Mean Value Theorems — Flashcards

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Step 1: Check continuity on [0, π]. - f(x) = tan x is not defined at x = π/2. - So the function is not continuous on the full interval. Step 2: Rolle's theorem needs continuity on [a, b], differentiab…

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Step 1: End values. - f(-1) = 1 and f(1) = 1. - So the endpoint condition is satisfied. Step 2: Check differentiability on (-1, 1). - At x = 0, the right derivative is 1 and the left derivative is -1.

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Step 1: Check continuity. - At x = 6, the left and right limits are different. - So the function is not continuous at x = 6. Step 2: Rolle's theorem needs continuity on [a, b]. Step 3: Since continuit…

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Geometrically, f'(x) is the slope of the tangent to the curve y = f(x). If f'(c) = 0, the tangent at x = c is parallel to the x-axis. So Rolle's theorem guarantees at least one point where the curve h…

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If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c ∈ (a, b) such that f'(c) = 0. Meaning of symbols: - a, b: end points - c: point inside the interval - f'(c…

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Step 1: Check conditions. - Polynomial functions are continuous and differentiable everywhere. - f(-√3) = (-√3)^3 - 3(-√3) = -3√3 + 3√3 = 0. - f(0) = 0. Step 2: Since f(-√3) = f(0), Rolle's theorem ap…

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Step 1: Check conditions. - sin x is continuous and differentiable on the interval. - f(0) = 0 and f(π) = 0. Step 2: Rolle's theorem applies. Step 3: Differentiate. - f'(x) = cos x Step 4: Set f'(c) =…

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Step 1: Let f(x) = cos x - 1. - f(0) = 0 and f(2π) = 0. - So Rolle's theorem applies. Step 2: Differentiate. - f'(x) = -sin x Step 3: Set f'(c) = 0. - -sin c = 0 - c = π Step 4: Find the point. - f(π)…