00:00Hi guys. Okay, so in
00:02the last video I can
00:03show you how or where
00:05the McLaren series comes from.
00:07In this video we are
00:08going to actually find a
00:09McLaren series using this formula.
00:13Now in the formula booklet
00:14we are given these five
00:17McLaren series, ETX, Ln of
00:191 plus X, Sine X,
00:20Cos X and Arc 10
00:22of X. This one is
00:23not in the formula booklet
00:24but it specifically says it
00:26in the guide so I'm
00:27going to do it
00:28lesson on this one in
00:29particular. This is one plus
00:30x to the bar of
00:31p where p is a
00:32rational number. So don't worry
00:34about that one for now.
00:36These ones are all given
00:36in the formula booklet. You
00:38can use them. Of course,
00:41when they're given in the
00:42formula booklet, but you need
00:44to be able to show
00:45them from this, like from
00:48the formula you need to
00:49be able to show where
00:50they come from. So my
00:51example, I'm only going to
00:53do one. I'm not going
00:53to do all of these
00:55five.
00:56But the concept is the
00:59same. You can find the
01:00McLaurin series for any function.
01:01In fact, not just these
01:02five functions using this method
01:04that I'm about to show
01:05in. It's pretty straightforward. It's
01:07simply just subbing things into
01:08this formula. But I want
01:10to show you how to
01:11lay it out and how
01:11to do a property. So
01:13the function is sine of
01:14x. And we want to
01:15show that the McLaurin series
01:16for sine of x is
01:17this. And it says the
01:20McLaurin series up to the
01:21term in x to the
01:244.
01:24Well, actually, I meant to
01:26say, in x to the
01:297, it's a bit annoying.
01:31Okay, in x to the
01:327, 4, sine x is
01:35given by this. Okay, so,
01:39look, the McLaren series goes
01:40on forever. So, in an
01:42exam, whenever you're asked to
01:43find a McLaren series, they're
01:44always going to tell you
01:45to stop somewhere. You can't
01:46write out the whole thing
01:47forever and ever and ever.
01:48So, I'm just, sometimes they
01:50say the first four non
01:52-zero
01:52terms, sometimes to say this,
01:53what I want to do
01:54is up to the term
01:55in x to the power
01:56of whatever x to our
01:577 is this. Okay. So
01:59what I do is I
02:00say f of x, I'm
02:04going to do f of
02:04x equals sin x, whatever
02:06the thing is. And then
02:08I need to find out
02:09how many derivatives do I
02:11need? Okay. So look, this
02:13is for, this term is
02:15the third derivative. This is
02:17the fourth derivative. Sorry, this
02:18is the fifth derivative and
02:19this is the seventh derivative.
02:20There are terms there with
02:22there are going to be
02:23zero terms that are zero
02:25like when I sub in
02:26zero here it becomes zero.
02:28So I actually need seven
02:30derivatives here. So the first
02:31derivative, the first derivative is
02:35cos x, the second derivative
02:38is minus sign, the third
02:43now to be fair. It's
02:44not often the dasquy to
02:45do seven derivatives. I just
02:48want to
02:48I just wanted to be
02:50very clear on how it's
02:52done. So minus cos x,
02:55the fourth derivative is back
02:59to sine. So then it
03:01just repeats itself. The fifth
03:03derivative is cos sixth derivative.
03:13You can write sigma or
03:14if you can't remember how
03:16write Roman numerals. It's okay
03:18to write six here in
03:19a bracket. So the sixth
03:21derivative is where am I
03:24minus sign and the seventh
03:27derivative is minus cos x.
03:36Okay, so I need all
03:37the derivatives. To complete the
03:40McLaren series, you need f
03:42of zero, f dash of
03:44zero, f
03:44dash dash of zero and
03:46all of these. So first
03:46we get the derivatives and
03:48then I get f, I
03:51get the value of them
03:53at zero. So f of
03:54zero is sine of zero,
03:57which is zero. f dash
04:00of zero is cos of
04:02zero, which is one. f
04:05dash dash of zero equals
04:07minus sine of zero, which
04:09is zero. Now notice, because
04:11of the way this,
04:12one, this one is it's
04:13sine cosine cosine cos, it's
04:15going to be 0, 1,
04:160, negative 1, etc. There's
04:18a pattern there. So it's
04:21f, third derivative at 0
04:23is negative 1, fourth derivative
04:25at 0 is 0, fifth
04:30derivative is 1, sixth derivative
04:35is 0 and the seventh
04:39derivative
04:40because negative one. Now once
04:43I have these, it's actually
04:46pretty straightforward. I'm just going
04:47to sub these into this
04:49McLaren series. Now let me
04:50just copy and paste this
04:53so I have it down
04:55here. Okay, so this is
05:01my McLaren series. I'm going
05:03to sub these into this.
05:06So I can say f
05:08of x
05:08x f of x is
05:12approximately equal to. So it
05:14is equal to when it
05:15goes on forever, but he's
05:16only asking me to find
05:18the term up to x
05:19to the power of seven.
05:20So I'm going to put
05:21in the approximately, approximately equals.
05:26Then it's f zero. I
05:27find my f zero. It's,
05:29let's put in zero plus
05:31x times f dash of
05:34zero, x times one, which
05:36is
05:36x times 1 plus x
05:42squared over 2 factorial times
05:450 plus and then I
05:50just keep going x cubed
05:53over 3 factorial times negative
05:561 plus x to the
05:594 over 4 factorial times
06:020 and do you
06:04You have to put in
06:05these zeros all the time.
06:06The fact that it was
06:08a show that question, I'm
06:12going to do it because
06:12I like to be crystal
06:13clear to my examiner what
06:15I'm doing. X to the
06:175, this is my fifth
06:18derivative. This is 1 plus
06:22let me just get all
06:25the space I have here.
06:27X to the 6 over
06:296 factorial times 0 plus
06:33x to the 7 over
06:347 factorial times minus 1.
06:38Now this is equal to
06:42x, so 0 plus so
06:44that's just my x. This
06:46is 0. There's a minus
06:471, so it's minus x
06:49cubed over 3 factorial. Then
06:52this is 0 plus x
06:54to the 5 over 5
06:56factorial plus 0 minus x
06:59to the 7 over 7.
07:01factorial, which is this exactly
07:05what they wanted us to
07:07show. So that's the McLaren
07:08Series. It's like the first
07:10video I showed you, I
07:13said this is complicated. It's
07:15hard to get your head
07:15around what's going on. But
07:17the actual writing out a
07:19McLaren Series isn't that difficult
07:21at all. All you do
07:23is write out the derivatives,
07:25then sub in zero and
07:26sub it into the formula,
07:27and that's it.
07:29Definitely, I'd like you to
07:32go and try and do
07:33E to the X, L
07:34and a 1 plus X,
07:35Cos X and our octana
07:36VX. Do them now, do
07:39them on your own. Look,
07:41you have these are the
07:42answers. This is what you're
07:42trying to get. Make sure
07:44you know how to do
07:44it. I'm going to do
07:45a lesson on this one
07:49after, and I'm also going
07:50to do lessons where we
07:53combine them and look at
07:54more complicated functions. Okay.
07:57see you in the next,
07:58see you in the next
07:58video.