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Use a graphing calculator to display the terms of the Fourier series given in the indicated example or answer for the indicated exercise. Compare with the sketch of the function. For each calculator display, use $\mathrm{Xmin}=-8$ and $\mathrm{Xmax}=8$Exercise 5

The Answer is Graph.

Calculus 2 / BC

Chapter 30

Expansion of Functions in Series

Section 6

Introduction to Fourier Series

Series

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Campbell University

University of Nottingham

Idaho State University

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and this problem, we're going to start by finding a sub zero, and this is going to be equal to of one over one of the limits. So I call these negative L toe L. That's so depending on what those values are within our, uh, bounce, then that's going to be won over high or l on. And then the integral from negative elta el, which is negative Pai tau pi of X squared. And so here we will take the integral and we'll end up getting with respect X here, one of getting too high squared over three and then the only other thing we need to keep track of this. If we were to plug it back in that these twos would cancel here. And so this blue to here, we could put that down here, my moves would cancel. So, really, our first term is just going to be Hi. It's weird over three here, and then we're going to evaluate a sub k and beasts. Okay, I'm actually going to go ahead, just jump to bees, okay? Since these are actually going to end up being zero, which we'll see in just a moment, so beasts. Okay, Is the same set up one over L in this case is pie. That's our upper limit. Integral, negative pi toe high. And we'll have more Uh uh X squared multiplied by sign of K X. And when we have these K's, you can evaluate these with a graphing, um uh, program. And you can replace K with one to start with and then k with two. But as you'll see if you do that, any of these will actually evaluate to zero. And so all of these terms at the end will be zero. And that makes sense because these air odd terms and our original function is even so, really, we're only going to keep the even function, which are be co sign terms here. And so next we'll go ahead and evaluate. Uh, it's a k very similar set up to what we had before. So it's OK is being a girl one over pi from negative pi to Pie X squared, well supplied by co side of X Co sign of K X. And this is where we'll go ahead and find the 1st 4 terms and eventually will find eight of them and Then there's a D X at the end of this as well. And evaluating for, uh, let's say, a sub one we were took pointed out, Something does Most werewolf ram or a graphing calculator. We're gonna end up getting that that whole integral evaluates to negative or and we do the same thing we change, cater to we integrate that term. But with co sign of two X on that would integrate a positive one the third term, but in a great too a negative. So you could see that these air alternating terms here negative for ninth and then finally a sub four would be positive 1 16 And so we now have the 1st 4 term. So we could graft this and see how close the actual graph is too X squared. And this is actually a great approximation, as you'll see in just a moment. And so we'll write out a whole function first here. So at this point, the four year Siri's approximation of X squared using sons and co signs would be our a subzero term, which the twos canceled, likely said. So it's high squared over three, and thats going Teoh allow it to kind of settle along the X axis here minus or co sign X Um, plus assigned to X since the term out front is ah one there. So just be one multiplied by co signed to accent. If you can see inside the parentheses, the constants are gonna increase by one each time. And then we've got a Step three, which is negative for ninth multiplied by co sign of three x Someone three X and then finally plus 1 16 before you can see the pattern here on 16 co sign of forex and graphing This we can determine from negative to pie to positive two pi What? This graph? How close it would be approximating, um X way I like. I said, If you do this, it's actually going to be pretty good approximation. So we'll graph this here. Kind of a little pixelated, I guess. So it's It's a good outline. It's a little more rigid. Must smooth them actual, uh, quadratic. So it would kind of look a little flatter on the bottle. I'm exaggerating it a little bit here. I would come back around again at high. Same on this side. So has the basic, uh, shape Oh X squared. But again, it it's it's kind of less smooth than you would want it to be. And, um, I could go ahead and include some of these other terms here. So, like, if you want to do a graph best on your own, it's of five would evaluate toe negative or 25th. So after this term here, we could say plus or minus negative for 25th co sign of five X and then a sub six. But evaluate too 1/9. So then we could say, plus 1/9 coastline a six x A seven is negative for 40 nights, so minus or 49th co signed seven X And then finally, it's a big would evaluate to 1 60 Ah, I just realized I was thinking that the oddest both of these were 1 16 This one here, that talent scroll up. That's what actually be 14 so we could go ahead and change this Here. 14 get rid of at 1 16 term. Okay, there we go. And so at this point now, if we were to graphic, that it would honestly would just be a better version of what we had over to the left would be a little bit smoother. You sort it still come up a pie down that negative high, rather, and then back around begin. So this point would be negative high. This would be positive by this would be to negative two pi to be positive, to buy basically a smoother, better version. Um, and obviously closer approximation for X squared here.

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The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

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