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Curriculum & Standards · Calculus (Intro)

Calculus (Intro) — the formal single-variable calculus course, calibrated to AP Calc BC + IB AA HL + Japan Math III + Korea Cálculo + freshman university.

From the ε-δ definition of limit to Taylor series with Lagrange remainder. ε-δ limits and the squeeze theorem, the five classical continuity theorems (IVT formal, EVT, Bolzano, MVT, Rolle, Darboux), the complete differential calculus (all rules + implicit + logarithmic + transcendentals + higher-order), curve sketching and modeling (optimization, related rates, Newton-Raphson, L'Hôpital formal), antiderivatives with u-substitution, integration by parts, partial fractions in three forms, trig substitution, reduction formulas, improper integrals, the two parts of the Fundamental Theorem of Calculus, applications (volumes by disks / washers / shells, arc length, surface of revolution, work), separable ODEs (exponential growth, Newton's cooling, logistic, slope fields), formal sequences and series with eleven convergence tests, absolute vs conditional convergence, power series with radius and interval, and Taylor and Maclaurin series with Lagrange remainder + classical series + binomial series — aligned to the union of AP Calculus AB+BC, IB Mathematics AA HL, Cambridge A2 9709 Pure 3, Singapore JC2 H2, Japan MEXT Math III, Korea KICE 2022 Revised Calculus, and freshman university calculus (MIT 18.01, UC Berkeley Math 1A-1B, Stanford MAT 19/20, UNAM Cálculo I-II).

94

Skills

13

Modules (hex)

9

Official standards benchmarked

17–19

Age coverage

Why Calculus (Intro) is the gate to a STEM degree

Formal calculus is the prerequisite for every quantitative degree at university: physics, engineering, computer science, economics, statistics, biomedical research, finance. The ε-δ definition of limit, the FTC and convergence tests are not optional — they are the language in which higher mathematics is written. Japan Math III, Korea KICE Calculus 2022, MIT 18.01 and UC Berkeley Math 1A all formalize at this level; AP Calculus BC and IB AA HL are the HS proxies. Rodybee tracks the union of all of them.

#CountryTIMSS Advanced score
1🇸🇬Singapur605
2🇹🇼Chinese Taipei597
3🇰🇷Corea596
4🇯🇵Japón595

Source: TIMSS Advanced 2015 Final Year Math ↗

Find your child's level

Pick an age and see exactly what we teach at that point — and what the most demanding standards in the world expect at the same age. There's no marketing fluff: each topic links to the official curriculum document.

Your child's age

Rodybee for your child · 7 años

What we teach at this age

No skills calibrated for this exact age yet — try an adjacent year.

Highest international bar at this age · G12 / JC2 Y2 / Y13 / Math III

🇯🇵Japón

ε-δ definition of limit (the salto from intuitive to formal), squeeze theorem, classical continuity theorems (Bolzano, IVT formal, EVT, MVT, Rolle, Darboux, inverse function continuity), derivative as a limit with all standard rules (power, product, quotient, chain, implicit, logarithmic), derivatives of all transcendentals (six trig, inverse trig, exp, log, hyperbolic, higher-order), and full curve analysis via the first and second derivative tests + curve sketching complete + MVT/Rolle, modeling with derivative (optimization with constraints, related rates, Newton-Raphson, linear approximation, L'Hôpital formal, differentials).

Grade-by-grade benchmark

Full table of what each major standard expects per grade. Every cell is sourced from the official ministry, board or framework document.

G12 / JC2 Y2 / Y13 / Math III

First third — formal limits + classical theorems + complete derivative (~17–18 años)

Highest bar · 🇯🇵 Japón

ε-δ definition of limit (the salto from intuitive to formal), squeeze theorem, classical continuity theorems (Bolzano, IVT formal, EVT, MVT, Rolle, Darboux, inverse function continuity), derivative as a limit with all standard rules (power, product, quotient, chain, implicit, logarithmic), derivatives of all transcendentals (six trig, inverse trig, exp, log, hyperbolic, higher-order), and full curve analysis via the first and second derivative tests + curve sketching complete + MVT/Rolle, modeling with derivative (optimization with constraints, related rates, Newton-Raphson, linear approximation, L'Hôpital formal, differentials).

