The Mean Value Theorem

Calculus 1 · Axiom Academy

Over any trip, there is an instant when your speed exactly equals your average speed — that is, some c where . 1. A Tangent That Matches the Average Take a function f that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) . The secant line through the endpoints has slope equal to the average rate of change, . The MVT promises at least one interior point c whose tangent is parallel to that secant — the instantaneous rate there equals the average. for some c strictly between a and b 2. Rolle's Theorem — the Level Case Suppose the endpoints sit at the same height , f(a) = f(b) . Then the secant is horizontal, so the MVT's guaranteed tangent must be horizontal too: there is a c with f'(c) = 0 — a peak or a valley. This special case is Rolle's Theorem , and it is the engine of the MVT's proof: tilt any MVT picture until the endpoints are level and you are looking at Rolle. Why both hypotheses are load-bearing Consider f(x) = |x| on [-1, 1] . It is continuous, and f(-1) = f(1) = 1 , so the secant slope is 0 — Rolle seems to promise a flat tangent. But f'(x) = -1 for and f'(x) = +1 for , and f'(0) does not exist . The slope is never 0 , so no such c exists . The single failure — not differentiable at the corner x=0 — is enough to kill the conclusion. (Continuity is equally required: a curve that jumps can dodge the guaranteed slope in the same way.) y = |x| : a corner at x = 0 means no tangent line there — and no horizontal tangent anywhere.

This is the written version of the interactive lesson above. See the full Calculus 1 course.