Use Simpson's rule with 7 ordinates (6 strips) to find an estimate for \(\int _ { 0 } ^ { 3 } 4 ^ { x } \mathrm {~d} x\).
A curve is defined by the equation \(y = 4 ^ { x }\). The curve intersects the line \(y = 8 - 2 x\) at a single point where \(x = \alpha\).
Show that \(\alpha\) lies between 1.2 and 1.3.
The equation \(4 ^ { x } = 8 - 2 x\) can be rearranged into the form \(x = \frac { \ln ( 8 - 2 x ) } { \ln 4 }\).
Use the iterative formula \(x _ { n + 1 } = \frac { \ln \left( 8 - 2 x _ { n } \right) } { \ln 4 }\) with \(x _ { 1 } = 1.2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
(2 marks)