00:00Hi everyone. So this is
00:02my first lesson on the
00:04McLaren Series. This is just
00:06going to be an introduction,
00:07explain what it is, where
00:08it comes from, et cetera,
00:10et cetera. The next lessons
00:11I'll actually do, expansions will
00:13derive McLaren Series for different
00:16functions. But yeah, as I
00:18say in this video, I
00:19just want to kind of
00:20get your head around what
00:22we're actually doing. I'm going
00:24to start by showing you
00:25something, something that when I
00:28saw this, I was like
00:30what on earth, how is
00:33this possible, I thought it
00:33was incredible, right? So I'm
00:35going to just show you,
00:36I can write sine of
00:38x as x minus x
00:42cubed over three factorial plus
00:46x to the five over
00:49five factorial minus x to
00:51the seven over seven factorial,
00:54etc. If I if this
00:56This goes on forever. I
00:58get sine x. This is
01:02exactly the same as sine
01:03x if it goes on
01:04for infinity. I can actually
01:07do the same for cos
01:08x and e to the
01:09x in various different functions
01:11like this using, by using
01:13the McLaren series. The way
01:16it works is, and the
01:17reason, look at these two,
01:19what these two functions have
01:20in common is, they have
01:23to be
01:24differentiable. So I can do
01:25this very well. Maclaurin can
01:28do it here. He did
01:29it for any function. That
01:31is infinitely differentiable at zero
01:35and we can find that
01:36derivative at zero. Now when
01:38I say infinitely differentiable, I
01:40mean I can find the
01:41first derivative, the second derivative,
01:43the third derivative, the fourth
01:44derivative, the 100th derivative, the
01:46millions derivative. And you know
01:48that's true for sine x
01:49because sine x is just,
01:51and just keep it, there's
01:52a
01:52pattern it'll be sine x,
01:53the derivative of sine x
01:54is cos x. So that's
01:56the first derivative of cos
01:57x. Second derivative is negative
01:58sine, third derivative is negative
02:00cos fourth derivative is back
02:02to sine. And then just
02:04repeat this up. And e
02:05to the x is even
02:06easier. First derivative is e
02:07to the x, second derivative
02:08is e to the x,
02:09100th derivative is e to
02:10the x. OK. And I
02:13know the value of that
02:14derivative at zero, because I
02:16just sub in zero. In
02:17fact, for e to the
02:17x, it's always one.
02:20Okay, now I'm gonna show
02:24you the end. I'm gonna
02:27gonna show you how it
02:28works because instead of kind
02:31of trying to derive the
02:32whole thing, I'm just gonna
02:33show you how it works
02:34and then hopefully that'll be
02:35enough for you to understand
02:36this. So basically what McCloran
02:40did now, I may as
02:42well introduce you to Taylor,
02:45Brooke Taylor invented the Taylor
02:47series.
02:48It was actually, well, McClaren,
02:50the McClaren series came from
02:53the Taylor series. So I
02:53think Brooke Taylor deserves the
02:56most credit, but we're not
02:58studying Taylor series in the
02:59IB. We're only going to
03:01look at the McClaren series.
03:02And then McClaren series is
03:03a very particular type of
03:04Taylor series, basically where it
03:05all happens at zero. So
03:07that's all I'll say about
03:08that. You can definitely advise
03:09you go and Google Taylor
03:12series and McClaren series to
03:14learn a bit more about
03:15it. But for now,
03:16I'm just going to talk
03:17to you about McClorne. So
03:20what Taylor or McClorne decided
03:23to do is they said,
03:25well, okay, we have some
03:26function that is infinitely differentiable.
03:30What if we got a
03:31polynomial who had the same
03:34first derivative and the same
03:36second derivative and the same
03:38third derivative and the same
03:39fourth derivative and the same
03:40fifth derivative and the same
03:42millionth derivative? Now just think
03:44about
03:44about that. If you have
03:45two functions, one is a,
03:47one is just a, you
03:48know, sin x, one, let's
03:50look at this one. One
03:51is, so this is sin
03:52x, and this is a
03:53polynomial. So you have the
03:54function, sin x, and some
03:57polynomial function. And the first
03:59derivative is exactly the same.
04:01And the second derivative is
04:02exactly the same. And the
04:04third derivative is exactly the
04:05same at zero. And the
04:07fourth derivative, fifth derivative, millionth
04:09derivative. What's gonna happen is
04:11you're just gonna end up
04:12with the same
04:12the same function. It has
04:14to be the same thing.
04:15I mean, how could you
04:16have the every single derivative
04:18is the same and it
04:19not be exactly the same
04:21function? So that's the kind
04:22of idea. I'm going to
04:23show you how for any
04:26for any function, for any
04:28infinitely differentiable function, this series,
04:33the way it's set up,
04:34will always give you the
04:35same first derivative of second
04:36and third derivative. And it'll
04:38actually be, so I'll start
04:39with f of zero.
