1 00:00:00,040 --> 00:00:01,940 ANNOUNCER: The following content is provided under a 2 00:00:01,940 --> 00:00:03,690 Creative Commons license. 3 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 4 00:00:06,630 --> 00:00:09,990 offer high quality educational resources for free. 5 00:00:09,990 --> 00:00:12,830 To make a donation or to view additional materials from 6 00:00:12,830 --> 00:00:17,010 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,010 --> 00:00:18,260 ocw.mit.edu. 8 00:00:29,720 --> 00:00:34,770 HERBERT GROSS: Hi, our lecture today is about curve plotting. 9 00:00:34,770 --> 00:00:38,710 And actually, I call it 'Curve Plotting with and without 10 00:00:38,710 --> 00:00:41,600 Calculus' to emphasize the fact that what we're 11 00:00:41,600 --> 00:00:45,340 interested in is curve plotting and that calculus, in 12 00:00:45,340 --> 00:00:49,460 particular, differentiation, gives us a powerful tool that 13 00:00:49,460 --> 00:00:52,960 is not available to us, at least in an accessible form 14 00:00:52,960 --> 00:00:54,820 without the calculus. 15 00:00:54,820 --> 00:00:58,550 Let's see what I mean by this by going back to a typical 16 00:00:58,550 --> 00:01:01,660 high school type analytical geometry problem. 17 00:01:01,660 --> 00:01:06,980 For example, sketch the curve 'y' equals 'x squared'. 18 00:01:06,980 --> 00:01:10,380 Now we all know that the graph of 'y' equals 'x squared' is 19 00:01:10,380 --> 00:01:12,080 something like this. 20 00:01:12,080 --> 00:01:13,660 And how'd we find that? 21 00:01:13,660 --> 00:01:16,480 You may recall that in the truer sense of the word, the 22 00:01:16,480 --> 00:01:21,420 pre-calculus approach really is curve plotting as opposed 23 00:01:21,420 --> 00:01:22,635 to curve sketching. 24 00:01:22,635 --> 00:01:23,950 And I hope I make that a little bit 25 00:01:23,950 --> 00:01:25,660 clearer as we go along. 26 00:01:25,660 --> 00:01:29,740 Namely, you look to see for certain values of 'x' what 27 00:01:29,740 --> 00:01:33,700 value of 'y' corresponds to that. 28 00:01:33,700 --> 00:01:36,960 And then we locate the corresponding point 'x' comma 29 00:01:36,960 --> 00:01:40,320 'y' in the plane of the blackboard. 30 00:01:40,320 --> 00:01:43,930 And what we then do is somehow or other, take a French curve, 31 00:01:43,930 --> 00:01:47,780 or whatever it happens to be, and we sketch a smooth curve 32 00:01:47,780 --> 00:01:49,310 through the given points. 33 00:01:49,310 --> 00:01:51,450 This is the usual technique. 34 00:01:51,450 --> 00:01:54,210 The question that comes up is, that as long as there are 35 00:01:54,210 --> 00:01:57,350 spaces between the points that you've sketched, how can you 36 00:01:57,350 --> 00:02:01,890 be sure that the curve that you've drawn isn't inaccurate? 37 00:02:01,890 --> 00:02:05,120 In other words, starting in reverse here, let's suppose 38 00:02:05,120 --> 00:02:08,080 that these are the points I've happened to sketch for the 39 00:02:08,080 --> 00:02:09,800 curve 'y' equals 'x squared'. 40 00:02:09,800 --> 00:02:12,910 And by the way, as is often the case in our course, don't 41 00:02:12,910 --> 00:02:15,210 be misled that we pick something as simple as 'y' 42 00:02:15,210 --> 00:02:16,400 equals 'x squared'. 43 00:02:16,400 --> 00:02:18,890 I simply, again, wanted to pick something that was 44 00:02:18,890 --> 00:02:22,350 straightforward enough so that we could concentrate our 45 00:02:22,350 --> 00:02:26,680 attention on what the mathematical implications were 46 00:02:26,680 --> 00:02:29,430 rather than to be bogged down by computation. 47 00:02:29,430 --> 00:02:30,510 But the idea is this. 48 00:02:30,510 --> 00:02:33,340 Going back to the problem 'y' equals 'x squared', suppose we 49 00:02:33,340 --> 00:02:36,490 had located these points and now we said, let's sketch a 50 00:02:36,490 --> 00:02:37,930 smooth curve through these. 51 00:02:37,930 --> 00:02:41,350 What would have been wrong with say, doing 52 00:02:41,350 --> 00:02:42,600 something like this? 53 00:02:45,080 --> 00:02:49,260 Why, for example, couldn't this have been the curve? 54 00:02:49,260 --> 00:02:54,290 Now again, notice that in terms of mathematical 55 00:02:54,290 --> 00:02:57,220 analysis, not necessarily calculus but in terms of 56 00:02:57,220 --> 00:03:00,210 mathematical analysis, we could immediately strike out 57 00:03:00,210 --> 00:03:01,410 something like this. 58 00:03:01,410 --> 00:03:05,590 For example, notice in terms of our input versus output, 59 00:03:05,590 --> 00:03:08,360 this says that for any real input since the square of a 60 00:03:08,360 --> 00:03:11,240 real number cannot be negative, the output can never 61 00:03:11,240 --> 00:03:12,120 be negative. 62 00:03:12,120 --> 00:03:15,910 And pictorially, this means that no portion of our diagram 63 00:03:15,910 --> 00:03:18,350 can be below the x-axis. 64 00:03:18,350 --> 00:03:20,980 In other words, without knowing anything about 65 00:03:20,980 --> 00:03:23,940 calculus but knowing a little bit about arithmetic, we can 66 00:03:23,940 --> 00:03:26,790 supplement our knowledge of how the points go by 67 00:03:26,790 --> 00:03:27,640 saying look-it . 68 00:03:27,640 --> 00:03:30,870 Something like this can't happen because in this region 69 00:03:30,870 --> 00:03:33,310 over here 'y' would be negative. 70 00:03:33,310 --> 00:03:35,430 And we know from the relationship that 'y' equals 71 00:03:35,430 --> 00:03:38,060 'x squared' that 'y' cannot be negative. 72 00:03:38,060 --> 00:03:42,820 Well you see, armed with this information, we could say, OK, 73 00:03:42,820 --> 00:03:45,750 given the same points here, why couldn't the curve go 74 00:03:45,750 --> 00:03:48,780 through something like this? 75 00:03:48,780 --> 00:03:50,240 That's what the question mark is in here for. 76 00:03:50,240 --> 00:03:52,530 Obviously, this is not the graph of 77 00:03:52,530 --> 00:03:53,470 'y' equals 'x squared'. 78 00:03:53,470 --> 00:03:57,280 But the question is, noticing that this curve doesn't go 79 00:03:57,280 --> 00:04:00,440 below the x-axis, what's wrong with this? 80 00:04:00,440 --> 00:04:04,710 And again, we can get bailed out by a pre-calculus 81 00:04:04,710 --> 00:04:06,710 knowledge of mathematics. 