1 00:00:00,000 --> 00:00:00,040 2 00:00:00,040 --> 00:00:02,400 The following content is provided under a Creative 3 00:00:02,400 --> 00:00:03,690 Commons license. 4 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 5 00:00:06,630 --> 00:00:09,980 offer high quality educational resources for free. 6 00:00:09,980 --> 00:00:12,830 To make a donation, or to view additional materials from 7 00:00:12,830 --> 00:00:16,760 hundreds of MIT courses, visit MIT OpenCourseWare at 8 00:00:16,760 --> 00:00:18,010 ocw.mit.edu. 9 00:00:18,010 --> 00:00:33,210 10 00:00:33,210 --> 00:00:33,900 PROFESSOR: Hi. 11 00:00:33,900 --> 00:00:37,390 Welcome once again to our Calculus Revisited lecture, 12 00:00:37,390 --> 00:00:41,750 where today we shall discuss the concept of infinitesimals, 13 00:00:41,750 --> 00:00:45,510 a rather elusive but very important concept. 14 00:00:45,510 --> 00:00:49,650 And because most textbooks illustrate this topic in terms 15 00:00:49,650 --> 00:00:53,150 of approximations, our topic today will be called 16 00:00:53,150 --> 00:00:55,650 approximations and infinitesimals. 17 00:00:55,650 --> 00:01:00,240 Now, how shall we introduce our subject in terms of topics 18 00:01:00,240 --> 00:01:02,220 that you may be more familiar with? 19 00:01:02,220 --> 00:01:05,870 Perhaps the easiest way is to go back to an elementary 20 00:01:05,870 --> 00:01:09,200 algebra course, to distance, rate and time problems, when 21 00:01:09,200 --> 00:01:14,100 one talked about distance equaling rate times time. 22 00:01:14,100 --> 00:01:17,750 The question, of course, is what rate do you use if the 23 00:01:17,750 --> 00:01:19,755 rate is not constant? 24 00:01:19,755 --> 00:01:22,430 You see, the question of distance equals rate times 25 00:01:22,430 --> 00:01:27,430 time presupposes that you are dealing with a constant rate. 26 00:01:27,430 --> 00:01:31,000 Now what does this mean, and how is it directly connected 27 00:01:31,000 --> 00:01:33,210 with the development of our calculus course? 28 00:01:33,210 --> 00:01:37,750 This shall be the subject of our investigation today. 29 00:01:37,750 --> 00:01:39,950 See, the idea is this. 30 00:01:39,950 --> 00:01:47,010 Let's consider the curve 'y' equals 'f of x', and let's 31 00:01:47,010 --> 00:01:48,640 suppose that the curve is smooth. 32 00:01:48,640 --> 00:01:51,490 That is, that it possesses a derivative, say, at the point 33 00:01:51,490 --> 00:01:53,780 'x' equals 'x1'. 34 00:01:53,780 --> 00:01:58,430 Let's draw the tangent line to the curve at 'x' equals 'x1'. 35 00:01:58,430 --> 00:02:01,120 36 00:02:01,120 --> 00:02:02,390 Now the idea is this. 37 00:02:02,390 --> 00:02:06,200 In general what we investigate in a calculus course is the 38 00:02:06,200 --> 00:02:09,660 concept known as 'delta y'. 39 00:02:09,660 --> 00:02:12,220 'Delta y' geometrically is what? 40 00:02:12,220 --> 00:02:15,650 It's how much 'y' has changed along the curve 41 00:02:15,650 --> 00:02:17,690 with respect to 'x'. 42 00:02:17,690 --> 00:02:21,260 It turns out that there is a simpler thing that we could 43 00:02:21,260 --> 00:02:22,670 have computed. 44 00:02:22,670 --> 00:02:26,960 Notice, if we look at this particular diagram, that since 45 00:02:26,960 --> 00:02:30,400 the tangent line never changes its slope-- and by the way, 46 00:02:30,400 --> 00:02:33,980 when I say tangent line I mean at the point 'x1'-- 47 00:02:33,980 --> 00:02:39,830 that once we leave the point 'x1' the tangent line, in a 48 00:02:39,830 --> 00:02:42,610 way, no longer resembles the curve. 49 00:02:42,610 --> 00:02:44,960 But the point that is important is that I can 50 00:02:44,960 --> 00:02:47,290 compute the change in 'y' to the tangent 51 00:02:47,290 --> 00:02:49,180 line here very easily. 52 00:02:49,180 --> 00:02:52,520 And the reason for this, you see, is rather apparent. 53 00:02:52,520 --> 00:02:55,900 Namely, the slope of the tangent line is, on the one 54 00:02:55,900 --> 00:03:00,650 hand, 'delta y-tan' divided by 'delta x'. 55 00:03:00,650 --> 00:03:05,210 'Delta y-tan' divided by 'delta x'. 56 00:03:05,210 --> 00:03:09,150 On the other hand, by definition of slope, the slope 57 00:03:09,150 --> 00:03:12,710 of the line 'L' is also equal to what? 58 00:03:12,710 --> 00:03:13,960 It's 'dy dx'. 59 00:03:13,960 --> 00:03:18,960 60 00:03:18,960 --> 00:03:23,750 It's 'dy dx' evaluated at the point, or the value, 'x' 61 00:03:23,750 --> 00:03:25,850 equals 'x1'. 62 00:03:25,850 --> 00:03:29,030 You see, in general the slope of the curve varies from point 63 00:03:29,030 --> 00:03:32,220 to point, so when we talk about the tangent line we must 64 00:03:32,220 --> 00:03:35,110 emphasize at what point on the curve we've 65 00:03:35,110 --> 00:03:36,385 drawn the tangent line. 66 00:03:36,385 --> 00:03:40,130 At any rate, from this particular diagram it is not 67 00:03:40,130 --> 00:03:44,150 difficult to see that to compute the change in 'y' to 68 00:03:44,150 --> 00:03:48,860 the tangent line, that this is nothing more than what? 69 00:03:48,860 --> 00:03:54,910 'dy dx' evaluated at 'x' equals 'x1' times 'delta x'. 70 00:03:54,910 --> 00:03:57,690 And you see this is not an approximation. 71 00:03:57,690 --> 00:04:03,280 This is precisely the value of 'delta y-tan'. 72 00:04:03,280 --> 00:04:11,530 The approximation seems to be when we say let 'delta y-tan' 73 00:04:11,530 --> 00:04:13,240 represent 'delta y'. 74 00:04:13,240 --> 00:04:16,529 In other words, we get the intuitive feeling that as 75 00:04:16,529 --> 00:04:20,709 'delta x' gets small, the difference between the true 76 00:04:20,709 --> 00:04:24,610 'delta y' and 'delta y-tan' also gets small. 77 00:04:24,610 --> 00:04:26,760 Another way of saying this is what? 78 00:04:26,760 --> 00:04:30,440 Our intuitive feeling is that, in a neighborhood of the point 79 00:04:30,440 --> 00:04:35,050 of tangency, the tangent line serves as a good approximation 80 00:04:35,050 --> 00:04:36,640 to the curve itself. 