# IB DP Mathematics HL and SL : Study Notes

Topic 1 – Core: Algebra

• Topic 1.1 :
• Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series,
• Sigma notation,
• Applications.
• Topic 1.2 :
• Exponents and logarithms,
• Laws of exponents; laws of logarithms,
• Change of base SL Paper 1
• Topic 1.3 :
• Counting principles, including permutations and combinations
• The binomial theorem: expansion of $${\left( {a + b} \right)^n}$$, $$n \in N$$
• Topic 1.4 :
• Proof by mathematical induction.
• Topic 1.5 :
•  Complex numbers: the number $${\text{i}} = \sqrt { – 1}$$ ; the terms real part, imaginary part, conjugate, modulus and argument.
• Cartesian form $$z = a + {\text{i}}b$$ .
• Sums, products and quotients of complex numbers.
• Topic 1.6 :
• Modulus–argument (polar) form $$z = r\left( {\cos \theta + {\text{i}}\sin \theta } \right) = r{\text{cis}}\theta = r{e^{{\text{i}}\theta }}$$. The complex plane.
• Sums, products and quotients of complex numbers.
• Topic 1.7 :
• Powers of complex numbers: de Moivre’s theorem.
•  $$n$$th roots of a complex number.
• Topic 1.8 :
• Conjugate roots of polynomial equations with real coefficients.
• Topic 1.9 :
• Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinity of solutions or no solution.

Topic 2 – Core: Functions and equations

• Topic 2.1
• Concept of function $$f:x \mapsto f\left( x \right)$$ : domain, range; image (value)
• Odd and even functions.
• Composite functions $$f \circ g$$
• Identity function.
• One-to-one and many-to-one functions.
• Inverse function $${f^{ – 1}}$$, including domain restriction. Self-inverse functions.
• Topic 2.2
• The graph of a function; its equation $$y = f\left( x \right)$$ .
• Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range.
• The graphs of the functions $$y = \left| {f\left( x \right)} \right|$$ and $$y = f\left( {\left| x \right|} \right)$$
• The graph of $$\frac{1}{{f\left( x \right)}}$$ given the graph of $$y = f(x)$$ .
• Topic 2.3
• Transformations of graphs: translations; stretches; reflections in the axes.
• The graph of the inverse function as a reflection in $$y = x$$.
• Topic 2.4
• The rational function $$x \mapsto \frac{{ax + b}}{{cx + d}}$$ and its graph.
• The function $$x \mapsto {a^x}$$ , $$a > 0$$ , and its graph.
• The function $$x \mapsto {\log _a}x$$ , $$x > 0$$ , and its graph
• Topic 2.5
• Polynomial functions and their graphs.
• The factor and remainder theorems.
• The fundamental theorem of algebra.
• Topic 2.6
• Use of the discriminant $$\Delta = {b^2} – 4ac$$ to determine the nature of the roots.
• Solving polynomial equations both graphically and algebraically.
• Sum and product of the roots of polynomial equations.
• Solution of $${a^x} = b$$ using logarithms.
• Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
• Topic 2.7
• Solutions of $$g\left( x \right) \geqslant f\left( x \right)$$ .
• Graphical or algebraic methods, for simple polynomials up to degree 3.
• Use of technology for these and other functions.

Topic 3 – Core: Circular functions and trigonometry

• Topic 3.1 :
• The circle: radian measure of angles.
• Length of an arc; area of a sector.
• Topic 3.2 :
• Definition of $$\cos \theta$$ , $$\sin \theta$$ and $$\tan \theta$$ in terms of the unit circle.
• Exact values of $$\sin$$, $$\cos$$ and $$\tan$$ of $$0$$, $$\frac{\pi }{6}$$, $$\frac{\pi }{4}$$, $$\frac{\pi }{3}$$, $$\frac{\pi }{2}$$ and their multiples.
• Definition of the reciprocal trigonometric ratios $$\sec \theta$$ , $$\csc \theta$$ and $$\cot \theta$$ .
• Pythagorean identities: $${\cos ^2}\theta + {\sin ^2}\theta = 1$$ ; $$1 + {\tan ^2}\theta = {\sec ^2}\theta$$ ; $$1 + {\cot ^2}\theta = {\csc ^2}\theta$$ .
• Topic 3.3 :
• Compound angle identities.
• Double angle identities
• Topic 3.4 :
• Composite functions of the form $$f(x) = a\sin (b(x + c)) + d$$ .
• Applications.
• Topic 3.5 :
• The inverse functions $$x \mapsto \arcsin x$$ , $$x \mapsto \arccos x$$ , $$x \mapsto \arctan x$$ ; their domains and ranges; their graphs.
• Topic 3.6 :
• Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorization.
• Topic 3.7 :
• The cosine rule.
• The sine rule including the ambiguous case.
• Area of a triangle as $$\frac{1}{2}ab\sin C$$ .
• Applications.

