2 A curve has equation \(y = 2 \ln ( 2 \mathrm { e } - x )\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
- Find an equation of the normal to the curve \(y = 2 \ln ( 2 \mathrm { e } - x )\) at the point on the curve where \(x = \mathrm { e }\).
[0pt]
[4 marks] - The curve \(y = 2 \ln ( 2 \mathrm { e } - x )\) intersects the line \(y = x\) at a single point, where \(x = \alpha\).
- Show that \(\alpha\) lies between 1 and 3 .
- Use the recurrence relation
$$x _ { n + 1 } = 2 \ln \left( 2 \mathrm { e } - x _ { n } \right)$$
with \(x _ { 1 } = 1\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
- Figure 1, on the opposite page, shows a sketch of parts of the graphs of \(y = 2 \ln ( 2 \mathrm { e } - x )\) and \(y = x\), and the position of \(x _ { 1 }\).
On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
[0pt]
[2 marks]
\section*{(c)(iii)}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-05_864_1284_1802_386}
\end{figure}