IB Math AI HL Syllabus
Complete breakdown of topics, assessment structure, and learning resources for IB Mathematics Applications & Interpretations Higher Level. Master sophisticated mathematical applications and advanced data analysis with our expert video tutorials and comprehensive study materials.
Topic 1: Number & Algebra
1.1 Operations with Numbers
- Numbers in the form a × 10k
- Where 1 ≤ a < 10 and k is an integer
1.2 Arithmetic Sequences and Series
- Formulae for nth term: un = u1 + (n − 1)d
- Sum of first n terms: Sn = n/2 (2u1 + (n − 1)d)
- Use of sigma (Σ) notation, e.g. Σk=1n (3k + 2)
- Applications: analysis, interpretation, prediction when models are not perfectly arithmetic
1.3 Geometric Sequences and Series
- Formulae for nth term: un = u1 rn−1
- Sum of first n terms: Sn = u1(rn − 1) / (r − 1)
- Sigma notation for sums of geometric sequences
- Identify first term and ratio; applications
1.4 Financial Applications
- Compound interest: FV = PV × (1 + r / (100k))nk
- Annual depreciation
1.5 Laws of Exponents and Logarithms
- Laws of exponents with integer exponents
- Introduction to logarithms: log10 x and ln x
- Numerical evaluation using technology
1.6 Approximation
- Decimal places and significant figures
- Upper and lower bounds of rounded numbers
- Percentage errors: ε = | (vA − vE) / vE | × 100%, estimation
1.7 Amortization and Annuities
- Use of technology to calculate payments and balances
1.8 Solving Equations with Technology
- Systems of linear equations (up to 3 variables)
- Polynomial equations
1.9 Laws of Logarithms
- loga(xy) = logax + logay
- loga(x/y) = logax − logay
- loga(xm) = m · logax
- Condition: a, x, y > 0
1.10 Simplifying Expressions
- Simplifying expressions involving rational exponents
1.11 Infinite Geometric Sequences
- Sum of infinite geometric sequences
- Formula: S∞ = u1 / (1 − r), where |r| < 1
1.12 Complex Numbers
- Cartesian form: z = a + bi
- Terms: real part, imaginary part, conjugate z*, modulus |z|, argument
- Operations: sums, differences, products, quotients, powers (Cartesian)
- Complex plane and solutions of quadratics ax2 + bx + c = 0 where Δ < 0
1.13 Polar and Exponential Forms
- Polar form: z = r(cos θ + isin θ) = r cis θ
- Exponential form: z = r eiθ
- Converting between Cartesian, polar, and exponential forms
- Products, quotients, and powers in polar/exponential forms
1.14 Matrices
- Definition: element, row, column, order
- Algebra: equality, addition, subtraction, scalar multiplication, matrix multiplication
- Determinants and inverses (2×2 by hand, larger with technology)
- Solving systems of equations Ax = b using inverse x = A−1b
1.15 Eigenvalues and Eigenvectors
- Characteristic polynomial of 2×2 matrices: det(A − λI) = 0
- Diagonalization with distinct real eigenvalues: A = PDP−1
- Applications to powers of 2×2 matrices
Topic 2: Functions
2.1 Equations of a Straight Line
- Different forms: gradient, intercepts
- Parallel lines: m1 = m2
- Perpendicular lines: m1 × m2 = −1
2.2 Concept of a Function
- Domain, range, graph, notation (f(x), v(t), C(n))
- Inverse function as reflection in y = x, notation f−1(x)
2.3 Graphing Functions
- Sketch from context or data; transfer from screen to paper
- Graph sums and differences using technology
2.4 Key Features of Graphs
- Determine points of intersection of curves/lines using technology
2.5 Modelling with Functions
- Linear: f(x) = mx + c
- Quadratic: f(x) = ax2 + bx + c, a ≠ 0; axis, vertex, zeros, intercepts
- Exponential: f(x) = k · ax + c, f(x) = k · erx + c; horizontal asymptote
- Direct/inverse variation: f(x) = a · xn, n ∈ ℤ
- Cubic: f(x) = ax3 + bx2 + cx + d
- Sinusoidal: f(x) = a sin(bx) + d, f(x) = a cos(bx) + d
2.6 Modelling Skills
- Develop, fit, test, reflect, and use models
- Select a reasonable domain
- Justify the choice of a model based on data, curve shape, and context
2.7 Composite and Inverse Functions
- Composite functions: (f ∘ g)(x) = f(g(x))
- Inverse functions with domain restriction; find inverse
2.8 Transformations
- Translations: y = f(x) + b, y = f(x − a)
- Reflections: y = −f(x), y = f(−x)
- Vertical stretch: y = p · f(x)
- Horizontal stretch: y = f(qx)
- Composite transformations
2.9 Logarithmic, Sinusoidal, Logistic & Piecewise
- Logarithmic model: f(x) = a + b · ln(x)
- Sinusoidal models:
- f(x) = a · sin(b·x − c) + d
- f(x) = a · cos(b·x − c) + d
