[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.66,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.66,0:00:02.93,Default,,0000,0000,0000,,In the last video, we\Ntook the Maclaurin series
Dialogue: 0,0:00:02.93,0:00:04.18,Default,,0000,0000,0000,,of cosine of x.
Dialogue: 0,0:00:04.18,0:00:06.67,Default,,0000,0000,0000,,We approximated it\Nusing this polynomial.
Dialogue: 0,0:00:06.67,0:00:08.56,Default,,0000,0000,0000,,And we saw this pretty\Ninteresting pattern.
Dialogue: 0,0:00:08.56,0:00:10.31,Default,,0000,0000,0000,,Let's see if we can\Nfind a similar pattern
Dialogue: 0,0:00:10.31,0:00:14.36,Default,,0000,0000,0000,,if we try to approximate sine\Nof x using a Maclaurin series.
Dialogue: 0,0:00:14.36,0:00:16.00,Default,,0000,0000,0000,,And once again, a\NMaclaurin series
Dialogue: 0,0:00:16.00,0:00:18.24,Default,,0000,0000,0000,,is really the same thing\Nas a Taylor series,
Dialogue: 0,0:00:18.24,0:00:20.80,Default,,0000,0000,0000,,where we are centering\Nour approximation
Dialogue: 0,0:00:20.80,0:00:23.74,Default,,0000,0000,0000,,around x is equal to 0.
Dialogue: 0,0:00:23.74,0:00:26.83,Default,,0000,0000,0000,,So it's just a special\Ncase of a Taylor series.
Dialogue: 0,0:00:26.83,0:00:29.87,Default,,0000,0000,0000,,So let's take f of\Nx in this situation
Dialogue: 0,0:00:29.87,0:00:31.41,Default,,0000,0000,0000,,to be equal to sine of x.
Dialogue: 0,0:00:31.41,0:00:36.38,Default,,0000,0000,0000,,
Dialogue: 0,0:00:36.38,0:00:38.89,Default,,0000,0000,0000,,And let's do the same thing\Nthat we did with cosine of x.
Dialogue: 0,0:00:38.89,0:00:40.60,Default,,0000,0000,0000,,Let's just take the\Ndifferent derivatives
Dialogue: 0,0:00:40.60,0:00:42.45,Default,,0000,0000,0000,,of sine of x really fast.
Dialogue: 0,0:00:42.45,0:00:46.24,Default,,0000,0000,0000,,So if you have the first\Nderivative of sine of x,
Dialogue: 0,0:00:46.24,0:00:48.49,Default,,0000,0000,0000,,is just cosine of x.
Dialogue: 0,0:00:48.49,0:00:51.16,Default,,0000,0000,0000,,The second derivative\Nof the sine of x
Dialogue: 0,0:00:51.16,0:00:55.88,Default,,0000,0000,0000,,is the derivative of cosine of\Nx, which is negative sine of x.
Dialogue: 0,0:00:55.88,0:00:58.79,Default,,0000,0000,0000,,The third derivative is going\Nto be the derivative of this.
Dialogue: 0,0:00:58.79,0:01:00.44,Default,,0000,0000,0000,,So I'll just write\Na 3 in parentheses
Dialogue: 0,0:01:00.44,0:01:02.27,Default,,0000,0000,0000,,there, instead of doing\Nprime, prime, prime.
Dialogue: 0,0:01:02.27,0:01:04.31,Default,,0000,0000,0000,,So the third derivative\Nis the derivative
Dialogue: 0,0:01:04.31,0:01:07.79,Default,,0000,0000,0000,,of this, which is\Nnegative cosine of x.
Dialogue: 0,0:01:07.79,0:01:11.57,Default,,0000,0000,0000,,The fourth derivative\Nis the derivative
Dialogue: 0,0:01:11.57,0:01:15.02,Default,,0000,0000,0000,,of this, which is\Npositive sine of x again.
Dialogue: 0,0:01:15.02,0:01:17.95,Default,,0000,0000,0000,,So you see, just like cosine\Nof x, it kind of cycles
Dialogue: 0,0:01:17.95,0:01:19.99,Default,,0000,0000,0000,,after you take the\Nderivative enough times.
