2. Consider the equation f(x) = x4 – 4x² + 4 and the problem of computing the positiveroot of f(x) = 0.-If possible, determine starting points a, b in the bisection method thata)will guarantee convergence to the root.

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Answered: 2. Consider the equation f(x) = x4 –…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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10. a) Approximate f(x) = √x by a Taylor polynomial with degree 2 at the number a = 8. b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x)= T₂ (x) when x lies in the interval 7 ≤ x ≤9.
) Consider the function f(x)=4x-1-sin(x) on the interval Osxs2. a) Perform 4 iterations of the Bisection Method on f(x) using the end points of the interval as initial data. Show the new estimate xn and f(xn) at each step. b) Perform 4 iterations of Newton's Method on f(x) with the initial guess x0=1. Show xn and f(xn) at each step. c) Perform 4 iterations of Secant Method on f(x) using the end points of the interval as initial data. Show the new estimate xn and f(xn) at each step.
Solve the question in the same way as a book and the Q is complete
Let x = a and x = b be the brackets for a root-finding problem for a scalar function f(x). If 8 = b-a is the size of the original bracket, the size of the interval containing the root after four iterations of the bisection method is (a) 8 (unchanged). (b) 8/4 (c) 6/8 (d) 8/16 (e) not knowable with the given information.
61. Given the fact that a root for the function f(x) = 2 - 3 lies in the interval 1, 1.5, use the Bisection Method twice to find an interval of length 1/8 containing the root
We are interested in the real-valued zeros for f(x) = x² - 2 within the interval [0,2]. (a) Perform two steps of the bisection method and provide the boundaries of the reduced interval, in which the zero must be located. (b) Use Newton's iteration method and perform one step. As a start value use xo = 1.0. (c) Rewrite the above search for zeros as a fixed-point iteration problem. Is there a guarantee that the iterative sequence converges to a fixed-point if one chooses an arbitrary start value out of the interval [0,2]? Justify your answer with a calculation.

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2. Consider the equation f(x) = x4 – 4.x² +4 and the problem of computing the positive root of f(x) = 0. a) will guarantee convergence to the root. If possible, determine starting points a, b in the bisection method that