Problem

Newton's method fails. Explain why the method fai…

06:17

Problem 45 Easy Difficulty

Newton's method fails. Explain why the method fails and, if possible, find a root by correcting the problem
$$\frac{4 x^{2}-8 x+1}{4 x^{2}-3 x-7}=0, x_{0}=-1$$.

Answer

$$1.8660$$

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Calculus 1 / AB

Calculus: Early Transcendental Functions

Chapter 3

Applications of Differentiation

Section 1

Linear Approximations and Newton's Method

Discussion

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Video Transcript

in Brooklyn. 44. When we used Newton's method, So get an approximate root for this equation. Using this initial guests, we have a problem. We want to find the cause off this problem. Let's find Do you call? What's Newton's method is Evidence method is based on iterations to get an approximate route for any equation or a function using an initial guests. The next truth X in the plus one equals the bravest route xn minus F of x n divided by F dash off xn. This means we need to define ah function for these. The Newton's method. The function is the left hand side off the equation only if if the right hand side equals zero, this means off equals the left hand side off the equation. We have four x squared minus eight x plus one divided by four X squared minus three X minus seven. Now we can define or get f dash of X by differentiating f of X, we have here a fraction. We square the denominator four X squared minus three x minus seven all squared and in the nominator we have denominators. Four X squared minus three x minus seven multiplied by the differential off the nominator. Well, tabloid boy, it x minus it minus. We have the nominator for X squared minus eight X plus one multiplied by the defection off the denominator. It x minus three. This is the referential off the function of X and by starting by ex node equals minus one. We can use Newton's method This new transmitted equation to get x one x one equals exclude minus if off x no, divided by F dash off X node. We have excellent given in the equation equals minus one minus f of x node. At this means we get we substitute by X equals minus one. In this equation, we get the value off one divided by zero or in number by the Bazil. If we we substitute by minus one, we have four minus eight, four plus eight, which is 12 plus one, which is 13 divided by zero 13 divided by zero. And we don't know we don't need to find what is if dash of X because we have a division by zero in the nominator here, which makes sister infinity and X one is undefined and we can't continue iterations to find X to an extreme and so on. This means that there is a problem during alterations off Newton's method, and this is the problem, and this is a cause off the problem. To solve this problem, we can start using any other value any other initial value off X note we can start using. For example, explode equals zero. And to make our calculations a clear, we can make a table. For all our calculations started by the iteration, the initial value X equals zero. It was X equals. When we substitute here by X equals zero, we get the value off minus 4.14 it 286 then get a dash of X in substitute in here by X equals zero, we get the value off 1.2 for all it. And by applying Newton's method, we can get X one, which equals zero minus. It was X divided by evidential. Fix equals 0.11864 as long as X n plus one is not equal to X, n can continue or iterations. Then we get X one by substituting here by X equals x warm, and it equals minus Abide all one, four, 68 and we get a dash of X one. By substituting here, my X equals ex warm, and it gives a 0.97 004 Using Newton's Method equation, we get exit to, which equals 0.133 78 Again. Next to is not equal to X one. We continue operations. We get f of x two. Every X two equals minus or point away. One nine. We get a dash off X two equals Oh point mind. 4548 Using the interns method, we can get Exit three, which equals 4.1339 seven five. We can see that we are getting close. Do the accurate number. We can get another iteration to make it more accurate. F of X three equals minus 3.1 off the boy by thin to the board of negative it. If they shove X equals 4.94517 Using Nathan's the method we can get X four, which equals 4.133 975 It's the same as Exit three X four equals X three. Now we can stop operations and this is the route off the equation given in the the problem

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