Topic 4 – Core: Vectors

• Topic 4.1
• Concept of a vector.
• Representation of vectors using directed line segments.
• Unit vectors; base vectors $$i$$, $$j$$, $$k$$.
• Components of a vector: $$v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k$$ .
• Algebraic and geometric approaches to the sum and difference of two vectors.
• Algebraic and geometric approaches to the zero vector $$0$$, the vector $$– v$$ .
• Algebraic and geometric approaches to multiplication by a scalar, $$kv$$ .
• Algebraic and geometric approaches to magnitude of a vector, $$\left| v \right|$$ .
• Algebraic and geometric approaches to position vectors $$\overrightarrow {OA} = a$$ .
• $$\overrightarrow {AB} = b – a$$ .
• Topic 4.2
• The definition of the scalar product of two vectors.
• Properties of the scalar product: $${\boldsymbol{v}} \cdot {\boldsymbol{w}} = {\boldsymbol{w}} \cdot {\boldsymbol{v}}$$ ; $${\boldsymbol{u}} \cdot \left( {{\mathbf{v}} + {\boldsymbol{w}}} \right) = {\boldsymbol{u}} \cdot {\boldsymbol{v}} + {\boldsymbol{u}} \cdot {\boldsymbol{w}}$$ ; $$\left( {k{\boldsymbol{v}}} \right) \cdot {\boldsymbol{w}} = k\left( {{\boldsymbol{v}} \cdot {\boldsymbol{w}}} \right)$$ ; $${\boldsymbol{v}} \cdot {\boldsymbol{v}} = {\left| {\boldsymbol{v}} \right|^2}$$ .
• The angle between two vectors.
• Perpendicular vectors; parallel vectors.
• Topic 4.3
• Vector equation of a line in two and three dimensions: $$r = a + \lambda b$$ .
• Simple applications to kinematics.
• The angle between two lines.
• Topic 4.4
• Coincident, parallel, intersecting and skew lines; distinguishing between these cases.
• Points of intersection.
• Topic 4.5
• The definition of the vector product of two vectors.
• Properties of the vector product: $${\text{v}} \times {\text{w}} = – {\text{w}} \times {\text{v}}$$ ; $${\text{u}} \times ({\text{v}} + {\text{w}}) = {\text{u}} \times {\text{v}} + {\text{u}} \times {\text{w}}$$ ; $$(k{\text{v}}) \times {\text{w}} = k({\text{v}} + {\text{w}})$$ ; $${\text{v}} \times {\text{v}} = 0$$ .
• Geometric interpretation of $${\text{v}} \times {\text{w}}$$ .
• Topic 4.6
• Vector equation of a plane $$r = a + \lambda b + \mu c$$ .
• Use of normal vector to obtain the form $$r \cdot n = a \cdot n$$ .
• Cartesian equation of a plane $$ax + by + cz = d$$ .
• Topic 4.7
• Intersections of: a line with a plane; two planes; three planes.
• Angle between: a line and a plane; two planes.

Topic 5 – Core: Statistics and probability

• Topic 5.1 :
• Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.
• Grouped data: mid-interval values, interval width, upper and lower interval boundaries.
• Mean, variance, standard deviation.
• Topic 5.2 :
• Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
• The probability of an event $$A$$ is $$P\left( A \right) = \frac{{n\left( A \right)}}{{n\left( U \right)}}$$
• The complementary events $$A$$ and $${A’}$$ (not $$A$$).
• Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve problems.
• Topic 5.3 :
• Combined events; the formula for $$P\left( {A \cup B} \right)$$ .
• Mutually exclusive events.
• Topic 5.4 :
• Conditional probability; the definition $$P\left( {\left. A \right|P} \right) = \frac{{P\left( {A\mathop \cap \nolimits B} \right)}}{{P\left( B \right)}}$$ .
• Independent events; the definition $$P\left( {\left. A \right|B} \right) = P\left( A \right) = P\left( {\left. A \right|B’} \right)$$ .
• Use of Bayes’ theorem for a maximum of three events.
• Topic 5.5 :
• Concept of discrete and continuous random variables and their probability distributions.
• Definition and use of probability density functions.
• Expected value (mean), mode, median, variance and standard deviation.
• Applications.
• Topic 5.6 :
• Binomial distribution, its mean and variance.
• Poisson distribution, its mean and variance.
• Topic 5.7 :
• Normal distribution.
• Properties of the normal distribution.
• Standardization of normal variables.

