44 00:03:40,960 --> 00:03:43,688 Because in the immediate 45 00:03:43,688 --> 00:03:46,688 vicinity. There is no bigger 46 00:03:46,688 --> 00:03:53,178 value. Now usually we forget about the local and we just call 47 00:03:53,178 --> 00:03:57,958 it a maximum. Similarly, this point is called a local. 48 00:04:00,740 --> 00:04:06,695 Minimum. Because again, in this particular region of it, the 49 00:04:06,695 --> 00:04:13,905 curve goes no lower, so this is the least value in this area are 50 00:04:13,905 --> 00:04:19,934 local minimum. Again, we forget the word local quite often. 51 00:04:19,934 --> 00:04:26,210 Unjust use minimum. In essence, this video is going to be about 52 00:04:26,210 --> 00:04:27,779 finding these points. 53 00:04:28,400 --> 00:04:34,640 Maxima and minima finding these two points Maxima and minima. 54 00:04:35,180 --> 00:04:38,284 And distinguishing between them because being able to 55 00:04:38,284 --> 00:04:43,328 find them isn't a great deal of use unless we can tell them 56 00:04:43,328 --> 00:04:47,596 apart. So we need to be able to distinguish between a 57 00:04:47,596 --> 00:04:51,476 maximum and the minimum. So let's begin by exploring what 58 00:04:51,476 --> 00:04:53,416 happens around a minimum point. 59 00:04:54,670 --> 00:04:56,440 Draw a minimum. 60 00:04:57,410 --> 00:05:00,618 Let's take the Tangent. 61 00:05:02,090 --> 00:05:07,161 At the minimum point, we know that the gradient of this 62 00:05:07,161 --> 00:05:13,154 tangent is 0 because it's parallel to the X axis. So DY by 63 00:05:13,154 --> 00:05:14,537 DX is 0. 64 00:05:15,170 --> 00:05:19,618 Let's take a point a little bit before. 65 00:05:21,670 --> 00:05:26,470 The minimum point and take a tangent at that point. And here 66 00:05:26,470 --> 00:05:28,470 we see that the Tangent. 67 00:05:30,250 --> 00:05:36,886 Is negative, it's got a negative slope and so D. Why by DX is 68 00:05:36,886 --> 00:05:41,626 negative. I'm using this as a shorthand for negative, not 69 00:05:41,626 --> 00:05:47,314 actually bothered about the value of the why by DX just it's 70 00:05:47,314 --> 00:05:52,054 sign. Let's take a point here a little bit after. 71 00:05:52,890 --> 00:05:57,858 What's happened there? Well, we can see we've got a line that's 72 00:05:57,858 --> 00:06:03,240 got a positive slope to it, which means that the why by the 73 00:06:03,240 --> 00:06:05,724 X has got to be positive. 74 00:06:06,700 --> 00:06:13,190 So what's happening as we pass through this point as X 75 00:06:13,190 --> 00:06:17,910 increases? What's happening to do, why by DX? 76 00:06:18,760 --> 00:06:24,326 Well, as X increases, the why by DX goes from negative. 77 00:06:25,480 --> 00:06:28,460 0. To positive. 78 00:06:29,620 --> 00:06:36,473 So divide by the ex is increasing as X increases, so 79 00:06:36,473 --> 00:06:42,703 let's write that down deep. Why by DX is increasing. 80 00:06:46,270 --> 00:06:48,820 As X. 81 00:06:51,300 --> 00:06:58,308 Increases. What does that mean? DY by DX is 82 00:06:58,308 --> 00:07:02,637 increasing well that means that the rate of change. 83 00:07:03,150 --> 00:07:05,660 Of the why by DX? 84 00:07:06,880 --> 00:07:11,910 Is positive. 85 00:07:13,670 --> 00:07:21,302 IE D2Y by DX squared is 86 00:07:21,302 --> 00:07:25,118 greater than 0. 87 00:07:27,470 --> 00:07:32,750 So if D2Y by the X squared is greater than 0. 88 00:07:33,550 --> 00:07:39,892 And divide by the X is equal to 0. We know that we have 89 00:07:39,892 --> 00:07:41,704 got a minimum point. 90 00:07:43,130 --> 00:07:47,267 Is that true for all minimum points? 91 00:07:48,500 --> 00:07:54,044 Little trick of logic here, but just write down what we've got 92 00:07:54,044 --> 00:07:57,706 if. DY by DX 93 00:07:57,706 --> 00:08:01,000 equals 0. And. 94 00:08:02,390 --> 00:08:08,011 D2Y by DX squared is greater than 95 00:08:08,011 --> 00:08:10,420 or positive then. 96 00:08:11,820 --> 00:08:14,880 We have 97 00:08:14,880 --> 00:08:19,520 a. Minimum. 98 00:08:20,650 --> 00:08:23,974 Does it work the other way around? If we have a minimum? 