By sketching a suitable pair of graphs, show that the equation
$$\sqrt { x } = \mathrm { e } ^ { x } - 3$$
has only one root.
Show by calculation that this root lies between 1 and 2 .
Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \ln \left( 3 + \sqrt { x _ { n } } \right)$$
converges, then it converges to the root of the equation in (a).
Use the iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.