82 00:04:06,710 --> 00:04:09,980 Among other things, notice the rather interesting property 83 00:04:09,980 --> 00:04:13,710 that if we replace 'x' by 'minus x' here, we get the 84 00:04:13,710 --> 00:04:15,890 same curve as before. 85 00:04:15,890 --> 00:04:19,790 To generalize this result, notice that in this particular 86 00:04:19,790 --> 00:04:23,580 case, 'f of x' gives us the same result as 87 00:04:23,580 --> 00:04:25,470 'f of 'minus x''. 88 00:04:25,470 --> 00:04:27,330 Now this will not happen in general. 89 00:04:27,330 --> 00:04:30,760 What does that mean if we know that 'f of x' equals 'f of 90 00:04:30,760 --> 00:04:31,560 'minus x''? 91 00:04:31,560 --> 00:04:33,370 Well, look it. 92 00:04:33,370 --> 00:04:36,950 The relationship between 'x' and 'minus x' is clear, they 93 00:04:36,950 --> 00:04:42,580 are located symmetrically with respect to the y-axis. 94 00:04:42,580 --> 00:04:46,555 In other words, if this is 'x1', this will be 'minus x1'. 95 00:04:46,555 --> 00:04:48,250 Now, here's the point. 96 00:04:48,250 --> 00:04:50,140 Let's suppose our curve happens to be 'y' 97 00:04:50,140 --> 00:04:51,190 equals 'f of x'. 98 00:04:51,190 --> 00:04:55,660 When the input is 'x1', the output will be 'f of x1'. 99 00:04:55,660 --> 00:04:59,390 When the input is 'minus x1', the output will be 'f of 100 00:04:59,390 --> 00:05:00,750 'minus x1''. 101 00:05:00,750 --> 00:05:05,560 To say that 'f of 'minus x1'' equals 'f of x1' is the same 102 00:05:05,560 --> 00:05:10,710 as saying that not only are these two coordinates, 'x1' 103 00:05:10,710 --> 00:05:14,490 and 'minus x1', symmetric with respect to the y-axis, but the 104 00:05:14,490 --> 00:05:16,010 outputs also are. 105 00:05:16,010 --> 00:05:18,830 In other words, to say that 'f of x1' equals 'f of 'minus 106 00:05:18,830 --> 00:05:25,140 x1'', says that not only are these two points the same left 107 00:05:25,140 --> 00:05:28,370 and right displacement for the y-axis, but they're also the 108 00:05:28,370 --> 00:05:30,380 same height above the x-axis. 109 00:05:30,380 --> 00:05:34,340 In short, this point is the mirror image of this point. 110 00:05:34,340 --> 00:05:38,090 In terms of a graph, if 'f of x' equals 'f of 'minus x'' for 111 00:05:38,090 --> 00:05:41,990 all 'x', the graph of 'y' equals 'f of x' is symmetric 112 00:05:41,990 --> 00:05:44,220 with respect to the y-axis. 113 00:05:44,220 --> 00:05:49,110 By the way, that's why we could rule out this type of 114 00:05:49,110 --> 00:05:50,330 possibility here. 115 00:05:50,330 --> 00:05:54,760 For example, here's 'x1' over here. 116 00:05:54,760 --> 00:05:56,240 Here's 'minus x1'. 117 00:05:56,240 --> 00:06:00,350 And notice that for this choice of 'x', the points on 118 00:06:00,350 --> 00:06:03,730 the curve are not mirror images of one another with 119 00:06:03,730 --> 00:06:05,380 respect to the y-axis. 120 00:06:05,380 --> 00:06:07,460 In short, even though we may not know what this curve is 121 00:06:07,460 --> 00:06:10,840 supposed to look like, the fact that 'f of x' equals 'f 122 00:06:10,840 --> 00:06:14,180 of 'minus x'' tells us that whatever the graph is, it 123 00:06:14,180 --> 00:06:16,670 should be symmetric with respect to the y-axis. 124 00:06:16,670 --> 00:06:19,860 This leads us to a rather interesting aside. 125 00:06:19,860 --> 00:06:22,720 It's something called 'even and odd functions' that we 126 00:06:22,720 --> 00:06:25,410 could technically leave out here, but which I think is a 127 00:06:25,410 --> 00:06:26,690 good place to bring in. 128 00:06:26,690 --> 00:06:29,660 And the fact that there are many, many places in our 129 00:06:29,660 --> 00:06:32,060 advanced treatment that will come up later where it's 130 00:06:32,060 --> 00:06:35,060 important to understand what these things mean, that 131 00:06:35,060 --> 00:06:39,300 perhaps this is a nice place to bring it into play. 132 00:06:39,300 --> 00:06:41,080 So I call this an aside and it's about 133 00:06:41,080 --> 00:06:43,440 even and odd functions. 134 00:06:43,440 --> 00:06:47,350 When we have the case that 'f of x' equals 'f of 'minus x'', 135 00:06:47,350 --> 00:06:50,040 that's the case where in terms of the graph you have symmetry 136 00:06:50,040 --> 00:06:53,880 with respect to the y-axis such a function is called an 137 00:06:53,880 --> 00:06:55,370 'even function'. 138 00:06:55,370 --> 00:06:57,710 And I'll come back in a moment and show you why it's called 139 00:06:57,710 --> 00:07:00,740 even, though it's not really important. 140 00:07:00,740 --> 00:07:03,520 The counterpart to an even function as you may guess 141 00:07:03,520 --> 00:07:05,870 almost from the association of ideas is 142 00:07:05,870 --> 00:07:08,290 called an 'odd function'. 143 00:07:08,290 --> 00:07:13,280 Now an odd function is one for which 'f of x' is the negative 144 00:07:13,280 --> 00:07:15,200 of 'f of 'minus x''. 145 00:07:15,200 --> 00:07:19,310 See for 'f' to be odd, 'f of x' has to be the negative of 146 00:07:19,310 --> 00:07:20,760 'f of 'minus x''. 147 00:07:20,760 --> 00:07:22,720 Now what does this mean pictorially? 148 00:07:22,720 --> 00:07:27,860 Again, 'x1' and 'minus x1' are symmetrically located with 149 00:07:27,860 --> 00:07:32,920 respect to the y-axis. 150 00:07:32,920 --> 00:07:37,610 This height would be called 'f of x1' and this height here 151 00:07:37,610 --> 00:07:42,000 would be called 'f of 'minus x1''. 152 00:07:42,000 --> 00:07:47,340 And to say that these two heights are equal in magnitude 153 00:07:47,340 --> 00:07:49,530 but opposite in sign means what? 154 00:07:49,530 --> 00:07:53,130 That these two lengths are equal but on opposite sides of 155 00:07:53,130 --> 00:07:54,050 the x-axis. 156 00:07:54,050 --> 00:07:57,960 In other words, here's 'f of x1'. 157 00:07:57,960 --> 00:07:59,770 'f of 'minus x1'' is the height 158 00:07:59,770 --> 00:08:01,330 corresponding to this point. 159 00:08:01,330 --> 00:08:04,720 And the fact that these must be equal in magnitude but 160 00:08:04,720 --> 00:08:08,370 opposite in sign says that one of the heights must be above 161 00:08:08,370 --> 00:08:11,630 the axis, the other one must be below the axis. 162 00:08:11,630 --> 00:08:15,530 And if that's the case from some very elementary geometry, 163 00:08:15,530 --> 00:08:18,360 if 'f' is odd, what it means with respect to 164 00:08:18,360 --> 00:08:19,780 its graph is this. 165 00:08:19,780 --> 00:08:25,420 If you put a ruler connecting the origin with any point on 166 00:08:25,420 --> 00:08:31,820 the curve, if you extend that line that line will meet the 167 00:08:31,820 --> 00:08:35,919 curve again, such that these distances here will be equal. 