81 00:04:36,640 --> 00:04:39,360 Now let's see what this means in terms 82 00:04:39,360 --> 00:04:41,920 of a specific example. 83 00:04:41,920 --> 00:04:44,740 I've taken the liberty of computing the 84 00:04:44,740 --> 00:04:48,510 cube of 4.01 in advance. 85 00:04:48,510 --> 00:04:53,990 It turns out to be 64.481201. 86 00:04:53,990 --> 00:04:57,680 And it's sort of arbitrary, like cubing this. 87 00:04:57,680 --> 00:05:00,330 If this if this doesn't look messy enough for you we could 88 00:05:00,330 --> 00:05:02,680 have taken this to the sixth power and then we could have 89 00:05:02,680 --> 00:05:03,310 squared this. 90 00:05:03,310 --> 00:05:04,860 But that part is irrelevant. 91 00:05:04,860 --> 00:05:07,990 A simple check shows that, more or less, this will be a 92 00:05:07,990 --> 00:05:09,100 correct statement. 93 00:05:09,100 --> 00:05:12,280 And what I would like to do, you see, is simply illustrate 94 00:05:12,280 --> 00:05:15,940 what our earlier comments mean in terms of 95 00:05:15,940 --> 00:05:17,540 this specific example. 96 00:05:17,540 --> 00:05:21,840 Let's suppose I want to find an approximation for 4.01 97 00:05:21,840 --> 00:05:23,800 fairly rapidly. 98 00:05:23,800 --> 00:05:25,460 The idea is this. 99 00:05:25,460 --> 00:05:31,240 What I do know is one number that's very easy to cube, 100 00:05:31,240 --> 00:05:34,420 which is near 4.01, is 4 itself. 101 00:05:34,420 --> 00:05:38,820 In other words, I know that 4 cubed is 64. 102 00:05:38,820 --> 00:05:41,540 And by the way, I've deliberately drawn this 103 00:05:41,540 --> 00:05:44,670 slightly distorted according to scale so that we can see 104 00:05:44,670 --> 00:05:46,390 what's happening over here. 105 00:05:46,390 --> 00:05:55,060 What 64.481201 represents is the actual change in height 106 00:05:55,060 --> 00:05:58,880 from here to here along the curve. 107 00:05:58,880 --> 00:06:02,550 In other words, it would be the length of the segment 108 00:06:02,550 --> 00:06:06,310 joining the point 'P' to the point 'Q' here. 109 00:06:06,310 --> 00:06:11,520 What I claim is that if I instead tried to find the 110 00:06:11,520 --> 00:06:16,190 length of 'PR', the change in 'y' not along the curve but 111 00:06:16,190 --> 00:06:21,030 along the line tangent to the curve at the point 4.64, this 112 00:06:21,030 --> 00:06:23,300 is what I can find fairly rapidly. 113 00:06:23,300 --> 00:06:28,350 In other words, what I am going to do is to work this 114 00:06:28,350 --> 00:06:33,060 same idea here with a special case. 115 00:06:33,060 --> 00:06:41,040 You see, I'm going to take 'x1' to equal 4 and 'delta x' 116 00:06:41,040 --> 00:06:43,410 to be 0.01. 117 00:06:43,410 --> 00:06:48,260 Now you see the curve is 'y' equals 'x cubed'. 118 00:06:48,260 --> 00:06:51,935 From this I can compute 'dy dx' rather quickly. 119 00:06:51,935 --> 00:06:54,860 120 00:06:54,860 --> 00:06:58,150 Now, I don't want 'dy dx' at any old point, I want to 121 00:06:58,150 --> 00:07:00,830 compute it when 'x' is 4 so I can find the 122 00:07:00,830 --> 00:07:02,080 slope of the line 'L'. 123 00:07:02,080 --> 00:07:07,890 124 00:07:07,890 --> 00:07:11,060 And when 'x' is 4 this, of course, simply is what? 125 00:07:11,060 --> 00:07:15,420 4 squared is 16, times 3 is 48. 126 00:07:15,420 --> 00:07:17,280 So what do I have? 127 00:07:17,280 --> 00:07:21,790 I have that the slope is 48. 128 00:07:21,790 --> 00:07:24,860 I also have that 'delta x' is 0.01. 129 00:07:24,860 --> 00:07:30,370 So according to my recipe, 'delta y-tan' is what? 130 00:07:30,370 --> 00:07:35,870 It's 'dy dx' evaluated at 'x' equals 4, which is 48, times 131 00:07:35,870 --> 00:07:38,870 'delta x', which is 0.01. 132 00:07:38,870 --> 00:07:42,960 And that's 0.48. 133 00:07:42,960 --> 00:07:45,850 See again, let's just juxtaposition these two. 134 00:07:45,850 --> 00:07:50,300 All I have done now is computed this recipe in the 135 00:07:50,300 --> 00:07:56,150 particular example of trying to find the cube of 4.01. 136 00:07:56,150 --> 00:08:01,530 And you see, now notice that this point 0.48 is exactly the 137 00:08:01,530 --> 00:08:04,680 length of the accented line here. 138 00:08:04,680 --> 00:08:06,670 It's the length of 'PR'. 139 00:08:06,670 --> 00:08:10,420 And what we do know is that the height from the x-axis to 140 00:08:10,420 --> 00:08:13,610 'R' is now exactly what? 141 00:08:13,610 --> 00:08:20,210 Well, it's the 64 plus the 0.48. 142 00:08:20,210 --> 00:08:22,470 This then is our approximation. 143 00:08:22,470 --> 00:08:24,960 And notice that this compares with what? 144 00:08:24,960 --> 00:08:32,270 The precise answer, which is 64.481201. 145 00:08:32,270 --> 00:08:35,159 In other words, notice what a small error we happen to have 146 00:08:35,159 --> 00:08:37,039 in this particular case. 147 00:08:37,039 --> 00:08:39,280 And this is the way the subject is usually brought up. 148 00:08:39,280 --> 00:08:43,559 It is not a very important thing from my point of view. 149 00:08:43,559 --> 00:08:46,480 In other words, I think it's rather easy to see that, first 150 00:08:46,480 --> 00:08:50,680 of all, this approximation is rather nebulous in the sense 151 00:08:50,680 --> 00:08:55,160 that it requires a knowledge of how fast the tangent line 152 00:08:55,160 --> 00:08:56,650 is separating from the curve. 153 00:08:56,650 --> 00:08:59,530 And this is a rather difficult topic in its own right. 154 00:08:59,530 --> 00:09:03,040 And secondly, this was a rather simple example. 155 00:09:03,040 --> 00:09:06,280 And we had the luxury here, you see, of being able to find 156 00:09:06,280 --> 00:09:09,520 the exact answer so we could compare our approximation with 157 00:09:09,520 --> 00:09:11,070 the exact answer. 158 00:09:11,070 --> 00:09:14,600 In many cases it is difficult or impossible to find the 159 00:09:14,600 --> 00:09:16,000 exact answer. 