HL Syllabus

Math HL

SYLLABUS OUTLINE

The course consists of the study of seven core topics and one option topic. Total 240 hrs

Core syllabus content 190 hrs

Requirements

All topics in the core are compulsory. Students must study all the sub-topics in each of the topics in the syllabus as listed in this guide. Students are also required to be familiar with the topics listed as presumed knowledge (PK).

Topic 1—Algebra 20 hrs

Topic 2—Functions and equations 26 hrs

Topic 3—Circular functions and trigonometry 22 hrs

Topic 4—Matrices 12 hrs

Topic 5—Vectors 22 hrs

Topic 6—Statistics and probability 40 hrs

Topic 7—Calculus 48 hrs

Option syllabus content 40 hrs

Requirements

Students must study all the sub-topics in one of the following options as listed in the syllabus details.

Topic 8—Statistics and probability 40 hrs

Topic 9—Sets, relations and groups 40 hrs

Topic 10—Series and differential equations 40 hrs

Topic 11—Discrete mathematics 40 hrs

Portfolio 10 hrs

Two pieces of work, based on different areas of the syllabus, representing the following two types of tasks:

• mathematical investigation

• mathematical modelling.

SYLLABUS DETAILS

Format of the syllabus

The syllabus to be taught is presented as three topics.

• Content: the first column lists, under each topic, the sub-topics to be covered.

• Amplifications/inclusions: the second column contains more explicit information on specific sub-topics listed in the first column. This helps to define what is required in terms of preparing for the examination.

• Exclusions: the third column contains information about what is not required in terms of preparing for the examination.

Although the mathematics HL course is similar in content to parts of the mathematics SL course, there are differences. In particular, students and teachers are expected to take a more sophisticated approach for mathematics HL, during the course and in the examinations. Where appropriate, guidelines are provided in the second and third columns of the syllabus details (as indicated by the phrase “See SL guide”).

Course of study

Teachers are required to teach all the sub-topics listed for the seven topics in the core, together with all the sub-topics in the chosen option. The topics in the syllabus do not need to be taught in the order in which they appear in this guide. Similarly, it is not necessary to teach all the topics in the core before starting to teach an option. Teachers should therefore construct a course of study that is tailored to the needs of their students and that integrates the areas covered by the syllabus, and, where necessary, the presumed knowledge (PK).

Integration of portfolio assignments

The two pieces of work for the portfolio, based on the two types of tasks (mathematical investigation and mathematical modeling), should be incorporated into the course of study, and should relate directly to topics in the syllabus. Full details of how to do this are given in the section on internal assessment.

Time allocation

The recommended teaching time for higher level courses is 240 hours. For mathematics HL, it is expected that 10 hours will be spent on work for the portfolio. The time allocations given in this guide are approximate, and are intended to suggest how the remaining 230 hours allowed for the teaching of the syllabus might be allocated. However, the exact time spent on each topic depends on a number of factors, including the background knowledge and level of preparedness of each student. Teachers should therefore adjust these timings to correspond to the needs of their students.

Use of calculators

Students are expected to have access to a graphic display calculator (GDC) at all times during the course. The minimum requirements are reviewed as technology advances, and updated information will be provided to schools. It is expected that teachers and schools monitor calculator use with reference to the calculator policy. Regulations covering the types of calculator allowed are provided in the Vade Mecum. Further information and advice is provided in the teacher support material. There are specific requirements for calculators used by students studying the statistics and probability option.

Mathematics HL information booklet

Because each student is required to have access to a clean copy of this booklet during the examination, it is recommended that teachers ensure students are familiar with the contents of this document from the beginning of the course. The booklet is provided by the IBO and is published separately.

Teacher support materials

A variety of teacher support materials will accompany this guide. These materials will include suggestions to help teachers integrate the use of GDCs into their teaching, guidance for teachers on the marking of portfolios, and specimen examination papers and markschemes. These will be distributed to all schools.

External assessment guidelines

It is recommended that teachers familiarize themselves with the section on external assessment guidelines, as this contains important information about the examination papers. In particular, students need to be familiar with notation the IBO uses and the command terms, as these will be used without explanation in the examination papers.

Presumed knowledge

General

Students are not required to be familiar with all the topics listed as PK before they start this course. However, they should be familiar with these topics before they take the examinations, because questions assume knowledge of them. Teachers must therefore ensure that any topics designated as PK that are unknown to their students at the start of the course are included at an early stage. They should also take into account the existing mathematical knowledge of their students to design an appropriate course of study for mathematics HL. This list of topics is not designed to represent the outline of a course that might lead to the mathematics HL course. Instead, it lists the knowledge, together with the syllabus content, that is essential to successful completion of the mathematics HL course. Students must be familiar with SI (Système International) units of length, mass and time, and their derived units.

Topics

Number and algebra

Routine use of addition, subtraction, multiplication and division using integers, decimals and fractions, including order of operations. Example: 2(3+4×7)=62.

Simple positive exponents. Examples: 2³ =8; (-3)³= -27; (-2)⁴=16.

Simplification of expressions involving roots (surds or radicals). Examples: .

Prime numbers and factors, including greatest common factors and least common multiples.

Simple applications of ratio, percentage and proportion, linked to similarity.

Definition and elementary treatment of absolute value (modulus), a .