168 00:08:35,919 --> 00:08:38,830 This is called symmetry with respect to the origin. 169 00:08:38,830 --> 00:08:41,750 By the way, an example of this kind of a curve is 170 00:08:41,750 --> 00:08:43,480 'y' equals 'x cubed'. 171 00:08:43,480 --> 00:08:48,360 You see, if I replace 'x' by 'minus x', this becomes what? 172 00:08:48,360 --> 00:08:50,550 'Minus x cubed'. 173 00:08:50,550 --> 00:08:55,490 And 'minus 'minus x cubed'' is 'x cubed'. 174 00:08:55,490 --> 00:08:57,940 By the way, all I'm saying now is what? 175 00:08:57,940 --> 00:09:02,060 If I were to take a ruler and place it here and let this 176 00:09:02,060 --> 00:09:05,110 line go from curve to curve. 177 00:09:05,110 --> 00:09:07,370 all I'm saying is that this portion 178 00:09:07,370 --> 00:09:08,890 would equal this portion. 179 00:09:08,890 --> 00:09:12,300 And by the way, maybe you can already guess where the words 180 00:09:12,300 --> 00:09:14,410 even and odd come from. 181 00:09:14,410 --> 00:09:17,420 Notice that when I dealt with 'y' equals 'x squared' I had 182 00:09:17,420 --> 00:09:18,990 an even function. 183 00:09:18,990 --> 00:09:20,470 The exponent was even. 184 00:09:20,470 --> 00:09:23,170 When I dealt with 'y' equals 'x cubed', 185 00:09:23,170 --> 00:09:24,740 I had an odd function. 186 00:09:24,740 --> 00:09:26,230 The exponent was odd. 187 00:09:26,230 --> 00:09:28,140 And by the way, there are other examples. 188 00:09:28,140 --> 00:09:32,710 But for the time being, notice something like say, 'y' equals 189 00:09:32,710 --> 00:09:35,420 'x to the fourth' plus 'x squared' say. 190 00:09:46,030 --> 00:09:49,100 If I replace 'x' by 'minus x', I get back the same thing. 191 00:09:49,100 --> 00:09:52,190 In other words, here's a case where if 'y' equals 'f of x', 192 00:09:52,190 --> 00:09:56,150 'f of x' is the same as 'f of 'minus x''. 193 00:09:56,150 --> 00:09:59,980 An example of an odd function might be 'y' equals 'x cubed' 194 00:09:59,980 --> 00:10:02,220 say, plus 'x'. 195 00:10:02,220 --> 00:10:03,720 See first power over here. 196 00:10:03,720 --> 00:10:08,080 If I replace 'x' by 'minus x', I have 'minus x 197 00:10:08,080 --> 00:10:10,840 cubed' plus 'minus x'. 198 00:10:10,840 --> 00:10:14,120 That of course, is just 'minus x cubed' minus 'x'. 199 00:10:14,120 --> 00:10:18,040 And that's minus the quantity 'x cubed + x'. 200 00:10:18,040 --> 00:10:21,240 That when I replace 'x' by 'minus x' all I do is change 201 00:10:21,240 --> 00:10:22,300 the sign here. 202 00:10:22,300 --> 00:10:26,070 In other words, that would be an example for which 'f of x' 203 00:10:26,070 --> 00:10:30,350 is minus 'f of 'minus x''. 204 00:10:30,350 --> 00:10:32,670 By the way, while we're dealing with examples like 205 00:10:32,670 --> 00:10:36,120 this, unlike the case with whole numbers where a whole 206 00:10:36,120 --> 00:10:41,000 number is either even or odd but not both, it's rather 207 00:10:41,000 --> 00:10:45,590 important to notice that a function need not be either 208 00:10:45,590 --> 00:10:47,550 even or odd. 209 00:10:47,550 --> 00:10:50,220 And by the way, if you think of our geometric definition, 210 00:10:50,220 --> 00:10:52,560 that's not too hard to see. 211 00:10:52,560 --> 00:10:55,370 Namely, there's no reason why a curve drawn at random should 212 00:10:55,370 --> 00:10:58,430 be symmetric either with respect to the y-axis or with 213 00:10:58,430 --> 00:10:59,390 respect to the origin. 214 00:10:59,390 --> 00:11:03,940 In fact, maybe the quickest way to see this is to put an 215 00:11:03,940 --> 00:11:07,060 odd power of 'x' in with an even power of 216 00:11:07,060 --> 00:11:08,580 'x' in the same diagram. 217 00:11:08,580 --> 00:11:14,420 If I now replace 'x' by 'minus x' you see 'minus x' cubed is 218 00:11:14,420 --> 00:11:15,990 of course, 'minus x cubed'. 219 00:11:15,990 --> 00:11:19,670 But 'minus x' quantity squared is just 'x squared'. 220 00:11:19,670 --> 00:11:22,750 And now you'll notice that if I compare these two, I don't 221 00:11:22,750 --> 00:11:25,040 get the same thing nor do I get the same 222 00:11:25,040 --> 00:11:26,850 thing with a sign change. 223 00:11:26,850 --> 00:11:30,020 You see, in other words, this is not either equal to this or 224 00:11:30,020 --> 00:11:32,020 to the negative of this. 225 00:11:32,020 --> 00:11:34,310 Obviously you understand that to be the negative of this, 226 00:11:34,310 --> 00:11:37,770 there would also have to be a minus sign over here. 227 00:11:37,770 --> 00:11:43,400 Or if we wanted to go into more detail about this, it is 228 00:11:43,400 --> 00:11:46,390 perhaps exciting to know that whereas it's not true that 229 00:11:46,390 --> 00:11:50,480 every function is either even or odd, every function can be 230 00:11:50,480 --> 00:11:53,880 written as the sum of two functions, one of which is 231 00:11:53,880 --> 00:11:56,750 even, one of which is odd. 232 00:11:56,750 --> 00:12:00,800 Just to give you an idea of what that means let me just 233 00:12:00,800 --> 00:12:02,180 write down something here. 234 00:12:02,180 --> 00:12:05,220 This will come back to be more important later on, but for 235 00:12:05,220 --> 00:12:09,060 the time being, just to show you a connecting thread here 236 00:12:09,060 --> 00:12:11,970 as long as we're talking about even and odd functions, all I 237 00:12:11,970 --> 00:12:14,480 want to see is the following identity. 238 00:12:14,480 --> 00:12:20,400 If you write ''f of x' plus 'f of 'minus x'' over 2. 239 00:12:20,400 --> 00:12:22,840 And don't worry about what motivates this, I just wanted 240 00:12:22,840 --> 00:12:25,820 to show you this to keep this fairly complete. 241 00:12:25,820 --> 00:12:31,000 Now suppose I add on to that ''f of x' minus 'f of 'minus 242 00:12:31,000 --> 00:12:33,690 x'' over 2. 243 00:12:33,690 --> 00:12:35,740 You see, notice that the expression on the right-hand 244 00:12:35,740 --> 00:12:38,750 side is just writing 'f of x' the hard way. 245 00:12:38,750 --> 00:12:41,720 See, here's half of 'f of x', here's half of 'f of x'. 