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Q1) (25p) Find an approximation to the root f (x) = ex +x = 0 using Secant method Perform 2 iterations and find the successive error for the last iterative value?...

Question

Q1) (25p) Find an approximation to the root f (x) = ex +x = 0 using Secant method Perform 2 iterations and find the successive error for the last iterative value?

Q1) (25p) Find an approximation to the root f (x) = ex +x = 0 using Secant method Perform 2 iterations and find the successive error for the last iterative value?



Answers

Use Newton's method to approximate the indicated zero of each function. Continue with the iteration until two successive approximations differ by less than 0.0001 . The zero of $f(x)=\sqrt{x+1}-x$ between $x=1$ and $x=2$

You're going to use both Newton's method and the second method to calculate a route for the no linear equation X Square plus two x plus y equals zero and for the nuisance math that we're going to use the initial gas X nautical swan. And for the second method, we're going to use thes same initial guests. It's not equals equals one. And as x one, he were going to use the first gas from News and Smith. So we're going to apply first Mittens method to obtain from that the first guess and to use that at the S one of the initial guesses for the second method and we're going to fix ah, the accuracy within three decimal places. So we're we want to have the approximation from both method within three decimal places. Then, um, we know that we're going to find a solution of thes known immigration. X equals negative one because X squared plus two eggs plus one sequel to eggs. My express want square. That is why the equation x square plus two x plus one equals zero school into the equation. Experts one square zero. So we have disagree well into eggs being able to nearly one. So this'd is the the only route of the linear equation. So we got to be approximating to tease value in. We use both methods, and so we're going to start with Newton's method. We know that, um, in essence methods he's extra ventless, want equals. It's a pen minus F of X upended by the way to the river. It eve off if at x seven for an driller and go to zero using initial guess x not. And the second method he reads ex van equals x seven minus Juan miners If exhibition minus one times fraction in generator it has eggs have been minus one minus exhibit minus tuned about about half of except in minus one minus F of x of n minus two. And here we and this is for end very that we go to two. So vehicle we begin by calculating X two from X zero in a more X not on independents want that x not an ex want are given. These are the two method, and in this case we know that the functions, if it's given us ever if of X equals X squared plus two x plus one and is a function we get used. Go in both methods is derivative is equal to two times X plus two. And this is the expression we get to you seen in Newton's method. We're seeing the second whether we only use expression of fifth. Then if we start me, then you Ethan. Intense method and the initial guess X not equals one, which is indicated in the statement of your problem. Then we, uh, calculated following it reads X one equals two zero x two equals to sue a negative 0.5 a three equals two negative 0.75 Here we're using equality because thes solutions are, uh, except in the sense that they have the few 1,000,000 decimals explore is equal to negative 0.8 in 75 it's five is equal to negative zero point nine three 75 and we continue this way and we get to except eight opportunity as sorry Equal to is negative. 0.9 nine two 18 75 In here we have and 29 70 that deport decimal point. So we are very close to the solution. Oh, closer to a solution and x nine is equal to They have the zero point and 996 zero 9375 and ah extend sequel two negatives. Your point 998 issue for 68 and five in eggs 11 Sequel to negative zero point 999 And here we have three decimal places of accuracy zero, 234 and the seas. And here we have convergence of the mutants method. So the route we're finding here with the three decimal places is negative 0.999 and, uh, and we have used in this case, 11 interrelations. So now we're gonna start the second method, ease and differs the same initial gas as we using. Then you'd since Mrs X, not people see equals one. And as we obtained the first guests from the Newton's method ex wanting consumer zero iss, the other guests, we get used to the second bit. So we're going to start. He's sick and method with the initial guesses. It's not people's one and eggs warning goes to zero. And with that, we I want to find a solution within three decimal places that we did, um, with the Newton Smith. So we're going to start calculating here. So we get X to approximately equal to this case. We're gonna have quality in fact, for X to see No, sorry, it's a proclamation because he's and find it. It's precious. We have presumably to They had the syrup 0.33 36 times and a 37 human illegal too negative. This case is equality. You have equality to take three. His native 0.6 explore is everything is equal to it's again equal fact Negative serial 0.75 takes five is everything of people negative 0.8 4615 for the next six Every 70 but negative zero point nine We have it. The 1st 9 here syrup for 762 And we continue this way until eggs 15 we should say approximately equal to near the 0.998 seven for bait and eggs 7 16 Approximately 12 negative. 0.99 here for 999 2 to 6. And we have here the three nines. So, uh, in these it trade the method secret meth converse. So we have the approximate solution They had lives here by 9992 to 6 with three decimal places. And for these, we have used Ah, 15 inspirations off the second myth. So is a little bit more than than hidden Smith and much, much more. So they're comparable in this case. And because we have used ah, one of the initial gases as the first gas off the unit since method and with that way have found solution in 15 situation with three, this whole place is

