6. The following four methods are proposed to compute 71/5 . Rank them in order, based on their apparent speed of convergence, assuming Po=1. a. Pn = Pn-11+ b. PnPn-1- C. PnPn-1 d. Pn = Pn-1 - 5 5 7-Pn-1 Pn-1-7 2 2 Pn-1 5 5 Pn-1 Pn-1-7 4 5pn-1 12 Pn-1-7 3

Consider the function f(x) = a* – 4 + 6x² – 4x + 1. (6.1) Determine the order of the root z = 1. (6.2) If the Newton-Raphson algorithm is used to approximate the root z = 1, what will the order of convergence be? Motivate your answer, referring to (6.1). (6.3) Assume that a, → 1 as n → oc, when r, is determined by the Newton-Raphson method. Confirm your answer in (6.2) by determining En+1 lim for the relevant value of r, without calculating iterations. Hints: e, = 1-In and we can factorize f and f' as follows: f(x) = (x – 1)' and f'(x) = 4(x – 1)°.

3. The following four methods are proposed to compute 213, Rank them in order, based on their apparent speed of convergence, assuming po = 1. 20p-1+21/p1 21 P1-21 3p-1 a. Pa = b. Pa = Pa-1 P- 21pa-1 Pi1– 21 1/2 Pn = Pn-1 - 21 d. P.= C. Pa-1

a. Find a real root of the equation    e\power{x}tanx=1by using iteration method   and also apply Aitken's Δ\power{2} -process to acceleratethe convergence. b Find the value of e\power{-1} where e=2.718 by using  newton raphson.