2025 IGCSE Additional Mathematics May/June 0606/23 Q8

It is given that \(y=\mathrm e^{3x+2}\tan x\).

Use calculus to find the approximate change in \(y\) as \(x\) increases from \(0.1\) to \(0.1+h\), where \(h\) is small.[5]

A curve is such that \(\dfrac{\mathrm dy}{\mathrm dx}=\sin(3x+\pi)\).

The curve passes through the point \(\left(\dfrac\pi9,\dfrac43\right)\).

Find the exact \(y\)-coordinate of the point on the curve where \(x=\dfrac{5\pi}{12}\).[4]