1 00:00:06,891 --> 00:00:07,390 Hi. 2 00:00:07,390 --> 00:00:08,850 Welcome back to recitation. 3 00:00:08,850 --> 00:00:11,450 In class, Professor Jerison and Professor Miller 4 00:00:11,450 --> 00:00:13,510 have taught you a little bit about Taylor series 5 00:00:13,510 --> 00:00:15,390 and some of the manipulations you can do with them, 6 00:00:15,390 --> 00:00:17,434 and have computed a bunch of examples for you. 7 00:00:17,434 --> 00:00:18,850 So I have three more examples here 8 00:00:18,850 --> 00:00:22,150 of functions whose Taylor series are nice to compute. 9 00:00:22,150 --> 00:00:24,460 So the first one is cosh x. 10 00:00:24,460 --> 00:00:26,050 That's the hyperbolic cosine. 11 00:00:26,050 --> 00:00:28,220 So just to remind you, this can be 12 00:00:28,220 --> 00:00:30,345 written in terms of the exponential function as e 13 00:00:30,345 --> 00:00:35,940 to the x plus e to the minus x over 2. 14 00:00:35,940 --> 00:00:37,600 The second one is the function 2 times 15 00:00:37,600 --> 00:00:40,525 sine of x times cosine of x, just your regular sine 16 00:00:40,525 --> 00:00:41,720 and cosine here. 17 00:00:41,720 --> 00:00:46,240 And the third one is x times the logarithm of the quantity 1 18 00:00:46,240 --> 00:00:47,620 minus x cubed. 19 00:00:47,620 --> 00:00:50,070 So why don't you pause the video, take some time 20 00:00:50,070 --> 00:00:52,815 to work out the Taylor series for these three functions, 21 00:00:52,815 --> 00:00:54,690 come back, and we can work them out together. 22 00:01:02,950 --> 00:01:05,961 So here we have three functions whose Taylor series 23 00:01:05,961 --> 00:01:06,960 we're trying to compute. 24 00:01:06,960 --> 00:01:09,150 Let's start with the first one and go from there. 25 00:01:09,150 --> 00:01:10,940 So this first one is the hyperbolic cosine 26 00:01:10,940 --> 00:01:13,525 that's given by the formula e to the x plus e to the minus 27 00:01:13,525 --> 00:01:14,351 x over 2. 28 00:01:14,351 --> 00:01:15,850 So there are a couple different ways 29 00:01:15,850 --> 00:01:17,030 you could go about this one. 30 00:01:17,030 --> 00:01:19,020 This is actually, the hyperbolic cosine 31 00:01:19,020 --> 00:01:21,420 is very susceptible to the method of just using 32 00:01:21,420 --> 00:01:23,440 the formula that you have. 33 00:01:23,440 --> 00:01:25,520 So if you remember, the derivative 34 00:01:25,520 --> 00:01:27,660 of the hyperbolic cosine is the hyperbolic sine. 35 00:01:27,660 --> 00:01:29,290 The derivative of the hyperbolic sine 36 00:01:29,290 --> 00:01:30,720 is the hyperbolic cosine again. 37 00:01:30,720 --> 00:01:33,750 So this function has very easy-to-understand derivatives, 38 00:01:33,750 --> 00:01:36,390 which you can see, you know, just by looking at its formula. 39 00:01:36,390 --> 00:01:40,130 It's easy to understand, because the exponential function has 40 00:01:40,130 --> 00:01:42,390 very simple derivatives, and e to the the minus 41 00:01:42,390 --> 00:01:44,440 x also has very simple derivatives. 42 00:01:44,440 --> 00:01:46,180 So you could do it like that. 43 00:01:46,180 --> 00:01:48,221 The other thing you could do, is that you already 44 00:01:48,221 --> 00:01:50,250 know the Taylor series for e to the x. 45 00:01:50,250 --> 00:01:52,500 And I believe you've also seen the Taylor series for e 46 00:01:52,500 --> 00:01:55,530 to the minus x, and even if you haven't, you can figure it out 47 00:01:55,530 --> 00:01:56,800 just by substitution. 48 00:01:56,800 --> 00:02:02,560 So if you remember, so e to the x is given by the sum from n 49 00:02:02,560 --> 00:02:07,940 equals 0 to infinity of x to the n over n factorial. 50 00:02:07,940 --> 00:02:10,410 I'm going to pull the 1/2 out in front. 51 00:02:10,410 --> 00:02:13,230 And e to the minus x is given by the same thing, 52 00:02:13,230 --> 00:02:16,250 if you put in minus x for x. 53 00:02:16,250 --> 00:02:20,740 So it's n equals 0 to infinity, so that works out to minus 1 54 00:02:20,740 --> 00:02:25,580 to the n x to the n or n factorial. 