 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/ib/paper-2/5d/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

Every sine term vanishes at zero, so $G (0) = 0$ . The one-sided values of the periodic extension of

f^{'}

are

f_{per}^{'} (0 -) = - 1

and

f_{per}^{'} (0 +) = 1

. The Fourier series converges at this jump to their midpoint,

G (0) = \frac{f _{per}^{'} ( 0 - ) + f _{per}^{'} ( 0 + )}{2} = 0.

This midpoint need not equal the arbitrarily assigned value of the periodic derivative at the endpoint.

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