246 00:12:41,720 --> 00:12:43,190 The sum is 'f of x'. 247 00:12:43,190 --> 00:12:45,470 And here's half of 'f of 'minus x''. 248 00:12:45,470 --> 00:12:48,250 And I'm subtracting half of 'f of 'minus x''. 249 00:12:48,250 --> 00:12:49,490 That drops out. 250 00:12:49,490 --> 00:12:55,080 The point is that this is always an even function and 251 00:12:55,080 --> 00:12:57,550 this is always an odd function. 252 00:12:57,550 --> 00:12:59,830 And just to review the definition so that you see 253 00:12:59,830 --> 00:13:03,890 what happens here, all I'm saying is if I replace 'x' by 254 00:13:03,890 --> 00:13:06,720 'minus x' here, what happens? 255 00:13:06,720 --> 00:13:11,340 If I replace 'x' by 'minus x' this becomes 'f of 'minus x''. 256 00:13:11,340 --> 00:13:15,390 And if I replace 'x' by 'minus x' here since minus minus is 257 00:13:15,390 --> 00:13:17,970 plus, this becomes 'f of x'. 258 00:13:17,970 --> 00:13:20,800 Now notice that when you add two terms, the sum is 259 00:13:20,800 --> 00:13:23,140 independent of the order in which you add them. 260 00:13:23,140 --> 00:13:27,170 'f of x' plus 'f of 'minus x'', therefore, is the same as 261 00:13:27,170 --> 00:13:29,680 'f of 'minus x'' plus 'f of x'. 262 00:13:29,680 --> 00:13:33,000 In other words, when I replace 'x' by 'minus x' in this 263 00:13:33,000 --> 00:13:36,970 bracketed function, I do not change the value of what's in 264 00:13:36,970 --> 00:13:38,070 the brackets. 265 00:13:38,070 --> 00:13:43,230 On the other hand, if I interchange 'x' with 'minus x' 266 00:13:43,230 --> 00:13:46,740 here, notice that since I'm subtracting look what happens. 267 00:13:46,740 --> 00:13:48,790 I replace 'x' by 'minus x'. 268 00:13:48,790 --> 00:13:50,770 This gives me an 'f of 'minus x''. 269 00:13:50,770 --> 00:13:52,810 Now I make the same replacement here. 270 00:13:52,810 --> 00:13:54,840 Minus minus is positive. 271 00:13:54,840 --> 00:13:58,130 But now notice that if I look at this expression here, I've 272 00:13:58,130 --> 00:13:59,440 changed the order. 273 00:13:59,440 --> 00:14:01,960 See 'f of x' minus 'f of 'minus x''. 274 00:14:01,960 --> 00:14:04,990 Here, 'f of 'minus x'' minus 'f of x'. 275 00:14:04,990 --> 00:14:08,650 And when you change the order you change the sign. 276 00:14:08,650 --> 00:14:11,750 In other words then, all I'm saying is that when we talk 277 00:14:11,750 --> 00:14:14,230 about even and odd functions, they play a very important 278 00:14:14,230 --> 00:14:19,130 role in calculus and in other mathematical analysis topics. 279 00:14:19,130 --> 00:14:22,590 That not every function is either even or odd, but every 280 00:14:22,590 --> 00:14:26,340 function that's defined on the appropriate domain is the sum 281 00:14:26,340 --> 00:14:29,020 of both an even and an odd function. 282 00:14:29,020 --> 00:14:31,730 But in terms of curve plotting, the main point is 283 00:14:31,730 --> 00:14:35,120 not so much these extra remarks as much as what? 284 00:14:35,120 --> 00:14:38,930 An even function is symmetric with respect to the y-axis and 285 00:14:38,930 --> 00:14:42,110 an odd function is symmetric with respect to the origin. 286 00:14:42,110 --> 00:14:46,000 At any rate, if we now go back to our curve 'y' equals 'x 287 00:14:46,000 --> 00:14:54,290 squared' the fact that 'f of x' equals 'x squared' is even 288 00:14:54,290 --> 00:14:57,670 means that whatever our graph looks like, it must be 289 00:14:57,670 --> 00:15:00,200 symmetric with respect to the y-axis. 290 00:15:00,200 --> 00:15:02,780 Now give or take how I've drawn this, this should be 291 00:15:02,780 --> 00:15:05,100 symmetric with respect to the y-axis. 292 00:15:05,100 --> 00:15:09,210 If it doesn't look that way, imagine that it is that way. 293 00:15:09,210 --> 00:15:10,570 And so the idea is what? 294 00:15:10,570 --> 00:15:13,650 Well, any knowledge of calculus whatsoever, what I 295 00:15:13,650 --> 00:15:16,880 was able to do here is show that whatever the graph of 'y' 296 00:15:16,880 --> 00:15:20,960 equals 'x squared' is, it must never dip below the x-axis. 297 00:15:20,960 --> 00:15:23,230 And whatever the curve looks like to the right of the 298 00:15:23,230 --> 00:15:26,900 y-axis, it must be the mirror image of that to the left of 299 00:15:26,900 --> 00:15:28,000 the y-axis. 300 00:15:28,000 --> 00:15:31,010 Again, this is how much one can do without calculus. 301 00:15:31,010 --> 00:15:34,100 And most of you who are practicing engineers, I'm sure 302 00:15:34,100 --> 00:15:37,140 not only understand this type of technique as far as the 303 00:15:37,140 --> 00:15:40,440 pre-calculus is concerned, but can probably draw curves much 304 00:15:40,440 --> 00:15:41,530 better than I can. 305 00:15:41,530 --> 00:15:44,260 In fact, even if you're not practicing engineers you can 306 00:15:44,260 --> 00:15:46,800 probably draw curves much better than I can. 307 00:15:46,800 --> 00:15:48,270 But that part is irrelevant. 308 00:15:48,270 --> 00:15:50,180 What I wanted to show up to now-- 309 00:15:50,180 --> 00:15:51,930 and this is what's important to stress-- 310 00:15:51,930 --> 00:15:55,360 is that to get as far as I've gotten so far, I did not have 311 00:15:55,360 --> 00:15:58,090 to have any knowledge of calculus. 312 00:15:58,090 --> 00:16:02,130 The way calculus comes in, as I say again, is a supplement 313 00:16:02,130 --> 00:16:04,000 to our previous techniques. 314 00:16:04,000 --> 00:16:06,630 For example, let's suppose we did have the 315 00:16:06,630 --> 00:16:09,270 curve drawn this way. 316 00:16:09,270 --> 00:16:13,150 From this, we certainly aren't contradicting the fact that 317 00:16:13,150 --> 00:16:15,140 'x' can't be negative. 318 00:16:15,140 --> 00:16:16,750 We're not contradicting the fact that 319 00:16:16,750 --> 00:16:18,110 'f' is an even function. 320 00:16:18,110 --> 00:16:20,120 But how do we know that this is the wrong picture? 321 00:16:20,120 --> 00:16:24,710 Well, given that 'y' equals 'x squared', we can easily verify 322 00:16:24,710 --> 00:16:29,400 that 'dy dx' is '2x.' Knowing that 'dy dx' is '2x', that 323 00:16:29,400 --> 00:16:32,400 tells us among other things that 'dy dx' and 'x' 324 00:16:32,400 --> 00:16:33,740 have the same sign. 325 00:16:33,740 --> 00:16:37,310 In other words, 'dy dx' is positive if 'x' is positive, 326 00:16:37,310 --> 00:16:40,690 'dy dx' is negative if 'x' is negative. 