Introduction to Calculus
What You Will Learn in Calculus
IB Mathematics: AA SL topic guide
Calculus
Calculus is a core part of IB Mathematics: AA SL. 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 4 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.
Limits and Differentiation from First Principles
Introduction to Limits · Differentiation from First Principles
The Derivative and Rates of Change
Derivative Rules for Standard Functions · Increasing/Decreasing Functions & the Second Derivative
Tangents, Normals and Optimisation
Tangents & Normals · Optimisation Problems
The Chain, Product and Quotient Rules
The Chain Rule · The Product & Quotient Rules
Curve Sketching: Concavity and Points of Inflexion
The Second Derivative Test & Points of Inflexion · Full Curve Sketching — Asymptotes and Intercepts
Kinematics via Calculus
Displacement, Velocity & Acceleration · Connected Rates of Change and Motion
Integration as Antidifferentiation
Basic Integration Rules · Area Under a Curve
Definite Integrals and Further Techniques
Integration by Substitution · The Trapezoidal Rule
Free worked preview
What You Will Learn in Calculus
This complete preview comes from the Introduction to Calculus 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.
Introduction to Calculus
Calculus is the mathematics of change. Where algebra describes static relationships, calculus describes how quantities grow, shrink, accelerate, accumulate, and approach limits. Invented — or discovered — independently by Newton and Leibniz in the 17th century, it remains the single most powerful mathematical tool in physics, engineering, economics, and biology. The AA course treats calculus with analytical depth: you will not only apply differentiation and integration rules, but derive them from first principles, prove that a function is continuous or differentiable at a point, and construct series approximations that turn transcendental functions like and into infinite polynomials. The section begins with the concept of a limit — the idea that makes calculus logically sound — and builds outward from there: differentiation (rates of change, optimisation, kinematics), integration (accumulation, areas, volumes), and at HL, differential equations and Maclaurin series that close the loop between functions and power series.
Section Map — What Is Covered
- Limits and Differentiation from First Principles (5.1): the limit as the foundation, and the derivative defined as
- The Derivative and Rates of Change (5.2): the power rule, derivatives of sin/cos/tan, , and ; and using the sign of to find where a function increases or decreases
- Tangents, Normals and Optimisation (5.3): equations of tangent and normal lines, and the systematic strategy for maximising or minimising quantities within given constraints
- The Chain, Product and Quotient Rules (5.4): differentiating composite functions, products, and quotients — the complete differentiation toolkit
- Curve Sketching (5.5): concavity, points of inflexion, and the full checklist for producing a labelled sketch from intercepts through stationary points to asymptotes
- Kinematics via Calculus (5.6): displacement → velocity → acceleration via differentiation, and the reverse via integration; the crucial distinction between total distance and net displacement when direction changes
- Integration as Antidifferentiation (5.7): reversing differentiation — the power rule for integration, standard integrals, and the definite integral as signed area between a curve and the x-axis
- Definite Integrals and Further Techniques (5.8): integration by substitution — the reverse chain rule — and the trapezoidal rule for numerical approximation when an exact antiderivative is unavailable
- Limits, Continuity and Differentiability (5.9, HL): why differentiability implies continuity but not the reverse; one-sided limits; and classifying discontinuities as removable, jump, or infinite
- Further Differentiation (5.10, HL): derivatives of arcsin, arctan, sec, csc, cot; implicit differentiation for curves not in the form ; and related rates problems
- Further Integration Techniques (5.11, HL): integration by parts — the reverse product rule — and volumes of revolution via the disc method
- Differential Equations (5.12, HL): separation of variables for first-order equations, particular solutions from initial conditions, slope fields as a visual tool, and Euler's method for numerical approximation
- Maclaurin Series (5.13, HL): expressing functions as infinite polynomials built from their derivatives at x = 0; standard series for ; and using truncation for approximation with error bounds
How to Approach This SectionCalculus is cumulative in a way that algebra is not — the chain rule appears in every topic that follows, and integration techniques depend directly on fluency with differentiation. If you fall behind on differentiation, integration will feel impossible not because integration is harder but because the prerequisite is shaky. The single most common error in calculus is the chain-rule mistake: forgetting to multiply by the derivative of the inner function. Train yourself to check for it every time you differentiate. For integration, the constant of integration (+c) is not a formality — forgetting it on an indefinite integral is marked as an error in every IB exam, and on a definite integral, leaving it in is equally wrong since the constants cancel. At HL, integration by parts follows a priority order for choosing u (logarithmic before inverse trig before algebraic before trig before exponential — LIATE), and Maclaurin series give you a new way to think about functions: not as opaque black boxes but as infinite polynomials whose first few terms already give excellent approximations near zero.
Reviewed by the Study to Learn editorial team · Updated 2026-07-24