160 00:09:16,000 --> 00:09:19,810 To emphasize this more abstractly and more generally, 161 00:09:19,810 --> 00:09:22,080 let's consider the following. 162 00:09:22,080 --> 00:09:27,060 Instead of trying to find the cube of 4.01, the 163 00:09:27,060 --> 00:09:28,980 generalization here is what? 164 00:09:28,980 --> 00:09:31,980 That we could take the curve 'y' equals 'x cubed'. 165 00:09:31,980 --> 00:09:34,990 The derivative of 'y' with respect to 'x' would then be 166 00:09:34,990 --> 00:09:36,450 '3 x squared'. 167 00:09:36,450 --> 00:09:40,130 Evaluated at an arbitrary point 'x' equals 'x1', we 168 00:09:40,130 --> 00:09:42,450 would get '3 x sub 1 squared'. 169 00:09:42,450 --> 00:09:47,820 In which case 'delta y-tan' would be '3 x1 squared' 170 00:09:47,820 --> 00:09:49,625 times 'delta x'. 171 00:09:49,625 --> 00:09:52,260 Could we have computed the exact value of 172 00:09:52,260 --> 00:09:54,560 'delta y' had we wished? 173 00:09:54,560 --> 00:09:56,290 And the answer, of course, is yes. 174 00:09:56,290 --> 00:09:59,690 Namely, what is the exact value of 'delta y'? 175 00:09:59,690 --> 00:10:04,510 Well, we want to compute this between 'x' equals 176 00:10:04,510 --> 00:10:06,290 'x1' plus 'delta x'. 177 00:10:06,290 --> 00:10:11,840 Well, what is the value of 'y' when 'x' is 'x1 plus delta x'? 178 00:10:11,840 --> 00:10:14,270 It's ''x1 plus delta x' cubed'. 179 00:10:14,270 --> 00:10:17,520 Then we subtract off 'x1 cubed'. 180 00:10:17,520 --> 00:10:19,850 And if we expand this, watch what happens. 181 00:10:19,850 --> 00:10:23,460 By the binomial theorem we get an 'x1 cubed' term here, which 182 00:10:23,460 --> 00:10:26,370 cancels with the 'x1 cubed' term over here. 183 00:10:26,370 --> 00:10:27,830 Then we get what? 184 00:10:27,830 --> 00:10:32,660 A '3 x1 squared delta x', so using the 185 00:10:32,660 --> 00:10:34,040 binomial theorem here. 186 00:10:34,040 --> 00:10:35,360 And then what else do we get? 187 00:10:35,360 --> 00:10:46,940 We get plus '3x1 delta x squared' plus 'delta x cubed'. 188 00:10:46,940 --> 00:10:51,140 And you see, what I'd like to have us view over here is if 189 00:10:51,140 --> 00:10:59,270 we look at just this much of the answer, this part here is 190 00:10:59,270 --> 00:11:03,780 precisely what we sought to be 'delta y-tan' before. 191 00:11:03,780 --> 00:11:08,050 See, this is 'delta y-tan'. 192 00:11:08,050 --> 00:11:12,070 And what's left over is the difference, of course, between 193 00:11:12,070 --> 00:11:13,990 'delta y-tan' and delta y. 194 00:11:13,990 --> 00:11:17,930 After all, delta y is just 'delta y-tan' plus this 195 00:11:17,930 --> 00:11:18,740 portion here. 196 00:11:18,740 --> 00:11:22,390 And and now if you look at this particular portion over 197 00:11:22,390 --> 00:11:26,540 here, observe that, as we expected, what the size of 198 00:11:26,540 --> 00:11:31,570 this thing is depends both on where we're measuring 'x1', 199 00:11:31,570 --> 00:11:34,910 and also on how big the size of the interval is, 200 00:11:34,910 --> 00:11:36,490 namely 'delta x'. 201 00:11:36,490 --> 00:11:38,040 And notice something that we're going to 202 00:11:38,040 --> 00:11:39,220 come back to here. 203 00:11:39,220 --> 00:11:42,820 Notice that the 'delta x' factor here appears at least 204 00:11:42,820 --> 00:11:44,230 to the second power. 205 00:11:44,230 --> 00:11:47,180 In other words, notice over here that as 'delta x' get 206 00:11:47,180 --> 00:11:52,450 small, both of these terms get small very rapidly. 207 00:11:52,450 --> 00:11:57,480 Because, you see, when 'delta x' is close to 0 the square of 208 00:11:57,480 --> 00:12:01,370 a number close to 0 is even closer to 0. 209 00:12:01,370 --> 00:12:03,290 But you see the important thing for now is to notice 210 00:12:03,290 --> 00:12:06,440 that this is the error in approximating 'delta y' by 211 00:12:06,440 --> 00:12:07,760 'delta y-tan'. 212 00:12:07,760 --> 00:12:12,800 And that this error depends both on 'xy' and 'delta x'. 213 00:12:12,800 --> 00:12:16,010 And the question is, can we state that in a little bit 214 00:12:16,010 --> 00:12:17,890 more of a mathematical language? 215 00:12:17,890 --> 00:12:20,620 The answer, of course, is that we can. 216 00:12:20,620 --> 00:12:23,800 In particular, what we're saying is let's take the 217 00:12:23,800 --> 00:12:28,200 difference between 'dy dx' at 'x' equals 'x1', namely the 218 00:12:28,200 --> 00:12:33,170 slope of the tangent line, and what the average change of 'y' 219 00:12:33,170 --> 00:12:36,400 with respect to 'x' is along the curve. 220 00:12:36,400 --> 00:12:38,630 Now, this is very typical in mathematics. 221 00:12:38,630 --> 00:12:41,220 And it's a very nice trick to learn, and a very 222 00:12:41,220 --> 00:12:42,450 quick trick to learn. 223 00:12:42,450 --> 00:12:46,200 We know that these two things are not equal and so all we do 224 00:12:46,200 --> 00:12:48,890 is tack on a correction factor. 225 00:12:48,890 --> 00:12:50,230 We tack on 'k'. 226 00:12:50,230 --> 00:12:51,370 What is 'k'? 227 00:12:51,370 --> 00:12:55,130 'k' is the difference between these two expressions. 228 00:12:55,130 --> 00:12:58,720 In other words, by definition I add on the difference of 229 00:12:58,720 --> 00:13:01,920 these two numbers and that makes this an equality. 230 00:13:01,920 --> 00:13:04,540 Notice, by the way, that 'k' is a variable. 231 00:13:04,540 --> 00:13:08,000 It depends on how big 'delta x' is. 232 00:13:08,000 --> 00:13:12,400 You see, notice that once 'x1' is chosen-- 233 00:13:12,400 --> 00:13:14,780 and this is an important observation. 234 00:13:14,780 --> 00:13:19,350 As messy as this thing looks, it's a fixed number for a 235 00:13:19,350 --> 00:13:21,040 particular value of 'x1'. 236 00:13:21,040 --> 00:13:23,570 Namely, we compute the derivative and evaluate it 237 00:13:23,570 --> 00:13:24,990 when 'x' equals 'x1'. 238 00:13:24,990 --> 00:13:26,110 This is a number. 239 00:13:26,110 --> 00:13:28,080 This, of course, clearly depends on the 240 00:13:28,080 --> 00:13:29,600 size of 'delta x'. 