1. (i) By sketching the graphs of y = arcsin (x - 1) and y = e-x on the same coordinate system, show that there is a root of the equation f(x) = arcsin (x - 1) - e-x = 0: (ii) Show that a zero of f(x) = 0 lies in the interval [1; 2]. (iii) Hence, use bisection method to perform four iterations to approximate the root of f(x) = 0 in the interval [1; 2]. (iv) The equation f(x) = 0 can be written in the form x = g(x) as x = g(x) = 1 + sin (e-x): Determine whether g has a fixed-point in the interval [0; 2]. Hence, find four approximate values of a root of the equation f(x) = 0 using x = 1 as the initial approximation. (v) By comparing the values in part (iii) and (iv), state which method will converge faster to the true value x = 1:27563754352669817

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 92E
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1. (i) By sketching the graphs of
y = arcsin (x - 1) and y = e-x
on the same coordinate system, show that there is a root of the equation
f(x) = arcsin (x - 1) - e-x = 0:
(ii) Show that a zero of f(x) = 0 lies in the interval [1; 2].
(iii) Hence, use bisection method to perform four iterations to approximate the root of f(x) = 0 in the interval [1; 2].
(iv) The equation f(x) = 0 can be written in the form x = g(x) as
x = g(x) = 1 + sin (e-x):
Determine whether g has a fixed-point in the interval [0; 2]. Hence,
find four approximate values of a root of the equation f(x) = 0 using
x = 1 as the initial approximation.
(v) By comparing the values in part (iii) and (iv), state which method will
converge faster to the true value x = 1:2756375435266981712.

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