Suppose f has period 2 and f(x) = x for 0 \leq x&lt.2. Find the fourth-degree Fourier polynomial and graph it on 0 \leq x&lt.2.

Line 2021-08-09 Answered

Suppose f has period 2 and f(x) = x for 0x<2. Find the fourth-degree Fourier polynomial and graph it on 0x<2

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Expert Answer

Brighton
Answered 2021-08-10 Author has 103 answers

Step 1
The given function
f(x)=x,0x<2
Since we know the general expression of Fourier series for f on [b2,b2] is
a0+k=1akcos(2πkxb)+k=1bksin(2πkxb)
Where constant expressed as
a0=1bb2b2f(x)dx,ak=2bb2b2f(x)cos(kx)dx, for k>0
bk=2bb2b2f(x)sin(kx)dx, for k>0
Now the value of constant
a0=12[02xdx]=12[x2202]=1
Furthermore
a1=42[02xcosπxdx]=2[(xsinπxπ+cosπxπ2)02]=0
And
b1=42[02xsinπxdx]=2[(xcosπxπ+sinπxπ2)02]=4π
Therefore the fourier polynomial of degree 1 is given by

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