Skip to content
LearnMathora

No grades. No wrong moves.

The playground

Eight stations, each one calculus idea made touchable. Drag things. Break nothing. Intuition built here makes every lesson land harder.

Station 01

Ride the wave

Slope is a living thing: positive on climbs, zero at crests, negative on descents.

InteractiveTangent explorer
0
At x = 0, the curve is changing at a rate of 1 — the slope of the tangent line.

Station 02

Shrink the gap

The derivative being born: averages over a closing gap settling on the instant.

InteractiveFrom average to instant
1.6
Average rate between the two points: 3.6. As the gap shrinks toward 0, it closes in on 2 — the instant rate at x = 1.

Station 03

Slice and stack

Riemann sums in your hands — rectangles melting into exact area.

InteractiveArea accumulator
4
8
Distance covered from t = 0 to 4: 9.06 — the area under the speed curve. More slices hug the curve more closely.

Station 04

Function vs. derivative

Two graphs, one truth: the bottom one's height is the top one's slope.

InteractiveA function and its derivative

The function

Its derivative — the slope, graphed

0.6
Slope of the top curve at x = 0.6 is 1.2 — and that is exactly the height of the bottom curve there.

Station 05

The moving cart

Position, velocity, acceleration — one slider, three linked stories.

InteractivePosition, velocity, acceleration

Position s(t) — where it is

Velocity v(t) = s′(t) — how fast

Acceleration a(t) = v′(t) — how the speed changes

0.6
At t = 0.6 the cart is moving forward — velocity 0.96, acceleration -2.8. Each graph is the slope of the one above it.

Station 06

Find the peak

Optimization by feel: walk uphill until the slope reads zero.

InteractiveOptimization: the best box
max 16 at x = 1
0.35
Cut squares of x = 0.35 from a 6×6 sheet → volume 9.8. Slope: 20.7. Uphill — a bigger cut still helps.

Station 07

The +C family

Infinitely many originals, one shared rate — why antiderivatives come with a constant.

InteractiveWhy +C exists
1
1
Every curve x² + C has slope 2 at x = 1 — the derivative can't tell them apart, so the antiderivative must carry a +C.

Station 08

The grand connection

Watch accumulation grow at exactly the rate of the curve above — the fundamental theorem, live.

InteractiveThe fundamental theorem, watched live

A rate f(t) — area shading in

Total so far A(x) — the area, graphed

3
Accumulated area so far: 6.49. The bottom curve grows at rate 1.64 — the exact height of the top curve. Accumulating undoes rate-taking.