The second derivative. Graphical behaviour of functions, including the relationship between the graphs of f , f ′ and f ″.

Exploring the IB Mathematics Analysis & Approaches syllabus provides a clear view of the skills and knowledge you'll acquire throughout the course. Join us as we unpack the key concepts of the syllabus, giving you a better understanding of what to expect on your learning journey.

Math AA

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 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 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 n th term and the sum of the first n terms of the sequence. Use of sigma notation for the sums of geometric sequences. Applications.

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

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

Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof. The symbols and notation for equality and identity.

Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof.

The symbols and notation for equality and identity.

Simple deductive proof

Laws of exponents with rational exponents. Laws of logarithms. loga xy = loga x + logay x loga = loga x − logay y m loga x = mloga x for a, x, y > 0 Change of base of a logarithm. logb x loga x = , for a, b, logba Solving exponential equations, including using logarithms.

Laws of exponents with rational exponents.

Laws of logarithms. loga xy = loga x + logay x loga = loga x − logay y m loga x = mloga x for a, x, y > 0 Change of base of a logarithm. logb x loga x = , for a, b, logba Solving exponential equations, including using logarithms.

Rational exponents and laws of logarithms

Sum of infinite convergent geometric sequences.

Infinite geometric sequences

The binomial theorem: expansion of (a + b) , n ∈ ℕ. Use of Pascal’s triangle and nCr.

The binomial theorem: expansion of (a + b) , n ∈ ℕ.

Binomial theorem

Counting principles, including permutations and combinations. Extension of the binomial theorem to fractional and negative indices, ie (a + b) , n ∈ ℚ.

Counting principles, including permutations and combinations.

Extension of the binomial theorem to fractional and negative indices, ie (a + b) , n ∈ ℚ.

Counting principles and binomial theorem

Partial fractions

Complex numbers: the number i, where i2 = − 1. Cartesian form z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument. The complex plane.

Complex numbers: the number i, where i2 = − 1.

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

Complex numbers in Cartesian form

Modulus–argument (polar) form: z = r(cosθ + isinθ) = rcisθ Euler form: z = reiθ Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.

Complex numbers in polar and Euler form

Complex conjugate roots of quadratic and polynomial equations with real coefficients. De Moivre’s theorem and its extension to rational exponents. Powers and roots of complex numbers.

Complex conjugate roots of quadratic and polynomial equations with real coefficients.

De Moivre’s theorem and its extension to rational exponents.

De Moivre’s theorem

Proof by mathematical induction. Proof by contradiction. Use of a counterexample to show that a statement is not always true.

Use of a counterexample to show that a statement is not always true.

Proof by induction, contradiction and counterexample

Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, infinite number of solutions or no solution.

Systems of linear equations

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. 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.

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

Composite functions. Identity function. Finding the inverse function −1 f (x).

Composite and inverse functions

The quadratic function f (x) = ax2 + bx + c: its graph, y -intercept (0, c). Axis of symmetry. The form f (x) = a(x − p)(x − q), x-intercepts (p, 0) and (q, 0). The form f (x) = a (x − h)2 + k, vertex (h, k).

The quadratic function f (x) = ax2 + bx + c: its graph, y -intercept (0, c).

The form f (x) = a(x − p)(x − q), x-intercepts (p, 0) and (q, 0).

The form f (x) = a (x − h)2 + k, vertex (h, k).

Quadratic functions

Solution of quadratic equations and inequalities. The quadratic formula. The discriminant Δ = b2 − 4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.

Solution of quadratic equations and inequalities.

The discriminant Δ = b2 − 4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.

Quadratic equations and inequalities

The reciprocal function f (x) = , x ≠ 0: its graph and self-inverse nature. Rational functions of the form f (x) = their graphs. Equations of vertical and horizontal asymptotes.

The reciprocal function f (x) = , x ≠ 0: its graph and self-inverse nature.

Rational functions of the form f (x) = their graphs.

Equations of vertical and horizontal asymptotes.

Reciprocal and rational functions

Exponential functions and their graphs: f (x) = ax, a > 0, f (x) = ex Logarithmic functions and their graphs: f (x) = loga x, x > 0, f (x) = lnx, x > 0.

Exponential and logarithmic functions

Solving equations, both graphically and analytically. Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach. Applications of graphing skills and solving equations that relate to real-life situations.

Solving equations, both graphically and analytically.

Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.

Applications of graphing skills and solving equations that relate to real-life situations.

Solving equations

Transformations of graphs. Translations: y = f (x) + b; y = f (x − a). Reflections (in both axes): y = − f (x); 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 both axes): y = − f (x); y = f ( − x).

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

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

Transformations of graphs

Polynomial functions, their graphs and equations; zeros, roots and factors. The factor and remainder theorems. Sum and product of the roots of polynomial equations.

Polynomial functions, their graphs and equations; zeros, roots and factors.

Sum and product of the roots of polynomial equations.

