1 It is given that \(\mathrm { f } ( x ) = x ^ { 2 } - \sin x\).
- The iteration \(x _ { n + 1 } = \sqrt { \sin x _ { n } }\), with \(x _ { 1 } = 0.875\), is to be used to find a real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\). Find \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving the answers correct to 6 decimal places.
- The error \(e _ { n }\) is defined by \(e _ { n } = \alpha - x _ { n }\). Given that \(\alpha = 0.876726\), correct to 6 decimal places, find \(e _ { 3 }\) and \(e _ { 4 }\). Given that \(\mathrm { g } ( x ) = \sqrt { \sin x }\), use \(e _ { 3 }\) and \(e _ { 4 }\) to estimate \(\mathrm { g } ^ { \prime } ( \alpha )\).