8
\includegraphics[max width=\textwidth, alt={}, center]{bdc467f6-105e-4429-95c6-701eaa43deff-10_549_495_258_824}
The diagram shows the curve with parametric equations
$$x = 2 - \cos 2 t , \quad y = 2 \sin ^ { 3 } t + 3 \cos ^ { 3 } t + 1$$
for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). The end-points of the curve are \(( 1,4 )\) and \(( 3,3 )\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 } \sin t - \frac { 9 } { 4 } \cos t\).
- Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures.
- Find the exact gradient of the normal to the curve at the point for which \(x = 2\).