99 00:08:24,950 --> 00:08:29,474 Do these two both apply? Well, certainly that one does that if 100 00:08:29,474 --> 00:08:34,752 we have a minimum DY by the X will be 0, but not necessarily 101 00:08:34,752 --> 00:08:41,842 this one. Let's have a look at the curve Y equals X to 102 00:08:41,842 --> 00:08:45,610 the 4th. If I sketch this curve. 103 00:08:47,030 --> 00:08:47,920 It looks. 104 00:08:51,010 --> 00:08:51,800 Like that? 105 00:08:53,830 --> 00:08:58,906 Here we've clearly got our turning point. Our minimum 106 00:08:58,906 --> 00:09:01,775 point. Let's have a look what's 107 00:09:01,775 --> 00:09:09,349 happening. DY by the X equals well. Why was X to the 4th so 108 00:09:09,349 --> 00:09:15,758 DY by DX is 4X cubed, and we know that for our stationary 109 00:09:15,758 --> 00:09:22,660 points we put it equal to 0, which tells us X equals 0 and 110 00:09:22,660 --> 00:09:29,562 X equals 0 tells us that Y equals 0, so we know that this 111 00:09:29,562 --> 00:09:31,041 is a stationary. 112 00:09:32,130 --> 00:09:33,110 Point 113 00:09:34,190 --> 00:09:39,536 What kind of stationary point is it? Let's do our second 114 00:09:39,536 --> 00:09:46,780 derivative. D2 why by DX squared is 12 X squared, but 115 00:09:46,780 --> 00:09:48,604 it equals 0. 116 00:09:49,170 --> 00:09:52,478 When X equals 0. 117 00:09:54,190 --> 00:09:56,788 So this doesn't work in reverse. 118 00:09:57,480 --> 00:10:02,744 If the why by DX is zero and Y2Y by the X squared is greater than 119 00:10:02,744 --> 00:10:05,047 not, then we know we have a 120 00:10:05,047 --> 00:10:09,620 minimum. But it doesn't work going the other way. What that 121 00:10:09,620 --> 00:10:13,635 means is that we have to be especially careful whenever D2, 122 00:10:13,635 --> 00:10:18,745 why? By DX squared comes out to be 0 when we're doing the test, 123 00:10:18,745 --> 00:10:19,840 so to speak. 124 00:10:20,610 --> 00:10:24,419 We need to look at it in a slightly different way. In fact, 125 00:10:24,419 --> 00:10:28,814 what we need to do is go back to basics and I'll show you an 126 00:10:28,814 --> 00:10:32,330 example later where we do go back to basics and how that 127 00:10:32,330 --> 00:10:38,122 works out. So having had a look at a minimum point and looked at 128 00:10:38,122 --> 00:10:42,815 it in some detail, let's now move on and look at a maximum 129 00:10:42,815 --> 00:10:44,620 point and see what happens 130 00:10:44,620 --> 00:10:48,480 there. So here's our maximum point. 131 00:10:49,940 --> 00:10:55,010 And at the point itself, we know that the tangent is parallel to 132 00:10:55,010 --> 00:10:57,740 the X axis and so its gradient. 133 00:10:58,280 --> 00:11:01,296 And hence the why by DX is 0. 134 00:11:02,350 --> 00:11:06,430 Let's take a point a little bit before that there. 135 00:11:07,670 --> 00:11:12,734 We can see that we've got a line which has a positive 136 00:11:12,734 --> 00:11:17,798 slope to it, and so DY by the X is positive there. 137 00:11:19,330 --> 00:11:23,826 Now let's take somewhere just past the point. 138 00:11:24,340 --> 00:11:27,148 And if we have a look at this week and see we've 139 00:11:27,148 --> 00:11:27,850 got a tangent. 140 00:11:29,860 --> 00:11:36,464 Where the slope is negative and so divide by the X is negative. 141 00:11:37,630 --> 00:11:43,114 So what's happening as we pass through this maximum point as X 142 00:11:43,114 --> 00:11:46,770 is increasing? What's happening today, why by DX? 143 00:11:47,990 --> 00:11:55,107 Well, divide by the axe goes from being positive to being 144 00:11:55,107 --> 00:12:02,224 zero to being negative. So as X increases, divided by DX 145 00:12:02,224 --> 00:12:06,198 decreases. As X increases. 146 00:12:07,480 --> 00:12:11,854 Why by 147 00:12:11,854 --> 00:12:14,041 DX 148 00:12:14,041 --> 00:12:21,420 decreases? Does that mean it must mean that the rate of 149 00:12:21,420 --> 00:12:24,534 change of the why by DX? 150 00:12:25,080 --> 00:12:28,500 Is. 151 00:12:30,130 --> 00:12:36,022 Negative. As X increases as we go through this point at this 152 00:12:36,022 --> 00:12:41,931 point. An the derivative divide by the X is negative. 153 00:12:41,931 --> 00:12:47,521 In other words, D2Y by the X squared is negative. 154 00:12:49,230 --> 00:12:53,523 So let's just write down again what we've got. 155 00:12:55,000 --> 00:13:01,150 If. Divide by DX equals 0. 