Taylor polynomials practice question 11

The error after $T_{2}$ is set by the next derivative at some unknown point between 0 and 0.5. Since you cannot know the point, what can you do with that derivative instead?

The remainder after $T_{2}$ is $\dfrac{f'''(c)}{3!}(0.5 - 0)^{3}$ for some $c$ between 0 and 0.5. Bound $|f'''(c)|$ on that interval.

$|\cos c| \leq 1$ and $4 + c^{2} \geq 4$, so $|f'''(c)| \leq \tfrac{1}{4}$.