For the following function, f(x) = 0

For the following function, f(x) = { 0 - 1 ? x < -1/2 sin(3?x) - 1/2 ? x < 1/2 0 1/2 ? x < 1 } a- Determine if the function is even, odd or neither. b- Find Fourier series. c- To what values will the Fourier Series converge at x = 5/2, x = 4 and x = 7?

Transcript

00:01 In this question we have two parts.

00:03 For part 1 we have fx is equal to 0, minus 1 is smaller than or equals to x is smaller than minus 1 upon 2.

00:16 Here sin 3 pi x minus 1 upon 2 is smaller than or equals to x is smaller than 1 upon.

00:25 Here at last we have 0, 1 upon 2 is smaller than or equals to x is smaller than 1.

00:31 Here f minus x is 0, minus 1 is smaller than or equals to minus x is smaller than minus 1 upon 2.

00:44 Here sin 3 pi minus x is minus 1 upon 2 is smaller than or equals to minus x is smaller than 1 upon 2.

00:55 Here 0, 1 upon 2 is smaller than or equals to minus x is smaller than 1.

01:02 Now here f minus x is equals to minus f x.

01:10 Hence f is odd function.

01:16 This is the answer for part 1.

01:20 Now we have part 2 for which as f is odd function so a n is equals to 0.

01:37 Now and b n is equals to 1 upon 1 minus 1 1 f x sin n pi x dx.

01:50 Here we are integrating it and we get minus 1 minus 1 upon 2 0 sin n pi x dx plus minus 1 upon 2 to 1 upon 2 sin 3 pi x sin n pi x dx plus 1 upon 2 1 0 sin n pi x dx.

02:30 Here we have b x b n is equals to minus 1 upon 2 to 1 upon 2 sin sin 3 pi x sin n pi x dx.

02:56 Now we are solving this then we then we get b n is equals to 2 minus 1 upon 2 to 1 upon 2 sin 3 pi x sin n pi x dx.

03:22 Here by integrating it we get b n is equals to 2 minus n minus 3 sin pi n plus 3 pi upon 2 plus minus n minus 3 sin n pi minus 3 pi upon 2 upon 2 pi n square minus 9...

Question

For the following function, f(x) = { 0 - 1 ? x < -1/2 sin(3?x) - 1/2 ? x < 1/2 0 1/2 ? x < 1 } a- Determine if the function is even, odd or neither. b- Find Fourier series. c- To what values will the Fourier Series converge at x = 5/2, x = 4 and x = 7?

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