- Logistic model: f(x) = L / (1 + C · e−k·x), where L, C, k > 0
- Piecewise models: defined for specific intervals
Topic 3: Geometry & Trigonometry
3.1 Distance, Midpoint, and Solids
- Distance between two points: d = √((x2−x1)2 + (y2−y1)2 + (z2−z1)2)
- Midpoint: M = ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)
- Volume and surface area: right pyramid, right cone, sphere, hemisphere, and combinations
- Angle between two intersecting lines or between a line and a plane
3.2 Trigonometry in Triangles
- Right-angled triangles (sine, cosine, tangent):
- sin θ = opp / hyp
- cos θ = adj / hyp
- tan θ = opp / adj
- Sine rule: a / sin A = b / sin B = c / sin C
- Cosine rule: c2 = a2 + b2 − 2ab cos C; cos C = (a2 + b2 − c2) / (2ab)
- Area of triangle: Area = 1/2 · a · b · sin C
3.3 Applications of Trigonometry
- Applications of right and non-right triangle including Pythagoras’ theorem: a2 + b2 = c2
- Angles of elevation and depression
- Constructing labelled diagrams from statements
3.4 The Circle
- Radian measure of angles
- Length of an arc: L = r · θ
- Area of a sector: A = 1/2 · r2 · θ
3.5 Equations of Perpendicular Bisectors
- Find perpendicular bisector of a line segment
- Given two points or the equation of a line segment, calculate midpoint
- Determine slope of perpendicular bisector
- Equation using point-slope form: y − y0 = m(x − x0)
3.6 Voronoi Diagrams
- Sites, vertices, edges, cells
- Adding a site to an existing diagram
- Nearest neighbour interpolation
- Applications such as the “toxic waste dump” problem
3.7 Radian Measure
- Definition of a radian
- Conversion between degrees and radians
- Using radians for sector area and arc length calculations
3.8 Unit Circle & Identities
- Definitions: cos θ, sin θ in unit circle
- Pythagorean identity: cos2 θ + sin2 θ = 1
- Tangent: tan θ = sin θ / cos θ
- Extension of sine rule to ambiguous case
- Graphs of f(x) = sin x and f(x) = cos x
- Graphical methods for solving trig equations in finite intervals
3.9 Geometric Transformations
- Transformations: reflections, horizontal/vertical stretches, enlargements, translations, rotations
- Matrix form: [a b; c d] · [x; y] + [e; f]
- Compositions of transformations
- Determinant interpretation: Area of image = |det A| × Area of object
3.10 Vectors – Concepts
- Vector vs scalar; representation using directed line segments
- Unit vectors; base vectors i, j, k
- Components: v = v1i + v2j + v3k
- Zero vector 0, negative vector −v
- Position vectors: OA = a
- Rescaling and normalizing vectors
3.11 Vector Equation of a Line
- Line in 2D and 3D: r = a + λb, where b is the direction vector
3.12 Kinematics (Vectors)
- Linear motion with constant velocity: r = r0 + vt
- Relative position: AB = B − A
- Motion with variable velocity in two dimensions
3.13 Scalar and Vector Products
- Scalar product: v · w = |v||w|cos θ (angle between vectors)
- Vector product: v × w = |v||w|sin θ (geometric interpretation of magnitude)
- Components of vectors
3.14 Graph Theory – Basic Concepts
- Graphs, vertices, edges, adjacent vertices/edges
- Degree of a vertex
- Types: simple, complete, weighted, directed (in/out degree)
- Subgraphs, trees
3.15 Adjacency Matrices and Walks
- Adjacency matrices and weighted adjacency tables
- Number of k-length walks between vertices
- Transition matrices for strongly-connected, undirected, or directed graphs
3.16 Tree and Cycle Algorithms
- Walks, trails, paths, circuits, cycles
- Eulerian trails and circuits
- Hamiltonian paths and cycles
- Minimum spanning tree: Kruskal’s and Prim’s algorithms
- Chinese postman problem: shortest route covering all edges
- Travelling salesman problem: nearest neighbour (upper bound) and deleted vertex (lower bound)
Topic 4: Statistics & Probability
4.1 Population and Sampling
- Concepts of population, sample, random sample, discrete and continuous data
- Reliability of data sources and bias in sampling
- Interpretation of outliers
- Sampling techniques and their effectiveness
4.2 Presentation of Data
- Frequency distributions (tables) for discrete and continuous data
- Histograms
- Cumulative frequency and cumulative frequency graphs; find median, quartiles, percentiles, range, interquartile range (IQR)
- Box-and-whisker diagrams
4.3 Measures of Central Tendency
- Mean, median, mode
- Estimation of mean from grouped data
