Single-variable calculus

Derivatives, integrals, Taylor — the working subset.

Calculus is the language of change. A derivative is the instantaneous rate at which one quantity responds to another; an integral is the accumulated total of a rate over an interval. Taylor expansion approximates a smooth function near a point by a polynomial in the displacement from that point, and linearisation — keeping only the first term — is the standard move that turns a nonlinear equation into a tractable one near equilibrium.

• 1.1 Derivatives — the limit definition, the four manipulation rules (linearity, product, chain, inverse), and the Newton-vs-Leibniz history that gave us the dual notations $\dot x$ and $dx/dt$.
• 1.2 Integrals — Riemann sums, the fundamental theorem of calculus, integration by parts, substitution, partial fractions, and RMS values.
• 1.3 Taylor series and linearisation — Taylor polynomials, the Maclaurin series of the standard functions, and the master move of physics: linearising nonlinear equations around an equilibrium.