Transition matrices. Powers of transition matrices. Regular Markov chains. Initial state probability matrices. Calculation of steady state and long-term probabilities by repeated multiplication of the transition matrix or by solving a system of linear equations.

Taking a look at the IB Mathematics Applications and Interpretaton (AI) syllabus gives you a good idea of the skills and knowledge you’ll pick up in the course. Come along as we break down the main concepts, helping you get a better feel for what to expect on your learning journey.

Math AI

In this topic, students explore the following concepts:

Number and algebra

Operations with numbers in the form a × 10k where 1 ≤ a < 10 and k is an integer.

Scientific notation

Arithmetic sequences and series. Use of the formulae for the nth term and the sum of used the first n terms of the sequence. Use of sigma notation for sums of arithmetic sequences. Applications. Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.

Use of the formulae for the nth term and the sum of used the first n terms of the sequence.

Use of sigma notation for sums of arithmetic sequences.

Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.

Arithmetic sequences and series

Geometric sequences and series. Use of the formulae for the nth term and the sum of used the first n terms of the sequence. Use of sigma notation for the sums of geometric sequences. Applications.

Use of sigma notation for the sums of geometric sequences.

Geometric sequences and series

Financial applications of geometric sequences and series: • compound interest • annual depreciation.

Financial applications of geometric sequences and series:

Financial applications of sequences

Laws of exponents with integer exponents. Introduction to logarithms with base 10 and e. Numerical evaluation of logarithms using technology.

Laws of exponents with integer exponents.

Introduction to logarithms with base 10 and e.

Numerical evaluation of logarithms using technology.

Integer exponents and logarithms

Approximation: decimal places, significant figures. Upper and lower bounds of rounded numbers. Percentage errors. Estimation.

Approximation: decimal places, significant figures.

Upper and lower bounds of rounded numbers.

Approximation and estimation

Amortization and annuities using technology.

Amortization and annuities

Use technology to solve: • Systems of linear equations in up to 3 variables be • Polynomial equations

Systems of linear equations in up to 3 variables be

Solving systems and polynomial equations

Laws of logarithms: loga xy = loga x + logay x loga = loga x … logay y loga xm = mloga x for a, x, y >

Laws of logarithms

Simplifying expressions, both numerically and algebraically, involving rational exponents.

Rational exponents

Infinite geometric sequences

Complex numbers: the number i such that i2 = … 1. Cartesian form: z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument. Calculate sums, differences, products, quotients, by hand and with technology. Calculating powers of complex numbers, in Cartesian form, with technology. The complex plane. Complex numbers as solutions to quadratic equations of the form ax + bx + c = 0, a ≠ 0, with real coefficients where b2 … 4ac < 0.

Complex numbers: the number i such that i2 = … 1.

Cartesian form: z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument.

Calculate sums, differences, products, quotients, by hand and with technology.

Calculating powers of complex numbers, in Cartesian form, with technology.

Complex numbers as solutions to quadratic equations of the form ax + bx + c = 0, a ≠ 0, with real coefficients where b2 … 4ac < 0.

Complex numbers in Cartesian form

Modulus–argument (polar) form: z = r cosθ + isinθ = rcisθ. Exponential form: iθ z = re . Conversion between Cartesian, polar and exponential forms, by hand and with technology. Calculate products, quotients and integer powers in polar or exponential forms. Adding sinusoidal functions with the same frequencies but different phase shift angles. Geometric interpretation of complex numbers.

Modulus–argument (polar) form: z = r cosθ + isinθ = rcisθ.

Conversion between Cartesian, polar and exponential forms, by hand and with technology.

Calculate products, quotients and integer powers in polar or exponential forms.

Adding sinusoidal functions with the same frequencies but different phase shift angles.

Geometric interpretation of complex numbers.

Complex numbers in polar and exponential form

Definition of a matrix: the terms element, row, column and order for m × n matrices. Algebra of matrices: equality; addition; subtraction; multiplication by a scalar for m × n matrices. Multiplication of matrices. Properties of matrix multiplication: associativity, distributivity and non-commutativity. Identity and zero matrices. Determinants and inverses of n × n matrices with technology, and by hand for 2 × 2 matrices. Awareness that a system of linear equations can be written in the form Ax = b. Solution of the systems of equations using inverse matrix.

Definition of a matrix: the terms element, row, column and order for m × n matrices.

Algebra of matrices: equality; addition; subtraction; multiplication by a scalar for m × n matrices.