55 00:02:25,580 --> 00:02:27,560 Now, when you add these two series together, 56 00:02:27,560 --> 00:02:30,110 what you see is that when n is even, over here, 57 00:02:30,110 --> 00:02:32,830 you have x to the n over n factorial, and over here, 58 00:02:32,830 --> 00:02:34,700 you have x to the n over n factorial. 59 00:02:34,700 --> 00:02:36,690 So what you get is, well, you get 2 x 60 00:02:36,690 --> 00:02:39,200 to the n over n factorial, and then you multiply by a half, 61 00:02:39,200 --> 00:02:41,900 so you just get x to the n over n factorial. 62 00:02:41,900 --> 00:02:44,920 When n is odd, here you have x to the n over n factorial, 63 00:02:44,920 --> 00:02:47,730 and here you have minus 1 x to the n over n factorial. 64 00:02:47,730 --> 00:02:49,630 So you add them and you get 0. 65 00:02:49,630 --> 00:02:51,820 So what happens is that this series looks just 66 00:02:51,820 --> 00:02:55,970 like the series for e to the x, except the odd terms have 67 00:02:55,970 --> 00:02:56,850 died off. 68 00:02:56,850 --> 00:03:03,530 So we're left with just 1 plus x squared over 2 factorial plus x 69 00:03:03,530 --> 00:03:09,050 to the fourth over 4 factorial plus x to the sixth over 6 70 00:03:09,050 --> 00:03:10,810 factorial, and so on. 71 00:03:10,810 --> 00:03:13,100 And if you wanted to write this in summation notation, 72 00:03:13,100 --> 00:03:15,710 you could write it as the sum from n 73 00:03:15,710 --> 00:03:24,870 equals 0 to infinity of x to the 2n over 2n factorial. 74 00:03:24,870 --> 00:03:29,880 So this is the Taylor series for the hyperbolic cosine function. 75 00:03:29,880 --> 00:03:32,891 Also, if you wanted, say, the hyperbolic sine function, 76 00:03:32,891 --> 00:03:34,390 you could do something very similar, 77 00:03:34,390 --> 00:03:37,120 or you could remember that the hyperbolic sine is 78 00:03:37,120 --> 00:03:38,890 the derivative of the hyperbolic cosine, 79 00:03:38,890 --> 00:03:41,870 and just take a derivative right from this expression. 80 00:03:41,870 --> 00:03:43,965 One other thing that you should notice 81 00:03:43,965 --> 00:03:46,830 is that this looks very similar to the expression of the Taylor 82 00:03:46,830 --> 00:03:48,010 series for cosine of x. 83 00:03:48,010 --> 00:03:51,730 So more of our sort of funny coincidences 84 00:03:51,730 --> 00:03:55,220 between regular trig functions and hyperbolic trig functions. 85 00:03:55,220 --> 00:03:55,720 All right. 86 00:03:55,720 --> 00:03:56,560 That's the first one. 87 00:03:56,560 --> 00:03:57,450 How about the second one? 88 00:03:57,450 --> 00:03:59,620 So here we have just some regular trig functions. 89 00:03:59,620 --> 00:04:01,460 We have 2 sine x cosine x. 90 00:04:01,460 --> 00:04:03,660 Let me see where I've got some space. 91 00:04:03,660 --> 00:04:05,390 I can do it right here. 92 00:04:05,390 --> 00:04:08,820 Let me box off a little space for myself. 93 00:04:08,820 --> 00:04:14,890 So 2 sine x cosine x-- there are a couple different ways 94 00:04:14,890 --> 00:04:18,430 you could proceed with this function. 95 00:04:18,430 --> 00:04:21,080 So one is, you know the Taylor series for sine x and cosine x 96 00:04:21,080 --> 00:04:21,880 already. 97 00:04:21,880 --> 00:04:24,670 So if all you wanted was a few terms of this Taylor series, 98 00:04:24,670 --> 00:04:27,800 one natural thing to do would be to take the series for sine x, 99 00:04:27,800 --> 00:04:29,970 take the series for cosine x, multiply them together 100 00:04:29,970 --> 00:04:31,470 like you would multiply polynomials, 101 00:04:31,470 --> 00:04:33,300 and what you would get is the Taylor series 102 00:04:33,300 --> 00:04:37,467 for this expression, for this function. 103 00:04:37,467 --> 00:04:38,550 That's one way to proceed. 104 00:04:38,550 --> 00:04:39,740 That works perfectly well. 105 00:04:39,740 --> 00:04:40,790 Another thing you could do, is you 106 00:04:40,790 --> 00:04:42,080 could try taking derivatives. 