Syllabus Map

AA HL Calculus Syllabus Points

The official IB syllabus content covering AHL points 5.9 through 5.16:

  1. 5.9–5.10Higher derivatives, continuity and differentiability, derivatives of reciprocal and inverse trigonometric functions ($\sec x, \arcsin x, \arctan x$)
  2. 5.11Implicit differentiation and related rates of change
  3. 5.12L'Hôpital's rule for indeterminate forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$
  4. 5.13Advanced integration techniques: integration by parts ($\int u\,dv = uv - \int v\,du$), integration by partial fractions, trigonometric substitution
  5. 5.14First-order differential equations: separable variables, linear differential equations with integrating factor $I(x) = e^{\int P(x)dx}$, and Euler's numerical step method
  6. 5.15Volumes of revolution about the $x$-axis ($V = \pi \int y^2 dx$) and $y$-axis ($V = \pi \int x^2 dy$)
  7. 5.16Maclaurin series expansions: definition $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n$, standard expansions ($e^x, \sin x, \cos x, \ln(1+x), (1+x)^p$), and Taylor series approximations

Study by topic

AHL Topic 5 Interactive Study Guides

Equipped with our standard 4-tab interface: Study Notes (theory, SVG curves & worked proofs), Practice Exercises (10 tiered problems with collapsible solutions), Formula Sheet, and Mini Test (MCQs & exam FRQs).

5.12 Advanced Limits & L’Hôpital’s Rule Indeterminate quotient limits $\frac{0}{0}$ and $\frac{\infty}{\infty}$, repeated application of L'Hôpital's rule, and converting indeterminate products $0 \times \infty$ into quotients. Open Interactive Guide 5.10, 5.11 Implicit Differentiation & Related Rates Differentiating implicit equations $F(x, y) = 0$, finding tangents and normals to conics, derivatives of inverse trig functions, and geometric multi-variable related rates of change. Open Interactive Guide 5.13 Advanced Integration: Parts & Partial Fractions Integration by parts using the LIATE hierarchy, repeated parts, looping cyclic integrals ($e^{ax}\sin bx$), partial fraction decomposition integration, and trig substitution. Open Interactive Guide 5.14 First-Order Differential Equations Analytical solutions via separation of variables and integrating factor $I(x) = e^{\int P(x)dx}$ for $\frac{dy}{dx} + P(x)y = Q(x)$, plus Euler's numerical step method approximation. Open Interactive Guide 5.15 Volumes of Revolution Definite integral formulas for revolving curves about the $x$-axis and the $y$-axis, disk method, washer method for regions between two curves, and modeling real-world solid vessels. Open Interactive Guide 5.16 Maclaurin & Taylor Series Expansions Deriving power series using repeated differentiation $f^{(n)}(0)$, applying standard booklet expansions ($e^x, \cos x, \sin x, \ln(1+x)$), differentiation and integration of series, and limits. Open Interactive Guide

Prerequisite Knowledge

Core SL Foundation Prerequisites

Review the 5 standard level Topic 5 guides:

5.1 Limits & First Derivatives 5.2–5.4 Derivative Rules, Tangents & Normals 5.5–5.6 Stationary Points & Optimization 5.7 Kinematics with Derivatives 5.8 Definite Integration & Area