156 00:13:02,150 --> 00:13:07,680 And. D2 why by DX squared. 157 00:13:08,330 --> 00:13:11,490 Is less than 0. 158 00:13:13,820 --> 00:13:20,828 Then we have a maximum point. 159 00:13:20,830 --> 00:13:24,724 Again, just the same argument applies. Does it work back the 160 00:13:24,724 --> 00:13:29,680 same way? Answer no, it doesn't. And the problem is what do we do 161 00:13:29,680 --> 00:13:33,928 about D2? Why? By DX squared being zero? Well, as I said, 162 00:13:33,928 --> 00:13:38,884 will show a way of dealing with that in a moment. It's time now 163 00:13:38,884 --> 00:13:43,840 to actually look at an example or two and see if we can make 164 00:13:43,840 --> 00:13:45,256 use of this test. 165 00:13:45,810 --> 00:13:52,818 In order to sort out the turning points on a curve. So 166 00:13:52,818 --> 00:13:59,826 the first example we're going to have a look at is Y 167 00:13:59,826 --> 00:14:03,330 equals X cubed, minus three X 168 00:14:03,330 --> 00:14:09,417 +2. The first thing we need to do is differentiated. 169 00:14:09,417 --> 00:14:15,060 Why by DX equals if we differentiate this we have three 170 00:14:15,060 --> 00:14:21,216 X squared. We multiply by the index and subtract 1 prom it 171 00:14:21,216 --> 00:14:26,859 minus the derivative of the three axis, just three, and this 172 00:14:26,859 --> 00:14:29,937 is equal to 0 for stationary. 173 00:14:31,320 --> 00:14:37,366 Points. We can factorize this. There's a common 174 00:14:37,366 --> 00:14:43,840 factor of three leaves us with X squared minus one, and this is 175 00:14:43,840 --> 00:14:48,820 now recognizable as the difference of two squares, so it 176 00:14:48,820 --> 00:14:53,302 factorizes us three times, X minus 1X plus one. 177 00:14:53,330 --> 00:14:54,840 And that's equal to 0. 178 00:14:55,430 --> 00:14:57,488 Now what we can see is that. 179 00:14:58,020 --> 00:15:04,642 Either this bracket is 0 or this bracket is 0, or they're both 0, 180 00:15:04,642 --> 00:15:09,845 so we have X equals 1 or X equals minus one. 181 00:15:10,910 --> 00:15:16,084 Now this is just given us the X coordinates of the points that 182 00:15:16,084 --> 00:15:21,258 were interested in what we need now or the Y coordinates, and so 183 00:15:21,258 --> 00:15:26,034 let's work those out, X equals 1, Y equals and will substitute 184 00:15:26,034 --> 00:15:32,402 it in on the very first line. So that's one cubed is just 1 - 3 185 00:15:32,402 --> 00:15:35,586 times by one is minus 3 + 2. 186 00:15:35,640 --> 00:15:38,370 Altogether, that gives us 0. 187 00:15:39,150 --> 00:15:46,506 And so our point, our stationary point is 1 zero. If we 188 00:15:46,506 --> 00:15:52,636 take X equals minus one and Y is equal to. 189 00:15:53,200 --> 00:15:59,427 And we put minus one in there well minus one cubed is just 190 00:15:59,427 --> 00:16:06,612 minus 1 - 3 times by minus one is plus three and plus two at 191 00:16:06,612 --> 00:16:14,276 the end, and altogether 3 + 2 is 5 takeaway, one is 4 and so our 192 00:16:14,276 --> 00:16:16,671 other stationary point is minus 193 00:16:16,671 --> 00:16:19,980 1 four. Now we need to decide. 194 00:16:20,640 --> 00:16:24,924 All these points, Maxima, minima or that funny one where it was 195 00:16:24,924 --> 00:16:30,279 flat. So in order to sort that out, what we need to have a look 196 00:16:30,279 --> 00:16:32,064 at is the 2nd derivative. 197 00:16:32,950 --> 00:16:37,262 So I'm going to write the equation of the curve down 198 00:16:37,262 --> 00:16:41,182 again, and then I'm going to write down this first 199 00:16:41,182 --> 00:16:44,710 derivative, then the points, and then the second derivative. 200 00:16:45,800 --> 00:16:53,490 So we've Y equals X cubed, minus three X +2. 