Sketch the graph of \(\mathrm { y } = \operatorname { coth } \mathrm { x }\) for \(x > 0\) and state the equations of the asymptotes.
Starting from the definitions of coth and cosech in terms of exponentials, prove that
$$\operatorname { coth } ^ { 2 } x - \operatorname { cosech } ^ { 2 } x = 1$$
The curve \(C\) has equation \(\mathrm { y } = \ln \operatorname { coth } \left( \frac { 1 } { 2 } \mathrm { x } \right)\) for \(x > 0\).
Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \operatorname { cosechx }\).
It is given that the arc length of \(C\) from \(\mathrm { x } = \mathrm { a }\) to \(\mathrm { x } = 2 \mathrm { a }\) is \(\ln 4\), where \(a\) is a positive constant.
Show that \(\cosh a = 2\) and find, in logarithmic form, the exact value of \(a\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.