A question that asks students to derive or prove an error bound for a numerical integration method (e.g., trapezoidal rule) using tools such as the Peano kernel, Taylor expansions, or bounding arguments.
We consider the trapezoidal rule on $I = [a,b]$ with associated error $e_n(f) = \int_a^b f(x)\,\mathrm{d}x - T_n(f)$, where $f$ is of class $\mathcal{C}^2$. Deduce the error bound $$\left|e_n(f)\right| \leqslant \frac{(b-a)^3}{12n^2} \sup_{x \in [a,b]} |f''(x)|.$$