1 The curve \(y = x ^ { 3 } - x - 7\) intersects the \(x\)-axis at the point where \(x = \alpha\).
- Show that \(\alpha\) lies between 2.0 and 2.1.
- Show that the equation \(x ^ { 3 } - x - 7 = 0\) can be rearranged in the form \(x = \sqrt [ 3 ] { x + 7 }\).
- Use the iteration \(x _ { n + 1 } = \sqrt [ 3 ] { x _ { n } + 7 }\) with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to three significant figures.