Aball is thrown upward from the top of a building. the function below shows the height of the ball in relation to sea level, f(t), in feet, at different times, t, in seconds: f(t) = −16t2 + 32t + 384 the average rate of change of f(t) from t = 4 seconds to t = 6 seconds is feet per second.

First, we differentiate the equation for the distance so to find the velocity in feet per second (ft/s).
                                f(t) = -16t² + 48t + 100
Differentiating,
                              df(t)/dt = -32t + 48
The value at t = 3 is,
                            df(t) / dt = -32(3) + 48 = -48 ft/s
The value at t = 5 is
                           df(t) / dt = -32(5) + 48 = -112 ft/s

The average of both values is -80 ft/s.  

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