3.
$$f ( x ) = 2 ^ { x - 1 } - 4 + 1.5 x \quad x \in \mathbb { R }$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as
$$x = \frac { 1 } { 3 } \left( 8 - 2 ^ { x } \right)$$
The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\), where \(\alpha = 1.6\) to one decimal place.
- Starting with \(x _ { 0 } = 1.6\), use the iteration formula
$$x _ { n + 1 } = \frac { 1 } { 3 } \left( 8 - 2 ^ { x _ { n } } \right)$$
to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
- By choosing a suitable interval, prove that \(\alpha = 1.633\) to 3 decimal places.