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Diploma Programme
Further mathematics HL guideFirst examinations 2014
Diploma Programme
Further mathematics HL guideFirst examinations 2014
Further mathematics HL guideDiploma Programme
International Baccalaureate, Baccalauréat International and Bachillerato Internacional are registered trademarks of the International Baccalaureate Organization.
Printed in the United Kingdom by Antony Rowe Ltd, Chippenham, Wiltshire
Published June 2012Updated August 2014
Updated May 2016
Published on behalf of the International Baccalaureate Organization, a not-for-profit educational foundation of 15 Route des Morillons, 1218 Le Grand-Saconnex, Geneva,
Switzerland by the
International Baccalaureate Organization (UK) LtdPeterson House, Malthouse Avenue, Cardiff Gate
Cardiff, Wales CF23 8GLUnited Kingdom
Phone: +44 29 2054 7777Fax: +44 29 2054 7778Website: www.ibo.org
© International Baccalaureate Organization 2012
The International Baccalaureate Organization (known as the IB) offers three high-quality and challenging educational programmes for a worldwide community of schools, aiming to create a better, more peaceful world. This publication is one of a range of materials produced to support these programmes.
The IB may use a variety of sources in its work and checks information to verify accuracy and authenticity, particularly when using community-based knowledge sources such as Wikipedia. The IB respects the principles of intellectual property and makes strenuous efforts to identify and obtain permission before publication from rights holders of all copyright material used. The IB is grateful for permissions received for material used in this publication and will be pleased to correct any errors or omissions at the earliest opportunity.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior written permission of the IB, or as expressly permitted by law or by the IB’s own rules and policy. See http://www.ibo.org/copyright.
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Phone: +44 29 2054 7746Fax: +44 29 2054 7779Email: sales@ibo.org
5041
IB mission statementThe International Baccalaureate aims to develop inquiring, knowledgeable and caring young people who help to create a better and more peaceful world through intercultural understanding and respect.
To this end the organization works with schools, governments and international organizations to develop challenging programmes of international education and rigorous assessment.
These programmes encourage students across the world to become active, compassionate and lifelong learners who understand that other people, with their differences, can also be right.
IB learner profileThe aim of all IB programmes is to develop internationally minded people who, recognizing their common humanity and shared guardianship of the planet, help to create a better and more peaceful world.
IB learners strive to be:
Inquirers They develop their natural curiosity. They acquire the skills necessary to conduct inquiry and research and show independence in learning. They actively enjoy learning and this love of learning will be sustained throughout their lives.
Knowledgeable They explore concepts, ideas and issues that have local and global significance. In so doing, they acquire in-depth knowledge and develop understanding across a broad and balanced range of disciplines.
Thinkers They exercise initiative in applying thinking skills critically and creatively to recognize and approach complex problems, and make reasoned, ethical decisions.
Communicators They understand and express ideas and information confidently and creatively in more than one language and in a variety of modes of communication. They work effectively and willingly in collaboration with others.
Principled They act with integrity and honesty, with a strong sense of fairness, justice and respect for the dignity of the individual, groups and communities. They take responsibility for their own actions and the consequences that accompany them.
Open-minded They understand and appreciate their own cultures and personal histories, and are open to the perspectives, values and traditions of other individuals and communities. They are accustomed to seeking and evaluating a range of points of view, and are willing to grow from the experience.
Caring They show empathy, compassion and respect towards the needs and feelings of others. They have a personal commitment to service, and act to make a positive difference to the lives of others and to the environment.
Risk-takers They approach unfamiliar situations and uncertainty with courage and forethought, and have the independence of spirit to explore new roles, ideas and strategies. They are brave and articulate in defending their beliefs.
Balanced They understand the importance of intellectual, physical and emotional balance to achieve personal well-being for themselves and others.
Reflective They give thoughtful consideration to their own learning and experience. They are able to assess and understand their strengths and limitations in order to support their learning and personal development.
© International Baccalaureate Organization 2007
Further mathematics HL guide
Contents
Introduction 1Purpose of this document 1
The Diploma Programme 2
Nature of the subject 4
Aims 8
Assessment objectives 9
Syllabus 10Syllabus outline 10
Approaches to the teaching and learning of further mathematics HL 11
Syllabus content 15
Glossary of terminology: Discrete mathematics 44
Assessment 46Assessment in the Diploma Programme 46
Assessment outline 48
Assessment details 49
Appendices 51Glossary of command terms 51
Notation list 53
Further mathematics HL guide 1
Purpose of this document
Introduction
This publication is intended to guide the planning, teaching and assessment of the subject in schools. Subject teachers are the primary audience, although it is expected that teachers will use the guide to inform students and parents about the subject.
This guide can be found on the subject page of the online curriculum centre (OCC) at http://occ.ibo.org, a password-protected IB website designed to support IB teachers. It can also be purchased from the IB store at http://store.ibo.org.
Additional resourcesAdditional publications such as teacher support materials, subject reports and grade descriptors can also be found on the OCC. Specimen and past examination papers as well as markschemes can be purchased from the IB store.
Teachers are encouraged to check the OCC for additional resources created or used by other teachers. Teachers can provide details of useful resources, for example: websites, books, videos, journals or teaching ideas.
First examinations 2014
2 Further mathematics HL guide
Introduction
The Diploma Programme
The Diploma Programme is a rigorous pre-university course of study designed for students in the 16 to 19 age range. It is a broad-based two-year course that aims to encourage students to be knowledgeable and inquiring, but also caring and compassionate. There is a strong emphasis on encouraging students to develop intercultural understanding, open-mindedness, and the attitudes necessary for them to respect and evaluate a range of points of view.
The Diploma Programme hexagonThe course is presented as six academic areas enclosing a central core (see figure 1). It encourages the concurrent study of a broad range of academic areas. Students study: two modern languages (or a modern language and a classical language); a humanities or social science subject; an experimental science; mathematics; one of the creative arts. It is this comprehensive range of subjects that makes the Diploma Programme a demanding course of study designed to prepare students effectively for university entrance. In each of the academic areas students have f lexibility in making their choices, which means they can choose subjects that particularly interest them and that they may wish to study further at university.
Studies in language and literature
Individualsand societies
Mathematics
The arts
Experimentalsciences
Languageacquisition
Group 2
Group 4
Group 6
Group 5
Group 1
Group 3
theo
ry o
f k
nowledge extended essay
creativity, action, service
TH
E IB LEARNER PRO
FILE
Figure 1Diploma Programme model
Further mathematics HL guide 3
The Diploma Programme
Choosing the right combinationStudents are required to choose one subject from each of the six academic areas, although they can choose a second subject from groups 1 to 5 instead of a group 6 subject. Normally, three subjects (and not more than four) are taken at higher level (HL), and the others are taken at standard level (SL). The IB recommends 240 teaching hours for HL subjects and 150 hours for SL. Subjects at HL are studied in greater depth and breadth than at SL.
At both levels, many skills are developed, especially those of critical thinking and analysis. At the end of the course, students’ abilities are measured by means of external assessment. Many subjects contain some element of coursework assessed by teachers. The courses are available for examinations in English, French and Spanish, with the exception of groups 1 and 2 courses where examinations are in the language of study.
The core of the hexagonAll Diploma Programme students participate in the three course requirements that make up the core of the hexagon. Reflection on all these activities is a principle that lies at the heart of the thinking behind the Diploma Programme.
The theory of knowledge course encourages students to think about the nature of knowledge, to reflect on the process of learning in all the subjects they study as part of their Diploma Programme course, and to make connections across the academic areas. The extended essay, a substantial piece of writing of up to 4,000 words, enables students to investigate a topic of special interest that they have chosen themselves. It also encourages them to develop the skills of independent research that will be expected at university. Creativity, action, service involves students in experiential learning through a range of artistic, sporting, physical and service activities.
The IB mission statement and the IB learner profileThe Diploma Programme aims to develop in students the knowledge, skills and attitudes they will need to fulfill the aims of the IB, as expressed in the organization’s mission statement and the learner profile. Teaching and learning in the Diploma Programme represent the reality in daily practice of the organization’s educational philosophy.
4 Further mathematics HL guide
Introduction
Nature of the subject
IntroductionThe nature of mathematics can be summarized in a number of ways: for example, it can be seen as a well-defined body of knowledge, as an abstract system of ideas, or as a useful tool. For many people it is probably a combination of these, but there is no doubt that mathematical knowledge provides an important key to understanding the world in which we live. Mathematics can enter our lives in a number of ways: we buy produce in the market, consult a timetable, read a newspaper, time a process or estimate a length. Mathematics, for most of us, also extends into our chosen profession: visual artists need to learn about perspective; musicians need to appreciate the mathematical relationships within and between different rhythms; economists need to recognize trends in financial dealings; and engineers need to take account of stress patterns in physical materials. Scientists view mathematics as a language that is central to our understanding of events that occur in the natural world. Some people enjoy the challenges offered by the logical methods of mathematics and the adventure in reason that mathematical proof has to offer. Others appreciate mathematics as an aesthetic experience or even as a cornerstone of philosophy. This prevalence of mathematics in our lives, with all its interdisciplinary connections, provides a clear and sufficient rationale for making the study of this subject compulsory for students studying the full diploma.
Summary of courses availableBecause individual students have different needs, interests and abilities, there are four different courses in mathematics. These courses are designed for different types of students: those who wish to study mathematics in depth, either as a subject in its own right or to pursue their interests in areas related to mathematics; those who wish to gain a degree of understanding and competence to understand better their approach to other subjects; and those who may not as yet be aware how mathematics may be relevant to their studies and in their daily lives. Each course is designed to meet the needs of a particular group of students. Therefore, great care should be taken to select the course that is most appropriate for an individual student.
In making this selection, individual students should be advised to take account of the following factors:
• their own abilities in mathematics and the type of mathematics in which they can be successful
• their own interest in mathematics and those particular areas of the subject that may hold the most interest for them
• their other choices of subjects within the framework of the Diploma Programme
• their academic plans, in particular the subjects they wish to study in future
• their choice of career.
Teachers are expected to assist with the selection process and to offer advice to students.
Further mathematics HL guide 5
Nature of the subject
Mathematical studies SLThis course is available only at standard level, and is equivalent in status to mathematics SL, but addresses different needs. It has an emphasis on applications of mathematics, and the largest section is on statistical techniques. It is designed for students with varied mathematical backgrounds and abilities. It offers students opportunities to learn important concepts and techniques and to gain an understanding of a wide variety of mathematical topics. It prepares students to be able to solve problems in a variety of settings, to develop more sophisticated mathematical reasoning and to enhance their critical thinking. The individual project is an extended piece of work based on personal research involving the collection, analysis and evaluation of data. Students taking this course are well prepared for a career in social sciences, humanities, languages or arts. These students may need to utilize the statistics and logical reasoning that they have learned as part of the mathematical studies SL course in their future studies.
Mathematics SLThis course caters for students who already possess knowledge of basic mathematical concepts, and who are equipped with the skills needed to apply simple mathematical techniques correctly. The majority of these students will expect to need a sound mathematical background as they prepare for future studies in subjects such as chemistry, economics, psychology and business administration.
Mathematics HLThis course caters for students with a good background in mathematics who are competent in a range of analytical and technical skills. The majority of these students will be expecting to include mathematics as a major component of their university studies, either as a subject in its own right or within courses such as physics, engineering and technology. Others may take this subject because they have a strong interest in mathematics and enjoy meeting its challenges and engaging with its problems.
Further mathematics HLThis course is available only at higher level. It caters for students with a very strong background in mathematics who have attained a high degree of competence in a range of analytical and technical skills, and who display considerable interest in the subject. Most of these students will expect to study mathematics at university, either as a subject in its own right or as a major component of a related subject. The course is designed specifically to allow students to learn about a variety of branches of mathematics in depth and also to appreciate practical applications. It is expected that students taking this course will also be taking mathematics HL.
Note: Mathematics HL is an ideal course for students expecting to include mathematics as a major component of their university studies, either as a subject in its own right or within courses such as physics, engineering or technology. It should not be regarded as necessary for such students to study further mathematics HL. Rather, further mathematics HL is an optional course for students with a particular aptitude and interest in mathematics, enabling them to study some wider and deeper aspects of mathematics, but is by no means a necessary qualification to study for a degree in mathematics.
