9.
\begin{figure}[h]
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\caption{Figure 2}
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\end{figure}
Figure 2 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where
$$\mathrm { f } ( x ) = 0.5 \mathrm { e } ^ { x } - x ^ { 2 } .$$
The curve \(C\) cuts the \(y\)-axis at \(A\) and there is a minimum at the point \(B\).
- Find an equation of the tangent to \(C\) at \(A\).
The \(x\)-coordinate of \(B\) is approximately 2.15 . A more exact estimate is to be made of this coordinate using iterations \(x _ { n + 1 } = \ln \mathrm { g } \left( x _ { n } \right)\).
- Show that a possible form for \(\mathrm { g } ( x )\) is \(\mathrm { g } ( x ) = 4 x\).
- Using \(x _ { n + 1 } = \ln 4 x _ { n }\), with \(x _ { 0 } = 2.15\), calculate \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). Give the value of \(x _ { 3 }\) to 4 decimal places.