241 00:13:29,600 --> 00:13:32,090 'Delta y' depends on 'delta x'. 242 00:13:32,090 --> 00:13:34,520 At any rate, here's what we do. 243 00:13:34,520 --> 00:13:39,330 We recognize that 'dy dx' is the limit of 'delta y' over 244 00:13:39,330 --> 00:13:42,080 'delta x' as 'delta x' approaches 0. 245 00:13:42,080 --> 00:13:46,420 So with this as a hint, we take the limit of both sides 246 00:13:46,420 --> 00:13:49,210 here as 'delta x' approaches 0. 247 00:13:49,210 --> 00:13:51,180 Notice the structure again. 248 00:13:51,180 --> 00:13:53,630 In our last series of lectures we 249 00:13:53,630 --> 00:13:55,110 talked about limit theorems. 250 00:13:55,110 --> 00:13:57,490 We talked about the limit of a sum being the sum of the 251 00:13:57,490 --> 00:13:59,100 limits, and things of this type. 252 00:13:59,100 --> 00:14:02,140 And now, you see, we're going to use these results. 253 00:14:02,140 --> 00:14:05,360 What we do is we take the limit of both sides here as 254 00:14:05,360 --> 00:14:06,830 'delta x' approaches 0. 255 00:14:06,830 --> 00:14:14,360 256 00:14:14,360 --> 00:14:16,220 The point being what? 257 00:14:16,220 --> 00:14:22,440 That the limit of a sum is the sum of the limits. 258 00:14:22,440 --> 00:14:26,060 So when I take the limit of this entire side I can express 259 00:14:26,060 --> 00:14:31,880 that as the sum of the limits. 260 00:14:31,880 --> 00:14:35,000 It is the limit of this term as 'delta x' approaches 0 plus 261 00:14:35,000 --> 00:14:38,380 the limit of this term as 'delta x' approaches 0. 262 00:14:38,380 --> 00:14:41,950 Now let's stop to think what these things mean. 263 00:14:41,950 --> 00:14:45,180 The limit of 'delta x' approaching 0 of 'delta y' 264 00:14:45,180 --> 00:14:49,070 divided by 'delta x' is precisely the definition of 265 00:14:49,070 --> 00:14:50,050 derivative. 266 00:14:50,050 --> 00:14:54,820 In other words, this term here is just the 'dy dx'. 267 00:14:54,820 --> 00:14:57,150 And keep in mind we've evaluated this 268 00:14:57,150 --> 00:14:58,610 at 'x' equals 'x1'. 269 00:14:58,610 --> 00:15:01,290 That's the point at which we're starting. 270 00:15:01,290 --> 00:15:07,480 Now the fact that this is a constant, and we've already 271 00:15:07,480 --> 00:15:10,600 learned that the limit of a constant as 'delta x' 272 00:15:10,600 --> 00:15:17,290 approaches 0 is that constant, this term here is also 'dy dx' 273 00:15:17,290 --> 00:15:20,940 evaluated at 'x' equals 'x1'. 274 00:15:20,940 --> 00:15:24,600 And of course this term here is just the limit of 'k' as 275 00:15:24,600 --> 00:15:26,810 'delta x' approaches 0. 276 00:15:26,810 --> 00:15:32,510 And now you see if we look at this, since all we have to do 277 00:15:32,510 --> 00:15:34,770 is observe that this term appears on both sides of the 278 00:15:34,770 --> 00:15:37,750 equation, canceling, or subtracting equals from 279 00:15:37,750 --> 00:15:43,570 equals, we wind up with the result that the limit of 'k' 280 00:15:43,570 --> 00:15:46,810 as 'delta x' approaches 0 is 0. 281 00:15:46,810 --> 00:15:50,300 282 00:15:50,300 --> 00:15:53,810 And we're going to talk more about that pictorially and 283 00:15:53,810 --> 00:15:55,100 analytically in a little while. 284 00:15:55,100 --> 00:16:01,760 But from this result, all I'm going to do now is go back to 285 00:16:01,760 --> 00:16:04,280 our first statement here. 286 00:16:04,280 --> 00:16:08,320 Remembering that 'delta x' is a nonzero number, I will 287 00:16:08,320 --> 00:16:14,210 simply multiply both sides of this equation by 'delta x'. 288 00:16:14,210 --> 00:16:17,550 And if I do that, what do I wind up with? 289 00:16:17,550 --> 00:16:25,840 I wind up with that 'delta y' is 'dy dx' evaluated at 'x' 290 00:16:25,840 --> 00:16:34,940 equals 'x1' times 'delta x' plus 'k delta x'. 291 00:16:34,940 --> 00:16:40,280 Now, if I keep in mind here that if I take the slope at a 292 00:16:40,280 --> 00:16:44,550 particular point and multiply that by the change in 'x', 293 00:16:44,550 --> 00:16:49,330 that that's precisely what we define to be 'delta y-tan'. 294 00:16:49,330 --> 00:16:52,240 If I put this all together I get what? 295 00:16:52,240 --> 00:16:57,050 That 'delta y' is equal to ''delta y-tan' plus 'k delta 296 00:16:57,050 --> 00:17:02,380 x'', where the limit of 'k' as 'delta x' approaches 0 is 0. 297 00:17:02,380 --> 00:17:06,020 Now you see, when we started this program and started to 298 00:17:06,020 --> 00:17:09,470 talk at the beginning of our lecture about approximations, 299 00:17:09,470 --> 00:17:12,349 we did something that was quite crude. 300 00:17:12,349 --> 00:17:14,119 The crude part was simply this. 301 00:17:14,119 --> 00:17:18,440 What we had said was, why don't we compute 'delta y' by 302 00:17:18,440 --> 00:17:21,640 computing 'delta y-tan' instead. 303 00:17:21,640 --> 00:17:24,900 In other words, this is a very easy thing to compute. 304 00:17:24,900 --> 00:17:28,270 It's just a derivative multiplied by a change in 'x'. 305 00:17:28,270 --> 00:17:31,430 This is a very simple thing to compute. 306 00:17:31,430 --> 00:17:34,190 And we'll use that as an approximation for the thing 307 00:17:34,190 --> 00:17:36,650 that we really want, which is 'delta y'. 308 00:17:36,650 --> 00:17:39,120 The question that we deliberately overlooked at 309 00:17:39,120 --> 00:17:43,930 this point was how big was the error when you do this? 310 00:17:43,930 --> 00:17:46,470 And what we discovered was something very interesting 311 00:17:46,470 --> 00:17:50,060 and, by the way, may be something which teaches us why 312 00:17:50,060 --> 00:17:53,440 the analysis is better than the picture even though the 313 00:17:53,440 --> 00:17:55,430 picture is easier to visualize. 314 00:17:55,430 --> 00:18:00,820 You see, we knew intuitively that as you picked a smaller 315 00:18:00,820 --> 00:18:02,850 neighborhood at the point of tangency. 