- 4.4.1 Explain the meaning of Rolle’s theorem.
- 4.4.2 Describe the significance of the Mean Value Theorem.
- 4.4.3 State three important consequences of the Mean Value Theorem.
The Mean Value Theorem is one of the most important theorems in calculus. We look at some of its implications at the end of this section. First, let’s start with a special case of the Mean Value Theorem, called Rolle’s theorem.
Informally, Rolle’s theorem states that if the outputs of a differentiable function are equal at the endpoints of an interval, then there must be an interior point where Figure 4.21 illustrates this theorem.
Let be a continuous function over the closed interval and differentiable over the open interval such that There then exists at least one such that
Let We consider three cases:
- for all
- There exists such that
- There exists such that
Case 1: If for all then for all
Case 2: Since is a continuous function over the closed, bounded interval by the extreme value theorem, it has an absolute maximum. Also, since there is a point such that the absolute maximum is greater than Therefore, the absolute maximum does not occur at either endpoint. As a result, the absolute maximum must occur at an interior point Because has a maximum at an interior point and is differentiable at by Fermat’s theorem,
Case 3: The case when there exists a point such that is analogous to case 2, with maximum replaced by minimum.
An important point about Rolle’s theorem is that the differentiability of the function is critical. If is not differentiable, even at a single point, the result may not hold. For example, the function is continuous over and but for any as shown in the following figure.
Let’s now consider functions that satisfy the conditions of Rolle’s theorem and calculate explicitly the points where
Using Rolle’s Theorem
For each of the following functions, verify that the function satisfies the criteria stated in Rolle’s theorem and find all values in the given interval where
Verify that the function defined over the interval satisfies the conditions of Rolle’s theorem. Find all points guaranteed by Rolle’s theorem.
The Mean Value Theorem and Its Meaning
Rolle’s theorem is a special case of the Mean Value Theorem. In Rolle’s theorem, we consider differentiable functions defined on a closed interval with . The Mean Value Theorem generalizes Rolle’s theorem by considering functions that do not necessarily have equal value at the endpoints. Consequently, we can view the Mean Value Theorem as a slanted version of Rolle’s theorem (Figure 4.25). The Mean Value Theorem states that if is continuous over the closed interval and differentiable over the open interval then there exists a point such that the tangent line to the graph of at is parallel to the secant line connecting and
Mean Value Theorem
Let be continuous over the closed interval and differentiable over the open interval Then, there exists at least one point such that
The proof follows from Rolle’s theorem by introducing an appropriate function that satisfies the criteria of Rolle’s theorem. Consider the line connecting and Since the slope of that line is
and the line passes through the point the equation of that line can be written as
Let denote the vertical difference between the point and the point on that line. Therefore,
Since the graph of intersects the secant line when and we see that Since is a differentiable function over is also a differentiable function over Furthermore, since is continuous over is also continuous over Therefore, satisfies the criteria of Rolle’s theorem. Consequently, there exists a point such that Since
we see that
Since we conclude that
In the next example, we show how the Mean Value Theorem can be applied to the function over the interval The method is the same for other functions, although sometimes with more interesting consequences.
Verifying that the Mean Value Theorem Applies
For over the interval show that satisfies the hypothesis of the Mean Value Theorem, and therefore there exists at least one value such that is equal to the slope of the line connecting and Find these values guaranteed by the Mean Value Theorem.
One application that helps illustrate the Mean Value Theorem involves velocity. For example, suppose we drive a car for 1 h down a straight road with an average velocity of 45 mph. Let and denote the position and velocity of the car, respectively, for h. Assuming that the position function is differentiable, we can apply the Mean Value Theorem to conclude that, at some time the speed of the car was exactly
Mean Value Theorem and Velocity
If a rock is dropped from a height of 100 ft, its position seconds after it is dropped until it hits the ground is given by the function
- Determine how long it takes before the rock hits the ground.
- Find the average velocity of the rock for when the rock is released and the rock hits the ground.
- Find the time guaranteed by the Mean Value Theorem when the instantaneous velocity of the rock is
Suppose a ball is dropped from a height of 200 ft. Its position at time is Find the time when the instantaneous velocity of the ball equals its average velocity.
Corollaries of the Mean Value Theorem
Let’s now look at three corollaries of the Mean Value Theorem. These results have important consequences, which we use in upcoming sections.
At this point, we know the derivative of any constant function is zero. The Mean Value Theorem allows us to conclude that the converse is also true. In particular, if for all in some interval then is constant over that interval. This result may seem intuitively obvious, but it has important implications that are not obvious, and we discuss them shortly.
Corollary 1: Functions with a Derivative of Zero
Let be differentiable over an interval If for all then constant for all
Since is differentiable over must be continuous over Suppose is not constant for all in Then there exist where and Choose the notation so that Therefore,
Since is a differentiable function, by the Mean Value Theorem, there exists such that
Therefore, there exists such that which contradicts the assumption that for all
From Corollary 1: Functions with a Derivative of Zero, it follows that if two functions have the same derivative, they differ by, at most, a constant.
Corollary 2: Constant Difference Theorem
If and are differentiable over an interval and for all then for some constant
Let Then, for all By Corollary 1, there is a constant such that for all Therefore, for all
The third corollary of the Mean Value Theorem discusses when a function is increasing and when it is decreasing. Recall that a function is increasing over if whenever whereas is decreasing over if whenever Using the Mean Value Theorem, we can show that if the derivative of a function is positive, then the function is increasing; if the derivative is negative, then the function is decreasing (Figure 4.29). We make use of this fact in the next section, where we show how to use the derivative of a function to locate local maximum and minimum values of the function, and how to determine the shape of the graph.
This fact is important because it means that for a given function if there exists a function such that then, the only other functions that have a derivative equal to are for some constant We discuss this result in more detail later in the chapter.
Corollary 3: Increasing and Decreasing Functions
Let be continuous over the closed interval and differentiable over the open interval
- If for all then is an increasing function over
- If for all then is a decreasing function over
We will prove i.; the proof of ii. is similar. Suppose is not an increasing function on Then there exist and in such that but Since is a differentiable function over by the Mean Value Theorem there exists such that
Since we know that Also, tells us that We conclude that
However, for all This is a contradiction, and therefore must be an increasing function over
Section 4.4 Exercises
Why do you need continuity to apply the Mean Value Theorem? Construct a counterexample.
Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.
When are Rolle’s theorem and the Mean Value Theorem equivalent?
If you have a function with a discontinuity, is it still possible to have Draw such an example or prove why not.
For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.
For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points such that
For the following exercises, use the Mean Value Theorem and find all points such that
For the following exercises, show there is no such that Explain why the Mean Value Theorem does not apply over the interval
(Hint: This is called the floor function and it is defined so that is the largest integer less than or equal to
For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval Justify your answer.
For the following exercises, consider the roots of the equation.
Show that the equation has exactly one real root. What is it?
Find the conditions for exactly one root (double root) for the equation
Find the conditions for to have one root. Is it possible to have more than one root?
For the following exercises, use a calculator to graph the function over the interval and graph the secant line from to Use the calculator to estimate all values of as guaranteed by the Mean Value Theorem. Then, find the exact value of if possible, or write the final equation and use a calculator to estimate to four digits.
At 10:17 a.m., you pass a police car at 55 mph that is stopped on the freeway. You pass a second police car at 55 mph at 10:53 a.m., which is located 39 mi from the first police car. If the speed limit is 60 mph, can the police cite you for speeding?
Two cars drive from one spotlight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.
Show that and have the same derivative. What can you say about
Show that and have the same derivative. What can you say about