00:01
In this question, we are asked to write down the third degree taylor polynomial for the function f centered at the point a.
00:09
In our case a is zero, so basically we are writing down the maclaurin polynomial.
00:14
First of all, recall that e to the negative 4x, well the series for e to the u is the series u to the n divided by n factorial and from zero to infinity.
00:29
If we replace u by 5x, sorry u by negative 4x, we'll get e to the negative 4x equals the series negative 4x to the n divided by n factorial and similarly sine u equals the series negative 1 to the n u to the 2n plus 1 divided by 2n plus 1 factorial and from zero to infinity.
01:06
Now replace u by 5x, we'll get sine 5x equals the series negative 1 to the n times 5x to the 2n plus 1 divided by 2n plus 1 factorial and from zero to infinity.
01:29
Now let's write down the first few terms.
01:34
For n equals zero, we are going to get 1.
01:36
For n equals 1, we'll get minus 4x over 1 factorial, which is 1.
01:43
For n equals 2, we'll get plus 16x squared over 2 factorial, which is 2.
01:50
And let's do the third degree minus 64x cubed divided by 3 factorial, which is 6 plus on.
02:03
And also let's write down the terms up to x cubed for sine 5x.
02:09
For n equals zero, we'll get 5x and for n equals 1, we'll get minus 125x cubed divided by 3 factorial, which is 6 plus on.
02:25
Now to get the third degree taylor polynomial, so we'll first multiply these two series, 64 over 6 equals 32 over 3 and multiply this by sine 5x.
03:06
And now let's multiply them and see what we are going to get...