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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-2/2g/iv/solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 2G / iv / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

The fundamental theorem of calculus gives

f_{n} (x) = 2 n \int_{x - 1/ n}^{x + 1/ n} h^{'} (t) d t .

(1)

Thus $f_{n} (x) \to 4 h^{'} (x)$ . Uniform continuity of $h^{'}$ makes the difference from $4 h^{'} (x)$ uniformly small over every interval of radius $1/ n$ , so convergence is uniform on all of $R$ .

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