5.
\begin{figure}[h]
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\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{bdb2b50d-0816-4ef2-adfa-aee3afe18582-3_515_739_228_534}
\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, \(\alpha = - 0.58\).