11 The equation of a curve is \(y = \sqrt { \tan x }\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
- Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan x\), and verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = \frac { 1 } { 4 } \pi\).
The value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is also 1 at another point on the curve where \(x = a\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{87be326f-f638-43e9-a654-b7b53d5141ef-18_605_492_1493_822} - Show that \(t ^ { 3 } + t ^ { 2 } + 3 t - 1 = 0\), where \(t = \tan a\).
- Use the iterative formula
$$a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 3 } \left( 1 - \tan ^ { 2 } a _ { n } - \tan ^ { 3 } a _ { n } \right) \right)$$
to determine \(a\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
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