Interactive lessons

Calculus I
limits & derivatives

Ten interactive lessons building from intuition first. Start with the physical feeling of a rate of change, then let the formalism arrive naturally. Each lesson has a draggable canvas — no passive reading.

Limits & Continuity

— 3 lessons
x → c L lim x→c f(x) ▶ drag
Limits — 01
The Idea of a Limit
Slide a point toward a hole from both sides and watch where the curve is headed. Left-hand, right-hand, and two-sided limits — all before a formal definition.
interactive · 2D · draggable→
L+ε L−ε ε-δ def ▶ sliders
Limits — 02
The ε-δ Definition
Tighten the epsilon band and challenge yourself to find a delta window that keeps f(x) inside. The formal definition becomes a visual game.
interactive · 2D · sliders→
jump hole continuous IVT
Limits — 03
Continuity & Discontinuities
Toggle between continuous curves, removable holes, and jump breaks. Close with the Intermediate Value Theorem: drag a target height and the continuous curve must cross it.
interactive · 2D · toggle→

The Derivative

— 4 lessons
s(t) graph ▶ velocity
Derivative — 04
Position & Velocity
A ball rolls on a track above a position-time graph. Shrink the time interval until the average velocity collapses to the instantaneous velocity — the derivative as physical intuition first.
interactive · 2D · animated→
f(x) f'(x) dual panel
Derivative — 05
The Slope Connection
Two panels, always in sync. Drag a probe on f(x) and watch f′(x) trace out on the right. Zeros of f′ land exactly at the peaks and troughs of f.
interactive · 2D · dual-panel→
Power Rule d/dx xⁿ = nxⁿ⁻¹ Product Rule (fg)' = f'g + fg' Chain Rule dy/dx = dy/du·du/dx Sum Rule (f+g)' = f' + g' step-through
Derivative — 06
Differentiation Rules
Four rules, four stages. Choose a power and watch the formula update live. The numerical check confirms each rule to machine precision — no faith required.
interactive · step-through · verifiable→
x² + y² = 4 dy/dx = −x/y ▶ drag 4 curves
Derivative — 07
Implicit Differentiation
Drag a point around a circle, ellipse, folium, and lemniscate. The tangent updates from the implicitly differentiated slope — no need to solve for y first.
interactive · 2D · 4 curves→

Applications of the Derivative

— 3 lessons
local max local min infl. f' & f'' concavity
Applications — 08
Curve Sketching
Toggle f′ and f″ overlays on a quartic. Watch critical points, inflection points, and concavity shading emerge directly from sign changes in the derivatives.
interactive · 2D · toggle layers→
w − 2x h − 2x x V(x) = x(w−2x)² max volume ▶ slider
Applications — 09
Optimization
Cut corners from a flat sheet and fold an open box. Drag the slider to find the cut size that maximises volume — then confirm with V′(x) = 0 and snap to the exact answer.
interactive · 2D · slider→
sin(x)/x = 1 0/0 L'Hôpital
Applications — 10
L'Hôpital's Rule
Four canonical 0/0 indeterminate forms. Drag the probe toward the problem point and watch the ratio converge. Zoom in on sin(x)/x to see why the rule works geometrically.
interactive · 2D · zoom→