State Rolle's Theorem.

Rolle's Theorem and Mean Value Theorem

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Mathematics
•
12th Grade
•
Hard
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1.
FLASHCARD QUESTION
Front
Back
Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and the function values at the endpoints are equal, then there exists at least one point c in the open interval (a, b) where the derivative of the function is zero.
2.
FLASHCARD QUESTION
Front
What is the difference between Mean Value Theorem and Rolle's Theorem?
Back
Mean Value Theorem states that there exists at least one point c in (a, b) where the derivative of the function is equal to the average rate of change of the function over the interval [a, b], while Rolle's Theorem states that there exists at least one point c in (a, b) where the derivative of the function is zero.
3.
FLASHCARD QUESTION
Front
Find the value of c that satisfies the Mean Value Theorem for the function f(x) = sin(x) on the interval [0, π/2].
Back
π/2
4.
FLASHCARD QUESTION
Front
State the Mean Value Theorem.
Back
If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in the interval (a, b) where the instantaneous rate of change (derivative) of the function is equal to the average rate of change of the function over the interval [a, b].
5.
FLASHCARD QUESTION
Front
What is the significance of the Mean Value Theorem?
Back
The Mean Value Theorem states that there exists at least one point in an interval where the derivative of a function is equal to the average rate of change of the function over that interval.
6.
FLASHCARD QUESTION
Front
Define continuity in the context of functions.
Back
A function is continuous on an interval if there are no breaks, jumps, or holes in the graph of the function on that interval.
7.
FLASHCARD QUESTION
Front
What does differentiable mean?
Back
A function is differentiable at a point if it has a defined derivative at that point, meaning the function has a tangent line at that point.
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