APM2611 Assignment 4 Solutions 2026

APM2611 Assignment 4 Solutions 2026

Full Solutions By TA tutor iQ level

UNISA

DUE DATE: 23-09-2026

, APM2611 Assignment 04 – Full Solutions



QUESTION 1

1.1 Fourier Series of f(x) = |x| on [−π, π]

The Fourier series is:
∞
𝑎0
𝑓(𝑥) = + ∑(𝑎𝑛 cos⁡ 𝑛𝑥 + 𝑏𝑛 sin⁡ 𝑛𝑥)
2
𝑛=1



Compute a₀:

1 𝜋 2 𝜋 2 𝑥2 𝜋 2 𝜋2
𝑎0 = ∫ ∣ 𝑥 ∣ 𝑑𝑥 = ∫ 𝑥 𝑑𝑥 = ⋅ ∣ = ⋅ =𝜋
𝜋 −𝜋 𝜋 0 𝜋 2 0 𝜋 2


Compute aₙ:

Since |x| is even, the product |x|cos(nx) is even, and |x|sin(nx) is odd.

1 𝜋 2 𝜋
𝑎𝑛 = ∫ ∣ 𝑥 ∣ cos⁡ 𝑛𝑥 𝑑𝑥 = ∫ 𝑥cos⁡ 𝑛𝑥 𝑑𝑥
𝜋 −𝜋 𝜋 0


Integrate by parts: let u = x, dv = cos(nx)dx → du = dx, v = sin(nx)/n
𝜋
2 𝑥sin⁡ 𝑛𝑥 𝜋 sin⁡ 𝑛𝑥
= [ 𝑛 ∣0 − ∫
𝜋 0
𝑛 𝑑𝑥]
2 1 cos⁡ 𝑛𝑥 𝜋 2 1
= [0 + ⋅ ∣0 ] = ⋅ 2 [cos⁡(𝑛𝜋) − 1]
𝜋 𝑛 𝑛 𝜋 𝑛
2
= 2 [(−1)𝑛 − 1]
𝜋𝑛


• If n is even: 𝑎𝑛 = 0
2 4
• If n is odd: 𝑎𝑛 = 𝜋𝑛2 (−2) = − 𝜋𝑛2

Compute bₙ:

Since |x|sin(nx) is odd:

1 𝜋
𝑏𝑛 = ∫ ∣ 𝑥 ∣ sin⁡ 𝑛𝑥 𝑑𝑥 = 0
𝜋 −𝜋