CountryExpected topicsOfficial source
🇯🇵JapónMEXT Math III (微分積分): definición ε-δ de límite explícita, derivada con todas las reglas, derivada implícita, MVT y Rolle formales, optimización y related rates.MEXT Senior High School Math III (Japan) ↗
🇰🇷CoreaKICE 2022 Revised — Calculus (미적분): definición formal de límite, continuidad formal, derivada completa con reglas, MVT (평균값 정리), análisis completo de funciones.KICE 2022 Revised Curriculum — Calculus (미적분) ↗
🇸🇬SingapurJC2 H2 9758: límites rigurosos (no ε-δ explícito pero formal), derivada con todas las reglas + implícita, MVT, optimización, related rates.Singapore JC2 H2 Mathematics 9758 (SEAB) ↗
🇺🇸Estados UnidosAP Calculus AB+BC: límites intuitivos, continuidad, derivada con todas las reglas, IVT/MVT/EVT/Rolle, optimización, related rates, L'Hôpital, Newton's method.AP Calculus AB & BC (College Board CED) ↗
🌐IB AA HLTopic 5: límites rigurosos, derivada completa con reglas, implícita, derivadas de orden superior, MVT, análisis de funciones completo.IB Diploma Mathematics: Analysis and Approaches HL ↗
🌐Cambridge A2 9709 P3Pure 3: derivadas paramétricas/implícitas, MVT, Newton-Raphson formal, aplicaciones a optimización.Cambridge International A Level Mathematics 9709 (Pure 3) ↗

Rodybee skills at this grade

ε-δ definition of limit · Limit existence proof (ε-δ) · Proof of limit properties · Squeeze (sandwich) theorem · Advanced continuity theorems · Uniform continuity (intro) · Bolzano's theorem · IVT formal with proof · Extreme Value Theorem (Weierstrass) · MVT for continuous functions · Darboux's property · Inverse function continuity theorem · Derivative — formal definition (full) · Power rule (real n) · Sum/difference rule · Product rule · Quotient rule · Chain rule · Implicit differentiation · Logarithmic differentiation · Derivatives of all 6 trig functions · Derivatives of inverse trig · Derivative of e^x and a^x · Derivative of ln x and log_a x · Derivatives of sinh, cosh, tanh · Higher-order derivatives · Derivative of an inverse function · Monotonicity from f' · Local vs global extrema · First derivative test classification · Concavity from f'' · Inflection points classification · Second derivative test · Complete curve sketching · Mean Value Theorem (full) · Rolle's theorem · Constrained optimization · Related rates · Linear approximation · Newton-Raphson method · L'Hôpital's rule (formal) · Estimation with differentials

G12 / JC2 Y2 / Math III / Univ Year 1

Second third — antiderivada + FTC + integral definida + técnicas avanzadas (~17–18 años)

Highest bar · 🇸🇬 Singapur

Antiderivada e integral indefinida (formulas básicas + transcendentales + tabla), u-substitution e integración por partes (LIATE/ILATE), suma de Riemann formal y definición rigurosa de integral, propiedades de la integral definida, las dos partes del Teorema Fundamental del Cálculo, valor promedio y MVT para integrales, áreas con signo. Técnicas avanzadas: fracciones parciales (lineales distintos, lineales repetidos, cuadráticas irreducibles), sustitución trigonométrica (3 casos), potencias trigonométricas, fórmulas de reducción, integrales impropias tipos I y II.

CountryExpected topicsOfficial source
🇸🇬SingapurJC2 H2: antiderivadas + FTC + técnicas completas (sustitución, partes, fracciones parciales), integrales impropias.Singapore JC2 H2 Mathematics 9758 (SEAB) ↗
🇯🇵JapónMath III: integral de Riemann formal + FTC partes I y II + técnicas (u-sub, partes, sustitución trig), integrales impropias.MEXT Senior High School Math III (Japan) ↗
🇰🇷CoreaCalculus 2022 Revised: 정적분 (definida) + 부정적분 (indefinida) + 부분적분 (partes) + 치환적분 (sustitución), 무한적분 (impropias).KICE 2022 Revised Curriculum — Calculus (미적분) ↗
🇺🇸Estados UnidosAP Calc BC: u-sub, partes, fracciones parciales, sustitución trig, potencias trig, integrales impropias.AP Calculus AB & BC (College Board CED) ↗
🌐IB AA HLTopic 5: FTC, técnicas completas, integración por partes con LIATE.IB Diploma Mathematics: Analysis and Approaches HL ↗
🌐Cambridge A2 9709 P3Pure 3: técnicas (sustitución, partes, fracciones parciales), incl. integrales impropias.Cambridge International A Level Mathematics 9709 (Pure 3) ↗
🇺🇸Estados UnidosMIT 18.01 / Berkeley Math 1A / Stanford MAT 19: cobertura formal completa equivalente.MIT OCW 18.01 Single Variable Calculus ↗

