(i)[3]
Sketch the curve $y = \ln(x + 1)$ and then, by drawing a second curve, show that the equation $x^3 + \ln(x + 1) = 40$ has exactly one real root. State the equation for the second curve.
(ii)[2]
Check by calculation that the root is between $3$ and $4$.
(iii)[3]
Apply the iterative formula $x_{n+1} = \sqrt[3]{40 - \ln(x_n + 1)}$, starting from a suitable initial value, in order to find the root correct to $3$ decimal places. Give each iterate to $5$ decimal places.
(iv)[2]
Deduce the root of $(e^y - 1)^3 + y = 40$, giving the answer to $2$ decimal places.