201 00:16:53,490 --> 00:17:01,180 We've got the first derivative D. Why by DX3 X 202 00:17:01,180 --> 00:17:08,870 squared minus three and the points that were interested in 203 00:17:08,870 --> 00:17:15,791 with the points one not and minus 1 four. 204 00:17:15,830 --> 00:17:21,989 So now. Let's take the second derivative D2Y by DX squared 205 00:17:21,989 --> 00:17:26,408 equals. So we differentiate this. The derivative of the 206 00:17:26,408 --> 00:17:32,300 first term we multiply by the index. So two, freezer 6 and 207 00:17:32,300 --> 00:17:38,683 take one of the index that's X and the derivative of three is 208 00:17:38,683 --> 00:17:44,712 just zero. So let's now look at the points and. It's the values 209 00:17:44,712 --> 00:17:47,456 of X we're interested in. So X 210 00:17:47,456 --> 00:17:54,990 equals 1. D2Y by DX squared equals 6. 211 00:17:56,360 --> 00:17:59,390 So this is 212 00:17:59,390 --> 00:18:06,690 positive. What do we know? We know that if divided by 213 00:18:06,690 --> 00:18:13,762 DX is zero and if D2, why by DX squared is positive, then 214 00:18:13,762 --> 00:18:19,746 the point we're looking at is a minimum turning point, so 215 00:18:19,746 --> 00:18:23,010 therefore 10 is a minimum mean 216 00:18:23,010 --> 00:18:30,120 for short. X equals minus one. This is our second point. 217 00:18:30,620 --> 00:18:36,100 D2Y by the X squared is equal to minus 6. 218 00:18:37,930 --> 00:18:41,998 And that is negative. The value of these doesn't 219 00:18:41,998 --> 00:18:45,614 matter. It's their sign that matters and therefore, 220 00:18:45,614 --> 00:18:51,038 what do we know? We know that if the Wi-Fi DX is 221 00:18:51,038 --> 00:18:56,010 zero and Y2Y by DX squared is negative, then we have 222 00:18:56,010 --> 00:19:00,530 got a maximum, so minus 1 four is a Max. 223 00:19:01,660 --> 00:19:06,820 One of the things that this does help us do is when we do know 224 00:19:06,820 --> 00:19:11,292 the turning points. We can use this to help us gain a picture 225 00:19:11,292 --> 00:19:15,764 of the graph to help us sketch a picture. So let's just plot 226 00:19:15,764 --> 00:19:16,796 these two points. 227 00:19:19,120 --> 00:19:24,230 And see if by plotting them we can build up a picture of the 228 00:19:24,230 --> 00:19:26,420 graph. Now this won't be an 229 00:19:26,420 --> 00:19:30,900 exact picture. The scales will be different. They'll be a 230 00:19:30,900 --> 00:19:34,410 little bit cramped in some cases, expanded in others 231 00:19:34,410 --> 00:19:38,700 because what we're trying to show the important points on the 232 00:19:38,700 --> 00:19:42,990 graph, so we've one and minus one. Where are two important 233 00:19:42,990 --> 00:19:45,720 values of X, and we know that 234 00:19:45,720 --> 00:19:52,128 one 0. Was one of our turning points and if we just go back, 235 00:19:52,128 --> 00:19:57,216 we know that one zero was a minimum turning point, so we 236 00:19:57,216 --> 00:20:01,456 know that at this point the curve looks like that. 237 00:20:02,410 --> 00:20:08,034 The other point was minus 1 four there. 238 00:20:09,030 --> 00:20:15,862 And we know that at that point the curve was like that. It was 239 00:20:15,862 --> 00:20:22,230 a maximum. The other thing we know is that there are no 240 00:20:22,230 --> 00:20:27,092 turning points anywhere else. These are the only ones, and so 241 00:20:27,092 --> 00:20:33,280 effectively we can join up the curve and so that we get a rough 242 00:20:33,280 --> 00:20:38,142 picture that it looks something like that. We can find this 243 00:20:38,142 --> 00:20:43,004 point of course by substituting X equals 0. Remember, our curve 244 00:20:43,004 --> 00:20:47,866 was Y equals X cubed minus 3X, plus 2X is 0. 245 00:20:47,900 --> 00:20:50,318 Then this must be 2 here. 246 00:20:50,840 --> 00:20:55,388 This we can find bike waiting why to zero and solving but 247 00:20:55,388 --> 00:20:59,936 we're not interested in that at the moment. It's the idea of 248 00:20:59,936 --> 00:21:04,105 using this maximum and minimum turning points in order to help 249 00:21:04,105 --> 00:21:07,895 us gain a picture of the shape of the curve. 250 00:21:08,990 --> 00:21:15,374 We're going to have a look at an example now where the method we 251 00:21:15,374 --> 00:21:21,758 use is one that enables us to go back to 1st principles, and this 252 00:21:21,758 --> 00:21:28,142 is also the method that we would use if D2. Why? By DX squared 253 00:21:28,142 --> 00:21:30,422 turned out to be 0. 254 00:21:30,960 --> 00:21:34,040 So it's a very useful method to have at your fingertips. 