Polynomial functions

Rational functions of the form ax + b f (x) = cx2 + dx + e

Rational functions

Odd and even functions. Finding the inverse function, f −1(x), including domain restriction. Self-inverse functions.

Finding the inverse function, f −1(x), including domain restriction.

Odd, even and inverse functions

Solutions of g(x) ≥ f (x), both graphically and analytically.

Solving inequalities

The graphs of the functions, y = | f (x)| and y = f (|x|), y = f (x) Solution of modulus equations and inequalities.

Absolute value functions

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’s theorem. Angles of elevation and depression. Construction of labelled diagrams from written statements.

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

Construction of labelled diagrams from written statements.

Applications of trigonometry

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

Radians, arcs and sectors

Definition of cosθ, sinθ in terms of the unit circle. Definition of tanθ as Exact values of trigonometric ratios of 0, , , , and their multiples. Extension of the sine rule to the ambiguous case.

Definition of cosθ, sinθ in terms of the unit circle.

Definition of tanθ as Exact values of trigonometric ratios of 0, , , , and their multiples.

Extension of the sine rule to the ambiguous case.

Unit circle and exact values

The Pythagorean identity cos2θ + sin2θ = 1. Double angle identities for sine and cosine. The relationship between trigonometric ratios.

The Pythagorean identity cos2θ + sin2θ = 1.

Double angle identities for sine and cosine.

The relationship between trigonometric ratios.

Trigonometric identities

The circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs Composite functions of the form f (x) = asin(b(x + c)) + d. Transformations. Real-life contexts.

The circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs Composite functions of the form f (x) = asin(b(x + c)) + d.

Circular functions and their graphs

Solving trigonometric equations in a finite interval, both graphically and analytically. Equations leading to quadratic equations in sinx, cosx or tanx.

Solving trigonometric equations in a finite interval, both graphically and analytically.

Equations leading to quadratic equations in sinx, cosx or tanx.

Trigonometric equations

Definition of the reciprocal trigonometric ratios secθ, cosecθ and cotθ. Pythagorean identities: The inverse functions f (x) = arcsinx, f (x) = arccosx, f (x) = arctanx; their domains and ranges; their graphs.

Definition of the reciprocal trigonometric ratios secθ, cosecθ and cotθ.

Pythagorean identities: The inverse functions f (x) = arcsinx, f (x) = arccosx, f (x) = arctanx; their domains and ranges; their graphs.

Reciprocal and inverse trigonometric functions

Compound angle identities. Double angle identity for tan.

Compound angle identities

Relationships between trigonometric functions and sin(π the symmetry properties of their graphs.

Symmetry of trigonometric graphs

Concept of a vector; position vectors; displacement vectors. Representation of vectors using directed line segments. Base vectors i, j, k. Components of a vector: v1 v = v2 = v1i + v2 j + v3k. v3 Algebraic and geometric approaches to the following: • the sum and difference of two vectors • the zero vector 0, the vector −v • multiplication by a scalar, kv, parallel vectors • magnitude of a vector, |v|; unit vectors, • position vectors OA = a, OB = b • displacement vector AB = b − a Proofs of geometrical properties using vectors.

Concept of a vector; position vectors; displacement vectors.

Representation of vectors using directed line segments.

Components of a vector: v1 v = v2 = v1i + v2 j + v3k. v3 Algebraic and geometric approaches to the following:

multiplication by a scalar, kv, parallel vectors

magnitude of a vector, |v|; unit vectors,

displacement vector AB = b − a Proofs of geometrical properties using vectors.

Introduction to vectors

The definition of the scalar product of two vectors. The angle between two vectors. Perpendicular vectors; parallel vectors.

The definition of the scalar product of two vectors.

Scalar product of vectors

Vector equation of a line in two and three dimensions: r = a + λb. The angle between two lines. Simple applications to kinematics.

Vector equation of a line in two and three dimensions: r = a + λb.

Vector equation of a line

Coincident, parallel, intersecting and skew lines, distinguishing between these cases. Points of intersection.

Coincident, parallel, intersecting and skew lines, distinguishing between these cases.

Intersecting, parallel and skew lines

The definition of the vector product of two vectors. Properties of the vector product. Geometric interpretation of | v × w |

The definition of the vector product of two vectors.

Vector product

Vector equations of a plane: r = a + λb + μc, where b and c are non-parallel vectors within the plane. r · n = a · n, where n is a normal to the plane and a the position vector of a point on the plane. Cartesian equation of a plane ax + by + cz = d.

Vector equations of a plane: r = a + λb + μc, where b and c are non-parallel vectors within the plane. r · n = a · n, where n is a normal to the plane and a the position vector of a point on the plane.

Cartesian equation of a plane ax + by + cz = d.

Equations of a plane

Intersections of: a line with a plane; two planes; three planes. Angle between: a line and a plane; two planes.

Intersections of: a line with a plane; two planes; three planes.

Angle between: a line and a plane; two planes.

Intersections of lines and planes

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.

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The second derivative

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