Calculus & Probability Lab — Drag the curve, see the derivative

The most useful kind lets you move something and see the math respond immediately — a tangent line that updates as you drag a point, or an integral that converges as you add rectangles — rather than showing a fixed, already-solved graph. Calculus & Probability Lab is built around that: 22 function families and true 3D z = f(x, y) surfaces you orbit and reshape, with a live tangent-line overlay and a Riemann sum overlay tied to 50 lessons across 4 worlds.

You pick a curve and a range, then slide a control from 1 up to 60 rectangles. At low rectangle counts the sum is visibly rough; as the count climbs, the total area of the rectangles converges on the exact definite integral given by the Fundamental Theorem of Calculus. Watching the convergence happen is the point — it connects a discrete approximation technique to the continuous answer it's estimating.

It's a function of two input variables, which produces a surface in three dimensions instead of a curve in two — a paraboloid, a saddle, a Gaussian hill, and similar shapes. Multivariable calculus concepts like partial derivatives and critical points are easier to build intuition for when you can orbit and reshape the actual surface, rather than only working through the algebra for a shape you've never seen.

The Probability Lab tosses physics-simulated dice repeatedly and builds a real empirical histogram from the results, overlaid against the exact theoretical distribution for that scenario. The Central Limit Theorem predicts that sums of independent random outcomes approach a normal distribution as trials increase — rolling enough times and watching the histogram visibly close in on that predicted bell curve turns an abstract theorem into something you watched happen.

Yes — enter an AP Calculus AB exam date and the app builds a day-by-day study plan counted back from it, alongside one AP-style free-response justification question per world that's graded point by point against a rubric by AI, and a 60-second warm-up drill weighted toward topics you've missed. AP is a trademark of the College Board, which is not affiliated with this app.

Tapping Explain on a surface, formula or dice distribution gets a plain-language breakdown tied to the specific numbers currently on screen — your coefficients, your slope at that point, your mean and standard deviation from your own dice rolls — rather than a generic definition pulled from a textbook.

The first 3 lessons of every world (12 of 50 total), 6 function families, 4 dice scenarios, a daily dice-roll allowance, a limited number of daily AI explanations and free-response gradings, and the study plan and warm-ups are free, with ads. Pro removes ads and unlocks all 50 lessons, all 22 function families, all 18 dice scenarios, unlimited rolls and AI explanations, and more daily free-response gradings.

Four worlds in order: Functions & Limits (function families, slope, limits, continuity), Derivative Lab (power, product, quotient and chain rules, partial derivatives, related rates, optimization, the Mean Value Theorem), Integral Lab (Riemann sums, the Fundamental Theorem of Calculus, area between curves, volumes of revolution), and Probability & Distributions (sample spaces, expected value, binomial and Poisson models, the normal curve).

Yes — World 1, Functions & Limits, doubles as a refresher covering what a function is, families of functions, slope, limits and exponentials and logs, before Derivative Lab and Integral Lab move into calculus proper. Because every lesson links to its matching 3D or probability lab, rebuilding the visual intuition doesn't require re-deriving every rule from scratch first.