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Can you explain Rolle's Theroem? How do you apply it in your work?
Explain Rolle's Theroem #math #theory #calculus #ap-calculus
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Abe’s Answer
One of the good explanations of Rolle's theorem is on Wikipedia (https://en.wikipedia.org/wiki/Rolle's_theorem). In a nutshell, the theorem says that for a continuous and differentiable function that has equal values at the end points of an interval [a,b], the derivative must be zero somewhere in that interval.
The key is continuity and differentiability. A function is continuous if intuitively you can draw the function without lifting the pen, and differentiable if the derivative is defined and varies smoothly within the interval.
The key is continuity and differentiability. A function is continuous if intuitively you can draw the function without lifting the pen, and differentiable if the derivative is defined and varies smoothly within the interval.
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Carlos’s Answer
Hey Jedric! Rolle's theorem is an incredibly useful theorem in regards to the foundations of math. Rolle's Theorem states that if you have a function f, that is continuous and differentiable on a closed interval [a,b] where f(a) = f(b), then for some point c in [a,b] there exists f'(c) = 0, or the derivative at some point is 0.
The significance to Rolle's Theorem is that is allows us to prove the Mean Value Theorem (https://en.wikipedia.org/wiki/Mean_value_theorem) which later let's prove many many things including many properties of integrals. Which are quite useful to many fields.
Overall, I would argue that probably no one uses Rolle's Theorem in their work. Since Rolle's Theorem ends up being a specific form of the Mean Value Theorem, the Mean Value Theorem ends up being used much more often. It is much more likely that you will use applications of the Mean Value Theorem in your work and most of these theorems.
Take a look into the Mean Value Theorem.
Look into how the Mean Value Theorem is critical to integrals.
Look at applications of integrals in fields that you are interested in!
The significance to Rolle's Theorem is that is allows us to prove the Mean Value Theorem (https://en.wikipedia.org/wiki/Mean_value_theorem) which later let's prove many many things including many properties of integrals. Which are quite useful to many fields.
Overall, I would argue that probably no one uses Rolle's Theorem in their work. Since Rolle's Theorem ends up being a specific form of the Mean Value Theorem, the Mean Value Theorem ends up being used much more often. It is much more likely that you will use applications of the Mean Value Theorem in your work and most of these theorems.
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