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Mean Value Theorem

Instantaneous rate matches the average · BC Calculus · greeting

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Owl · A subway train covers 2.4 km in exactly 3 minutes between two stations. Average speed: 0.8 km/min. Must the speedometer have read exactly 0.8 km/min at some moment in between? If the train's position function is continuous on the closed interval and differentiable on the open interval, the Mean Value Theorem says yes — some c where s'(c) equals the secant slope. MVT guarantees existence; it does not always hand you c on a silver platter. And without continuity (or differentiability), the promise can fail. Sketch the secant from start to stop, then imagine a tangent parallel to that secant. Where does that picture live in the hypotheses?

Owl: A subway train covers 2.4 km in exactly 3 minutes between two stations. Average speed: 0.8 km/min. Must the speedometer have read exactly 0.8 km/min at some moment in between? If the train's position function is continuous on the closed interval and differentiable on the open interval, the Mean Value Theorem says yes — some c where s'(c) equals the secant slope. MVT guarantees existence; it does not always hand you c on a silver platter. And without continuity (or differentiability), the promise can fail. Sketch the secant from start to stop, then imagine a tangent parallel to that secant. Where does that picture live in the hypotheses?

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