Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 12 / a

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- Solution a

Solution

The periodic extension is an even triangular wave: peaks of height one occur at the even integers

x = 2 k

and valleys of height zero at the odd integers

x = 2 k + 1

. Adjacent peaks and valleys are joined by straight segments of slopes

\pm 1

. The function is continuous and has corners at every integer.

Figure 1.
Period-two triangular wave and its first odd cosine harmonics
. Original graph of the periodic extension of $1 - ∣ x ∣$ , together with the first one and four nonconstant terms of its Fourier cosine series. Details.

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