Find at least the first four nonzero terms of the series expansion for the solution to y 2ry sin(2) JatT.For reference: sin(z) = 1...

Find the first four nonzero terms in the Maclaurin series for the functions. $$\left(\tan ^{-1} x\right)^{2}$$

For this problem, we want to find the first four non zero terms of the taylor series. About zero for the function arc tanne, also known as tan inverse, are 10 of R squared using known taylor series. So over to the side here, I have some known taylor series and we can see you that's cheating. Can't use that one. Um mm But I believe at least for the purpose of this problem, intention because I say we can't use this one because of the textbook doesn't actually give this to you. I found this on the internet. Okay, So the intention for this I believe is that is to see, okay, this isn't a known taylor series according to the textbook, but if we look at a derivative table, we should be able to see that the derivative of arc tanne X. It's going to be 1/1 plus X squared, which we can make the necessary adjustments. We find that the derivative of arc tanne of R squared should be d by D R, R R squared. It's going to give us to our over one plus R to the power of four. And now this doesn't immediately look like it but it's actually comparable to this form up here. The 1/1 minus X where the necessary changes would be. We can write this as two times are over our two times are times 1/1 minus negative R. To the power of four. Where we clearly have that X equals negative R to the power of four. So having that we can do, yeah, essentially what we'll do is do the taylor expansion of the derivative, then integrate. So the tailored expansion of the derivative. I'm just going to write it down here with the equal sign. That's the taylor expansion of the derivative. Arkan, R squared 20 to our times. That will be one minus R. To the power of four. It would be plus R to the power of eight then it would be minus R. To the power of 16 not 16. Excuse me? Art of our of 12. Yeah. One second here. Okay. Just wanted to double check. So we're playing in that to our we'd have to ar minus two R. To the power of five plus two R. To the power of nine minus two R. To the power of 13. So now to get our taylor series of arc tanne of R squared, we integrate this. So we want to integrate two ar minus dot dot dot. I'm just going to write that down there. Expedience. So integral of two. R is going to be our two R squared over two. So that's just going to go to R squared and we have minus two R. To the power of 6/6 plus two R. To the power of 10/10 minus two R. To the power of 14 over 14 plus a constant plus dot dot dot. So the constant since we're expanding about R equals zero, we need arc 10 of zero. Let's take care. The concept of integration would be our 10 of zero, but that's just zero. So the constant zero. So we can simplify daisy can simplify this down. The final result that we should get is just going to be R squared minus R to the power of 6/3 plus R to the power of 10 over five minus r to the power of 14. I was 14/7 plus dot dot dot.

For non zero terms of the taylor series. About zero for tangent of T plus pi over four. So F F T. Is the tangent of T plus pi over four. Uh distributive 2nd. T Plus Pyro four squared or c get squared. That double prime ft is something squared. So it's derivative as to something to the one power times the derivative of the something which is seeking tangents are seeking T plus power for tangent. T plus fire for So that's to seek it. T plus power for Square Tangent. T. Plus Pyro 4th. Okay, one more derivative. This time I have to use the product rule. So I'm just gonna put the two out front here. So it's the 1st, 2nd, square t plus power. four times the drift of of the second which is seeking squared T plus primer for it was the second times the derivative of the first is something squared. So it's derivative is to something to the one power times the derivative of the something which will be seeking tangent. Okay, so it's two. Seek it to the fourth. T plus pi over four plus two tangent squared. Seacon square. Okay, so that's two second squared T plus pira. Four times second squared T plus, however, for plus two tangent squared cheap. Oh, I forgot I'm gonna plug a number. I don't have to simplify that much. Okay, so we're doing it about zero. So F of zero is tangent five or 4. Which is one if prime is the secret squared Pi over four. Which is to F double prime of zero is two times 2 times one which is four After Triple Partners. zero. Let's see two times two Times two Plus 2, four times four. 16. Okay, so the taylor series will be one x minus 0 to 0/0 factorial plus two X 0 to the 1/1 factorial plus four X minus zero squared over two factorial Plus 16 x zero Cubed over three factorial, which is one plus two X Plus two x squared Plus 8/3 execute.

Alright for this problem we are asked to find the first four non zero terms the taylor series. About zero for the function lawn of one minus two. Y. Using known taylor series. So what we can do here is use the fact that we have that the taylor series of lawn of X about zero is equal to x minus one minus x minus one squared over two plus x minus one cubed over three minus x minus one. The power of four over four. Uh dot dot dot. So that is in terms of X. We want um about why equal 01 2nd here. We want to make sure that this expansion converges for the range that we want so long of X. That expansion converges for zero. Less than X. Less than or equal to two. So that would then mean that we need to have one minus two. Why? To be less or zero? To be less than one minus two. Why? Which would then have to be less than or equal to two? So okay that is then essentially going to mean that if y equals zero, if we're expanding about y equals zero then we have zero is less than one. Less than or equal to two. So we are good. So having that what we can do now is substitute in X equals one minus two. Why? So we have that lawn of one minus two Y. It's going to equal what? We'd have one minus two Y minus one. So that's just going to give us minus two. Why? And we'll have negative one minus two y minus one. So we'll have negative two Y squared over two. Then we have plus negative two. Y cubed over three minus negative two Y to the power of four over four plus dot dot dot. Yeah. So I'll pause and simplify this down. A little bit. Should arrive that the first four terms in the expansion will be negative two Y plus two Y squared minus 8/3 Y cubed minus four Y. To the power of floor.

In terms of the taylor series, about equal zero for f of y equals to cuba to one minus y. Or one minus y to the one third power. Yeah. Okay so we need the function. Well that's why the one third power and we need some derivative. So the prime of life one third one minus y to the minus two thirds times the derivative of one minus y which is minus one so minus one third one minus y to the minus two thirds. Okay. Next derivative two nights, 1 horse, two nights one minus y to the minus five thirds times minus one. Next derivative uh 10 27 times y minus one minus y to the minus eight thirds times minus one. one More. Just in case 80 over 81 times one minus Y to the minus 11 3rd Times 2 -1. Okay now we need the drift but now we need all of these values at a equals zero. So f of zero is 1. Have prime of zero minus one third Have double promo. 0 -2 nights Have triple prime of zero -10. We don't need the last one. So F of X. It's FF- zero was 1. Okay. one x 0 to the zero power over zero factorial minus one third X minus zero to the one Power over one factorial minus two nights, Explain a 0 to the to power over two factorial -10. Expanded 0 to the 3rd over three factorial. Okay so that's 1 -1 3rd. X minus 19 X squared minus five over 54. Execute. Let's see. That's 10 27 times six. Two goes in there. 5 2 goes into three. Oh, not, not do for 27 times three, which is 81. Oops.