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In Problems, a function $f(t)$ defined on an interval $0<t<L$ is given. Find the Fourier cosine and sine series of $f$ and sketch the graphs of the two extens...

Question

In Problems, a function $f(t)$ defined on an interval $0<t<L$ is given. Find the Fourier cosine and sine series of $f$ and sketch the graphs of the two extensions of $f$ to which these two series converge.$$f(t)=sin t, 0<t<pi$$

In Problems, a function $f(t)$ defined on an interval $0<t<L$ is given. Find the Fourier cosine and sine series of $f$ and sketch the graphs of the two extensions of $f$ to which these two series converge. $$ f(t)=sin t, 0<t<pi $$



Answers

Find the Maclaurin series of $ f $ (by any method)
and its radius of convergence. Graph $ f $ and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and $ f? $

$ f(x) = \cos (x^2) $

This video, we're gonna go through the answer to question number 28 from chapter 10.3. So first, impart a were testifying the Fourier Siri's for FX where next sequence a X squared on the domain and mine is pie too high. So do that. We need to work out all of the Korea coefficients. So first up, um, yeah, this part a first that we're gonna find a zero, which is given by one of the pie into it reminds pie pie off X squared the X So it's gonna be one of a pi times into of X squared is 1/3 x cubed C minus pi pi. This is gonna be a well, the third just come out that's gonna become 2/3 because to you, um, evaluations of pie mine's pie you're gonna have together it's gonna be 2/3 pi cubed, divided by pi says take those by square. Next up a ends for and not equal to zero is gonna be one of a pi times into three minus pi pi Back squared caused an ex So we do this. We need to integrate my past twice. It's a fester one of one of a pie. Are friends leaving the X squared? Is that integrating? Because annex that's gonna become one of the end sign and X, Do you mind? Is pie pie minus into Grozny minus pi pi of ex squares? Excuse me. We need to know differential X word That's gonna be two X. Let's put the two out front, then integrating the cause. That's gonna be one of the end with another friend. That sign and x d x and then close that bracket. Okay, uh, sign off. End pie and sign of minors. Empire, Any Instru end is gonna be zeros. That's gonna be zero. So we're gonna have minus taking the minus two over and out. The front, my two of and hi, then. For this guy, we need to do integration by parts again. Leaving the exit is integrate. Sign gonna get minus one of the end cause and X Do you mind? Is pine pi minus in school? Which reminds pie and pie off deficient. Different actually. Get zero. Sorry. Different ex. You get one integrate sign. We're gonna get minus one over and cause and ex the ex. Yes, We're gonna have minus two and covered in pie. Um, eso this guy when we put in the pie and minus pi that's gonna become while we can take out a factor minus pi over and and then we're going to get cause of n pi, which just might just want to the end. I'm gonna get two of those. So that's minus to high end times just once. Then this guy that's just gonna integrate to be, like one over and we're gonna make it one of n squared times by, ah, sign an ex between minus pi and pie and a sign of end women substrate in those limits side of any pie sound of mind sent 50 So that is always gonna be there. So just close the back there and that leaves us with this pie. This party gonna cancel that month, Ones they're gonna cancel before Oh, that and squared times minus one too. Then next to be ends. Now that's a bit easier because X squared is never function. Sign an ex is an art function in our function times and even function is not function obsession with Bennett the X here so integrating and odd function between minus pi pi. That's just gonna be zero for any. So therefore, FX it's gonna be equal to Was gonna be ableto half a zero is there was 2/3 pi squared so it's just gonna be pi squared over two That's quite over three. In fact, plus some n equals one divinity of a ends Times cause an ex and ends before months Wants the end over and squared Yeah, because and thanks be interest zero So all the sign in's not gonna be involved. So that is our for a series Full exc wet. Okay, Bobby. Okay, so in part, we were us to use part a to show. Um, the thea some aa minus wants the end plus one over and squared is equal. Thio hi. Squared over 12 s. So to do that, let's just have a look back. It, um I've looked like our Fourier series. So if we look at this and think about what happens if we substitute in X equals zero well, we know wife of zero is right, because every backs is just X squared. So effort zero is just there. X squared, which is zero. But we can also serve a shoo in as there are in this side and see what happens there. We know that that's gonna be equal. Thio. It could be 1/3 pi squared close Cem and it's equal to one to infinity of we have four minus wants the end Ah, and sweat. And then it was also most quite by cause and X but it was strictly an X equals zero the cause of an X equals What? So we now have this an equation so we can rearrange the equation? Uh, I put in this guy on this side that's gonna give us another minus one, which can go to make this end B and close one. And we're left with Cem and is equal to one to infinity. It was about four over and squared, But now we've got minus wants the end times minus one which is minds once the end plus one on dawn the right inside We still have hi sweat. So what we do here is by both sides by four that because that was one and this becomes that's well, which is what was wanted. But see, I must do the same thing now, But think about what happens. Ah, X equals pi. So F pie is was just pi squared. So she needs substituting in the right inside. Though we need to think a little bit carefully about this now when you think what the limit is gonna be. So let's see what f looks like Looks like this between minus pi pi. So that limit the below the limit is gonna be pi squared off limits, you know, pi squared. So, um, the limit off the Fourier series for this function is gonna be the mean of those two numbers, the meaning pi square by square pie squared. So that's great. We know that this can be equated to the furrier, the Fourier series form of dysfunction evaluated at pie. So I got the third pi squared, plus soon from n equals one to infinity. Ah, for minus wants the end of n squared times by cause of end Pye street in pine X because of empires minus one to the end. Miners, once the end times by minus, wants the end. It's just explore Shakespeare. So now we have this equation, which we need to rearrange. Um, we could do that by putting this guy on the side and we're left with is and some. And it was one to Tennessee of four. Over and squared. Yes, that's gonna equal. Gonna be with the pi squared minus 1/3 place where that's gonna be put. 2/3. Hi squares. We can fight beside before that becomes one. This becomes one over six on, guys. Ah!

