(b) Given f (x) = x – e-², use the Bisection Method to approximate a root of f on [0, 1], accurate to 4 decimal digits. If no root exists on the interval, then write "No root exists."
(b) Given f (x) = x – e-², use the Bisection Method to approximate a root of f on [0, 1], accurate to 4 decimal digits. If no root exists on the interval, then write "No root exists."
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(a) Given f (x) = x-e* 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) = x – e=", use the Bisection Method to approximate a root of f on
[0, 1], accurate to 4 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 r € 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 roER.
(d) Apply the fixed point iteration to cos a – x = -1 four times, with xo = }. Use
an accuracy of 6 decimal digits throughout.
(e) Determine if the fixed point iteration applied in the previous question will con-
verge.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d75d102-9d9d-4bc5-abcd-58b375b4956a%2Fcb78df73-fe63-4105-a568-8bf18d0dee55%2Fq2pyx_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Given f (x) = x-e* 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) = x – e=", use the Bisection Method to approximate a root of f on
[0, 1], accurate to 4 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 r € 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 roER.
(d) Apply the fixed point iteration to cos a – x = -1 four times, with xo = }. 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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