- Modal class
- Interquartile range (IQR), standard deviation, variance
- Effect of constant changes on data
4.4 Linear Correlation
- Scatter diagrams; lines of best fit (by eye through mean point)
- Pearson’s correlation coefficient r
- Regression line of y on x: y = ax + b; interpret parameters a and b
- Use of regression for prediction
4.5 Probability – Basic Concepts
- Trial, outcome, equally likely outcomes, relative frequency, sample space U, event
- Probability: P(A) = n(A) / n(U)
- Complementary events: P(A′) = 1 − P(A)
- Expected number of occurrences
4.6 Probability – Diagrams and Rules
- Venn diagrams, tree diagrams, sample space diagrams, tables of outcomes
- Combined events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Mutually exclusive: P(A ∩ B) = 0
- Conditional probability: P(A|B) = P(A ∩ B) / P(B)
- Independent events: P(A ∩ B) = P(A) · P(B)
4.7 Discrete Random Variables
- Probability distributions
- Expected value (mean) E(X)
- Applications
4.8 Binomial Distribution
- General notation: X ~ B(n, p)
- Mean formula: μ = np
- Variance formula: σ2 = np(1 − p)
4.9 Normal Distribution
- General notation: X ~ N(μ, σ2)
- Properties of normal curve and diagrammatic representation
- Normal probability calculations using technology
- Inverse normal calculations
4.10 Spearman’s Rank Correlation
- Spearman’s rank correlation coefficient rs
- Awareness of Pearson vs Spearman correlation
- Effect of outliers
4.11 Hypothesis Testing
- Null and alternative hypotheses: H0 and H1
- Significance levels, p-values
- Chi-square test (χ2): independence, goodness of fit
- t-test: comparing two population means
- One-tailed and two-tailed tests
4.12 Data Collection and Design
- Surveys, questionnaires, selecting relevant variables
- Categorizing data in χ2 tables and degrees of freedom
- Reliability and validity tests
4.13 Non-linear Regression
- Least squares regression curves using technology
- Sum of square residuals (SSres) as measure of fit
- Coefficient of determination R2
4.14 Linear Transformation of Variables
- Expected value: E(aX + b) = aE(X) + b
- Variance: Var(aX + b) = a2Var(X)
- Linear combinations of n independent random variables
- Unbiased estimates: x̄ = Σxi / n, sn−12 = Σ(xi − x̄)2 / (n − 1)
4.15 Normal Variables and CLT
- Linear combination of independent normal variables is normal
- Central Limit Theorem: X̄ ~ N(μ, σ2/n)
4.16 Confidence Intervals
- Confidence intervals for the mean of a normal population
4.17 Poisson Distribution
- General notation: X ~ Po(m)
- Mean and variance: E(X) = Var(X) = m
- Sum of independent Poisson distributions is Poisson
4.18 Hypothesis Testing – Advanced
- Critical values and regions
- Test for population mean, proportion, Poisson mean
- Testing population correlation ρ = 0
- Type I and II errors, probability calculations
4.19 Markov Chains
- Transition matrices, powers of matrices
- Regular Markov chains, initial state probabilities
- Steady state and long-term probabilities
Topic 5: Calculus
5.1 Introduction to Limits and Derivatives
- Concept of a limit
- Derivative interpreted as gradient function and rate of change
5.2 Increasing and Decreasing Functions
- Graphical interpretation: f′(x) > 0, f′(x) = 0, f′(x) < 0
5.3 Derivatives of Polynomial Functions
- f(x) = axn ⇒ f′(x) = anxn−1, n ∈ ℤ
- Derivative of f(x) = axn + bxm + …, all integer exponents
5.4 Tangents and Normals
- Tangent and normal lines at a given point
- Equations of tangents and normals
5.5 Introduction to Integration
- Anti-differentiation of f(x) = axn + bxm + …, n ∈ ℤ, n ≠ −1
- Notation: ∫ f(x) dx
- Anti-differentiation with boundary condition
- Definite integrals using technology
- Area of a region enclosed by y = f(x) and the x-axis where f(x) > 0
5.6 Stationary Points
- Values of x where gradient is zero: f′(x) = 0
- Local maximum and minimum points
5.7 Optimisation Problems
- Solving context-based optimisation problems
5.8 Approximating Areas
- Trapezoidal rule for approximating areas under curves
5.9 Derivatives of Standard Functions
- Derivatives:
- d/dx (sin x) = cos x
- d/dx (cos x) = −sin x
- d/dx (tan x) = sec2 x
- d/dx (ex) = ex
- d/dx (ln x) = 1/x
- d/dx (xn) = nxn−1, n ∈ ℚ
- Rules: Chain rule, product rule, quotient rule
- Related rates of change
5.10 Second Derivative
- Notation: d2y/dx2 and f″(x)
- Second derivative test for maxima and minima