So this is this one trying to find the MacLaurin series, and we're gonna stay the radius of corporations off dysfunction. And then after that, we're just gonna, uh, So it was gonna use Thea, uh, table. Right. What is the territories? Will co sign X to use you for co sign X go sign eggs is a summation. And from zero to infinity, negative one, right? Did he in, uh, exited to end over so extra two and forward to and factorial. Right, So So this is the, uh, the MacLaurin series for cosa. So to find for co sign X Square, which is gonna replace every instance of x with X squared. So this is gonna be co signed X squared equals. Ah, this negative one to the power in. Now there's gonna be X squared right in order to end because we're replacing every instance of x with X squared, right, And then over to in factorial. So finally, this is gonna be, uh, summation. And from zero Teoh infinity. Negative one. Right to the end, Native one city an, uh so they have negative one to the in. This is gonna be exited for in over two in Factorial. Right. So this is what you have. S So what is the radius of convergence where we're still gonna use the ratio test? Right. So we used the ratio test. What has happened? We have the limits and approaches Infinity A n plus one over a in a What is that gonna be? You know, this is the A N, right? This this part, this is the sequence. So a n plus one means that wherever we see any and we're gonna put in those one and then the, uh, the absolute value is gonna take care of those native it's gonna make it positive. Right? So we've been doing this throughout, so this isn't before a n plus four, right? If we placed here by n plus one and then we foil, we're gonna get for and plus four. I mean, been doing this a lot of time, and we place here by M plus one. We foil, we're gonna get two n plus two, right? And then we multiply by a in which is exactly this guy right here. And that is a reciprocal right, because it is division. So when you change most application it becomes a reciprocal. Right? So this is, uh, business factorial. And then over Exeter four in. Right. So you have this eso, once you have, this is gonna do some cancellations, right? This one is gonna cancel this. This one will cancel this. Leaving, uh, is gonna leave to end plus two to end post one, right with new of this one throughout. They're gonna get excited for this For is gonna be left here, Right? Good. So and then you take in the limits, as in approaches infinity of that, as in approaches infinity of that. So the limit as an approaches infinity of this one is just cereal, right? This approach zero, because there is no and enumerators only had a denominator. And as it's number two gets larger and larger, uh, limit is in a good zero. So in the ultimate zero, then it means that from any ex, those particular sequence converges. And if it converges, then for any X than the radius of convergence is infinity, right? Because for any eggs, the sequence is gonna converge. Okay, so there's areas of convergence. Uh, now we're gonna try this kitsch if you Oh, the part of new meals. Eso When you have this one the first the function itself is co sign X squared, right? Sucrose and ex. Weird. Uh, rough. Skittish is gonna be like that. Uh, isn't it? We like this. Me? It's gonna be like this. Thank you. No. Is that a way for him like that? And in the next one, right next Taylor series. Uh, polynomial they see, uh, maybe let's see to you for what is t four t four iss? Uh, t four is one, uh, minus X 4/2 factorial e believe from here. Gets here to four. Right. So once you have that, then what is happening? I can sketch that one as well as this is still gonna be a, uh It's gonna be a, uh it's gonna be something like those a and, you know, t t eight t 12. I mean, we just taking the so So you can you can do with t three t two to you one. All of it. Right. So this is some of the few sketches. Some We're gonna be like that, and so I'm gonna be like this. Okay, So this is the original function ffx. This is some of the taylor polynomial this is to you for and this is t 12. I mean, you can have t three. You know that, right? So some of in a v uh, wait for miss, I'm just gonna be like, uh, uh, it can cave down function.

