1 The function $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = \sqrt { 4 - x ^ { 2 } }$ for $- 2 \leqslant x \leqslant 2$.
\begin{enumerate}[label=(\roman*)]
\item Show that the curve $y = \sqrt { 4 - x ^ { 2 } }$ is a semicircle of radius 2 , and explain why it is not the whole of this circle.

Fig. 9 shows a point $\mathrm { P } ( a , b )$ on the semicircle. The tangent at P is shown.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{9e68f5e0-3394-4962-acd9-25bb31f09f2b-1_628_935_728_657}
\captionsetup{labelformat=empty}
\caption{Fig. 9}
\end{center}
\end{figure}
\item (A) Use the gradient of OP to find the gradient of the tangent at P in terms of $a$ and $b$.\\
(B) Differentiate $\sqrt { 4 - x ^ { 2 } }$ and deduce the value of $\mathrm { f } ^ { \prime } ( a )$.\\
(C) Show that your answers to parts (A) and (B) are equivalent.

The function $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 3 \mathrm { f } ( x - 2 )$, for $0 \leqslant x \leqslant 4$.
\item Describe a sequence of two transformations that would map the curve $y = \mathrm { f } ( x )$ onto the curve $y = \mathrm { g } ( x )$.

Hence sketch the curve $y = \mathrm { g } ( x )$.
\item Show that if $y = \mathrm { g } ( x )$ then $9 x ^ { 2 } + y ^ { 2 } = 36 x$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI C3  Q1 [18]}}