Further mathematics HL—course detailsThe nature of the subject is such that it focuses on different branches of mathematics to encourage students to appreciate the diversity of the subject. Students should be equipped at this stage in their mathematical progress to begin to form an overview of the characteristics that are common to all mathematical thinking, independent of topic or branch.
Further mathematics HL guide6
Nature of the subject
All categories of student can register for mathematics HL only or for further mathematics HL only or for both. However, students registering for further mathematics HL will be presumed to know the topics in the core syllabus of mathematics HL and to have studied one of the options, irrespective of whether they have also registered for mathematics HL.
Examination questions are intended to be comparable in difficulty with those set on the four options in the mathematics HL course. The challenge for students will be to reach an equivalent level of understanding across all topics. There is no internal assessment component in this course. Although not a requirement, it is expected that students studying further mathematics HL will also be studying mathematics HL and therefore will be required to undertake a mathematical exploration for the internal assessment component of that course.
Prior learningMathematics is a linear subject, and it is expected that most students embarking on a Diploma Programme (DP) mathematics course will have studied mathematics for at least 10 years. There will be a great variety of topics studied, and differing approaches to teaching and learning. Thus students will have a wide variety of skills and knowledge when they start the further mathematics HL course. Most will have some background in arithmetic, algebra, geometry, trigonometry, probability and statistics. Some will be familiar with an inquiry approach, and may have had an opportunity to complete an extended piece of work in mathematics.
As previously stated, students registering for further mathematics HL will be presumed to know the topics in the core syllabus of mathematics HL and to have studied one of the options.
Links to the Middle Years ProgrammeThe prior learning topics for the DP courses have been written in conjunction with the Middle Years Programme (MYP) mathematics guide. The approaches to teaching and learning for DP mathematics build on the approaches used in the MYP. These include investigations, exploration and a variety of different assessment tools.
A continuum document called Mathematics: The MYP–DP continuum (November 2010) is available on the DP mathematics home pages of the OCC. This extensive publication focuses on the alignment of mathematics across the MYP and the DP. It was developed in response to feedback provided by IB World Schools, which expressed the need to articulate the transition of mathematics from the MYP to the DP. The publication also highlights the similarities and differences between MYP and DP mathematics, and is a valuable resource for teachers.
Mathematics and theory of knowledgeThe Theory of knowledge guide (March 2006) identifies four ways of knowing, and it could be claimed that these all have some role in the acquisition of mathematical knowledge. While perhaps initially inspired by data from sense perception, mathematics is dominated by reason, and some mathematicians argue that their subject is a language, that it is, in some sense, universal. However, there is also no doubt that mathematicians perceive beauty in mathematics, and that emotion can be a strong driver in the search for mathematical knowledge.
As an area of knowledge, mathematics seems to supply a certainty perhaps missing in other disciplines. This may be related to the “purity” of the subject that makes it sometimes seem divorced from reality. However, mathematics has also provided important knowledge about the world, and the use of mathematics in science and technology has been one of the driving forces for scientific advances.
Further mathematics HL guide 7
Nature of the subject
Despite all its undoubted power for understanding and change, mathematics is in the end a puzzling phenomenon. A fundamental question for all knowers is whether mathematical knowledge really exists independently of our thinking about it. Is it there “waiting to be discovered” or is it a human creation?
Students’ attention should be drawn to questions relating theory of knowledge (TOK) and mathematics, and they should be encouraged to raise such questions themselves, in mathematics and TOK classes. This includes questioning all the claims made above. Examples of issues relating to TOK are given in the “Links” column of the syllabus. Teachers could also discuss questions such as those raised in the “Areas of knowledge” section of the TOK guide.
Mathematics and the international dimensionMathematics is in a sense an international language, and, apart from slightly differing notation, mathematicians from around the world can communicate within their field. Mathematics transcends politics, religion and nationality, yet throughout history great civilizations owe their success in part to their mathematicians being able to create and maintain complex social and architectural structures.
Despite recent advances in the development of information and communication technologies, the global exchange of mathematical information and ideas is not a new phenomenon and has been essential to the progress of mathematics. Indeed, many of the foundations of modern mathematics were laid many centuries ago by Arabic, Greek, Indian and Chinese civilizations, among others. Teachers could use timeline websites to show the contributions of different civilizations to mathematics, but not just for their mathematical content. Illustrating the characters and personalities of the mathematicians concerned and the historical context in which they worked brings home the human and cultural dimension of mathematics.
The importance of science and technology in the everyday world is clear, but the vital role of mathematics is not so well recognized. It is the language of science, and underpins most developments in science and technology. A good example of this is the digital revolution, which is transforming the world, as it is all based on the binary number system in mathematics.
Many international bodies now exist to promote mathematics. Students are encouraged to access the extensive websites of international mathematical organizations to enhance their appreciation of the international dimension and to engage in the global issues surrounding the subject.
Examples of global issues relating to international-mindedness (Int) are given in the “Links” column of the syllabus.
8 Further mathematics HL guide
Introduction
Aims
Group 5 aimsThe aims of all mathematics courses in group 5 are to enable students to:
1. enjoy mathematics, and develop an appreciation of the elegance and power of mathematics
2. develop an understanding of the principles and nature of mathematics
3. communicate clearly and confidently in a variety of contexts
4. develop logical, critical and creative thinking, and patience and persistence in problem-solving
5. employ and refine their powers of abstraction and generalization
6. apply and transfer skills to alternative situations, to other areas of knowledge and to future developments
7. appreciate how developments in technology and mathematics have influenced each other
8. appreciate the moral, social and ethical implications arising from the work of mathematicians and the applications of mathematics
9. appreciate the international dimension in mathematics through an awareness of the universality of mathematics and its multicultural and historical perspectives
10. appreciate the contribution of mathematics to other disciplines, and as a particular “area of knowledge” in the TOK course.
Further mathematics HL guide 9
Introduction
Assessment objectives
Problem-solving is central to learning mathematics and involves the acquisition of mathematical skills and concepts in a wide range of situations, including non-routine, open-ended and real-world problems. Having followed a DP mathematics HL course, students will be expected to demonstrate the following.
1. Knowledge and understanding: recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts.
2. Problem-solving: recall, select and use their knowledge of mathematical skills, results and models in both real and abstract contexts to solve problems.
3. Communication and interpretation: transform common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation.
4. Technology: use technology, accurately, appropriately and efficiently both to explore new ideas and to solve problems.
5. Reasoning: construct mathematical arguments through use of precise statements, logical deduction and inference, and by the manipulation of mathematical expressions.
6. Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving organizing and analysing information, making conjectures, drawing conclusions and testing their validity.
10 Further mathematics HL guide
Syllabus outline
Syllabus
Syllabus componentTeaching hours
HL
All topics 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 all of the core topics in mathematics HL.
Topic 1
Linear algebra
48
Topic 2
Geometry
48
Topic 3
Statistics and probability
48
Topic 4
Sets, relations and groups
48
Topic 5
Calculus
48
Topic 6
Discrete mathematics
48
Note: One of topics 3–6 will be assumed to have been taught as part of the mathematics HL course and therefore the total teaching hours will be 240 not 288.
Total teaching hours 240
Further mathematics HL guide 11
Syllabus
Approaches to the teaching and learning of further mathematics HL
Throughout the DP further mathematics HL course, students should be encouraged to develop their understanding of the methodology and practice of the discipline of mathematics. The processes of mathematical inquiry, mathematical modelling and applications and the use of technology should be introduced appropriately. These processes should be used throughout the course, and not treated in isolation.
Mathematical inquiryThe IB learner profile encourages learning by experimentation, questioning and discovery. In the IB classroom, students should generally learn mathematics by being active participants in learning activities rather than recipients of instruction. Teachers should therefore provide students with opportunities to learn through mathematical inquiry. This approach is illustrated in figure 2.
Explore the context
Make a conjecture
Extend
Prove
Accept
RejectTest the conjecture
Figure 2
Further mathematics HL guide12
Approaches to the teaching and learning of further mathematics HL
Mathematical modelling and applicationsStudents should be able to use mathematics to solve problems in the real world. Engaging students in the mathematical modelling process provides such opportunities. Students should develop, apply and critically analyse models. This approach is illustrated in figure 3.
Pose a real-world problem
Develop a model
Extend
Reflect on and apply the model
Accept
RejectTest the model
Figure 3
TechnologyTechnology is a powerful tool in the teaching and learning of mathematics. Technology can be used to enhance visualization and support student understanding of mathematical concepts. It can assist in the collection, recording, organization and analysis of data. Technology can increase the scope of the problem situations that are accessible to students. The use of technology increases the feasibility of students working with interesting problem contexts where students reflect, reason, solve problems and make decisions.
As teachers tie together the unifying themes of mathematical inquiry, mathematical modelling and applications and the use of technology, they should begin by providing substantial guidance, and then gradually encourage students to become more independent as inquirers and thinkers. IB students should learn to become strong communicators through the language of mathematics. Teachers should create a safe learning environment in which students are comfortable as risk-takers.
Teachers are encouraged to relate the mathematics being studied to other subjects and to the real world, especially topics that have particular relevance or are of interest to their students. Everyday problems and questions should be drawn into the lessons to motivate students and keep the material relevant; suggestions are provided in the “Links” column of the syllabus.
Further mathematics HL guide 13
Approaches to the teaching and learning of further mathematics HL
For further information on “Approaches to teaching a DP course”, please refer to the publication The Diploma Programme: From principles into practice (April 2009). To support teachers, a variety of resources can be found on the OCC and details of workshops for professional development are available on the public website.
Format of the syllabus• Content: this column lists, under each topic, the sub-topics to be covered.
• Further guidance: this column contains more detailed information on specific sub-topics listed in the content column. This clarifies the content for examinations.
• Links: this column provides useful links to the aims of the further mathematics HL course, with suggestions for discussion, real-life examples and ideas for further investigation. These suggestions are only a guide for introducing and illustrating the sub-topic and are not exhaustive. Links are labelled as follows.
Appl real-life examples and links to other DP subjects
Aim 8 moral, social and ethical implications of the sub-topic
Int international-mindedness
TOK suggestions for discussion
Note that any syllabus references to other subject guides given in the “Links” column are correct for the current (2012) published versions of the guides.
Notes on the syllabus• Formulae are only included in this document where there may be some ambiguity. All formulae required
for the course are in the mathematics HL and further mathematics HL formula booklet.
• The term “technology” is used for any form of calculator or computer that may be available. However, there will be restrictions on which technology may be used in examinations, which will be noted in relevant documents.
• The terms “analysis” and “analytic approach” are generally used when referring to an approach that does not use technology.
Course of studyThe content of topics 1 and 2 as well as three out of the four remaining topics in the syllabus must be taught, although not necessarily in the order in which they appear in this guide. Teachers are expected to construct a course of study that addresses the needs of their students and includes, where necessary, the topics noted in prior learning.
Further mathematics HL guide14
Approaches to the teaching and learning of further mathematics HL
Time allocationThe recommended teaching time for higher level courses is 240 hours. The time allocations given in this guide are approximate, and are intended to suggest how the 240 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 calculatorsStudents 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 calculators allowed in examinations are provided in the Handbook of procedures for the Diploma Programme. Further information and advice is provided in the Mathematics HL/SL: Graphic display calculators teacher support material (May 2005) and on the OCC.
Mathematics HL and further mathematics HL formula bookletEach 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. It is the responsibility of the school to download a copy from IBIS or the OCC, check that there are no printing errors, and ensure that there are sufficient copies available for all students.
Command terms and notation listTeachers and students need to be familiar with the IB notation and the command terms, as these will be used without explanation in the examination papers. The “Glossary of command terms” and “Notation list” appear as appendices in this guide.
Further mathematics HL guide 15
Sylla
bus
Sylla
bus
cont
ent
Top
ic 1
—Li
near
alg
ebra
48
hou
rs
The
aim
of
this
sec
tion
is t
o in
trodu
ce s
tude
nts
to t
he p
rinci
ples
of
mat
rices
, vec
tor
spac
es a
nd l
inea
r al
gebr
a, i
nclu
ding
eig
enva
lues
and
geo
met
rical
in
terp
reta
tions
.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
1.1
Def
initi
on o
f a m
atrix
: the
term
s ele
men
t, ro
w,
colu
mn
and
orde
r for
m ×
n m
atric
es.