Topic 6 – Core: Calculus

• Topic 6.1
• Informal ideas of limit, continuity and convergence.
• Definition of derivative from first principles as $$f’\left( x \right) = \mathop {\lim }\limits_{h \to 0} {\frac{{f\left( {x + h} \right) – f\left( x \right)}}{h}}$$.
• The derivative interpreted as a gradient function and as a rate of change.
• Finding equations of tangents and normals.
• Identifying increasing and decreasing functions.
• The second derivative.
• Higher derivatives.
• Topic 6.2
• Derivatives of $${x^n}$$ , $$\sin x$$ , $$\cos x$$ , $$\tan x$$ , $${{\text{e}}^x}$$ and \$$\ln x$$ .
• Differentiation of sums and multiples of functions.
• The product and quotient rules.
• The chain rule for composite functions.
• Related rates of change.
• Implicit differentiation.
• Derivatives of $$\sec x$$ , $$\csc x$$ , $$\cot x$$ , $${a^x}$$ , $${\log _a}x$$ , $$\arcsin x$$ , $$\arccos x$$ and $$\arctan x$$ .
• Topic 6.3
• Local maximum and minimum values.
• Optimization problems.
• Points of inflexion with zero and non-zero gradients.
• Graphical behaviour of functions, including the relationship between the graphs of $$f$$ , $${f’}$$ and $${f”}$$ .
• Topic 6.4
• Indefinite integration as anti-differentiation.
• Indefinite integral of $${x^n}$$ , $$\sin x$$ , $$\cos x$$ and $${{\text{e}}^x}$$ .
• Other indefinite integrals using the results from 6.2.
• The composites of any of these with a linear function.
• Topic 6.5
• Anti-differentiation with a boundary condition to determine the constant of integration.
• Definite integrals.
• Area of the region enclosed by a curve and the $$x$$-axis or $$y$$-axis in a given interval; areas of regions enclosed by curves.
• Volumes of revolution about the $$x$$-axis or $$y$$-axis.
• Topic 6.6
• Kinematic problems involving displacement $$s$$, velocity $$v$$ and acceleration $$a$$.
• Total distance travelled.
• Topic 6.7
• Integration by substitution.
• Integration by parts.

### Optional

Topic 7 – Option: Statistics and probability

• Topic 7.1
• Cumulative distribution functions for both discrete and continuous distributions.
• Geometric distribution.
• Negative binomial distribution.
• Probability generating functions for discrete random variables.
• Using probability generating functions to find mean, variance and the distribution of the sum of $$n$$ independent random variables.
• Topic 7.2
• Linear transformation of a single random variable.
• Mean of linear combinations of $$n$$ random variables.
• Variance of linear combinations of $$n$$ independent random variables.
• Expectation of the product of independent random variables.
• Topic 7.3
• Unbiased estimators and estimates.
• Comparison of unbiased estimators based on variances.
• $${\bar X}$$ as an unbiased estimator for $$\mu$$ .
• $${S^2}$$ as an unbiased estimator for $${\sigma ^2}$$ .
• Topic 7.4
• A linear combination of independent normal random variables is normally distributed. In particular, $$X{\text{ ~ }}N\left( {\mu ,{\sigma ^2}} \right) \Rightarrow \bar X{\text{ ~ }}N\left( {\mu ,\frac{{{\sigma ^2}}}{n}} \right)$$ .
• The central limit theorem.
• Topic 7.5
• Confidence intervals for the mean of a normal population.
• Topic 7.6
• Null and alternative hypotheses, $${H_0}$$ and $${H_1}$$ .
• Significance level.
• Critical regions, critical values, $$p$$-values, one-tailed and two-tailed tests.
• Type I and II errors, including calculations of their probabilities.
• Testing hypotheses for the mean of a normal population.
• Topic 7.7
• Introduction to bivariate distributions.
• Covariance and (population) product moment correlation coefficient $$\rho$$.
• Proof that $$\rho = 0$$ in the case of independence and $$\pm 1$$ in the case of a linear relationship between $$X$$ and $$Y$$.
• Definition of the (sample) product moment correlation coefficient $$R$$ in terms of n paired observations on $$X$$ and $$Y$$.
• Its application to the estimation of $$\rho$$.Informal interpretation of $$r$$, the observed value of $$R$$. Scatter diagrams.
• Topics based on the assumption of bivariate normality: use of the $$t$$-statistic to test the null hypothesis $$\rho = 0$$ .
• Topics based on the assumption of bivariate normality: knowledge of the facts that the regression of $$X$$ on $$Y$$ ($${E\left. {\left( X \right)} \right|Y = y}$$) and $$Y$$ on $$X$$ ($${E\left. {\left( Y \right)} \right|X = x}$$) are linear.
• Topics based on the assumption of bivariate normality: least-squares estimates of these regression lines (proof not required).
• Topics based on the assumption of bivariate normality: the use of these regression lines to predict the value of one of the variables given the value of the other.