316 00:18:02,850 --> 00:18:04,920 the tangent line became a better and better 317 00:18:04,920 --> 00:18:07,710 approximation to the curve itself. 318 00:18:07,710 --> 00:18:11,640 This boxed-in result tells us much more than that. 319 00:18:11,640 --> 00:18:16,360 What this tells us is, look, if this term by itself became 320 00:18:16,360 --> 00:18:19,720 small as 'delta x' approached 0, this would still say the 321 00:18:19,720 --> 00:18:20,780 same thing. 322 00:18:20,780 --> 00:18:22,760 This says much more than that. 323 00:18:22,760 --> 00:18:24,620 You see, what this says is-- 324 00:18:24,620 --> 00:18:27,010 and watch, this is very, very profound-- 325 00:18:27,010 --> 00:18:33,500 as 'delta x' approaches 0, 'k' is also approaching 0. 326 00:18:33,500 --> 00:18:38,230 In other words, the term 'k' times 'delta x' seems to be 327 00:18:38,230 --> 00:18:42,820 approaching 0 much more rapidly than 'delta x' itself. 328 00:18:42,820 --> 00:18:47,060 That, by the way, in as simple a way as I know how, defines 329 00:18:47,060 --> 00:18:49,190 what an infinitesimal is. 330 00:18:49,190 --> 00:18:54,450 See, 'k' times 'delta x' is called an infinitesimal 331 00:18:54,450 --> 00:18:58,790 because it goes to 0 faster than 'delta x' itself. 332 00:18:58,790 --> 00:19:01,650 In other words, anything that approaches 0 faster than 333 00:19:01,650 --> 00:19:05,640 'delta x' itself approaches 0 is called an infinitesimal 334 00:19:05,640 --> 00:19:08,070 with respect to 'delta x'. 335 00:19:08,070 --> 00:19:11,270 Now I'm not going to go into a long philosophical discussion 336 00:19:11,270 --> 00:19:12,140 about that. 337 00:19:12,140 --> 00:19:15,682 Rather, I'm going to let the actions speak louder than the 338 00:19:15,682 --> 00:19:19,560 words, and try to show why this is such a crucial idea. 339 00:19:19,560 --> 00:19:21,220 In other words, again let me emphasize that. 340 00:19:21,220 --> 00:19:25,820 What's crucial here is not so much that the error goes to 0, 341 00:19:25,820 --> 00:19:29,850 it's that the error goes to 0 much faster than 342 00:19:29,850 --> 00:19:31,490 'delta x' goes to 0. 343 00:19:31,490 --> 00:19:32,620 See, what is the error? 344 00:19:32,620 --> 00:19:34,790 The error is 'k' times 'delta x'. 345 00:19:34,790 --> 00:19:37,810 That's the difference between these two things. 346 00:19:37,810 --> 00:19:41,080 Again, if you want to see what this thing means pictorially, 347 00:19:41,080 --> 00:19:44,370 and I wish I knew a better way of doing this but I don't, 348 00:19:44,370 --> 00:19:49,540 notice that 'k' times 'delta x' was defined to be the 349 00:19:49,540 --> 00:19:53,220 difference between 'delta y' and 'delta y-tan'. 350 00:19:53,220 --> 00:19:58,610 In other words, since this length here is 'delta y-tan', 351 00:19:58,610 --> 00:20:01,600 and since this entire length would be called 'delta y', 352 00:20:01,600 --> 00:20:06,890 again the accented line, this length here, is 'k' 353 00:20:06,890 --> 00:20:09,160 times 'delta x'. 354 00:20:09,160 --> 00:20:13,490 And what does it mean to say that 'k' approaches 0 as 355 00:20:13,490 --> 00:20:15,020 'delta x' approaches 0? 356 00:20:15,020 --> 00:20:16,640 What it means is this. 357 00:20:16,640 --> 00:20:22,400 It means that not only does this vertical difference get 358 00:20:22,400 --> 00:20:29,490 small as 'delta x' goes to 0, but more importantly, it means 359 00:20:29,490 --> 00:20:34,350 that this vertical distance gets small very rapidly 360 00:20:34,350 --> 00:20:36,610 compared to how 'delta x' get small. 361 00:20:36,610 --> 00:20:37,840 Now how can I prove that? 362 00:20:37,840 --> 00:20:40,550 Well I'm not even going to try to prove that. 363 00:20:40,550 --> 00:20:42,690 What I'm going to just try to do is to have you see from the 364 00:20:42,690 --> 00:20:43,870 picture what's happening here. 365 00:20:43,870 --> 00:20:47,080 See, here is a fairly large 'delta x'. 366 00:20:47,080 --> 00:20:50,030 And the difference between 'delta y' and 'delta y-tan' 367 00:20:50,030 --> 00:20:53,670 for that large 'x' is this length over here. 368 00:20:53,670 --> 00:20:56,150 Now suppose we take a smaller 'delta x'. 369 00:20:56,150 --> 00:20:58,530 In other words, let's take 'delta x' to be 370 00:20:58,530 --> 00:21:00,420 this length over here. 371 00:21:00,420 --> 00:21:03,920 Notice that the length of 'delta x' here is still quite 372 00:21:03,920 --> 00:21:04,790 significant. 373 00:21:04,790 --> 00:21:07,110 I think if you look at this you can see it's a fairly 374 00:21:07,110 --> 00:21:08,770 significant length. 375 00:21:08,770 --> 00:21:12,510 On the other hand, notice what the difference between 'delta 376 00:21:12,510 --> 00:21:14,780 y' and 'delta y-tan' is now. 377 00:21:14,780 --> 00:21:17,755 It's just this little tiny thing over here. 378 00:21:17,755 --> 00:21:22,160 In other words, notice that as 'delta x' gets small, 379 00:21:22,160 --> 00:21:25,840 ratio-wise as 'delta x' get small, the vertical difference 380 00:21:25,840 --> 00:21:29,600 between the tangent line and the curve is getting small at 381 00:21:29,600 --> 00:21:31,070 a much faster rate. 382 00:21:31,070 --> 00:21:34,270 Now of course the question is why is that so important? 383 00:21:34,270 --> 00:21:36,740 And the answer will always come back to our 384 00:21:36,740 --> 00:21:40,440 old friend of 0/0. 385 00:21:40,440 --> 00:21:46,070 In general, if you tell a person that the numerator of a 386 00:21:46,070 --> 00:21:51,200 fraction is small, he jumps to the conclusion that the size 387 00:21:51,200 --> 00:21:53,380 of the fraction must be small. 388 00:21:53,380 --> 00:21:56,980 But you see, the trouble is when the denominator is also 389 00:21:56,980 --> 00:22:00,970 small then we cannot conclude that the 390 00:22:00,970 --> 00:22:03,120 fraction itself is small. 