255 00:21:34,630 --> 00:21:35,938 So let's begin. 256 00:21:36,720 --> 00:21:43,683 We're going to be looking at finding the turning points of 257 00:21:43,683 --> 00:21:45,582 this particular curve. 258 00:21:45,590 --> 00:21:49,040 X minus one all squared over 259 00:21:49,040 --> 00:21:56,792 X. Well it's a U over a V. It's a quotient, so we need 260 00:21:56,792 --> 00:22:02,600 to be able to differentiate as though it were a quotient. So 261 00:22:02,600 --> 00:22:08,408 let's remember how we do that. We take the bottom the X. 262 00:22:09,020 --> 00:22:14,660 And we multiply X by the derivative of U what's on top? 263 00:22:14,660 --> 00:22:17,950 So that's two times X minus one. 264 00:22:18,500 --> 00:22:23,135 Times the derivative of what's inside a one and 265 00:22:23,135 --> 00:22:29,315 then minus. And we take UX minus one 4 squared and we 266 00:22:29,315 --> 00:22:33,950 multiply it by the derivative of E. What's on 267 00:22:33,950 --> 00:22:38,585 the bottom? And that's the derivative of X. That's 268 00:22:38,585 --> 00:22:43,220 just one, and then it's all over X squared. 269 00:22:44,390 --> 00:22:50,132 And for our stationary points we put this equal to 0. 270 00:22:51,560 --> 00:22:52,230 Now. 271 00:22:53,920 --> 00:22:58,795 Let's have a look at this. We need to tidy it up. It looks a 272 00:22:58,795 --> 00:23:03,020 bit of a mess, so let's have a look for some common factors. 273 00:23:03,020 --> 00:23:07,570 Well, here we've got an X minus one and here as well. So let's 274 00:23:07,570 --> 00:23:12,120 take that X minus one out as a factor. Let's have a look what 275 00:23:12,120 --> 00:23:16,345 we're left with. Well, here we've got two and X giving us 2X 276 00:23:16,345 --> 00:23:18,295 and one times by one. That's 277 00:23:18,295 --> 00:23:23,816 still 2X. Minus and here we've got one times by now we just 278 00:23:23,816 --> 00:23:28,834 took out X minus one as a factor, so it's minus X minus 279 00:23:28,834 --> 00:23:33,466 one and notice I've kept it in the bracket because it's the 280 00:23:33,466 --> 00:23:36,168 whole of X minus one that I'm 281 00:23:36,168 --> 00:23:43,350 taking away. Close that bracket all over X squared so this 282 00:23:43,350 --> 00:23:46,600 is still DY by DX. 283 00:23:47,530 --> 00:23:54,535 Now I can simplify what's in the bracket and have DY by the X is 284 00:23:54,535 --> 00:24:01,073 equal to X minus one times by now have two X takeaway X that's 285 00:24:01,073 --> 00:24:08,078 just an X, but then I've also to take away minus one and so that 286 00:24:08,078 --> 00:24:09,946 gives me plus one. 287 00:24:10,010 --> 00:24:15,758 All over X squared and this is to be equal to 0 288 00:24:15,758 --> 00:24:17,674 for my stationary points. 289 00:24:18,730 --> 00:24:23,662 Now when we have an expression like that that's equal to 0, 290 00:24:23,662 --> 00:24:27,772 it's only the numerator that we're interested in. It's only 291 00:24:27,772 --> 00:24:31,060 the numerator that will make this expression 0. 292 00:24:31,700 --> 00:24:37,844 So got is X minus 1X 293 00:24:37,844 --> 00:24:40,916 plus one equals 294 00:24:40,916 --> 00:24:47,030 0. And that tells us that either this bracket is equal 295 00:24:47,030 --> 00:24:53,204 to 0 or that bracket is equal to 0. In other words, X minus 296 00:24:53,204 --> 00:24:59,378 one equals 0 or X Plus one equals 0. So X equals 1. Four 297 00:24:59,378 --> 00:25:01,142 X equals minus one. 298 00:25:02,310 --> 00:25:08,327 Now it's the points where after so let's now calculate the 299 00:25:08,327 --> 00:25:15,438 values of Y, remembering that Y is equal to X minus one 4 300 00:25:15,438 --> 00:25:17,626 squared all over X. 301 00:25:18,690 --> 00:25:24,710 So let's take X equals 1 and see what that gives us for why? 302 00:25:25,420 --> 00:25:31,885 We put one in there. We have 1 - 1 all squared over one. Well, 303 00:25:31,885 --> 00:25:38,781 that's just zero because 1 - 1 is 0 square at that still 0 / 1. 