Rolle's Theorem & mean value theorem

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What is Rolle's Theorem?

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Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and F(a) = F(b), then there exists at least one c in (a, b) such that F'(c) = 0.

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What are the conditions for applying Rolle's Theorem?

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1. The function must be continuous on the closed interval [a, b]. 2. The function must be differentiable on the open interval (a, b). 3. The function values at the endpoints must be equal, i.e., F(a) = F(b).

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What is the Mean Value Theorem?

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The Mean Value Theorem states that 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 c in (a, b) such that F'(c) = (F(b) - F(a)) / (b - a).

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How does Rolle's Theorem relate to the Mean Value Theorem?

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Rolle's Theorem is a special case of the Mean Value Theorem where F(a) = F(b). In this case, the average rate of change is zero, leading to at least one point where the derivative is also zero.

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What does it mean for a function to be continuous on an interval?

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A function is continuous on an interval if there are no breaks, jumps, or holes in the graph of the function within that interval.

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What does it mean for a function to be differentiable on an interval?

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A function is differentiable on an interval if it has a derivative at every point in that interval, meaning it has a defined slope at every point.

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Provide an example of a function that satisfies the conditions of Rolle's Theorem.

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An example is F(x) = x^2 - 4 on the interval [2, 2]. Here, F(2) = F(2) = 0, and the function is continuous and differentiable.

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What is the significance of the value 'c' in Rolle's Theorem?

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The value 'c' represents a point in the interval (a, b) where the instantaneous rate of change (the derivative) of the function is zero, indicating a horizontal tangent.

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How do you find the value of 'c' in Rolle's Theorem?

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To find 'c', you first find the derivative of the function, set it equal to zero, and solve for 'c' within the interval (a, b).

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What is the derivative of the function f(x) = x^2 - 5x + 4?

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The derivative f'(x) = 2x - 5.