Single-variable calculus

Derivatives, integrals, Taylor — the working subset.

This chapter is a quick refresher of the single-variable calculus we actually use across the other books. If derivatives, the chain rule, integration by parts, and Taylor series are second nature, skip it. Otherwise the three lessons below cover the working subset — derivatives, integrals, and Taylor / linearisation — in three short passes, with interactive visualisations of each idea and worked-example collapsibles where the algebra is non-trivial.

• 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.