Rodybee skills at this grade

Antiderivative definition · Basic antiderivative formulas · Antiderivatives of transcendentals · u-substitution · Integration by parts · Using an integral table · Riemann integral (formal) · Riemann sum → definite integral · Properties of definite integral · FTC Part I (g(x) = ∫_a^x f) · FTC Part II (∫_a^b f = F(b) − F(a)) · Average value of a function · Signed area (full) · Partial fractions — distinct linear · Partial fractions — repeated linear · Partial fractions — irreducible quadratic · Trigonometric substitution · Powers of trig (sin^m cos^n) · Reduction formulas · Improper integral type I (∞ limits) · Improper integral type II (asymptote)

G12 / Univ Year 1

Final third — aplicaciones de la integral + ODEs separables + series formales + Taylor (~18–19 años)

Highest bar · 🇺🇸 Estados Unidos

Aplicaciones de la integral (área entre curvas, volúmenes por discos/anillos/cascarones, longitud de arco, área de superficie de revolución, trabajo con fuerza variable). Ecuaciones diferenciales separables (ODE separable formal, crecimiento/decaimiento exponencial formal, ley de Newton del enfriamiento, logística intro, slope fields). Sucesiones y series formales con todos los tests de convergencia (divergencia, geométrica, p-series, comparación directa, comparación al límite, razón D'Alembert, raíz Cauchy, integral, alternantes Leibniz). Convergencia absoluta vs condicional. Series de potencias con radio e intervalo. Series de Taylor y Maclaurin con resto de Lagrange y series clásicas (e^x, sin, cos, ln(1+x), 1/(1−x), arctan, binomial).

CountryExpected topicsOfficial source
🇺🇸Estados UnidosAP Calculus BC: aplicaciones de la integral (volúmenes, arc length, work), ODEs separables + slope fields, series con todos los tests + Taylor con resto de Lagrange + series clásicas.AP Calculus AB & BC (College Board CED) ↗
🌐IB AA HLTopic 5: ODEs separables, Maclaurin/Taylor, series clásicas, resto de Lagrange.IB Diploma Mathematics: Analysis and Approaches HL ↗
🇸🇬SingapurJC2 H2: ODEs separables incl. Newton cooling, integración aplicada (volúmenes, arc length), series con tests (no formal completo a HS).Singapore JC2 H2 Mathematics 9758 (SEAB) ↗
🇯🇵JapónMath III: aplicaciones (área, volumen, longitud), ODEs separables, sucesiones formales (数列). Series formales con tests es freshman universitario.MEXT Senior High School Math III (Japan) ↗
🇰🇷CoreaCalculus 2022 Revised: aplicaciones completas, ODEs separables, sucesiones rigurosas. Series formales en universidad.KICE 2022 Revised Curriculum — Calculus (미적분) ↗
🇺🇸Estados UnidosMIT 18.01 / Berkeley Math 1B / Stanford MAT 20: aplicaciones + ODEs + series formales + Taylor con resto. Cobertura completa universitaria.UC Berkeley Math 1A / 1B ↗
🇲🇽MéxicoUNAM Cálculo I-II: límites ε-δ, derivada formal, integral con técnicas, series con todos los tests + Taylor + binomial, todo a primer año universitario.UNAM Cálculo Diferencial e Integral I-II ↗

Rodybee skills at this grade

Area between curves · Volume — disk method · Volume — washer method · Volume — shell method · Arc length · Surface area of revolution · Work (pumping, spring) · Separable ODE — solve · Exponential growth/decay ODE · Newton's law of cooling ODE · Logistic ODE (intro) · Slope fields (intro) · Sequence convergence (formal) · Series convergence (formal) · Divergence test · Geometric series (formal) · p-series test · Direct comparison test · Limit comparison test · Ratio test (D'Alembert) · Root test (Cauchy) · Integral test for convergence · Alternating series test (Leibniz) · Absolute vs conditional convergence · Power series — radius and interval · Power series — termwise diff/integration · Taylor series definition · Taylor polynomials · Maclaurin classics (e^x, sin, cos, ln…) · Lagrange remainder · Binomial series

Full research document

Methodology, gap analysis, recommendations and the complete list of sources are in the underlying research markdown.

Read the full benchmark document →
Calculus (Intro) Curriculum & Standards — Rodybee