The Mean Value Theorem

Real Analysis · Axiom Academy

Somewhere on the curve, the instantaneous rate of change must equal the average rate — the tangent runs parallel to the secant. Suppose f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) . The MVT promises a point c strictly between a and b where the derivative equals the slope of the secant line. Worked example. Take f(x) = x^3 on [0,3] . The secant climbs from f(0)=0 to f(3)=27 , so its slope is the average rate Since f'(x) = 3x^2 , the MVT point solves 3c^2 = 9 , i.e. (we reject , which lies outside (0,3) ): 2. Rolle's Theorem — the Flat Special Case The MVT is the slanted version of an older, simpler fact. Rolle's theorem is exactly the MVT when the two endpoints sit at the same height , so the secant is horizontal and the guaranteed tangent is flat. Geometrically: if a differentiable curve leaves and returns to the same height, it must turn around somewhere in between — and at that turning point the tangent is horizontal. For f(x) = x^3 - 3x on we have . Then f'(x) = 3x^2 - 3 = 0 gives — two interior points with a horizontal tangent. Here is the geometric heart of the MVT. Fix the secant through and . Now slide a tangent along the curve. The MVT says its slope must match the secant's at least once — and it can happen more than once. The average rate — a single fixed number for the interval. The instantaneous rate f'(x) — it changes as the point slides along the curve.

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