00:01
So we want to determine the first four non -zero terms of the taylor series, which we ultimately know that what we'll have as a result is the first term of our taylor series.
00:20
We know that a taylor series is f to the n minus 1 times 2 times x minus 2.
00:36
And this is f to the n minus 1 prime of 2 times x minus 2 to the n minus 1 all over all going to be over n minus 1 factorial so with this in mind what we'll do is plug in different values so the first term that we have will be a 0 so over 1 here so we just have f of 2, f of 2 x minus 2 to the 0, so that's just going to give us 1.
01:41
So f of 2 over 0 factorial.
01:48
We know that 0 factorial is just 1, and f of 2 is 1 3rd.
01:55
So we get 1 3rd times 2, or 1 3rd over 1, which is just going to give us 1 3rd.
02:06
And for the second term, what we'll have is we know that f prime of two, since that's what we'll have, will give us a negative one -ninth times.
02:25
In this case, we have x minus 2.
02:29
So x minus 2 to the 1, and this will be just one factorial down here.
02:39
So what we'll end up getting as a result is going to be equal to a negative 1 9th times x minus 2.
02:54
And for our third term of the taylor series, what we have is this will be f double prime of 2, which we know to be 2 over 27...