2024 ACJC P2 Q1

The diagram shows part of the graph \(y = x^{\cos 2x}\), for \(0 \le x \le 5\), which represents the path of a roller coaster. The horizontal distance travelled by the roller coaster is denoted by \(x\) units and its vertical distance travelled is denoted by \(y\) units.

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = x^{\cos 2x} \left[ \frac{\cos 2x}{x} - (2 \sin 2x) \ln x \right]\).[2]
2. At the point on the graph where \(x = \pi\), find the rate at which the roller coaster is moving vertically when it is moving horizontally at a rate of \(8\) units per hour.[2]
3. Find the acute angle that the tangent to the graph where \(x = \pi\) makes with the horizontal.[1]