Sketch the graph of \(y = \operatorname { coth } x\), and give the equations of any asymptotes.
It is given that \(\mathrm { f } ( x ) = x \tanh x - 2\). Use the Newton-Raphson method, with a first approximation \(x _ { 1 } = 2\), to find the next three approximations \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) to a root of \(\mathrm { f } ( x ) = 0\). Give the answers correct to 4 decimal places.
If \(\mathrm { f } ( x ) = 0\), show that \(\operatorname { coth } x = \frac { 1 } { 2 } x\). Hence write down the roots of \(\mathrm { f } ( x ) = 0\), correct to 4 decimal places.