1.
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\caption{Figure 1}
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Figure 1 shows part of the curve with equation \(y = - x ^ { 3 } + 2 x ^ { 2 } + 2\), which intersects the \(x\)-axis at the point \(A\) where \(x = \alpha\).
To find an approximation to \(\alpha\), the iterative formula
$$x _ { n + 1 } = \frac { 2 } { \left( x _ { n } \right) ^ { 2 } } + 2$$
is used.
- Taking \(x _ { 0 } = 2.5\), find the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
Give your answers to 3 decimal places where appropriate.
- Show that \(\alpha = 2.359\) correct to 3 decimal places.