Course Specifications IB Math AI HL

Course Specifications for IB Math AI HL

This page covers the full IB Math AI HL syllabus. Practise each topic with the AI HL question bank. Download the AI HL formula sheets and use past papers alongside this guide.

1. Number & Algebra

  • Number Toolkit

    • Standard Form

    • Approximation

    • Upper & Lower Bounds

    • Percentage Error

    • Accuracy & Estimation

    • Solving Equations using a GDC

  • Exponentials & Logs

    • Laws of Indices

    • Introduction to Logarithms

    • Laws of Logarithms

  • Sequences & Series

    • Language of Sequences & Series

    • Sigma Notation

    • Arithmetic Sequences & Series

    • Geometric Sequences & Series

    • Applications of Sequences & Series

  • Financial Applications

    • Compound Interest & Depreciation

    • Amortisation

    • Annuities

  • Complex Numbers

    • Introduction to Complex Numbers

    • Operations with Complex Numbers

    • Complex Roots of Quadratics

    • Modulus & Argument

    • Introduction to Argand Diagrams

  • Further Complex Numbers

    • Geometry of Complex Numbers

    • Modulus–Argument (Polar) Form

    • Exponential (Euler’s) Form

    • Conversion between Forms of Complex Numbers

    • Frequency & Phase of Trig Functions

  • Matrices

    • Introduction to Matrices

    • Operations with Matrices

    • Determinants & Inverses

    • Solving Systems of Linear Equations with Matrices

  • Eigenvalues & Eigenvectors

    • The Characteristic Polynomial, Eigenvalues & Eigenvectors

    • Diagonalisation & Powers of Matrices

2. Functions

  • Linear Functions & Graphs

    • Equations of a Straight Line

    • Parallel & Perpendicular Lines

  • Further Functions & Graphs

    • Functions & Mappings

    • Graphing Functions & Their Key Features

    • Intersecting Graphs

    • Quadratic Functions & Graphs

    • Cubic Functions & Graphs

    • Exponential Functions & Graphs

    • Sinusoidal Functions & Graphs

  • Modelling with Functions

    • Linear Models

    • Quadratic Models

    • Cubic Models

    • Exponential Models

    • Direct & Inverse Variation

    • Sinusoidal Models

    • Strategy for Modelling Functions

  • Functions Toolkit

    • Composite Functions

    • Inverse Functions

  • Transformations of Graphs

    • Translations of Graphs

    • Reflections of Graphs

    • Stretches of Graphs

    • Composite Transformations of Graphs

  • Modelling with Logarithmic, Logistic & Piecewise Functions

    • Natural Logarithmic Models

    • Logistic Models

    • Piecewise Models

3. Geometry & Trigonometry

  • Geometry Toolkit

    • Coordinate Geometry

    • Perpendicular Bisectors

    • Arcs & Sectors Using Degrees

    • Radian Measure

    • Arcs & Sectors Using Radians

  • Geometry of 3D Shapes

    • 3D Coordinate Geometry

    • Volume & Surface Area

  • Trigonometry

    • Pythagoras & Right-Angled Trigonometry

    • Sine Rule, Cosine Rule & Area of a Triangle

    • Angles of Elevation & Depression

    • Bearings & Constructions

  • Trigonometric Identities & Equations

    • The Unit Circle

    • Simple Identities

    • Graphs of Trigonometric Functions

    • Solving Equations Using Trigonometric Graphs

  • Voronoi Diagrams

    • Drawing Voronoi Diagrams

    • Interpreting Voronoi Diagrams

    • Toxic Waste Dump Problem

  • Matrix Transformations

    • Matrix Transformations

    • Matrices of Geometric Transformations

    • Matrices of Composite Transformations

    • Determinant of a Transformation Matrix

  • Vector Properties

    • Introduction to Vectors

    • Parallel Vectors

    • Adding & Subtracting Vectors

    • Position & Displacement Vectors

    • Magnitude of a Vector & Unit Vectors

    • The Scalar Product

    • Angle Between Two Vectors & Perpendicular Vectors

    • The Vector Product

    • Areas using the Vector Product

    • Components of Vectors

    • Geometric Proof with Vectors

  • Vector Equations of Lines

    • Equation of a Line in Vector Form

    • Equation of a Line in Parametric Form

    • Angle Between Two Lines

    • Shortest Distance Between a Point and a Line

    • Shortest Distance Between Two Lines

  • Modelling with Vectors

    • Kinematics with Vectors

    • Constant & Variable Velocity

  • Graph Theory

    • Introduction to Graph Theory

    • Walks & Adjacency Matrices

    • Minimum Spanning Trees (Kruskal’s Algorithm)

