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Point) Write the Maclaurin serles for f(r) 3cos(3r" ) as Cer" Find Ine following coelticlentsNote: You can earn partal credt on nz DrodienPrevicw My Answe...

Question

Point) Write the Maclaurin serles for f(r) 3cos(3r" ) as Cer" Find Ine following coelticlentsNote: You can earn partal credt on nz DrodienPrevicw My AnswersSubmi AnswersYou have allempied this problem tmies You have unlmited atempts femainingEmail insuuctor

point) Write the Maclaurin serles for f(r) 3cos(3r" ) as Cer" Find Ine following coelticlents Note: You can earn partal credt on nz Drodien Previcw My Answers Submi Answers You have allempied this problem tmies You have unlmited atempts femaining Email insuuctor



Answers

Find the Maclaurin series for the functions.
$x \sin x$

All right, we're gonna find a general term equation for TN for the function. F X is equal to coast X squared, and we're gonna use a is equal to zero. So to make this one a little bit easier to solve for it, I'm actually gonna use substitution. So I'm gonna let a b x squared so that this is actually what I'm solving. Um, just so that I have a much easier derivative to find, and this is a derivative we've used before. So it's a little bit nicer to try to solve for it because we know then that we kind of repeat after every four. So our first derivative goes to negative sign. Maybe I'll use instead of a ll use. Um, we'll use you. You might be a little bit nicer. You was X squared. Not that it really changes anything. Just I might remember to write you for all of these. You you sign you. Oh, our 2nd 1 goes to negative coasts of you. Third derivative goes to sign of you and then our fourth derivative, we're back to coast of you. So we've repeated by our 4th 1 So going through this Our 1st 1 is the same as our 4th 1 which is going to be one our first derivative zero. Our second derivative is negative one. Our third derivative is zero. And our fourth drivel be said is already backups to start, which is one So finding our tailor polynomial terms or coefficients, our constant term would be one. Our ex term would be zero Our X squared over two term would be X squared over to factorial except not X in this case, Sorry to do because we replaced um with use you You squared over two factorial. Our 3rd 1 it's going to be zero. And our 4th 1 We're just right at the bottom. Would be, um you We're back to this one's negative. Positive you for over four factorial. So the nice part about this is now I can just substitute back in my X squared is this becomes a negative x four over to factorial and this becomes an ex h over four factorial. And this one substituting doesn't change. It is one this one substituting doesn't change it. Zero this one substituting doesn't change. It is eight. So right now I'm sitting at my tailor Polana will being one minus x four over to factorial plus X eight over four factorial. And we want to try to write a pattern for this so we can kind of see And if we done it, um, our next one we're going to get is you six, because five would be zero. So you six would be it was six factorial. It's gonna alternate and b negative and it's going to switch in the back and would be a negative x 12 over six factorial so we can add that one on If we wanted to try to find a pattern so we can kind of see that the tops are going up by fours as a power. So 48 12 Our next one is probably going to be 16. Um, and our bottoms are going by r factorial sze 246 or even factorial sze So right now we can con a say that to find them it will just draw some dots here dr dot our bottom is going by are even factorial Sze our top we said are going by fours and we're alternating positive Negative, positive, Negative. So I'm gonna write my negative one in there and it's second and third terms are negative. First and third terms that are positive. So to do that one so that it works out, um, we'd want to write it as well. Second answer. Third answer. Negative. First and third are positive. Um, we wanna make sure that one is positive. So decorated as and minus one. So two minus one is one that would make that negative three months. One is to that. Make that positive. Four minus one is three. So that makes that posit. Now the thing is, though, this will find us our next answer. But there's a bunch of answers in between that we skipped so we can't write. This is tien anymore, because we're not doing every answer. We've skipped out the zeros and the zeros using the you form happen every other answer using the co sign formula, it's repeating every four. So this pattern is happening every four. So what I'm actually gonna rate this has instead is every 4th 1 and we can see that one will work out for this one, because to find our second answer or sorry, what we wanted to find her fourth answer, That X 12/6 we'd have to put in and is one. Oh, because we're gonna do that. That's actually going to change this guy. That would change. This gonna end if we use for n and this one for this one. Um, and then we'd be putting in four times one for this one. So that is for and two factories that would actually find me my 4th 1 here. And then this one will be putting in too. So 842 positive. This one would be putting in three, 12 6 Negative. It's our next when we be putting in four. So that would be 16 on the top sheet on the bottom, and it would work out to a positive. So our general term for this guy here with the X squared term would be that guy right there. Uh, naked wants the park and Ford of power for end over two and factorial

For the glory series of the function. Sign acts over, too. We'll use the result. MacLaurin series off Sighel X is equal to some from zero to infinity. Make you want to party in acts to party to end plus one over Shoot and blast Juan pictorial then replaced X by Axel or two we have McAra. Siri's off the function. Sign off, Axel. Where to? Is he going to some From zero to infinity Makes you want Thio axe to to end. Plus one off shoe to the power. You damn Paswan much. Bye. Shoot and plus one factorial.

We'll find the McLaren series for this affects the Xanax uh Science critics. So then first we have a zero is equal to zero here. So take the trade into this sequel to to sign X. Cosign X. So that one also equals zero. All right, well, Take the 2nd derivative here. The Eagles native to science critics was to co sign its critics by product role. And so that The second right of zero is equal to two Taking the 3rd derivative. So that's according to a negative eight cynics council tax. So that's by chain role there. And so Have trouble. Private zero is equal to zero. So then the next one, fourth relative is equal to sp eight. That's critics minus eight sense critics. So that's byproduct. Well, again here, since we have the Alternative of zero is equal to -8. Sort of a pattern here ends up being that it's going to multiply but negative four each time. And so that we end up with Next one will be Cyril 10 32 And then zero. The -128. Okay, so that means we'll have so it's starting at the second derivative going into the fourth. So the even numbered derivatives. Okay, trends. Yeah, two square it over two factorial and it's eight X. To the fourth. Four factorial Plus. X. to the 6th over six victoria. Since 128. Save over antibacterial. Getting this empower serious representation. And so we have chemical 02 infinity of 2 to 2 K minus one next to the UK over to K. Victoria. That's -1. OK, so that's where I'm Clarence series presentation there.

Objective here is to get the MacLaurin series for co cenex by ticket. The integral of MacLaurin series were signed. So in order to your Okay so to do that here so that's sine of X. Is equal to it's X minus. It's the third. Overthrew pictorial Plus X to the 5th by a pictorial. The 6 to the 7th or sent Victoria here. Okay, integral of this here. We've got think cosine X. It's equal to X squared over two sex to fourth over four times three. Factorial access to 1/6 over six times five. Factorial sexted. Uh He's eight times 7 Victoria. Right? So then we'll do from here is multiply everything by -1. And so we've got co sign X. Close negative x squared over two Plus next to the 4th or or pictorial And a 6 to the 6th over six factorial. The six today. The range back to earlier. And so then that's quick. And to that's the summation of kegels 12 infinity here, -1 to the K. The two K. Over. Okay, pictorial. So we can see how that's Cosine of X. from K. Equals one. That would be the same thing going from Okay, cool. Zero -1 to the K. It's okay over check victoria. We started at zero here. Scott set. That's so just stick with this one here is going to be the Chlorine expansion. If we started with the co sign that chemicals one here so we have


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