A particle moves along a line with velocity given by v(t)=t2e−t meters per second for 0≤t≤4. The interval is split into eight equal subintervals of length Δt=0.5. On each subinterval, a rectangle is drawn whose height equals the velocity at the time halfway through that subinterval.
(a) Write an expression that approximates the average velocity of the particle on [0,4] by using the rectangles described. Your expression should be a single numerical sum multiplied by an appropriate constant factor.
(b) Evaluate the approximation from part (a) to three decimal places.
(c) Determine whether the approximation in part (b) is an overestimate or an underestimate of the true average velocity. Justify your answer by discussing the concavity of v on [0,4].