7 Fig. 8 shows part of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \mathrm { e } ^ { - \frac { 1 } { 5 } x } \sin x\), for all \(x\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8350e810-3ceb-4876-a7a8-249e17616057-4_645_1100_461_516}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{figure}
- Sketch the graphs of
(A) \(y = \mathrm { f } ( 2 x )\),
(B) \(y = \mathrm { f } ( x + \pi )\). - Show that the \(x\)-coordinate of the turning point P satisfies the equation \(\tan x = 5\).
Hence find the coordinates of P .
- Show that \(\mathrm { f } ( x + \pi ) = \mathrm { e } ^ { - \frac { 1 } { 5 } \pi } \mathrm { f } ( x )\). Hence, using the substitution \(u = x - \pi\), show that
$$\int _ { \pi } ^ { 2 \pi } \mathrm { f } ( x ) \mathrm { d } x = \mathrm { e } ^ { - \frac { 1 } { 5 } \pi } \int _ { 0 } ^ { \pi } \mathrm { f } ( u ) \mathrm { d } u .$$
Interpret this result graphically. [You should not attempt to integrate \(\mathrm { f } ( x )\).]