5.11 Integration of Standard Functions
- Definite and indefinite integrals of xn, sin x, cos x, 1/cos2x, ex
- Includes n ∈ ℚ, n ≠ −1
- Integration by inspection or substitution: ∫ f(g(x)) g′(x) dx
5.12 Areas and Volumes
- Area of region enclosed by a curve and the x- or y-axis
- Volumes of revolution about the x-axis or y-axis
5.13 Kinematics and Calculus
- Displacement s, velocity v, acceleration a
- Formulas: v = ds/dt, a = dv/dt = d2s/dt2 = v dv/ds
- Displacement: ∫t1t2 v(t) dt
- Total distance: ∫t1t2 |v(t)| dt
- Speed = magnitude of velocity
5.14 Modelling with Differential Equations
- Setting up models/differential equations from context
- Solving by separation of variables
5.15 Slope Fields
- Sketching slope fields and diagrams for differential equations
5.16 Euler’s Method
- Numerical solutions for first-order differential equations: dy/dx = f(x, y)
- Numerical solutions for coupled systems: dx/dt = f1(x, y, t), dy/dt = f2(x, y, t)
5.17 Phase Portraits
- Solutions of coupled differential equations: dx/dt = ax + by, dy/dt = cx + dy
- Qualitative analysis for distinct, real, complex, imaginary eigenvalues
- Sketching trajectories, identifying equilibrium points, stable populations, and saddle points
- Solving d2x/dt2 = f(x, dx/dt, t) by Euler’s method
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IB Math AI HL covers 5 main topics: Topic 1 - Number and Algebra (advanced financial mathematics, sequences, logarithms), Topic 2 - Functions (advanced modeling, regression analysis, functions), Topic 3 - Geometry and Trigonometry (complex geometry, advanced trigonometry, graph theory), Topic 4 - Statistics and Probability (advanced descriptive statistics, probability distributions, hypothesis testing, chi-squared tests), and Topic 5 - Calculus (differential and integral calculus with applications). The HL course includes additional depth and complexity compared to SL.
Assessment consists of four components: Paper 1 (calculator allowed, 120 minutes, 30% of final grade), Paper 2 (calculator allowed, 120 minutes, 30% of final grade), Paper 3 (calculator allowed, 60 minutes, 20% of final grade), and Internal Assessment (mathematical exploration, 20% of final grade). All external papers test advanced knowledge across curriculum topics with complex problem-solving and extended response questions.
Math AI HL includes all SL content plus additional advanced topics and greater depth. HL students study more complex statistical methods, advanced calculus applications, additional geometric concepts, and sophisticated modeling techniques. The HL course has an additional Paper 3 exam and requires deeper mathematical understanding. AI HL is more suitable for students planning careers requiring advanced mathematical applications, such as economics, psychology research, or data science.
Students need proficiency with graphing calculators (TI-84, Casio fx-CG50, or similar) for all three exam papers. Essential skills include statistical calculations, regression analysis, graphing functions, solving equations, and working with matrices. Students should also be comfortable with statistical software and spreadsheet applications for the Internal Assessment. Practice with technology should begin early in the course as it's integral to success in all assessments.
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Reviewing the IB Math AI HL Syllabus helps students clearly understand all required topics, learning objectives, and assessment formats. By aligning their study plan with the syllabus, they can prioritize important areas, manage time effectively, and avoid missing key concepts tested in the final examination.
The IB Math AI HL Syllabus may be revised periodically by the IB to reflect updated teaching methods and modern mathematical applications. Students should always check the latest syllabus to ensure they are studying the most relevant material, practicing current question types, and preparing with confidence.
The IB Math AI HL Syllabus includes all the main topics students need to study, such as algebra, functions, calculus, probability, and statistics. The IB Math AI HL Syllabus also explains how exams are structured and what technology skills are required. By reviewing the IB Math AI HL Syllabus regularly, students can focus on the right content and prepare effectively for assessments.