Calculus
∫ Core Calculus
The essential sequence from limits through integration — the ideas that power all of modern science and engineering.
- 1→
Limits
The foundational idea of calculus — what a function approaches as its input gets arbitrarily close to a value.
- 2→
Real Analysis Basics
The rigorous proof-based foundation underneath calculus — epsilon-delta definitions of limits and continuity, and what it actually means for a sequence to converge.
- 3→
Lebesgue Integral
A more powerful way to integrate than Riemann's — slice the range instead of the domain, making sense of integrals that the Riemann integral can't handle.
- 4→
Derivatives
The instantaneous rate of change of a function — defined as the limit of the difference quotient and interpreted as the slope of a tangent line.
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Chain Rule
How to differentiate composite functions — the most frequently used rule in calculus, underpinning substitution in integration.
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Differentiation Rules
The power rule, product rule, quotient rule, and chain rule — systematic shortcuts for computing derivatives without going back to the limit definition.
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Applications of Derivatives
Using derivatives to find maxima and minima, describe curve shape, solve optimisation problems, and analyse rates of change.
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Integrals
The integral as accumulated area — Riemann sums, definite integrals, and the antiderivative as an operation that reverses differentiation.
- 9→
Fundamental Theorem of Calculus
The bridge connecting differentiation and integration — why antiderivatives compute areas and why the two operations are inverses of each other.
- 10→
Applications of Integration
Computing areas between curves, volumes of solids of revolution, arc lengths, and other accumulated quantities.