Dialogue: 0,0:01:19.99,0:01:22.55,Default,,0000,0000,0000,,And we care-- in order to do\Nthe Maclaurin, series-- we care
Dialogue: 0,0:01:22.55,0:01:26.84,Default,,0000,0000,0000,,about evaluating the function,\Nand each of these derivatives
Dialogue: 0,0:01:26.84,0:01:28.45,Default,,0000,0000,0000,,at x is equal to 0.
Dialogue: 0,0:01:28.45,0:01:30.22,Default,,0000,0000,0000,,So let's do that.
Dialogue: 0,0:01:30.22,0:01:32.51,Default,,0000,0000,0000,,So for this, let me do this\Nin a different color, not
Dialogue: 0,0:01:32.51,0:01:34.03,Default,,0000,0000,0000,,that same blue.
Dialogue: 0,0:01:34.03,0:01:36.32,Default,,0000,0000,0000,,I'll do it in this purple color.
Dialogue: 0,0:01:36.32,0:01:38.81,Default,,0000,0000,0000,,So f-- that's hard\Nto see, I think
Dialogue: 0,0:01:38.81,0:01:40.93,Default,,0000,0000,0000,,So let's do this\Nother blue color.
Dialogue: 0,0:01:40.93,0:01:45.62,Default,,0000,0000,0000,,So f of 0, in this\Nsituation, is 0.
Dialogue: 0,0:01:45.62,0:01:50.28,Default,,0000,0000,0000,,And f, the first derivative\Nevaluated at 0, is 1.
Dialogue: 0,0:01:50.28,0:01:52.88,Default,,0000,0000,0000,,Cosine of 0 is 1.
Dialogue: 0,0:01:52.88,0:01:57.30,Default,,0000,0000,0000,,Negative sine of 0\Nis going to be 0.
Dialogue: 0,0:01:57.30,0:02:01.21,Default,,0000,0000,0000,,So f prime prime, the second\Nderivative evaluated at 0 is 0.
Dialogue: 0,0:02:01.21,0:02:06.34,Default,,0000,0000,0000,,The third derivative\Nevaluated at 0 is negative 1.
Dialogue: 0,0:02:06.34,0:02:08.41,Default,,0000,0000,0000,,Cosine of 0 is 1.
Dialogue: 0,0:02:08.41,0:02:09.66,Default,,0000,0000,0000,,You have a negative out there.
Dialogue: 0,0:02:09.66,0:02:10.85,Default,,0000,0000,0000,,It is negative 1.
Dialogue: 0,0:02:10.85,0:02:14.85,Default,,0000,0000,0000,,And then the fourth\Nderivative evaluated at 0
Dialogue: 0,0:02:14.85,0:02:16.59,Default,,0000,0000,0000,,is going to be 0 again.
Dialogue: 0,0:02:16.59,0:02:18.38,Default,,0000,0000,0000,,And we could keep going,\Nbut once again, it
Dialogue: 0,0:02:18.38,0:02:19.59,Default,,0000,0000,0000,,seems like there's a pattern.
Dialogue: 0,0:02:19.59,0:02:21.13,Default,,0000,0000,0000,,0, 1, 0, negative\N1, 0, then you're
Dialogue: 0,0:02:21.13,0:02:22.92,Default,,0000,0000,0000,,going to go back to positive 1.
Dialogue: 0,0:02:22.92,0:02:24.75,Default,,0000,0000,0000,,So on and so forth.
Dialogue: 0,0:02:24.75,0:02:28.23,Default,,0000,0000,0000,,So let's find its\Npolynomial representation
Dialogue: 0,0:02:28.23,0:02:29.51,Default,,0000,0000,0000,,using the Maclaurin series.
Dialogue: 0,0:02:29.51,0:02:31.38,Default,,0000,0000,0000,,And just a reminder,\Nthis one up here,
Dialogue: 0,0:02:31.38,0:02:34.21,Default,,0000,0000,0000,,this was approximately\Ncosine of x.
Dialogue: 0,0:02:34.21,0:02:36.22,Default,,0000,0000,0000,,And you'll get closer and\Ncloser to cosine of x.