D
ata
stor
age
and
man
ipul
atio
n, e
g st
ock
cont
rol.
Alg
ebra
of m
atric
es: e
qual
ity; a
dditi
on;
subt
ract
ion;
mul
tiplic
atio
n by
a sc
alar
for m
× n
m
atric
es.
Incl
udin
g us
e of
the
GD
C.
Mat
rix o
pera
tions
to h
andl
e or
pro
cess
in
form
atio
n.
TO
K: G
iven
the
man
y ap
plic
atio
ns o
f mat
rices
as
seen
in th
is c
ours
e, c
onsid
er th
e fa
ct th
at
mat
hem
atic
ians
mar
vel a
t som
e of
the
deep
co
nnec
tions
bet
wee
n di
spar
ate
parts
of t
heir
subj
ect.
Is th
is e
vide
nce
for a
sim
ple
unde
rlyin
g m
athe
mat
ical
real
ity?
Mul
tiplic
atio
n of
mat
rices
.
Prop
ertie
s of m
atrix
mul
tiplic
atio
n:
asso
ciat
ivity
, dis
tribu
tivity
.
Iden
tity
and
zero
mat
rices
.
Tran
spos
e of
a m
atrix
incl
udin
g T
A n
otat
ion:
T
TT
()=
AB
BA
.
Stud
ents
shou
ld b
e fa
mili
ar w
ith th
e no
tatio
n I
and
0.
St
uden
ts sh
ould
be
fam
iliar
with
the
defin
ition
of
sym
met
ric a
nd sk
ew-s
ymm
etric
mat
rices
.
Further mathematics HL guide16
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
1.2
Def
initi
on a
nd p
rope
rties
of t
he in
vers
e of
a
squa
re m
atrix
: 1
11
()−
−−
=A
BB
A,
T1
1T
()
()
−−
=A
A,
11
()
()
nn
−−
=A
A.
The
term
s sin
gula
r and
non
-sin
gula
r mat
rices
.
Cal
cula
tion
of1−
A.
Use
of e
lem
enta
ry ro
w o
pera
tions
to fi
nd
1−A
.
Form
ulae
for t
he in
vers
e an
d de
term
inan
t of a
2
× 2
mat
rix a
nd th
e de
term
inan
t of a
3 ×
3
mat
rix.
Usi
ng a
GD
C.
The
resu
lt de
t()
det
det
=A
BA
B.
1.3
Elem
enta
ry ro
w a
nd c
olum
n op
erat
ions
for
mat
rices
.
Scal
ing,
swap
ping
and
piv
otin
g.
Cor
resp
ondi
ng e
lem
enta
ry m
atric
es.
Row
redu
ced
eche
lon
form
.
Row
spac
e, c
olum
n sp
ace
and
null
spac
e.
Row
rank
and
col
umn
rank
and
thei
r equ
ality
.
Pivo
ting
is th
e 1
2R
aR+
met
hod.
1.4
Solu
tions
of m
line
ar e
quat
ions
in n
unk
now
ns:
both
aug
men
ted
mat
rix m
etho
d, le
adin
g to
re
duce
d ro
w e
chel
on fo
rm m
etho
d, a
nd in
vers
e m
atrix
met
hod,
whe
n ap
plic
able
.
Incl
ude
the
non-
exis
tent
, uni
que
and
infin
itely
m
any
case
s. In
t: A
lso
know
n as
Gau
ssia
n el
imin
atio
n af
ter
the
Ger
man
mat
hem
atic
ian
CF
Gau
ss (1
777–
1855
).
Aim
8: L
inea
r opt
imiz
atio
n w
as d
evel
oped
by
Geo
rge
Dan
tzig
in th
e 19
40s a
s a w
ay o
f al
loca
ting
scar
ce re
sour
ces.
Further mathematics HL guide 17
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
1.5
The
vect
or sp
ace
n
.
Line
ar c
ombi
natio
ns o
f vec
tors
.
Span
ning
set.
Line
ar in
depe
nden
ce o
f vec
tors
.
A
im 8
: Lin
k to
Fou
rier a
naly
sis f
or w
aves
, w
hich
has
man
y us
es in
phy
sics
(Phy
sics
4.5
) an
d en
gine
erin
g.
Bas
is a
nd d
imen
sion
for a
vec
tor s
pace
.
Subs
pace
s.
Incl
ude
linea
r dep
ende
nce
and
use
of
dete
rmin
ants
.
Stud
ents
shou
ld b
e fa
mili
ar w
ith th
e te
rm
orth
ogon
al.
1.6
Line
ar tr
ansf
orm
atio
ns:
T(u
+ v)
= T
(u) +
T(v
), T(
ku) =
kT(
u).
Com
posi
tion
of li
near
tran
sfor
mat
ions
.
A
im 8
: Lor
enz
trans
form
atio
ns a
nd th
eir u
se in
re
lativ
ity a
nd q
uant
um m
echa
nics
(Phy
sics
13
.1).
Dom
ain,
rang
e, c
odom
ain
and
kern
el.
The
kern
el o
f a li
near
tran
sfor
mat
ion
is th
e nu
ll sp
ace
of it
s mat
rix re
pres
enta
tion.
R
esul
t and
pro
of th
at th
e ke
rnel
is a
subs
pace
of
the
dom
ain.
Res
ult a
nd p
roof
that
the
rang
e is
a su
bspa
ce o
f th
e co
dom
ain.
Ran
k-nu
llity
theo
rem
(pro
of n
ot re
quire
d).
Dim
(dom
ain)
= D
im(r
ange
) + D
im(k
erne
l)
whe
re
Dim
(ran
ge) =
rank
and
Dim
(ker
nel)
= nu
llity
.
Further mathematics HL guide18
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
1.7
Res
ult t
hat a
ny li
near
tran
sfor
mat
ion
can
be
repr
esen
ted
by a
mat
rix, a
nd th
e co
nver
se o
f th
is re
sult.
Res
ult t
hat t
he n
umbe
rs o
f lin
early
inde
pend
ent
row
s and
col
umns
are
equ
al, a
nd th
is is
the
dim
ensi
on o
f the
rang
e of
the
trans
form
atio
n (p
roof
not
requ
ired)
.
App
licat
ion
of li
near
tran
sfor
mat
ions
to
solu
tions
of s
yste
m o
f equ
atio
ns.
Solu
tion
of A
x =
b.
Usi
ng
(par
ticul
ar so
lutio
n) +
(any
mem
ber o
f the
nul
l sp
ace)
.
1.8
Geo
met
ric tr
ansf
orm
atio
ns re
pres
ente
d by
2
× 2
mat
rices
incl
ude
gene
ral r
otat
ion,
gen
eral
re
flect
ion
in
(tan
)y
xα
=, s
tretc
hes p
aral
lel t
o ax
es, s
hear
s par
alle
l to
axes
, and
pro
ject
ion
onto
(ta
n)
yx
α=
.
Com
posi
tions
of t
he a
bove
tran
sfor
mat
ions
.
A
im 8
: Com
pute
r gra
phic
s in
thre
e-di
men
sion
al m
odel
ling.
Geo
met
ric in
terp
reta
tion
of d
eter
min
ant.
New
are
ade
tol
d ar
ea.
A=
×
Further mathematics HL guide 19
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
1.9
Eige
nval
ues a
nd e
igen
vect
ors o
f 2 ×
2
mat
rices
.
Cha
ract
eris
tic p
olyn
omia
l of 2
× 2
mat
rices
.
Dia
gona
lizat
ion
of 2
× 2
mat
rices
(res
trict
ed to
th
e ca
se w
here
ther
e ar
e di
stin
ct re
al
eige
nval
ues)
.
App
licat
ions
to p
ower
s of 2
× 2
mat
rices
.
Geo
met
ric in
terp
reta
tion.
St
ocha
stic
pro
cess
es.
Stoc
k m
arke
t val
ues a
nd tr
ends
.
Inva
riant
stat
es.
Rep
rese
ntat
ion
of c
onic
s.
TOK
: “W
e ca
n us
e m
athe
mat
ics s
ucce
ssfu
lly
to m
odel
real
-wor
ld p
roce
sses
. Is t
his b
ecau
se
we
crea
te m
athe
mat
ics t
o m
irror
the
wor
ld o
r be
caus
e th
e w
orld
is in
trins
ical
ly
mat
hem
atic
al?”
For e
xam
ple,
the
natu
ral f
requ
ency
of a
n ob
ject
ca
n be
cha
ract
eriz
ed b
y th
e ei
genv
alue
of
smal
lest
mag
nitu
de (1
940
Taco
ma
Nar
row
s B
ridge
dis
aste
r).
Aim
8: D
ampi
ng n
oise
in c
ar d
esig
n, te
st fo
r cr
acks
in so
lid o
bjec
ts, o
il ex
plor
atio
n, th
e G
oogl
e Pa
geR
ank
form
ula,
and
“th
e $2
5 bi
llion
dol
lar e
igen
vect
or”.
Mar
kov
chai
ns, l
ink
with
gen
etic
s (B
iolo
gy 4
).
Further mathematics HL guide20
Syllabus content
Top
ic 2
—G
eom
etry
48
hou
rs
The
aim
of t
his s
ectio
n is
to d
evel
op st
uden
ts’ g
eom
etric
intu
ition
, vis
ualiz
atio
n an
d de
duct
ive
reas
onin
g.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
2.1
Sim
ilar a
nd c
ongr
uent
tria
ngle
s.
Eucl
id’s
theo
rem
for p
ropo
rtion
al se
gmen
ts in
a
right
-ang
led
trian
gle.
T
OK
, Int
: The
influ
ence
of E
uclid
’s a
xiom
atic
ap
proa
ch o
n ph
iloso
phy
(Des
carte
s) a
nd
polit
ics (
Jeff
erso
n: A
mer
ican
Dec
lara
tion
of
Inde
pend
ence
).
TOK
: Hip
pasu
s’ e
xist
ence
pro
of fo
r irr
atio
nal
num
bers
and
impa
ct o
n se
para
te d
evel
opm
ent
of n
umbe
r and
geo
met
ry.
TOK
: Cris
is o
ver n
on-E
uclid
ean
geom
etry
pa
ralle
ls w
ith th
at o
f Can
tor’
s set
theo
ry.
2.2
Cen
tres o
f a tr
iang
le: o
rthoc
entre
, inc
entre
, ci
rcum
cent
re a
nd c
entro
id.
The
term
s alti
tude
, ang
le b
isec
tor,
perp
endi
cula
r bis
ecto
r, m
edia
n.
Proo
f of c
oncu
rren
cy th
eore
ms.
The
term
s ins
crib
ed a
nd c
ircum
scrib
ed.
App
l: ce
ntre
of m
ass,
trian
gula
tion.
2.3
Circ
le g
eom
etry
. A
ngle
at c
entre
theo
rem
and
cor
olla
ries.
Tang
ents
; arc
s, ch
ords
and
seca
nts.
In a
cyc
lic q
uadr
ilate
ral,
oppo
site
ang
les a
re
supp
lem
enta
ry, a
nd th
e co
nver
se.
The
tang
ent–
seca
nt a
nd se
cant
–sec
ant
theo
rem
s and
inte
rsec
ting
chor
ds th
eore
ms.
2.4
Ang
le b
isec
tor t
heor
em; A
pollo
nius
’ circ
le
theo
rem
, Men
elau
s’ th
eore
m; C
eva’
s the
orem
; Pt
olem
y’s t
heor
em fo
r cyc
lic q
uadr
ilate
rals
.
Proo
fs o
f the
se th
eore
ms a
nd c
onve
rses
.
The
use
of th
ese
theo
rem
s to
prov
e fu
rther
re
sults
.
Further mathematics HL guide 21
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
2.5
Find
ing
equa
tions
of l
oci.
Coo
rdin
ate
geom
etry
of t
he c
ircle
.
Tang
ents
to a
circ
le.
The
equa
tions
2
22
()
()
xh
yk
r−
+−
= a
nd
22
0x
ydx
eyf
++
++
=.
TOK
: Con
sequ
ence
s of D
esca
rtes’
uni
ficat
ion
of a
lgeb
ra a
nd g
eom
etry
.
2.6
Con
ic se
ctio
ns.
The
para
bola
, elli
pse
and
hype
rbol
a, in
clud
ing
rect
angu
lar h
yper
bola
.
