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QUESTION Estimate the function f (x) = 4.8714e3x 9828x + 1.9845 atx = 0.1MN by using quadratic interpolation ol the given step size h = 0.05487.For this question; Y...

Question

QUESTION Estimate the function f (x) = 4.8714e3x 9828x + 1.9845 atx = 0.1MN by using quadratic interpolation ol the given step size h = 0.05487.For this question; YOu are required t0 determine the values of MN in the by using 4he Jst twa (2dieits OL vur student IL Example [: SUKDI234567, M = 6 and N = 7,-x=0.1167 Example 2: SUKDI234508,M = 0 and N = 8,~x = 0.1108 (Calculate in 5 decimal places)

QUESTION Estimate the function f (x) = 4.8714e3x 9828x + 1.9845 atx = 0.1MN by using quadratic interpolation ol the given step size h = 0.05487. For this question; YOu are required t0 determine the values of MN in the by using 4he Jst twa (2dieits OL vur student IL Example [: SUKDI234567, M = 6 and N = 7,-x=0.1167 Example 2: SUKDI234508,M = 0 and N = 8,~x = 0.1108 (Calculate in 5 decimal places)



Answers

Estimate $f(x)$ for $x=2,4,6,$ using the given values of $f^{\prime}(x)$ and the fact that $f(0)=50$. $$\begin{array}{l|r|r|r|r}\hline x & 0 & 2 & 4 & 6 \\\hline f^{\prime}(x) & 17 & 15 & 10 & 2 \\\hline\end{array}$$

Here we are even the linear approximation and will be I am X y If we go to the F Thanks Times the X minus X zero plus have y times the woman a squad is zero Therefore, we should get now ever x in his ICO judo five times X minus. Hey will have end upon one and then plus one were given a coach in a minus three times window y minus. Why would you let you? Therefore we have this. We coaching a five x minus three Why you have a minus five Ah bless six day Cocina Bless one here. So this will be that exploit Selenia Brush mission and that Would you estimate the F off the one ponza one and 1.99 and therefore we should be approximated and I am 1.11 point nine I and then we get equal to the five times 1.1 minus three times one on 99 plus one And then we get Echo Ju five times 1.1 minus three times 1.99 Then 1st 1 again, Coach Izzo on 08 Here

In this problem, we need to take our linear approximation formula substituted known values to get our linear approximation equation for this set up and then a substitute in 5.1 into this equation to get our answer of fo 5.1. So let's start by substituting in our known values in this case, A is going to equal five because that is the given value. We haven't that is given value for this problem. So our Ella Vex is going to be F five plus of prime of five times ex my size now each effort five and a terrified of we know what those are 10 and negative two, respectively. So we can rewrite this as l've X equals 10 minus two times X minus five, I wrote minus students that have plus negative, too, because the negative just carries over the positive. So it's really just 10 minus two times X minus five and simplifying this down. We get our final at l A X equation, which is negative two X plus 20. So this is our final of X, and what we can now do have solved for F of 5.1 is Since this is the linear approximation at f of five, we can substitute 5.1 and for X and get our final answer. So L a five point warned equals F of 5.1, which equals negative two times 5.1, uh, plus 20. And this equals 9.8, which is our final answer.

With a given problem, we want to estimate F prime of one considering that f of X equals two X plus X to the negative first power, such as can be one over X. With that in mind we're gonna have F of 1.01 -5 of one, All divided by 1.01 -1. Yeah. That ended giving us a slope of about one. And the more the smaller we make. H we see that it's getting very, very close to the value of one. So that's what we predict our slope to be. And we can even confirm this because if we look at two times at close H Plus one over experts, H all minus this term right here, provided by H and we send the limit to zero, we see that we end up getting a slope of one, so that's our final answer.

Really given problem, we want to substitute values and we have ffx is equal to x squared plus X plus one. And we want to find the slope At f prime of one so that this is gonna look like is Um f of 1.01. Finance f of one Divided by 1.01, my next one. So we get a value very close to three. We could increase the number of Zeros to make each even smaller. And again, it looks like we're approaching this value of three algebraic lee. We can show that this is the case by taking the limit because we would have X plus H squared plus X plus H plus one minus R F of X and that's going to be over H Islamic O c zero. We end up finding that the slope is in fact going to be three. So I have the prime of one is equal to three.


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