Dialogue: 0,0:02:36.22,0:02:37.96,Default,,0000,0000,0000,,I'm not rigorously\Nshowing you how
Dialogue: 0,0:02:37.96,0:02:40.76,Default,,0000,0000,0000,,close, in that it's definitely\Nthe exact same thing as cosine
Dialogue: 0,0:02:40.76,0:02:42.71,Default,,0000,0000,0000,,of x, but you get closer\Nand closer and closer
Dialogue: 0,0:02:42.71,0:02:44.98,Default,,0000,0000,0000,,to cosine of x as you\Nkeep adding terms here.
Dialogue: 0,0:02:44.98,0:02:46.35,Default,,0000,0000,0000,,And if you go to\Ninfinity, you're
Dialogue: 0,0:02:46.35,0:02:48.92,Default,,0000,0000,0000,,going to be pretty\Nmuch at cosine of x.
Dialogue: 0,0:02:48.92,0:02:51.49,Default,,0000,0000,0000,,Now let's do the same\Nthing for sine of x.
Dialogue: 0,0:02:51.49,0:02:52.96,Default,,0000,0000,0000,,So I'll pick a new color.
Dialogue: 0,0:02:52.96,0:02:54.98,Default,,0000,0000,0000,,This green should be nice.
Dialogue: 0,0:02:54.98,0:02:56.82,Default,,0000,0000,0000,,So this is our new p of x.
Dialogue: 0,0:02:56.82,0:02:58.59,Default,,0000,0000,0000,,So this is approximately\Ngoing to be
Dialogue: 0,0:02:58.59,0:03:01.60,Default,,0000,0000,0000,,sine of x, as we add\Nmore and more terms.
Dialogue: 0,0:03:01.60,0:03:06.82,Default,,0000,0000,0000,,And so the first term here, f\Nof 0, that's just going to be 0.
Dialogue: 0,0:03:06.82,0:03:09.05,Default,,0000,0000,0000,,So we're not even going\Nto need to include that.
Dialogue: 0,0:03:09.05,0:03:11.09,Default,,0000,0000,0000,,The next term is\Ngoing to be f prime
Dialogue: 0,0:03:11.09,0:03:13.80,Default,,0000,0000,0000,,of 0, which is 1, times x.
Dialogue: 0,0:03:13.80,0:03:15.88,Default,,0000,0000,0000,,So it's going to be x.
Dialogue: 0,0:03:15.88,0:03:18.45,Default,,0000,0000,0000,,Then the next term is f\Nprime, the second derivative
Dialogue: 0,0:03:18.45,0:03:21.22,Default,,0000,0000,0000,,at 0, which we see here is 0.
Dialogue: 0,0:03:21.22,0:03:23.02,Default,,0000,0000,0000,,Let me scroll down a little bit.
Dialogue: 0,0:03:23.02,0:03:24.34,Default,,0000,0000,0000,,It is 0.
Dialogue: 0,0:03:24.34,0:03:26.61,Default,,0000,0000,0000,,So we won't have\Nthe second term.
Dialogue: 0,0:03:26.61,0:03:29.59,Default,,0000,0000,0000,,This third term right\Nhere, the third derivative
Dialogue: 0,0:03:29.59,0:03:32.92,Default,,0000,0000,0000,,of sine of x evaluated\Nat 0, is negative 1.
Dialogue: 0,0:03:32.92,0:03:36.83,Default,,0000,0000,0000,,So we're now going\Nto have a negative 1.
Dialogue: 0,0:03:36.83,0:03:39.49,Default,,0000,0000,0000,,Let me scroll down\Nso you can see this.
Dialogue: 0,0:03:39.49,0:03:42.24,Default,,0000,0000,0000,,Negative 1-- this is\Nnegative 1 in this case--
Dialogue: 0,0:03:42.24,0:03:45.50,Default,,0000,0000,0000,,times x to the third\Nover 3 factorial.
Dialogue: 0,0:03:45.50,0:03:50.88,Default,,0000,0000,0000,,
Dialogue: 0,0:03:50.88,0:03:52.86,Default,,0000,0000,0000,,And then the next\Nterm is going to be 0,
Dialogue: 0,0:03:52.86,0:03:55.85,Default,,0000,0000,0000,,because that's the\Nfourth derivative.
