9. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where
$$\mathrm { f } ( x ) = x + 2 \ln ( \mathrm { e } - x )$$
- Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by
$$y = \left( \frac { \mathrm { e } } { 2 - \mathrm { e } } \right) x + 2$$
- Find the exact area enclosed by the normal and the coordinate axes.
Fully justify your answer.
- The equation \(\mathrm { f } ( x ) = 0\) has one positive root, \(\alpha\).
- Show that \(\alpha\) lies between 2 and 3
Fully justify your answer.
- Show that the roots of \(\mathrm { f } ( x ) = 0\) satisfy the equation
$$x = \mathrm { e } - \mathrm { e } ^ { - \frac { x } { 2 } }$$
[2 marks]
- Use the recurrence relation
$$x _ { n + 1 } = \mathrm { e } - \mathrm { e } ^ { - \frac { x _ { n } } { 2 } }$$
with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\) giving your answers to three decimal places.
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[2 marks] - Figure 1 below shows a sketch of the graphs of \(y = e - e ^ { - \frac { x } { 2 } }\) 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.
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\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{75e73922-4324-441c-b942-c709a71d9025-22_1236_1566_1519_360}
\end{figure}
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