Topic 8 – Option: Sets, relations and groups

• Topic 8.1
• Finite and infinite sets.
• Subsets.
• Operations on sets: union; intersection; complement; set difference; symmetric difference.
• De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).
• Topic 8.2
• Ordered pairs: the Cartesian product of two sets.
• Relations: equivalence relations; equivalence classes.
• Topic 8.3
• Functions: injections; surjections; bijections.
• Composition of functions and inverse functions.
• Topic 8.4
• Binary operations.
• Operation tables (Cayley tables).
• Topic 8.5
• Binary operations: associative, distributive and commutative properties.
• Topic 8.6
• The identity element $$e$$.
• The inverse $${a^{ – 1}}$$ of an element $$a$$.Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse.
• Proofs of the uniqueness of the identity and inverse elements.
• Topic 8.7
• The definition of a group $$\left\{ {G, * } \right\}$$ .
• The operation table of a group is a Latin square, but the converse is false.
• Abelian groups.
• Topic 8.8
• Example of groups: $$\mathbb{R}$$, $$\mathbb{Q}$$, $$\mathbb{Z}$$ and $$\mathbb{C}$$ under addition.
• Example of groups: integers under addition modulo $$n$$.
• Example of groups: non-zero integers under multiplication, modulo $$p$$, where $$p$$ is prime.
• Symmetries of plane figures, including equilateral triangles and rectangles.
• Invertible functions under composition of functions.
• Topic 8.9
• The order of a group.
• The order of a group element.
• Cyclic groups.
• Generators.
• Proof that all cyclic groups are Abelian.
• Topic 8.10
• Permutations under composition of permutations.
• Cycle notation for permutations.
• Result that every permutation can be written as a composition of disjoint cycles.
• The order of a combination of cycles.
• Topic 8.11
• Subgroups, proper subgroups.
• Use and proof of subgroup tests.
• Definition and examples of left and right cosets of a subgroup of a group.
• Lagrange’s theorem.
• Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange’s theorem.)
• Topic 8.12
• Definition of a group homomorphism.
• Definition of the kernel of a homomorphism.
• Proof that the kernel and range of a homomorphism are subgroups.
• Proof of homomorphism properties for identities and inverses.
• Isomorphism of groups.
• The order of an element is unchanged by an isomorphism.

Topic 9 – Option: Calculus

• Topic 9.1
• Infinite sequences of real numbers and their convergence or divergence.
• Topic 9.2
• Convergence of infinite series.
• Tests for convergence: comparison test; limit comparison test; ratio test; integral test.
• The $$p$$-series, $$\mathop \sum \nolimits \frac{1}{{{n^p}}}$$ .Series that converge absolutely.
• Series that converge conditionally.
• Alternating series.
• Power series: radius of convergence and interval of convergence. Determination of the radius of convergence by the ratio test.
• Topic 9.3
• Continuity and differentiability of a function at a point.
• Continuous functions and differentiable functions.
• Topic 9.4
• The integral as a limit of a sum; lower and upper Riemann sums.
• Fundamental theorem of calculus.
• Improper integrals of the type $$\int\limits_a^\infty {f\left( x \right){\text{d}}} x$$ .
• Topic 9.5
• First-order differential equations.
• Geometric interpretation using slope fields, including identification of isoclines.
• Numerical solution of $$\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {x,y} \right)$$ using Euler’s method.
• Variables separable.
• Homogeneous differential equation $$\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {\frac{y}{x}} \right)$$ using the substitution $$y = vx$$ .Solution of $$y’ + P\left( x \right)y = Q\left( x \right)$$, using the integrating factor.
• Topic 9.6
• Rolle’s theorem.
• Mean value theorem.
• Taylor polynomials; the Lagrange form of the error term.
• Maclaurin series for $${{\text{e}}^x}$$ , $$\\sin x$$ , $$\cos x$$ , $$\ln (1 + x)$$ , $${(1 + x)^p}$$ , $$P \in \mathbb{Q}$$ .
• Use of substitution, products, integration and differentiation to obtain other series.
• Taylor series developed from differential equations.
• Topic 9.7
• The evaluation of limits of the form $$\mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right)}}{{g\left( x \right)}}$$ and $$\mathop {\lim }\limits_{x \to \infty } \frac{{f\left( x \right)}}{{g\left( x \right)}}$$ .
• Using l’Hôpital’s rule or the Taylor series.

