There are some theorems or ideas in Calculus that may seem rather obvious. Rolle's Theorem is one such theorem. Let's say you leave your house to go for a walk. After your walk, you return home. Rolle's Theorem says that because you started and ended at the same place, you must have made a turn at some point during your walk. Though this fact seems evident, Rolle's Theorem is a significant discovery in Calculus.
Explore our app and discover over 50 million learning materials for free.
Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken
Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.
Jetzt kostenlos anmeldenThere are some theorems or ideas in Calculus that may seem rather obvious. Rolle's Theorem is one such theorem. Let's say you leave your house to go for a walk. After your walk, you return home. Rolle's Theorem says that because you started and ended at the same place, you must have made a turn at some point during your walk. Though this fact seems evident, Rolle's Theorem is a significant discovery in Calculus.
To be able to use Rolle's Theorem, a few conditions must be met. The function should be:
Now that we've gone over the conditions for Rolle's Theorem, let's look at what this theorem says.
Rolle's Theorem states that if a function is:
then there exists at least one number in such that .
Geometrically speaking, if a function meets the requirements listed above, then there is a point on the function where the slope of the tangent line is 0 (the tangent line is horizontal).
In our walking example, Rolle's Theorem says that since we started and ended at the same place, there must have been a movement where we made a turn (the derivative is 0).
Recall the Mean Value Theorem, which states that if a function is:
then there is a number c such that and
Rolle's Theorem is a "special case" of the Mean Value Theorem. Rolle's Theorem says that if the requirements are met and there are points a and b such that , or , then there is a point where . If we plug in to the Mean Value Theorem equation for , we get . So, Rolle's Theorem is the case of the Mean Value Theorem where .
Let's assume that a function f is continuous on the interval [a, b], differentiable on the interval , and . Thus, the requirements of Rolle's Theorem are met. We must prove that the function has a point where . In other words, the point where occurs is either a maximum or minimum value (extrema) on the interval.
We know that our function will have extrema per the Extreme Value Theorem, which says that if a function is continuous, it is guaranteed to have a maximum value and a minimum value on the interval.
There are two cases:
The function is a constant value (a horizontal line segment).
The function is not a constant value.
Every point on the function meets the Rolle's Theorem requirements as everywhere.
Because the function is not a constant value, it must change direction to start and end at the same function value. So, somewhere inside the graph, the function will either have a minimum, a maximum, or both.
We must prove that the minimum or maximum (or both) occur when the derivative equals 0.
Extrema cannot occur when because when , the function is increasing. At an extrema value, the function cannot be increasing. At a maximum point, the function cannot be increasing because we are already at the maximum value. At a minimum point, the function cannot be increasing because the function was a little smaller to the left of where we are now. Since we're at the minimum value, cannot be any smaller than it is now.
Extrema cannot occur when because when , the function is decreasing. At an extrema value, the function cannot be decreasing. At a maximum point, the function cannot be increasing because which means was larger a little to the left of where we are now. Since we're at the maximum value, cannot be any larger than it is now. At a minimum point, the function cannot be decreasing because we are already at the minimum value.
Since isn't less than 0 or greater than 0, must equal 0.
While no explicit formula is associated with Rolle's Theorem, there is a step-by-step process to find the point .
1. ensure that the function meets Rolle's Theorem: continuous on the closed interval and differentiable on the open interval .
2. plug a and b into the function to guarantee that .
3. If the function meets all requirements of Rolle's Theorem, then we know that we are guaranteed at least one point where .
4. To find , we can set the first derivative equal to 0 and solve for .
Show through Rolle's Theorem that over has at least one value such that . Then, find the maximum or minimum value of the function over the interval.
By nature, we know that the cosine function is continuous and differentiable everywhere.
Plugging in 0 and into
Since , we can apply Rolle's Theorem.
By Rolle's Theorem, we are guaranteed at least one point where . So we can find and set it equal to 0.
Using our knowledge of trigonometry and the unit circle, we know the the sine function equals 0 when and multiples of . However, the only multiples of within our interval are and . So, in our interval, when .
has a maximum value of 3 at and a minimum value of 1 at
Let . Does Rolle's Theorem guarantee a value where over the interval ? Explain why or why not.
To check if we can apply Rolle's Theorem, we must ensure that the requirements are met.
We know that is continuous over the given interval because it is a polynomial. We also know that is differentiable over the interval:
When we plug in , we get . When we plug in , we get .
Since, is continuous over , differentiable over , and , then Rolle's Theorem tells us that there exists a number such that .
Rolle's Theorem is a special case of the Mean Value Theorem where
Rolle's Theorem states that if a function is:
Rolle's Theorem is a special case of the Mean Value Theorem that states that if a function is continuous over the closed interval [a, b], differentiable over the open interval (a, b), and f(a) = f(b), then there exists at least one number c in (a, b) such that f'(c) = 0.
Essentially, Rolle's Theorem is the same as MVT. It is a special case of the MVT where f(b) - f(a) = 0.
An example of Rolle's Theorem is the function f(x) = cos(x) + 2 over the interval [0, 2pi]. Rolle's Theorem states that because this function meets the theorem's requirements, there exists at least one value c such that f'(c) = 0.
Assume that the requirements of Rolle's Theorem hold for a function f. We can prove Rolle's Theorem by considering the two cases: the function is a constant value and the function is not a constant value. If the function is a constant value, then f(a) = f(b) everywhere and Rolle's Theorem applies over the entire interval (a, b). If it is not a constant value, then we know the function must change direction in order to start and end at the same function value. So, somewhere inside the graph, the function will either have a minimum, a maximum, or both. Minimum and maximum values occur when f'(x) = 0.
Rolle's Theorem states that the value c where f'(c) = 0 is in the open interval (a, b). Thus, endpoints are not included.
State Rolle's Theorem.
Rolle's Theorem says that if a function f is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there is at least one value c where f'(c) = 0.
Rolle's Theorem is a special case of the Mean Value Theorem where...
f(b) - f(a) = 0 or f(b) = f(a)
How can we interpret Rolle's Theorem geometrically?
If a function meets the requirements of Rolle's Theorem, then there is a point on the function between the endpoints where the tangent line is horizontal, or the slope of the tangent line is 0.
Already have an account? Log in
Open in AppThe first learning app that truly has everything you need to ace your exams in one place
Sign up to highlight and take notes. It’s 100% free.
Save explanations to your personalised space and access them anytime, anywhere!
Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of Vaia.
Already have an account? Log in
Already have an account? Log in
The first learning app that truly has everything you need to ace your exams in one place
Already have an account? Log in