Mean Value Theorem Explorer

Function & Interval

Left endpoint a -4.0

Right endpoint b 3.0

Point c (drag manually) -0.5

Adjust a, b, or search for c.

Live Values

Secant slope (f(b)−f(a))/(b−a)

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Tangent slope f'(c)

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|f'(c) − secant slope|

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How it works

The Mean Value Theorem (Lagrange) says: if f is continuous on [a,b] and differentiable on (a,b), then there exists at least one point c ∈ (a,b) where the instantaneous rate of change equals the average rate of change over the whole interval:

f'(c) = (f(b) − f(a)) / (b − a)

Secant line (orange) — the straight chord connecting (a, f(a)) and (b, f(b)); its slope is the average rate of change.
Tangent line (cyan) — drawn at the current point c; its slope is f'(c), computed from the exact derivative of the chosen function.
Find c by bisection — scans (a,b) for a sign change of g(x) = f'(x) − secant slope, then bisects to locate a root to high precision, exactly the constructive argument behind the theorem's proof (apply Rolle's theorem to h(x) = f(x) − [secant line]).
Drag c yourself to see the tangent line rotate independently of the secant — the theorem guarantees at least one c makes the two lines parallel, but away from that point they generally are not.

This is the same principle behind Rolle's theorem (the special case f(a) = f(b), where the guaranteed point has a horizontal tangent) and underlies the proof of the Fundamental Theorem of Calculus and error bounds in numerical methods.

Curve f(x) Secant line Tangent at c

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