AP Calculus AB and BC glossary
Mean Value Theorem
Also called: MVT
The Mean Value Theorem says that if a function is continuous on a closed interval and differentiable on the open interval, then somewhere inside there is a point where the instantaneous rate of change equals the average rate of change across the whole interval.
Geometrically, some tangent line is parallel to the secant line joining the endpoints. In a motion context it says that if you average 60 miles per hour over a trip, at some instant your speedometer read exactly 60.
The two hypotheses differ deliberately: continuity is required on the closed interval including endpoints, while differentiability is only required on the open interior. Stating both is required for credit.
The mistake
Applying it to a function with a corner inside the interval. The absolute value function on has average rate zero but no point where the derivative is zero, because it is not differentiable at the origin.
Appears in: Unit 5: Analytical Applications