304 00:25:38,781 --> 00:25:44,815 The answer is still 0. So one of our points is one Nord. That's 305 00:25:44,815 --> 00:25:49,987 one of our stationary points. Now let's do X equals minus one. 306 00:25:50,590 --> 00:25:58,446 Why is equal to minus 1 - 1 all squared all over minus 1 - 1 307 00:25:58,446 --> 00:26:05,811 - 1 is minus 2 and we square it to that's four but we divide 308 00:26:05,811 --> 00:26:12,685 by minus one, so that is minus four. So other point is minus 1 309 00:26:12,685 --> 00:26:18,194 - 4. So we've got two points which are stationary 310 00:26:18,194 --> 00:26:24,408 points and we need to find out are they turning points and if 311 00:26:24,408 --> 00:26:28,710 they are turning points, are they Maxima or minima? 312 00:26:29,560 --> 00:26:34,636 So let's write down again. Why do you? Why 313 00:26:34,636 --> 00:26:37,456 by DX and our points. 314 00:26:38,680 --> 00:26:45,780 So we had why that was X minus one or 315 00:26:45,780 --> 00:26:47,910 squared over X. 316 00:26:48,500 --> 00:26:55,611 We had DY by the X&YYY DX was over X squared and let's 317 00:26:55,611 --> 00:26:59,440 just go back so we can see 318 00:26:59,440 --> 00:27:06,660 what. The numerator was it was X minus one times by X plus one. 319 00:27:07,370 --> 00:27:14,240 X minus one times by X Plus One and our 320 00:27:14,240 --> 00:27:21,250 two points. Are the points one note and minus 321 00:27:21,250 --> 00:27:23,440 1 - 4? 322 00:27:24,410 --> 00:27:25,590 Now. 323 00:27:26,640 --> 00:27:31,619 In order to sort out if these are turning points and if they 324 00:27:31,619 --> 00:27:36,598 are, which is a maximum and which is a minimum, then we need 325 00:27:36,598 --> 00:27:40,428 to differentiate this again and that is very very fearsome. 326 00:27:41,420 --> 00:27:42,920 Very fearsome indeed. 327 00:27:43,790 --> 00:27:45,980 Is it really worth differentiating 328 00:27:45,980 --> 00:27:49,046 differentiating it again? We might make a 329 00:27:49,046 --> 00:27:51,674 mistake. The algebra might go wrong. 330 00:27:52,740 --> 00:27:57,833 So rather than differentiating it again, let's do what we did 331 00:27:57,833 --> 00:28:02,926 before. Let's look at the gradient a little bit before X 332 00:28:02,926 --> 00:28:09,871 equals 1. At X equals 1 and a little bit after X equals 1 and 333 00:28:09,871 --> 00:28:15,890 form a picture from doing that of whether this is a maximum or 334 00:28:15,890 --> 00:28:21,446 minimum point, and this method that we're going to have a look 335 00:28:21,446 --> 00:28:22,835 at also works. 336 00:28:22,890 --> 00:28:27,776 Wendy to why? By DX squared is 0, so hence it's a good method 337 00:28:27,776 --> 00:28:31,964 and it's well worth adding it to your Armory of techniques for 338 00:28:31,964 --> 00:28:36,152 looking at stationary points. So let's set this up and the way 339 00:28:36,152 --> 00:28:38,944 we're going to set it up is too. 340 00:28:39,890 --> 00:28:43,238 Have a little table. 341 00:28:43,240 --> 00:28:47,025 So here's the Y by 342 00:28:47,025 --> 00:28:53,580 The X. And here is the point where interested in at 343 00:28:53,580 --> 00:28:54,990 X equals 1. 344 00:28:56,150 --> 00:29:02,562 I want to look a little bit before X equals 1, so I'm going 345 00:29:02,562 --> 00:29:08,974 to say this is X equals 1 minus Epsilon and all. I mean by 346 00:29:08,974 --> 00:29:11,722 Ipsilon is a little positive bit 347 00:29:11,722 --> 00:29:17,007 of X. North Point One North Point North one if you like, and 348 00:29:17,007 --> 00:29:19,226 I'm also going to look at the 349 00:29:19,226 --> 00:29:25,714 gradient. A little bit after X equals 1, so let me call that 350 00:29:25,714 --> 00:29:27,115 one plus Epsilon. 351 00:29:28,470 --> 00:29:35,036 Now I'm not interested in the value of the why by DX. I'm only 352 00:29:35,036 --> 00:29:39,726 interested in its sign whether it's positive or negative. I 353 00:29:39,726 --> 00:29:44,885 know that when X equals 1 divided by DX is 0. 354 00:29:46,290 --> 00:29:53,970 What happens if I am a little bit before X equals 1? 