Rounding, decimal approximations and significant figures, including appreciation of errors.

Expression of numbers in standard form (scientific notation), that is, a ×10^k , � .

Concept and notation of sets, elements, universal (reference) set, empty (null) set, complement, subset, equality of sets, disjoint sets. Operations on sets: union and intersection. Commutative, associative and distributive properties. Venn diagrams.

Number systems: natural numbers; integers, 􀁝; rationals, 􀁟, and irrationals; real numbers, 􀁜.

Intervals on the real number line using set notation and using inequalities. Expressing the solution set

of a linear inequality on the number line and in set notation.

The concept of a relation between the elements of one set and between the elements of one set and

those of another set. Mappings of the elements of one set onto or into another, or the same, set.

Illustration by means of tables, diagrams and graphs.

Basic manipulation of simple algebraic expressions involving factorization and expansion.

Examples: ab+ac=a(b+c) ; (a±b)2=a2+b2±2ab; a2−b2=(a−b)(a+b);

3x2+5x+2=(3x+2)(x+1); xa−2a+xb−2b=(x−2)(a+b) .

Rearrangement, evaluation and combination of simple formulae. Examples from other subject areas,

particularly the sciences, should be included.

The linear function x 􀀶 ax + b and its graph, gradient and y-intercept.

Addition and subtraction of algebraic fractions with denominators of the form ax + b .

Example: 2 3 1

3 1 2 4

x x

x x

+ +

− +

The properties of order relations: <, ≤, >, ≥.

Examples: a>b,c>0⇒ac>bc; a>b,c<0 ⇒ac<bc.

Solution of equations and inequalities in one variable, including cases with rational coefficients.

Example: 3 2 1(1 ) 5

7 5 2 7

− x= −x ⇒x= .

Solution of simultaneous equations in two variables.

SYLLABUS DETAILS

12 © International Baccalaureate Organization 2006

Geometry

Elementary geometry of the plane including the concepts of dimension for point, line, plane and space.

Parallel and perpendicular lines, including m1=m2, and 1 2 mm = −1. Geometry of simple plane

figures. The function x 􀀶 ax + b : its graph, gradient and y-intercept.

Angle measurement in degrees. Compass directions and bearings. Right-angle trigonometry. Simple

applications for solving triangles.

Pythagoras’ theorem and its converse.

The Cartesian plane: ordered pairs (x, y) , origin, axes. Mid-point of a line segment and distance

between two points in the Cartesian plane.

Simple geometric transformations: translation, reflection, rotation, enlargement. Congruence and

similarity, including the concept of scale factor of an enlargement.

The circle, its centre and radius, area and circumference. The terms “arc”, “sector”, “chord”, “tangent”

and “segment”.

Perimeter and area of plane figures. Triangles and quadrilaterals, including parallelograms,

rhombuses, rectangles, squares, kites and trapeziums (trapezoids); compound shapes.

Statistics

Descriptive statistics: collection of raw data, display of data in pictorial and diagrammatic forms (for

example, pie charts, pictograms, stem and leaf diagrams, bar graphs and line graphs).

Calculation of simple statistics from discrete data, including mean, median and mode.

Core syllabus content

Topic 1—Core: Algebra 20 hrs

Aims

The aim of this section is to introduce students to some basic algebraic concepts and applications.

Details

Content Amplifications/inclusions Exclusions

Arithmetic sequences and series; sum of finite

arithmetic series; geometric sequences and

series; sum of finite and infinite geometric

series.

Examples of applications: compound interest

and population growth.

1.1

Sigma notation.

Exponents and logarithms. Elementary treatment only is required.

Laws of exponents; laws of logarithms.

1.2

Change of base. log log

log

c

b

c

a a

b

= .

Counting principles, including permutations and

combinations.

Simple applications only.

The formula for

n

r

⎛ ⎞

⎜ ⎟

⎝ ⎠

also denoted by nC

r .

Formula for nP

r .

Permutations where some objects are identical.

The binomial theorem: expansion of

(a+b)n , n∈􀁠 .

1.3

See SL guide

Topic 1—Core: Algebra (continued)

Content Amplifications/inclusions Exclusions

1.4 Proof by mathematical induction. Proof of binomial theorem.

Forming conjectures to be proved by

mathematical induction.

Complex numbers: the number i= −1; the

terms real part, imaginary part, conjugate,

modulus and argument.

Cartesian form z=a+ib.

Modulus–argument form z=r(cosθ+isinθ). Awareness that z=r(cosθ+isinθ)can be

written as z=reiθ and z=rcisθ .

1.5

The complex plane. The complex plane is also known as the Argand

diagram.

Loci in the complex plane.

1.6 Sums, products and quotients of complex

numbers.

1.7 De Moivre’s theorem. Proof by mathematical induction for n∈􀁝+ .

Powers and roots of a complex number.

1.8 Conjugate roots of polynomial equations with

real coefficients.

Equations with complex coefficients.

Topic 2—Core: Functions and equations 26 hrs

Aims

The aims of this section are to explore the notion of function as a unifying theme in mathematics, and to apply functional methods to a variety of mathematical

situations. It is expected that extensive use will be made of a GDC in both the development and the application of this topic.

