\end{figure}
Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
(A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
(B) Find the area of the region bounded by the line and the curve.
You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
(A) Complete this identity for \(\mathrm { f } ( x + h )\).
$$f ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$
(B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
(C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
(D) State what this limit represents.