a-find-the-taylor-polynomials-up-to-degree-5-for-f-x-sin-x-centered-at-a-0-graph-f-and-these-polynom

Question

(a) Find the Taylor polynomials up to degree 5 for $ f (x) = sin x $ centered at $ a = 0. $ Graph $ f $ and these polynomials on a common screen.

(b) Evaluate $ f $ and these polynomials at $ x = \pi/4, \pi/2, $ and $ \pi $.

(c) Comment on how the Taylor polynomials converge to $ f(x). $

Doruk Isik · Accepted Answer

in this problem where us to explain side effects using Taylor&#x27;s full normal, up to degree off five vino that, uh, Taylor&#x27;s suspension of the Siri&#x27;s is off form like this, and we can approximate a given function of effects around the point using De Ritter&#x27;s and actually high order derivatives. So since we are needing to expend it up to or drove five, we need, um, if prime of X after double prime effects, F three put prime of X A four off eggs as well as F five ex. All right, we know that affects a sign effects or the first tear it off that would be co sign effects. David, of course, and would be negative side effects. There is a negative sign is negative. Go sign effects. There were talking. I get it consigned its side effects, and it is it has a psychic pattern is you can see now the critical 50 Every two would again become side effects. May also less a lead dysfunction, um, Iran&#x27;s 20.0. So one A is zero. We have f prime of zero as one f double problem. Zero is zero f triple prime of syrup is negative one F four off zero is zero since sign 00 in Cosenza is one F five off, 0/5 orbiter off F What else would be? Well, all right now we have pretty much everything that we need. All we need to do is to take these values and employ it in to this question. Damn, he can estimate approximate sign effects. Pass. Um, sign of zero plus f prime of Syria&#x27;s wants. So one times X minus zero plus if secondary to zero. So I had zero plus 30. Ritter is negative on so that it&#x27;s negative. One off X minus zero cube divide by three Factorial 40 with zero Soviet plus zero plus 50 Ritter. Oh yes, one. So we have one times X minus 0 to 25 divided by five sectorial. You know that signs here is also zero. So we can ride this one s x minus X cube over three factorial plus exit effect or five factorial From this, what we see is this We see that t zero so&#x27;s your order is equal to zero. First and second order approximations are equal and that is equal to this term or disturb so that is equal to X. Also, third and fort order approximations are also equal, and that is equal to X minus execute plus three. Factorial said this is up to this part now. We&#x27;re also asked to graft dysfunctions. You could use any OLAND function, Porter, if you do so, we see that Miss Blaker is the origin of election of effects, which is side effects, that the first sort of rubber information gives us that straight line. That is the red line. This green car is the third order or second order, since they&#x27;re equal approximation and the blue one is the future order approximation. The idea is, that s we increase the order or as we had more and more terms, the approximation gets closer and closer to the board, you know, function. So let&#x27;s write it here as, um en increases be approximation gets better. All right. We&#x27;re also asked to you will a function and the approximations at few given points And those points are so you wanna have Brooks Mitt or every light effects T bone off x t three of fax 95 for fax at points pi over four, power two and pie So let&#x27;s make a table. All right? Signed pile before s screwed of two or two. That is a good point. 7071 Signed. Pirate too is equal to want and sign a pi zero. So let&#x27;s fill the table forward for Starter Perks Mission. Since our first order approximation is here, we can see it won&#x27;t as X uh, whatever value be, have whatever are going to be had for the angle that will return to someone. So, um, first order approximation of what that pyre for will be equal to fire before. And that is able to 0.79 That will be good to pirate too, which is? Five star 1.57 and first order approximation. What excess pi is equal to apply and week, unless write this one as 3.14 No. For third order of purse mission. Uh, this is the expression and using that, uh, playing in the values, we find out what excess pull over for Turtle wrote Turk. Older folks mission is 0.704 when expire to third Order gives us 0.92 And when exes pie 30 or perks mission gives us later. Two points your 26 Um, the whole thing is thief you further approximation. So let&#x27;s look at that. Does this t fire for facts using this implanting the X X values we see that when expire over for youth order approximation is 0.7 sierra one for prior to that is almost one and four pi. It is even in point of fact before, as you can see, especially for pyro for and pie over to fifth order Brooks mission is against pretty much the same results as orginal function. However, when X is equal to pi, B should terrifically get 04 to function. However, both heard and Ford are approximations are far away from real answer reason. We actually can see it graphically. Here, Um, we should close the X axis right here. However, fifth order approximation overshoots the answer and we can see that it is positive and third order approximation under shoots The answer. So, in order to get our approximate that function at that point accurately, we would need to add more points

