- A curve has equation \(y = \mathrm { f } ( x )\), where
$$\mathrm { f } ( x ) = \frac { 7 x \mathrm { e } ^ { x } } { \sqrt { \mathrm { e } ^ { 3 x } - 2 } } \quad x > \ln \sqrt [ 3 ] { 2 }$$
- Show that
$$\mathrm { f } ^ { \prime } ( x ) = \frac { 7 \mathrm { e } ^ { x } \left( \mathrm { e } ^ { 3 x } ( 2 - x ) + A x + B \right) } { 2 \left( \mathrm { e } ^ { 3 x } - 2 \right) ^ { \frac { 3 } { 2 } } }$$
where \(A\) and \(B\) are constants to be found.
- Hence show that the \(x\) coordinates of the turning points of the curve are solutions of the equation
$$x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }$$
The equation \(x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }\) has two positive roots \(\alpha\) and \(\beta\) where \(\beta > \alpha\)
A student uses the iteration formula
$$x _ { n + 1 } = \frac { 2 \mathrm { e } ^ { 3 x _ { n } } - 4 } { \mathrm { e } ^ { 3 x _ { n } } + 4 }$$
in an attempt to find approximations for \(\alpha\) and \(\beta\)
Diagram 1 shows a plot of part of the curve with equation \(y = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }\) and part of the line with equation \(y = x\)
Using Diagram 1 on page 42 - draw a staircase diagram to show that the iteration formula starting with \(x _ { 1 } = 1\) can be used to find an approximation for \(\beta\)
Use the iteration formula with \(x _ { 1 } = 1\), to find, to 3 decimal places,
- the value of \(x _ { 2 }\)
- the value of \(\beta\)
Using a suitable interval and a suitable function that should be stated
- show that \(\alpha = 0.432\) to 3 decimal places.
Only use the copy of Diagram 1 if you need to redraw your answer to part (c).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-42_736_812_372_143}
\captionsetup{labelformat=empty}
\caption{Diagram 1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-42_738_815_370_1114}
\captionsetup{labelformat=empty}
\caption{copy of Diagram 1}
\end{figure}