Johanna jogs along a straight path. For $0 \leq t \leq 40$, Johanna's velocity is given by a differentiable function $v$. Selected values of $v ( t )$, where $t$ is measured in minutes and $v ( t )$ is measured in meters per minute, are given in the table below.
| \begin{tabular}{ c } $t$ |
| (minutes) |
& 0 & 12 & 20 & 24 & 40 \hline
| $v ( t )$ |
| (meters per minute) |
& 0 & 200 & 240 & - 220 & 150 \hline \end{tabular}
(a) Use the data in the table to estimate the value of $v ^ { \prime } ( 16 )$.
(b) Using correct units, explain the meaning of the definite integral $\int _ { 0 } ^ { 40 } | v ( t ) | \, dt$ in the context of the problem. Approximate the value of $\int _ { 0 } ^ { 40 } | v ( t ) | \, dt$ using a right Riemann sum with the four subintervals indicated in the table.
(c) Bob is riding his bicycle along the same path. For $0 \leq t \leq 10$, Bob's velocity is modeled by $B ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 300$, where $t$ is measured in minutes and $B ( t )$ is measured in meters per minute. Find Bob's acceleration at time $t = 5$.
(d) Based on the model $B$ from part (c), find Bob's average velocity during the interval $0 \leq t \leq 10$.