355 00:29:55,200 --> 00:30:00,645 X is a little bit less than one, so I have something a little bit 356 00:30:00,645 --> 00:30:02,823 less than one, and I'm taking 357 00:30:02,823 --> 00:30:08,710 one away. So the answer to that is negative. This bracket here 358 00:30:08,710 --> 00:30:13,178 is negative. However, I've got something that's a little bit 359 00:30:13,178 --> 00:30:15,374 less than one plus one that's 360 00:30:15,374 --> 00:30:19,756 positive. So I have a negative times by a 361 00:30:19,756 --> 00:30:23,976 positive and I've got X squared on the bottom here 362 00:30:23,976 --> 00:30:28,196 and X squared is positive, so I'm negative times a 363 00:30:28,196 --> 00:30:31,572 positive divided by a positive is a negative. 364 00:30:33,030 --> 00:30:38,084 Let's have a look when X is a little bit more than one little 365 00:30:38,084 --> 00:30:42,416 bit bigger than one 1.911. Take away one. Well, a little bit 366 00:30:42,416 --> 00:30:47,109 more than one. We set. Take away one that's positive. One plus a 367 00:30:47,109 --> 00:30:49,275 little bit more than one that's. 368 00:30:49,890 --> 00:30:53,256 Positive again, and this is definitely positive in the 369 00:30:53,256 --> 00:30:57,744 denominator, so we've a positive times by a positive divided by a 370 00:30:57,744 --> 00:30:59,614 positive gives us a positive. 371 00:31:00,180 --> 00:31:01,928 So if we sketch. 372 00:31:02,540 --> 00:31:08,656 The slopes of the tangents. We have a negative slope aflat 373 00:31:08,656 --> 00:31:15,328 slope under positive slope and So what we can see is that 374 00:31:15,328 --> 00:31:22,000 this defines the shape of a minimum and so we can conclude 375 00:31:22,000 --> 00:31:25,336 that one zero is a minimum. 376 00:31:25,370 --> 00:31:31,337 Point notice what we've done is go back to basics. This is how 377 00:31:31,337 --> 00:31:36,386 we arrived at our test for D2. Why? By DX squared. 378 00:31:37,340 --> 00:31:43,376 Now let's do it again for this point, minus 1 - 4. 379 00:31:44,850 --> 00:31:52,134 So we've got DYIDX is equal to X minus one times by 380 00:31:52,134 --> 00:31:58,811 X Plus one all over X squared, and the point we're 381 00:31:58,811 --> 00:32:06,095 having a look at is minus 1 - 4. So let's set 382 00:32:06,095 --> 00:32:08,523 up our table again. 383 00:32:09,960 --> 00:32:14,960 We're looking at the sign of DY by The X. 384 00:32:15,490 --> 00:32:21,301 My looking at it at the point X equals minus one. As we 385 00:32:21,301 --> 00:32:26,665 pass through that point as X increases. So we want to know 386 00:32:26,665 --> 00:32:30,688 what's happening alittle bit before that minus one minus 387 00:32:30,688 --> 00:32:31,135 Epsilon. 388 00:32:32,180 --> 00:32:38,076 And we want to know what's happening alittle bit after that 389 00:32:38,076 --> 00:32:40,220 minus one plus Epsilon. 390 00:32:41,070 --> 00:32:47,625 Well, we know that when X is minus one DY by DX is 0. Because 391 00:32:47,625 --> 00:32:49,810 this factor here is 0. 392 00:32:51,440 --> 00:32:57,356 What happens if X is a little bit less than minus one? 393 00:32:57,356 --> 00:32:59,328 Something like minus 1.1? 394 00:33:00,080 --> 00:33:05,000 Well you got minus 1.1 takeaway. One that's definitely negative. 395 00:33:06,200 --> 00:33:11,840 Minus 1.1 at on. One that's minus .1. That's definitely 396 00:33:11,840 --> 00:33:13,532 negative as well. 397 00:33:14,300 --> 00:33:18,810 However, on the bottom here, we've got a number and we're 398 00:33:18,810 --> 00:33:23,320 squaring it, so it must be positive. So we've a negative 399 00:33:23,320 --> 00:33:27,830 times a negative, which is a positive divided by a positive. 400 00:33:27,830 --> 00:33:30,290 So the answer must be positive. 401 00:33:31,260 --> 00:33:36,945 Now let's have a look a little bit after X equals minus 1 - 1, 402 00:33:36,945 --> 00:33:38,461 plus a little bit. 403 00:33:39,460 --> 00:33:45,438 So the little bit might be not .1 and minus one plus a little 404 00:33:45,438 --> 00:33:52,270 bit would then be minus N .9. So we've got minus 9.9 - 1 - 1.9 405 00:33:52,270 --> 00:33:57,730 it's negative. I've got minus not .9 plus one. Well, that's 406 00:33:57,730 --> 00:34:03,130 not .1, it's positive, so have a negative times a positive and 407 00:34:03,130 --> 00:34:08,080 this term X squared is always positive 'cause it's a square. 