00:01
In this question we are asked to find the fourth degree taylor polynomial about x equals 0.
00:08
So we need to calculate p for x and that equals to the sum n from 0 to 4 of f n of 0 divided by n factorial multiplied by x to the n where f n of 0 is the nth derivative at 0.
00:36
We need to calculate since n is from 0 to 4 we need to calculate the zeroth derivative which is simply f of x and f of 0 and f of 0 equals to 0.
00:50
We need to calculate f prime of x f prime of x equals to the derivative of x times sine x and by the product rule this equals to sine x plus x cos x when x equals 0 f prime of 0 equals to 0 plus 0 so it's also 0.
01:19
Now let's calculate f double prime.
01:22
In total we need four derivatives f double prime equals to the derivative of sine x plus x cos x that's going to be cos x plus cos x plus minus minus x sine x at 0.
01:56
Since cos x cos 0 equals to 1 we are going to get 2.
02:02
Let's get the third derivative f triple prime of x equals to the derivative of 2 cos x minus x sine x that's going to be negative 2 sine x minus sine x minus x cos x which is simply negative 3 sine x minus x cos x and at 0 that's going to be sine x sine 0 is 0 and 0 times cos 0 is also 0 so it's 0 minus 0 equals to 0.
03:01
Now let's calculate the fourth derivative.
03:09
The fourth derivative is going to be negative 3 cos x minus cos x minus x sine x and after simplification that's going to be negative 4 cos x plus x sine x...