04:40at zero, this function, so
04:44imagine this is your sine
04:46x and this is your
04:47polynomial. So at zero f
04:50of zero equals f of
04:53zero plus these all become
04:59zero because I'm subbing in
05:00zero for x. So everything
05:02here to the right is
05:03zero. So it's plus zero
05:05if you like, or just
05:07f of zero,
05:08equals f of zero, great.
05:11They're the same. Now let's
05:13get the first derivative. The
05:16first derivative now, let me
05:18just go back to this.
05:20The first derivative of this
05:22is equal to, now remember
05:25f of zero, f dash
05:27of zero, f dash dash
05:29dash of zero, f dash
05:30dash dash of zero, these
05:31are just values. These are
05:34a constant. So it could
05:36be
05:36behave like a constant. So
05:38it's like something x. So
05:40what's the derivative of this?
05:41Well, the derivative of a
05:43constant is 0. So that
05:44disappears. What's the derivative of
05:46this? Well, it's something x.
05:48So the derivative is just
05:49f dash of 0. Plus,
05:53what's the derivative of this?
05:55Well, again, these are constants,
05:56but I've got a x
05:57squared here. So it's going
05:59to be 2 times x,
06:012x, over 2 factorial. Let's
06:03stay there.
06:04f dash dash of zero.
06:07So that's the derivative of
06:08this. Just two x times
06:10the constant that's there. Plus,
06:14what's the derivative of this?
06:15Well, it's 3x squared over
06:193 factorial times this. This
06:22is our third derivative at
06:24zero plus 4x cubed over
06:284 factorial. The fourth derivative
06:31at zero.
06:330 plus etc. Now let's
06:38get f dash of 0.
06:40So f dash of 0
06:42is equal to f dash
06:45of 0 is going to
06:46give me f dash of
06:480. So I have f
06:50dash of 0 plus. So
06:56I'm just subbing in 0
06:57for x. This is there's
06:59no x here. So this
06:59just stays the same.
07:01This becomes zero, this becomes
07:02zero, this becomes zero, everything
07:04else is zero. So I'm
07:05left with f dash of
07:07zero equals f dash of
07:08zero. So the first derivative
07:10of this thing is equal
07:11to the first derivative of
07:13this polynomial. Let's keep going.
07:16What about the, what about
07:19the second derivative? So the
07:21second derivative is I'm gonna
07:22now differentiate this thing. This
07:24goes to zero. This turns
07:26to two over two factorial
07:28is one
07:29So, um, and I have
07:31an x here, so it's
07:331x, the derivative of x
07:34is just 1, so I'm
07:35left with f dash dash
07:37of 0. Plus, what's the
07:41derivative of this? Well, it's
07:426x, 2 times 3 is
07:456, 6x, f dash dash
07:49of 0 over 3 factorial,
07:53plus 3 times 4 is
07:5412, 12,
07:57over 4 factorial, sorry, 12x
08:00squared, f, fourth derivative, that's
08:040 plus dot dot dot.
08:07Hopefully you're starting to see
08:09the pattern here. So what
08:11is f dash dash of
08:130, the second derivative of
08:14it's 0. Well, it's f
08:16dash dash of 0 plus
08:180, 0, everything else is
08:240. So I'm just left
08:25with f dash dash of
08:26zero. So the second derivative
08:27is the same. So now
08:29you can see, and you
08:30can see hopefully why it's
08:32factorial, because every time I
08:34do this, I'll just do
08:35it once more, when I
08:37get the third derivative of
08:39x, this disappears, and I'll
08:42get six over three factorial.
08:48I'll put this six over
08:50three factorial. The x disappears,
08:53f,
08:53dash dash dash zero. But
08:55the six over three factorial
08:57is just going to be
08:58one. They cancel. Similarly here,
09:01when I get the next
09:03derivative, this is going to
09:04be 24 over four factorial.
09:07This is going to, this
09:10is going to cancel because
09:1124 over four factorial is
09:14one. But this is going
09:16to be 24 over four
09:17factorial. X, F, four
09:21derivative at zero plus dot
09:24dot dot. And now I
09:26get the third derivative at
09:29zero is equal to the
09:30cancel f dash dash dash
09:33dash of zero plus zero.
09:36They're the same. So look,
09:38f of zero is the
09:39same for my let's say
09:40sine of x and my
09:41polynomial. The first derivative at
09:43zero is the same. The
09:45second derivative at zero is
09:46the same. The third derivative
09:48at zero is the same.
