I recently have been going over Power Series and Taylor Series (and their theorems) but ran into the following multiple choice problem. I am not allowed to use a calculator so I am unsure as to how we find the interval of accuracy.
On which interval does the polynomial $y=-\cfrac{3\pi}{2}+ x - \cfrac{(2x-3\pi)^3}{48}$ best approximates the function $y=\cos (x)$.
By playing with the given function I did find that it can be re-written as $y=\bigg(x-\cfrac{3\pi}{2}\bigg) - \cfrac{8(x-\frac{3\pi}{2})^3}{48}$ and this leads me to the fact that this is a Taylor Series for $\sin (x)$ centered at $\cfrac{3\pi}{2}$.
By graphing it I find the solution is roughly the interval [3,6], but I am not sure how to do this without calculator. Any help is appreciated. Thank you!