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Can the Taylor polynomial in problem 4 be used to approximate the function at I = 2? If s0, find an approximation and an upper bound for the error: If not; why not?...

Question

Can the Taylor polynomial in problem 4 be used to approximate the function at I = 2? If s0, find an approximation and an upper bound for the error: If not; why not?Find the third-degree Tkylor polynomial for the function +* fl#)ceptered a: €Include the romainder term

Can the Taylor polynomial in problem 4 be used to approximate the function at I = 2? If s0, find an approximation and an upper bound for the error: If not; why not? Find the third-degree Tkylor polynomial for the function +* fl#) ceptered a: € Include the romainder term



Answers

Compute the Taylor polynomial indicated and use the Error Bound to find the maximum possible size of the error. Verify your result with a calculator. $f(x)=x^{-1 / 2}, \quad a=4 ; \quad\left|f(4.3)-T_{3}(4.3)\right|$

In this exercise equals four and functioning. FDR cycles X. to the one cup. So if Crimeans equals Mhm. A half times extra miners a half in F. Double primary sickles. 1 4th times out to the -3/2. And their function values at 4. 2. one. Force miners one or sorry to respect him has the second taylor upon um No page two because two Plus 1 4th that's -4 miners while or 64 Year It's Miners Full Square. And its value at 4.06. Yes, approximately 2.1 Sorry 2.01494. And the next we want to find and capital L. Such that it is a creator then the absolute value of. Yeah. Travel cram nuts. And they know that have triple growing Eriks that is a third order derogatory of of this function. He calls 3/8 X. To the power of five house enhance capital on just equals. It's right away out of four which echoes three Over eight times 32 and it has the procedure. Four points there were six is smarter than capital and um, over spray factorial times 4.06 -4 to a third power, which is approximately four points 218, 8 Times 10 to the -7.

We're going to approximate you today. Syrah 0.3 five. Were you gonna do? You can do a couple taylor approximations are you can do on approximation Taylor approximation. Well, you know, approximation. Uh, near zero. It's very bold way. So we take a dysfunction Me three x on approximate near zero. So, for these fun, shall we have well in the city today? If I thanks. Yeah, that's you. The relative off their little bit function is just gonna be What did the X on? Also second, they're ready because the ato be three X is either the X and also a third already on the fourth. So well evaluated at zero All the narratives below that you will be one The signal narrative physical to the signal Purity of heart With that zero is equal to the function. Well, we got zero. So these one you have expansion? Uh, one plus, uh, syrah 0.35 Patties are corresponding eggs here. Uh, plus syrup 0.35 squared over to Victorio Blast syrup. 135 Um, que or three factorial on. So one. But we could do also do another taylor approximation. We could say Well, uh, because you they're not Probably the most to Well, this number is about, uh 0.3, 69 So? So it is sort of like in the way, uh, like 0.3 Children. Three is almost you re paying zero and natural away too. You have 13 five is like somewhere around there. So the girls within an approximation near I actually didn't want to. So that approximation would be a function of natural removed too, plus the value of the irritable, the function of natural loyal to times syrup 0.35 When it's not really available to plus the money off the second, the red over the function have not relevant to divided by two times or in 35 minus natural available. Two squared a lot of these tools to factorial, plus their duty for the function at Matara living Roku divided by three factorial I'm syrup 135 minus not follow edible toe que no collateral and so on. So more terms. But there is little the function I'm not really move to. It's just gonna be and you narrative Why don't they are not related to will be just You did not run away and look to which is equal to two the's expansion musical Joe do blast through our booms. She went 35 What does that really want? To plus, um to e verify to factorial I'm syrah 0.35 It is not available to squared. Ah, plus two five a thief factorial 135 minus natural available to him and so on. So well, we could comparable the approximations. And for this first approximation or dark first approximation the error he would truncate up to the re end year will be heat the sierra 0.35 That is about for the Ah, yeah, they do for the end plus one, if we gonna see there, this is there about food and they're ready for sea. We're see lies between C is eating cereal and 0.35 so they would be this bound times. Um, 0.35 today and plus one divided by plus one Victorio Onda Uh, well, where we consider just the case and it goes to one which is the linear approximation, which is that Ah, the approximation is 1.35 So the ever there in this case playing into these formula. The error is more learn. Um, he is more than 0.86 on. Then we compute. What is the actual error from like flooding in the value? It's like this always cheeses morning and that So this is for the first approximation. Now, for the second approximation, you re truncate up to the end. We're gonna have or are bound for these Nazi but see no lies between it's even. I was today interval from C after naturally rooftop so that these corresponding body Oh, will we eat of the natural of it off Because the function is secrecy dimes, um, 0.35 miners natural areal through will. Absolutely. This pretty m plus one. If I impose one factorial on a side, you can see uh, this bound the number this number here is to So this bomb is bigger. So this one should be more. More I created is that it doesn't smaller. Oh, so we go just up to agree to You would have thought that approximation is too Plus two times 0.35 miners natural really do all that number, but number is about that number. Ah, you send there? I see your point 1053 on the playing in this formula for n equals two one, you have that That should be smarter than C 0.11 sitting. So as you can see Well, this is the real error, which is this ever is bigger. Them that ever So these one is more ism is a more accurate approximation. Toe to the the approximation around zero for exclamation around Syria is, uh, it's better. Is this sense that has, uh, smaller error, as you can see from these.

This problem, we are asked to let t to be the taylor polynomial of fx equals X centered at a equals four. We are asked to apply the error bound to find the maximum possible value of the error of between the function at 3.9 and the taylor polynomial at 3.9. So to begin, we want to figure out what K is going to be, where K is the absolute value of the this is going to be the absolute value of the third derivative for some point. You Where we are going to want to figure out essentially what the maximum value of the third derivative is going to be on the interval between 3.9 and four. So to do that, we'll need to find the third derivative. So we know that our first derivative is going to be one half X power of negative 1/2. Our second derivative is going to be negative 1/4 times X. To the power of negative 3/2. So our 3rd derivative is going to be equal to positive 3/8 times X. To the power of negative 5/2. Having that then can note that our function is going to be strictly um It's going to be strictly decreasing with respect to X there. So our maximum value is going to be at 3.9. So we can say that K is going to be equal to 3/8 times 3.9 to the power of negative 5/2 then means that we can determine the error bound. I'm just going to write error being less than or equal to. It's going to be 3/8 times 3.9 to the power of negative 5/2 times we have x minus A. So that's going to be or absolute value of x minus A. So it's going to be 0.1 to the power of M plus one. So it's going to be 0.1 to the power of three Divided by three factorial. I'm going to pause and calculate that offscreen. All right. So that comes out to an incredibly small value. Specifically, that is coming out to about 2.08 times 10 to the power of -6.

In this exercise a call zero and function is X plus one to the one cuff. So it's derivative of the second order derivative. Half tax +12 miners. Aha. And minus one falls. Yes. Plus one to the minus three hugs. And their function values of the aurora one a half and miners one force. So the 2nd Taylor Party nominal. Yes one plus a half X miners 18 X squared. And its value at .1. Because 1.0 4875. Next we want to find a bound for the arrow of this approximation first we need to find a capital M. Such that it is greater than or equal to the absolute value of F. Triple prime acts and the third other derivative as three over eight x. Just one choose them Power of five House. And when X equals zero this several auto derogative red transits maximum in the hands of vehicles which is 3/8. hence the residual and .1 Because Captain Ryan or worse, three factorial Times .1 Cubed It tickles 6.25 times 10 to empower miners. Far


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