107 00:04:42,080 --> 00:04:44,040 You could have a situation where every time you 108 00:04:44,040 --> 00:04:45,980 take a derivative, you apply product rule. 109 00:04:45,980 --> 00:04:47,813 It's going to get more and more complicated. 110 00:04:47,813 --> 00:04:48,780 It still works. 111 00:04:48,780 --> 00:04:51,950 It's a little complicated to do it that way, 112 00:04:51,950 --> 00:04:53,719 if you wanted more than just a few terms. 113 00:04:53,719 --> 00:04:56,260 The other thing you could do, is you could remember your trig 114 00:04:56,260 --> 00:04:57,400 identities. 115 00:04:57,400 --> 00:04:58,960 So if you look at this expression, 116 00:04:58,960 --> 00:05:02,210 this should be familiar to you, because it's just sine of 2x. 117 00:05:06,780 --> 00:05:08,740 So once you realize that this is sine of 2x, 118 00:05:08,740 --> 00:05:10,152 there's a much, much shorter path 119 00:05:10,152 --> 00:05:12,610 available to you, which is that you already know the Taylor 120 00:05:12,610 --> 00:05:14,710 series for sine of x, so what you can do, 121 00:05:14,710 --> 00:05:17,290 is you can just plug in 2x into that Taylor series. 122 00:05:17,290 --> 00:05:20,550 So sine of x is-- well, so OK. 123 00:05:20,550 --> 00:05:23,560 So sine of x is x-- so in this case, 124 00:05:23,560 --> 00:05:27,850 that's going to be 2x-- then minus, so in sine of x, 125 00:05:27,850 --> 00:05:30,050 we have x cubed over 3 factorial. 126 00:05:30,050 --> 00:05:37,350 So here we're going to have 2x cubed over 3 factorial plus-- 127 00:05:37,350 --> 00:05:37,850 OK. 128 00:05:37,850 --> 00:05:39,730 So then, you know, and so on. 129 00:05:39,730 --> 00:05:45,560 So here we'll have 2x to the fifth over 5 factorial 130 00:05:45,560 --> 00:05:46,527 minus-- so on. 131 00:05:46,527 --> 00:05:48,610 If you wanted to write this in summation notation, 132 00:05:48,610 --> 00:05:56,570 you could write it as the sum from n equals 0 to infinity. 133 00:05:56,570 --> 00:06:01,500 Well, the denominator has got to be 2n plus 1 factorial, 134 00:06:01,500 --> 00:06:03,310 because we want it to go through the odds. 135 00:06:03,310 --> 00:06:10,140 And then we've got minus 1 to the n times 2 to the 2n plus 1 136 00:06:10,140 --> 00:06:13,360 times x to the 2n plus 1. 137 00:06:13,360 --> 00:06:14,625 So this is 2x. 138 00:06:14,625 --> 00:06:17,340 What we've got here, if you didn't have the 2's there, 139 00:06:17,340 --> 00:06:20,371 that would just be the series for the regular sine. 140 00:06:20,371 --> 00:06:20,870 OK. 141 00:06:20,870 --> 00:06:24,570 So this is the series for this function, 2 sine x cosine x. 142 00:06:24,570 --> 00:06:28,039 And I'll go over here to do the third one. 143 00:06:28,039 --> 00:06:29,080 So what is the third one? 144 00:06:29,080 --> 00:06:35,880 It's x ln 1 minus x cubed. 145 00:06:35,880 --> 00:06:39,400 Well, what can we do with this series? 146 00:06:39,400 --> 00:06:43,780 The x out front is just multiplying this logarithm 147 00:06:43,780 --> 00:06:44,290 part. 148 00:06:44,290 --> 00:06:46,190 That's something we can save until the end. 149 00:06:46,190 --> 00:06:48,820 If we can figure out what the Taylor series for the ln of 1 150 00:06:48,820 --> 00:06:52,382 minus x cubed part is, then we just multiply x into it, 151 00:06:52,382 --> 00:06:54,840 and that'll give us the Taylor series for this whole thing. 152 00:06:54,840 --> 00:06:58,010 So the x out front is pretty simple. 153 00:06:58,010 --> 00:07:01,460 So now what about this ln of 1 minus x cubed stuff? 154 00:07:01,460 --> 00:07:04,251 Well, a thing to remember is, does it remind you 155 00:07:04,251 --> 00:07:05,500 of anything we've done before? 156 00:07:05,500 --> 00:07:07,821 Well, we have a Taylor series for a logarithm function, 157 00:07:07,821 --> 00:07:08,320 right? 158 00:07:08,320 --> 00:07:12,820 We've already seen in lecture, I believe, 159 00:07:12,820 --> 00:07:17,240 we've seen that ln of 1 plus x is 160 00:07:17,240 --> 00:07:26,610 equal to x minus x squared over 2 plus x cubed over 3, 161 00:07:26,610 --> 00:07:32,810 minus x to the fourth over four, and so on, alternating signs. 