A water tank drains so that the water level (height) h(t), measured in centimeters, is a differentiable function of time t (in minutes). For 0≤t≤8, the height is modeled by the quadratic expression h(t)=25−0.5t2.
(a) Calculate the average rate at which the water level changes over the interval [2,6]. Include units.
(b) Show algebraically that the instantaneous rate of change of h at t=4 minutes is equal to the average rate found in part (a).
(c) Explain why the equality demonstrated in part (b) would hold for any quadratic function of the form p(t)=at2+bt+c whenever the midpoint of the interval is used. Provide a concise but complete justification.