Properties of matrix multiplication: associativity, distributivity and non-commutativity.

Determinants and inverses of n × n matrices with technology, and by hand for 2 × 2 matrices.

Awareness that a system of linear equations can be written in the form Ax = b.

Solution of the systems of equations using inverse matrix.

Matrices

Eigenvalues and eigenvectors. Characteristic polynomial of 2 × 2 matrices. Diagonalization of 2 × 2 matrices (restricted to the case where there are distinct real eigenvalues). Applications to powers of 2 × 2 matrices.

Characteristic polynomial of 2 × 2 matrices.

Diagonalization of 2 × 2 matrices (restricted to the case where there are distinct real eigenvalues).

Applications to powers of 2 × 2 matrices.

Eigenvalues and eigenvectors

Functions

Different forms of the equation of a straight line. Gradient; intercepts. Lines with gradients m1 and m2 Parallel lines m1 = m2. Perpendicular lines m1 × m2 = … 1.

Different forms of the equation of a straight line.

Lines with gradients m1 and m2 Parallel lines m1 = m2.

Equations of a straight line

Concept of a function, domain, range and graph. Function notation, for example f (x) , v(t) , C(n). The concept of a function as a mathematical model. A Informal concept that an inverse function reverses or undoes the effect of a function. Inverse function as a reflection in the line y = x, and the notation f …1(x).

Concept of a function, domain, range and graph.

Function notation, for example f (x) , v(t) , C(n).

The concept of a function as a mathematical model.

A Informal concept that an inverse function reverses or undoes the effect of a function.

Inverse function as a reflection in the line y = x, and the notation f …1(x).

Functions, domain and range

The graph of a function; its equation y = f (x). Creating a sketch from information given or a context, including transferring a graph from screen to paper. Using technology to graph functions including their sums and differences.

The graph of a function; its equation y = f (x).

Creating a sketch from information given or a context, including transferring a graph from screen to paper.

Using technology to graph functions including their sums and differences.

Graphs of functions

Determine key features of graphs. Finding the point of intersection of two curves or lines using technology.

Finding the point of intersection of two curves or lines using technology.

Key features of graphs

Modelling with the following functions: Linear models. f x = mx + c. Quadratic models. f x = ax + bx + c ; a ≠ 0. Axis of symmetry, vertex, zeros and roots, intercepts on the x-axis and y -axis. Exponential growth and decay models. x f (x) = ka + c f (x) = ka…x + c (for a > 0) f (x) = kerx + c Equation of a horizontal asymptote. Direct/inverse variation: f x = axn, n ∈ ℤ The y-axis as a vertical asymptote when n < 0. Cubic models: f x = ax3 + bx2 + cx + d. Sinusoidal models: f x = asin(bx) + d,

Modelling with the following functions: Linear models. f x = mx + c.

Quadratic models. f x = ax + bx + c ; a ≠ 0.

Axis of symmetry, vertex, zeros and roots, intercepts on the x-axis and y -axis.

Exponential growth and decay models. x f (x) = ka + c f (x) = ka…x + c (for a > 0) f (x) = kerx + c Equation of a horizontal asymptote.

Direct/inverse variation: f x = axn, n ∈ ℤ The y-axis as a vertical asymptote when n < 0.

Modelling with functions

Modelling skills: Use the modelling process described in the “mathematical modelling” section to create, fit and use the theoretical models in section SL2.5 and their graphs. Develop and fit the model: Given a context recognize and choose an appropriate model and possible parameters. Determine a reasonable domain for a model. Find the parameters of a model. Test and reflect upon the model: Comment on the appropriateness and reasonableness of a model. Justify the choice of a particular model, based on the shape of the data, properties of the curve and/or on the context of the situation. Use the model: Reading, interpreting and making predictions based on the model.

Modelling skills: Use the modelling process described in the “mathematical modelling” section to create, fit and use the theoretical models in section SL2.5 and their graphs.

Develop and fit the model: Given a context recognize and choose an appropriate model and possible parameters.

Determine a reasonable domain for a model.

Test and reflect upon the model: Comment on the appropriateness and reasonableness of a model.

Justify the choice of a particular model, based on the shape of the data, properties of the curve and/or on the context of the situation.

Use the model: Reading, interpreting and making predictions based on the model.

Modelling skills

Composite functions in context. The notation ( f ∘ g)(x) = f (g(x)). …1 Inverse function f Finding an inverse function.

The notation ( f ∘ g)(x) = f (g(x)). …1 Inverse function f Finding an inverse function.

