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Mathematics: the step-by-step guide

33 steps from the algebra you need before calculus, through every unit of AP Calculus AB and BC, to the extra topics of Cambridge 9709 Paper 3 and IB Analysis and Approaches HL. Each step has worked examples, graphs, six practice questions with worked answers and a quick self-check.

Before Calculus: FoundationsAP prerequisites · Cambridge P1 · IB SL topics 1–3
  1. Indices, Surds and Polynomials
  2. Quadratics and Inequalities
  3. Functions and Inverses
  4. Lines and Circles
  5. Exponentials and Logarithms
  6. Sequences, Series and the Binomial Theorem
  7. Trigonometry: Radians, Graphs and Equations
  8. Trigonometric Identities and the Harmonic Form
Unit 1: Limits and ContinuityAP Calculus Unit 1 (AB 10–15%, BC 5–10% of multiple choice)
  1. Limits from Graphs, Tables and Algebra
  2. Continuity, Asymptotes and the Intermediate Value Theorem
Unit 2: Differentiation — Definition and Basic RulesAP Calculus Unit 2 (AB 10–15%, BC 5–10% of multiple choice)
  1. The Derivative from First Principles
  2. Power, Product and Quotient Rules
Unit 3: Differentiation — Composite, Implicit and Inverse FunctionsAP Calculus Unit 3 (AB 5–10%, BC 5–10% of multiple choice)
  1. The Chain Rule
  2. Implicit Differentiation and Inverse Functions
Unit 4: Contextual Applications of DifferentiationAP Calculus Unit 4 (AB 10–15%, BC 5–10% of multiple choice)
  1. Rates of Change, Motion and Related Rates
  2. Linearisation and L'Hôpital's Rule
Unit 5: Analytical Applications of DifferentiationAP Calculus Unit 5 (AB 15–20%, BC 10–15% of multiple choice)
  1. The Mean Value Theorem and Extreme Values
  2. Stationary Points, Concavity and Curve Sketching
  3. Optimisation
Unit 6: Integration and Accumulation of ChangeAP Calculus Unit 6 (AB 15–20%, BC 15–20% of multiple choice)
  1. Riemann Sums and the Definite Integral
  2. The Fundamental Theorem of Calculus
  3. Integration by Substitution
  4. Integration by Parts and Partial Fractions
Unit 7: Differential EquationsAP Calculus Unit 7 (AB 5–10%, BC 5–10% of multiple choice)
  1. Separable Differential Equations and Slope Fields
  2. Exponential and Logistic Models and Euler's Method
Unit 8: Applications of IntegrationAP Calculus Unit 8 (AB 10–15%, BC 5–10% of multiple choice)
  1. Area Between Curves and Average Value
  2. Volumes of Revolution and Arc Length
Unit 9: Parametric, Polar and Vector-Valued FunctionsAP Calculus BC only (10–15% of multiple choice)
  1. Parametric, Polar and Vector-Valued Functions
Unit 10: Infinite Sequences and SeriesAP Calculus BC only (15–20% of multiple choice)
  1. Series and Convergence Tests
  2. Taylor and Maclaurin Series
Beyond AP: Cambridge and IB ExtrasCambridge 9709 Paper 3 · IB AA Higher Level
  1. Numerical Methods: Roots and Iteration
  2. Vectors and Lines in 3D
  3. Complex Numbers

Which steps does my course need?

CourseSteps
AP Calculus AB1–8 as needed, then 9–27 (skip anything marked BC only)
AP Calculus BC1–8 as needed, then 9–30
Cambridge 9709 Paper 11–4, 6, 7, 11–13, 18, 19, 21, 26, 27
Cambridge 9709 Papers 2 and 31, 2, 5, 6, 8, 12–14, 20–24, 31–33
IB Maths AA SL1–7, 9, 11–13, 15, 17–21, 26
IB Maths AA HLEverything for SL, plus 8, 10, 14, 16, 22–25, 27, 30, 32, 33

Each exam guide lists every official unit or topic of your syllabus and links it to the right step.

The books behind the guide

Each step is a short version of a chapter of Mathematics Made Simple, which gives the full explanation, more worked examples and every graph, from indices and quadratics through to series, vectors and complex numbers.

Mathematics Made Simple textbook coverEvery step in full: Mathematics Made Simple covers algebra, functions and calculus for AP, Cambridge 9709 and IB.Get the textbook · $12Mathematics Made Simple Practice Questions coverPractise until it sticks: Mathematics Made Simple Practice Questions has 1,565 questions, each with a fully worked answer.Get the practice book · $9Mathematics Made Simple Complete Bundle coverThe textbook and the practice book together: the Mathematics Made Simple bundle for $16 (was $21).Both books · $16

Studying more than one subject? The Complete Book Bundle includes every Economics, Statistics, Mathematics, Econometrics and Research Methods book.