So we have this one and straight away we're gonna look at, uh, table eat. City X is given as summation and from zero to ready. Um, you know, exit E an over in factorial, right? You know this one? Eso we need to replace every instance of eggs with this one first of all. But we also know that cause an X is also summation. And from zero to infinity negative one to the end Exited to end over to an all factorial No, this one right. We did it in, uh, the tutorial capacitor before this one. So is there eso you need to replace this every instance of X with X squared. Right? Because we have eatery X, I wouldn't find e to the X squared. So so e to the negative. X squared is gonna be summation Hand from zero toe infinity. Negative X squares to the end right over and factorial And later added to the summation. The co sign one right. Causes close CO signed X. So this is also gonna be negative. One x to the two in right over two in factorial. Okay, So what is happening? We're gonna you know, here you can separate our negative one to the right. You have negative excluded Teoh. And you can separate out negative one to the end. So if you separate that one, this was whenever another native wanted in here. So so either negative ones where plus coups and eggs is I'm gonna combine a summation says summation is just one. Ah, no, I have ah into the So this is gonna be negative. One to the an, right. I'm separating that out and then I think is the same. So it's gonna be twice, twice off negative one to the end times exit to end. Right? Because this is the same as this. That's what I'm trying to say. So it's just twice of it. So once you have that twice of it, then you can have Ah, So actually, this let's make you like this This Mr Ah Goose step by step. So this is gonna be, uh you know, I'm combining these two, right? I said this is gonna be negative. One to the end, exit itude And over in factorial. Right? Plus, uh, negative one to the end. Exit itu in over to end all factorial. Right? So that's what. This what I have. So once I have this one now, I wanna make some combination. So when I make some combination, this is gonna be summation. No, I'm gonna pull all this out because it is coming. Right? So when I pulled that out, what am I gonna get? I'm gonna get negative one to the end, Exited to an right. And this is gonna be one over and factorial uh, plus 1/2 in factorial. You can leave you answer like that. It's not a problem. Or if one you can decide Teoh Makesem manipulations and simplify further at his new problem. Eso Once you have that, you can do your, uh into off radius of convergence. Right? So what is here? Radius of convenience. There is a convergence is, uh, infinity. Because we know if you watched the last eternal before this one, we showed that videos of convergence for Costa and experts infinity and is the same thing for for science for each of our X. Because the function, the function convergence or the sequence converges for both easy power X and co sign X rammasun that in previous tutorials therefore the radius of convergence is infinity, right? So if you have a three year, then you can see that, uh, this function e to the negative X square plus co side X. It's this one. And then it's radius off Convergences infinity. Right. So now, if you wanna sketch if you of the, uh, Taylor series terms, there is no problem. Let me see if I can give his tree and much straighter, uh, search. Try to find to the line. Yeah. Yes, this is better. Today, uh, put this one here. Yep. And then I, uh Let's catch a little bit. So we're trying to sketch if you have a graphing calculator, and when a sketch, I mean, label here to you, you know? So, uh, this one can come from Ah, year. Right. And you can have ah, many outer functions and projections. Right. So you can have some like this, uh, well, passing to passing through like that, and we can have several other projections. Right? Summer steeper than others? No. So that is your sketching some of the terms of the tenor para normal. You know, this is this is a Siri's, so you can generate this series for an equal zero and, of course, one and that you can sketch each other. Terms would be a first few of the terms, right? Can sketch each of, um, little by little. I mean, someone gonna also come like this from the bottom like that, you know?

So if we have F is some periodic function. And quite typically we're looking at sign and co sign when we talk about periodic functions and we know that the period is P. If we look at f of t plus T, this means that the graph of the periodic function is going to be shifted left mhm p units, but it will just end up looking like the same graph. It's just it will start a wavelength later, so we know that why equals sine of X and co Sign of X are both periodic functions and they have a period of two pi radiance and they have an amplitude, both of one. And if we want to graph those we know for the sine function, it's said to grab it from 0 to 2. Pi. Here is two pi here's pie and then the other tick marks. That's pi over two, and that's three pi over two goes up to a high of one down to a low of negative one, and the sine function will be like so, like, so so like, so like so So there's the graph of the sine function. Now I'm going to go on the same set of axes and I'm going to grab the coastline function. But I'll do the cosine function in red. The coastline function will start here. Come here. Come here. Come here. Come here. Let's see if I can kind of smooth that in and make it look like it's not too bad. So there's co sign and read and sign in black. Yeah, Mhm.


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