(a) Given f (x) = cos r – re on (0, 1], with reference to the Bisection Method,determine the minimum number of iterations required to have an accuracy of10-9 in the root. If no root exists on the interval, then write "No root exists."(b) Given f (x) = cosr – re", use the Bisection Method to approximate a root off on [0, 1], accurate to 3 decimal digits. If no root exists on the interval, thenwrite “No root exists."(c) Given f: R → R, suppose that f is differentiable and |f' (x)| < 1 for all z e R.Show that the sequence generated by the fixed point iteration method appliedto f converges to a fixed point of f for any initial ro E R.(d) Apply the fixed point iteration to sin r –r = -1 four times, with ro = . Usean accuracy of 6 decimal digits throughout.(e) Determine if the fixed point iteration applied in the previous question will con-verge.

“Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

(a) Given f (x) = cos r – re on [0, 1], with reference to the Bisection Method, determine the minimum number of iterations required to have an accuracy of 10-9 in the root. If no root exists on the interval, then write "No root exists."

(a) Given f (x) = cos r – re on [0, 1], with reference to the Bisection Method, determine the minimum number of iterations required to have an accuracy of 10-9 in the root. If no root exists on the interval, then write "No root exists."

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 91E

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Question

(a) Given f (x) = cos r – re on (0, 1], with reference to the Bisection Method,
determine the minimum number of iterations required to have an accuracy of
10-9 in the root. If no root exists on the interval, then write "No root exists."
(b) Given f (x) = cosr – re", use the Bisection Method to approximate a root of
f on [0, 1], accurate to 3 decimal digits. If no root exists on the interval, then
write “No root exists."
(c) Given f: R → R, suppose that f is differentiable and |f' (x)| < 1 for all z e R.
Show that the sequence generated by the fixed point iteration method applied
to f converges to a fixed point of f for any initial ro E R.
(d) Apply the fixed point iteration to sin r –r = -1 four times, with ro = . Use
an accuracy of 6 decimal digits throughout.
(e) Determine if the fixed point iteration applied in the previous question will con-
verge.

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