IB Mathematics: AA HL topic guide

Calculus

Calculus is a core part of IB Mathematics: AA HL. This guide connects the syllabus ideas behind Introduction to Calculus, Limits and Differentiation from First Principles, The Derivative and Rates of Change, Tangents, Normals and Optimisation, The Chain, Product and Quotient Rules and 9 more units, shows how they appear in worked problems, and points you to the formulas and full lessons needed for exam revision.

What you will learn

Calculus syllabus outline

The units below follow the structure used in the full Study to Learn course. Use the outline to identify exactly which idea needs attention, then work through the public example before continuing to the complete lesson path.

5.0

Introduction to Calculus

What You Will Learn in Calculus

5.1

Limits and Differentiation from First Principles

Introduction to Limits · Differentiation from First Principles

5.2

The Derivative and Rates of Change

Derivative Rules for Standard Functions · Increasing/Decreasing Functions & the Second Derivative

5.3

Tangents, Normals and Optimisation

Tangents & Normals · Optimisation Problems

5.4

The Chain, Product and Quotient Rules

The Chain Rule · The Product & Quotient Rules

5.5

Curve Sketching: Concavity and Points of Inflexion

The Second Derivative Test & Points of Inflexion · Full Curve Sketching — Asymptotes and Intercepts

5.6

Kinematics via Calculus

Displacement, Velocity & Acceleration · Connected Rates of Change and Motion

5.7

Integration as Antidifferentiation

Basic Integration Rules · Area Under a Curve

5.8

Definite Integrals and Further Techniques

Integration by Substitution · The Trapezoidal Rule

5.9

Limits, Continuity and Differentiability

Continuity, Differentiability & Indeterminate Forms · One-Sided Limits and Removable Discontinuities

5.10

Further Differentiation

Derivatives of Further Functions · Implicit Differentiation & Related Rates

5.11

Further Integration Techniques

Integration by Parts · Areas Between Curves & Volumes of Revolution

5.12

Differential Equations

Separation of Variables & Slope Fields · Euler's Method

5.13

Maclaurin Series

Maclaurin Series for Standard Functions · Approximation and Error Bounds in Maclaurin Series

Free worked preview

Continuity, Differentiability & Indeterminate Forms

This complete preview comes from the Limits, Continuity and Differentiability unit. It introduces the core language, shows the method in context, and gives you a real example of the lesson quality before you create an account.

Continuity and Differentiability

A function can be continuous (no breaks, jumps, or holes) without being differentiable —

f(x)=|x|

$ is continuous everywhere but has a sharp corner at x = 0, where the derivative does not exist. Differentiability is a stronger condition: if a function is differentiable at a point, it must be continuous there, but the reverse is false. Indeterminate forms like 0/0 require algebraic manipulation to resolve. By the end you will be able to determine whether a function is continuous or differentiable at a point, explain why differentiability implies continuity, and resolve indeterminate limit forms through algebraic simplification.

Continuity at a Point

f is continuous at x=a if: f(a) exists, $\lim_{x\to a}f(x)$ exists, and they're equal. Informally, the graph has no break, jump, or hole at a.

Differentiability

f is differentiable at a only if it's continuous there and the graph has no sharp corner or vertical tangent — differentiability is a stronger condition than continuity (differentiable implies continuous, but not the reverse — e.g. $f(x)=|x|$ is continuous everywhere but not differentiable at x=0, due to the sharp corner).

Indeterminate Forms

When direct substitution into a limit gives $0/0$ or $\infty/\infty$ , the result is undetermined by that form alone — further algebraic manipulation (factoring, rationalizing, or informally recognizing the limit as a derivative definition) is needed to resolve it.

Common ErrorAn indeterminate form like 0/0 is not automatically equal to 1 or 0 — it genuinely could resolve to any value (or not exist at all) depending on the specific functions involved, which is exactly why further work is required.

Worked Example Determine whether

f(x)=|x-2|

is differentiable at x=2.

1For x<2, |x−2| = −(x−2)

\text{Left of } x=2: f(x)=-(x-2),\ \text{gradient} -1

2For x>2, |x−2| = x−2

\text{Right of } x=2: f(x)=x-2,\ \text{gradient} +1

\text{The gradient jumps from } -1 \text{ to } +1 \text{ at } x=2 \text{ — not differentiable there (a sharp corner)}

Practice QuestionExplain why

f(x)=\sqrt{x}

is continuous but not differentiable at x=0.

Reviewed by the Study to Learn editorial team · Updated 2026-07-24