GEOMETRIC (chord):
f is convex on [a,b] iff for all λ∈[0,1]: f(λa+(1−λ)b) ≤ λf(a)+(1−λ)f(b) i.e. chord lies ON or ABOVE the curve
DIFFERENTIAL (2nd deriv):
f convex on I ↔ f″(x) ≥ 0 ∀x∈I
f concave on I ↔ f″(x) ≤ 0 ∀x∈I strictly convex if f″(x) > 0 everywhere
SUPPORTING LINE:
f convex ↔ tangent at every pt lies below or on the curve: f(x) ≥ f(c) + f′(c)(x−c) ∀x
CONCAVITY vs CONVEXITY: Concave up = convex (same thing) Concave down = concave (same thing) Convention differs by field!
🎵 SPECIAL f′(x) VALUE — triggers alarm tone
when f′(x) =
tolerance ±
When you click a point where f′(x) matches this value,
a special alarm tone + pink highlight triggers.
At f′=0 → lowest audible pitch. Higher f′ → higher pitch.
—
FREQUENCY MAP (continuous)
f′(x) → frequency (continuous, not fixed)
f′ = 0 → 110 Hz (A₂, audible bass)
f′ = + → rises up to ~1400 Hz
f′ = − → falls down to ~60 Hz
Waveform shape morphs with frequency:
low → sine | mid → triangle | high → sawtooth
⑬ ALL SOUNDS — f, f′, f″ SONIFICATION— choose a function and click a type —
x₀ (global)
1.00
wave shape
MASTER WAVEFORM
Chord: —
PITCH MAPPING — y=0 always = 220 Hz · frequency rises/falls with y value
y = 0 → 220 Hz (always audible base) y > 0 → pitch rises above 220 Hz y < 0 → pitch falls below 220 Hz Waveform morphs: low=sine · mid=triangle · high=sawtooth · very high=square
x² f′=2x → rises with x · f″=2 (const 440Hz+) x³ f′=3x² → always ≥0, low at x=0 sin x f′=cos x → oscillates ±1 around 220Hz 2x+1 f′=2 always → constant note (flat slope!)
Sweep range: x = −20 → +20
Each function sweeps its full behavior — steep functions show dramatic pitch changes. Set a special f′ value above to hear an alarm when the sweep hits that slope.
SWEEP PLAYER — hear f, f′, f″ sweep across x from left to right
sweep type
sweep speed
🔴 SPECIAL f(x) VALUE
f(x) =
±
→ 🔴 RED FLASH
—
🔵 SPECIAL f′(x) VALUE
f′(x) =
±
→ 🔵 CYAN FLASH
—
🟢 SPECIAL f″(x) VALUE
f″(x) =
±
→ ✦ GOLDEN FLASH
—
☀ Brightness
OFF
☀♪ Brightness + Audio
OFF
♪
— Hz
function
driven by
color
GRAPH — drag cursor to scrub x
x
0.00
LIVE VALUES
f(x) =
—
f′(x) =
—
f″(x) =
—
BRIGHTNESS
0%
mapped from normalised value of selected derivative
How it works:
Toggle ON → screen flashes to show the chosen derivative's value at x. Bright = large positive. Dark = near zero. Medium = negative (inverted).
Drag x-slider to scrub. Try f″ on sin x — watch brightness pulse!
AUTO SWEEP — brightness animates as x sweeps left → right
speed
sweep type
—
⑰ CONVERGENCE RACE — same Δx applied to both curves simultaneously
Ax₀1.00
vs
Bx₀1.00
Δx (shared)3.000
Quick presets:
💡 The Journey to dx:
For linear functions (2x+1), the secant line is always the tangent — zero error at any Δx. The journey is instant.
For x·sin(1/x), the curve oscillates so wildly near x₀ that the secant keeps changing direction — it needs an extremely tiny Δx before it settles.
This is the maximum possible gap between Δx and dx.
🦴WHY INTEGRATION MUST SUBTRACT — CAVEMAN UNDERSTAND
Curve:
a
-1.0
b
2.0
F(b) alone
—
F(a) junk
—
F(b)−F(a) ✓
—
∫ₐᵇ f dt
—
F(b)−F(a) = ∫ₐᵇ f dt ✓
error: —
🦣 UGH. CAVE-MAN EXPLAIN WHY SUBTRACT.
1
CAVE-MAN FIND ANTIDERIVATIVE F(x).
F(x) like bucket. It hold accumulated stuff since beginning of time (x=−∞).
F(x) not know where cave-man want to start counting. F always carry old junk from before.
2
∫ₐᵇ f(t) dt = AREA BETWEEN a AND b ONLY.
Cave-man say: "UGH! Me not care about stuff before a. Me only care about area from a to b."
But F(b) has junk from before a inside it. F(a) is exactly the same junk.
Same junk! So cave-man SUBTRACT F(a) to cancel junk.
3
F(b) − F(a) = JUNK CANCEL. ONLY GOOD AREA REMAIN. F(b) = stuff up to b ·
F(a) = same old junk up to a ·
F(b)−F(a) = only the area between a and b.
Pure! No junk! This why we always subtract.
🪨 THE ROCK-SOLID RULE: ∫ₐᵇ f(t) dt = F(b) − F(a) It not matter which antiderivative F you pick.
All antiderivatives differ only by constant C.
→ [F(b)+C] − [F(a)+C] = F(b) − F(a) ← C vanish! POOF! Cave-man very happy. Constant C always die in subtraction. 🦴
SIMPLE CAVE-BRAIN ANALOGY:
You start at position F(a) on number line. You walk to position F(b). Distance walked = F(b) − F(a).
F(b) alone tell cave-man WHERE he end up, not HOW FAR he walk. Subtraction give distance. Area = distance F travel.
∫ₐᵇ f dt = F walks from a to b = F(b) − F(a). ALWAYS SUBTRACT. NO EXCEPTION. CAVE-MAN LAW.
Stage 1 — Accumulating area: Drag the x slider left/right.
The gold shaded region is the definite integral ∫₀ˣ f(t)dt.
Notice: when f(x) is positive, A(x) increases; when negative, it decreases.
The slope of A(x) at any x equals f(x) — that's the Fundamental Theorem.