Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Continuous mathematics / 5 / d

Write the computed function values as $f(x\pm h)+e_{\pm }$, where $|e_{\pm }|\le \delta$. The computed central finite difference $\tilde{d}$ satisfiesThe two contributions are respectively evaluation absolute error and truncation error. For $\delta >0$, step-size choice for a noisy central derivative minimizes their sum: its derivative is $h/3-\delta /h^2$, negative before and positive afterIf $\delta =0$, the bound tends to zero as $h\downarrow 0$ and has no positive minimizing step.