Train $A$ runs back and forth on an east-west section of railroad track. Train A's velocity, measured in meters per minute, is given by a differentiable function $v _ { A } ( t )$, where time $t$ is measured in minutes. Selected values for $v _ { A } ( t )$ are given in the table below.
| \begin{tabular}{ c } $t$ |
| (minutes) |
& 0 & 2 & 5 & 8 & 12 \hline
| $v _ { A } ( t )$ |
| (meters/minute) |
& 0 & 100 & 40 & - 120 & - 150 \hline \end{tabular}
(a) Find the average acceleration of train $A$ over the interval $2 \leq t \leq 8$.
(b) Do the data in the table support the conclusion that train $A$'s velocity is $-100$ meters per minute at some time $t$ with $5 < t < 8$? Give a reason for your answer.
(c) At time $t = 2$, train $A$'s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train $A$, in meters from the Origin Station, at time $t = 12$. Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time $t = 12$.
(d) A second train, train $B$, travels north from the Origin Station. At time $t$ the velocity of train $B$ is given by $v _ { B } ( t ) = - 5 t ^ { 2 } + 60 t + 25$, and at time $t = 2$ the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train $A$ and train $B$ is changing at time $t = 2$.