3.71: Small increments and integrating a sine derivative

0606/23/M/J/25 — Question 8 · 9 marks

(a)
Given y=e3x+tan⁡2xy=\mathrm{e}^{3x}+\tan^{2}x, find the approximate change in yy as xx increases from 0.10.1 to 0.1+h0.1+h.
[5]
(b)
A curve has dydx=π+3sin⁡x\dfrac{\mathrm{d}y}{\mathrm{d}x}=\pi+3\sin x and passes through a given point. Find the exact yy-coordinate where x=5π12x=\dfrac{5\pi}{12}.
[4]