391 00:22:03,120 --> 00:22:06,190 And therefore in trying to cancel out a term as being 392 00:22:06,190 --> 00:22:08,650 insignificant, when we're dealing with something like 393 00:22:08,650 --> 00:22:13,010 this, it is no longer enough to say but the numerator is 394 00:22:13,010 --> 00:22:14,150 going to 0. 395 00:22:14,150 --> 00:22:17,900 Let me jump ahead to the topic that will be covered in our 396 00:22:17,900 --> 00:22:22,650 next lecture just to use this as an insight as to what we 397 00:22:22,650 --> 00:22:27,060 have to see and what's going on over here. 398 00:22:27,060 --> 00:22:29,910 Let me just show you what I mean by this. 399 00:22:29,910 --> 00:22:34,220 let's go back to our recipe, which says 'delta y' is this 400 00:22:34,220 --> 00:22:37,310 thing over here where what? 401 00:22:37,310 --> 00:22:40,560 This is important to put in here, the limit of 'k' as 402 00:22:40,560 --> 00:22:44,410 'delta x' approaches 0 is 0. 403 00:22:44,410 --> 00:22:46,440 Now all I'm going to do something like this. 404 00:22:46,440 --> 00:22:50,540 Let's suppose we're dealing in a situation where 'y' and 'x' 405 00:22:50,540 --> 00:22:54,320 happen to be functions, say of, 't' as well, some third 406 00:22:54,320 --> 00:22:55,460 variable 't'. 407 00:22:55,460 --> 00:22:59,240 And what we really want to find is 'dy dt'. 408 00:22:59,240 --> 00:23:02,380 You see, what we'd be tempted to do is simply say what? 409 00:23:02,380 --> 00:23:06,870 Let's divide through by 'delta t'. 410 00:23:06,870 --> 00:23:09,340 We do this, we divide through by 'delta t'. 411 00:23:09,340 --> 00:23:12,130 Then we take the limit of this thing as 'delta 412 00:23:12,130 --> 00:23:13,800 t' approaches 0. 413 00:23:13,800 --> 00:23:14,950 Now what happens? 414 00:23:14,950 --> 00:23:18,280 As 'delta t' approaches 0, this is a sum. 415 00:23:18,280 --> 00:23:20,840 The limit of a sum is the sum of the limits. 416 00:23:20,840 --> 00:23:23,590 Each of the terms in the sum is a product. 417 00:23:23,590 --> 00:23:26,590 And the limit of a product is the product of the limits. 418 00:23:26,590 --> 00:23:30,010 So working out the regular limit theorems, this is what? 419 00:23:30,010 --> 00:23:33,230 The limit, 'delta t' approaches 0, 'dy dx' 420 00:23:33,230 --> 00:23:36,830 evaluated at 'x' equals 'x1', times the limit of 'delta x' 421 00:23:36,830 --> 00:23:40,250 divided by 'delta t' as 'delta t' approaches 0, plus the 422 00:23:40,250 --> 00:23:43,730 limit of 'k' as 'delta t' approaches 0, times the limit 423 00:23:43,730 --> 00:23:48,060 of 'delta x' divided by 'delta t' as 'delta t' approaches 0. 424 00:23:48,060 --> 00:23:52,670 Now, if we remember what this thing here means, this is just 425 00:23:52,670 --> 00:23:56,110 by definition 'dy dt'. 426 00:23:56,110 --> 00:23:58,940 For the sake of brevity I will leave out subscripts now, but 427 00:23:58,940 --> 00:24:01,700 we'll just talk about this. 428 00:24:01,700 --> 00:24:06,170 This is a constant, and the limit of constant as 'delta t' 429 00:24:06,170 --> 00:24:08,440 approaches 0 is just that constant, which 430 00:24:08,440 --> 00:24:10,520 I'll call 'dy dx'. 431 00:24:10,520 --> 00:24:13,700 It's understood this is evaluated at 'x' equals 'x1'. 432 00:24:13,700 --> 00:24:19,120 By definition the limit of 'delta x' divided by 'delta t' 433 00:24:19,120 --> 00:24:25,020 as 'delta t' approaches 0, that's called 'dx dt'. 434 00:24:25,020 --> 00:24:27,020 Now we come to this term over here. 435 00:24:27,020 --> 00:24:28,340 And here's the key point. 436 00:24:28,340 --> 00:24:31,460 Many people say, well, as 'delta t' goes to 0 437 00:24:31,460 --> 00:24:32,860 so does 'delta x'. 438 00:24:32,860 --> 00:24:36,150 That makes the numerator here 0 and that makes the whole 439 00:24:36,150 --> 00:24:37,220 fraction 0. 440 00:24:37,220 --> 00:24:39,080 But the point is that's not true. 441 00:24:39,080 --> 00:24:42,440 What is true here is that both the numerator and denominator 442 00:24:42,440 --> 00:24:43,760 are going to 0. 443 00:24:43,760 --> 00:24:47,680 This does not make this 0, but rather it makes it what? 444 00:24:47,680 --> 00:24:48,530 'dx dt'. 445 00:24:48,530 --> 00:24:50,870 That's the definition of 'dx dt'. 446 00:24:50,870 --> 00:24:52,870 The important point is what? 447 00:24:52,870 --> 00:24:56,630 That as 'delta t' approaches 0 so does 'delta x'. 448 00:24:56,630 --> 00:24:59,230 And what property does 'k' have? 449 00:24:59,230 --> 00:25:03,110 'k' has the property that the limit of 'k' as 'delta x' 450 00:25:03,110 --> 00:25:08,140 approaches 0 is 0 itself. 451 00:25:08,140 --> 00:25:11,900 In other words, the topics that we'll be talking about in 452 00:25:11,900 --> 00:25:15,460 our next lecture concerns this recipe. 453 00:25:15,460 --> 00:25:19,260 And in the development of this recipe, the reason that the 454 00:25:19,260 --> 00:25:23,170 error term drops out is not because the numerator of this 455 00:25:23,170 --> 00:25:28,310 term was small, it was because the number multiplying limit 456 00:25:28,310 --> 00:25:30,610 happened to be 0. 457 00:25:30,610 --> 00:25:34,240 Now, we will talk about that in more detail next time. 458 00:25:34,240 --> 00:25:38,690 But the message I want you to see for right now is how we 459 00:25:38,690 --> 00:25:43,630 get hung up on this 0/0 bit, and why we must always be 460 00:25:43,630 --> 00:25:46,700 careful in how we handle something like this. 461 00:25:46,700 --> 00:25:49,460 Now, by the way, there is something rather interesting 462 00:25:49,460 --> 00:25:50,820 that does happen over here. 463 00:25:50,820 --> 00:25:54,210 If you look at this thing, you get the feeling that it's 464 00:25:54,210 --> 00:25:58,020 almost as if you could cancel a common factor from both the 465 00:25:58,020 --> 00:26:00,240 numerator and the denominator. 466 00:26:00,240 --> 00:26:02,390 We have to be very, very careful. 467 00:26:02,390 --> 00:26:04,560 And I would like to introduce the language which you will be 468 00:26:04,560 --> 00:26:07,700 reading in the text in this assignment, called 469 00:26:07,700 --> 00:26:09,250 'Differentials Now'. 