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II Use Bisection Method with [ao, L bo] [0,3],to estimate 5 to two significant digits by teating itas te positive root of g(x) =x"Error3 1...

Question

II Use Bisection Method with [ao, L bo] [0,3],to estimate 5 to two significant digits by teating itas te positive root of g(x) =x"Error3 1

II Use Bisection Method with [ao, L bo] [0,3],to estimate 5 to two significant digits by teating itas te positive root of g(x) =x" Error 3 1



Answers

Use the bisection method to approximate the solution in Problem 51 to an accuracy of two decimal places.

To solve this problem, we need to choose positive numbers that create an interval where the root lies. So I'm going to choose one and two for this one. So we're going to start by plugging those numbers in so f of one schools one cubed, minus one squared, minus two, which is negative two. Then we're gonna plug in f of two. So two cubed minus two squared minus two, which is positive, too. And that is the correct interval, since one of them is negative and one of them is positive. So now we're going to use the dissection method to find the midpoint between one and two, which is 1.5, and now we can see we have to intervals one between one point one and 1.5 and then 1.5 and two. Not to figure out which interval lies in. Now we're gonna plug in the middle number, so plug in 1.5. So 1.5 cubed minus 1.5 squared minus two, which gives you negative 0.875 And since that one's negative, the interval will be between 1.5 and two. So again, we're going to use the by section method and find the midpoint of 1.5 and to, and we'll get 1.75 So our new intervals air 1.5 to 1.75 and 1.75 to to So then let's plug in 1.75 So 1.75 cube minus 1.75 squared minus two, which is 0.297 rounded up. And since that's a positive number are interval will be between 1.5 and 1.75 So let's find the midpoint. 1.5 plus 1.75 divided by two is 1.6 to 5. So our two intervals are 1.5 to 1.6 to five and then to 1.75 so we'll plug in. It's 1.625 now 1.6 to 5 cubed minus 1.625 squared minus two, which is negative. Negative 0.350 And since it's negative, our new interval is 1.625 to 1 point 75 So taking the midpoint of those two, we have 1.6 to 5 plus 1.75 divided by two, and that's 1.6875 So our new intervals are 1.625 to 1.6875 to 1.75 Then if we plug in F of 1.6875 we'll have 1.6875 cube minus 1.6875 squared minus two, which is negative. 0.4 to 6. And since that's negative, our new interval is this one here and we can actually stop right there. Since we're rounding to the nearest tent, anything in this interval that rounds will round to 1.7.