Details

Content Amplifications/inclusions Exclusions

Concept of function f :x􀀶f(x) : domain,

range; image (value).

On examination papers: if the domain is the set

of real numbers then the statement “ x∈􀁜 ” will

be omitted.

The term “codomain”.

Composite functions f 􀁄 g ; identity function. The composite function ( f 􀁄g)(x) is defined as

f(g(x)).

Inverse function f −1 . Distinction between one-to-one and

many-to-one functions. Domain restriction.

2.1

See SL guide

The graph of a function; its equation y= f(x) . On examination papers: questions may be set

that require the graphing of functions that do not

explicitly appear on the syllabus.

Function graphing skills:

use of a GDC to graph a variety of functions

investigation of key features of graphs Identification of asymptotes.

2.2

solutions of equations graphically. May be referred to as roots of equations, or

zeros of functions.

Topic 2—Core: Functions and equations (continued)

Content Amplifications/inclusions Exclusions

Transformations of graphs: translations;

stretches; reflections in the axes.

Translations: y=f(x)+b; y=f(x−a) .

Stretches: y=pf(x); y=f(x q).

Reflections (in both axes):

y= −f(x); y=f(−x).

Examples: y=x2 used to obtain y=3x2+2 by

a stretch of scale factor 3 in the y-direction

followed by a translation of

⎛ ⎞

⎜ ⎟

⎝ ⎠

y= sinx used to obtain y= 3sin 2x by a

stretch of scale factor 3 in the y-direction and a

stretch of scale factor 1

in the x-direction.

The graph of y= f−1(x)as the reflection in the

line y=x of the graph of y= f(x) .

The graph of ( )

y 1

f x

= from y= f(x) .

2.3

The graphs of the absolute value functions,

y= f(x) and y= f(x).

2.4

The reciprocal function x 1, x 0

􀀶 ≠ : its

graph; its self-inverse nature.

Topic 2—Core: Functions and equations (continued)

Content Amplifications/inclusions Exclusions

The quadratic function x 􀀶 ax2+bx+c : its

graph.

Real coefficients only.

Axis of symmetry

x b

a

= − .

The form x 􀀶 a(x−h)2+k .

2.5

The form x 􀀶 a(x−p)(x−q) .

The solution of ax2+bx+c=0, a≠0. On examination papers: questions requiring

elaborate factorization techniques will not be set.

The quadratic formula.

2.6

Use of the discriminant Δ =b2 −4ac.

The function: x 􀀶 ax , a >0.

The inverse function loga x 􀀶 x, x >0. log x

aa =x;aloga x =x, x >0.

Graphs of y=ax and loga y= x.

2.7

Solution of ax =b using logarithms.

Topic 2—Core: Functions and equations (continued)

Content Amplifications/inclusions Exclusions

The exponential function x 􀀶 ex .

The logarithmic function x 􀀶 ln x, x >0. ax = exlna .

2.8

Examples of applications: compound interest,

growth and decay.

Inequalities in one variable, using their

graphical representation.

Use of the absolute value sign in inequalities. On examination papers: questions requiring

elaborate manipulation will not be set.

2.9

Solution of g(x) ≥f(x) , where f, g are linear or

quadratic.

Analytical solution for simple cases.

2.10 Polynomial functions. The graphical significance of repeated roots.

The factor and remainder theorems, with

application to the solution of polynomial

equations and inequalities.

Topic 3—Core: Circular functions and trigonometry 22 hrs

Aims

The aims of this section are to explore the circular functions, to introduce some important trigonometric identities and to solve triangles using trigonometry.

Details

Content Amplifications/inclusions Exclusions

3.1 The circle: radian measure of angles; length of

an arc; area of a sector.

Radian measure may be expressed as multiples

of π, or decimals.

3.2 Definition of cosθ and sinθ in terms of the

unit circle.

Definition of tanθ as sin

cos

Definition of secθ, cscθ and cotθ .

Pythagorean identities: cos2θ+sin2θ=1;

1+tan2θ=sec2θ; 1+cot2θ=csc2θ.

See SL guide

Compound angle identities. Proof of compound angle identities: sin(A±B) ,

cos(A±B) , tan(A±B) .

Double angle identities. Proof of double angle identities.

Given sinθ , finding possible values of other

ratios (for example sin 2θ ) without finding θ.

3.3

See SL guide

Topic 3—Core: Circular functions and trigonometry (continued)

Content Amplifications/inclusions Exclusions

The circular functions sin x , cos x and tan x ;

their domains and ranges; their periodic nature;

their graphs.

On examination papers: radian measure should

be assumed unless otherwise indicated, for

example, by x 􀀶 sin x° .

Composite functions of the form

f (x)=asin(b(x+c))+d.

Example: f(x)=3tan(4(x−2)) +1.

The inverse functions x 􀀶 arcsin x ,

x 􀀶 arccos x, x 􀀶 arctan x ; their domains and

ranges; their graphs.

On examination papers: questions requiring

elaborate analytical treatment of inverse

trigonometric functions will not be set.

Examples of applications: height of tide; Ferris

wheel.

3.4

See SL guide

Solution of trigonometric equations in a finite

interval.

Examples:

2sin x = 3cos x , 0≤x≤2π .

2sin 2x = 3cos x , 0° ≤ x ≤180° .