Jacquelinne S. Mejia Sandoval · Answer

Hello, everyone. We&#x27;re dealing with serious to him. So, Kourtney, we know the what? A floor serious core. Relax, which is co sign a lax. Is it going to recall to the summation from any super zero to a community Oh, negative one to, um, Times X to the two and all over to end her told Sold My ignores MacLaurin series was that it&#x27;s centered at zero. We&#x27;re trying to find the Parnell&#x27;s, but other polling ALS. So let&#x27;s start with the sixth one&#x27;s t 06 X is going to recall the circulation because you put us here 23 So now I&#x27;m using I instead of oh negative one to the time next to ah, go over to i fact tour. So this is going to be negative 120 next to the two times your oh, over to time zero factorial plus negative one to the one next to the two times one all over to times one. Her tour plus native one squared thanks to the two times two divided about two times two factorial plus native one cubed next to the two tons three all over, two times three toil. Now, when you make sure, so we know that. Negative 12 an odd longer. He&#x27;s going to be native, one in the negative one too uneven. It&#x27;s fun is going to steal one. So we&#x27;re going to use this right here. No, Um, let&#x27;s find the 16 paranormal. So t six x has been illegal to one minus x squared or to fact oil plus extra fourth or whore factorial minus X to the six are lower six factorial. And now we&#x27;re gonna know that was rolled R t to the zero ax is gonna go to tea one of X, which is gonna equals one. And we&#x27;re also gonna know the tea to Rex is going to court to t through your rex Physical one minus x squared or two. And finally, the T 24 x is in equal to T T T 05 X, which is a little one minus X square or two plus external force over 24. Now we have There are for no nos Well here. So this winery here is our t to the zoo. Next searches t to the one and then this one&#x27;s are to to the two packs which is equal to teach both two years. And then this was going on with me. Teoh, to works or just go to two five writes, is a rear t success. And why is this the case? Well, as you can see, the degree break here, it&#x27;s just a constant. So that&#x27;s wise. It&#x27;s zero or one, and then the highest degree right here is gonna be the two. So that&#x27;s why it is the second teller pardonable, but is also the three, Because was you move on to this on the highest polynomial were the highs degrees before. So then has to be the fourth opinion and the five and then over here, the highest degree six. And we&#x27;re here. Your right, as you can see, is a graph. Um, as you can tell, the green one is looking better. Ah, as ah approximation Um, Cosan of X show. This is a port, So now we&#x27;re moving to part B. So in part B, where do you want us to know what is Ah, power for power to imp? I, um, value she takes into each of the cosign of eggs in the all the six still appointed earlier. So as you&#x27;re gonna see cause I know car fours. You know, the 0.707 co sign a part 20 and cultural surprising one when ah, tip on. It was just one thing. Part of four. Power to empire. It&#x27;s always going to be one in our second, um, Taylor pulling on you. So when you plug in power for for one minus X squared over two, you&#x27;re gonna get your 20.6 mine. We do the same thing for part two. You&#x27;re gonna get negatives. You&#x27;re a 20.223 And when you pull your pie, you&#x27;ll get negative 3.3 and then there&#x27;s rou won re here It&#x27;s missing. The war is actually okay, So this one would be the the fifth tiller partner over you. So then, yes, in the fourth. So when you play in part of 441 minus x squared, you ready by to post X to the fourth or 24 you&#x27;re gonna get your 240.707 When you play in part two, you&#x27;re gonna get zero points your war nine approximately and we&#x27;re putting in pi going to gives your corn 123 and now loss when you pull you in her for power to and pie into the sixties Parnham you you&#x27;re going to get your 0.7 negative 8.94 times Tentena before in negative 1.21 respectively. And now we move on to card. See? So you card. See, as you can see this purple. But I learned right here. So this one is, um, coastline of X. So that&#x27;s what we&#x27;re trying to approximate, uh, with a taylor polynomial is drawn. Um, So as you can see, the 61 which is the green one is looking kind of better. Um, this one is kind of a better approximation than all the other ones. Um, so you can say the to call in a muse, forgetting closer to the function coastline elex When you let and go to insanely

