3.
$$\mathrm { f } ( x ) = x + \tan \left( \frac { 1 } { 2 } x \right) \quad \pi < x < \frac { 3 \pi } { 2 }$$
Given that the equation \(\mathrm { f } ( x ) = 0\) has a single root \(\alpha\)
- show that \(\alpha\) lies in the interval [3.6, 3.7]
- Find \(\mathrm { f } ^ { \prime } ( x )\)
- Using 3.7 as a first approximation for \(\alpha\), apply the Newton-Raphson method once to obtain a second approximation for \(\alpha\). Give your answer to 3 decimal places.