162 00:07:32,810 --> 00:07:36,070 Notice that the denominators, when you have a logarithm, 163 00:07:36,070 --> 00:07:37,330 these are not factorials. 164 00:07:37,330 --> 00:07:40,240 These are just the integer 2, the integer 3, the integer 4, 165 00:07:40,240 --> 00:07:44,100 unlike for exponentials and trig functions. 166 00:07:44,100 --> 00:07:46,990 So this is what log of 1 plus x-- 167 00:07:46,990 --> 00:07:49,454 this is the Taylor series for log of 1 plus x. 168 00:07:49,454 --> 00:07:50,620 Well, how does that help us? 169 00:07:50,620 --> 00:07:55,100 Well, log of 1 minus x cubed we can get from log of 1 plus x, 170 00:07:55,100 --> 00:07:56,780 with the appropriate substitution. 171 00:07:56,780 --> 00:07:58,550 So in particular, we just have to put 172 00:07:58,550 --> 00:08:01,430 minus x cubed in for x here. 173 00:08:01,430 --> 00:08:05,130 So what does that give us? 174 00:08:05,130 --> 00:08:12,090 It gives us the ln of 1 minus x cubed is equal to-- well, 175 00:08:12,090 --> 00:08:20,510 minus x cubed minus, so we put minus x cubed in here, 176 00:08:20,510 --> 00:08:22,870 we square it, and we just get x to the sixth. 177 00:08:22,870 --> 00:08:25,214 x to the sixth over 2. 178 00:08:25,214 --> 00:08:25,880 Then, all right. 179 00:08:25,880 --> 00:08:29,660 So minus x cubed quantity cubed is minus x to the ninth. 180 00:08:29,660 --> 00:08:37,690 So minus x to the ninth over 3, minus x to the twelfth over 4, 181 00:08:37,690 --> 00:08:38,710 and so on. 182 00:08:38,710 --> 00:08:48,840 And so finally, x ln of 1 minus x cubed, 183 00:08:48,840 --> 00:08:52,660 we just get by multiplying this whole expression through by x. 184 00:08:52,660 --> 00:08:56,592 So this is equal to minus x to the fourth minus x 185 00:08:56,592 --> 00:09:02,203 to the seventh over 2 minus x-- whoops, not ten-- minus 186 00:09:02,203 --> 00:09:09,504 x to the tenth over 3, minus x to the 13 over 4, and so on. 187 00:09:09,504 --> 00:09:11,170 And I'll leave it as an exercise for you 188 00:09:11,170 --> 00:09:13,850 to figure out how to write this in summation notation, 189 00:09:13,850 --> 00:09:14,497 if you wanted. 190 00:09:14,497 --> 00:09:17,080 So just quickly to summarize, we had these three power series, 191 00:09:17,080 --> 00:09:21,029 these three functions that we started out with, 192 00:09:21,029 --> 00:09:22,820 and we used a bunch of different techniques 193 00:09:22,820 --> 00:09:25,236 that we've learned in order to compute their power series. 194 00:09:25,236 --> 00:09:29,520 So over here, we took the function that we'd seen, 195 00:09:29,520 --> 00:09:33,010 and we knew a formula for it in terms of other functions 196 00:09:33,010 --> 00:09:35,760 that we already knew, and so we plugged in those power series, 197 00:09:35,760 --> 00:09:38,447 and used our addition rule for power series. 198 00:09:38,447 --> 00:09:41,030 We could have also done this one directly from the definition, 199 00:09:41,030 --> 00:09:42,130 if we had wanted to. 200 00:09:42,130 --> 00:09:44,670 For the second one, for the 2 sine x cosine x, 201 00:09:44,670 --> 00:09:46,965 we recognized that as something that 202 00:09:46,965 --> 00:09:50,750 is susceptible to a substitution, although also, 203 00:09:50,750 --> 00:09:52,290 with a little more, work, we could 204 00:09:52,290 --> 00:09:54,220 have done it by a couple of different methods. 205 00:09:54,220 --> 00:09:56,810 For example, by multiplying two power series together. 206 00:09:56,810 --> 00:09:59,420 And finally, for this third one, for the x ln 207 00:09:59,420 --> 00:10:02,586 of 1 minus x cubed, we first saw the substitution here 208 00:10:02,586 --> 00:10:04,960 that we could make, and then we just did a multiplication 209 00:10:04,960 --> 00:10:07,050 by a polynomial, which is a relatively easy thing 210 00:10:07,050 --> 00:10:09,970 to do for power series. 211 00:10:09,970 --> 00:10:11,980 So that's what we did in this recitation, 212 00:10:11,980 --> 00:10:14,114 and I'll leave it at that.