327 00:16:40,690 --> 00:16:43,760 In terms of geometry, that means what? 328 00:16:43,760 --> 00:16:47,020 That the curve is always rising for positive values of 329 00:16:47,020 --> 00:16:50,730 'x' and always falling for negative values of 'x'. 330 00:16:50,730 --> 00:16:54,060 Well, you see with that as a hint, I say ah-ha. 331 00:16:54,060 --> 00:16:55,480 This can't happen. 332 00:16:55,480 --> 00:16:57,340 Because look what's happening over here. 333 00:16:57,340 --> 00:16:58,840 Or for that matter, over here. 334 00:16:58,840 --> 00:17:03,740 Here the curve is falling for positive values of 'x'. 335 00:17:03,740 --> 00:17:06,280 And that contradicts the fact that the curve must always be 336 00:17:06,280 --> 00:17:08,240 rising when 'x' is positive. 337 00:17:08,240 --> 00:17:11,660 In a similar way, we know that the curve can't be rising when 338 00:17:11,660 --> 00:17:16,319 'x' is negative, yet get over here and here too, we've drawn 339 00:17:16,319 --> 00:17:17,730 the curve to be rising. 340 00:17:17,730 --> 00:17:22,200 That again is contradicted by this diagram. 341 00:17:22,200 --> 00:17:24,099 So you see the knowledge of the first 342 00:17:24,099 --> 00:17:26,410 derivative does what? 343 00:17:26,410 --> 00:17:29,420 It tells us where the curve is rising or falling, and that 344 00:17:29,420 --> 00:17:31,590 gives us another way of checking whether the graph 345 00:17:31,590 --> 00:17:33,610 we've drawn is accurate or not. 346 00:17:33,610 --> 00:17:35,690 By the way, it's not all quite that simple. 347 00:17:35,690 --> 00:17:39,570 And notice again, subtlety how step by step we strengthen our 348 00:17:39,570 --> 00:17:41,290 procedures each time. 349 00:17:41,290 --> 00:17:44,690 For example, now knowing that the curve must always be 350 00:17:44,690 --> 00:17:48,390 rising when 'x' is positive and always falling when 'x' is 351 00:17:48,390 --> 00:17:51,810 negative, how about this possibility for the graph of 352 00:17:51,810 --> 00:17:53,770 'y' equals 'x squared'. 353 00:17:53,770 --> 00:17:56,260 You see this curve is always fallen for negative values of 354 00:17:56,260 --> 00:17:59,630 'x', it's always rising for positive values of 'x'. 355 00:18:02,380 --> 00:18:04,720 Let's take a look now at what the second derivative means. 356 00:18:04,720 --> 00:18:08,470 If 'y' equals 'x squared', obviously as we saw before, 357 00:18:08,470 --> 00:18:14,150 'dy dx' is '2x' and the second derivative of 'y' with respect 358 00:18:14,150 --> 00:18:16,450 to 'x' is 2. 359 00:18:16,450 --> 00:18:19,280 And 2 is a constant, which is always positive. 360 00:18:19,280 --> 00:18:20,210 This says what? 361 00:18:20,210 --> 00:18:23,110 That the second derivative is always positive. 362 00:18:23,110 --> 00:18:24,740 Now what is the second derivative? 363 00:18:24,740 --> 00:18:27,230 The second derivative is the first derivative of the first 364 00:18:27,230 --> 00:18:28,270 derivative. 365 00:18:28,270 --> 00:18:31,780 That means the rate of change of the rate of change. 366 00:18:31,780 --> 00:18:34,250 Well, the rate of change of the rate of change is called 367 00:18:34,250 --> 00:18:36,850 acceleration. 368 00:18:36,850 --> 00:18:39,600 So if the rate of change of the rate of change is positive 369 00:18:39,600 --> 00:18:42,010 that means that the curve must be accelerating, or the 370 00:18:42,010 --> 00:18:43,470 function is accelerating. 371 00:18:43,470 --> 00:18:45,810 And if it's negative, function is decelerating. 372 00:18:45,810 --> 00:18:48,290 What does that mean in terms of a picture? 373 00:18:48,290 --> 00:18:50,530 And the author of the text uses a very 374 00:18:50,530 --> 00:18:51,980 descriptive phrase here. 375 00:18:51,980 --> 00:18:56,010 He talks about 'holding water' and 'spilling water'. 376 00:18:56,010 --> 00:19:00,040 Notice, for example, here the curve would tend to collect 377 00:19:00,040 --> 00:19:03,180 water, whereas here if water were poured on it, the curve 378 00:19:03,180 --> 00:19:05,070 would tend to spill water. 379 00:19:05,070 --> 00:19:08,140 Holding water represents acceleration. 380 00:19:08,140 --> 00:19:10,680 You see that not only is the curve rising here, but it's 381 00:19:10,680 --> 00:19:13,710 rising at a faster and faster rate. 382 00:19:13,710 --> 00:19:15,860 Again, more primitively, in terms of 383 00:19:15,860 --> 00:19:17,780 slopes, notice that what? 384 00:19:17,780 --> 00:19:20,330 Not only is the slope positive, but as you move 385 00:19:20,330 --> 00:19:24,350 along this portion of the curve, the slope increases as 386 00:19:24,350 --> 00:19:26,020 you move along. 387 00:19:26,020 --> 00:19:28,900 And what typifies this portion? 388 00:19:28,900 --> 00:19:32,500 That even though the slope is always positive, it decreases 389 00:19:32,500 --> 00:19:34,010 as you move along the curve. 390 00:19:34,010 --> 00:19:36,480 In other words, 'holding water' corresponds to the 391 00:19:36,480 --> 00:19:39,320 second derivative being positive, 'spilling water' 392 00:19:39,320 --> 00:19:42,410 corresponds to the second derivative being negative. 393 00:19:42,410 --> 00:19:45,540 Returning then to our original problem, the fact that this 394 00:19:45,540 --> 00:19:52,160 thing here is greater than 0 for all 'x' says that this 395 00:19:52,160 --> 00:19:54,080 curve could never spill water. 396 00:19:54,080 --> 00:19:57,650 And that rules out this portion in here. 397 00:19:57,650 --> 00:19:58,990 In other words, now we put everything 398 00:19:58,990 --> 00:20:00,670 together, we know what? 399 00:20:00,670 --> 00:20:03,660 The curve can never go below the x-axis. 400 00:20:03,660 --> 00:20:06,960 It's symmetric with respect to the y-axis. 401 00:20:06,960 --> 00:20:09,350 It's always rising when 'x' is positive and 402 00:20:09,350 --> 00:20:11,700 always holding water. 403 00:20:11,700 --> 00:20:15,160 Now you see this is what I call curve sketching versus 404 00:20:15,160 --> 00:20:16,080 curve plotting. 