Home▸Mathematics▸Mean Value Theorem Explorer (2D)

Mean Value Theorem Explorer (2D)

Interactive 2D real-analysis simulator: drag the interval endpoints a and b on a differentiable curve, pan/zoom the graph, and watch an animated bisection search locate the guaranteed point c where the tangent slope f'(c) exactly matches the secant slope.

Mathematics2DModerate60 FPS📱 Mobile-adapted⇄ 3D version

Details

CategoryMathematics

RenderingCanvas 2D

DifficultyModerate

Target FPS60

ControlsPick one of three functions from a dropdown, drag the a and b sliders (or drag the orange dots directly on the graph) to set the interval, drag the c slider or the cyan dot to scrub the tangent point by hand, drag the graph background to pan and scroll/pinch to zoom, toggle Rolle's-theorem mode to auto-set b so f(b)=f(a), adjust the animation-speed slider, and click 'Animate bisection' to watch the residual panel converge on the theorem's guaranteed point.

real analysiscalculusmean value theoremderivativesmathematicsbisection method

Related simulations

Calculus in 3D: Tangent Planes & Solids of RevolutionInteractive 3D calculus lab: watch a tangent plane track the gradient across a surface…

The Derivative as a LimitInteractive 3D curve where dragging a secant line's second point closer to a fixed point…

Mathematics Calculus SimulationThis simulation demonstrates calculus concepts such as derivatives and integrals through… ⊞ All 323 in Mathematics →

A hands-on look at one of real analysis's cornerstone results: the Mean Value Theorem. Pick from three differentiable functions, drag the interval endpoints a and b directly on the graph, and watch the secant line connect its endpoints. A second panel plots the residual g(x) = f'(x) − secant slope and animates a bisection search — the same constructive technique behind the theorem's proof — hunting for the guaranteed point c where the tangent line is exactly parallel to the secant, with live numeric readouts for the secant slope, the exact derivative f'(c), and the gap between them.

⚙ Under the hood

Drag the endpoints of an interval directly on a 2D function graph and watch a second panel plot the residual g(x) = f'(x) minus the secant slope, animating a bisection search that locates the guaranteed point where the tangent runs exactly parallel to the secant — the constructive heart of the Mean Value Theorem. Pan and zoom the graph, drag a, b and c by hand, or flip on Rolle's-theorem mode.

real analysiscalculusmean value theoremderivativesmathematicsbisection method

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install