Assuming no friction, describe the motion of a simple pendulum released from rest at t=0 at amplitude A? Provide information about its speed and position at characteristic times during one period. [The 1D equation of motion is described by a cosine]

The displacement of the bob of mass m is given by the equation x(t)=A cos(w*t), with no phase offset as given by the boundary conditions (zero speed at t=0). By differentiating this equation twice the first and second derivatives of displacement, i.e. speed and acceleration as a function of displacement can be obtained. By finding maxima of these quantities by looking at peaks of higher order derivatives, one can find the times t at which speed and acceleration are maximised and plot the graphs for one period.
Either we can treat this mathematically or provide physical insight into what should happen to the pendulum. As the pendulum is released from rest, the initial speed is zero. Due to the tangential component of the gravitational force, the bob of mass m is accelerated until it reaches a maximum speed at zero height. As the mass continues to move due to inertia and the gravitational force acting now opposite to its motion, it will slow down again and reach the same height as initially (assuming no air resistance etc.).