Mean Value Theorem Calculator

How it works

If f is continuous on [a, b] and differentiable on (a, b), there is a c in (a, b) with f′(c) = [f(b) − f(a)] / (b − a) · solve f′(c) − m = 0 and keep the solutions strictly between a and b · Rolle’s theorem: f(a) = f(b) ⇒ f′(c) = 0

The Mean Value Theorem says that somewhere inside the interval the instantaneous rate of change equals the average rate of change: at least one tangent line is parallel to the secant line joining the endpoints. Polynomials are continuous and differentiable everywhere, so the theorem always applies. The calculator computes the secant slope exactly, then solves f′(c) = slope and keeps only the roots strictly inside (a, b) (OpenStax Calculus Vol. 1 §4.4). If f(a) = f(b) the slope is 0 — that special case is Rolle’s theorem.

Worked example

f(x) = x³ − 3x + 2 on [0, 2]: f(0) = 2 and f(2) = 4, so the average rate of change is (4 − 2)/(2 − 0) = 1. f′(x) = 3x² − 3, and 3c² − 3 = 1 gives c² = 4/3, c = ±1.154701. Only c = 1.154701 lies in (0, 2). The secant line is y = x + 2 and the tangent at c is parallel to it.

Sources: OpenStax Calculus Vol. 1 — 4.4 The Mean Value Theorem · Paul’s Online Math Notes — The Mean Value Theorem

Frequently asked questions