Math concepts & snippets

Friendly article-style notes that match what you see in the First Principles game and graph. For first principles thinking & business (reasoning from fundamentals vs analogy — often associated with Elon Musk’s public builder rhetoric, tied to this game’s curve / derivative metaphors), see that standalone page and the Math tips & snippets reader in-app.

For a dedicated engineering lens, see Engineering math — including § Aerospace / aerodynamics and matching in-game Aerospace: levels.

How this ties to the game

  • Gold / curve → the function (f(x)).
  • Derivative curve → (f’(x)) (slope). In many levels, where the derivative is “high enough,” you get platforms; otherwise, gaps / hazards.
  • Stages → story beats and visual “pops” on the derivative as you move right.
  • Math tips / concepts overlay → from Level select open “Math tips & snippets”, or during Game tap “Math concepts” (top-right) — same scrollable reader, with an opening section that explains how the visuals tie to f(x), f′(x), platforms, Riemann shading, and stages.

1. Derivative = slope and rate

In one sentence: the derivative measures how fast (f(x)) changes when (x) nudges forward.

  • On a graph, it is the slope of the tangent line.
  • In applications, it is often a rate: velocity (position vs time), growth rate, heat flow per unit length, etc.

Why the game cares: the engine uses the sign and size of the derivative to shape safe ground—so you literally walk on the calculus.

1b. Differentiation rules — power, product, quotient, chain (game playbook)

The Math tips / Math concepts reader includes a section “Differentiation rules — your skill tree”: same formulas as class, written with player / stage metaphors (buffs, dual meters, nested stages, quotient “lanes”). It’s the quick in-run version.

For the longer GitHub Pages strip with tables and mnemonics: Differentiation rules (playbook).

2. Parabolas (power / quadratic)

A quadratic like (y = a(x-h)^2 + k) is a single smooth bump or bowl. Classic uses:

  • Projectile motion (ideal intro model).
  • Optimization (max/min story problems).

The vertex is where the slope flattens to zero—a key calculus idea (critical point).

3. Sine, cosine, and waves

Sine and cosine describe repeating behavior: rotations, vibrations, AC signals.

  • They are the same wave with a phase shift.
  • Amplitude = how tall; frequency = how packed the waves are.

Game link: Waves of Sine and Shadows of Cosine.

4. Absolute value and corners

( x ) and ( f(x) ) can create kinks where the graph bends sharply. Derivatives aren’t defined at those ideal corners in the pure math sense—but numerically we still sample nearby slopes, which is why games and labs need care at nonsmooth points.

5. Taylor / Maclaurin series

Taylor: approximate a nice function near a point using a polynomial whose derivatives match. Maclaurin means “expand around 0.”

Intuition: low-degree terms capture local behavior; adding terms usually improves the fit near the expansion point (where the series converges).

Game link: Maclaurin: e^x, Maclaurin: sin(x).

6. Geometric series

Sums like (1 + u + u^2 + \cdots) appear everywhere: probability, DSP, economics models. When ( u < 1), the infinite sum converges to a neat closed form—the same “tail shrinks” mood as stability.

Game link: Series: geometric tail.

7. Multivariable “slices”

A surface (z = f(x,y)) can be cut by fixing (y = y_0). You then see a 1D curve in (x)—exactly how many engineers reduce dimension to reason about a bigger model.

Game link: Saddle slice, Paraboloid slice.

8. Integrals and Riemann sums

Definite vs indefinite (quick)

  • Indefinite (\int f(x)\,dx) → a family of antiderivatives (F(x)+C) with (F’=f) (“+ (C)” because derivatives kill constants).
  • Definite (\int_a^b f(x)\,dx) → a number: signed area from (a) to (b); no “+ (C)” in the final value.
  • FTC link: if (F’=f), then (\int_a^b f(x)\,dx = F(b)-F(a)).

In-app Metaphor block: “Integrals: definite vs indefinite — score vs loadout” in the Math reader. Longer table + notes: Definite vs indefinite integrals.