Doruk Isik · Answer

in this problem were asked to expand contingent effects earned zero using Taylor&#x27;s polynomial up to order off three. And we know that, um, Taylor&#x27;s pull normal. A tailor&#x27;s expansion of order of three will be off this form, so we first need to lead the They&#x27;re beautiful 1st 2nd and third order that less fried aftereffects again, if effective, Secret detention off X and prime. A fax is the Nico to seek on skirt of X F double prime professor dirt off sequined skirt effects is to seek and skirt if x terms attention to FX and they&#x27;re better off. That would be our We could find it using product Roland that is equal to four seek and skirt of ex Tensions skirt effects plus to seek and forth off X All right, unless you let the function in this their pictures at a given point. So let&#x27;s find fr zero f prime of zero F double primal zero as well as F triple prime of zero. If he&#x27;d plug zero intentions here, we know that zero second zero is one so she can skirt of fear would want We have tension here. So zero attention 00 again and this part when plugging in zero, this part would be zero. But see, Candice won&#x27;t. So this would be tickled to know we have everything that we need. So we could just plug this in to this given equation and find t three effects of third order approximation. Um, effects as zero plus one times x minus zero plus zero. This is the function itself. This is the first service. This is the second narrative term. Plus, the turkey with the term narrative is too. We have X minus zero cube divided by three Factual. That&#x27;s six. So that&#x27;s three. So from this week, right? This one s, um X plus X cube over three. And what we see here is this We already know what turn order approximation. Looks like we know that is the order approximation as this turned. And that is zero. And we hear See that t one off backs off. First order approximation is equal second order approximation. And that would be this part only. And that is equal to X or it. Next thing we RST of late this, um, function at given points implode it. Well, plotting it, you could use any online function. Porter deployed it, plotting it. We see that as the order Oprah Approximation gets larger. So as we include more terms, the approximation gets closer and closer to the real function. So let&#x27;s ride it here as, uh, and and meaning the order Cofer approximation. As end gets larger, our approximation gets better. So as we include more terms, it means that we will be or be answer will be closer and closer to the real answer. Now we are asked to await the function. Is there a tow or D? Taylor approximations off dysfunction at key points? Let&#x27;s form a table ffx. Do you want effects? Anti three effects and again, points are pi over sex pie or four and pie over three. We have our table. Now, if you plug these values in knowing, uh, that front&#x27;s musical do tension defects and these are the expressions for the upper informations plugging then, and we see that functions well. You won&#x27;t explain or sex is equal to 0.5774 but excess pi over four Tension is one, and the nexus pie were three. Tangent is 1.732 Now again, these and these expressions that plugging into values you can find the approximate while used for to function. And let&#x27;s just fill it out. We have 0.5 to 3. We have 30.5 714 when expire before the first order approximation gives 0.785 and third order Brooks Mission Gaze 0.946 when expired. Three. First order. Brooks Mission gets 1.472 and third order approx. Mission gets one point for three. The idea is what we see here, usually as you in Crete. More terms says you go from first order turtle approximation. The results get closer to the actual function Mellie&#x27;s.