470 00:26:09,250 --> 00:26:12,370 And what that means is simply this. 471 00:26:12,370 --> 00:26:18,120 Notice that 'dy dx' is one symbol it is not 'dy' divided 472 00:26:18,120 --> 00:26:23,090 by 'dx' because these things have not been defined 473 00:26:23,090 --> 00:26:23,420 separately. 474 00:26:23,420 --> 00:26:27,400 In other words, you can't just operate with symbols the way 475 00:26:27,400 --> 00:26:28,600 you might like to. 476 00:26:28,600 --> 00:26:31,800 Here's an interesting little aside that you may find 477 00:26:31,800 --> 00:26:35,570 entertaining if it has no other value at all: the idea 478 00:26:35,570 --> 00:26:38,870 of being told that you can cancel common factors from 479 00:26:38,870 --> 00:26:40,710 both numerator and denominator. 480 00:26:40,710 --> 00:26:43,660 The uninitiated may say, you know, here's a 6 481 00:26:43,660 --> 00:26:44,700 and here's a 6. 482 00:26:44,700 --> 00:26:46,890 I'll cancel them out. 483 00:26:46,890 --> 00:26:49,070 Now this is not what cancellation really meant, 484 00:26:49,070 --> 00:26:51,810 even though you had the same number both upstairs and 485 00:26:51,810 --> 00:26:53,310 downstairs. 486 00:26:53,310 --> 00:26:54,925 This is not what we meant by cancellation. 487 00:26:54,925 --> 00:26:59,730 Yet, notice that you happen to, by accident, get the right 488 00:26:59,730 --> 00:27:01,650 answer in this particular case. 489 00:27:01,650 --> 00:27:03,180 You see, if you do cancel this, this does 490 00:27:03,180 --> 00:27:06,120 happen to be 1/4. 491 00:27:06,120 --> 00:27:09,650 This lecture has been going on in kind of a dull way for you. 492 00:27:09,650 --> 00:27:13,120 Just in case you'd like some comic relief, there are a few 493 00:27:13,120 --> 00:27:15,860 other examples that work the same way. 494 00:27:15,860 --> 00:27:18,860 And you can amaze your friends at the next party with them. 495 00:27:18,860 --> 00:27:23,980 I mean 19/95 happens to be 1/5. 496 00:27:23,980 --> 00:27:30,410 26/65 happens to be 2/5. 497 00:27:30,410 --> 00:27:31,970 And I think there's one more. 498 00:27:31,970 --> 00:27:36,570 49/98 happens to be 4/8. 499 00:27:36,570 --> 00:27:39,880 But these, I believe, are the only four fractions 500 00:27:39,880 --> 00:27:41,340 that this works for. 501 00:27:41,340 --> 00:27:43,490 In other words, the mere fact that it looks like something 502 00:27:43,490 --> 00:27:46,700 is going to work is no guarantee that you can do 503 00:27:46,700 --> 00:27:48,980 these things without running into trouble. 504 00:27:48,980 --> 00:27:52,380 On the other hand, it's fair to assume that the men who 505 00:27:52,380 --> 00:27:57,310 invented differential calculus must've been clever enough not 506 00:27:57,310 --> 00:27:59,650 to have invented a symbolism that would have 507 00:27:59,650 --> 00:28:01,370 gotten us into trouble. 508 00:28:01,370 --> 00:28:03,450 That, by the way, is a big assumption to make, that they 509 00:28:03,450 --> 00:28:04,620 must have been clever enough. 510 00:28:04,620 --> 00:28:08,010 We will find, in places during our course, that this was not 511 00:28:08,010 --> 00:28:09,400 always the case. 512 00:28:09,400 --> 00:28:10,870 The point is this though. 513 00:28:10,870 --> 00:28:16,950 That when you write 'dy dx' in this way, if it were not going 514 00:28:16,950 --> 00:28:21,010 to be somehow identifiable with a quotient such as 'dy' 515 00:28:21,010 --> 00:28:24,430 divided by 'dx', the chances are we would never have 516 00:28:24,430 --> 00:28:27,100 invented this notation in the first place. 517 00:28:27,100 --> 00:28:28,580 And so what I'd like to just point out 518 00:28:28,580 --> 00:28:30,860 briefly now is the following. 519 00:28:30,860 --> 00:28:34,780 Can we, in fact we should, how shall we-- 520 00:28:34,780 --> 00:28:35,940 let's put it this way-- 521 00:28:35,940 --> 00:28:42,160 how shall we define separate symbols dy and dx so that when 522 00:28:42,160 --> 00:28:46,030 we write down 'dy dx' it will make no difference whether you 523 00:28:46,030 --> 00:28:48,770 think of this as being the derivative or whether you 524 00:28:48,770 --> 00:28:52,230 think of it as being the quotient of two numbers? 525 00:28:52,230 --> 00:28:54,730 And by the way, you see the answer to this question will 526 00:28:54,730 --> 00:28:57,520 not be that difficult if we stop to think for just a 527 00:28:57,520 --> 00:28:59,610 moment over here. 528 00:28:59,610 --> 00:29:01,650 We've already answered this question except that we 529 00:29:01,650 --> 00:29:03,950 haven't concentrated on the fact that 530 00:29:03,950 --> 00:29:05,220 we solved the problem. 531 00:29:05,220 --> 00:29:07,350 See, let's go back to this line 'L' again. 532 00:29:07,350 --> 00:29:10,190 What is the slope of the line 'L'? 533 00:29:10,190 --> 00:29:13,130 On the one hand we've looked at slope as the 534 00:29:13,130 --> 00:29:15,580 derivative 'dy dx'. 535 00:29:15,580 --> 00:29:18,560 On the other hand, just by looking at this little 536 00:29:18,560 --> 00:29:23,140 accentuated triangle here, the slope of the line 'L' is also 537 00:29:23,140 --> 00:29:30,710 given by 'delta y-tan' divided by 'delta x'. 538 00:29:30,710 --> 00:29:37,690 And now one of the best ways to define 'dy' and 'dx' 539 00:29:37,690 --> 00:29:41,690 separately, it would seem to me, is what? 540 00:29:41,690 --> 00:29:45,880 Notice that this number divided by this number yields 541 00:29:45,880 --> 00:29:47,800 the symbol 'dy dx'. 542 00:29:47,800 --> 00:29:56,460 Therefore why not define this number to be 'dy' and define 543 00:29:56,460 --> 00:29:59,660 this number to be 'dx'? 544 00:29:59,660 --> 00:30:02,950 In other words the symbol 'dx' will just be a fancy 545 00:30:02,950 --> 00:30:04,510 name for 'delta x'. 546 00:30:04,510 --> 00:30:09,110 On the other hand, the symbol 'dy' will not be a fancy name 547 00:30:09,110 --> 00:30:10,020 for 'delta y'. 