7 The curve with equation
$$y = \frac { x } { \cosh x }$$
has one stationary point for \(x > 0\).
- Show that the \(x\)-coordinate of this stationary point satisfies the equation \(x \tanh x - 1 = 0\).
The positive root of the equation \(x \tanh x - 1 = 0\) is denoted by \(\alpha\).
- Draw a sketch showing (for positive values of \(x\) ) the graph of \(y = \tanh x\) and its asymptote, and the graph of \(y = \frac { 1 } { x }\). Explain how you can deduce from your sketch that \(\alpha > 1\).
- Use the Newton-Raphson method, taking first approximation \(x _ { 1 } = 1\), to find further approximations \(x _ { 2 }\) and \(x _ { 3 }\) for \(\alpha\).
- By considering the approximate errors in \(x _ { 1 }\) and \(x _ { 2 }\), estimate the error in \(x _ { 3 }\).