09:49And if I
09:49I kept doing this forever
09:51and ever and ever. What
09:52I would get is the
09:53same first derivative, the same
09:5410th derivative, the same 100th
09:56derivative, and ultimately the same
09:58function. That is the McCleurin
10:00series. That's what the whole
10:02thing is all about. Okay,
10:05so, as I said, in
10:08the next videos, we will
10:13actually derive some McCleurin series.
10:15We'll do the, we'll derive
10:16sine x and
10:17I will look at e
10:18to the x and we'll
10:19do some some other ones.
10:21But before I do that,
10:22I just want to show
10:23you how this how this
10:25looks visually because this is
10:26the this is the next
10:27thing that when I saw
10:29it, I was like, my
10:30word. This is incredible. So
10:32let me just come back
10:34to this. Okay. So here
10:38I have here I have,
10:41let's see, my little bit.
10:43This is
10:45the function, sine x. Now,
10:50if I just, this is
10:52actually the expansion that I
10:54showed you of sine x,
10:55it's x minus x cubed
10:58of three -vittorial plus x
10:59to the five -vittorial, forever
11:00and ever and ever. So,
11:03if I just put x
11:04like this, so what happens
11:05here is this has the
11:08same first sign of x,
11:11it's first derivative at zero,
11:13is 1 because its first
11:15derivative is cos x, cos
11:16of 0 is 1. So
11:18his first derivative is 1
11:20at 0 and an x
11:22is first derivative is 1
11:23at 0. So they have
11:24the same first derivative. But
11:26they don't have the same
11:26second derivative because his second
11:29derivative is actually, well actually
11:32they maybe they do the
11:32same second derivative. But they
11:33certainly don't have the same
11:34third, fourth, fifth, sixth, and
11:35seventh derivative. But this guy,
11:38now if we look at
11:39this curve, what happens
11:41is, he's got the same
11:44first derivative and the same.
11:46So the reason there's a
11:48zero term here, the x
11:51squared term is the same.
11:52So he's got the same
11:53first derivative, second derivative and
11:56third derivative. So look what
11:59happens. It starts to look
12:02a bit more like the
12:04sine x function. Let me
12:06draw another one or add
12:08another term. So we
12:09start to see, it actually
12:11starts to hug the curve
12:15like this. It gets closer
12:17and close. Each polynomial gets
12:19closer and closer and closer.
12:22And actually, if you draw
12:26this is the series in,
12:30this is the series in
12:32sigma notation. It's the sum
12:36from 0 to n.
12:37the negative one makes a
12:39minus plus minus plus minus
12:40minus. And I'm going to
12:41do a, I'm going to
12:41show you how to do
12:42this in a bit more
12:43detail. But here it's x
12:44to the power of 2k
12:45plus 1 over 2k plus
12:471 factorial. Because that's my
12:493, 5, 7 sequence. So
12:53if I show you this,
12:55now this is when n
12:57is 0, when n is
12:591, 2, 3. And what's
13:02this? It just starts hogging
13:05the curve
13:06and the more the higher
13:08I go up the more
13:10it looks like sine X.
13:12Now if you actually zoom
13:13in here you zoom right
13:16in here it looks look
13:17it's not actually touching. It
13:21looks like it is. Well
13:23there might be a few
13:24places where it does actually
13:26meet but the the only
13:28place where it's that you
13:30know it's going to be
13:31exactly the same is that
13:33zero. The rest of
13:34but it's just going to
13:34be really, really, really close.
13:36But even if, like, if
13:37I zoomed in, even I'd
13:38say here, I don't know,
13:40like, it'll be so close.
13:43I don't think even Desmos
13:45can go close enough there.
13:50That's, that's the limit of
13:51how, how zoomed in Desmos
13:53can go. And I can't
13:57even see the difference. Even
14:00though there probably is some
14:02time
14:02tiny, tiny difference and that's
14:04why it's called an approximation.
14:07Here, certainly I'll be able
14:10to see here. So the
14:11Taylor series is an approximation
14:13of the function, but if
14:14it goes on front to
14:15infinity, it's to be, it's
14:18theoretically the exact same function.
14:22Okay, hopefully that makes some
14:25bit of sense. As I
14:26say, the first time I
14:27saw this, I was both
14:28amazed and confused.
14:30So definitely I would recommend
14:35going watching the next videos,
14:38practicing some of the Cloron
14:40series questions, deriving some series
14:44and then come back to
14:45this to kind of hopefully
14:47it'll be easier for you
14:49to then get your head
14:50around what McClurene was doing
14:53or Taylor were doing. Taylor
14:55was doing when they came
14:57up with these
14:58But as I say, the
15:01fact that you can write
15:02these functions that you're very
15:03familiar with as a sum
15:05of terms like this is
15:09pretty incredible. Okay, so I'll
15:10see you in the next
15:11few, the next few videos.