Sketch the curve \(y = \ln ( x + 1 )\) and hence, by sketching a second curve, show that the equation
$$x ^ { 3 } + \ln ( x + 1 ) = 40$$
has exactly one real root. State the equation of the second curve.
Verify by calculation that the root lies between 3 and 4 .
Use the iterative formula
$$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 40 - \ln \left( x _ { n } + 1 \right) \right)$$
with a suitable starting value, to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
Deduce the root of the equation
$$\left( \mathrm { e } ^ { y } - 1 \right) ^ { 3 } + y = 40$$
giving the answer correct to 2 decimal places.