The graph at right shows the height of a soccer ball after being kicked straight up into the air.

1. Explain why the slope of the tangent at point A will determine the velocity of the ball at that point.

   Hint (a):

   Since the slope of a secant determines the Average Velocity (between two points),the slope of a tangent will determine the Instantaneous Velocity (at one point).

2. At which of the labeled points is the velocity the greatest? How can you tell?

   Hint (b):

   Be sure to consider the difference between velocity and speed.

3. At what point does the ball momentarily stop? What is the velocity at this point?

   Hint (c):

   Since velocity changes from positive to negative continuously, and since $0$ is in between positive and negative values, the Intermediate Value Theorem guarantees that velocity will (at one moment) be $0$; therefore, the ball will momentarily stop.

4. The distance from the ground can be described by the function $s(t) = -16t^2 + 76.8t + 5$. Find $v(t) = s'(t)$.

   Hint (d):

   Use the Power rule.

5. Find the instantaneous velocity at $t = 2, 3,$ and $4$ seconds.

   Hint (e):

   Use your derivative function in part (d) to find $v(2)$, $v(3)$ and $v(4)$.

6. What does negative velocity represent in this problem?

   Hint (f):

   What is happening to the soccer ball when the velocity is negative?