Here we see a key theorem of calculus.

Here are some interesting questions involving derivatives:
Suppose you toss a ball into the air and then catch it. Must the ball’s vertical velocity have been zero at some point?
Suppose you drive a car from toll booth on a toll road to another toll booth miles away in half of an hour. Must you have been driving at miles per hour at some point?
Suppose two different functions have the same derivative. What can you say about the relationship between the two functions?

While these problems sound very different, it turns out that the problems are very closely related. We’ll start simply:

We can now answer our first question above.

Rolle’s Theorem is a special case of a more general theorem.

We can now answer our second question above.

Now we will address the unthinkable: could there be a continuous function on whose derivative is zero on that is not constant? As we will see, the answer is “no.”

Now let’s answer our third question.

Finally, let us investigate two young mathematicians who run to class.