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Mathematical Expression Editor

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:

Rolle’s Theorem Suppose that is differentiable on the interval , continuous on the
interval , and that .

Then for some in the open interval .

If the function happens to be a constant, then for all points in the open interval
.

If is not a constant, then it must attain both global extrema in the interval , by the
Extreme Value Theorem. One of these two different extrema is attained in the open
interval at some point , and the function has a critical point there. Since is
differentiable on , it follows that .

We can now answer our first question above.

Suppose you toss a ball upward into the air and then catch it at the same point
where you tossed it from. Must the ball’s velocity have been zero at some point?

Let
be the position of the ball (on the vertical line) at time . Our time interval in
question will be We may assume that is continuous on and differentiable on
. Since , we may now apply Rolle’s Theorem and conclude that at some
time in the given time interval, . Hence the velocity must be zero at some
point.

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

Mean Value Theorem Suppose that has a derivative on the interval and is
continuous on the interval .

Then for some in .

We can now answer our second question above.

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?

Let is the position of the car at time , and hours is the starting time with hours
being the final time, where , and . We can assume that is continuous on and
differentiable on . Now the Mean Value Theorem states that there is a time such
that Since the derivative of position is velocity, this says that the car must have been
driving at miles per hour at some point.

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.”

If for all in an interval , then is constant on .

Let be two points in . Since is
continuous on and differentiable on , by the Mean Value Theorem we know for some
in the interval . Since we see that . Moreover, since and were arbitrarily chosen,
must be the constant function.

Now let’s answer our third question.

Suppose two different functions have the same derivative on some interval . What can
you say about the relationship between the two functions?

Set , so . Now on the interval . This means that , for all in , where is some constant.
Hence, for all in

Describe all functions whose derivative is .

One such function is , so all such functions
have the form ,

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

Two students Devyn and Riley raced to class down a straight hall. They start at the
same time and finish in a tie. Was there a point during the race that Devyn and
Riley were running at exactly the same velocity?

Let represent Devyn’s position
with respect to time, and let represent Riley’s position with respect to time. Let be
the starting time of the race, and be the end of the race. Set Note, we may assume
that and are continuous on and that they are differentiable on . Hence the
same is true for . Since both runners start and finish at the same place,
In fact, this shows us that the average rate of change of is . Hence by the
mean value theorem, there is a point with . However, Hence at , this
means that there was a time when they were running at exactly the same
velocity.

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 point 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 point 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 point 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.