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Exercise 8: Find the curve fit y = P(x) =(Ax+B)-1 by using the transformation X=xY= forthe data and calculate Ez(P)13.453.010.670.15Exercise 9: #) Use the Newton po...

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Exercise 8: Find the curve fit y = P(x) =(Ax+B)-1 by using the transformation X=xY= forthe data and calculate Ez(P)13.453.010.670.15Exercise 9: #) Use the Newton polynomial to derive formula for f"(x) based on the points Xrto,+ where a # Are arbitrary rcal numbers b) Obtain the forword formula f'(r) = ~ft4h-h particular case of the formula obtainedin part 4) Exercise 10) Derive the numerical diflerentiation formula using Taylor Series 3f (x)-4f( h)+ f(x 2h) and define the order of appr

Exercise 8: Find the curve fit y = P(x) =(Ax+B)-1 by using the transformation X=xY= for the data and calculate Ez(P) 13.45 3.01 0.67 0.15 Exercise 9: #) Use the Newton polynomial to derive formula for f"(x) based on the points Xrto,+ where a # Are arbitrary rcal numbers b) Obtain the forword formula f'(r) = ~ft4h-h particular case of the formula obtained in part 4) Exercise 10) Derive the numerical diflerentiation formula using Taylor Series 3f (x)-4f( h)+ f(x 2h) and define the order of approximation Exercise 13) Consider f (x)=e use the following formula to answer the given questions f"6) =-6+l6f-30f+16f--h 12h? h"' f( (c) Erc (f ,h) = Approximate f"(1.2) with h-0.05 Find the best value of h to approximate f"(x) over [1,2] with rounding error bounded by 5 =10 ' and evaluate f"(.2) using the best value of h:



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Approximating square roots Let $p_{1}$ and $q_{1}$ be the first-order Taylor polynomials for $f(x)=\sqrt{x},$ centered at 36 and $49,$ respectively. a. Find $p_{1}$ and $q_{1}$. b. Complete the following table showing the errors when using $p_{1}$ and $q_{1}$ to approximate $f(x)$ at $x=37,39,41,43,45,$ and 47 Use a calculator to obtain an exact value of $f(x)$. $$\begin{array}{|c|c|c|} \hline x & \left|\sqrt{x}-p_{1}(x)\right| & \left|\sqrt{x}-q_{1}(x)\right| \\ \hline 37 & & \\ \hline 39 & & \\ \hline 41 & & \\ \hline 43 & & \\ \hline 45 & & \\ \hline 47 & & \\ \hline \end{array}$$ c. At which points in the table is $p_{1}$ a better approximation to $f$ than $q_{1}$ ? Explain this result.

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We're going to compute the local linear approximation of F f P and use that to approximate the value of FX que And then from there we will take the absolute value of our local linear approximation. Evaluated a Q minus f evaluated at Kew and divide that by the distance between P and Q. So to get started, let's find the partial derivative of F with respect to acts on the partial derivative of F with respect toe wine. So we know that f of X Y is equal to the quantity X squared plus y squared to the minus one half. So the partial derivative of F with respect to X is equal to minus one half times of quantity. X squared plus y squared to the minus three halfs times two acts. And now that we have this, we can evaluate the partial derivative of F with respect to X. At the point, P and P is the 0.4 comma three. And when we do that, we get that this is equal to minus four all over 125 and similarly, when we compute the partial derivative of F with respect to why we get that This is minus one half times a quantity X squared plus y squared to the minus three halfs this time multiplied by two y. And when we evaluate this at P, we get that this is equal to minus three over 125. So now from here, we know that our local linear approximation of F at the point p that's equal to F evaluated at P plus the partial derivative of F with respect to X evaluated at p times a quantity x minus x not which in this case is for plus the partial derivative of f with respect to why evaluated at P times of quantity, why minus why not, which in this case is three but we solved for these values above. So all we have to do is just plug in that information we get. That local linear approximation is equal to 1/5 minus 4/1 25 times the quantity X minus four minus 3/1 25 times the quantity. Why minus three. So that completes part one. And now for part two, we just plug in. Q. We plug Q into our local linear approximation and into F and take that difference and then dividing by the distance between P and Q. So when we do that, we get that. That's the absolute value of 1/5 minus 4/1 25. And then in this time, instead of writing X well, right? The X coordinate of Q, which is 3.92 And then this time instead of writing why? Well, right, the Y coordinate of Q, which is 3.1 And then from here we want to subtract F evaluated at Kew. So that's just one over the square root of 3.92 squared, plus 3.1 squared. And then we want to divide that by the distance between P and Q. Which is the square? The sum of the square of these numbers above. So we get that. That's 3.92 minus form squared, plus the quantity 3.1 minus three squared. And then once we compute the value of this, we get that that's approximately 0.0 17 66 036 and that completes the problem. But more importantly, our reason for computing. This value was to show that the air in our approximation is much smaller than the distance between are two points p and Q. Thanks for watching.

It was a human dimension. F thanks. Why? It could do the trail minus apart. X squared minus eight more square and appointment is one for And we did you estimate that I have on the one is one for the find trip on 95 and recorded for the rainy approximation, you know. Thanks. Why with Nikola Judah have eggs. Zero y zero plus f x x moments eg zero Does f y y minus Y zero Andi universe them. When did you find a bash? Nearly with you. Now you know, we go join the, uh minus made. Thanks f why you go to the minus 16. Why? And from here we get thinkable again. They really approximation explain why you thought you if excel was so it would because you, uh trail minus 88 minus uh, a times sixth in many Jenna minus quantity virginity. And we have the FX. You go, June, uh, less eight. Yeah. Why express Kwan on Brian with Nico chilled my house 64. Why might? That's fun way. Somebody find it when you get acquitted of eight. Thanks. By the 64. Why on and here we have, uh and e uh, Let's go on a 24. That's community the approximation. And from what you estimate and have minus one bones 05 on five way a prostrated in the mid minus 105 day on 95 way Coach of any times minus 1.5. Buying a 64 times trip on my five plus 1 44 Get it? Could you wait, go to minus one?

Any approximation function X y by a formula with a call to the function and upon x zero y zero. Then we plus the personally Rippetoe respected The X Times X minus x zero plus F Y Times Square minus y zero on a in this question were given defection. Have extra fly Nico Judi Trail minutes for X squared minus eight by square and appointments one for and the first time we need to find a brand on the f X and f y here. So the person Eri with respect to that next a good U minus eight Thanks something for them for the f y, including minus 16. Why Therefore we can attend the Rainier brush mission now X y with Buddhist follow here into the function that we're getting that around minus hand with the minus far and then minus No, I Thames 16. So we're gonna go to the minus 120 we have the f x where you go Junior plus eight x minus, plus one. Here on den, we have the FBI Ego ju minus 16 teams are cut your 64 why minus far and the next time. Once a purse made a function and the party off the miners went from 05 Trip on 95 Citizen impersonated by an early near function when being minus 1.5 Trip on 95. Then we get Nico Teoh minus 120 plus many times now minus one punch. If I bless one Bay Minister Bonds of 5 64 times three bonds 95 minus four. Co chairman has no plans over five, and then we seem to find his one. We have minus 100 gente and then minus times airport himself five and then class 64 times upon. Fine. So getting coached you the minus 117 point you.


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