Topic 10 – Option: Discrete mathematics

• Topic 10.1
• Strong induction.
• Pigeon-hole principle.
• Topic 10.2
• $$\left. a \right|b \Rightarrow b = na$$ for some $$n \in \mathbb{Z}$$ .
• The theorem $$\left. a \right|b$$ and $$a\left| {c \Rightarrow a} \right|\left( {bx \pm cy} \right)$$ where $$x,y \in \mathbb{Z}$$ .
• Division and Euclidean algorithms.
• The greatest common divisor, gcd($$a$$,$$b$$), and the least common multiple, lcm($$a$$,$$b$$), of integers $$a$$ and $$b$$.
• Prime numbers; relatively prime numbers and the fundamental theorem of arithmetic.
• Topic 10.3
• Linear Diophantine equations $$ax + by = c$$ .
• Topic 10.4
• Modular arithmetic.
• The solution of linear congruences.
• Solution of simultaneous linear congruences (Chinese remainder theorem).
• Topic 10.5
• Representation of integers in different bases.
• Topic 10.6
• Fermat’s little theorem.
• Topic 10.7
• Graphs, vertices, edges, faces.
• Degree of a vertex, degree sequence.
• Handshaking lemma.
• Simple graphs; connected graphs; complete graphs; bipartite graphs; planar graphs; trees; weighted graphs, including tabular representation.
• Subgraphs; complements of graphs
• .Euler’s relation: $$v – e + f = 2$$ ; theorems for planar graphs including $$e \leqslant 3v – 6$$ , $$e \leqslant 2v – 4$$ , leading to the results that $${\kappa _5}$$ and $${\kappa _{3,3}}$$ are not planar.
• Topic 10.8
• Walks, trails, paths, circuits, cycles.
• Eulerian trails and circuits.
• Hamiltonian paths and cycles.
• Topic 10.9
• Graph algorithms: Kruskal’s; Dijkstra’s.
• Topic 10.10
• Chinese postman problem.
• Travelling salesman problem.
• Nearest-neighbour algorithm for determining an upper bound.
• Deleted vertex algorithm for determining a lower bound.
• Topic 10.11
• Recurrence relations.
• Initial conditions, recursive definition of a sequence.
• Solution of first- and second-degree linear homogeneous recurrence relations with constant coefficients.
• The first-degree linear recurrence relation $${u_n} = a{u_{n – 1}} + b$$ .
• Modelling with recurrence relations.

### New DP mathematics courses

Two new courses will become part of the Diploma Programme (DP) in 2019, both taught at the Higher level (HL) and Standard level (SL). The first is Mathematics: analysis and approaches and the second is Mathematics: applications and interpretation. Each course approaches topics at varying levels of teaching hours. This guide will provide some insights into which course will be the best fit for your school and students.

The courses are separated by how they approach mathematics, described generally by the table below:

#### Mathematics: analysis and approaches

• Emphasis on algebraic methods
• Develop strong skills in mathematical thinking
• Real and abstract mathematical problem solving
• For students interested in mathematics, engineering physical sciences, and some economics

#### Mathematics: applications and interpretation

• Emphasis on modelling and statistics
• Develops strong skills in applying mathematics to the real-world
• Real mathematical problem solving using technology
• For students interested in social sciences, natural sciences, medicine, statistics, business, engineering, some economics, psychology and design

#### Subject breakdown

All of these courses (SL and HL in each) cover the same 5 topics within mathematics but with varying emphasis in each area: number and algebra, functions, geometry and trigonometry, statics and probability, and calculus. The chart below may help you select the right course based on the amount of time dedicated to a given topic.