Dialogue: 0,0:03:55.85,0:03:59.66,Default,,0000,0000,0000,,The fourth derivative evaluated\Nat 0 is the next coefficient.
Dialogue: 0,0:03:59.66,0:04:03.05,Default,,0000,0000,0000,,We see that that is going to be\N0, so it's going to drop off.
Dialogue: 0,0:04:03.05,0:04:04.51,Default,,0000,0000,0000,,And what you're\Ngoing to see here--
Dialogue: 0,0:04:04.51,0:04:06.82,Default,,0000,0000,0000,,and actually maybe I haven't\Ndone enough terms for you,
Dialogue: 0,0:04:06.82,0:04:08.30,Default,,0000,0000,0000,,for you to feel good about this.
Dialogue: 0,0:04:08.30,0:04:10.33,Default,,0000,0000,0000,,Let me do one more\Nterm right over here.
Dialogue: 0,0:04:10.33,0:04:12.69,Default,,0000,0000,0000,,Just so it becomes clear.
Dialogue: 0,0:04:12.69,0:04:15.45,Default,,0000,0000,0000,,f of the fifth\Nderivative of x is
Dialogue: 0,0:04:15.45,0:04:17.44,Default,,0000,0000,0000,,going to be cosine of x again.
Dialogue: 0,0:04:17.44,0:04:20.21,Default,,0000,0000,0000,,The fifth derivative-- we'll\Ndo it in that same color,
Dialogue: 0,0:04:20.21,0:04:27.18,Default,,0000,0000,0000,,just so it's consistent-- the\Nfifth derivative evaluated at 0
Dialogue: 0,0:04:27.18,0:04:29.57,Default,,0000,0000,0000,,is going to be 1.
Dialogue: 0,0:04:29.57,0:04:33.47,Default,,0000,0000,0000,,So the fourth derivative\Nevaluated at 0 is 0,
Dialogue: 0,0:04:33.47,0:04:36.86,Default,,0000,0000,0000,,then you go to the fifth\Nderivative evaluated at 0,
Dialogue: 0,0:04:36.86,0:04:38.81,Default,,0000,0000,0000,,it's going to be positive 1.
Dialogue: 0,0:04:38.81,0:04:40.73,Default,,0000,0000,0000,,And if I kept doing this,\Nit would be positive
Dialogue: 0,0:04:40.73,0:04:44.44,Default,,0000,0000,0000,,1-- I have to write the 1\Nas the coefficient-- times x
Dialogue: 0,0:04:44.44,0:04:47.77,Default,,0000,0000,0000,,to the fifth over 5 factorial.
Dialogue: 0,0:04:47.77,0:04:50.53,Default,,0000,0000,0000,,So there's something\Ninteresting going on here.
Dialogue: 0,0:04:50.53,0:04:55.50,Default,,0000,0000,0000,,For cosine of x, I had 1,\Nessentially 1 times x to the 0.
Dialogue: 0,0:04:55.50,0:04:58.42,Default,,0000,0000,0000,,Then I don't have x\Nto the first power.
Dialogue: 0,0:04:58.42,0:05:00.24,Default,,0000,0000,0000,,I don't have x to the\Nodd powers, actually.
Dialogue: 0,0:05:00.24,0:05:02.83,Default,,0000,0000,0000,,And then I just essentially have\Nx to all of the even powers.
Dialogue: 0,0:05:02.83,0:05:06.65,Default,,0000,0000,0000,,And whatever power it is, I'm\Ndividing it by that factorial.
Dialogue: 0,0:05:06.65,0:05:09.40,Default,,0000,0000,0000,,And then the sines\Nkeep switching.
Dialogue: 0,0:05:09.40,0:05:12.30,Default,,0000,0000,0000,,I shouldn't say this is an even\Npower, because 0 really isn't.
Dialogue: 0,0:05:12.30,0:05:14.44,Default,,0000,0000,0000,,Well, I guess you can\Nview it as an even number,
Dialogue: 0,0:05:14.44,0:05:17.54,Default,,0000,0000,0000,,because-- well I won't\Ngo into all of that.
Dialogue: 0,0:05:17.54,0:05:22.44,Default,,0000,0000,0000,,But it's essentially 0, 2,\N4, 6, so on and so forth.
