7
\includegraphics[max width=\textwidth, alt={}, center]{074597e7-5bb1-4249-9cfa-784974a6fd2b-3_531_1065_1208_539}
The line \(y = x\) and the curve \(y = 2 \ln ( 3 x - 2 )\) meet where \(x = \alpha\) and \(x = \beta\), as shown in the diagram.
- Use the iteration \(x _ { n + 1 } = 2 \ln \left( 3 x _ { n } - 2 \right)\), with initial value \(x _ { 1 } = 5.25\), to find the value of \(\beta\) correct to 2 decimal places. Show all your working.
- With the help of a 'staircase' diagram, explain why this iteration will not converge to \(\alpha\), whatever value of \(x _ { 1 }\) (other than \(\alpha\) ) is used.
- Show that the equation \(x = 2 \ln ( 3 x - 2 )\) can be rewritten as \(x = \frac { 1 } { 3 } \left( \mathrm { e } ^ { \frac { 1 } { 2 } x } + 2 \right)\). Use the NewtonRaphson method, with \(\mathrm { f } ( x ) = \frac { 1 } { 3 } \left( \mathrm { e } ^ { \frac { 1 } { 2 } x } + 2 \right) - x\) and \(x _ { 1 } = 1.2\), to find \(\alpha\) correct to 2 decimal places. Show all your working.
- Given that \(x _ { 1 } = \ln 36\), explain why the Newton-Raphson method would not converge to a root of \(\mathrm { f } ( x ) = 0\).