Question

A function is defined over (0, 4) by f(x) = { 1/4 x 0 < x and x <= 2 { 0 2 < x and x < 4 We then extend it to an odd periodic function of period 8 and its graph is displayed below. y 1 0.8 0.6 0.4 0.2 -8 -6 -4 -2 0 2 4 6 8 10 12 -0.2 -0.4 -0.6 -0.8 -1 x The function may be approximated by the Fourier series f(x) = a0 + sum_{n=1}^{infinity} (an cos(n pi x / L) + bn sin(n pi x / L)), where L is the half-period of the function. Use the fact that f(x) and f(x) cos(n pi x / L) are odd functions, enter the value of an in the box below. an = [ ] , for n = 0, 1, 2, ... Hence the Fourier series made up entirely of sines. Calculate the following coefficients of the Fourier series and enter them below in Maple syntax. b1 = [ ] b2 = [ ] b3 = [ ]

          A function is defined over (0, 4) by
f(x) = { 1/4 x  0 < x and x <= 2
       { 0    2 < x and x < 4
We then extend it to an odd periodic function of period 8 and its graph is displayed below.
y
1
0.8
0.6
0.4
0.2
-8 -6 -4 -2 0 2 4 6 8 10 12
-0.2
-0.4
-0.6
-0.8
-1
x
The function may be approximated by the Fourier series
f(x) = a0 + sum_{n=1}^{infinity} (an cos(n pi x / L) + bn sin(n pi x / L)),
where L is the half-period of the function.
Use the fact that f(x) and f(x) cos(n pi x / L) are odd functions, enter the value of an in the box below.
an = [ ] , for n = 0, 1, 2, ...
Hence the Fourier series made up entirely of sines.
Calculate the following coefficients of the Fourier series and enter them below in Maple syntax.
b1 = [ ]
b2 = [ ]
b3 = [ ]

Show more…

A function is defined over (0, 4) by
f(x) = 1/4 x 0 < x and x <= 2
0 2 < x and x < 4
We then extend it to an odd periodic function of period 8 and its graph is displayed below.
y
1
0.8
0.6
0.4
0.2
-8 -6 -4 -2 0 2 4 6 8 10 12
-0.2
-0.4
-0.6
-0.8
-1
x
The function may be approximated by the Fourier series
f(x) = a0 + sumn=1^infinity (an cos(n pi x / L) + bn sin(n pi x / L)),
where L is the half-period of the function.
Use the fact that f(x) and f(x) cos(n pi x / L) are odd functions, enter the value of an in the box below.
an = [ ] , for n = 0, 1, 2, ...
Hence the Fourier series made up entirely of sines.
Calculate the following coefficients of the Fourier series and enter them below in Maple syntax.
b1 = [ ]
b2 = [ ]
b3 = [ ]

Added by Gordon D.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

04/27/2023

Step 1

Since the function is odd, we know that all the cosine terms will be zero, i.e., $a_n = 0$ for all $n$. Now, we need to find the sine coefficients $b_n$. The formula for $b_n$ is given by: $$b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) Show more…

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A function is defined over (0, 4) by f(x) = { 1/4 x, 0 < x and x <= 2; 0, 2 < x and x < 4 We then extend it to an odd periodic function of period 8 and its graph is displayed below. The function may be approximated by the Fourier series f(x) = a0 + sum_{n=1}^{infinity} (an cos(n pi x / L) + bn sin(n pi x / L)), where L is the half-period of the function. Use the fact that f(x) and f(x) cos(n pi x / L) are odd functions, enter the value of an in the box below. an = [ ] , for n = 0, 1, 2, ... Hence the Fourier series made up entirely of sines. Calculate the following coefficients of the Fourier series and enter them below in Maple syntax. b1 = [ ] b2 = [ ] b3 = [ ]

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00:01 Hi, in this question, given that the function f of x is defined in the interval 0, 4 and it is 1 by 4 x, 0 less than x and x less than or equal to 2 and here 0 when 2 less than x and x less than 4.

00:22 So, here the function is approximated by the fourier series f of x equals a naught plus summation n equals 1 to infinity a n cos n pi x divided by l plus b n sin n pi x divided by l.

00:42 So, here we need to use the fact f of x and f of x cos n pi x divided by l are odd function.

00:50 So, that we can conclude that the value of a n equals 0 for n equals 0, 1, 2 and etc.

01:01 Next, we have to find the coefficients of b n.

01:07 We know that b n equals 2 by l integral over 0 to l f of x sin n pi x divided by l into dx.

01:14 On substituting l in this, then we get 2 by 4 integral over 0 to 4 f of x sin n pi x divided by l into dx.

01:21 On substituting the corresponding functions, then we get 1 by 2 integral over 0 to 2, 1 by 4 x sin n pi x divided by 4 dx integral over 2 to 4 0 sin n pi x divided by 4 into dx.

01:33 So, that we can write it as 1 by 8 integral over 0 to 2 x sin n pi x divided by 4 into dx.

01:40 We need to solve this integration.

01:42 So, that we have to use the formula u dv.

01:46 So, consider u equals x and integral dv equals integral sin n pi x divided by 4 into dx.

01:52 U dash equals 1 and u double dash equals 0 and v1 equals minus 4 by n pi cos n pi x divided by 4...

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