2sin x = cos2x , −π≤x≤π.

3.5 The general solution of trigonometric equations.

Use of trigonometric identities and factorization

to transform equations.

Both analytical and graphical methods required.

Topic 3—Core: Circular functions and trigonometry (continued)

Content Amplifications/inclusions Exclusions

Solution of triangles.

The cosine rule: c2=a2+b2−2abcosC.

The sine rule:

sin sin sin

a b c

A B C

= = . The ambiguous case of the sine rule.

Area of a triangle as 1 sin .

ab C

3.6

Applications to real-life situations in two

dimensions, and simple cases in three

dimensions, for example, navigation.

Topic 4—Core: Matrices 12 hrs

Aims

The aim of this section is to provide an elementary introduction to matrices, a fundamental concept of linear algebra.

Details

Content Amplifications/inclusions Exclusions

4.1 Definition of a matrix: the terms element, row,

column and order.

Use of matrices to store data. Use of matrices to represent transformations.

Algebra of matrices: equality; addition;

subtraction; multiplication by a scalar.

Matrix operations to handle or process

information.

Multiplication of matrices.

4.2

Identity and zero matrices.

Determinant of a square matrix. The terms singular and non-singular matrices.

Calculation of 2× 2 and 3×3 determinants. The result det AB = det Adet B . Cofactors and minors.

4.3

Inverse of a matrix: conditions for its existence. Obtaining the inverse of a 3×3 matrix using a

GDC.

Other methods for finding the inverse of a 3×3

matrix.

Solution of systems of linear equations (a

maximum of three equations in three

unknowns).

4.4

Conditions for the existence of a unique

solution, no solution and an infinity of solutions.

These cases can be investigated using row

reduction, including the use of augmented

matrices. Unique solutions can also be found

using inverse matrices.

Topic 5—Core: Vectors 22 hrs

Aims

The aim of this section is to introduce the use of vectors in two and three dimensions, and to facilitate solving problems involving points, lines and planes.

Details

Content Amplifications/inclusions Exclusions

Vectors as displacements in the plane and in

three dimensions.

Distance between points in three dimensions.

Components of a vector; column representation

2 1 2 3

v

v v v v

v

⎛ ⎞

=⎜⎜ ⎟⎟= + +

⎜ ⎟

⎝ ⎠

v i j k .

Components are with respect to the unit vectors

i, j, k (standard basis).

Algebraic and geometric approaches to the

following topics:

the sum and difference of two vectors; the zero

vector, the vector −v ;

The difference of v and w is v−w=v+(−w) .

multiplication by a scalar, kv ;

magnitude of a vector, v ;

unit vectors; base vectors i, j, k;

5.1

position vectors OA

→

= a. AB OB OA

→ → →

= − =b−a.

Topic 5—Core: Vectors (continued)

Content Amplifications/inclusions Exclusions

The scalar product of two vectors,

v⋅w= v w cosθ ; 1 1 2 2 3 3 v⋅w=v w +v w +v w .

The scalar product is also known as the “dot

product” or “inner product”.

Projections.

Algebraic properties of the scalar product.

Perpendicular vectors; parallel vectors. For non-zero perpendicular vectors v⋅w=0 ;

for non-zero parallel vectors v⋅w=±v w .

5.2

The angle between two vectors.

Vector equation of a line r=a+λb. Lines in the plane and in three-dimensional

space.

Knowledge of the following forms for equations

of lines.

Parametric form:

0 x=x +λl, 0 y=y +λm, 0 z=z +λn.

Cartesian form: 0 0 0 x x y y z z

l m n

− = − = − .

The angle between two lines.

5.3

See SL guide

Coincident, parallel, intersecting and skew lines,

distinguishing between these cases.

5.4

Points of intersection.

Topic 5—Core: Vectors (continued)

Content Amplifications/inclusions Exclusions

The vector product of two vectors, v×w. The vector product is also known as the cross

product.

The determinant representation.

5.5

Geometric interpretation of v×w . Areas of triangles and parallelograms.

Vector equation of a plane r=a+λb+μc.

Use of normal vector to obtain the form

r⋅n=a⋅n.

5.6

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

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

three planes.

Inverse matrix method and row reduction for

finding the intersection of three planes.

5.7

Angle between: a line and a plane; two planes. Awareness that three planes may intersect in a

point, or in a line, or not at all.

Topic 6—Core: Statistics and probability 40 hrs

Aims

The aim of this section is to introduce basic concepts. It may be considered as three parts: manipulation and presentation of statistical data (6.1–6.4), the laws

of probability (6.5–6.8), and random variables and their probability distributions (6.9–6.11). It is expected that most of the calculations required will be done

on a GDC. The emphasis is on understanding and interpreting the results obtained.

Details

Content Amplifications/inclusions Exclusions

6.1 Concepts of population, sample, random sample

and frequency distribution of discrete and

continuous data.

Elementary treatment only.

Presentation of data: frequency tables and

diagrams, box and whisker plots.

Treatment of both continuous and discrete data.

Grouped data: mid-interval values, interval

width, upper and lower interval boundaries,

6.2

frequency histograms. A frequency histogram uses equal class

intervals.

Histograms based on unequal class intervals.

