2024 SAJC H1 Q4

The curve \(D\) has equation \(y = \ln(1 + 2x^2)\).

1. Use a non-calculator method to find the equation of the tangent to \(D\) at the point where \(x = 2\), giving your answer in the form \(ax + by = c + d \ln 3\) where \(a\), \(b\), \(c\) and \(d\) are integers.[3]
2. Sketch the graphs of \(D\) and the tangent found in part (a) on a single diagram, stating the coordinates of the point(s) of intersection of the two graphs and the points where the tangent meets the axes.[3]
3. Find the area of the finite region bounded by the curve \(D\), the \(x\)-axis and the tangent line. Give your answer correct to \(3\) significant figures.[3]

A point \(P(x, y)\) moves along the curve \(D\) in such a way that its \(x\) coordinate is increasing at a constant rate of \(3\) units per second. Find the rate of change of its \(y\) coordinate at the instant when \(x=1\).[2]