Here we see a key theorem of calculus.

Here are some interesting questions involving derivatives:
(a)
Suppose you toss a ball upward into the air and then catch it. Must the ball’s velocity have been zero at some point?
(b)
Suppose you drive a car on a straight stretch of highway from toll booth to another toll booth miles away in half of an hour. Must you have been driving at miles per hour at some point?
(c)
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.

In conclusion, the Mean Value Theorem relates the function and its derivative, . Since the derivative has many interpretations, e.g. instantaneous rate of change, slope of the tangent line, velocity, it is no surprise that we can use the MVT in different contexts.

Therefore, if the the function is continuous on and differentiable on , then the MVT guarantees that

(a)
there is a number in where the instantaneous rate of change of is equal to the average rate of change of over the interval ;

or

(b)
there is a number in where the slope of the tangent line to the curve is equal to the slope of the secant line through the points and ;

or

(c)
there is a number in the time interval where the instantaneous velocity is equal to the average velocity over the time interval . Here, we assume that is the position function for the object moving along a straight line.