Dialogue: 0,0:05:22.44,0:05:24.20,Default,,0000,0000,0000,,So this is\Ninteresting, especially
Dialogue: 0,0:05:24.20,0:05:25.44,Default,,0000,0000,0000,,when you compare to this.
Dialogue: 0,0:05:25.44,0:05:26.76,Default,,0000,0000,0000,,This is all of the odd powers.
Dialogue: 0,0:05:26.76,0:05:29.06,Default,,0000,0000,0000,,This is x to the first\Nover 1 factorial.
Dialogue: 0,0:05:29.06,0:05:30.30,Default,,0000,0000,0000,,I didn't write it here.
Dialogue: 0,0:05:30.30,0:05:32.58,Default,,0000,0000,0000,,This is x to the\Nthird over 3 factorial
Dialogue: 0,0:05:32.58,0:05:34.48,Default,,0000,0000,0000,,plus x to the fifth\Nover 5 factorial.
Dialogue: 0,0:05:34.48,0:05:35.81,Default,,0000,0000,0000,,Yeah, 0 would be an even number.
Dialogue: 0,0:05:35.81,0:05:39.83,Default,,0000,0000,0000,,Anyway, my brain is in a\Ndifferent place right now.
Dialogue: 0,0:05:39.83,0:05:40.89,Default,,0000,0000,0000,,And you could keep going.
Dialogue: 0,0:05:40.89,0:05:43.19,Default,,0000,0000,0000,,If we kept this process\Nup, you would then
Dialogue: 0,0:05:43.19,0:05:44.28,Default,,0000,0000,0000,,keep switching sines.
Dialogue: 0,0:05:44.28,0:05:48.20,Default,,0000,0000,0000,,X to the seventh over\N7 factorial plus x
Dialogue: 0,0:05:48.20,0:05:49.53,Default,,0000,0000,0000,,to the ninth over 9 factorial.
Dialogue: 0,0:05:49.53,0:05:51.11,Default,,0000,0000,0000,,So there's something\Ninteresting here.
Dialogue: 0,0:05:51.11,0:05:55.28,Default,,0000,0000,0000,,You once again see this\Nkind of complimentary nature
Dialogue: 0,0:05:55.28,0:05:56.93,Default,,0000,0000,0000,,between sine and cosine here.
Dialogue: 0,0:05:56.93,0:05:58.63,Default,,0000,0000,0000,,You see almost\Nthis-- they're kind
Dialogue: 0,0:05:58.63,0:06:00.96,Default,,0000,0000,0000,,of filling each\Nother's gaps over here.
Dialogue: 0,0:06:00.96,0:06:03.22,Default,,0000,0000,0000,,Cosine of x is all\Nof the even powers
Dialogue: 0,0:06:03.22,0:06:05.68,Default,,0000,0000,0000,,of x divided by that\Npower's factorial.
Dialogue: 0,0:06:05.68,0:06:08.31,Default,,0000,0000,0000,,Sine of x, when you take its\Npolynomial representation,
Dialogue: 0,0:06:08.31,0:06:12.47,Default,,0000,0000,0000,,is all of the odd powers of\Nx divided by its factorial,
Dialogue: 0,0:06:12.47,0:06:14.10,Default,,0000,0000,0000,,and you switch sines.
Dialogue: 0,0:06:14.10,0:06:16.64,Default,,0000,0000,0000,,In the next video,\NI'll do e to the x.
Dialogue: 0,0:06:16.64,0:06:18.66,Default,,0000,0000,0000,,And what's really\Nfascinating is that e
Dialogue: 0,0:06:18.66,0:06:22.31,Default,,0000,0000,0000,,to the x starts to look like\Na little bit of a combination
Dialogue: 0,0:06:22.31,0:06:24.05,Default,,0000,0000,0000,,here, but not quite.
Dialogue: 0,0:06:24.05,0:06:25.79,Default,,0000,0000,0000,,And you really do\Nget the combination
Dialogue: 0,0:06:25.79,0:06:28.31,Default,,0000,0000,0000,,when you involve\Nimaginary numbers.
Dialogue: 0,0:06:28.31,0:06:32.86,Default,,0000,0000,0000,,And that's when it starts to\Nget really, really mind blowing.