00:01
In this question, we are given the third degree taylor polynomial around 0 for the function sine x, and we are asked to find the error between the function, sine x and its tailor polynomial at x equals to 0 .6.
00:17
And we are going to use a formula which says that the error between the function and its nth degree taylor polynomial is less equal than some constant m times the absolute value of x to the n plus 1 power divided by n plus 1 factorial, where the constant m bounds the n plus 1 derivative of the function f at t for t in the interval around the 0 .6.
00:58
For example, we can take the interval to b from 0 .5 to 0 .7.
01:06
But for us it's not important because what is the n plus, so in our case n equals to n equals to 3 because it's the third degree tailor polynomial.
01:24
Therefore what we are doing here is we are trying to find the difference between sign 0 .6 minus p3 of 0 .6 right and this must be less equal now than m over 4 factorial because n equals to multiplied by the absolute value 0 .6 to the 4th power.
01:59
So in this formula we replaced x by 0 .6 and we replaced n by 3.
02:09
Now we need to determine the constant m.
02:15
Determine the constant m, we need to calculate the fourth derivative of the function f, right? 4 of x equals to the fourth derivative of sine x.
02:33
Of sine x.
02:35
The first derivative of sine x is cos x, right? cosine x.
02:45
The second derivative of sine x is the derivative of cosine x is negative sine x.
02:54
The third derivative is negative cosine x and the fourth derivative equals to sine x...