Question

Find the Fourier cosine series for the function f(x)=4−x x∈(0,4)

defined on the interval x∈(0,4) in the form
[infinity]
f(x)=A0+∑ An cos(πnx/4)
n=1
Enter the symbolic formulae for the Fourier coefficients below. They should depend on the index n.
A0=
An=

Answer 1

f(x) = 8 + Σ (-4/nπ) * sin(nπ) * cos(nπx/4) on (0, 4). Fourier cosine series for f(x) = 4 - x.

Ao = 8

An = (-4/nπ) * sin(nπ)

Let’s find the Fourier cosine series for the function f(x) = 4 - x defined on the interval (0, 4).

Here's how we can solve this problem:

1. Find the Fourier coefficients:

Ao = (2/pi) * ∫f(x) dx from 0 to 4

An = (1/pi) * ∫f(x) * cos(nπx/4) dx from 0 to 4, for n ≥ 1

2. Substitute the function f(x) and integrate:

Ao = (2/pi) * ∫(4 - x) dx from 0 to 4

An = (1/pi) * ∫(4 - x) * cos(nπx/4) dx from 0 to 4

3. Simplify the integrals and solve for Ao and An:

By solving these integrals, we get:

Ao = 8

An = (-4/nπ) * sin(nπ)

4. Write the Fourier cosine series:

f(x) = Ao + ∑ An cos(nπx/4)

Replace Ao and An with the values we found:

f(x) = 8 + ∑ (-4/nπ) * sin(nπ) * cos(nπx/4)

Therefore, the Fourier cosine series for the function f(x) = 4 - x defined on the interval (0, 4) is:

f(x) = 8 + ∑ (-4/nπ) * sin(nπ) * cos(nπx/4)