Jacquelinne S. Mejia Sandoval · Answer

Hello, Ron. We&#x27;re working on a problem over to Okay, So first, um, because you&#x27;re working with this one of us. But we can also write this down if you bring the X up as I extra native one show you who are clogging wherever a cnx uplevel one. We&#x27;re just going to get now. The primal is going to be later. Horn, actually later, too. And from one just going to be in native more and a frying the whole time. Both ex Is it going to be to x two? Then a three and after book row warns going to Ah, Triple Crown off X Spun to the nature six x elated four and you&#x27;re elevated. How one is going to be new six? No, we know the tailor. Siri&#x27;s centred up. One centred that horn. Yes, a homer US registry for tour function One over us, would you see? Put to the summation as end goes from zero to infinity of a negative one to the end and factorial times X one is one two and over in factorial. Okay, so you would say Well, what&#x27;s going Well, look, the formal. It&#x27;s always gonna be in factorial here on the bottom. And it&#x27;s going to be X one is whatever your senator. So it will be X one is one to the and don&#x27;t a different part here. Is this part right here? The in factorial and a NATO want to. Then well, I got it from for is going. Do you see that on the first derivative? Well, we&#x27;re gonna get a positive, right? And then we&#x27;re going to get a new do. So if I do negative one to whatever in is. So for example, in this case, where it&#x27;s, ah, where there is no derivative uh, then it&#x27;s just gonna be negative. 120 which is interview positive number. It&#x27;s just going to be one. And they were never if I put a negative 1 to 1. No, it&#x27;s gonna be a native. So that&#x27;s home getting up, then. Negative. Want to Then now to get the the in factorial over here, right? Ah, well, because you could see the same thing. Ah, see the numbers beneath it, So one. Right. So you start with one. So this is just, um zero factorial, right? It&#x27;s just one. And then one factorial is one right. And then to Factorial is too. And then through factorial were six right there and the negative was already taking care of, um, as I said before. Okay, Now, if you were to simplify this right, So scared of the in fact Orioles, right? So then Now well, we&#x27;re gonna have left over is going to be the summation, as in is, he couldn&#x27;t zero. You&#x27;re infinity. Uh, negative. One to the end, X minus one to the end. Okay. And like, what was this whole thing about? Well, we&#x27;re trying to find the ah, third. Ah, degree off the table, polynomial. So now let&#x27;s find it. So t the ranks because this is 1/3. The telephone is going to be negative. Want to the do you grow? And then x one is one too. The zero. So right now we&#x27;re reusing list. Right here. Um, and then plus negative 1 to 1. Ex miners, One to the one close. Negative. One squared. Next one is one squared haas native. One cubed ex run is one Q. It was just going to be good. A one minus x minus one plus X minus one squared minus X summer is one cube. Okay, so this is the third degree to report on you. Okay, um, and then from here, right, we could also get on the second and the first. Ah, so every reverberated down. Ah, this one right here. So it&#x27;s much this one is gonna be t one off X. Is the first taylor planning on you and then, uh, soaking one. It&#x27;s gonna be this on t two X and then the 3rd 1 skin that me, uh, t three x, um, because you could see, uh, wonder graph here on the right. Um, so the the red is just line the blue, uh, looks like a correval. And then ah, the teaching three X Well, second yes. Shaped her. Um, Okay, so now this is for and now it would look for part B. Right? Um, as you could see, Yeah, they want to find those that Ah, for the functions as report nine and 1.3. Um, so just with the original function one of Rex, um, we&#x27;re gonna get when it&#x27;s excess your 0.9, we&#x27;re gonna get 1.11 and then when X is 1.3, we&#x27;re gonna is your points of 69. And then, uh, we do it for the first degree, which is when X is your 0.9, Uh, always gonna be 1.1. But when X is 1.3 or wise going to use your 0.7 and then, Ah, for the lunar here, which is the second degree. So, uh, see this one right here, ISS that one. The blue warn us this one. Actually, it&#x27;s not that one. It&#x27;s this one. And then the green one right here. It&#x27;s going to be that one. Okay, so then, uh, when you plug in zero point 9/4 second degree And so your players, your point into two money sex plus explained is one quantity squared. We&#x27;re gonna get 1.11. You pour you 1.3, you&#x27;re going to use your 0.79 And when you do it for the, um, their degree polynomial, uh, the play 0.9 into to my sex plus exporters, will quantity squared minus X one is one quantity cubed. You&#x27;re gonna get 1.11 when we played in 1.3 years against your points at 63 So as you can see the numbers are getting closer. Tour trying to approximate. Okay, so now we will want to part C. So, for part C Ah, let&#x27;s look at his breath. So Ah, the purple Right here. Show this right here. Um, this is one over X, and the other one&#x27;s was overturned approximately, so they&#x27;re very you&#x27;re getting closer to it. Um, as a green you can see is just looks like it&#x27;s getting kind of curry at this point. Well, we look goes like, around here, and he goes this way. So this part right here, I was say, uh, right here. It&#x27;s getting close to what would one? Um, but so we can say has look Taylor polynomial kids close to one or X as and approaches to infinity.