Composite and inverse functions

Transformations of graphs. Translations: y = f (x) + b ;y = f (x … a). Reflections: in the x axis y = … f (x), and in the y axis y = f ( … x). Vertical stretch with scale factor p: y = p f (x). Horizontal stretch with scale factor : y = f (qx) Composite transformations.

Translations: y = f (x) + b ;y = f (x … a).

Reflections: in the x axis y = … f (x), and in the y axis y = f ( … x).

Vertical stretch with scale factor p: y = p f (x).

Horizontal stretch with scale factor : y = f (qx) Composite transformations.

Transformations of graphs

In addition to the models covered in the SL content the AHL content extends this to include modelling with the following functions: Exponential models to calculate half-life. Natural logarithmic models: f (x) = a + blnx Sinusoidal models: f x = asin b x … c + d Logistic models: L f x = ; L, C, k > 0 1 + Ce…kx Piecewise models.

In addition to the models covered in the SL content the AHL content extends this to include modelling with the following functions: Exponential models to calculate half-life.

Natural logarithmic models: f (x) = a + blnx Sinusoidal models: f x = asin b x … c + d Logistic models: L f x = ; L, C, k > 0 1 + Ce…kx Piecewise models.

Further modelling with functions

Scaling very large or small numbers using logarithms. Linearizing data using logarithms to determine if the data has an exponential or a power relationship using best-fit straight lines to determine parameters. Interpretation of log-log and semi-log graphs.

Scaling very large or small numbers using logarithms.

Linearizing data using logarithms to determine if the data has an exponential or a power relationship using best-fit straight lines to determine parameters.

Interpretation of log-log and semi-log graphs.

Logarithmic scaling and linearization

Geometry and trigonometry

The distance between two points in three-dimensional space, and their midpoint. Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids. The size of an angle between two intersecting lines or between a line and a plane.

The distance between two points in three-dimensional space, and their midpoint.

Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids.

The size of an angle between two intersecting lines or between a line and a plane.

3D geometry

Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles. a The sine rule: = sinA sinB sinC The cosine rule: c2 = a2 + b2 … 2abcosC a2 + b2 … c2 cosC = 2ab Area of a triangle as absinC.

Trigonometry in triangles

Applications of right and non-right angled trigonometry, including Pythagoras’ theorem. Angles of elevation and depression. Construction of labelled diagrams from written statements.

Applications of right and non-right angled trigonometry, including Pythagoras’ theorem.

Construction of labelled diagrams from written statements.

Applications of trigonometry

The circle: length of an arc; area of a sector.

Arcs and sectors

Perpendicular bisectors

Voronoi diagrams: sites, vertices, edges, cells. Addition of a site to an existing Voronoi diagram. Nearest neighbour interpolation. Applications of the “toxic waste dump” problem.

Voronoi diagrams: sites, vertices, edges, cells.

Addition of a site to an existing Voronoi diagram.

Applications of the “toxic waste dump” problem.

Voronoi diagrams

The definition of a radian and conversion between degrees and radians. Using radians to calculate area of sector, length of arc.

The definition of a radian and conversion between degrees and radians.

Using radians to calculate area of sector, length of arc.

Radians

The definitions of cosθ and sinθ in terms of the unit circle. The Pythagorean identity: cos2θ + sin2θ = 1 Definition of tanθ as Extension of the sine rule to the ambiguous case. Graphical methods of solving trigonometric equations in a finite interval.

The definitions of cosθ and sinθ in terms of the unit circle.

The Pythagorean identity: cos2θ + sin2θ = 1 Definition of tanθ as Extension of the sine rule to the ambiguous case.

Graphical methods of solving trigonometric equations in a finite interval.

Unit circle and trigonometric equations

Geometric transformations of points in two dimensions using matrices: reflections, horizontal and vertical stretches, enlargements, translations and rotations. Compositions of the above transformations. Geometric interpretation of the determinant of a transformation matrix.

Geometric transformations of points in two dimensions using matrices: reflections, horizontal and vertical stretches, enlargements, translations and rotations.

Compositions of the above transformations.

Geometric interpretation of the determinant of a transformation matrix.

Matrix transformations

Concept of a vector and a scalar. Representation of vectors using directed line segments. Unit vectors; base vectors i, Components of a vector; column representation; v1 v = v2 = v1i + v2 j + v3k v3 The zero vector 0, the vector …v. → Position vectors OA = a. Rescaling and normalizing vectors.

Representation of vectors using directed line segments.

