What Does Rolle's Theorem Mean for Differentiable Functions on Closed Intervals?

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In summary: The statement of Rolle's theorem is: If a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b) and f(a)=f(b), then there exists a number c in (a,b) such that f'(c)=0. In summary, the 3 hypotheses of Rolle's theorem must be verified on the given interval [-1,1]. The conclusion of the theorem is that there exists at least one number c in (-1,1) such that the derivative of the function f(x) is equal to 0. To find all numbers c that satisfy this conclusion, we need to solve the equation f'(c)=0, which in this case is
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afcwestwarrior
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verify that the 3 hypothesis of rolle's theorem on the given interval . then find all the numbers c that satisfy the conclusion of rolle's theorem.

f(x)=sin2piex [-1,1]

i found the derivative cos 2pie x, but what do i do, and what does the theorem mean when f is differentiable on the open interval (a,b)
 
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what do i do
 
  • #3
First you go back and read the statement of Rolle's theorem? You are asked to "verify that the 3 hypothesis of rolle's theorem [are true] on the given interval". What are the 3 hypotheses? Being differentiable on the interior of the interval is one of them.

You are also asked to "find all the numbers c that satisfy the conclusion of rolle's theorem." Okay, what is the conclusion of Rolle's theorem?
 
  • #4
well all these are true, so how do u find all numbers c
 
  • #5
afcwestwarrior said:
well all these are true, so how do u find all numbers c

[itex]f'(c)=0[/itex]

10 chars...
 
  • #6
Again, what is the conclusion of Rolle's theorem?
 
  • #7
afcwestwarrior said:
verify that the 3 hypothesis of rolle's theorem on the given interval . then find all the numbers c that satisfy the conclusion of rolle's theorem.

f(x)=sin2piex [-1,1]

i found the derivative cos 2pie x, but what do i do, and what does the theorem mean when f is differentiable on the open interval (a,b)

No The derivative of [itex] \sin (2\pi x ) [/itex] is not that. Using the chain rule it is [itex]2\pi \cos (2\pi x)[/itex]
 

FAQ: What Does Rolle's Theorem Mean for Differentiable Functions on Closed Intervals?

What is Rolle's Theorem?

Rolle's Theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b), then there exists at least one point c in (a,b) where the derivative of the function is equal to 0.

How is Rolle's Theorem different from the Mean Value Theorem?

While Rolle's Theorem guarantees the existence of a point where the derivative is 0, the Mean Value Theorem guarantees the existence of a point where the derivative is equal to the slope between the endpoints. In other words, the Mean Value Theorem is a generalization of Rolle's Theorem.

What is the significance of Rolle's Theorem in calculus?

Rolle's Theorem is significant because it allows us to prove the existence of a critical point on a function without having to find it explicitly. It is also a key tool in proving other important theorems, such as the First Derivative Test for finding local extrema.

How do you apply Rolle's Theorem to solve a problem?

To apply Rolle's Theorem, you must first check that the given function satisfies the conditions of the theorem: it must be continuous on the closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b). Then, you can use the theorem to prove the existence of a point c in (a,b) where the derivative is equal to 0.

Can Rolle's Theorem be used to find the exact value of a point on a function?

No, Rolle's Theorem only guarantees the existence of a point where the derivative is 0. It does not provide a method for finding the exact value of this point. To find the exact value, you would need to use other techniques, such as finding the roots of the derivative function.

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