How fast was the pen going?

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Two young mathematicians think about the rate of change of periodic motion.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Riley, do you remember your pen bouncing on a spring?
Riley: Sure! I still have the graph of it’s height that we made. It looked like this:
Devyn: Did you notice that at the top, the graph looks like it has horizontal tangent lines?
Riley: And at the bottoms, too!
Devyn: Right! That means the pen was at rest at those instants. What about at other points, though?
Riley: The steepest slopes look like they happen when the graph crossed the $t$ -axis. That means the pen was moving fastest at those times.
Devyn: How fast was it going though? And what about at other times?
Riley: Hmmmm. I’m not sure yet…

Let’s put some labels on the graph so we can talk about it.

At which of these points does the pen have the highest velocity?

$A$ $B$ $C$ $D$ None of these.

At which of these points does the pen have the lowest speed?

$A$ $B$ $C$ $D$ None of these.

Use slopes of tangent lines to plot the graph of the derivative. Does it look familiar?