Homework Answers
ANSWER: Since the graph of resulting function is symmetrical about Y-axis hence the given function has been extended oeriodically to an even function. Thus
![b_n=\frac{1}{L}\int_{-L}^Lf(x)\sin\left ( n\pi x/L \right )dx\\ \implies {\color{Blue} b_n=0} \\ {\color{Red} (\because f(x)\sin\left ( n\pi x/L \right )\ is\ odd\ function)}\\ \text{and}\\ {\color{Magenta} a_0=\frac{1}{L}\int_{-L}^Lf(x)dx}\\ \implies a_0=\frac{1}{3}\int_{-3}^3f(x)dx\\ \implies a_0=\frac{1}{3}\int_{-3}^3\left ( \frac{x}{4}+1 \right )dx=\color{blue}\frac{11}{4}\\ {\color{Magenta} a_n=\frac{1}{L}\int_{-L}^Lf(x)\cos\left ( n\pi x/L \right )dx}\\ \implies a_n=\frac{1}{3}\int_{-3}^3(x/4+1)\cos(n\pi x/3)dx\\ \implies a_n=\frac{3}{2n^2\pi^2}\left [ 1-(-1)^n \right ]\\ \text{Put n=2k+1}\\ a_{2k+1}=\frac{3}{2(2k+1)^2\pi^2}\left [ 1-(-1)^{2k+1} \right ]=\frac{3}{(2k+1)^2\pi^2}\\ \text{Put n=2k}\\ a_{2k}=0](http://img.homeworklib.com/questions/a38a90e0-e001-11ea-b36b-1d3ef997d704.png?x-oss-process=image/resize,w_560)
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A function is defined over (0,3) by f(3) = 12 +1. We then extend it to...
A function is defined over (0,6) by 0 <and I <3 f(1) = - { 3<;...
A function is defined over (0,6) by 0 <and I <3 f(1) = - { 3<; and <6 We then extend it to an odd periodic function of period 12 and its graph is displayed below. N y 1 0 -10 5 5 10 15 X The function may be approximated by the Fourier series f (t) = a0 + 1 (an cos (021 ) + bn sin ( 122 )), where L is the half-period of the function. Use...
function is defined over (0,6) by f(x)={14x00<xandx≤33<xandx<6. We then extend it to an odd periodic function of...
function is defined over (0,6) by
f(x)={14x00<xandx≤33<xandx<6.
We then extend it to an odd periodic function of period 12
and its graph is displayed below.
calculate b1,b2,b3,b4, Thanks so much
A function is defined over (0,6) by 0<x and x <3 f (x) = 3<x and x < 6 We then extend it to an odd periodic function of period 12 and its graph is displayed below. 1.5 1 у 0.5 -10 5 10. 15 -1 -1.5 The function may be...
0 3 and z s 6 We then extend it to an odd periodic function of period 12 and its graph is display...
0 3 and z s 6 We then extend it to an odd periodic function of period 12 and its graph is displayed below 2 y 1 -105 5 10 15 2 The function may be approximated by the Fourier series where L is the half-period of the function Use the fact that J(e) and fe)cL) are odd functions, enter the value of en in the box below f(z) cos an 0 for n 0,1,2,... Hence the Fourier series made...
Given the periodic function 5 f(1) = { 1 f (+4) 0<i and I<2 2 <r...
Given the periodic function 5 f(1) = { 1 f (+4) 0<i and I<2 2 <r and I<4 otherwise and its graph is displayed below. 6 5 4 y 3 2 1 0 -2 2 4 6 00+ x The function may be approximated by the Fourier series f(t) = 40 + 1 (an cos ( 172 ) + bn sin where L is the half-period of the function. + bn sin ne :)), L Calculate the coefficients of the...
4. Recall that if f(x) is a function defined on (-7, that converges to its' Fourier...
4. Recall that if f(x) is a function defined on (-7, that converges to its' Fourier Series then f(3) =" + ] (a, cos nz + by sin n2) where an = = ſs(z) cos(n2) dz for n = 0,1,2,..., and bn = "S(2) sin(n2) d2 for n = 1,2,.. Show that the Fourier Series above can be expressed in the following alternative form: S(=) = :slads + ŽIs(5) coaln(5 – 7 ) ds.
Question 6 Consider the function defined over the interval 0<x<L. Extend f(x) as a function of...
Question 6 Consider the function defined over the interval 0<x<L. Extend f(x) as a function of period 2L by using an odd half-range expansion 1) Sketch the extended function over the interval -3L<XS3L. 2) Calculate the coefficients for the Fourier Series representation of the extended function. 3) Write the first 5 non-zero terms of the Fourier Series. (10 marks)
Given the function f(x) -3x + 1 defined on the interval (0, 5], denote by fe the even extension o...
Given the function f(x) -3x + 1 defined on the interval (0, 5], denote by fe the even extension on [-5, 5] off. the Fourier series expansion of fe Find feF, + bn sin / - n-l that is, find the coefficients a , an , and bn , with n 1 . ao = anF
Given the function f(x) -3x + 1 defined on the interval (0, 5], denote by fe the even extension on [-5, 5] off. the...
Condsider the ODE d2 x () + 32 x (t) = F (t) where the forcing function is given by the Fourier s...
Condsider the ODE d2 x () + 32 x (t) = F (t) where the forcing function is given by the Fourier series with co -1, c18, Assuming a particular solution of the form find and enter the exact values of an and bn requested below Cn cos (n t), 3p (t)-a0 + Σο.1 (an cos (n ) + bn sin (n t))
Condsider the ODE d2 x () + 32 x (t) = F (t) where the forcing function...