Carlos Pinilla · Answer

Yes, we have is a function ex people&#x27;s science fix. Fine. Looks we want to do that. Uh, a lure. Explosion, You know, explosion around. It was a fight over four. You have to do with interest. Expansion of the equals, Cyril. So the vision star, you know that&#x27;s gonna be Did you know that is just a function by what&#x27;s, um, societal by over for she put too. So that would be this first expansion. Constant are Gozo the expression for crows? One, this is the union. Look. Sedition up. Well, you know everyone. It&#x27;s gonna body over the function of fine or or plus the ready for function at that point from six miles by waas. So really writing off these functions, they sign. Oh, that is a good sign on the course of by over four course by before it is over to so that these don&#x27;t know you want will be Screw it off. Two for two, plus squared off over. I am so X miles fight over four on, um, the second to expansion the thirties interview. I use that word, the function or I for six minus pi over four, plus the body off the second day with the country I find for I would like to, um, snakes minus pi over four. Um, split. So it is that, you know. So you have these bodies, like these bodies. We&#x27;ll go. He&#x27;s down there. Just go. Go, go. Same sex minus find over four for the second, the people the second, you know, shook our function Gonna be really a course sign that it&#x27;s minus site. So these number will be figured my no sign by or for musically great toe. So we&#x27;re gonna get, uh, he&#x27;s mine us minus questions Still divided by some city Or so it is, uh, so that&#x27;s not the one. And so, uh, trying to see how it looks like. See you here. The X waxes sign effects step function the books. Uh, but this point there, that is house on the There should be five spore. So this is not, you know, this is gonna be This goes the same body that the function pigs there. That is a line. Why? Because the una approximation should you? It is. Neither will she much there. So it is. The line squares off. We don&#x27;t mock. Plus Sprinkle kub X minus pi for that is, uh, But why you will see that that is one year expansion. This is, you know, this year of expansion and then four day visit is when you&#x27;re supporting young over Looks down. So he said, putting on No, that&#x27;s that. These don&#x27;t worry, we&#x27;ll keep parabola. But it&#x27;s, uh I swear it&#x27;ll photo my plus, uh, school home. Six. My spy number four glass was full of football before I&#x27;m six months by four. Where? So this is, uh, on the whole weeks. Well, these functions agree at that point five before.

Jack Chen · Answer

for this program, we need to find, um, the first of the narrative and the second of curative off if at the beginning. So if Prime Michels to co sign X half double prime because to minors Sykes seems we consider the tailor porn Omino center that a equals two pi over four. So if off half of a because to root off to divide by two the first of the narrative at Exit Costa is also route up to divide by two. The second of purity is, ah minus root off to divide by two. So with this information, we have p two because two information and from the altitude the end of the purity of that A. If I buy in factorial times six minus a to the end, which is root off to over to class route off to over two times X minus umpire full minus root of tour Full X minus pyro for square. And the way we can reduce this result of P one, which is just the 1st 2 terms. So it&#x27;s a linear function. Now we can grab both p one p two, combined with the original function gave in one coordinate assistant will see the difference between them. So, um, Mr Reagan line, we will be the original function safe. The blue line is a straight line with which is P one. It&#x27;s a linear function. The purple wise p two. It&#x27;s a quadratic function. So off them off, these three agree and appoint pyre of four so near powerful off this Ryker start close to each other.

Problem

(a) Find the Taylor polynomials up to degree 3 fo…

Problem 1 Medium Difficulty

(a) Find the Taylor polynomials up to degree 5 for $ f (x) = sin x $ centered at $ a = 0. $ Graph $ f $ and these polynomials on a common screen.

(b) Evaluate $ f $ and these polynomials at $ x = \pi/4, \pi/2, $ and $ \pi $.

(c) Comment on how the Taylor polynomials converge to $ f(x). $

Answer

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