Okay. Using Newton's method, we know that we have and plus one equals X event minus the original over the derivative. So minus 1/2 xa van. This is equivalent to 1/2 accident, right? We just got one minus 1/2 was looking at the coefficients hoardings attraction. Okay, so now we have to look at all our values. Acts of 1 1/2 x zero, minus 1/2 tons. One gives us 1/2 access to is 1/4 acts of three is 1/8. So four is 1 16 five is one over 32. At this point, you can probably see a pattern. We're dividing the denominator by remote climate denominator by two each time. In other words, were making the fraction smaller. So, as you can see, two times two is 44 times two is eight times to 16 and then 32. So, given this, we know that ex of 11 is gonna be 10 to 4 times to, which is 20 for a, which is approximately 0.0 for 88281 So we know that ex of 11 equals one over 204 is a root of the function now, using the secret method we have except on plus one pools and excellent minus one. Your exhibition plus on minus one, you know X zeros. One ex of one is 1/2. It's one of the exit two equals one times 1/2 over one plus 1/2 which is 1/3 acts of three is 1/5 except for is 1/8 ex of So we could go a lip to exit 11 is one over 233 all the way up until X of 16 is one over 2584 And this is when we get the value. That is 0.0 387 or 69 so we can see after 15 generations, there's the root ex of 16 equals one over 2584 which is approximately equivalent to zero, um, with the decimal accuracy. So 11 iterations with Newton's method and 15 generations with the secret method

Okay, We're using the approximation of proper vex. The derivatives is half of exposed H minus off of axe, divided by age, and H is less than one. Okay, so now solving this, we know that either the X minus two equals zero. This means each of the AKs equals two. Therefore, X equals natural log of two. So considering the initial values of X zero equals one exit one equals zero. Plugging into the formula Newton's formula X of one minutes off of X one over X minimized x zero of affects one minus affect zero. We get access to equal 0.58 acts of three equals 0.74 except four equals 0.69 So five equals 0.69 So we can see that natural log of two minus x the minus x 50.693 is approximately 0.5 So after four it orations we have the root. So the answer is Axe equal 0.693093

If half of axe equals e to the X minus one, then af prime of acts equals E to the X. Therefore, writing this out in Newton's formula we have X to the M plus one equals X to the n minus. Okay, now Newton's formula X zero equals two. We end up with ex of one equals two minus one plus one over. E Squared is 1.135 ex of 20.4 sex ex of 30.9 except for is 0.39 except five 0.73 times 10 to the negative sex. So uneven, smaller number. Okay, now use in the form of a secret method with exit vehicles to ex of one equals one plus one over East. Where did we end up with X two equals 0.7083 acts of three equals 0.3, except for equal 0.9 Exit five equals 0.12 ex of six 3.6 times 10 to the negative sex. So we can see that for Newton's method. For Newton's method, we had five iterations because that was the point up to 7.73452 temps into negative six on them for the secret method. It took five generations as well to get this 3.63 times 10 to the negative sex, so it took five iterations for both of these method.


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