Unit vectors; base vectors i, Components of a vector; column representation; v1 v = v2 = v1i + v2 j + v3k v3 The zero vector 0, the vector …v. → Position vectors OA = a.

Introduction to vectors

Vector equation of a line in two and three dimensions: r = a + λb, where b is a direction vector of the line.

Vector equation of a line

Vector applications to kinematics. Modelling linear motion with constant velocity in two and three dimensions. Motion with variable velocity in two dimensions.

Modelling linear motion with constant velocity in two and three dimensions.

Motion with variable velocity in two dimensions.

Vector applications to kinematics

Definition and calculation of the scalar product of two vectors. The angle between two vectors; the acute angle between two lines. Definition and calculation of the vector product of two vectors. Geometric interpretation of | v × w |. Components of vectors.

Definition and calculation of the scalar product of two vectors.

The angle between two vectors; the acute angle between two lines.

Definition and calculation of the vector product of two vectors.

Scalar and vector products

Graph theory: Graphs, vertices, edges, adjacent vertices, adjacent edges. Degree of a vertex. Simple graphs; complete graphs; weighted graphs. Directed graphs; in degree and out degree of a directed graph. Subgraphs; trees.

Graph theory: Graphs, vertices, edges, adjacent vertices, adjacent edges.

Simple graphs; complete graphs; weighted graphs.

Directed graphs; in degree and out degree of a directed graph.

Graph theory

Adjacency matrices. Walks. Number of k -length walks (or less than k -length walks) between two vertices. Weighted adjacency tables. Construction of the transition matrix for a strongly-connected, undirected or directed graph.

Number of k -length walks (or less than k -length walks) between two vertices.

Construction of the transition matrix for a strongly-connected, undirected or directed graph.

Adjacency matrices and walks

Tree and cycle algorithms with undirected graphs. Walks, trails, paths, circuits, cycles. Eulerian trails and circuits. Hamiltonian paths and cycles. Minimum spanning tree (MST) graph algorithms: Kruskal’s and Prim’s algorithms for finding minimum spanning trees. Chinese postman problem and algorithm for solution, to determine the shortest route around a weighted graph with up to four odd vertices, going along each edge at least once. Travelling salesman problem to determine the Hamiltonian cycle of least weight in a weighted complete graph. Nearest neighbour algorithm for determining an upper bound for the travelling salesman problem. Deleted vertex algorithm for determining a lower bound for the travelling salesman problem.

Tree and cycle algorithms with undirected graphs.

Minimum spanning tree (MST) graph algorithms: Kruskal’s and Prim’s algorithms for finding minimum spanning trees.

Chinese postman problem and algorithm for solution, to determine the shortest route around a weighted graph with up to four odd vertices, going along each edge at least once.

Travelling salesman problem to determine the Hamiltonian cycle of least weight in a weighted complete graph.

Nearest neighbour algorithm for determining an upper bound for the travelling salesman problem.

Deleted vertex algorithm for determining a lower bound for the travelling salesman problem.

Tree and cycle algorithms

Statistics and probability

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.

Concepts of population, sample, random sample, discrete and continuous data.

Reliability of data sources and bias in sampling.

Sampling techniques and their effectiveness.

Sampling

Presentation of data (discrete and continuous): frequency distributions (tables). Histograms. Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR). Production and understanding of box and whisker diagrams.

Presentation of data (discrete and continuous): frequency distributions (tables).

Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR).

Production and understanding of box and whisker diagrams.

Presentation of data

Measures of central tendency (mean, median and mode). Estimation of mean from grouped data. Modal class. Measures of dispersion (interquartile range, standard deviation and variance). Effect of constant changes on the original data. Quartiles of discrete data.

Measures of central tendency (mean, median and mode).

Measures of dispersion (interquartile range, standard deviation and variance).

Effect of constant changes on the original data.

Measures of central tendency and dispersion

Linear correlation of bivariate data. Pearson’s product-moment correlation coefficient, r. hand Scatter diagrams; lines of best fit, by eye, passing through the mean point. Equation of the regression line of y on x. Use of the equation of the regression line for prediction purposes. Interpret the meaning of the parameters, a and b, in • a linear regression y = ax + b.

Pearson’s product-moment correlation coefficient, r. hand Scatter diagrams; lines of best fit, by eye, passing through the mean point.

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Physics

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Markov chains

Syllabus concepts to know

Study & practice Markov chains

Practice Markov chains in Question Bank

Read Notes for Markov chains

Review Markov chains Flashcards