3.23: Trigonometric motion; AP product; exponential GP

A particle $P$ moves in a straight line such that, $t$ seconds after leaving a fixed point $O$, its displacement $s$ metres is given by $s=4t-4\mathrm{cos}\,2t+4$.

Find the velocity, $v$, of $P$ at time $t$.

On the axes, sketch the velocity–time graph for $P$ for $0\leqslant t\leqslant\pi$, stating the intercepts with the axes in exact form.

Find the acceleration of $P$ at time $t$.

Find the times when the acceleration of $P$ is zero for $0\leqslant t\leqslant\pi$.

An AP has first three terms summing to $42$ and product $-6720$. Show $a(a+2d)=-480$.

Given that $a>0$, find the value of $a$ and of $d$.

A geometric progression has third term $\dfrac{\mathrm{e}^{4x}}{4}$ and tenth term $\dfrac{\mathrm{e}^{11x}}{512}$. Find the first term and the common ratio.