17. The curve \(C\) with equation \(y = p + q \mathrm { e } ^ { x }\), where \(p\) and \(q\) are constants, passes through the point \(( 0,2 )\). At the point \(P ( \ln 2 , p + 2 q )\) on \(C\), the gradient is 5 .
- Find the value of \(p\) and the value of \(q\).
The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).
- Show that the area of \(\triangle O L M\), where \(O\) is the origin, is approximately 53.8.
\section*{18.}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{d0c23635-3b9b-4666-9cb4-21b931fb3719-08_487_695_259_683}
\end{figure}
Figure 1 shows a sketch of the curve with equation \(y = \mathrm { e } ^ { - x } - 1\). - Copy Fig. 1 and on the same axes sketch the graph of \(y = \frac { 1 } { 2 } | x - 1 |\). Show the coordinates of the points where the graph meets the axes.
The \(x\)-coordinate of the point of intersection of the graph is \(\alpha\).
- Show that \(x = \alpha\) is a root of the equation \(x + 2 \mathrm { e } ^ { - x } - 3 = 0\).
- Show that \(- 1 < \alpha < 0\).
The iterative formula \(x _ { \mathrm { n } + 1 } = - \ln \left[ \frac { 1 } { 2 } \left( 3 - x _ { n } \right) \right]\) is used to solve the equation \(x + 2 \mathrm { e } ^ { - x } - 3 = 0\).
- Starting with \(x _ { 0 } = - 1\), find the values of \(x _ { 1 }\) and \(x _ { 2 }\).
- Show that, to 2 decimal places, \(\alpha = - 0.58\).