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The curve \(y = x \sqrt { \sin x }\) has one stationary point in the interval \(0 < x < \pi\), where \(x = a\) (see diagram).
- Show that \(\tan a = - \frac { 1 } { 2 } a\).
- Verify by calculation that \(a\) lies between 2 and 2.5.
- Show that if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula \(x _ { n + 1 } = \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } \right)\) converges, then it converges to \(a\), the root of the equation in part (a). [2]
- Use the iterative formula given in part (c) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.