    • Minimum Spanning Trees (Prim’s Algorithm)

    • Chinese Postman Problem

    • Travelling Salesman Problem

    • Nearest Neighbour & Deleted Vertex Algorithms

4. Statistics & Probability

  • Statistics Toolkit

    • Sampling

    • Reliability & Validity of Data Collection Methods

    • Measures of Central Tendency

    • Measures of Dispersion

    • Frequency Tables

    • Linear Transformations of Data

    • Outliers

    • Box & Whisker Diagrams

    • Cumulative Frequency Graphs

    • Histograms

    • Interpreting Data

  • Correlation & Regression

    • Scatter Diagrams & Correlation

    • Pearson’s Product-Moment Correlation Coefficient

    • Spearman’s Rank Correlation Coefficient

    • Comparison of Correlation Coefficients

    • Linear Regression

  • Non-linear Regression

    • Least Squares Regression Curves & Coefficient of Determination

    • Logarithmic Scales

    • Linearising using Logarithms

  • Probability

    • Probability & Types of Events

    • Independent & Mutually Exclusive Events

    • Conditional Probability

    • Venn Diagrams

    • Tree Diagrams

  • Probability Distributions

    • Discrete Probability Distributions

    • Expected Values

  • Random Variables

    • Linear Combinations of Random Variables

    • Unbiased Estimates

  • Binomial Distribution

    • The Binomial Distribution

    • Calculating Binomial Probabilities

  • Normal Distribution

    • The Normal Distribution

    • Calculations with Normal Distribution

  • Combinations of Normal Distributions & Sample Mean Distributions

    • Sample Mean Distribution

    • Central Limit Theorem

    • Confidence Interval for the Mean

  • Poisson Distribution

    • Poisson Distribution

    • Calculating Poisson Probabilities

  • Hypothesis Testing using the Chi-squared Distribution

    • Introduction to Hypothesis Testing

    • Chi-squared Test for Independence

    • Goodness of Fit Test

  • Hypothesis Testing for Population Parameters

    • Hypothesis Testing for Mean (One Sample)

    • Hypothesis Testing for Mean (Two Sample)

    • Binomial Hypothesis Testing

    • Poisson Hypothesis Testing

    • Hypothesis Testing for Correlation

    • Type I & Type II Errors

  • Transition Matrices & Markov Chains

    • Markov Chains

    • Transition Matrices

    • Powers of Transition Matrices

    • Steady State & Long-term Probabilities

5. Calculus

    • Differentiation

      • Introduction to Derivatives

      • Differentiating Powers of x

      • Gradients, Tangents & Normals

      • Increasing & Decreasing Functions

      • Local Minimum & Maximum Points

      • Modelling with Differentiation

    • Further Differentiation

      • Differentiating Special Functions

      • Chain Rule

      • Product Rule

      • Quotient Rule

      • Related Rates of Change

      • Second Order Derivatives

      • Stationary Points

      • Concavity & Points of Inflection

    • Integration

      • Numerical Integration using the Trapezoidal Rule

      • Introduction to Integration

      • Integrating Powers of x

      • Finding the Constant of Integration

      • Finding Areas Using a GDC

    • Further Integration

      • Integrating Special Functions

      • Reverse Chain Rule

      • Integration by Substitution

      • Definite Integrals

      • Negative Integrals

      • Area Between Curve & y-axis

      • Area Between a Curve and a Line

      • Volumes of Revolution

    • Kinematics

      • Displacement, Velocity & Acceleration

      • Calculus for Kinematics

    • Differential Equations

      • Separation of Variables

      • Modelling with Differential Equations

      • Slope Fields

      • Approximate Solutions to Differential Equations using Euler’s Method

    • Coupled & Second Order Differential Equations

      • Coupled Differential Equations

      • Phase Portraits

      • Equilibrium Points

      • Sketching Solution Trajectories

      • Second Order Differential Equation

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