By sketching a suitable pair of graphs, show that the equation \(2 + \mathrm { e } ^ { - 0.2 x } = \ln ( 1 + x )\) has only one root.
Show by calculation that this root lies between 7 and 9 .
\includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-08_2716_40_109_2009}
Use the iterative formula
$$x _ { n + 1 } = \exp \left( 2 + \mathrm { e } ^ { - 0.2 x _ { n } } \right) - 1$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\(\left[ \exp ( x ) \right.\) is an alternative notation for \(\left. \mathrm { e } ^ { x } .\right]\)
\includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-10_481_789_262_639}
The diagram shows the curve \(y = \sin 2 x ( 1 + \sin 2 x )\), for \(0 \leqslant x \leqslant \frac { 3 } { 4 } \pi\), and its minimum point \(M\). The shaded region bounded by the curve that lies above the \(x\)-axis and the \(x\)-axis itself is denoted by \(R\).