Mean Value Theorem Explorer (2D)

Function & Interval

Left endpoint a -4.0

Right endpoint b 3.0

Point c (drag manually) -0.5

Rolle's theorem mode (auto-set b so f(b) = f(a))

Bisection Search

Animation speed 120 ms/step

Adjust a, b, or animate the search.

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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Bisection step

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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.
Bottom panel plots g(x) = f'(x) − secant slope over [a,b]. Its root is exactly the guaranteed point c — the theorem is really a claim that g must cross zero somewhere on the interval (an application of Rolle's theorem to h(x) = f(x) − [secant line]).
Animate bisection brackets a sign change of g and repeatedly halves it, shading the shrinking interval [lo, hi] on the residual panel — the same constructive argument behind the theorem's proof.
Drag c yourself (slider or directly on the curve) to see the tangent line rotate independently of the secant — away from the guaranteed point the two lines generally are not parallel.
Rolle's theorem mode auto-searches for a b with f(b) = f(a), the special case where the guaranteed point has a perfectly horizontal tangent.

Drag the graph to pan, scroll/pinch to zoom, and drag the orange/cyan dots directly to move a, b, or c. This underlies the proof of the Fundamental Theorem of Calculus and error bounds in numerical methods.

Curve f(x) Secant line Tangent at c

drag to pan · scroll to zoom

g(x) = f'(x) − secant slope