Topic 6—Core: Statistics and probability (continued)

Content Amplifications/inclusions Exclusions

Mean, median, mode; quartiles, percentiles. Awareness that the population mean, μ, is

generally unknown, and that the sample mean, x ,

serves as an unbiased estimate of this quantity.

Estimation of the mode from a histogram.

Formal treatment of unbiased estimation.

Range; interquartile range; variance, standard

deviation.

Awareness of the concept of dispersion and an

understanding of the significance of the

numerical value of the standard deviation.

Obtain the standard deviation (and indirectly the

variance) from a GDC and by other methods.

Awareness that the population variance, σ2, is

generally unknown, and that 2 2

n1 1n

s n s

− n =

−

serves as an unbiased estimate of σ2.

6.3

See SL guide

6.4 Cumulative frequency; cumulative frequency

graphs; use to find median, quartiles,

percentiles.

Concepts of trial, outcome, equally likely

outcomes, sample space (U) and event.

The probability of an event A as P( ) ( )

( )

A n A

n U

= .

The calculation of n(A) and n(U) may involve

counting principles.

6.5

The complementary events A and A′ (not A);

P(A)+P(A′)=1 .

Topic 6—Core: Statistics and probability (continued)

Content Amplifications/inclusions Exclusions

Combined events, the formula:

P(A∪B)=P(A)+P(B)−P(A∩B) .

6.6 Appreciation of the non-exclusivity of “or”.

P(A∩B)=0 for mutually exclusive events. Use of P(A∪B)=P(A)+P(B) for mutually

exclusive events.

Conditional probability; the definition:

P( | ) P( )

P( )

A B A B

B

= ∩ .

Independent events; the definition:

P(A|B)=P(A)=P(A|B′).

The term “independent” is equivalent to

“statistically independent”. Use of

P(A∩B)=P(A)P(B) for independent events.

6.7

Use of Bayes’ theorem for two events. ( ) ( )

( ) ( )

P( )P |

P |

P( )P | P( )P |

B A B

B A

B A B B A B

+ ′ ′

6.8 Use of Venn diagrams, tree diagrams and tables

of outcomes to solve problems.

Topic 6—Core: Statistics and probability (continued)

Content Amplifications/inclusions Exclusions

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.

Knowledge and use of the formulae for E(X )

and Var(X ) .

Applications of expectations, for example,

games of chance.

6.9

See SL guide

Binomial distribution, its mean and variance.

Poisson distribution, its mean and variance.

Conditions under which random variables have

these distributions.

6.10 Formal proof of means and variances.

See SL guide

Normal distribution. Normal approximation to the binomial

distribution.

Properties of the normal distribution. Appreciation that the standardized value (z)

gives the number of standard deviations from

the mean.

6.11

Standardization of normal variables. Use of calculator (or tables) to find normal

probabilities; the reverse process.

Topic 7—Core: Calculus 48 hrs

Aims

The aim of this section is to introduce students to the basic concepts and techniques of differential and integral calculus and their application.

Details

Content Amplifications/inclusions Exclusions

Informal ideas of limit and convergence. Only an informal treatment of limit and

convergence, including the result

limsin 1

→ θ

= .

Definition of derivative as

( ) lim ( ) ()

h

f x f x h f x

→ h

′ = ⎛⎜ + − ⎞⎟

⎝ ⎠

Use of this definition for differentiation of

polynomials, and for justification of other

derivatives.

Familiarity with both forms of notation, d

y

x

and

f ′(x), for the first derivative.

Derivative of xn (n∈􀁟), sin x , cos x , tan x ,

ex and ln x .

On examination papers: students will not be

required to prove these results.

Derivative interpreted as a gradient function and

as rate of change.

Finding equations of tangents and normals.

Identifying increasing and decreasing functions.

Derivatives of reciprocal circular functions.

Derivatives of ax and loga x . Derivatives of

arcsin x , arccos x , arctan x .

7.1

See SL guide

Topic 7—Core: Calculus (continued)

Content Amplifications/inclusions Exclusions

Differentiation of a sum and a real multiple of

the functions in 7.1.

The chain rule for composite functions.

Application of chain rule to related rates of

change.

The product and quotient rules.

The second derivative. Familiarity with both forms of notation,

y

x

and f ′′(x) , for the second derivative.

Awareness of higher derivatives. Familiarity with the notations d

n

n

y

x

, f (n)(x).

7.2

See SL guide

Local maximum and minimum points. Testing for the maximum or minimum using

change of sign of the first derivative and using

sign of second derivative.

7.3

Use of the first and second derivative in

optimization problems.

Examples of applications: profit, area, volume.

Topic 7—Core: Calculus (continued)

Content Amplifications/inclusions Exclusions

Indefinite integration as anti-differentiation. Indefinite integral interpreted as a family of

curves.

Indefinite integral of xn (n≠ −1) , sin x , cos x ,

ex , 1

1dx lnx C

∫ = + .

The composites of any of these with the linear

function ax + b .

Examples:

f′(x)=cos(2x+3) ( ) 1sin(2 3)

⇒ f x= x+ +C .

7.4

See SL guide

Anti-differentiation with a boundary condition to

determine the constant term. Example: if d 3 2

y x x

x

= + and y =10 when

x = 0, then 3 1 2 10

y=x + x + .

Definite integrals.

