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\includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_515_508_1105_815}
The diagram shows part of the curve with parametric equations
$$x = 2 \ln ( t + 2 ) , \quad y = t ^ { 3 } + 2 t + 3$$
- Find the gradient of the curve at the origin.
- At the point \(P\) on the curve, the value of the parameter is \(p\). It is given that the gradient of the curve at \(P\) is \(\frac { 1 } { 2 }\).
(a) Show that \(p = \frac { 1 } { 3 p ^ { 2 } + 2 } - 2\).
(b) By first using an iterative formula based on the equation in part (a), determine the coordinates of the point \(P\). Give the result of each iteration to 5 decimal places and each coordinate of \(P\) correct to 2 decimal places.