4 The diagram shows a sketch of the curves with equations \(y = 2 x ^ { \frac { 3 } { 2 } }\) and \(y = 8 x ^ { \frac { 1 } { 2 } }\).
\includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-3_433_720_1452_644} The curves intersect at the origin and at the point \(A\), where \(x = 4\).

1. 1. For the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 4\).
      (2 marks)
   2. Find an equation of the normal to the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\) at the point \(A\).
2. 1. Find \(\int 8 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
   2. Find the area of the shaded region bounded by the two curves.
3. Describe a single geometrical transformation that maps the graph of \(y = 2 x ^ { \frac { 3 } { 2 } }\) onto the graph of \(y = 2 ( x + 3 ) ^ { \frac { 3 } { 2 } }\).
   (2 marks)