Area between a curve and the x-axis or y-axis in

a given interval, areas between curves. d b

∫ y x and d b

∫ x y .

Volumes of revolution. Revolution about the x-axis or the y-axis.

π 2d b

V= ∫ y x, π 2d b

V= ∫ x y.

7.5

See SL guide

Topic 7—Core: Calculus (continued)

Content Amplifications/inclusions Exclusions

Kinematic problems involving displacement, s,

velocity, v, and acceleration, a.

v s

t

= ,

d d d

a v s vv

t t s

= = = . Area under

velocity–time graph represents distance.

7.6

See SL guide

Graphical behaviour of functions: tangents and

normals, behaviour for large x ;

Included: both “global” and “local” behaviour.

asymptotes. Oblique asymptotes.

The significance of the second derivative;

distinction between maximum and minimum

points.

Use of the terms “concave up” for f′′(x) > 0 ,

“concave down” for f′′(x) < 0 .

Points of inflexion with zero and non-zero

gradients.

At a point of inflexion f′′(x) = 0 and f ′′(x)

changes sign (concavity change). f′′(x) = 0 is

not a sufficient condition for a point of inflexion:

for example, y=x4 at (0,0) .

Points of inflexion where f ′′(x) is not

defined, for example, y=x1 3 at (0,0) .

7.7

See SL guide

7.8 Implicit differentiation.

Topic 7—Core: Calculus (continued)

Content Amplifications/inclusions Exclusions

Further integration: Limit changes in definite integrals. Integration using partial fractions.

integration by substitution On examination papers: unusual substitutions

may be given.

7.9

integration by parts. Examples: ∫ xsinxdx and ∫ln xdx .

Repeated integration by parts:

examples: ∫ x2exdx and ∫ex sin xdx .

Reduction formulae.

7.10 Solution of first order differential equations by

separation of variables.

Option syllabus content

Topic 8—Option: Statistics and probability 40 hrs

Aims

The aims of this option are to allow students the opportunity to approach statistics in a practical way; to demonstrate a good level of statistical understanding;

and to understand which situations apply and to interpret the given results. It is expected that GDCs will be used throughout this option and that the minimum

requirement of a GDC will be to find pdf, cdf, inverse cdf, p-values and test statistics including calculations for the following distributions: binomial, Poisson,

normal, t and chi-squared. Students are expected to set up the problem mathematically and then read the answers from the GDC, indicating this within their

written answers. Calculator-specific or brand-specific language should not be used within these explanations.

Details

Content Amplifications/inclusions Exclusions

Expectation algebra. E(aX+b)=aE(X )+b ;

Var(aX+b)=a2Var(X ) .

Linear transformation of a single random

variable.

8.1

Mean and variance of linear combinations of

two independent random variables.

Extension to linear combinations of n

independent random variables.

1 1 2 2 1 1 2 2 E(a X±a X )=a E(X )±a E(X ) ;

2 2

1 1 2 2 1 1 2 2 Var(a X±a X )=a Var(X )+a Var(X ) .

Topic 8—Option: Statistics and probability (continued)

Content Amplifications/inclusions Exclusions

Cumulative distribution functions. Formal treatment of proof of means and

variances.

Discrete distributions: uniform, Bernoulli,

binomial, negative binomial, Poisson,

geometric, hypergeometric.

Probability mass functions, means and

variances.

8.2

Continuous distributions: uniform, exponential,

normal.

Probability density functions, means and

variances.

Distribution of the sample mean. Sampling without replacement.

The distribution of linear combinations of

independent normal random variables. In

particular ( ) 2

X~ N , 2 X~ N ,

μ σ μ σ

⎛ ⎞

⇒ ⎜⎟⎝⎠.

A linear combination of independent normally

distributed random variables is also normally

distributed.

The central limit theorem. Proof of the central limit theorem.

8.3

The approximate normality of the proportion of

successes in a large sample.

The extension of these results for large samples

to distributions that are not normal, using the

central limit theorem.

Distributions that do not satisfy the central limit

theorem.

Topic 8—Option: Statistics and probability (continued)

Content Amplifications/inclusions Exclusions

Finding confidence intervals for the mean of a

population.

Use of the normal distribution when σ is known

and the t-distribution when σ is unknown

(regardless of sample size). The case of paired

samples (matched pairs) could be tested as an

example of a single sample technique.

The difference of means and the difference of

proportions.

8.4

Finding confidence intervals for the proportion

of successes in a population.

Significance testing for a mean. Significance

testing for a proportion.

Use of the normal distribution when σ is known

and the t-distribution when σ is unknown. The

case of paired samples (matched pairs) could be

tested as an example of a single sample

technique.

The difference of means and the difference of

proportions.

Null and alternative hypotheses 0 H and 1 H .

Type I and Type II errors.

8.5

Significance levels; critical region, critical

values, p-values; one-tailed and two-tailed tests.

Topic 8—Option: Statistics and probability (continued)

Content Amplifications/inclusions Exclusions

The chi-squared distribution: degrees of

freedom, υ.

The χ 2 statistic, ( )2

2 o e

calc

e

f f

f

−

= Σ .

Awareness of the fact that 2

calc χ is a measure of

the discrepancy between observed and expected

values.

