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Q2. Determine the number of iterations necessary to solve f(x) = e-= cos(x) = 0 with accuracy 10-` using the Bisection Method on the interval [2,7]....

Question

Q2. Determine the number of iterations necessary to solve f(x) = e-= cos(x) = 0 with accuracy 10-` using the Bisection Method on the interval [2,7].

Q2. Determine the number of iterations necessary to solve f(x) = e-= cos(x) = 0 with accuracy 10-` using the Bisection Method on the interval [2,7].



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 $f(x)=x^{2}-x-3$ between $x=2$ and $x=3$

For this problem we're going to be graphing the function or using the by section method to approximate the real root of the given equation. So given that our interval is 12 and our graph is going to be too cocaine X line assignments, we can change our window of lives. That way we consider graph a little bit easier. So we go from negative 10 10. Doing this and looking from 1 to 2, we see that we can make our window um such that we have a value here that's positive. We have value down here that's negative. So when we evaluate these, that can be used in our by section method, we can also use a graph to estimate it to be about 1.11 That would be our final answer. Regardless of what method we use.

We will use both Newton's method and the second method to calculate Root for Dina Linear regression. Sign of X equals zero As first guess for the Newton's method for getting use, X note equals one and as, ah, the other guests to the second minute of. Besides these ex nautical one, he's you know we gotta use your first gas from Newton Smith. So we get to apply first the nuisance method to attain its first guess and used that value besides the year. But you won us. The first guess is for the secret. And now here we know that the Newton's method is given by expend bliss wanting Bulls X seven minus effects of Ben, divided by the derivative of F and X seven for and greater than or equal to zero and eggs suit. It's not given, and for the second method, we have eggs, um, and equals x seven minus one minus f of x seven minus one times a fraction. Whose new territories, X seven minus one minus X seven minus two. You gotta buy gifts of eggs, some in minus one, minus stiff at except in minus two, and that for angry. They're equal to because we could begin a cook for hey, start calculating X two from egg zero next one which are given and, um, then for the second method, Illini. The function F and for the Newton's method being in the function have and it's the rivet and the function. Have you know, in this case, he's equal to sign of X? He's a really TV's co signed off eggs. So this already to function sweet years in the Newton's method. And these is for sure. We gotta using the second method to write a computer problem or an updating rule in a calculator. So we're going to use and the initial gas X nautical swan for the Newton's method. If you do that, we're going to find the following it treats so it's not equals one. Then we have X one, approximately two negative zero point five 57 for Ciro eight. Then we have X to aboriginal people to see your 0.6 five nine 36 and x three ever team illegal too native, nine point 57 to times 10 to the negative five. And here we have convergence of the mutants method between three of this whole place is to the approximate solution. Negative 0.0 Sierra 095 or 96 if you want. And here we have three. This whole place is so figures. Um, this is because the solution of the aggression we know that the solution of his equation Year one is cereal. So we have for the sine function that devalue one is close to the rude ex eagle suit. That is why we're going to, uh, we are approaching the sirah value here. And this is the solution with three. That's only in fact, with four there's no place is. And for these we have used in the neatness method three iterations suites were wrapping convergence. And for thes second myth, we use then the say the same initial guess us The Newton's method it's not equals one on the other guests we gotta using. The second method is going to be the first guests obtained from the Newton's method disease ex wanna proceed with equal to Nancy syrup on 557 for 08 We do that. We calculate then x two from second method us Ah, the Emperor approximate value 0.0 a 0.0 for 3741 x three and personally too negative. Zero point Sirrah, Sirrah to 157 And, uh, it's war opportunity to six point 54 zero times tend to the native seven. And here we have convergence of the second method to the abruption made Rude 6.6 is he's going five foreign times, 10 to the negative seven? Well, yes, which has more than three. This whole place is secure, See? And for these we have used three inspirations of the second, but she's a very rapid convergence. And that is due in part because we have used these older gas for the rood us the first guest from the intense Smith. So this is very good. Instructed you to find the, uh, the other gas for the secret method that is to have a first guest near the route and the other guests from the one integration of mutants

Here the function is eric's cube Plus X -3 And its derivative as three X square plus form. And hence we have this little radio formula here. The numerator is just F accent and General Motors have prime medicine Without the initial battle to one. And if they substitute X end with one again X. One, and if a subsidy to the X and with X one gets to, we can repeat this process And into the 4th iteration, this value does not change. The value is 1.21341.

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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