2 The curve with equation \(y = x ^ { x }\), where \(x > 0\), intersects the line \(y = 5\) at a single point, where \(x = \alpha\).
- Show that \(\alpha\) lies between 2 and 3 .
- Show that the equation \(x ^ { x } = 5\) can be rearranged into the form
$$x = \mathrm { e } ^ { \left( \frac { \ln 5 } { x } \right) }$$
- Use the iterative formula
$$x _ { n + 1 } = \mathrm { e } ^ { \left( \frac { \ln 5 } { x _ { n } } \right) }$$
with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
- Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to
$$\int _ { 0.5 } ^ { 1.7 } \left( 5 - x ^ { x } \right) \mathrm { d } x$$
giving your answer to three significant figures.
- Hence find an approximation to \(\int _ { 0.5 } ^ { 1.7 } x ^ { x } \mathrm {~d} x\).