9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\). The endpoints of the curve are \(\mathrm { P } ( - \pi , 1 )\) and \(\mathrm { Q } ( \pi , 3 )\), and \(\mathrm { f } ( x ) = a + \sin b x\), where \(a\) and \(b\) are constants.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82825739-6239-4afd-9621-538d35c09f28-4_663_1265_386_440}
\captionsetup{labelformat=empty}
\caption{Fig. 9}
\end{figure}
- Using Fig. 9, show that \(a = 2\) and \(b = \frac { 1 } { 2 }\).
- Find the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 0,2 )\).
Show that there is no point on the curve at which the gradient is greater than this.
- Find \(\mathrm { f } ^ { - 1 } ( x )\), and state its domain and range.
Write down the gradient of \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(( 2,0 )\).
- Find the area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \pi\).