405 00:20:16,080 --> 00:20:18,900 With the information that I have from calculus, I know 406 00:20:18,900 --> 00:20:22,010 what's going on for each point, not just for the 407 00:20:22,010 --> 00:20:25,590 isolated points that I happened to have plotted in 408 00:20:25,590 --> 00:20:26,840 the data for. 409 00:20:28,970 --> 00:20:31,910 See, the calculus fills in the missing 410 00:20:31,910 --> 00:20:33,600 data very, very nicely. 411 00:20:33,600 --> 00:20:38,230 Now you see this does not mean I'm going to replace this by-- 412 00:20:38,230 --> 00:20:39,980 my previous analysis by calculus. 413 00:20:39,980 --> 00:20:42,950 It means I'm going to add calculus as one of my bags of 414 00:20:42,950 --> 00:20:43,890 tools here. 415 00:20:43,890 --> 00:20:47,970 Notice again that there is a very nice relationship between 416 00:20:47,970 --> 00:20:49,810 pictures and analysis. 417 00:20:49,810 --> 00:20:51,450 And I'm not going to belabor that point. 418 00:20:51,450 --> 00:20:53,820 All I'm saying is that if you add to our previous 419 00:20:53,820 --> 00:20:54,810 identifications-- 420 00:20:54,810 --> 00:20:56,430 what identifications? 421 00:20:56,430 --> 00:20:58,780 Like increase means rising. 422 00:20:58,780 --> 00:21:01,060 I mean this, if a derivative is positive, 423 00:21:01,060 --> 00:21:02,240 the curve is rising. 424 00:21:02,240 --> 00:21:04,970 If a derivative is negative, the curve is falling. 425 00:21:04,970 --> 00:21:07,450 If the second derivative is positive, the 426 00:21:07,450 --> 00:21:08,770 curve is holding water. 427 00:21:08,770 --> 00:21:10,920 If the second derivative is negative, the curve is 428 00:21:10,920 --> 00:21:11,820 spilling water. 429 00:21:11,820 --> 00:21:15,360 And again, we have ample exercises and portions of this 430 00:21:15,360 --> 00:21:17,630 in the reading material to illustrate the 431 00:21:17,630 --> 00:21:19,030 computational aspects. 432 00:21:19,030 --> 00:21:22,090 But again, all I want you to get from this lecture is 433 00:21:22,090 --> 00:21:24,740 what's happening here conceptually. 434 00:21:24,740 --> 00:21:28,640 Let's look at this, a few more applications. 435 00:21:28,640 --> 00:21:31,990 We're going to find in subsequent lectures as well as 436 00:21:31,990 --> 00:21:34,280 other portions of the course, we're going to be interested, 437 00:21:34,280 --> 00:21:37,810 for example, in things called stationary points. 438 00:21:37,810 --> 00:21:40,540 A 'stationary point' is a point at which the curve is 439 00:21:40,540 --> 00:21:42,230 neither rising nor falling. 440 00:21:42,230 --> 00:21:44,830 In other words, if the curve happens to be smooth, it's 441 00:21:44,830 --> 00:21:48,520 characterized by the fact that 'dy dx' is 0 it such a point. 442 00:21:48,520 --> 00:21:51,840 In terms of the language of functions to say that the 443 00:21:51,840 --> 00:21:55,100 curve is smooth means that the function is differentiable and 444 00:21:55,100 --> 00:21:58,910 what we're saying is to be stationary at 'x' equals 'x1', 445 00:21:58,910 --> 00:22:03,460 it must be that 'f prime of x1' equals 0. 446 00:22:03,460 --> 00:22:06,370 And the importance of stationary points can be seen 447 00:22:06,370 --> 00:22:09,020 in terms of a physical interpretation. 448 00:22:09,020 --> 00:22:11,970 We haven't used our freely falling body for quite a 449 00:22:11,970 --> 00:22:14,610 while, let's go back to such an example. 450 00:22:14,610 --> 00:22:20,750 Suppose a particle is projected vertically upward in 451 00:22:20,750 --> 00:22:23,880 the absence of air resistance, et cetera, with an initial 452 00:22:23,880 --> 00:22:26,690 speed of 160 feet per second. 453 00:22:26,690 --> 00:22:30,560 It can then be shown that the height 's' to which the ball 454 00:22:30,560 --> 00:22:33,520 rises in feet at time 't' is given by 455 00:22:33,520 --> 00:22:37,100 '160t - 16't squared''. 456 00:22:37,100 --> 00:22:43,460 A very natural question to ask is, when will the ball be at 457 00:22:43,460 --> 00:22:45,450 its maximum height? 458 00:22:45,450 --> 00:22:47,360 And I'm sure you can see in terms of this physical 459 00:22:47,360 --> 00:22:50,870 example, the ball will be at its maximum height when the 460 00:22:50,870 --> 00:22:53,020 velocity is 0. 461 00:22:53,020 --> 00:22:57,840 In other words, since this is a smooth type of motion, if 462 00:22:57,840 --> 00:23:02,910 the velocity is not 0, the ball is still rising. 463 00:23:02,910 --> 00:23:06,010 If the velocity is positive, the ball is still rising. 464 00:23:06,010 --> 00:23:10,340 If the velocity is negative, the ball is already falling. 465 00:23:10,340 --> 00:23:13,060 Consequently, if the speed is smooth, which it is in this 466 00:23:13,060 --> 00:23:16,430 case, the only way it can go from rising to falling is to 467 00:23:16,430 --> 00:23:20,550 first level off and the velocity must be 0. 468 00:23:20,550 --> 00:23:22,760 But you see, what is the velocity? 469 00:23:22,760 --> 00:23:26,440 The velocity is the derivative of the displacement. 470 00:23:26,440 --> 00:23:28,500 In other words, without solving this problem, which is 471 00:23:28,500 --> 00:23:32,220 not important here, to find the time at which you have the 472 00:23:32,220 --> 00:23:34,960 maximum height, you simply do what? 473 00:23:34,960 --> 00:23:37,410 Set the derivative equal to 0. 474 00:23:37,410 --> 00:23:40,870 In other words, stationary points tell us where we have 475 00:23:40,870 --> 00:23:42,740 high and low points for functions. 476 00:23:42,740 --> 00:23:45,480 And knowing where we have high and low points is a very 477 00:23:45,480 --> 00:23:47,440 important portion in curve plotting. 478 00:23:47,440 --> 00:23:50,580 We'll talk about this in a future lecture very shortly in 479 00:23:50,580 --> 00:23:51,460 more detail. 480 00:23:51,460 --> 00:23:55,670 But coming back here to what we were talking about before, 481 00:23:55,670 --> 00:23:59,980 what we're saying in terms of curve plotting is that where 482 00:23:59,980 --> 00:24:05,600 the derivative is 0 gives us a good candidate to have either 483 00:24:05,600 --> 00:24:09,300 a low point on the curve or to have a high 484 00:24:09,300 --> 00:24:11,940 point on the curve. 485 00:24:11,940 --> 00:24:14,990 However, we should not read more into this than what's 486 00:24:14,990 --> 00:24:16,340 already there. 