Focu
s–di
rect
rix d
efin
ition
s.
Tang
ents
and
nor
mal
s.
The
stan
dard
form
s 2
4y
ax=
, 2
2
22
1x
ya
b+
=,
22
22
1x
ya
b−
= a
nd th
eir t
rans
latio
ns.
App
l: Sa
telli
te d
ish,
hea
dlig
ht, o
rbits
, pr
ojec
tiles
(Phy
sics
9.1
).
TOK
: Kep
ler’
s diff
icul
ties a
ccep
ting
that
an
orbi
t was
not
a “
perf
ect”
circ
le.
2.7
Para
met
ric e
quat
ions
.
Para
met
ric d
iffer
entia
tion.
Tang
ents
and
nor
mal
s.
The
stan
dard
par
amet
ric e
quat
ions
of t
he
circ
le, p
arab
ola,
elli
pse,
rect
angu
lar h
yper
bola
, hy
perb
ola.
cos
xa
θ=
, si
ny
aθ
=,
2x
at=
, 2
yat
=,
cos
xa
θ=
, si
ny
bθ
=,
xct
=,
cy
t=
,
sec
xa
θ=
, ta
ny
bθ
=.
2.8
The
gene
ral c
onic
2
22
0ax
bxy
cydx
eyf
++
++
+=
, an
d th
e qu
adra
tic fo
rm
T2
22
axbx
ycy
=+
+x
Ax
.
Dia
gona
lizin
g th
e m
atrix
A w
ith th
e ro
tatio
n m
atrix
P a
nd re
duci
ng th
e ge
nera
l con
ic to
st
anda
rd fo
rm.
The
gene
ral c
onic
can
be
rota
ted
to g
ive
the
form
22
12
0x
ydx
eyf
λλ
++
++
=,
whe
re
1λ a
nd
2λ a
re th
e ei
genv
alue
s of t
he
mat
rix A
in th
e qu
adra
tic fo
rm
T xA
x.
Further mathematics HL guide22
Syllabus content
Geometry theorems—clarification of theorems used in topic 2Teachers and students should be aware that some of the theorems mentioned in this section may be known by other names, or some names of theorems may be associated with different statements in some textbooks. To avoid confusion, on examination papers, theorems that may be misinterpreted are defined below.
Euclid’s theorem for proportional segments in a right-angled triangleThe proportional segments p and q satisfy the following:
h2 = pq,
a2 = pc,
b2 = qc.
Angle at centre theoremThe angle subtended by an arc at the circumference is half that subtended by the same arc at the centre.
CorollariesAngles subtended at the circumference by the same arc are equal.
ab
h
c pq
2x
O
x
x x
Further mathematics HL guide 23
Syllabus content
The angle in a semicircle is a right angle.
The alternate segment theorem: The angle between a tangent and a chord is equal to the angle subtended by the chord in the alternate segment.
The tangent is perpendicular to the radius at the point of tangency.
The intersecting chords theoremab = cd
O
x
x
O
a
d
c
b
Further mathematics HL guide24
Syllabus content
The tangent–secant and secant–secant theoremsPT2 = PA × PB = PC × PD
Apollonius’ circle theorem (circle of Apollonius)If A and B are two fixed points such that PA
PB is a constant not equal to one, then the locus of P is a circle. This
is called the circle of Apollonius.
Included: the converse of this theorem.
Menelaus’ theoremIf a transversal meets the sides [BC], [CA] and [AB] of a triangle at D, E and F respectively, then
BDDC
CEEA
AFFB
× × = −1.
Converse: if D, E and F are points on the sides [BC], [CA] and [AB], respectively, of a triangle such that BDDC
CEEA
AFFB
× × = −1, then D, E and F are collinear.
TP
C
D
B
A
F
B
E
C D
EF
A
B C D
Further mathematics HL guide 25
Syllabus content
Ceva’s theoremIf three concurrent lines are drawn through the vertices A, B and C of a triangle ABC to meet the opposite sides at D, E and F respectively, thenBDDC
� � ��CEEA
AFFB
1.
Converse: if D, E and F are points on [BC], [CA] and [AB], respectively, such that BDDC
� � ��CEEA
AFFB
1, then [AD], [BE] and [CF] are concurrent.
Note on Ceva’s theorem and Menelaus’ theoremThe statements and proofs of these theorems presuppose the idea of sensed magnitudes. Two segments [AB] and [PQ] of the same or parallel lines are said to have the same sense or opposite senses (or are sometimes called like or unlike) according to whether the displacements A → B and P → Q are in the same or opposite directions. The idea of sensed magnitudes may be used to prove the following theorem:
If A, B and C are any three collinear points, then AB BC CA 0, where AB, BC and CA denote sensed magnitudes.
Ptolemy’s theoremIf a quadrilateral is cyclic, the sum of the products of the two pairs of opposite sides equals the products of the diagonals. That is, for a cyclic quadrilateral ABCD, AB CD BC DA AC BD .
A
D
OE
C
F
B
A
E
CDB
F
O
A
D
B
C
Further mathematics HL guide26
Syllabus content
Angle bisector theoremThe angle bisector of an angle of a triangle divides the side of the triangle opposite the angle into segments proportional to the sides adjacent to the angle.
If ABC is the given triangle with (AD) as the bisector of angle BAC intersecting (BC) at point D, then BDDC
ABAC
� for internal bisectors
and
BDDC
ABAC
�� for external bisectors.
BÂD CÂD CÂD EÂD
Included: the converse of this theorem.
A
B D C
E
DCB
A
Further mathematics HL guide 27
Syllabus content
Top
ic 3
—St
atis
tics
and
prob
abili
ty
48 h
ours
Th
e ai
ms o
f thi
s top
ic a
re to
allo
w st
uden
ts th
e op
portu
nity
to a
ppro
ach
stat
istic
s in
a pr
actic
al w
ay; t
o de
mon
stra
te a
goo
d le
vel o
f sta
tistic
al u
nder
stan
ding
; an
d to
und
erst
and
whi
ch s
ituat
ions
app
ly a
nd t
o in
terp
ret
the
give
n re
sults
. It
is e
xpec
ted
that
GD
Cs
will
be
used
thr
ough
out
this
opt
ion
and
that
the
m
inim
um r
equi
rem
ent
of a
GD
C w
ill b
e to
fin
d th
e pr
obab
ility
dis
tribu
tion
func
tion
), cu
mul
ativ
e di
strib
utio
n fu
nctio
n (c
df),
inve
rse
cum
ulat
ive
dist
ribut
ion
func
tion,
p-v
alue
s an
d te
st s
tatis
tics,
incl
udin
g ca
lcul
atio
ns f
or t
he f
ollo
win
g di
strib
utio
ns:
bino
mia
l, Po
isso
n, n
orm
al a
nd t
. St
uden
ts a
re
expe
cted
to se
t up
the
prob
lem
mat
hem
atic
ally
and
then
read
the
answ
ers
from
the
GD
C, i
ndic
atin
g th
is w
ithin
thei
r writ
ten
answ
ers.
Cal
cula
tor-s
peci
fic o
r br
and-
spec
ific
lang
uage
shou
ld n
ot b
e us
ed w
ithin
thes
e ex
plan
atio
ns.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
3.1
Cum
ulat
ive
dist
ribut
ion
func
tions
for b
oth
disc
rete
and
con
tinuo
us d
istri
butio
ns.
Geo
met
ric d
istri
butio
n.
Neg
ativ
e bi
nom
ial d
istri
butio
n.
Prob
abili
ty g
ener
atin
g fu
nctio
ns fo
r dis
cret
e ra
ndom
var
iabl
es.
()
E()
()
Xx
xG
tt
PX
xt
==
=∑
. In
t: A
lso
know
n as
Pas
cal’s
dis
tribu
tion.
Usi
ng p
roba
bilit
y ge
nera
ting
func
tions
to fi
nd
mea
n, v
aria
nce
and
the
distr
ibut
ion
of th
e su
m
of n
inde
pend
ent r
ando
m v
aria
bles
.
A
im 8
: Sta
tistic
al c
ompr
essi
on o
f dat
a fil
es.
3.2
Line
ar tr
ansf
orm
atio
n of
a si
ngle
rand
om
varia
ble.
Mea
n of
line
ar c
ombi
natio
ns o
f n ra
ndom
va
riabl
es.
Var
ianc
e of
line
ar c
ombi
natio
ns o
f n
inde
pend
ent r
ando
m v
aria
bles
.
E()
E()
aXb
aX
b+
=+
, 2
Var
()
Var
()
aXb
aX
+=
.
Expe
ctat
ion
of th
e pr
oduc
t of i
ndep
ende
nt
rand
om v
aria
bles
. E(
)E(
)E(
)XY
XY
=.
Further mathematics HL guide28
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
3.3
Unb
iase
d es
timat
ors a
nd e
stim
ates
.
Com
paris
on o
f unb
iase
d es
timat
ors b
ased
on
varia
nces
.
T is
an
unbi
ased
est
imat
or fo
r the
par
amet
er
θ if
E(
)T
θ=
.
1T is
a m
ore
effic
ient
esti
mat
or th
an
2T if
12
Var
()
Var
()
TT
<.
TO
K: M
athe
mat
ics a
nd th
e w
orld
. In
the
abse
nce
of k
now
ing
the
valu
e of
a p
aram
eter
, w
ill a
n un
bias
ed e
stim
ator
alw
ays b
e be
tter
than
a b
iase
d on
e?
X a
s an
unbi
ased
est
imat
or fo
r µ
.
2 S a
s an
unbi
ased
est
imat
or fo
r 2
σ.
1ni
i
XX
n=
=∑
.
()2
2
11
ni
i
XX
Sn
=
−=
−∑
.
3.4
A li
near
com
bina
tion
of in
depe
nden
t nor
mal
ra
ndom
var
iabl
es is
nor
mal
ly d
istri
bute
d. In
pa
rticu
lar,
22
~N
(,
)~
N,
XX
nσµσ
µ
⇒
.
The
cent
ral l
imit
theo
rem
.
A
im 8
/TO
K: M
athe
mat
ics a
nd th
e w
orld
. “W
ithou
t the
cen
tral l
imit
theo
rem
, the
re c
ould
be
no
stat
istic
s of a
ny v
alue
with
in th
e hu
man
sc
ienc
es.”
TO
K: N
atur
e of
mat
hem
atic
s. Th
e ce
ntra
l lim
it th
eore
m c
an b
e pr
oved
mat
hem
atic
ally
(f
orm
alis
m),
but i
ts tr
uth
can
be c
onfir
med
by
its a
pplic
atio
ns (e
mpi
ricis
m).
Further mathematics HL guide 29
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
3.5
Con
fiden
ce in
terv
als f
or th
e m
ean
of a
nor
mal
po
pula
tion.
U
se o
f the
nor
mal
dis
tribu
tion
whe
n σ
is
know
n an
d us
e of
the
t-dis
tribu
tion
whe
n σ
is
unkn
own,
rega
rdle
ss o
f sam
ple
size
. The
cas
e of
mat
ched
pai
rs is
to b
e tre
ated
as a
n ex
ampl
e of
a si
ngle
sam
ple
tech
niqu
e.
TO
K: M
athe
mat
ics a
nd th
e w
orld
. Cla
imin
g br
and
A is
“be
tter”
on
aver
age
than
bra
nd B
ca
n m
ean
very
littl
e if
ther
e is
a la
rge
over
lap
betw
een
the
conf
iden
ce in
terv
als o
f the
two
mea
ns.
App
l: G
eogr
aphy
.
3.6
Nul
l and
alte
rnat
ive
hypo
thes
es,
0H
and
1
H.
Sign
ifica
nce
leve
l.
Crit
ical
regi
ons,
criti
cal v
alue
s, p-
valu
es, o
ne-
taile
d an
d tw
o-ta
iled
test
s.
Type
I an
d II
erro
rs, i
nclu
ding
cal
cula
tions
of
thei
r pro
babi
litie
s.
Test
ing
hypo
thes
es fo
r the
mea
n of
a n
orm
al
popu
latio
n.
Use
of t
he n
orm
al d
istri
butio
n w
hen σ
is
know
n an
d us
e of
the
t-dis
tribu
tion
whe
n σ
is
unkn
own,
rega
rdle
ss o
f sam
ple
size
. The
cas
e of
mat
ched
pai
rs is
to b
e tre
ated
as a
n ex
ampl
e of
a si
ngle
sam
ple
tech
niqu
e.
