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Nov 2020 p31 q5
1901
(a) By sketching a suitable pair of graphs, show that the equation \(\csc x = 1 + e^{-\frac{1}{2}x}\) has exactly two roots in the interval \(0 < x < \pi\).
(b) The sequence of values given by the iterative formula \(x_{n+1} = \pi - \sin^{-1}\left( \frac{1}{e^{-\frac{1}{2}x_n} + 1} \right)\), with initial value \(x_1 = 2\), converges to one of these roots. Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
Solution
(a) To show that the equation \(\csc x = 1 + e^{-\frac{1}{2}x}\) has exactly two roots in the interval \(0 < x < \pi\), sketch the graphs of \(y = \csc x\) and \(y = 1 + e^{-\frac{1}{2}x}\).
The graph of \(y = \csc x\) is U-shaped, roughly symmetrical about \(x = \frac{\pi}{2}\), with \(y\left(\frac{\pi}{2}\right) = 1\), and the domain is at least \(\left(\frac{\pi}{6}, \frac{5\pi}{6}\right)\).
The graph of \(y = 1 + e^{-\frac{1}{2}x}\) is an exponential curve with \(y(0) = 2\), a negative gradient, always increasing, and \(y(\pi) > 1\). Mark the intersections with dots or crosses to show the roots.
(b) Use the iterative formula \(x_{n+1} = \pi - \sin^{-1}\left( \frac{1}{e^{-\frac{1}{2}x_n} + 1} \right)\) with \(x_1 = 2\).
Calculate the iterations:
\(x_2 = 2.3217\)
\(x_3 = 2.2760\)
\(x_4 = 2.2824\)
The sequence converges to 2.28 when rounded to 2 decimal places.
