Numerical AnalysisPlease follow all the steps in the given solution and use the same method. Do NOT skip any steps. Question 1 solution: We need to check the conditions:1. f € C[0, 1]2. f(x) = [0, 1] for all x = [0, 1]3. f'(x) exists on (0, 1)4. There exists a constant k € (0, 1) such that f'(x)| ≤ k for all x € (0, 1).If all the above are met then, by the Fixed Point Theorem, the sequence will convergeto a unique fixed point of f.For (a), (c)- clearly f is continuous, and f'(x) = -e* throughout (0, 1).For (b), we can see that f(x) is decreasing throughout the interval by checking thederivative: f'(x) < 0 for all x € (0, 1). Therefore f(x) is maximised at x = 0 andminimised at x = 1, so0 <<= 1 € = = f(1) ≤ f(x) ≤ f(0) = 1hence f(x) = [0, 1] for all x € [0, 1].For (d),(²^(x)) = ²1 e-tx;page 1if x € [0, 1],hence we can choose k = 1.Therefore all the conditions are satisfied and hence the sequence will converge to aunique fixed point. Let f(x) = e-0.5x on [0, 1].Check the conditions of the fixed point theorem and subsequently concludewhether or not the sequence (Po, P1, P2,...) defined byPn = 8(Pn-1), n≥1with po = 0.7 will converge to a unique fixed point in [0, 1].

To determine whether the sequence (p0,p1,p2,…) defined by the iterative process pn=g(pn−1), with…

Let f(x) = e-0.5x on [0, 1]. Check the conditions of the fixed point theorem and subsequently conclude whether or not the sequence (Po, P₁, P2,...) defined by Pn = 8(Pn-1), n≥1 with po = 0.7 will converge to a unique fixed point in [0, 1].

Let f(x) = e-0.5x on [0, 1]. Check the conditions of the fixed point theorem and subsequently conclude whether or not the sequence (Po, P₁, P2,...) defined by Pn = 8(Pn-1), n≥1 with po = 0.7 will converge to a unique fixed point in [0, 1].

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

Numerical Analysis

Please follow all the steps in the given solution and use the same method. Do NOT skip any steps.

Question 1 solution: We need to check the conditions:
1. f € C[0, 1]
2. f(x) = [0, 1] for all x = [0, 1]
3. f'(x) exists on (0, 1)
4. There exists a constant k € (0, 1) such that f'(x)| ≤ k for all x € (0, 1).
If all the above are met then, by the Fixed Point Theorem, the sequence will converge
to a unique fixed point of f.
For (a), (c)- clearly f is continuous, and f'(x) = -e* throughout (0, 1).
For (b), we can see that f(x) is decreasing throughout the interval by checking the
derivative: f'(x) < 0 for all x € (0, 1). Therefore f(x) is maximised at x = 0 and
minimised at x = 1, so
0 <
<= 1 € = = f(1) ≤ f(x) ≤ f(0) = 1
hence f(x) = [0, 1] for all x € [0, 1].
For (d),
(²^(x)) = ²1 e-tx;
page 1
if x € [0, 1],
hence we can choose k = 1.
Therefore all the conditions are satisfied and hence the sequence will converge to a
unique fixed point.

Let f(x) = e-0.5x on [0, 1].
Check the conditions of the fixed point theorem and subsequently conclude
whether or not the sequence (Po, P1, P2,...) defined by
Pn = 8(Pn-1), n≥1
with po = 0.7 will converge to a unique fixed point in [0, 1].

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The maximum absolute value of g'(x) in the interval [0,1] is approximately 0.39

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