TO
K: M
athe
mat
ics a
nd th
e w
orld
. In
prac
tical
te
rms,
is sa
ying
that
a re
sult
is si
gnifi
cant
the
sam
e as
sayi
ng th
at it
is tr
ue?
TO
K: M
athe
mat
ics a
nd th
e w
orld
. Doe
s the
ab
ility
to te
st o
nly
certa
in p
aram
eter
s in
a po
pula
tion
affe
ct th
e w
ay k
now
ledg
e cl
aim
s in
the
hum
an sc
ienc
es a
re v
alue
d?
App
l: W
hen
is it
mor
e im
porta
nt n
ot to
mak
e a
Type
I er
ror a
nd w
hen
is it
mor
e im
porta
nt n
ot
to m
ake
a Ty
pe II
err
or?
Further mathematics HL guide30
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
3.7
Intro
duct
ion
to b
ivar
iate
dis
tribu
tions
. In
form
al d
iscu
ssio
n of
com
mon
ly o
ccur
ring
situ
atio
ns, e
g m
arks
in p
ure
mat
hem
atic
s and
st
atis
tics e
xam
s tak
en b
y a
clas
s of s
tude
nts,
sala
ry a
nd a
ge o
f tea
cher
s in
a ce
rtain
scho
ol.
The
need
for a
mea
sure
of a
ssoc
iatio
n be
twee
n th
e va
riabl
es a
nd th
e po
ssib
ility
of p
redi
ctin
g th
e va
lue
of o
ne o
f the
var
iabl
es g
iven
the
valu
e of
the
othe
r var
iabl
e.
App
l: G
eogr
aphi
c sk
ills.
Aim
8: T
he c
orre
latio
n be
twee
n sm
okin
g an
d lu
ng c
ance
r was
“di
scov
ered
” us
ing
mat
hem
atic
s. Sc
ienc
e ha
d to
just
ify th
e ca
use.
Cov
aria
nce
and
(pop
ulat
ion)
pro
duct
mom
ent
corr
elat
ion
coef
ficie
nt ρ
. C
ov(
,)
E[(
)
,
()]
E()
xy
xy
XY
XY
XYµ
µµ
µ
=−
−
=−
whe
re
E(),
E()
xy
XY
µµ
==
. C
ov(
,)
Var
()V
ar(
)X
YX
Yρ=
.
App
l: U
sing
tech
nolo
gy to
fit a
rang
e of
cur
ves
to a
set o
f dat
a.
Proo
f tha
t ρ =
0 in
the
case
of i
ndep
ende
nce
and ±1
in th
e ca
se o
f a li
near
rela
tions
hip
betw
een
X an
d Y.
The
use
of ρ
as a
mea
sure
of a
ssoc
iatio
n be
twee
n X
and
Y, w
ith v
alue
s nea
r 0 in
dica
ting
a w
eak
asso
ciat
ion
and
valu
es n
ear +
1 or
nea
r –1
indi
catin
g a
stro
ng a
ssoc
iatio
n.
TOK
: Mat
hem
atic
s and
the
wor
ld. G
iven
that
a
set o
f dat
a m
ay b
e ap
prox
imat
ely
fitte
d by
a
rang
e of
cur
ves,
whe
re w
ould
we
seek
for
know
ledg
e of
whi
ch e
quat
ion
is th
e “t
rue”
m
odel
?
Aim
8: T
he p
hysi
cist
Fra
nk O
ppen
heim
er
wro
te: “
Pred
ictio
n is
dep
ende
nt o
nly
on th
e as
sum
ptio
n th
at o
bser
ved
patte
rns w
ill b
e re
peat
ed.”
Thi
s is t
he d
ange
r of e
xtra
pola
tion.
Th
ere
are
man
y ex
ampl
es o
f its
failu
re in
the
past
, eg
shar
e pr
ices
, the
spre
ad o
f dis
ease
, cl
imat
e ch
ange
.
(con
tinue
d)
Def
initi
on o
f the
(sam
ple)
pro
duct
mom
ent
corr
elat
ion
coef
ficie
nt R
in te
rms o
f n p
aire
d ob
serv
atio
ns o
n X
and
Y. It
s app
licat
ion
to th
e es
timat
ion
of ρ
.
1
22
11
1
22
22
1
()(
)
()
()
.
n
ii
in
n
ii
ii
n
ii
i
n
ii
i
XX
YY
RX
XY
Y
XY
nXY
XnX
YnY
=
==
=
=
−−
=
−−
−=
−
−
∑ ∑∑
∑
∑∑
Further mathematics HL guide 31
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
Info
rmal
inte
rpre
tatio
n of
r, th
e ob
serv
ed v
alue
of
R. S
catte
r dia
gram
s. V
alue
s of r
nea
r 0 in
dica
te a
wea
k as
soci
atio
n be
twee
nX
and
Y, a
nd v
alue
s nea
r 1
indi
cate
a
stro
ng a
ssoc
iatio
n.
(see
not
es a
bove
)
The
follo
win
g to
pics
are
bas
ed o
n th
e as
sum
ptio
n of
biv
aria
te n
orm
ality
. It
is e
xpec
ted
that
the
GD
C w
ill b
e us
ed
whe
reve
r pos
sibl
e in
the
follo
win
g w
ork.
Use
of t
he t-
stat
istic
to te
st th
e nu
ll hy
poth
esis
= 0.
22
1nR
R h
as th
e St
uden
t’s t-
dist
ribut
ion
with
(2)
n
deg
rees
of f
reed
om.
Kno
wle
dge
of th
e fa
cts t
hat t
he re
gres
sion
of X
onY
E(
|)
XY
y
and
Y o
n X
E(
|)
YX
x
are
linea
r.
Leas
t-squ
ares
est
imat
es o
f the
se re
gres
sion
lin
es (p
roof
not
requ
ired)
.
The
use
of th
ese
regr
essi
on li
nes t
o pr
edic
t the
va
lue
of o
ne o
f the
var
iabl
es g
iven
the
valu
e of
th
e ot
her.
1
2
1
1
22
1()(
) (
)(
)
(),
n
ii
in
ii
n
ii
in
ii
xx
yy
xx
yy
yy
xy
nxy
yy
yny
1
2
1
1
22
1()(
) (
)(
)
().
n
ii
in
ii
n
ii
in
ii
xx
yy
yy
xx
xx
xy
nxy
xx
xnx
Further mathematics HL guide32
Syllabus content
Top
ic 4
—Se
ts, r
elat
ions
and
gro
ups
48 h
ours
Th
e aim
s of t
his t
opic
are t
o pr
ovid
e the
opp
ortu
nity
to st
udy
som
e im
porta
nt m
athe
mat
ical
conc
epts,
and
intro
duce
the p
rinci
ples
of p
roof
thro
ugh
abstr
act a
lgeb
ra.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
4.1
Fini
te a
nd in
finite
sets
. Sub
sets
.
Ope
ratio
ns o
n se
ts: u
nion
; int
erse
ctio
n;
com
plem
ent;
set d
iffer
ence
; sym
met
ric
diff
eren
ce.
TO
K: C
anto
r the
ory
of tr
ansf
inite
num
bers
, R
usse
ll’s p
arad
ox, G
odel
’s in
com
plet
enes
s th
eore
ms.
De
Mor
gan’
s law
s: d
istri
butiv
e, a
ssoc
iativ
e an
d co
mm
utat
ive
law
s (fo
r uni
on a
nd in
ters
ectio
n).
Illus
tratio
n of
thes
e la
ws u
sing
Ven
n di
agra
ms.
Stud
ents
may
be
aske
d to
pro
ve th
at tw
o se
ts
are
the
sam
e by
est
ablis
hing
that
AB
⊆ a
nd
BA
⊆.
App
l: Lo
gic,
Boo
lean
alg
ebra
, com
pute
r ci
rcui
ts.
4.2
Ord
ered
pai
rs: t
he C
arte
sian
pro
duct
of t
wo
sets
.
Rel
atio
ns: e
quiv
alen
ce re
latio
ns; e
quiv
alen
ce
clas
ses.
An
equi
vale
nce
rela
tion
on a
set f
orm
s a
parti
tion
of th
e se
t. A
ppl,
Int:
Sco
ttish
cla
ns.
4.3
Func
tions
: inj
ectio
ns; s
urje
ctio
ns; b
iject
ions
. Th
e te
rm c
odom
ain.
Com
posi
tion
of fu
nctio
ns a
nd in
vers
e fu
nctio
ns.
Kno
wle
dge
that
the
func
tion
com
posi
tion
is n
ot
a co
mm
utat
ive
oper
atio
n an
d th
at if
f is
a
bije
ctio
n fr
om se
t A to
set B
then
1
f− e
xist
s an
d is
a b
iject
ion
from
set B
to se
t A.
4.4
Bin
ary
oper
atio
ns.
A b
inar
y op
erat
ion ∗
on a
non
-em
pty
set S
is a
ru
le fo
r com
bini
ng a
ny tw
o el
emen
ts
,ab
S∈
to
giv
e a
uniq
ue e
lem
ent c
. Tha
t is,
in th
is
defin
ition
, a b
inar
y op
erat
ion
on a
set i
s not
ne
cess
arily
clo
sed.
Ope
ratio
n ta
bles
(Cay
ley
tabl
es).
Further mathematics HL guide 33
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
4.5
Bin
ary
oper
atio
ns: a
ssoc
iativ
e, d
istri
butiv
e an
d co
mm
utat
ive
prop
ertie
s. Th
e ar
ithm
etic
ope
ratio
ns o
n
and
.
Exam
ples
of d
istri
butiv
ity c
ould
incl
ude
the
fact
that
, on
, mul
tiplic
atio
n is
dis
tribu
tive
over
add
ition
but
add
ition
is n
ot d
istri
butiv
e ov
er m
ultip
licat
ion.
TO
K: W
hich
are
mor
e fu
ndam
enta
l, th
e ge
nera
l mod
els o
r the
fam
iliar
exa
mpl
es?
4.6
The
iden
tity
elem
ent e
.
The
inve
rse
1a−
of a
n el
emen
t a.
Proo
f tha
t lef
t-can
cella
tion
and
right
-ca
ncel
latio
n by
an
elem
ent a
hol
d, p
rovi
ded
that
a h
as a
n in
vers
e.
Proo
fs o
f the
uni
quen
ess o
f the
iden
tity
and
inve
rse
elem
ents
.
Bot
h th
e rig
ht-id
entit
y a
ea
∗=
and
left-
iden
tity
ea
a∗
= m
ust h
old
if e
is a
n id
entit
y el
emen
t.
Bot
h 1
aa
e−
∗=
and
1
aa
e−∗
= m
ust h
old.
4.7
The
defin
ition
of a
gro
up {
,}
G∗
.
The
oper
atio
n ta
ble
of a
gro
up is
a L
atin
sq
uare
, but
the
conv
erse
is fa
lse.
For t
he se
t G u
nder
a g
iven
ope
ratio
n ∗:
• G
is c
lose
d un
der ∗;
• ∗
is a
ssoc
iativ
e;
• G
con
tain
s an
iden
tity
elem
ent;
• ea
ch e
lem
ent i
n G
has
an
inve
rse
in G
.
App
l: Ex
iste
nce
of fo
rmul
a fo
r roo
ts o
f po
lyno
mia
ls.
App
l: G
aloi
s the
ory
for t
he im
poss
ibili
ty o
f su
ch fo
rmul
ae fo
r pol
ynom
ials
of d
egre
e 5
or
high
er.
Abe
lian
grou
ps.
ab
ba
∗=
∗, f
or a
ll ,ab
G∈
.
Further mathematics HL guide34
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
4.8
Exam
ples
of g
roup
s:
•
, ,
and
u
nder
add
ition
;
• in
tege
rs u
nder
add
ition
mod
ulo
n;
• no
n-ze
ro in
tege
rs u
nder
mul
tiplic
atio
n,
mod
ulo
p, w
here
p is
prim
e;
Cro
ss-to
pic
ques
tions
may
be
set i
n fu
rther
m
athe
mat
ics e
xam
inat
ions
, so
ther
e m
ay b
e qu
estio
ns o
n gr
oups
of m
atric
es.
