Approaching the boundary. The formal definition of a limit, continuity, and why these ideas are the foundation all of calculus is built on.
Volume 5 — Analysis and proof
Volume 5
Analysis and Proof
Up until now, you have relied on assumptions about how numbers behave between points. This volume strips away those assumptions and rebuilds mathematics on a rigorous foundation of limits, instantaneous rates of change, and continuous accumulation.
By the end you will understand how to freeze a changing system precisely at a moment in time, how to add up an infinite number of infinitely small pieces, and how to construct a logical argument that proves something must be true — not just probably, but necessarily.
6 chapters
Grade 11–12
Needs Volume 4
Chapter Map
Measuring instantaneous change. Derivatives as limits of difference quotients, differentiation rules, and the chain rule — with geometric and physical interpretation throughout.
Accumulating continuous amounts. The definite integral as a limit of Riemann sums, the Fundamental Theorem, and integration techniques including substitution and parts.
Establishing absolute certainty. Direct proof, contradiction, induction — the logical structures that separate "this always works" from "this happened to work."
Approximating functions and finding zeros. Taylor and Maclaurin series, convergence tests, Newton's method — because most real problems don't have closed-form solutions.
Centroids, second moments, and RMS values. Integration applied to the physical quantities that structural, electrical, and mechanical engineering depend on.