2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x ^ { 6 } + \sqrt { x }\).
- Find the equation of the tangent to the curve \(y = x ^ { 4 }\) at the point where \(x = 2\). Give your answer in the form \(y = m x + c\).
- Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 4 }\) where \(x = 2\) and \(x = 2.1\).
- (A) Expand \(( 2 + h ) ^ { 4 }\).
(B) Simplify \(\frac { ( 2 + h ) ^ { 4 } - 2 ^ { 4 } } { h }\).
(C) Show how your result in part (iii) (B) can be used to find the gradient of \(y = x ^ { 4 }\) at the point where \(x = 2\). - Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 2 } - 7\) for which \(x = 3\) and \(x = 3.1\).
- Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 7\), find and simplify \(\frac { \mathrm { f } ( 3 + h ) - \mathrm { f } ( 3 ) } { h }\).
- Use your result in part (ii) to find the gradient of \(y = x ^ { 2 } - 7\) at the point where \(x = 3\), showing your reasoning.
- Find the equation of the tangent to the curve \(y = x ^ { 2 } - 7\) at the point where \(x = 3\).
- This tangent crosses the \(x\)-axis at the point P . The curve crosses the positive \(x\)-axis at the point Q . Find the distance PQ , giving your answer correct to 3 decimal places.
- \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8ff8b67d-1489-4cb1-bcd2-b32db674e29f-3_651_770_242_737}
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\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.