11.
$$y = \left( 2 x ^ { 2 } - 3 \right) \tan \left( \frac { 1 } { 2 } x \right) , \quad 0 < x < \pi$$
- Find the exact value of \(x\) when \(y = 0\)
Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = \alpha\),
- show that
$$2 \alpha ^ { 2 } - 3 + 4 \alpha \sin \alpha = 0$$
The iterative formula
$$x _ { n + 1 } = \frac { 3 } { \left( 2 x _ { n } + 4 \sin x _ { n } \right) }$$
can be used to find an approximation for \(\alpha\).
- Taking \(x _ { 1 } = 0.7\), find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving each answer to 4 decimal places.
- By choosing a suitable interval, show that \(\alpha = 0.7283\) to 4 decimal places.
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