2024 RVHS H1 Q4

The curve \(C\) has equation \(y=3-2\ln(1+x)\).

1. Sketch the graph of \(C\), stating clearly the exact coordinates of the axial intercepts and equations of the asymptotes (if any).[3]
2. Find the exact equation of the tangent to \(C\) at the point where \(x=\frac{1}{2}\), giving your answer in the form \(y=ax+b\), where \(a\) and \(b\) are constants to be determined.[4]
3. On the graph in part (i), sketch the graph of the tangent to \(C\) at \(x=\frac{1}{2}\), stating clearly the coordinates of the axial intercepts.[2]

The region bounded by the curve \(C\), the tangent found in part (ii) and the \(x\)-axis is denoted by \(R\).

4. Find the numerical value of the area of \(R\).[3]