CalcLens early prototype

Interactive demonstrations for Calculus 1 — limits, derivatives and integrals — built the same way as StatLens: no build step, keyboard-first, readable on a phone, and aimed squarely at the places students actually get stuck.

Cards without a link are sketched, not built. URL parameters for the working tools are frozen — see the URL API.

Integrals

Accumulation Function Slide x and watch A(x) = ∫ₐˣ f build itself. The height of f is the slope of A — the Fundamental Theorem, on screen. Worked Integral An antiderivative one rule at a time, with the substitution written out — which u, what du is, and what made it the one to try. The choice of u is the whole technique, so it is never skipped over.
Riemann Sums Left, right, midpoint and trapezoid rectangles with an n slider and a live error table.
u-Substitution, geometrically The picture behind the algebra Worked Integral writes out: watch the region itself transform when the variable changes, and see why du is not decoration.

Derivatives

Secant to Tangent Hold P still, slide Q in, and watch the difference quotient converge — with a table that approaches from either side. Works for any function, and strips down to a lecture figure. Derivative Builder Drag a point along f, drop each tangent slope onto a second axis, and watch f′ draw itself. Includes the secant-to-tangent limit. Sketch from Signs The sign charts are given; you pick one of four shapes per interval and the sketch assembles itself. Says so specifically when a shape and the given values disagree. Implicit Differentiation dy/dx for a curve that is not a function, four moves at a time. The one step students miss — that y is a function of x, so every y picks up a y′ — is named on the line where it happens, every time. Tilt and Bend The tangent gets the tilt right and cannot bend. Add one term and you have the best parabola, which gets both — and subtracting the tangent away leaves the bending on its own, which is the sharpest picture of f″ there is. Worked Derivative The derivative one rule at a time, with the rule named on the line that used it — so you can find the step where your own work went a different way, which is rarely the last one.
Local Linearity Zoom Zoom until the curve looks straight — with |x| at the origin as the control case that never does.
Match the Triple Given three graphs, decide which is f, which is f′ and which is f″.
Related Rates An animated ladder, cone or shadow with every rate read out live as the picture moves.

Limits

Sketch from Conditions Given one-sided limits, values and end behaviour, build a graph that has all of them. Nothing is compared to a model answer — there are infinitely many — so each condition is checked against what you drew and told back to you on its own. Derivative from the Definition The difference quotient worked out one move at a time — factor and cancel, common denominator, or the conjugate — with the margin saying which move was made and why cancelling h is allowed. Squeezing π Trap the circle between polygons drawn inside and outside it, as Archimedes did, then add sides. The same figure, read one wedge at a time, is where sin(x)/x → 1 comes from. Squeeze Theorem Three curves and a closing trap. Zoom in and watch the outer two pinch until g has nowhere to go — without ever evaluating g at the point.
Continuity Classifier Removable, jump, or infinite — sort discontinuities by what the one-sided limits do.

Tools

Grapher Up to three functions, with f′, f″, critical and inflection points, a draggable tangent, and asymptotes it refuses to draw through. Any spare letter becomes a slider; every graph is a URL you can embed. Function Evaluator Type a function and the values you want it at. Closes in on a point from both sides for a limit table, and says undefined where the function has no value rather than hiding the row. Check My Answer Type a derivative or antiderivative and find out if it is right — by the same differentiate-and-compare method you should use on paper.

About this prototype

CalcLens has its own small computer algebra system — about 1,700 lines in js/expr.js, with the step-by-step working in four modules beside it — rather than a general library. Calculus 1 needs a narrow catalogue of functions, but it needs the answers printed the way the course prints them: ln x rather than log(x), x³⁄3 rather than 0.3333x³. Owning the printer is the point, and it is also what lets a tool show the working rather than only the answer.

Student answers are checked by evaluating both sides at a spread of points, not by symbolic comparison. That never rejects a correct answer for being written in an unexpected form, and it hands back a specific counterexample when the answer is genuinely wrong.

The pages that show working — the derivative, the integral, the difference quotient, implicit differentiation — all obey one rule: only genuine equalities go after an =. Writing lim … = 0/0 is false, and it is the exact error students reproduce on exams, so an observation like “substituting gives 0/0” goes in the margin instead. The same reasoning puts + C on the line where the last integral sign disappears rather than at the end as an afterthought.

Where a technique is out of reach, the page names it — integration by parts, partial fractions, trigonometric substitution — and says it is not a mistake on the student's part. A tool honest about its edge is worth more than one that is vague about it.

Notes on the architecture, the supported syntax and what is still missing are in calclens/README.md. Page chrome, charts and maths rendering come from the shared kit/.