(i) Show, using calculus (meaning do not simply use graphs) and the Fixed Point Theo- rem, that the function g(x) = - 1 (x − 2)³ +2 16 has a unique fixed point in the interval [1,4]. (ii) Use the first corollary to the Fixed Point Theorem given in Lecture 4 to calculate how many iterations of the fixed point method, starting with po 3.2, would be required to guarantee that the absolute error p - Pn| < 10-6. Do NOT actually use the fixed point method.
(i) Show, using calculus (meaning do not simply use graphs) and the Fixed Point Theo- rem, that the function g(x) = - 1 (x − 2)³ +2 16 has a unique fixed point in the interval [1,4]. (ii) Use the first corollary to the Fixed Point Theorem given in Lecture 4 to calculate how many iterations of the fixed point method, starting with po 3.2, would be required to guarantee that the absolute error p - Pn| < 10-6. Do NOT actually use the fixed point method.
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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Question
![(i) Show, using calculus (meaning do not simply use graphs) and the Fixed Point Theo-
rem, that the function
1
g(x) = ——/(x-
16
has a unique fixed point in the interval [1,4].
(x − 2)³ +2
(ii) Use the first corollary to the Fixed Point Theorem given in Lecture 4 to calculate
how many iterations of the fixed point method, starting with po 3.2, would be
required to guarantee that the absolute error |p — Pn| < 10–6. Do NOT actually use
the fixed point method.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38e693cf-3870-4eab-9920-bd211e6a8ee6%2F8ea69e69-5e0e-4496-94e6-34bb0224c7e0%2Fwirdz8e_processed.png&w=3840&q=75)
Transcribed Image Text:(i) Show, using calculus (meaning do not simply use graphs) and the Fixed Point Theo-
rem, that the function
1
g(x) = ——/(x-
16
has a unique fixed point in the interval [1,4].
(x − 2)³ +2
(ii) Use the first corollary to the Fixed Point Theorem given in Lecture 4 to calculate
how many iterations of the fixed point method, starting with po 3.2, would be
required to guarantee that the absolute error |p — Pn| < 10–6. Do NOT actually use
the fixed point method.
=
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