Tangent, normal and triangle area

12

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-5_652_764_269_733} \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.
2. 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.

4.A curve \(C\) has equation \(y = \mathrm { f } ( x )\) with \(\mathrm { f } ^ { \prime } ( x ) > 0\) .The \(x\)-coordinate of the point \(P\) on the curve is \(a\) .The tangent and the normal to \(C\) are drawn at \(P\) .The tangent cuts the \(x\)-axis at the point \(A\) and the normal cuts the \(x\)-axis at the point \(B\) .

1. Show that the area of \(\triangle A P B\) is $$\frac { 1 } { 2 } [ \mathrm { f } ( a ) ] ^ { 2 } \left( \frac { \left[ \mathrm { f } ^ { \prime } ( a ) \right] ^ { 2 } + 1 } { \mathrm { f } ^ { \prime } ( a ) } \right)$$
2. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 5 x }\) and the area of \(\triangle A P B\) is \(\mathrm { e } ^ { 5 a }\) ,find and simplify the exact value of \(a\) .