(a)[2]
Using a suitable pair of sketches, show that the equation $\sqrt{x} = e^x - 3$ has just one root.
(b)[2]
Show by calculation that this root lies between $1$ and $2$.
(c)[1]
Show that, if a sequence generated by the iterative formula $x_{n+1} = \ln(3 + \sqrt{x_n})$ converges, then it converges to the root of the equation in (a).
(d)[3]
Use the iterative method to calculate the root correct to 2 decimal places. Give each iteration value to 4 decimal places.