WEBVTT
1
00:00:01.840 --> 00:00:03.439 A:middle L:90%
So for this problem, we want to use the
2
00:00:03.439 --> 00:00:07.290 A:middle L:90%
multiplication of division of power Siri's in order to find
3
00:00:07.290 --> 00:00:09.380 A:middle L:90%
the first three, none zero terms in the MacLaurin
4
00:00:09.380 --> 00:00:12.890 A:middle L:90%
series. So we're given the function y equals e
5
00:00:12.890 --> 00:00:18.059 A:middle L:90%
to the negative X squared coastline X. So we
6
00:00:18.059 --> 00:00:24.379 A:middle L:90%
already know that co sign X McLaurin Siri's were given
7
00:00:24.379 --> 00:00:27.109 A:middle L:90%
it already in table one. If you want to
8
00:00:27.109 --> 00:00:30.890 A:middle L:90%
refer to that and it's negative one to the N
9
00:00:31.539 --> 00:00:35.429 A:middle L:90%
of X to the to end time over two n
10
00:00:35.939 --> 00:00:39.630 A:middle L:90%
factorial. So with that in mind, we also
11
00:00:39.630 --> 00:00:46.869 A:middle L:90%
have e to the negative X squared, Equalling the
12
00:00:46.869 --> 00:00:51.570 A:middle L:90%
sum of an equal zero to infinity of negative X
13
00:00:51.570 --> 00:00:59.520 A:middle L:90%
squared to the end over in factorial. So with
14
00:00:59.520 --> 00:01:03.140 A:middle L:90%
that in mind, another way we could write this
15
00:01:03.149 --> 00:01:07.079 A:middle L:90%
would be if we want to rewrite it. We
16
00:01:07.079 --> 00:01:12.019 A:middle L:90%
could write it as, um, negative one to
17
00:01:12.019 --> 00:01:17.329 A:middle L:90%
the end. Since that's often preferred notation Negative one
18
00:01:17.329 --> 00:01:22.420 A:middle L:90%
to the end times X to the two n over
19
00:01:22.420 --> 00:01:27.010 A:middle L:90%
n factorial. So that being said, we see
20
00:01:27.010 --> 00:01:30.969 A:middle L:90%
that four terms of each will probably be enough.
21
00:01:32.340 --> 00:01:34.310 A:middle L:90%
So the first four terms of this will be one
22
00:01:34.310 --> 00:01:40.099 A:middle L:90%
minus x squared over two factorial put us back into
23
00:01:40.099 --> 00:01:44.829 A:middle L:90%
the boards over four factorial minus X to the sixth
24
00:01:44.829 --> 00:01:47.909 A:middle L:90%
over six sectorial. And that obviously keeps going on
25
00:01:47.909 --> 00:01:49.819 A:middle L:90%
. But we don't need any more. And then
26
00:01:49.819 --> 00:01:52.849 A:middle L:90%
this is gonna be one minus X squared over two
27
00:01:52.849 --> 00:01:57.549 A:middle L:90%
factorial plus X to the fourth over two factorial.
28
00:01:57.239 --> 00:02:01.750 A:middle L:90%
Um, actually, this will be just X squared
29
00:02:02.340 --> 00:02:07.210 A:middle L:90%
, Um, minus X to the sixth over three
30
00:02:07.210 --> 00:02:12.830 A:middle L:90%
factorial and then plus go on. So what we'll
31
00:02:12.830 --> 00:02:15.719 A:middle L:90%
do is we'll multiply these terms. So what we're
32
00:02:15.719 --> 00:02:28.550 A:middle L:90%
gonna end up getting as a result is this right
33
00:02:28.550 --> 00:02:44.159 A:middle L:90%
here and then we're gonna multiply this Bye. This
34
00:02:44.159 --> 00:02:49.560 A:middle L:90%
right here. So when we do that, what
35
00:02:49.560 --> 00:02:53.949 A:middle L:90%
we're gonna end up getting as a result, is
36
00:02:54.639 --> 00:03:00.020 A:middle L:90%
simplifying it further. We'll get about one minus X
37
00:03:00.030 --> 00:03:07.159 A:middle L:90%
squared over to minus two X squared over two.
38
00:03:07.539 --> 00:03:10.449 A:middle L:90%
So these will combine to give us a negative three
39
00:03:10.449 --> 00:03:14.469 A:middle L:90%
x squared over two so we can do that right
40
00:03:14.469 --> 00:03:20.180 A:middle L:90%
there, and then we'll get plus X to the
41
00:03:20.189 --> 00:03:24.729 A:middle L:90%
fourth over 24 plus two x to the fourth over
42
00:03:24.729 --> 00:03:30.629 A:middle L:90%
to turning this into 24 will make this into 24
43
00:03:30.889 --> 00:03:36.509 A:middle L:90%
. So we'll end up getting 25 x to the
44
00:03:36.509 --> 00:03:42.150 A:middle L:90%
fourth over 24 so those would be our first three
45
00:03:42.159 --> 00:03:46.289 A:middle L:90%
non zero terms within the MacLaurin series for the specific
46
00:03:46.289 --> A:middle L:90%
function.