Question 1 - A-Level Maths

\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1a6d059d-8ab8-41e0-8bf3-54e248f820e4-1_650_759_252_762} \captionsetup{labelformat=empty} \caption{Fig. 12}
\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 $\mathrm { f } ( x + h )$. $$\mathrm { f } ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (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.