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Question 110 ptsA function flt) is defined as,flt) =T-t for 0 < t < TWrite down the even extension of ft) for -T < t < 0.Determine the Fourier cosine se...

Question

Question 110 ptsA function flt) is defined as,flt) =T-t for 0 < t < TWrite down the even extension of ft) for -T < t < 0.Determine the Fourier cosine series, and hence, calculate the Fourier series approximation for ft) up to the Sth harmonics when t = 1.54. Use as least 6 decimal places for T. Give your answer to 3 decimal places_

Question 1 10 pts A function flt) is defined as, flt) =T-t for 0 < t < T Write down the even extension of ft) for -T < t < 0. Determine the Fourier cosine series, and hence, calculate the Fourier series approximation for ft) up to the Sth harmonics when t = 1.54. Use as least 6 decimal places for T. Give your answer to 3 decimal places_



Answers

7.52* Consider the periodic function defined over one wavelength by
ƒ(x) =
sin kx
0< kx < pi






0





pi< kx < 2pi
Determine the Fourier series representation of ƒ(x). Draw a diagram of
ƒ(x).

So the commission says in problem from the 40 cities affair for the Cuban interval. So let's sold this a note equals to one x 5 integral minus pi to pi fo. Fax de X. D X. One by five Integral -9 to 0 zero. I scared D. X plus one by pi Integral 0- five. Bye scared minus X. Is scared The X equals five x 3. I scared Bye Bye. three pages scared and equals A. N equals N. A N equals food N equals one by pi integral minus pi to pi fo facts course and X. Dx. Similarly we will solve this so that we will get what we will get to buy and scare minus one. Race to power and plus one. Similarly we get to be any question best replaced this by sine X. Sign an X. Sign sign an X. So this will becomes what this will become. This will become so what this will become somewhat Our five x N -1 raised to the Power N. And plus to buy and cube By 1 -1 is to the power and So therefore x equals two 55 is scaled by six plus. Submission Any questions 1 to Infinity to buy and scared -1 and plus one host and pie X. Plus bye bye an minus one raised to the power N plus two, 1 -1 raised to the Pardon by and cue by sign and X.

So let's all these questions the question says in programs find the fourier series of fon given extra belt. Apple fax, apple fax. Is it close to what This is close to three minus 3 -2 eggs And access greater than minus, paying less than five X. greater than -5 and less than by. So let's hold this question. So let's hold this question. Let's solve this question. So we have a note, a note. It goes to one by pi one by pi integration minus pi to pi minus pi to pi fo, facts. DX. So this is a costume one by pi minus by bye minus pi to pi three minus two. X. D. X equals to six. So A. N. N. Is equals to one by by one. Bye bye minus pi to pi fo fax horse and X. And D. X. So this is a quest to zero. Be any quest to one x 5 -5. Bye By 3 -2 x sign and X The X. This is a custom four by end -1 raised to power and therefore fight sequence to vote three plus four times Summation. n equals to one to infinity -1 raised to the power and by again signed and X.

So in a given problem we have to integrate the value in England zero to be E. X square Let's B to the power one x 2. Yeah week Simpson grew Simpson. Uh huh. So the value of Abyan and are given. So putting this value this integral. Yes to well X squared plus 3 to the power one x 2 in the years. Now this function Solved by Simpson door and they're the very unease equals two three. So how we can proceed since the this is the fun sign It is starting from zero and ending at three and we have three substrates and his so one two and 1 and two and entry. So starting limited zero and limited three. No how we will solve this problem is I mean the Simpson formula is integral of why the X. From A to B. Can we the dinners actually by three. This is the subject this one which is this is H. So in this case which is one that's quite three. Why zero plus right and Start limit plus and limit plus four times off. Why one plus y. three and so on. These are not limited. Yes that's right. Last two times off. Why two less wife or even limits and so on. So this all will be inside expert. Okay so we'll use this here. So what is our way here our white well expert the street to the powerhouse So 0 to 3 who will accept spare Let's ST to the power half dx It is equally true that by three is 1 x three. Why zero. So first of all we write down all the way Y zero Y. One Y two K. Of this from right zero in France. And our main concern is well X squared less three with the power half. So firstly you like all the values why zero is political power half Why one is To help us stay 15 to the powerhouse. Why two years two square four into one. That is 48 plus three 51 to the power half. and the right three is nine into 12. That is 108 In plus three. That is one of 11 to the power. No we'll use the health is what was our formula integral y ideas here. It will be That's right three with this one x 3 bracket. Why zero plus y. And that is would sleep plus Ryan is 111 less four times off Even value. That is why one. That is rude. 15 Plus two times of Advil or well that is why 0 51. So this is our final answer and we will calculate calculated by calculator and then the final answer will be so the final value will be 14 point 01. Nearly

