E.5 Higher-Level CalculusHL
The HL-only rigour-and-techniques chapter of calculus. Lesson 1 builds the foundation: continuity at a point (three-condition test), differentiability via the limit definition (with the canonical failure modes — corner, vertical tangent, prior discontinuity), and higher-order derivatives f^(n)(x) (polynomial termination, e^x as a fixed point, sin/cos period-4 cycle). Future lessons cover l'Hôpital's rule, implicit differentiation, related rates, advanced integration techniques, differential equations, Maclaurin series, and volumes of revolution.
Continuity, differentiability, and higher-order derivativesHLSign up
Continuity at a point · Differentiability — and where it fails · Higher-order derivatives
Limit evaluation — l'Hôpital's ruleHLSign up
l'Hôpital's rule · Convergence vs divergence
Implicit differentiation, related rates, and constrained optimisationHLSign up
Implicit differentiation · Related rates · Constrained optimisation
Further derivatives, integrals, and partial fractionsHLSign up
Further derivatives · Further integrals · Partial fractions
Integration techniques — substitution and partsHLSign up
Integration by substitution · Integration by parts
Volumes of revolution and area to the y-axisHLSign up
Area to the y-axis · Volume about x-axis · Volume about y-axis
First-order differential equationsHLSign up
Euler's method · Separation of variables · Homogeneous DEs (y = vx) · Integrating factor
Maclaurin seriesHLSign up
The five standard Maclaurin series · Building new series via sub/int/diff