4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{42aff260-e734-48ff-a92a-674032cb0377-12_595_930_219_603}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curve \(C\) with equation
$$y = \mathrm { e } ^ { - 2 x } + x ^ { 2 } - 3$$
The curve \(C\) crosses the \(y\)-axis at the point \(A\).
The line \(l\) is the normal to \(C\) at the point \(A\).
- Find the equation of \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
The line \(l\) meets \(C\) again at the point \(B\), as shown in Figure 1 .
- Show that the \(x\) coordinate of \(B\) is a solution of
$$x = \sqrt { 1 + \frac { 1 } { 2 } x - \mathrm { e } ^ { - 2 x } }$$
Using the iterative formula
$$x _ { n + 1 } = \sqrt { 1 + \frac { 1 } { 2 } x _ { n } - \mathrm { e } ^ { - 2 x _ { n } } }$$
with \(x _ { 1 } = 1\)
- find \(x _ { 2 }\) and \(x _ { 3 }\) to 3 decimal places.