548 00:30:10,020 --> 00:30:12,200 In fact, let's emphasize that. 549 00:30:12,200 --> 00:30:17,720 It's not equal to 'delta y', it is equal to 'delta y-tan'. 550 00:30:17,720 --> 00:30:22,480 In other words, if I allow 'dy' to stand for 'delta 551 00:30:22,480 --> 00:30:27,740 y-tan' and 'dx' to stand for 'delta x', I can now treat 'dy 552 00:30:27,740 --> 00:30:30,260 dx' as if it were what? 553 00:30:30,260 --> 00:30:33,850 'dy' divided by 'dx'. 554 00:30:33,850 --> 00:30:39,160 Well, that's easy to show in terms of an example just 555 00:30:39,160 --> 00:30:40,670 mechanically. 556 00:30:40,670 --> 00:30:43,240 Remember, given 'y' equals 'x cubed' we've 557 00:30:43,240 --> 00:30:44,730 already found what? 558 00:30:44,730 --> 00:30:47,550 That 'dy dx' is '3 x squared'. 559 00:30:47,550 --> 00:30:51,320 This thing practically begs to be treated like a fraction. 560 00:30:51,320 --> 00:30:54,350 We would like to be able to say, hey, if this divided by 561 00:30:54,350 --> 00:30:59,700 this is this, why isn't this equal to this times this? 562 00:30:59,700 --> 00:31:04,420 In other words, why isn't 'dy' equal to '3 x squared' dx'? 563 00:31:04,420 --> 00:31:08,290 And the answer is that since we've identified or defined 564 00:31:08,290 --> 00:31:13,130 'dy' to be 'delta y-tan', and since we've defined 'dx' to be 565 00:31:13,130 --> 00:31:17,230 'delta x', and since we already know that this recipe 566 00:31:17,230 --> 00:31:20,530 is correct, it means, in particular, that we can now 567 00:31:20,530 --> 00:31:24,590 write things like this without having to worry about whether 568 00:31:24,590 --> 00:31:26,150 it's proper or not. 569 00:31:26,150 --> 00:31:29,710 Now in later lectures this is going to play a 570 00:31:29,710 --> 00:31:31,320 very important role. 571 00:31:31,320 --> 00:31:34,890 This is what is known as the language of differentials. 572 00:31:34,890 --> 00:31:39,390 And differentials are the backbone off both differential 573 00:31:39,390 --> 00:31:41,390 and integral calculus. 574 00:31:41,390 --> 00:31:45,500 But the thing that I want you to really get into our minds 575 00:31:45,500 --> 00:31:50,100 today is the basic overall recipe. 576 00:31:50,100 --> 00:31:54,270 And I've taken the liberty of boxing this off over here. 577 00:31:54,270 --> 00:31:56,530 I want you to practice with approximations. 578 00:31:56,530 --> 00:32:00,403 I want you to think carefully about how you can get quick 579 00:32:00,403 --> 00:32:01,110 approximations. 580 00:32:01,110 --> 00:32:03,580 I don't want you to come away with the feeling that the 581 00:32:03,580 --> 00:32:05,680 approximation is what was important. 582 00:32:05,680 --> 00:32:07,660 What was important was what? 583 00:32:07,660 --> 00:32:11,910 That the change in 'y', the true 'delta y', is ''dy dx' 584 00:32:11,910 --> 00:32:16,760 times 'delta x'' plus 'k delta x', where the limit of 'k' as 585 00:32:16,760 --> 00:32:21,140 'delta x' approaches 0 is 0. 586 00:32:21,140 --> 00:32:24,070 And by the way, again, most books write it this way. 587 00:32:24,070 --> 00:32:28,450 I prefer that we emphasize that the derivative is 588 00:32:28,450 --> 00:32:32,680 evaluated or taken at a particular value of 'x'. 589 00:32:32,680 --> 00:32:36,840 And finally what I'd like to point out is that again, and 590 00:32:36,840 --> 00:32:39,400 we've done this many, many times, whereas the language 591 00:32:39,400 --> 00:32:44,240 we're used to in terms of intuition is geometry, that 592 00:32:44,240 --> 00:32:48,310 these results make perfectly good sense without reference 593 00:32:48,310 --> 00:32:50,190 to any diagram at all. 594 00:32:50,190 --> 00:32:54,280 In other words, then, the same result stated in analytic 595 00:32:54,280 --> 00:32:58,960 language simply says this: if 'f' is a function of 'x' and 596 00:32:58,960 --> 00:33:02,940 is differentiable when 'x' equals 'x1', then 'f of 'x1 597 00:33:02,940 --> 00:33:05,830 plus delta x'' minus 'f of x1'-- 598 00:33:05,830 --> 00:33:08,450 you see that's geometrically what corresponds 599 00:33:08,450 --> 00:33:10,970 to your 'delta y'. 600 00:33:10,970 --> 00:33:15,860 That's ''f prime of x1' times 'delta x'' plus 'k delta x' 601 00:33:15,860 --> 00:33:21,510 where the limit of 'k' as 'delta x' approaches 0 is 0. 602 00:33:21,510 --> 00:33:27,840 Now these two recipes summarize precisely what it is 603 00:33:27,840 --> 00:33:30,930 that we are interested in when we deal with the subject 604 00:33:30,930 --> 00:33:32,780 called infinitesimals. 605 00:33:32,780 --> 00:33:35,600 As I said before, and I can't emphasize this thing strongly 606 00:33:35,600 --> 00:33:39,540 enough, when it comes time to make approximations we will 607 00:33:39,540 --> 00:33:43,120 find better ways of getting approximations than by the 608 00:33:43,120 --> 00:33:46,570 method known as differentials and 'delta y-tan'. 609 00:33:46,570 --> 00:33:49,630 What is crucial is that you use the language of 610 00:33:49,630 --> 00:33:53,780 approximations enough so as you can see pictorially what's 611 00:33:53,780 --> 00:33:58,190 going on and then cement down the final recipe, which I've 612 00:33:58,190 --> 00:33:59,260 boxed in here. 613 00:33:59,260 --> 00:34:03,080 This will be the building block, as we shall see you 614 00:34:03,080 --> 00:34:06,480 next time, in the lecture which develops the derivative 615 00:34:06,480 --> 00:34:08,389 of composite functions. 616 00:34:08,389 --> 00:34:10,860 But more about that next time. 617 00:34:10,860 --> 00:34:12,260 And until next time, goodbye. 618 00:34:12,260 --> 00:34:15,250 619 00:34:15,250 --> 00:34:18,449 Funding for the publication of this video was provided by the 620 00:34:18,449 --> 00:34:22,500 Gabriella and Paul Rosenbaum Foundation. 621 00:34:22,500 --> 00:34:26,679 Help OCW continue to provide free and open access to MIT 622 00:34:26,679 --> 00:34:30,870 courses by making a donation at ocw.mit.edu/donate. 623 00:34:30,870 --> 00:34:35,626