408 00:34:08,080 --> 00:34:13,930 So we've got a negative times by a positive and divided by a 409 00:34:13,930 --> 00:34:16,180 positive. So I'll answer. Must 410 00:34:16,180 --> 00:34:23,044 be negative. So let's sketch the shape. We've got a positive 411 00:34:23,044 --> 00:34:29,004 slope, aflat slope and negative slope, and this defines the 412 00:34:29,004 --> 00:34:36,156 shape of a maximum, and so we can say minus 1 - 413 00:34:36,156 --> 00:34:38,540 4 is a maximum. 414 00:34:38,540 --> 00:34:45,790 Finally, let's just see how this helps us to sketch 415 00:34:45,790 --> 00:34:52,715 the curve. We recall the equation of the curve 416 00:34:52,715 --> 00:34:59,765 Y equals X minus one all squared over X, and 417 00:34:59,765 --> 00:35:06,110 we had a maximum point at minus 1 four. 418 00:35:07,020 --> 00:35:12,096 And we had a minimum point at one 0. 419 00:35:14,100 --> 00:35:20,946 So some axes, let's mark the .1 and the point minus one goes on 420 00:35:20,946 --> 00:35:28,281 the X axis. Now 10 is here and we know it's a minimum, so we 421 00:35:28,281 --> 00:35:34,638 know it looks something like that there minus 1 - 4 down here 422 00:35:34,638 --> 00:35:40,506 somewhere. And we know that that's a maximum, so it kind of 423 00:35:40,506 --> 00:35:41,973 looks like that. 424 00:35:43,430 --> 00:35:47,533 How can we join these up? Well, clearly there's something odd 425 00:35:47,533 --> 00:35:52,382 going on around 0 here because the curve seems to go that way 426 00:35:52,382 --> 00:35:56,485 and that way it seems to go in opposite directions. What's 427 00:35:56,485 --> 00:36:00,588 happening here? What's going on? Well, let's just have a little 428 00:36:00,588 --> 00:36:03,945 think about this as X approaches 0 from above. 429 00:36:04,620 --> 00:36:07,896 And this is always going to be positive, but this is going to 430 00:36:07,896 --> 00:36:12,380 be. Positive as well, and were divided by something very very 431 00:36:12,380 --> 00:36:17,138 small. Our answer is going to be very, very big, so we're going 432 00:36:17,138 --> 00:36:21,896 to have something up there when X is negative but near to 0. 433 00:36:21,896 --> 00:36:25,556 This is still going to be positive because we're squaring 434 00:36:25,556 --> 00:36:29,582 it, but we're dividing it by something which is getting very 435 00:36:29,582 --> 00:36:34,340 very small, but negative. So the answer is going to be very big, 436 00:36:34,340 --> 00:36:39,098 but negative, and so it will be somewhere down here. So what we 437 00:36:39,098 --> 00:36:42,673 can see. Is that the curve is going to look something? 438 00:36:43,240 --> 00:36:44,200 Like 439 00:36:45,930 --> 00:36:50,324 That and they will in fact be this almost what looks like a 440 00:36:50,324 --> 00:36:55,056 sort of hole in the middle of it. Bang on this value X equals 441 00:36:55,056 --> 00:37:02,914 0. So what have we done? We have found out how to find 442 00:37:02,914 --> 00:37:07,109 stationary points. Stationary points occur on a curve. Wendy 443 00:37:07,109 --> 00:37:08,944 why by DX is 0. 444 00:37:09,990 --> 00:37:13,458 If the why by DX is 445 00:37:13,458 --> 00:37:19,495 0? Then some of those points are what we call turning points, 446 00:37:19,495 --> 00:37:20,830 Maxima or minima. 447 00:37:21,870 --> 00:37:23,650 We can find those. 448 00:37:24,170 --> 00:37:31,123 In one of two ways, if the why by DX is zero and D2, why by DX 449 00:37:31,123 --> 00:37:36,031 squared is greater than not, then we know for sure that we 450 00:37:36,031 --> 00:37:38,076 have got a minimum point. 451 00:37:39,130 --> 00:37:45,310 If you divide by the X is zero and D2 why by DX squared is 452 00:37:45,310 --> 00:37:50,254 negative, then we know for sure we've got a maximum point. If 453 00:37:50,254 --> 00:37:55,610 however, D2 why by DX squared is 0, then we have no information 454 00:37:55,610 --> 00:38:00,142 whatsoever and we need to look very closely using the methods 455 00:38:00,142 --> 00:38:02,202 that I've just shown here. 456 00:38:03,470 --> 00:38:07,122 In order to determine what kind of stationary point we've got, 457 00:38:07,122 --> 00:38:10,774 whether we've got a maximum, whether we've got a minimum or 458 00:38:10,774 --> 00:38:14,758 whether we've got one of those odd ones that looked a little 459 00:38:14,758 --> 00:38:17,082 bit flat. A kink in the curve.