487 00:24:16,340 --> 00:24:20,960 Namely, it's possible that you have what? 488 00:24:20,960 --> 00:24:25,950 A situation like this, in which the derivative is 0 489 00:24:25,950 --> 00:24:28,930 here, but the curve is rising every place. 490 00:24:28,930 --> 00:24:33,790 And secondly, there is the possibility that if the 491 00:24:33,790 --> 00:24:35,975 derivative doesn't exist, for example, if 492 00:24:35,975 --> 00:24:38,490 there's a sharp corner. 493 00:24:38,490 --> 00:24:41,170 Where you have a sharp corner notice that-- see this is a 494 00:24:41,170 --> 00:24:43,410 straight line, this is a straight line. 495 00:24:43,410 --> 00:24:45,490 In this special case, the derivative is the slope of 496 00:24:45,490 --> 00:24:46,870 this straight line. 497 00:24:46,870 --> 00:24:48,940 Derivative here is the slope of this straight line. 498 00:24:48,940 --> 00:24:51,700 Therefore, the derivative is positive on this line, 499 00:24:51,700 --> 00:24:53,370 negative on this line. 500 00:24:53,370 --> 00:24:55,440 Yet the point is what? 501 00:24:55,440 --> 00:24:58,300 At their junction, there's a discontinuity. 502 00:24:58,300 --> 00:25:01,050 The function is continuous, but the derivative isn't. 503 00:25:01,050 --> 00:25:03,400 And at that particular point notice that you have a high 504 00:25:03,400 --> 00:25:06,580 point even though the derivative doesn't exist at 505 00:25:06,580 --> 00:25:07,840 that particular point. 506 00:25:07,840 --> 00:25:08,970 All we're saying is what? 507 00:25:08,970 --> 00:25:14,410 That for a smooth curve if there is to be a high point or 508 00:25:14,410 --> 00:25:17,780 a low point, a maximum or a minimum, and we'll talk about 509 00:25:17,780 --> 00:25:21,750 this as I say, in a future lecture, it must be that at 510 00:25:21,750 --> 00:25:25,190 particular point the derivative is 0. 511 00:25:25,190 --> 00:25:28,110 On the other hand, conversely, if the derivative is 0, you 512 00:25:28,110 --> 00:25:29,960 may not have a high or low point. 513 00:25:29,960 --> 00:25:32,280 It may be what we call a saddle point, the curve just 514 00:25:32,280 --> 00:25:35,410 levels off after rising and then rises again. 515 00:25:35,410 --> 00:25:38,440 And secondly, if the function isn't differentiable, or the 516 00:25:38,440 --> 00:25:41,050 curve isn't smooth at that particular point, you can have 517 00:25:41,050 --> 00:25:43,790 a high or a low point regardless of what the 518 00:25:43,790 --> 00:25:46,740 knowledge about the derivative is. 519 00:25:46,740 --> 00:25:49,130 Just a little buckshot here to give you an idea of how we're 520 00:25:49,130 --> 00:25:51,250 going to use this material. 521 00:25:51,250 --> 00:25:56,030 A very related topic that's also quite important here is 522 00:25:56,030 --> 00:25:58,740 something called points of inflection. 523 00:25:58,740 --> 00:26:02,540 Points of inflection are, in a way, to the second derivative 524 00:26:02,540 --> 00:26:05,930 what stationary points are to the first derivative. 525 00:26:05,930 --> 00:26:09,020 In many cases, we are interested in knowing, where 526 00:26:09,020 --> 00:26:11,090 does the curve change its concavity? 527 00:26:11,090 --> 00:26:13,770 Where does it go, in other words, from holding water to 528 00:26:13,770 --> 00:26:15,510 spelling water? 529 00:26:15,510 --> 00:26:19,780 And by the way, again, in terms of a geometrical 530 00:26:19,780 --> 00:26:23,860 interpretation, there's a very what I call exciting answer to 531 00:26:23,860 --> 00:26:24,780 this question. 532 00:26:24,780 --> 00:26:26,320 It almost results in what looks 533 00:26:26,320 --> 00:26:27,810 like an optical illusion. 534 00:26:27,810 --> 00:26:32,710 You see, if a curve is holding water, the tangent line lies 535 00:26:32,710 --> 00:26:34,020 below the curve. 536 00:26:34,020 --> 00:26:37,680 If the curve is spilling water, the tangent line to the 537 00:26:37,680 --> 00:26:40,210 curve at a point lies above the curve. 538 00:26:40,210 --> 00:26:42,760 Consequently, at a 'point of inflection', meaning where the 539 00:26:42,760 --> 00:26:47,280 curve changes concavity, the tangent line on one side must 540 00:26:47,280 --> 00:26:50,410 be above the curve, on the other side below the curve. 541 00:26:50,410 --> 00:26:53,350 And all I'm saying over here is that you recognize a point 542 00:26:53,350 --> 00:26:57,680 of inflection by what? 543 00:26:57,680 --> 00:27:00,120 It's the situation in which the tangent line 544 00:27:00,120 --> 00:27:01,970 to the curve appears-- 545 00:27:01,970 --> 00:27:03,440 in fact, it actually does in the manner of 546 00:27:03,440 --> 00:27:06,750 speaking, cross the curve. 547 00:27:06,750 --> 00:27:07,950 I think we talked about this in the previous 548 00:27:07,950 --> 00:27:09,130 lecture, I'm not sure. 549 00:27:09,130 --> 00:27:11,110 But we talked about the idea that a tangent line 550 00:27:11,110 --> 00:27:12,720 can cross the curve. 551 00:27:12,720 --> 00:27:15,250 And where can it cross the curve and still be a tangent 552 00:27:15,250 --> 00:27:16,220 line at that point? 553 00:27:16,220 --> 00:27:18,160 At a point of inflection. 554 00:27:18,160 --> 00:27:21,660 By the way, if this is a smooth type of thing in 555 00:27:21,660 --> 00:27:24,480 general, what we're saying is that for a point of inflection 556 00:27:24,480 --> 00:27:29,890 to occur, the second derivative must be 0. 557 00:27:29,890 --> 00:27:31,070 You see because that means what? 558 00:27:31,070 --> 00:27:32,850 The curve is neither-- 559 00:27:32,850 --> 00:27:34,810 it's going from holding to spilling, so it goes through a 560 00:27:34,810 --> 00:27:37,980 transition where it's doing neither. 561 00:27:37,980 --> 00:27:41,875 In the same as with first derivatives, the mere fact 562 00:27:41,875 --> 00:27:44,500 that the second derivative is 0 does not allow us to 563 00:27:44,500 --> 00:27:46,620 conclude that we have a point of inflection. 564 00:27:46,620 --> 00:27:49,570 In fact, let me close with this particular illustration. 565 00:27:49,570 --> 00:27:52,200 Let's take the curve 'y' equals 'x to the fourth'. 566 00:27:52,200 --> 00:27:54,280 The first derivative is '4 x cubed'. 567 00:27:54,280 --> 00:27:56,230 The second derivative is '12 x squared'. 568 00:27:56,230 --> 00:27:57,900 The curve is symmetric with respect to 569 00:27:57,900 --> 00:28:00,280 the y-axis, et cetera. 