Formal single-variable calculus: ε-δ definition of limit, classical continuity theorems (IVT formal, EVT, Bolzano, MVT, Rolle), derivative with all rules (power, product, quotient, chain, implicit, logarithmic), derivatives of transcendentals (trig, inverse trig, exp, log, hyperbolic, higher-order), curve analysis (monotonicity, concavity, inflection, full sketching), modeling (optimization, related rates, Newton-Raphson, L'Hôpital formal), antiderivatives + u-substitution + integration by parts, definite integral via Riemann sum + FTC parts I and II, advanced techniques (partial fractions in three forms, trig substitution, trig powers, reduction formulas, improper integrals types I and II), applications (area between curves, volumes by disks/washers/shells, arc length, surface area of revolution, work), separable ODEs (exponential growth formal, Newton's cooling, logistic intro, slope fields), formal sequences and series with all convergence tests (divergence, geometric, p-series, comparison, limit comparison, ratio, root, integral, alternating Leibniz), absolute vs conditional convergence, power series with radius and interval, Taylor and Maclaurin series with Lagrange remainder, classical Maclaurin series (e^x, sin, cos, ln(1+x), 1/(1−x), arctan), binomial series — calibrated to AP Calculus AB+BC + IB AA HL Y2 + Cambridge A2 9709 (Pure 3) + Singapore JC2 H2 + Japan SHS Math III + Korea KICE 2022 Revised Calculus + freshman university (MIT 18.01 / Stanford / UC Berkeley Math 1A-1B / UNAM Cálculo I-II).
1. Calc 1 — Limits (ε-δ formal)
17–18ε-δ definition of limit · Limit existence proof (ε-δ) · Proof of limit properties · Squeeze (sandwich) theorem · +2
2. Calc 2 — Continuity classical theorems
17–18Bolzano's theorem · IVT formal with proof · Extreme Value Theorem (Weierstrass) · MVT for continuous functions · +2
3. Calc 3 — Derivative + rules
17–18Derivative — formal definition (full) · Power rule (real n) · Sum/difference rule · Product rule · +4
4. Calc 4 — Derivatives of transcendentals
17–18Derivatives of all 6 trig functions · Derivatives of inverse trig · Derivative of e^x and a^x · Derivative of ln x and log_a x · +3
5. Calc 5 — Curve analysis (f', f'', sketch, MVT)
17–18Monotonicity from f' · Local vs global extrema · First derivative test classification · Concavity from f'' · +5
6. Calc 6 — Modeling with derivative
17–18Constrained optimization · Related rates · Linear approximation · Newton-Raphson method · +2
7. Calc 7 — Antiderivative + indefinite integral
17–18Antiderivative definition · Basic antiderivative formulas · Antiderivatives of transcendentals · u-substitution · +2
8. Calc 8 — Definite integral + FTC
17–18Riemann integral (formal) · Riemann sum → definite integral · Properties of definite integral · FTC Part I (g(x) = ∫_a^x f) · +3
9. Calc 9 — Advanced integration techniques
18–19Partial fractions — distinct linear · Partial fractions — repeated linear · Partial fractions — irreducible quadratic · Trigonometric substitution · +4
10. Calc 10 — Integral applications
18–19Area between curves · Volume — disk method · Volume — washer method · Volume — shell method · +3
11. Calc 11 — Separable ODEs
18–19Separable ODE — solve · Exponential growth/decay ODE · Newton's law of cooling ODE · Logistic ODE (intro) · +1
12. Calc 12 — Sequences & series formal
18–19Sequence convergence (formal) · Series convergence (formal) · Divergence test · Geometric series (formal) · +7
13. Calc 13 — Power series & Taylor
18–19Absolute vs conditional convergence · Power series — radius and interval · Power series — termwise diff/integration · Taylor series definition · +4