What polynomial degree should be taken in the function $f(x) = \sin(x)$ so that the largest modulus of the difference between the value of the Taylor polynomial and the value of the $\sin(x)$ function on the $[0, \frac{\pi}{2}]$ interval is not greater than $10^{-17}$?
I'm not quite sure how to do this task. I started by finding the general form of the Taylor series for $\sin(x)$. Then I found the general form of the remainder of the Taylor series as: $R_n(x) = \frac{\sin(c + \frac{n\pi}{2}) \cdot x^{n+1}}{(n+1)!}$. Then I constrained this function by $\frac{x^{n+1}}{(n+1)!}$. I think I need to calculate this inequality $\left|\frac{x^{n+1}}{(n+1)!}\right| < 10^{-17}$, but I have no idea what to take for $x$? The extreme values of the interval?
Best Answer
For a given $n$, your expression attains its maximum value on the interval when $x$ is equal to $\pi/2$. So yes, this is the value you should take for $x$.