570 00:28:00,280 --> 00:28:02,980 Using all of the given data you know the second derivative 571 00:28:02,980 --> 00:28:05,250 is always positive, what have you. 572 00:28:05,250 --> 00:28:06,940 We can sketch this curve. 573 00:28:06,940 --> 00:28:10,860 And again, in fact the uninitiated say this curve 574 00:28:10,860 --> 00:28:12,190 looks like a parabola. 575 00:28:12,190 --> 00:28:13,890 What do you mean it looks like a parabola? 576 00:28:13,890 --> 00:28:16,610 Well, he says, the parabola does something like this too. 577 00:28:16,610 --> 00:28:19,340 Well, what do we mean by something like this? 578 00:28:19,340 --> 00:28:21,370 I want to mention a few points here. 579 00:28:21,370 --> 00:28:24,110 One is, of course, that actually, the parabola 'y' 580 00:28:24,110 --> 00:28:25,360 equals 'x squared'. 581 00:28:30,780 --> 00:28:32,325 These are going to crisscross very shortly here. 582 00:28:32,325 --> 00:28:33,520 It doesn't make any difference. 583 00:28:33,520 --> 00:28:36,620 The parabola 'y' equals 'x squared' has the same general 584 00:28:36,620 --> 00:28:38,760 shape but with a few different properties, which we'll 585 00:28:38,760 --> 00:28:39,880 mention in a little while. 586 00:28:39,880 --> 00:28:42,640 But the point that I wanted to mention here first of all is 587 00:28:42,640 --> 00:28:43,610 simply this. 588 00:28:43,610 --> 00:28:48,790 At the value 'x' equals 0, 'y double prime' is 0. 589 00:28:48,790 --> 00:28:51,850 So you notice that the second derivative is 0 over here. 590 00:28:51,850 --> 00:28:54,930 Yet even though the second derivative is 0, notice that 591 00:28:54,930 --> 00:28:57,530 the curve does not change concavity. 592 00:28:57,530 --> 00:29:00,400 The curve here is always holding water. 593 00:29:00,400 --> 00:29:03,450 The concluding remark that I wanted to make is, what is the 594 00:29:03,450 --> 00:29:05,680 relationship between 'y' equals 'x squared' and 'y' 595 00:29:05,680 --> 00:29:06,670 equals 'x to the fourth'? 596 00:29:06,670 --> 00:29:08,580 Or how about 'y' equals 'x to the sixth'? 597 00:29:08,580 --> 00:29:10,720 Or 'y' equals 'x to the 12th'? 598 00:29:10,720 --> 00:29:13,240 Notice that any curve in that family will look 599 00:29:13,240 --> 00:29:15,160 something like this. 600 00:29:15,160 --> 00:29:15,840 Only what? 601 00:29:15,840 --> 00:29:19,570 As the exponent goes up, the curve becomes broader in the 602 00:29:19,570 --> 00:29:20,790 neighborhood of 0. 603 00:29:20,790 --> 00:29:25,010 And then once 'x' passes 1, the curve rises more sharply. 604 00:29:25,010 --> 00:29:27,600 See what we're saying is, if the magnitude of 'x' is less 605 00:29:27,600 --> 00:29:30,400 than 1, the higher a power you raise it to, 606 00:29:30,400 --> 00:29:31,750 the smaller 'y' is. 607 00:29:31,750 --> 00:29:34,250 On the other hand, if the absolute value of 'x' is 608 00:29:34,250 --> 00:29:40,020 greater than 1, the higher a power you raise it to, the 609 00:29:40,020 --> 00:29:41,640 bigger the output becomes. 610 00:29:41,640 --> 00:29:43,160 But the idea is this. 611 00:29:43,160 --> 00:29:47,470 Notice that the exponent-- 612 00:29:47,470 --> 00:29:50,540 in other words, how many derivatives are 0, gives you a 613 00:29:50,540 --> 00:29:54,080 way of getting into a problem that will become very, very 614 00:29:54,080 --> 00:29:55,960 crucial as this course continues. 615 00:29:55,960 --> 00:29:59,905 And it's the idea of, can one curve be more tangent to a 616 00:29:59,905 --> 00:30:01,380 line than another curve? 617 00:30:01,380 --> 00:30:06,720 You see, all of these curves are tangent to the line what? 618 00:30:06,720 --> 00:30:09,840 The x-axis, 'y' equals 0 at 'x' equals 0. 619 00:30:09,840 --> 00:30:12,610 How do we distinguish between these curves? 620 00:30:12,610 --> 00:30:17,370 Well, it seems that some of these curves fit the x-axis 621 00:30:17,370 --> 00:30:20,390 better than others in a neighborhood of the 622 00:30:20,390 --> 00:30:21,940 point 'x' equals 0. 623 00:30:21,940 --> 00:30:24,730 See the point that I want to bring out as to 'y' curve 624 00:30:24,730 --> 00:30:27,510 plotting tells us things that we don't learn in the ordinary 625 00:30:27,510 --> 00:30:28,970 physics class is this. 626 00:30:28,970 --> 00:30:32,290 If you study calculus the way it comes up in most physics 627 00:30:32,290 --> 00:30:35,310 courses, we essentially don't go past the second derivative. 628 00:30:35,310 --> 00:30:35,960 Why? 629 00:30:35,960 --> 00:30:38,880 Because in many cases, we're studying distance. 630 00:30:38,880 --> 00:30:41,210 And the derivative of distance is velocity. 631 00:30:41,210 --> 00:30:43,190 The second derivative of distance, namely the 632 00:30:43,190 --> 00:30:47,380 derivative of velocity is acceleration. 633 00:30:47,380 --> 00:30:49,690 And we don't usually talk physically beyond 634 00:30:49,690 --> 00:30:50,780 acceleration. 635 00:30:50,780 --> 00:30:54,530 But notice that in terms of curve plotting the third, 636 00:30:54,530 --> 00:30:59,440 fourth, fifth, sixth, seventh, tenth derivatives all have a 637 00:30:59,440 --> 00:31:02,030 meaning that gives you more information 638 00:31:02,030 --> 00:31:03,630 than what came before. 639 00:31:03,630 --> 00:31:06,970 Don't be deceived by the fact that in other applications 640 00:31:06,970 --> 00:31:09,490 that you never go past the second derivative means that 641 00:31:09,490 --> 00:31:13,610 there is no value to knowing how higher order derivatives 642 00:31:13,610 --> 00:31:15,240 are related to plotting curves. 643 00:31:15,240 --> 00:31:18,010 At any rate, I think this is enough of an introduction to 644 00:31:18,010 --> 00:31:20,710 the topic of curve plotting and curve sketching. 645 00:31:20,710 --> 00:31:23,530 We'll pursue these topics further in our next lectures. 646 00:31:23,530 --> 00:31:25,020 So until next time, goodbye. 647 00:31:28,290 --> 00:31:30,820 ANNOUNCER: Funding for the publication of this video was 648 00:31:30,820 --> 00:31:35,540 provided by the Gabriella and Paul Rosenbaum Foundation. 649 00:31:35,540 --> 00:31:39,710 Help OCW continue to provide free and open access to MIT 650 00:31:39,710 --> 00:31:43,910 courses by making a donation at ocw.mit.edu/donate.