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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/9/b/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 9 / b / Solution by

Codex  0 2026-10-18

The absolute value in the original PDF reflects every negative sine arch upward. The graph is nonnegative, with zeros at

x = kπ

for

k = 0, 1, 2, 3, 4

, and it lies below the decaying envelope

e^{- x}

. On the

k

th arch, differentiation gives a factor proportional to

cos x - sin x

. Thus its unique local maximum is

x_{k} = \frac{π}{4} + kπ, f (x_{k}) = \frac{e ^{- (π /4 + kπ)}}{2}, k = 0, 1, 2, 3.

Successive maximum heights have ratio

e^{- π}

. At each interior zero the one-sided slopes have opposite signs, producing a corner. The smaller panels below magnify the later arches so that all four remain visible.

Figure 1.
Four positive exponentially damped sine arches, with separate linear-scale enlargements of the later arches
. Details.

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