[9/ pa Peint ]DETALSPREVIOUS ANSWERS ROGACALcET2 0.4010Calculate the Tylor polynomials In(0)centendthe given funttion andwvalte DraIn( 4)192[~/2.08 Points]DETAILSRO...

$11 - 12$ Use a computer algebra system to find the Taylor polynomials $T _ { n }$ centered at a for $n = 2,3,4,5 .$ Then graph these polynomials and $f$ on the same screen.
$$f ( x ) = \cot x , \quad a = \pi / 4$$

Yeah, we want to find the taylor polynomial of and equals 234 centered around X equals zero. For equal zero. For f of x equals one plus extra P. Where P is a constant, remembers and taylor polynomial for F of X and degree and is P A and F X equals today. And that kind of exam next month A plus all the way up to the son of the derivative of F A. Divided and factorial times excellent. As a TPM. So we have to find the derivatives before we saw the PS. So let's find the road. It's F prime is P times one plus X. P minus one. F double prime is P times q minus one times one plus x minus two X triple prime. And F four are defined analogous lee. So we have that are polynomial czar for P 21 plus p x plus 10 to minus one X squared over two. P three is then simply one plus PX plus p 10 to minus one X squared over two plus P 10 p minus one times minus two X cubed over three factorial and before is one plus PX plus t times two minus one, export over two plus P 10 p minus one times minus to execute over three factorial plus P times q minus one times minus two and minus three X to the fourth over four factorial.

All right. We want to find a tip. Taylor polynomial is T two and T. Three for this function at equal zero. So we're gonna need some derivatives. All right. The first derivative will be negative. E to the negative x -2. E. to the -2. X second derivative yeah. E. To the minus X plus four E. To the minus two X. And then the 3rd derivative minus. Eat the minus x -8. Eat the -2 x. Now we need these functions and derivatives at zero. So f of zero would be one plus one. Which is to If primer zero would be -1 -2 minus three. A double prime one plus four. She's five mm add F triple prime. Yeah -1 -8 which is -9. So then we get um t seven two x -0-0-0 factorial -3. X 0 to the 1/1 factorial plus five X minus zero squared over two factorial minus nine X minus zero squared over three factorial plus. What's that? One was cubed plus dot dot dot Until you get to the inThe one which we don't know what it is so we'll just leave it like that. Maybe I better take that off. Okay so if you want t to then it's the first three out to here. So that'll be 2 -3 x plus 5/2 factorial. X square. Okay. Okay don't forget. zero factorial is one by definition. Okay so even though it looks weird it's really just equal the one so don't worry about it But I always write it like that to start because it's easy to remember the pattern that way, C3 will be the same thing with the added bonus of minus nine X. Cubed over three factorial.

For this problem, we are asked to calculate the 2nd and 3rd order taylor polynomial is for the function F. Of X equals 10 of x. And the value A equals pi by four. So to begin, we want to take our derivatives of tan X. First derivative of tan X is seek squared X. And find the second derivative by applying the chain rule is going to give us to seek squared X times tan X. And we can find the third derivative by applying chain rule and product rule that's going to give us negative two times seek X power of four times coasts of two X minus two. Now having those, we can calculate uh each value or the value of each. Rather when x equals pi by four. For Tan X, we just get one for sikhs where two X. We are going to get to For to seek squared x tan x of high by four. Or when X equals pi by four, we're going to get a result of four. And for the 3rd derivative there Evaluated when x equals pi by four. I'm going to give us a final result of 16. So having that we can write our 2nd and 3rd order taylor polynomial. The second order is going to be one plus two X or two times X minus pi by four I should say plus for over two factorial. So that's going to be plus two times X minus pi by four. All squared. Then the third order polynomial is going to be the same up until that X minus pi by four squared term. After which we are going to include our third order to term. Not to excuse me. So that's going to be plus 16/3 factorial times X minus pi by four. All cute.

For this problem we are asked to calculate the 2nd and 3rd order taylor polynomial is for the function F. Of X equals E. To the power of two X. And the value A equals lawn of two. We'll begin by taking our derivatives. So the first order derivative or first derivative brother is going to be to either of two X. Second derivative is going to be four E. To the power of two X. And the 3rd derivative is going to be eight E. to the power of two x. He then wants to go through and evaluate these when X is taking on the appropriate value there one of two. I'm going to pause and evaluate those off screen. Alright, so f of Lawn two is going to be four F prime of loan to Is going to be eight F double prime of lawn too is going to be 16 and F triple prime Of Lawn to it's going to be 32. Having that we can write our 2nd and 3rd order polynomial. Our second order is going to be four plus eight times X minus lawn too. Plus 16/2 factorial. So that's just going to be eight times X minus lawn of two. All squared. Then our third order polynomial is going to be very similar up to that squared term. I'm just gonna copy and paste it there. We'll have T3 equal to that. Up to that second Order term. Then we'll add on plus 32/3 factorial times X- Lawn of two. All cute.