App
l: R
ubik
’s c
ube,
tim
e m
easu
res,
crys
tal
stru
ctur
e, sy
mm
etrie
s of m
olec
ules
, stru
t and
ca
ble
cons
truct
ions
, Phy
sics
H2.
2 (s
peci
al
rela
tivity
), th
e 8-
fold
way
, sup
ersy
mm
etry
.
• sy
mm
etrie
s of p
lane
figu
res,
incl
udin
g eq
uila
tera
l tria
ngle
s and
rect
angl
es;
• in
verti
ble
func
tions
und
er c
ompo
sitio
n of
fu
nctio
ns.
The
com
posi
tion
21
TT
den
otes
1T fo
llow
ed
by2T.
4.9
The
orde
r of a
gro
up.
The
orde
r of a
gro
up e
lem
ent.
Cyc
lic g
roup
s.
Gen
erat
ors.
Proo
f tha
t all
cycl
ic g
roup
s are
Abe
lian.
A
ppl:
Mus
ic c
ircle
of f
ifths
, prim
e nu
mbe
rs.
Further mathematics HL guide 35
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
4.10
Pe
rmut
atio
ns u
nder
com
posi
tion
of
perm
utat
ions
.
Cyc
le n
otat
ion
for p
erm
utat
ions
.
Res
ult t
hat e
very
per
mut
atio
n ca
n be
writ
ten
as
a co
mpo
sitio
n of
dis
join
t cyc
les.
The
orde
r of a
com
bina
tion
of c
ycle
s.
On
exam
inat
ion
pape
rs: t
he fo
rm
12
33
12
p
=
or i
n cy
cle
nota
tion
(132
) will
be u
sed
to re
pres
ent t
he p
erm
utat
ion
13
→,
21
→, 3
2.→
App
l: C
rypt
ogra
phy,
cam
pano
logy
.
4.11
Su
bgro
ups,
prop
er su
bgro
ups.
A p
rope
r sub
grou
p is
neith
er th
e gro
up it
self
nor
the s
ubgr
oup
cont
aini
ng o
nly
the i
dent
ity el
emen
t.
Use
and
pro
of o
f sub
grou
p te
sts.
Supp
ose
that
{,
}G∗
is a
gro
up a
nd H
is a
no
n-em
pty
subs
et o
f G. T
hen
{,
}H
∗ is
a
subg
roup
of {
,}
G∗
if
1a
bH
−∗
∈ w
hene
ver
,ab
H∈
.
Supp
ose
that
{,
}G∗
is a
fini
te g
roup
and
H is
a
non-
empt
y su
bset
of G
. The
n {
,}
H∗
is a
su
bgro
up o
f {,
}G∗
if H
is c
lose
d un
der ∗.
Def
initi
on a
nd e
xam
ples
of l
eft a
nd ri
ght c
oset
s of
a su
bgro
up o
f a g
roup
.
Lagr
ange
’s th
eore
m.
Use
and
pro
of o
f the
resu
lt th
at th
e or
der o
f a
finite
gro
up is
div
isib
le b
y th
e or
der o
f any
el
emen
t. (C
orol
lary
to L
agra
nge’
s the
orem
.)
A
ppl:
Prim
e fa
ctor
izat
ion,
sym
met
ry b
reak
ing.
Further mathematics HL guide36
Syllabus content
Sylla
bus c
onte
nt
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
4.12
Def
initi
on o
f a g
roup
hom
omor
phis
m.
Infin
ite g
roup
s as w
ell a
s fin
ite g
roup
s.
Let {
,*}
Gan
d {
,}
H
be g
roup
s, th
en th
e fu
nctio
n:fG
H→
is a
hom
omor
phis
m if
(*
)(
)(
)f
ab
fa
fb
=
for a
ll,ab
G∈
.
Def
initi
on o
f the
ker
nel o
f a h
omom
orph
ism
.
Proo
f tha
t the
ker
nel a
nd ra
nge
of a
ho
mom
orph
ism
are
subg
roup
s.
If :fG
H→
is a
gro
up h
omom
orph
ism
, the
n K
er(
)fis
the
set o
f a
G∈
such
that
()
Hf
ae
=.
Proo
f of h
omom
orph
ism
pro
perti
es fo
r id
entit
ies a
nd in
vers
es.
Iden
tity:
let
Gean
d He
be th
e id
entit
y el
emen
ts
of {
,*}
Gan
d{
,}
H
,res
pect
ivel
y, th
en(
)G
Hf
ee
=.
Inve
rse:
(
)11
()
()
fa
fa
−−
=fo
r all
aG
∈.
Isom
orph
ism
of g
roup
s.In
finite
gro
ups a
s wel
l as f
inite
gro
ups.
The
hom
omor
phis
m
:fG
H→
is a
n is
omor
phis
m if
fis
bije
ctiv
e.
The
orde
r of a
n el
emen
t is u
ncha
nged
by
an
isom
orph
ism
.
Fur
ther
mat
hem
atic
s HL
guid
e22
Further mathematics HL guide 37
Syllabus content
Sy
llabu
s con
tent
Fur
ther
mat
hem
atic
s HL
guid
e 23
Topic 5—Calculus
48 hours
The
aim
s of t
his t
opic
are
to in
trodu
ce li
mit
theo
rem
s and
con
verg
ence
of s
erie
s, an
d to
use
cal
culu
s res
ults
to so
lve
diffe
rent
ial e
quat
ions
.
5.1
Infin
ite se
quen
ces o
f rea
l num
bers
and
thei
r co
nver
genc
e or
div
erge
nce.
In
form
al tr
eatm
ent o
f lim
it of
sum
, diff
eren
ce,
prod
uct,
quot
ient
; squ
eeze
theo
rem
.
Div
erge
nt is
take
n to
mea
n no
t con
verg
ent.
TOK
: Zen
o’s p
arad
ox, i
mpa
ct o
f inf
inite
se
quen
ces a
nd li
mits
on
our u
nder
stand
ing
of
the
phys
ical
wor
ld.
5.2
Conv
erge
nce
of in
finite
serie
s.
Tests
for c
onve
rgen
ce: c
ompa
rison
test;
lim
it co
mpa
rison
test;
ratio
test;
inte
gral
test.
The
sum
of a
serie
s is t
he li
mit
of th
e se
quen
ce
of it
s par
tial s
ums.
Stud
ents
shou
ld b
e aw
are
that
if li
mn
nx
= 0
then
the
serie
s is n
ot n
eces
saril
y co
nver
gent
, bu
t if
limn
nx
0,
the
serie
s div
erge
s.
TOK
: Eul
er’s
idea
that
1 2
11
11
.
Was
it a
mist
ake
or ju
st an
alte
rnat
ive
view
?
The
p-se
ries,
1 p n
. 1 p n
is
con
verg
ent f
or
1p
and
div
erge
nt
othe
rwise
. Whe
n1
p
, thi
s is t
he h
arm
onic
se
ries.
Serie
s tha
t con
verg
e ab
solu
tely
.
Serie
s tha
t con
verg
e co
nditi
onal
ly.
Cond
ition
s for
con
verg
ence
.
Alte
rnat
ing
serie
s.
Pow
er se
ries:
radi
us o
f con
verg
ence
and
in
terv
al o
f con
verg
ence
. Det
erm
inat
ion
of th
e ra
dius
of c
onve
rgen
ce b
y th
e ra
tio te
st.
The
abso
lute
val
ue o
f the
trun
catio
n er
ror i
s le
ss th
an th
e ne
xt te
rm in
the
serie
s.
Further mathematics HL guide38
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.3
Con
tinui
ty a
nd d
iffer
entia
bilit
y of
a fu
nctio
n at
a
poin
t. Te
st fo
r con
tinui
ty:
()
()
()
limlim
xa–
xa+
fx
=f
a=
fx
→→
.
Con
tinuo
us fu
nctio
ns a
nd d
iffer
entia
ble
func
tions
. Te
st fo
r diff
eren
tiabi
lity:
f is
con
tinuo
us a
t a a
nd
()
0
()
lim h
fa
h–
fa
h→
−
+ a
nd
()
0
()
lim h+
fa
h–
fa
h→
+ e
xist
and
are
equ
al.
Stud
ents
shou
ld b
e aw
are
that
a fu
nctio
n m
ay
be c
ontin
uous
but
not
diff
eren
tiabl
e at
a p
oint
, eg
()
fx
=x
and
sim
ple
piec
ewis
e fu
nctio
ns.
5.4
The
inte
gral
as a
lim
it of
a su
m; l
ower
and
up
per R
iem
ann
sum
s.
Int:
How
clo
se w
as A
rchi
med
es to
inte
gral
ca
lcul
us?
Int:
Con
tribu
tion
of A
rab,
Chi
nese
and
Indi
an
mat
hem
atic
ians
to th
e de
velo
pmen
t of c
alcu
lus.
Aim
8: L
eibn
iz v
ersu
s New
ton
vers
us th
e “g
iant
s” o
n w
hose
shou
lder
s the
y st
ood—
who
de
serv
es c
redi
t for
mat
hem
atic
al p
rogr
ess?
TOK
: Con
side
r
1f
x=
x, 1
x≤
≤∞
.
An
infin
ite a
rea
swee
ps o
ut a
fini
te v
olum
e.
Can
this
be
reco
ncile
d w
ith o
ur in
tuiti
on?
Wha
t do
es th
is te
ll us
abo
ut m
athe
mat
ical
kn
owle
dge?
Fund
amen
tal t
heor
em o
f cal
culu
s. d
()d
()
d
x a
fy
y=
fx
x
∫
.
Impr
oper
inte
gral
s of t
he ty
pe
()d
a
fx
x∞ ∫
.
Further mathematics HL guide 39
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.5
Firs
t-ord
er d
iffer
entia
l equ
atio
ns.
Geo
met
ric in
terp
reta
tion
usin
g sl
ope
field
s, in
clud
ing
iden
tific
atio
n of
isoc
lines
.
A
ppl:
Rea
l-life
diff
eren
tial e
quat
ions
, eg
New
ton’
s law
of c
oolin
g,
popu
latio
n gr
owth
,
carb
on d
atin
g.
Num
eric
al so
lutio
n of
d(
,)
dy=
fx
yx
usin
g Eu
ler’
s met
hod.
Var
iabl
es se
para
ble.
Hom
ogen
eous
diff
eren
tial e
quat
ion
d dyy
=f
xx
usin
g th
e su
bstit
utio
n y
= vx
.
Solu
tion
of
y′ +
P(x)
y =
Q(x
),
usin
g th
e in
tegr
atin
g fa
ctor
.
1(
,)
nn
nn
yy
hfx
y+=
+,
1n
nx
xh
+=
+,
whe
re h
is a
con
stan
t.
Further mathematics HL guide40
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.6
Rol
le’s
theo
rem
.
Mea
n va
lue
theo
rem
.
In
t, TO
K: I
nflu
ence
of B
ourb
aki o
n un
ders
tand
ing
and
teac
hing
of m
athe
mat
ics.
Int:
Com
pare
with
wor
k of
the
Ker
ala
scho
ol.
Tayl
or p
olyn
omia
ls; t
he L
agra
nge
form
of t
he
erro
r ter
m.
App
licat
ions
to th
e ap
prox
imat
ion
of fu
nctio
ns;
form
ula
for t
he e
rror
term
, in
term
s of t
he v
alue
of
the
(n +
1)th
der
ivat
ive
at a
n in
term
edia
te
poin
t.
Mac
laur
in se
ries f
or e
x, s
inx
, cos
x,
ln(1
)x+
, (1
)px
+,
p∈
.
Use
of s
ubst
itutio
n, p
rodu
cts,
inte
grat
ion
and
diff
eren
tiatio
n to
obt
ain
othe
r ser
ies.
Tayl
or se
ries d
evel
oped
from
diff
eren
tial
equa
tions
.
Stud
ents
shou
ld b
e aw
are
of th
e in
terv
als o
f co
nver
genc
e.
5.7
The
eval
uatio
n of
lim
its o
f the
form
()
()
lim xa
fx
gx
→ a
nd
()
()
lim x
fx
gx
→∞
.
The
inde
term
inat
e fo
rms
0 0 a
nd ∞ ∞
.
Usi
ng l’
Hôp
ital’s
rule
or t
he T
aylo
r ser
ies.
Rep
eate
d us
e of
l’H
ôpita
l’s ru
le.
