IB Mathematics: Analysis and Approaches HL
Free IB Mathematics: Analysis and Approaches HL course: clear theory, worked examples and past IB exam questions by type and level, with full solutions.
Block 1 · Number and algebra
Algebra review and mathematical induction
Algebra for IB Analysis and Approaches HL: powers, polynomials, quadratic equations, inequalities, induction and the binomial theorem, with IB questions.
- Real numbers, powers and radicals
- Polynomials: remainder and factor theorems
- Quadratic equations and the discriminant
- Inequalities and absolute value
- Proof and mathematical induction
- Binomial theorem
- IB exam questions: topic 1
Exponential equations and logarithms
Logarithms for IB Analysis and Approaches HL: definition, laws, change of base and exponential and logarithmic equations, with IB exam questions.
- Exponential equations
- Logarithms: definition, properties and change of base
- Logarithmic equations
- IB exam questions: topic 2
Arithmetic and geometric sequences
Arithmetic and geometric sequences for IB Analysis and Approaches HL: general term, sums, infinite sum, compound interest, growth and worked IB questions.
- Arithmetic sequences: general term, sum and sigma notation
- Geometric sequences and infinite sums
- Simple and compound interest, growth and depreciation
- IB exam questions: topic 3
Complex numbers
Complex numbers for IB Analysis and Approaches HL: Cartesian, polar and Euler forms, De Moivre, n-th roots and polynomials, with IB exam questions.
- Cartesian form and operations
- Modulus–argument and Euler form
- De Moivre's theorem and n-th roots
- Polynomials with complex roots
- IB exam questions: topic 6
Systems of linear equations
Systems of linear equations for IB Analysis and Approaches HL: Gaussian elimination, classifying systems, systems with a parameter and IB exam questions.
- Linear equations and systems
- Gaussian elimination method
- Systems with a parameter
- IB exam questions: topic 7
Block 2 · Geometry and trigonometry
Trigonometry
Trigonometry for IB Analysis and Approaches HL: radians, ratios, identities, trigonometric equations, sine and cosine rules, with IB exam questions.
- Trigonometric ratios in right-angled triangles
- Radians and the unit circle
- Sum, double-angle and half-angle identities
- Trigonometric equations
- Sine and cosine rules; area of a triangle
- IB exam questions: topic 4
Plane geometry: vectors and lines
Vectors in the plane for IB Analysis and Approaches HL: operations, scalar product, angles, equations of lines and distances, with IB exam questions.
- Vectors in the plane
- Scalar (dot) product and angle between vectors
- Equations of a line, intersections and distances
- IB exam questions: topic 5
Geometry in space
Geometry in space for IB Analysis and Approaches HL: vectors in R3, scalar and vector products, lines, planes, angles and distances.
- Vectors in space
- Scalar product and the angle between vectors in space
- Vector product, areas and volumes
- Equations of lines and planes in space
- Angles and distances in space
- Relative positions and intersections of lines and planes
- Volumes and surface areas of solids
- IB exam questions: topic 8
Block 3 · Functions and calculus
Functions
Functions for IB Analysis and Approaches HL: domain, graphs, transformations, inverses, limits, continuity, exponential, logarithmic and circular functions.
- Function, domain and range
- Graphs and elementary functions
- Quadratic function
- Intersection of graphs
- Features of the graph of a function
- Transformations of the graph of a function
- Combining functions and composition
- Inverse function
- Limit of a function: intuitive idea
- Calculating limits and asymptotes
- Continuity of a function
- Exponential and logarithmic functions
- Applications of exponentials and logarithms
- Circular functions
- Applications of circular functions
- IB exam questions on functions
Differential calculus
Differential calculus for IB Analysis and Approaches HL: derivative rules, tangents, extrema, optimisation, rates of change and series, with IB exam questions.
- Derivative of a function at a point
- Differentiability and continuity; operations with derivatives
- Calculating derivatives
- Chain rule
- Derivative of the inverse function and of the inverse trigonometric functions
- Derivatives of exponential and logarithmic functions
- Increasing, decreasing and extrema
- Optimisation: maxima and minima in context
- Rate of change problems
- Concavity and points of inflexion
- Rolle's and mean value theorems; L'Hôpital's rule
- Taylor and Maclaurin expansions
- Rectilinear motion: position, velocity and acceleration
- Implicit differentiation
- IB exam questions on differential calculus
Integral calculus
Integral calculus for IB Analysis and Approaches HL: antiderivatives, substitution, parts, differential equations, Euler's method and definite integrals.
- Antiderivatives and the indefinite integral
- Integration by substitution
- Integration by parts
- Integrating rational functions
- Separable differential equations
- Homogeneous differential equations
- Linear first-order differential equations
- Euler's method
- Definite integral and signed area under a curve
- Fundamental theorem of calculus and Barrow's rule
- Calculating areas with integrals
- Volumes of solids of revolution
- Improper integrals
- IB exam questions on integral calculus
Block 4 · Statistics and probability
Descriptive statistics
Descriptive statistics for IB Analysis and Approaches HL: tables, graphs, centre and spread, and the effect of changing data, with IB questions.
- Data, frequencies and tables
- Graphical representations of data
- Measures of central tendency
- Measures of dispersion
- Effect of changing the data
- IB exam questions: topic 12
Regression and correlation
Regression and linear correlation for IB Analysis and Approaches HL: scatter diagrams, Pearson's r, regression lines and predictions, with IB questions.
- Bivariate distributions and scatter diagrams
- Linear correlation and Pearson's r coefficient
- Regression line and predictions
- IB exam questions: topic 13
Probability
Probability for IB Analysis and Approaches HL: counting, events, conditional probability, independence, tree diagrams and Bayes' theorem, with exam questions.
- Combinatorics and the fundamental counting principle
- Random experiments, events and set operations
- Probability and its properties: Laplace, complement and union
- Conditional probability and independence of events
- Total probability, Bayes' theorem and tree diagrams
- IB exam questions: topic 14
Discrete random variables and the binomial distribution
Discrete random variables and the binomial distribution for IB Analysis and Approaches HL, with worked IB exam questions.
- Discrete random variable and its probability distribution
- Expected value, variance and fair games
- The binomial distribution B(n, p)
- Binomial: cumulative probabilities, expected value and finding n or p
- IB exam questions: topic 15
Continuous random variables and the normal distribution
Continuous random variables and the normal distribution in IB Analysis and Approaches HL: density, areas, inverse normal and z-scores.
- Continuous random variable and probability density function
- The normal distribution
- Normal calculations and the inverse normal
- Standardisation and finding μ and σ
- IB exam questions: topic 16
This is the course for Mathematics: Analysis and Approaches, Higher Level of the International Baccalaureate (IB) Diploma Programme. Each topic has a short explanation with worked examples and common mistakes, and a page with past IB exam questions for that topic, grouped by type and ordered from easiest to hardest, with step-by-step solutions.
How it is organised
- Brief, rigorous theory, with the key idea at the start and what the exam asks for.
- Worked examples, step by step, from easier to harder.
- Common mistakes that keep recurring in the papers.
- Past IB exam questions (Paper 1, without calculator, and Paper 2, with calculator) grouped by type and by level (★ basic, ★★ medium, ★★★ hard), with the solution one click away.
The Analysis and Approaches HL exam
The external assessment at HL has three papers: Paper 1 (no calculator), Paper 2 (with graphic display calculator) and Paper 3, with longer problem-solving questions. If you are preparing for the IB online, the academy offers online IB classes.
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