In many engineering applications, you will come across algebraic or transcendental equations of the form f(x) = 0, where it is not possible to find the roots using direct methods. The roots can be approximated using iterative methods. One such method to solve such equations is Newton's Method, which can be applied to differentiable functions based on the recursive sequence, f(xn) Xn+1 = Xn f'(xn) for n2 1 Consider the algebraic equation, tanx – 1 = 0 (1) If f(x) = tanx – 1, find f'(x) (ii) Find the recursive relation for {x„} to approximate the root, by taking the initial approximation as, x1 = 2 Generate the sequence {x„} by writing at least the first 10 terms.
In many engineering applications, you will come across algebraic or transcendental equations of the form f(x) = 0, where it is not possible to find the roots using direct methods. The roots can be approximated using iterative methods. One such method to solve such equations is Newton's Method, which can be applied to differentiable functions based on the recursive sequence, f(xn) Xn+1 = Xn f'(xn) for n2 1 Consider the algebraic equation, tanx – 1 = 0 (1) If f(x) = tanx – 1, find f'(x) (ii) Find the recursive relation for {x„} to approximate the root, by taking the initial approximation as, x1 = 2 Generate the sequence {x„} by writing at least the first 10 terms.
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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In many engineering applications, you will come across algebraic or transcendental
equations of the form f(x) = 0, where it is not possible to find the roots using
direct methods. The roots can be approximated using iterative methods. One such
method to solve such equations is Newton's Method, which can be applied to
differentiable functions based on the recursive sequence,
f(xn)
for n >1
f'(xn)
Xn+1 = Xn
Consider the algebraic equation,
tanx – 1 = 0
(i)
If f(x) = tanx – 1, find f'(x)
(ii)
Find the recursive relation for {x„} to approximate the root, by taking
the initial approximation as, x1 = 2
(ii)
Generate the sequence {xn} by writing at least the first 10 terms.
(iv)
Does the sequence converge? If so, find the limit of the sequence.
(v)
Comment on the results achieved.
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