Uh So we need to find the four year series of a function. Given effects is equals to uh I'll write this as a for simplicity already. Uh just to t So 50 Cindy and for where he is between 0 to pie and it is zero for 30 between two pi to five. For simplicity, I'm considering variability. So we need to find the four year series. The information of this now. Uh the four year, The four year cuisine cities for signing the court. The four year course in cities for cynics on 0 to 5 is given is given by Sin X. Is equals to Sin X. Is equals to 80 upon two plus submission and goes from one to N for night. IAN cause annex this is the four years equals N. Series of sinners. And we we need to find the variable which are unknown over here is zero. And when I do all this so we need to find this value. So there A N. Is equal to two upon pi Integral. sin eggs in due course and X. Dx. So we need to put the value of an and find value of a when you do it. Uh put some values we need to find and weekend general later we can generalize. So for, so after solving this is zero comes out to be zero. If I take end to be zero. So zero comes out to be uh zero comes out to be 4.5 and even comes out to be zero. Even considered to be zero. Now for general consider anger didn't equals to two And get an equal to two. Then a N. Is equal to two by five, zero to pi sign X. Into cosine X. Dx. So now so now this is equals, I can write this as uh I'll take you inside over years so I'll be having to sign X cause nx. So I'll be using formula techno magic formula to changes. So uh so now one upon pi 0 to Pi I can write this as sign End of n. plus one into eggs minus sign And -1 into X. Well but I can be so I used the trick no magic formula to change to sign X. Cosign X. Into this form. So now uh so I can write this as. So now this is the integral of sin X would be minus core second. And similarly so the identical so it would be one upon pi whole bracket cause of and plus one in two X Upon. And plus one minus cause of. And miners went into eggs Upon N -1 Hold Bracket and are limited from 0 to 5. So after solving this, so this comes out to be equals two -1 upon five. So cause of uh N plus one pile. So cause of endless one by his minus one, raise to n plus one minus one upon And plus 1 -1 For a sign it would be -1, raise to N -1 -1 upon and -1 -1 Which is equals to 1.5, 1.5 -1 and plus 1 -1, two Upon and square minus after solving this. So finally when any sword. So now two kids are here Ladies and Gentleman. So two case arise when N is odd bananas or then this N plus one would be even and this entire time would be when then this will get to zero. So when Aeneas or then a M would be zero And when N is even when N is even, Then this would be -1. So it would be -4 upon pi Into one. A point -1 upon pi into one upon when a bond and square minus slim. This is the case when Annie's even and this is the case when anne's heart. So, So from here also we can prove that 8080 is so zero. Um hmm. So I can write this over a year. So it would be 4.5 and four az goes to one. It would be zero since one is odd. So, so the four year cities for the even four years, he's forgiven function is Synnex equals to zero upon two plus submission and goes from 1 20 point night in cost. And next There is zero is equal to 4.5. Even in zero and E N For N greater than equal to two. N is equals to -1. So anything equals two. The energy goes to -4. The energy was 2-, wrote upon. Bye and Square- and Square -1. For and to be even And the music goes to zero for and to be on. And the last part is we need to plot the function effects. So so it would be a sign graph. So let's say this is zero sum varies over years by and somewhere we're used to buy, this is a rough ground diagram. So the function is The function is Sine X for when X is from 0 to Pi. So from 0 to buy it would be cynics. Sorry. So and this is where it would be one. So it's from zero to buy this sign XO0 by signing now and from pi to two pi zero by 22 by the zero. Let me depend it with the color. So this is the from 0 to 50 to five. It would be cynics and from 5 to 2 pi it would be zero. So this is the graph of fx I hope you understand the solution


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