The χ 2 goodness of fit test. Test for goodness of fit for all of the above

distributions; the requirement to combine classes

with expected frequencies of less than 5.

8.6

Contingency tables: the χ 2 test for the

independence of two variables.

Yates’ continuity correction for υ =1.

Topic 9—Option: Sets, relations and groups 40 hrs

Aims

The aims of this option are to provide the opportunity to study some important mathematical concepts, and introduce the principles of proof through abstract

algebra.

Details

Content Amplifications/inclusions Exclusions

Finite and infinite sets. Subsets. Operations on

sets: union; intersection; complement, set

difference, symmetric difference.

9.1

De Morgan’s laws; distributive, associative and

commutative laws (for union and intersection).

Illustration of these laws using Venn diagrams. Proofs of these laws.

9.2 Ordered pairs: the Cartesian product of two sets.

Relations; equivalence relations; equivalence

classes.

An equivalence relation on a set induces a

partition of the set.

9.3 Functions: injections; surjections; bijections. The term “codomain”.

Composition of functions and inverse functions. Knowledge that the function composition is not

a commutative operation and that if f is a

bijection from set A onto set B then f −1 exists

and is a bijection from set B onto set A.

Topic 9—Option: Sets, relations and groups (continued)

Content Amplifications/inclusions Exclusions

Binary operations. A binary operation ∗ on a non-empty set S is a

rule for combining any two elements a,b∈S to

give a unique element c. That is, in this

definition, a binary operation is not necessarily

closed.

On examination papers: candidates may be

required to test whether a given operation

satisfies the closure condition.

9.4

Operation tables (Cayley tables). Operation tables with the Latin square property

(every element appears once only in each row

and each column).

9.5 Binary operations with associative, distributive

and commutative properties.

The arithmetic operations in 􀁜 and 􀁞; matrix

operations.

The identity element e. Both the right-identity a∗e=a and left-identity

e∗a=a must hold if e is an identity element.

The inverse a−1 of an element a. Both a∗a−1=e and a−1∗a=e must hold.

Proof that left-cancellation and right-cancellation

by an element a hold, provided that a has an

inverse.

9.6

Proofs of the uniqueness of the identity and

inverse elements.

Topic 9—Option: Sets, relations and groups (continued)

Content Amplifications/inclusions Exclusions

The axioms of a group {G,∗}. For the set G under a given operation ∗:

• G is closed under ∗

• ∗ is associative

• G contains an identity element

• each element in G has an inverse in G.

9.7

Abelian groups. a∗b=b∗a, for alla,b∈G.

The groups:

• 􀁜, 􀁟, 􀁝 and 􀁞 under addition

• matrices of the same order under addition

• 2× 2 invertible matrices under

multiplication

• integers under addition modulo n

• groups of transformations The composition 1 2 TT denotes 2 T followed by 1 T .

• symmetries of an equilateral triangle,

rectangle and square

• invertible functions under composition of

functions

9.8

• permutations under composition of

permutations.

On examination papers: the form

1 2 3

3 1 2

⎛ ⎞

=⎜ ⎟

⎝ ⎠

will be used to represent the

mapping 1→3, 2→1, 3→2.

Topic 9—Option: Sets, relations and groups (continued)

Content Amplifications/inclusions Exclusions

9.9 Finite and infinite groups. Latin square property of a group table.

The order of a group element and the order of a

group.

9.10 Cyclic groups. Generators.

Proof that all cyclic groups are Abelian.

Subgroups, proper subgroups.

Use and proof of subgroup tests. Suppose G is a group and H is a non-empty

subset of G. H is a subgroup of G if ab−1∈H

whenever a,b∈H .

Suppose G is a finite group and H is a non-empty

subset of G. H is a subgroup of G if H is closed

under the group operation.

9.11

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.)

On examination papers: questions requiring the

proof of Lagrange’s theorem will not be set.

Topic 9—Option: Sets, relations and groups (continued)

Content Amplifications/inclusions Exclusions

Isomorphism of groups. Infinite groups as well as finite groups.

Two groups {G,􀁄} and {H,•} are isomorphic if

there exists a bijection f :G→H such that

f(a􀁄b)=f(a)•f(b) for all a,b∈G.

The function f :G→H is an isomorphism.

9.12

Proof of isomorphism properties for identities

and inverses.

Identity: let 1 e and 2 e be the identity elements

of G, H respectively, then 1 2 f(e)=e.

Inverse: ( )f(a 1) f(a) 1 − = − for all a∈G.

Topic 10—Option: Series and differential equations 40 hrs

Aims

The aims of this option are to introduce limit theorems and convergence of series, and to use calculus results to solve differential equations.

Details

Content Amplifications/inclusions Exclusions

Infinite sequences of real numbers.

Limit theorems as n approaches infinity. Limit of sum, difference, product, quotient;

squeeze theorem.

Limit of a sequence. Formal definition: the sequence { } n u converges

to the limit L, if for any ε > 0 , there is a positive

integer N such that n u −L<ε , for all n>N.

Improper integrals of the type ( )d

f x x ∞ ∫ .

10.1

The integral as a limit of a sum; lower sum and

upper sum.

Topic 10—Option: Series and differential equations (continued)

Content Amplifications/inclusions Exclusions

Convergence of infinite series. The sum of a series is the limit of the sequence

of its partial sums.