Riemann sums → definite integral

The definite integral (\int_a^b f(x)\,dx) is (for nice functions) the signed area under the curve from (a) to (b).

Riemann sum recipe:

  1. Split ([a,b]) into many small intervals of width (\Delta x).
  2. In each interval, pick a sample (x^*) (left end, right end, or midpoint).
  3. Add up (f(x^*)\,\Delta x).

More rectangles → usually closer to the true integral.

Game link: Area under the curve, Riemann: left / right / midpoint.

9. Exam preparation — separate guides

These are standalone prep pages (plus competition, AMC 10/12, and engineering below). Each uses official exam materials for real questions; the repo only provides unofficial topic maps and game cross-links.

Guide Audience
Competition math Contest lens — AMC/AIME-style bounding, concavity, (\ln) tricks (not affiliated with MAA)
AMC 10 & 12 — prep MAA AMC multiple-choice map (unofficial); calculus not required; algebra/geo/NT focus
TMUA — calculus UK TMUA — two-paper multiple choice; calculus fluency & elimination
MAT — calculus & reasoning UK MAT (Oxford, etc.) — careful reasoning & multi-step work
AP Calculus BC — prep US AP BC — series, polar/parametric/vector, DEs, FRQ/MC pairing
AP Physics C — prep US Physics C — calculus-first mechanics & E&M hooks

Used heavily in AP BC and in-game Polar: stages. Quick recap: (x=r\cos\theta), (y=r\sin\theta); area (\frac12\int r^2\,d\theta); arc length (\int \sqrt{r^2+(dr/d\theta)^2}\,d\theta). Full BC context: see AP Calculus BC — prep.

11. Where engineering math fits

Engineering math is still “calculus + algebra + models,” but the goal is systems you can build: circuits, structures, controls, signals. See the companion page:

→ Engineering math →

Economics “index chart” stages (qualitative)

Level select includes two stylized curves meant to evoke broad equity-index narratives: a late‑1990s / dot‑com run-up and unwind, and a 2007–09 global financial crisis arc (housing, credit, Lehman-era history-book framing). The graphs are smooth teaching splines, not downloaded ticker data. For contest/concavity notes with (\ln), see Competition math (still in docs; the old dedicated (\ln) contest stage was replaced by these economics moods in the catalog).

BOSS: Mandelbrot escape slice (final stage)

The Mandelbrot set is the set of complex numbers c for which (z \mapsto z^{2} + c) (starting from (z = 0)) stays bounded. The famous boundary is infinitely detailed; deep zooms are heavy.

In First Principles the boss stage plots a cheap 1D slice: fix Re(c) and let Im(c) vary along the graph’s horizontal parameter. Height encodes how many iterations occur before ** z > 2** (escape) or the iteration cap — a normalized escape-time picture.
Efficiency trick: the map is symmetric under c ↦ c̄ (reflect across the real axis): escape times match for c and its conjugate. The implementation iterates using ** Im(c) **, which is valid for that count and avoids redundant branch work when paired samples mirror each other.

This is a teaching view (moderate iteration cap, wide sample step), not a deep-zoom renderer.

In-game vs docs

Where What
Level select → Math tips & snippets Short TMP article (incl. differentiation rules + definite vs indefinite integrals) + first principles thinking & business + prep blocks (AMC, competition, TMUA, MAT, AP BC, AP Physics C).
This site (math-concepts.md) Game concepts + index to exam prep + derivative rules + definite vs indefinite integrals + first principles thinking & business + competition math + AMC 10 & 12 — prep.
amc-10-12.md, tmua-calculus.md, mat-calculus.md, ap-calculus-bc.md, ap-physics-c.md Unofficial standalone prep notes (not past papers).
engineering-math.md Damped motion, phasors, transforms, linear algebra hooks.

Aligned with the First Principles curriculum: primer through integrals, engineering stages, optional first principles business lens, and separate unofficial prep tracks for AMC 10/12, TMUA, MAT, AP Calculus BC, and AP Physics C.