Mean Value Theorem

If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then at least one interior point c must have tangent slope equal to the secant slope between the endpoints. Geometrically, the graph cannot go from one endpoint to the other without somewhere matching the overall rise-over-run, because the derivative measures instantaneous change while the fraction (f(b)-f(a))/(b-a) measures average change. This result is a precise bridge between local behavior and global behavior, and it is proved by applying Rolle’s Theorem to a suitably adjusted function.

Use this when you know a function’s values at two endpoints and want to guarantee an interior point where the instantaneous rate equals the average rate. It is especially useful in proving existence statements, estimating motion/velocity, and showing that a function must have at least one critical point with a specified slope.

• Forgetting the function must be continuous on [a, b] and differentiable on (a, b)
• Using endpoint values c = a or c = b, even though c must lie strictly inside the interval
• Confusing the average slope (secant line) with the instantaneous slope (tangent line)