Partial fractions and telescoping series (method

of differences).

Simple linear non-repeated denominators.

Tests for convergence: comparison test; limit

comparison test; ratio test; integral test.

Students should be aware that if lxim n 0

→∞

= then

the series is not necessarily convergent, but if

lim 0 x n

→∞

≠ , the series diverges.

The p-series, 1

np Σ . 1

np Σ is convergent for p >1 and divergent

otherwise. When p =1, this is the harmonic

series.

10.2

Use of integrals to estimate sums of series.

Series that converge absolutely.

Series that converge conditionally.

10.3

Alternating series. Conditions for convergence. The absolute value

of the truncation error is less than the next term

in the series.

10.4 Power series: radius of convergence and interval

of convergence. Determination of the radius of

convergence by the ratio test.

Topic 10—Option: Series and differential equations (continued)

Content Amplifications/inclusions Exclusions

Taylor polynomials and series, including the

error term.

Applications to the approximation of functions;

formulae for the error term, both in terms of the

value of the (n +1)th derivative at an

intermediate point, and in terms of an integral of

the (n +1)th derivative.

Proof of Taylor’s theorem.

Differentiation and integration of series (valid

only on the interval of convergence of the initial

series).

Use of products and quotients to obtain other

series.

Maclaurin series for ex , sin x , cos x , arctan x ,

ln(1+ x), (1 )p + x . Use of substitution to obtain

other series.

Intervals of convergence for these Maclaurin

series.

Example: 2 ex .

10.5

The evaluation of limits of the form ( )

( ) lim

x a

f x

→ g x

using l’Hôpital’s Rule and/or the Taylor series.

Cases where the derivatives of f (x) and g(x)

vanish for x = a .

Proof of l’Hôpital’s Rule.

Topic 10—Option: Series and differential equations (continued)

Content Amplifications/inclusions Exclusions

First order differential equations: geometric

interpretation using slope fields;

numerical solution of d ( , )

y f x y

x

= using

Euler’s method.

1 ( , ) n n n n y y h f x y + = + × ; n1 n x x h + = + , where h

is a constant.

Homogeneous differential equation d

y f y

x x

= ⎛⎜ ⎞⎟

⎝ ⎠

using the substitution y=vx.

10.6

Solution of y′+P(x)y=Q(x) , using the

integrating factor.

Topic 11—Option: Discrete mathematics 40 hrs

Aims

The aim of this option is to provide the opportunity for students to engage in logical reasoning, algorithmic thinking and applications.

Details

Content Amplifications/inclusions Exclusions

Division and Euclidean algorithms. The theorem a|b and a|c⇒a|(bx±cy)

where x,y∈Z .

The division algorithm a=bq+r, 0≤r<b.

The greatest common divisor, gcd(a,b) , and

the least common multiple, lcm(a,b) , of

integers a and b.

The Euclidean algorithm for determining the

greatest common divisor of two integers.

11.1

Relatively prime numbers; prime numbers and

the fundamental theorem of arithmetic.

Proof of the fundamental theorem of arithmetic.

11.2 Representation of integers in different bases. On examination papers: questions that go

beyond base 16 are unlikely to be set.

11.3 Linear diophantine equations ax+by=c . General solutions required and solutions subject

to constraints. For example, all solutions must

be positive.

11.4 Modular arithmetic. Linear congruences.

Chinese remainder theorem.

11.5 Fermat’s little theorem. ap ≡a(modp) where p is prime. On examination papers: questions requiring

proof of the theorem will not be set.

Topic 11—Option: Discrete mathematics (continued)

Content Amplifications/inclusions Exclusions

Graphs, vertices, edges. Adjacent vertices,

adjacent edges.

Two vertices are adjacent if they are joined by

an edge. Two edges are adjacent if they have a

common vertex.

Simple graphs; connected graphs; complete

graphs; bipartite graphs; planar graphs, trees,

weighted graphs.

Subgraphs; complements of graphs.

Euler’s relation: v−e+ f =2 ; theorems for

planar graphs including e≤3v−6, e≤2v−4 ,

5 κ and 3,3 κ are not planar.

11.6

Graph isomorphism. Simple graphs only for isomorphism.

11.7 Walks, trails, paths, circuits, cycles.

Hamiltonian paths and cycles; Eulerian trails

and circuits.

A connected graph contains a Eulerian circuit if

and only if every vertex of the graph is of even

degree.

Dirac’s theorem for Hamiltonian cycles.

Adjacency matrix. Applications to isomorphism and of the powers

of the adjacency matrix to number of walks.

11.8

Cost adjacency matrix.

11.9 Graph algorithms: Prim’s; Kruskal’s;

Dijkstra’s.

These are examples of “greedy” algorithms.

Topic 11—Option: Discrete mathematics (continued)

Content Amplifications/inclusions Exclusions

“Chinese postman” problem (“route inspection”). To determine the shortest route around a

weighted graph going along each edge at least

once (route inspection algorithm).

Graphs with more than two vertices of odd

degree.

“Travelling salesman” problem. To determine the Hamiltonian cycle of least

weight in a weighted complete graph.

11.10

Algorithms for determining upper and lower

bounds of the travelling salesman problem.

Graphs in which the triangle inequality is not

satisfied