Ch22 Discussion

Discussion Guide

Prompt 1 — Why does symmetry help? Have students derive, on the board, the forward and central differences from the Taylor expansions of $f(x\pm h)$. Ask them to point to the exact term that cancels in the central case and survives in the forward case. Then ask: is there a backward-looking formula that is also $O(h^2)$? (Yes — a one-sided second-order formula using $f(x), f(x-h), f(x-2h)$ with weights $3, -4, 1$ over $2h$; useful at a boundary where you cannot step forward. Good bridge to Chapter 24's boundary handling.)

Prompt 2 — The floor, argued from Chapter 20. Pose: "Truncation error says use a tiny $h$. Round-off says don't. Where's the optimum, and why can't you beat $\sqrt\varepsilon$ with a forward difference?" Let them reconstruct the $\tfrac{h}{2}|f''| + \varepsilon|f|/h$ model and minimize it. Connect to Case Study 1's inherited bug. Ask: which change helps more — a smaller $h$ or a better formula? (The formula: central moves the floor down and out.)

Prompt 3 — When is the fancy rule the wrong rule? Present two tasks: integrate an analytic $f$ you can call anywhere, versus integrate a fixed table of detector samples. Ask which rule fits each and why Gauss is unavailable for the second. Draw out the principle: the binding constraint is who chose the nodes. (CS-02.)

Prompt 4 — Reading order from data. Give the group a table of errors at $h, h/2, h/4$ with ratios $3.9, 3.98, 4.0$ and another with $2.1, 2.03, 2.0$. Have them state each method's order and name a rule that fits. Then give a "broken" table where the ratio is $2$ for a method that should be $O(h^2)$ — ask what bug that implies (a one-sided formula, or a wrong denominator).

Mini group activity (30–40 min): "Validate a mystery integrator." Hand each group a working but unlabeled quadrature function (secretly one of trapezoid / Simpson / Gauss-2) as a black box they may only call. Task: determine its order of accuracy and its exactness degree experimentally — by integrating $1, x, x^2, x^3, x^4$ on $[0,1]$ (which does it get exact?) and by running the halving test on a function it does not integrate exactly (what is the error ratio?). Groups report which rule they think it is and the evidence. This rehearses the entire chapter — order, exactness degree, the halving test, and the scientific habit of characterizing a routine you did not write before trusting it (the CS-02 lesson, and the Chapter 21 "call the library, then verify" instinct).