Mean Value Theorem

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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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What is the average rate of change of a function?

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The average rate of change of a function f(x) over an interval [a, b] is given by (f(b) - f(a)) / (b - a).

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How do you find the derivative of a function?

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To find the derivative of a function, apply the rules of differentiation (power rule, product rule, quotient rule, etc.) to determine the rate of change of the function with respect to its variable.

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What is the power rule for differentiation?

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The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).

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What is the product rule for differentiation?

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The product rule states that if u(x) and v(x) are two functions, then the derivative of their product is given by (uv)' = u'v + uv'.

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What is the quotient rule for differentiation?

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The quotient rule states that if u(x) and v(x) are two functions, then the derivative of their quotient is given by (u/v)' = (u'v - uv') / v^2.

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How do you determine if a function is continuous?

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A function is continuous at a point x = c if the following three conditions are met: 1) f(c) is defined, 2) lim (x -> c) f(x) exists, and 3) lim (x -> c) f(x) = f(c).

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What does it mean for a function to be differentiable?

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A function is differentiable at a point if it has a derivative at that point, which means the function is smooth and has no sharp corners or vertical tangents.

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

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

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How do you apply the Mean Value Theorem?

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To apply the Mean Value Theorem, verify that the function is continuous and differentiable on the given interval, then calculate the average rate of change and find the point(s) where the derivative equals this rate.