Q#4: (24 points) Given the function y-f(x) shown below: na f(r) -3 (a) Calculate the period of fu...
Q#4: (24 points) Given the function y-f(x) shown below: na f(r) -3 (a) Calculate the period of function, T and frequency, (2TT)/T (b) Calculate the Fourier Coefficients Ao. An, and Bn of the Fourier series expansion of function, y-f(x). Here n 1, 2, 3,... (integers) (c) Write the Fourier series approximation of function, yf(x), in terms of numbers, n & x only
Q#4: (24 points) Given the function y-f(x) shown below: na f(r) -3 (a) Calculate the period of function,...
1. The Fourier series expansion of the function f(x) which is defined over one period by 3_ z, f(z) = 쓸 +Σ1@n cos...
1. The Fourier series expansion of the function f(x) which is defined over one period by 3_ z, f(z) = 쓸 +Σ1@n cos nπχ +h,sin ηπΖ] is 1<zc2. Find the coefficients an r sin ax cosar x cos ar dr = We were unable to transcribe this image
1. The Fourier series expansion of the function f(x) which is defined over one period by 3_ z, f(z) = 쓸 +Σ1@n cos nπχ +h,sin ηπΖ] is 1
A function is defined over (0,6) by 0 <and I <3 f(1) = - { 3<; and <6 We then extend it to an odd periodic function of period 12 and its graph is displayed below. N y 1 0 -10 5 5 10 15 X The function may be approximated by the Fourier series f (t) = a0 + 1 (an cos (021 ) + bn sin ( 122 )), where L is the half-period of the function. Use...
function is defined over (0,6) by
f(x)={14x00<xandx≤33<xandx<6.
We then extend it to an odd periodic function of period 12
and its graph is displayed below.
calculate b1,b2,b3,b4, Thanks so much
A function is defined over (0,6) by 0<x and x <3 f (x) = 3<x and x < 6 We then extend it to an odd periodic function of period 12 and its graph is displayed below. 1.5 1 у 0.5 -10 5 10. 15 -1 -1.5 The function may be...
0 3 and z s 6 We then extend it to an odd periodic function of period 12 and its graph is displayed below 2 y 1 -105 5 10 15 2 The function may be approximated by the Fourier series where L is the half-period of the function Use the fact that J(e) and fe)cL) are odd functions, enter the value of en in the box below f(z) cos an 0 for n 0,1,2,... Hence the Fourier series made...
Given the periodic function 5 f(1) = { 1 f (+4) 0<i and I<2 2 <r and I<4 otherwise and its graph is displayed below. 6 5 4 y 3 2 1 0 -2 2 4 6 00+ x The function may be approximated by the Fourier series f(t) = 40 + 1 (an cos ( 172 ) + bn sin where L is the half-period of the function. + bn sin ne :)), L Calculate the coefficients of the...
4. Recall that if f(x) is a function defined on (-7, that converges to its' Fourier Series then f(3) =" + ] (a, cos nz + by sin n2) where an = = ſs(z) cos(n2) dz for n = 0,1,2,..., and bn = "S(2) sin(n2) d2 for n = 1,2,.. Show that the Fourier Series above can be expressed in the following alternative form: S(=) = :slads + ŽIs(5) coaln(5 – 7 ) ds.
Question 6 Consider the function defined over the interval 0<x<L. Extend f(x) as a function of period 2L by using an odd half-range expansion 1) Sketch the extended function over the interval -3L<XS3L. 2) Calculate the coefficients for the Fourier Series representation of the extended function. 3) Write the first 5 non-zero terms of the Fourier Series. (10 marks)
Given the function f(x) -3x + 1 defined on the interval (0, 5], denote by fe the even extension on [-5, 5] off. the Fourier series expansion of fe Find feF, + bn sin / - n-l that is, find the coefficients a , an , and bn , with n 1 . ao = anF
Given the function f(x) -3x + 1 defined on the interval (0, 5], denote by fe the even extension on [-5, 5] off. the...
Condsider the ODE d2 x () + 32 x (t) = F (t) where the forcing function is given by the Fourier series with co -1, c18, Assuming a particular solution of the form find and enter the exact values of an and bn requested below Cn cos (n t), 3p (t)-a0 + Σο.1 (an cos (n ) + bn sin (n t))
Condsider the ODE d2 x () + 32 x (t) = F (t) where the forcing function...
Q#4: (24 points) Given the function y-f(x) shown below: na f(r) -3 (a) Calculate the period of function, T and frequency, (2TT)/T (b) Calculate the Fourier Coefficients Ao. An, and Bn of the Fourier series expansion of function, y-f(x). Here n 1, 2, 3,... (integers) (c) Write the Fourier series approximation of function, yf(x), in terms of numbers, n & x only
Q#4: (24 points) Given the function y-f(x) shown below: na f(r) -3 (a) Calculate the period of function,...
1. The Fourier series expansion of the function f(x) which is defined over one period by 3_ z, f(z) = 쓸 +Σ1@n cos nπχ +h,sin ηπΖ] is 1<zc2. Find the coefficients an r sin ax cosar x cos ar dr = We were unable to transcribe this image
1. The Fourier series expansion of the function f(x) which is defined over one period by 3_ z, f(z) = 쓸 +Σ1@n cos nπχ +h,sin ηπΖ] is 1
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