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Use Newton's method to approximate the given numb…

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Problem 11 Medium Difficulty

Use Newton's method to approximate the given number correct to eight decimal places.
$ \sqrt[4]{75} $

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Julianna C.

November 4, 2020

Use Newton’s method to approximate the given number correct to eight decimal places. 4?75

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Video Transcript

Using Newton's method. We will approximate the fourth root of 75, Correent to eight decimal places. So remember Newton's method is a method with which we calculate are approximately route of no linear equation of the forum F of X equals zero. That is a general nonlinear expression equated to zero. And this method help us a person made solutions to this equation in the idea behind the method is that we start with some initial gas. It's not. And if we have something like this, this is a function. This route we want to approximate. And we started with, let's say some initial guess it's not relatively close to the root in some way. So we take the tangent line to the curve at that initial guess if he takes you take the tunnel line at this point, let's say a tangent line really got the X axis at some point which we which will be the next iterate. And we repeat the process will take tiny line at that point. And we continue that way. If we can see in this small sketch, we're going to be hopefully we will be approximated to the word or brushing the root of long nonlinear equation. That's equivalent to saying that we're calculating the zero of a function. So here we are asked to present the numbers of the question is which is the integration to apply Newton's method. But we're gonna say we gotta do before applying this method is uh state to following. We have the value X is the number 4th Root of 75. It means that X to the fourth is equal to 75. And then Eggs to the 4th 75 is equal to zero. And now we have an equation and only near creation where we can apply this in medicine. And what happens if we found an X. Which satisfies this equation? Well we do the reverse process, we have this equation and then this employed this. We can take in this step here. From here. We can take the fourth word because 75 is positive number. And so they're these equations are equivalent are the same. That is we can go from one to the other. So it means that if we solve this nonlinear equation we will have a value X. Which must be equal to the 4th root of 75. Sorry, just put this year's. Okay. Okay. So we at the souls the non linear equation we find to value Mhm. Force Group of 75. And the previous year the two equations that are equivalent. Well that's it. Now we know that our function F. of X. X. to the 4th -75. It's derivative is for X cube and derivative is needed because this idea have explained here. We made the calculations of the tender line at the root or point of intersection of the detainee in line with the X axis. We get something like this. The new eatery is equal to the previous one minus the function the that previous iterate over to the relative at that trade. So this is a way of advancing the illiterates using the previous one. We use this formula where we use the function and its derivative and doing the calculations, we get the next citrate and we repeat the process as we said in the graph. And that for any greater than or equal to zero, it means we can have an initial gas for the rule we can obtain the guests maybe in the licenses equation or the function or having a graph of the function in some way. So for our case here as we are calculating The 4th Root of 75. And the idea behind Newton's method is that the initial gas be close to the root in some way. What we can see here is that we can think about an exact false route. The number Which is close to 75 for example, 81, is exactly three to the fourth. That is The 4th Root of 81. He's three and 81 is not Far from 75 6 units away. It means that if we start with this valuable, we should be close to the four food of 75 sell the good initial guests for a problem is three. And now you applied Newton's method in this case and some to Medicare to be programmed some language to the calculations the iterations are Yeah, updating the traits using this formula. That's all. But there are some things we got to consider that is when the method is going to stop generally we used to um conditions to stop the iterations. One of these conditions can be two consecutive illiterates are separate from small distance. That is these conditions here. This number is small in some way which we get to precise in the code. Another way is that because we are solving the nonlinear equation F of x equals zero. We can shake also this if the function evaluated at an illiterate is also small but we can combine both of those conditions and with these conditions we stopped the iterations and here for the quantity to be small is where we decide how many decimals would get and putting a small number as the accuracy here we get more correct decimals. The other things we got to check in our code is that this denominator here is not no. In the sense of numerical note that is it is a small number in terms of the calculations in the computer. Now, not as a mathematical zero. Well if we It started three with small, Curious E that is put for example, 10- 1913 or 12. We'll get Almost 15 decim of security. And when newton smith converged converges. Okay, quadratic lee that is, it's a rapid convergence and here with and a heresy of let's say 10 to the -14. That is we you said number to compare these quantities and initial guess three. We Need four Iterations. In fact converges very quickly for alterations to obtain an approximation of four through 75 with 13 correct decimals. It is more much more than was asked if you want And the value is approximately equal to 2.9 four 28 28 30 95 638. Okay, 27 12 In four iterations that is a very fast convergence of this method For these function X to the 4th mine and 75 An initial guess of three with an accuracy for double precision of 10-2014 we have here we have double precision meaning that we have around 14 13 14 15 not fisting my women between 13 and 14, correct. In fact I have checked with calculator different ways and we have all 15 correct decimals here. And the important fact is that we have calculated that you've seen a very simple function in this case a polynomial function X to the fourth minus 75 with very clear need derivative for X cube and uh it means that it's a very effective way of calculating numbers and uh of course in the case where the method converges because in that case it converges quite dramatically that he is very fast

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