Compare how distance and velocity are related with these two scenarios:

1. As an arrow flies through the air, the distance it has traveled in feet at time $t$ is given by $s ( t ) = 4 \sqrt { t }$. Without your calculator, determine the velocity, $s^\prime\left(t\right)$, at times $t = 1, 4,$ and $16$ seconds. Explain what concepts of calculus you applied in order to solve this problem.

   Hint (a):

   Three big concepts to choose from are limits, derivatives and integrals.

2. As a train travels past a station, its velocity, measured in miles per hour, is $v\left(t\right) = 9t + 32$. If the train is directly across from the station when $t = 0$, determine the position of the train at $t = 1$ hour. Explain what concepts of calculus you applied in order to solve this problem.

   Hint (b):

   Refer to hint in part (a).

3. Parts (a) and (b) both involve distance and velocity. However, each required a different method or approach. Describe the relationship between distance and velocity, as well as the derivative and area under a curve.

   Hint (c):

   Think back to the Freeway Fatality problem 1-1.