Hello, everyone. So, in this problem we have a polynomial of degree five and we have been given an interval zero and one. And between them, we will be having one off the zero of this political. So if to approximately ah zero by using the by section method. Okay, so the by section method is the method in which we find a number which is just in the middle of them. So middle off zero and one which is given by will be given by C equals two A plus people a plus B by two. So if I consider is zero and be as one Okay, so then see, it'll be just in the middle. Uh, order ever is mean off the two numbers. Okay, Then we will find FC and then consider the sign off. Ever see if it's positive? Then we piss, choose between and be whichever is negative. So every if it is ah, negative on every it's positive. Then we produce the next candidate for nb this one and see Okay. So in this way, we have to always make sure that whatever the two numbers were choosing the function value are off opposite saying Okay, so let me do it. You'll understand Much better. So Ah, the first candidates are zero and one. So if I put a equals 20 and because toe one I will get Sequels to 0.5 by using this formula No, If you see f a would be. If you put zero in the function, you'll get effort goes to one, which is positive. Okay, so f is one now it is positive. Fine. And FB will be negative one, which is negative. Find. So now we will put FC. We'll find FFC. So that is f of 0.5. So F C 0.65 6 to 5. OK, this is also positive. Okay, So if I have to take next nb so I will choose those two numbers which will give me opposite sense since the FCS positive. Okay, so that is the next candidate for a in would be 0.5 and be, that is one B's B was one because everybody's negative. Okay, so the next thing there would be 0.5 and one and f off 0.5 is ah positive. 0.656 to 5 and f off. One is negative one as it is. As you can see here it is positive. Negative. So these are the appropriate candidates. Now see, in this case will be the main off This too. So they're meanies. Cedar 0.75 OK, so let me write 0.75 here. So this is our next candidate. Sit upon 75 Now, if you find fo 0.75 you'll see it is negative. So the value is negative. 0.2 83 Okay, so since that is name negative, I will put it. Didn't be so. Next be will visit a 0.75 and it that is 0.5. Which one is positive? So 0.656 to 5 and negative 0.283 So our more what he will be always to keep this two off opposite sense. Okay, See, we started with zero and one Now in the third step, we are between 0.5 and 0.75 The interval is getting smaller. Okay, so no, our next task would be to find C for this too. So the sea will be the average of this to which will be 0.6 to 5. Okay, now, if you'll find f off If you find f of ah zero born 6 to 5, you will get the value s 0.36294 This is positive. Okay, So positive. I will. Right here. Singapore in 36 294 So our next interval would be zero point 6 to 5 to 0.75 as because f of 0.75 is still negative. Okay. Says you can see the range became more network. Okay, so in this case Ah, the sea that is their mean would be zero point 6875 Okay, Now, if you'll find air 40.6875 you'll get the value, which is 0.17873 So this one is positive. So here we will put the positive value zero point. Once I went 873 and the negative one from the previous, which is this one. So that means next our candidate for a word with 0.68 75 Okay, so in a we have those candidates whose the function value are always positive. And under B, we have those candidates whose function well Oh, is negative. Okay, so you can see the gap became more narrow between the two numbers and be OK. So in this way will reach 21 approximately zero. Okay. And now 0.6875 and 0.75 So they're mean is zero point 71 88 Okay, fine. Now f off. If you find the function villa for 0.71 Double eight, you will get 0.778 So this is positive. So I will write the positive one over here. Okay? And the negative will be, as usual, the older one. So that is 0.283 So in C, we will have the previous one, but it will change. So now a 71 80. Okay, so that means our interval became more narrower. Okay, so we're in 0.71 82 0.75 So now they're mean. If you find see again By taking the average of this too, you'll find C is equal stew zero point seven three 44 OK, fine. CML f off zero point 7344 is equals to 0.2 five four. Okay, so this is a positive value. So I'll write it here. 0.254 And the negative will be from the previous on already too. So that means our you will change and it will be the new one. That is 0.73 Double four and we will be as it is. Report 75 Okay, fine. No, the last we can do is we can find. See, that is the average of this too. And that is Ah, 0.7 four. Go to. Okay, so this value the function Well, you off this C O B equals two to to this was to to so 0.74 double do. Okay. 0.74 double toe. The function value is negative. 1.284 285 multiply. Do tend to the bar minus two. So there's a very small value almost near to zero. Okay, so in this step, we can stop because the function value is half off. This is this one negative. 1.285 into tender ministry. So we can say that this is an approximately zero off this. Ah polynomial. Okay, 0.74 double too. Second date Ah zero is 0.74 double toe. Now we can approximate to two decimal places, so that will give us 0.74 Okay, so the zero off double animal given polynomial P x, which is equals to actually both five miners, three x cubed, plus one. So this is the zero which will give us nearly bx nearly goldstone zero. Okay, so this was the by section method. I hope you have understood. Uh, thank you for watching.

So for this problem, what we're looking at here. So it's going from this equation method of by section ready, isolated half So -2 which equals a negative number And then half of -1. She calls a positive number. Okay? So from there Take the halfway mark between them. We're going to try f take it up 1.5. That's equal to a negative number. So that means we're going to go from negative 1.5 two negative one. And then let's continue half remark between that is half of Get up to five. She goes to positive number. We now have from thinking of 1.52, You know, 1.25. Mhm. So continue from here. Got to have a .375. She goes to negative number. So now we have it from negative 1.375 two, So going halfway between those. Yeah, that's the negative 1.3125 which equals a positive number. We have from negative 1.3752. You go for three 25 So next one is 1.34 3 75. It's equal to a native number. So now we have it from all these was good princess. Okay, so this next one would be from 8 1.3 for 375 You know, open 3125. Kind of. So continuing along in this fashion here you would get that. It would lock into At 1.326171 875 1.3 to 4 to 18 75 To mean that the answer to two decimal places would be Make it at 1.32. That's right through there.

For the problem here we are asked to find the root of the equation, X cubed minus X plus one equals zero, accurate to two decimal places. So we have X cubed bus acts ar minus X plus one, two. And we see that if we don't get to zero, End up getting one answer and that's a negative 1.35. Um That is one way we can do it. Another way we can solve it is by letting this graph equals zero. So then we solve for X. Um We solve for X through this way we can also solve for x graphically by plotting it equal to zero and Jimmy. And so there's many ways that we can go about doing this. We see it's about a negative 1.3 25 But regardless, we see that ultimately When we graph it like this, our answer is going to be negative 1.3-5. That's our final answer.


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