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What is the Mean Value Theorem (MVT)?

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The Mean Value Theorem states that if a function f is continuous on the 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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What are the conditions required for the Mean Value Theorem to apply?

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The function must be continuous on a closed interval [a, b] and differentiable on the open interval (a, b).

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How do you find the value of c in the Mean Value Theorem?

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To find c, you need to calculate the average rate of change of the function over the interval [a, b] and then solve the equation f'(c) = (f(b) - f(a)) / (b - a).

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What is the Extreme Value Theorem?

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The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], then it attains both a maximum and a minimum value at some points in that interval.

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What is a discontinuous function?

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A discontinuous function is one that is not continuous at one or more points in its domain, meaning there are breaks, jumps, or asymptotes in the graph.

Tags

CCSS.HSF-IF.C.7D

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Give an example of a function that is discontinuous on the interval [-1, 1].

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An example is y = log(x), which is undefined for x ≤ 0, making it discontinuous on the interval [-1, 1].

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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 over that interval.

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How do you verify the Mean Value Theorem for a given function?

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To verify the MVT, check that the function is continuous on the closed interval and differentiable on the open interval, then find c such that f'(c) equals the average rate of change.

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What is the significance of the value c in the Mean Value Theorem?

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The value c represents a point in the interval where the instantaneous rate of change (slope of the tangent) equals the average rate of change over the interval.

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What is the derivative of a function?

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The derivative of a function measures the rate at which the function's value changes as its input changes, representing the slope of the tangent line to the graph of the function.