Further mathematics HL guide 41
Syllabus content
Top
ic 6
—D
iscr
ete
mat
hem
atic
s 48
hou
rs
The
aim
of t
his t
opic
is to
pro
vide
the
oppo
rtuni
ty fo
r stu
dent
s to
enga
ge in
logi
cal r
easo
ning
, alg
orith
mic
thin
king
and
app
licat
ions
.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
6.1
Stro
ng in
duct
ion.
Pige
on-h
ole
prin
cipl
e.
For e
xam
ple,
pro
ofs o
f the
fund
amen
tal
theo
rem
of a
rithm
etic
and
the
fact
that
a tr
ee
with
n v
ertic
es h
as n
– 1
edg
es.
TOK
: Mat
hem
atic
s and
kno
wle
dge
clai
ms.
The
diff
eren
ce b
etw
een
proo
f and
con
ject
ure,
eg
Gol
dbac
h’s c
onje
ctur
e. C
an a
mat
hem
atic
al
stat
emen
t be
true
befo
re it
is p
rove
n?
TOK
: Pro
of b
y co
ntra
dict
ion.
6.2
|a
bb
na⇒
= fo
r som
e n∈
.
The
theo
rem
|
ab
and
|
|()
ac
abx
cy⇒
±
whe
re
,xy∈
. Th
e di
visi
on a
lgor
ithm
abq
r=
+, 0
rb
≤<
.
Div
isio
n an
d Eu
clid
ean
algo
rithm
s.
The
grea
test
com
mon
div
isor
, gcd
(,
)a
b, a
nd
the
leas
t com
mon
mul
tiple
, lcm
(,
)a
b, o
f in
tege
rs a
and
b.
Prim
e nu
mbe
rs; r
elat
ivel
y pr
ime
num
bers
and
th
e fu
ndam
enta
l the
orem
of a
rithm
etic
.
The
Eucl
idea
n al
gorit
hm fo
r det
erm
inin
g th
e gr
eate
st c
omm
on d
ivis
or o
f tw
o in
tege
rs.
Int:
Euc
lidea
n al
gorit
hm c
onta
ined
in E
uclid
’s
Elem
ents
, writ
ten
in A
lexa
ndria
abo
ut
300
BC
E.
Aim
8: U
se o
f prim
e nu
mbe
rs in
cry
ptog
raph
y.
The
poss
ible
impa
ct o
f the
dis
cove
ry o
f po
wer
ful f
acto
rizat
ion
tech
niqu
es o
n in
tern
et
and
bank
secu
rity.
6.3
Line
ar D
ioph
antin
e eq
uatio
ns a
xby
c+
=.
Gen
eral
solu
tions
requ
ired
and
solu
tions
su
bjec
t to
cons
train
ts. F
or e
xam
ple,
all
solu
tions
mus
t be
posi
tive.
Int:
Des
crib
ed in
Dio
phan
tus’
Ari
thm
etic
a w
ritte
n in
Ale
xand
ria in
the
3rd c
entu
ry C
E.
Whe
n st
udyi
ng A
rith
met
ica,
a F
renc
h m
athe
mat
icia
n, P
ierr
e de
Fer
mat
(160
1–16
65)
wro
te in
the
mar
gin
that
he
had
disc
over
ed a
si
mpl
e pr
oof r
egar
ding
hig
her-o
rder
D
ioph
antin
e eq
uatio
ns—
Ferm
at’s
last
theo
rem
.
Further mathematics HL guide42
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
6.4
Mod
ular
arit
hmet
ic.
The
solu
tion
of li
near
con
grue
nces
.
Solu
tion
of si
mul
tane
ous l
inea
r con
grue
nces
(C
hine
se re
mai
nder
theo
rem
).
Int:
Dis
cuss
ed b
y C
hine
se m
athe
mat
icia
n Su
n Tz
u in
the
3rd c
entu
ry C
E.
6.5
Rep
rese
ntat
ion
of in
tege
rs in
diff
eren
t bas
es.
On
exam
inat
ion
pape
rs, q
uest
ions
that
go
beyo
nd b
ase
16 w
ill n
ot b
e se
t. In
t: Ba
bylo
nian
s dev
elop
ed a
base
60
num
ber
syste
m an
d th
e May
ans a
bas
e 20
num
ber s
yste
m.
6.6
Ferm
at’s
littl
e th
eore
m.
(mod
)p a
ap
=, w
here
p is
prim
e.
TOK
: Nat
ure
of m
athe
mat
ics.
An
inte
rest
may
be
pur
sued
for c
entu
ries b
efor
e be
com
ing
“use
ful”
.
6.7
Gra
phs,
verti
ces,
edge
s, fa
ces.
Adj
acen
t ve
rtice
s, ad
jace
nt e
dges
.
Deg
ree
of a
ver
tex,
deg
ree
sequ
ence
.
Han
dsha
king
lem
ma.
Two
verti
ces a
re a
djac
ent i
f the
y ar
e jo
ined
by
an e
dge.
Tw
o ed
ges a
re a
djac
ent i
f the
y ha
ve a
co
mm
on v
erte
x.
Aim
8: S
ymbo
lic m
aps,
eg M
etro
and
U
nder
grou
nd m
aps,
stru
ctur
al fo
rmul
ae in
ch
emis
try, e
lect
rical
circ
uits
.
TOK
: Mat
hem
atic
s and
kno
wle
dge
clai
ms.
Proo
f of t
he fo
ur-c
olou
r the
orem
. If a
theo
rem
is
pro
ved
by c
ompu
ter,
how
can
we
clai
m to
kn
ow th
at it
is tr
ue?
Sim
ple
grap
hs; c
onne
cted
gra
phs;
com
plet
e gr
aphs
; bip
artit
e gr
aphs
; pla
nar g
raph
s; tr
ees;
w
eigh
ted
grap
hs, i
nclu
ding
tabu
lar
repr
esen
tatio
n.
Subg
raph
s; c
ompl
emen
ts o
f gra
phs.
It sh
ould
be
stre
ssed
that
a g
raph
shou
ld n
ot b
e as
sum
ed to
be
sim
ple
unle
ss sp
ecifi
cally
stat
ed.
The
term
adj
acen
cy ta
ble
may
be
used
.
Aim
8: I
mpo
rtanc
e of
pla
nar g
raph
s in
cons
truct
ing
circ
uit b
oard
s.
Eule
r’s r
elat
ion:
2
ve
f−
+=
; the
orem
s for
pl
anar
gra
phs i
nclu
ding
3
6e
v≤
−,
24
ev
≤−
, le
adin
g to
the
resu
lts th
at
5κ a
nd
3,3
κ a
re n
ot
plan
ar.
If th
e gr
aph
is si
mpl
e an
d pl
anar
and
3
v≥
, th
en
36
ev
≤−
.
If th
e gr
aph
is si
mpl
e, p
lana
r, ha
s no
cycl
es o
f le
ngth
3 a
nd
3v≥
, the
n 2
4e
v≤
−.
TOK
: Mat
hem
atic
s and
kno
wle
dge
clai
ms.
App
licat
ions
of t
he E
uler
cha
ract
eris
tic
()
ve
f−
+ to
hig
her d
imen
sion
s. Its
use
in
unde
rsta
ndin
g pr
oper
ties o
f sha
pes t
hat c
anno
t be
vis
ualiz
ed.
Further mathematics HL guide 43
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
6.8
Wal
ks, t
rails
, pat
hs, c
ircui
ts, c
ycle
s.
Eule
rian
trails
and
circ
uits
. A
con
nect
ed g
raph
con
tain
s an
Eule
rian
circ
uit
if an
d on
ly if
eve
ry v
erte
x of
the
grap
h is
of
even
deg
ree.
Int:
The
“Br
idge
s of K
önig
sber
g” p
robl
em.
Ham
ilton
ian
path
s and
cyc
les.
Sim
ple
treat
men
t onl
y.
6.9
Gra
ph a
lgor
ithm
s: K
rusk
al’s
; Dijk
stra
’s.
6.10
C
hine
se p
ostm
an p
robl
em.
Not
req
uire
d:
Gra
phs w
ith m
ore
than
four
ver
tices
of o
dd
degr
ee.
To d
eter
min
e th
e sh
orte
st ro
ute
arou
nd a
w
eigh
ted
grap
h go
ing
alon
g ea
ch e
dge
at le
ast
once
.
Int:
Pro
blem
pos
ed b
y th
e C
hine
se
mat
hem
atic
ian
Kw
an M
ei-K
o in
196
2.
Trav
ellin
g sa
lesm
an p
robl
em.
Nea
rest
-nei
ghbo
ur a
lgor
ithm
for d
eter
min
ing
an u
pper
bou
nd.
Del
eted
ver
tex
algo
rithm
for d
eter
min
ing
a lo
wer
bou
nd.
To d
eter
min
e th
e H
amilt
onia
n cy
cle
of le
ast
wei
ght i
n a
wei
ghte
d co
mpl
ete
grap
h.
TOK
: Mat
hem
atic
s and
kno
wle
dge
clai
ms.
How
long
wou
ld it
take
a c
ompu
ter t
o te
st a
ll H
amilt
onia
n cy
cles
in a
com
plet
e, w
eigh
ted
grap
h w
ith ju
st 3
0 ve
rtice
s?
6.11
R
ecur
renc
e re
latio
ns. I
nitia
l con
ditio
ns,
recu
rsiv
e de
finiti
on o
f a se
quen
ce.
TO
K: M
athe
mat
ics a
nd th
e w
orld
. The
co
nnec
tions
of s
eque
nces
such
as t
he F
ibon
acci
se
quen
ce w
ith a
rt an
d bi
olog
y.
Solu
tion
of fi
rst-
and
seco
nd-d
egre
e lin
ear
hom
ogen
eous
recu
rren
ce re
latio
ns w
ith
cons
tant
coe
ffic
ient
s.
The
first
-deg
ree
linea
r rec
urre
nce
rela
tion
1n
nu
aub
−=
+.
Incl
udes
the
case
s whe
re a
uxili
ary
equa
tion
has
equa
l roo
ts o
r com
plex
root
s.
Mod
ellin
g w
ith re
curr
ence
rela
tions
. So
lvin
g pr
oble
ms s
uch
as c
ompo
und
inte
rest
, de
bt re
paym
ent a
nd c
ount
ing
prob
lem
s.
44 Further mathematics HL guide
Syllabus
Glossary of terminology: Discrete mathematics
IntroductionTeachers and students should be aware that many different terminologies exist in graph theory, and that different textbooks may employ different combinations of these. Examples of these are: vertex/node/junction/point; edge/route/arc; degree/order of a vertex; multiple edges/parallel edges; loop/self-loop.
In IB examination questions, the terminology used will be as it appears in the syllabus. For clarity, these terms are defined below.
Terminology
Bipartite graph A graph whose vertices can be divided into two sets such that no two vertices in the same set are adjacent.
Circuit A walk that begins and ends at the same vertex, and has no repeated edges.
Complement of a graph G
A graph with the same vertices as G but which has an edge between any two vertices if and only if G does not.
Complete bipartite graph
A bipartite graph in which every vertex in one set is joined to every vertex in the other set.
Complete graph A simple graph in which each pair of vertices is joined by an edge.
Connected graph A graph in which each pair of vertices is joined by a path.
Cycle A walk that begins and ends at the same vertex, and has no other repeated vertices.
Degree of a vertex The number of edges joined to the vertex; a loop contributes two edges, one for each of its end points.
Disconnected graph A graph that has at least one pair of vertices not joined by a path.
Eulerian circuit A circuit that contains every edge of a graph.
Eulerian trail A trail that contains every edge of a graph.
Graph Consists of a set of vertices and a set of edges.
Graph isomorphism between two simple graphs G and H
A one-to-one correspondence between vertices of G and H such that a pair of vertices in G is adjacent if and only if the corresponding pair in H is adjacent.
Hamiltonian cycle A cycle that contains all the vertices of the graph.
Hamiltonian path A path that contains all the vertices of the graph.
Further mathematics HL guide 45
Glossary of terminology: Discrete mathematics
Loop An edge joining a vertex to itself.
Minimum spanning tree
A spanning tree of a weighted graph that has the minimum total weight.
Multiple edges Occur if more than one edge joins the same pair of vertices.
Path A walk with no repeated vertices.
