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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ia/paper-2/5a/b/iii/solution

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Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 2 / 5A / b / iii / Solution by

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

Each smooth segment follows Newton's law of cooling:

T^{'} = - k (T - T_{0}) < 0

and

T^{''} = k^{2} (T - T_{0}) > 0

. Thus the graph decreases and is convex on both sides of

t_{1}

. It starts at

α T_{0}

, falls to

α T_{0} /2

, jumps upward by

β

, and then decreases asymptotically to

T_{0}

The right-hand limit at the jump is above the original temperature, because

β > α T_{0} /2

implies

T (t_{1} +) > α T_{0}

. The vertical dotted segment in the sketch denotes an instantaneous jump, not a continuous evolution through intermediate temperatures. The example uses

α = 4

β / T_{0} = 3

; its qualitative shape holds for all the stipulated parameters.

Figure 1.
Cooling interrupted by an instantaneous temperature increase
.

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