Planar graph A graph that can be drawn in the plane without any edge crossing another.
Simple graph A graph without loops or multiple edges.
Spanning tree of a graph
A subgraph that is a tree, containing every vertex of the graph.
Subgraph A graph within a graph.
Trail A walk in which no edge appears more than once.
Tree A connected graph that contains no cycles.
Walk A sequence of linked edges.
Weighted graph A graph in which each edge is allocated a number or weight.
Weighted tree A tree in which each edge is allocated a number or weight.
46 Further mathematics HL guide
Assessment in the Diploma Programme
Assessment
GeneralAssessment is an integral part of teaching and learning. The most important aims of assessment in the Diploma Programme are that it should support curricular goals and encourage appropriate student learning. Both external and internal assessment are used in the Diploma Programme. IB examiners mark work produced for external assessment, while work produced for internal assessment is marked by teachers and externally moderated by the IB.
There are two types of assessment identified by the IB.
• Formative assessment informs both teaching and learning. It is concerned with providing accurate and helpful feedback to students and teachers on the kind of learning taking place and the nature of students’ strengths and weaknesses in order to help develop students’ understanding and capabilities. Formative assessment can also help to improve teaching quality, as it can provide information to monitor progress towards meeting the course aims and objectives.
• Summative assessment gives an overview of previous learning and is concerned with measuring student achievement.
The Diploma Programme primarily focuses on summative assessment designed to record student achievement at or towards the end of the course of study. However, many of the assessment instruments can also be used formatively during the course of teaching and learning, and teachers are encouraged to do this. A comprehensive assessment plan is viewed as being integral with teaching, learning and course organization. For further information, see the IB Programme standards and practices document.
The approach to assessment used by the IB is criterion-related, not norm-referenced. This approach to assessment judges students’ work by their performance in relation to identified levels of attainment, and not in relation to the work of other students. For further information on assessment within the Diploma Programme, please refer to the publication Diploma Programme assessment: Principles and practice.
To support teachers in the planning, delivery and assessment of the Diploma Programme courses, a variety of resources can be found on the OCC or purchased from the IB store (http://store.ibo.org). Teacher support materials, subject reports, internal assessment guidance, grade descriptors, as well as resources from other teachers, can be found on the OCC. Specimen and past examination papers as well as markschemes can be purchased from the IB store.
Further mathematics HL guide 47
Assessment in the Diploma Programme
Methods of assessmentThe IB uses several methods to assess work produced by students.
Assessment criteriaAssessment criteria are used when the assessment task is open-ended. Each criterion concentrates on a particular skill that students are expected to demonstrate. An assessment objective describes what students should be able to do, and assessment criteria describe how well they should be able to do it. Using assessment criteria allows discrimination between different answers and encourages a variety of responses. Each criterion comprises a set of hierarchically ordered level descriptors. Each level descriptor is worth one or more marks. Each criterion is applied independently using a best-fit model. The maximum marks for each criterion may differ according to the criterion’s importance. The marks awarded for each criterion are added together to give the total mark for the piece of work.
MarkbandsMarkbands are a comprehensive statement of expected performance against which responses are judged. They represent a single holistic criterion divided into level descriptors. Each level descriptor corresponds to a range of marks to differentiate student performance. A best-fit approach is used to ascertain which particular mark to use from the possible range for each level descriptor.
MarkschemesThis generic term is used to describe analytic markschemes that are prepared for specific examination papers. Analytic markschemes are prepared for those examination questions that expect a particular kind of response and/or a given final answer from the students. They give detailed instructions to examiners on how to break down the total mark for each question for different parts of the response. A markscheme may include the content expected in the responses to questions or may be a series of marking notes giving guidance on how to apply criteria.
48 Further mathematics HL guide
Assessment
Assessment outline
First examinations 2014
Assessment component Weighting
External assessment (5 hours)
Paper 1 (2 hours 30 minutes)Graphic display calculator required.
Compulsory short- to medium-response questions based on the whole syllabus.
50%
Paper 2 (2 hours 30 minutes)Graphic display calculator required.
Compulsory medium- to extended-response questions based on the whole syllabus.
50%
Further mathematics HL guide 49
Assessment
Assessment details
External assessment
Papers 1 and 2These papers are externally set and externally marked. The papers are designed to allow students to demonstrate what they know and what they can do.
Markschemes are used to assess students in both papers. The markschemes are specific to each examination.
CalculatorsPapers 1 and 2Students must have access to a GDC at all times. However, not all questions will necessarily require the use of the GDC. Regulations covering the types of GDC allowed are provided in the Handbook of procedures for the Diploma Programme.
Mathematics HL and further mathematics HL formula bookletEach student must have access to a clean copy of the formula booklet during the examination. It is the responsibility of the school to download a copy from IBIS or the OCC and to ensure that there are sufficient copies available for all students.
Awarding of marksMarks may be awarded for method, accuracy, answers and reasoning, including interpretation.
In paper 1 and paper 2, full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations (in the form of, for example, diagrams, graphs or calculations). Where an answer is incorrect, some marks may be given for correct method, provided this is shown by written working. All students should therefore be advised to show their working.
Paper 1Duration: 2 hours 30 minutesWeighting: 50%• This paper consists of short- to medium-response questions. A GDC is required for this paper, but not
every question will necessarily require its use.
Syllabus coverage• Knowledge of all topics in the syllabus is required for this paper. However, not all topics are necessarily
assessed in every examination session.
Further mathematics HL guide50
Assessment details
Mark allocation• This paper is worth 150 marks, representing 50% of the final mark.
• Questions of varying levels of difficulty and length are set. Therefore, individual questions may not necessarily be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of the question.
• The intention of this paper is to test students’ knowledge across the breadth of the syllabus.
Question type• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.
Paper 2Duration: 2 hours 30 minutesWeighting: 50%• This paper consists of medium- to extended-response questions. A GDC is required for this paper, but
not every question will necessarily require its use.
Syllabus coverage• Knowledge of all topics in the core of the syllabus is required for this paper. However, not all topics are
necessarily assessed in every examination session.
Mark allocation• This paper is worth 150 marks, representing 50% of the final mark.
• Questions of varying levels of difficulty and length are set. Therefore, individual questions may not necessarily each be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of the question.
• The intention of this paper is to test students’ knowledge and understanding across the breadth of the syllabus.
Question type• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.
Internal assessmentThere is no internal assessment component in this course.
Further mathematics HL guide 51
Appendices
Glossary of command terms
Command terms with definitionsStudents should be familiar with the following key terms and phrases used in examination questions, which are to be understood as described below. Although these terms will be used in examination questions, other terms may be used to direct students to present an argument in a specific way.
Calculate Obtain a numerical answer showing the relevant stages in the working.
Comment Give a judgment based on a given statement or result of a calculation.
Compare Give an account of the similarities between two (or more) items or situations, referring to both (all) of them throughout.
Compare and contrast
Give an account of the similarities and differences between two (or more) items or situations, referring to both (all) of them throughout.
Construct Display information in a diagrammatic or logical form.
Contrast Give an account of the differences between two (or more) items or situations, referring to both (all) of them throughout.
Deduce Reach a conclusion from the information given.
Demonstrate Make clear by reasoning or evidence, illustrating with examples or practical application.
Describe Give a detailed account.
Determine Obtain the only possible answer.
Differentiate Obtain the derivative of a function.
Distinguish Make clear the differences between two or more concepts or items.
Draw Represent by means of a labelled, accurate diagram or graph, using a pencil. A ruler (straight edge) should be used for straight lines. Diagrams should be drawn to scale. Graphs should have points correctly plotted (if appropriate) and joined in a straight line or smooth curve.
Estimate Obtain an approximate value.
Explain Give a detailed account, including reasons or causes.
Find Obtain an answer, showing relevant stages in the working.
Hence Use the preceding work to obtain the required result.
Hence or otherwise It is suggested that the preceding work is used, but other methods could also receive credit.
Further mathematics HL guide52
Glossary of command terms
Identify Provide an answer from a number of possibilities.
Integrate Obtain the integral of a function.
Interpret Use knowledge and understanding to recognize trends and draw conclusions from given information.
Investigate Observe, study, or make a detailed and systematic examination, in order to establish facts and reach new conclusions.
Justify Give valid reasons or evidence to support an answer or conclusion.
Label Add labels to a diagram.
List Give a sequence of brief answers with no explanation.
Plot Mark the position of points on a diagram.
Predict Give an expected result.
Prove Use a sequence of logical steps to obtain the required result in a formal way.
Show Give the steps in a calculation or derivation.
Show that Obtain the required result (possibly using information given) without the formality of proof. “Show that” questions do not generally require the use of a calculator.
Sketch Represent by means of a diagram or graph (labelled as appropriate). The sketch should give a general idea of the required shape or relationship, and should include relevant features.
Solve Obtain the answer(s) using algebraic and/or numerical and/or graphical methods.
State Give a specific name, value or other brief answer without explanation or calculation.
Suggest Propose a solution, hypothesis or other possible answer.
Verify Provide evidence that validates the result.
Write down Obtain the answer(s), usually by extracting information. Little or no calculation is required. Working does not need to be shown.
Further mathematics HL guide 53
Appendices
Notation list
Notation list
Further mathematics HL guide 1
Of the various notations in use, the IB has chosen to adopt a system of notation based on the recommendations of the International Organization for Standardization (ISO). This notation is used in the examination papers for this course without explanation. If forms of notation other than those listed in this guide are used on a particular examination paper, they are defined within the question in which they appear.
Because students are required to recognize, though not necessarily use, IB notation in examinations, it is recommended that teachers introduce students to this notation at the earliest opportunity. Students are not allowed access to information about this notation in the examinations.
Students must always use correct mathematical notation, not calculator notation.
the set of positive integers and zero, {0,1, 2, 3, ...}
the set of integers, {0, 1, 2, 3, ...}
the set of positive integers, {1, 2, 3, ...}
the set of rational numbers
the set of positive rational numbers, { | , 0}x x x
the set of real numbers
the set of positive real numbers, { | , 0}x x x
the set of complex numbers, { i | , }a b a b
i 1
z a complex number
z the complex conjugate of z
z the modulus of z
arg z the argument of z
Re z the real part of z
Im z the imaginary part of z
cis ic s ino s
1 2{ , , ...}x x the set with elements 1 2, , ...x x
( )n A the number of elements in the finite set A
{ | }x the set of all x such that
is an element of
is not an element of
the empty (null) set
U the universal set
union
Further mathematics HL guide54
Notation list
Notation list
Further mathematics HL guide 2
intersection
is a proper subset of
is a subset of
A the complement of the set A
A B the Cartesian product of sets A and B (that is,
{( , ) , }A B a b a A b B )
|a b a divides b
1/ na , n a a to the power of 1n
, thn root of a (if 0a then 0n a )
1/ 2a , a a to the power 12
, square root of a (if 0a then 0a )
x the modulus or absolute value of x, that is for 0,
for 0, x x xx x x
identity
is approximately equal to
is greater than
is greater than or equal to
is less than
is less than or equal to
is not greater than
is not less than
,a b the closed interval a x b
,a b the open interval a x b
nu the thn term of a sequence or series
d the common difference of an arithmetic sequence
r the common ratio of a geometric sequence
nS the sum of the first n terms of a sequence, 1 2 ... nu u u
S the sum to infinity of a sequence, 1 2 ...u u
1
n
ii
u 1 2 ... nu u u
1
n
ii
u 1 2 ... nu u u
nr
!
!( )!n
r n r
Further mathematics HL guide 55
Notation listNotation list
Further mathematics HL guide 3
!n ( 1)( 2) 3 2 1n n n
:f A B f is a function under which each element of set A has an image in set B
:f x y f is a function under which x is mapped to y
( )f x the image of x under the function f
1f the inverse function of the function f
f g the composite function of f and g
lim ( )x a
f x
the limit of ( )f x as x tends to a
dd
yx
the derivative of y with respect to x
( )f x the derivative of ( )f x with respect to x
2
2
dd
yx
the second derivative of y with respect to x
( )f x the second derivative of ( )f x with respect to x
dd
n
n
yx
the thn derivative of y with respect to x
( )nf x the thn derivative of ( )f x with respect to x
dy x the indefinite integral of y with respect to x
db
ay x
the definite integral of y with respect to x between the limits x a and x b
ex the exponential function of x
loga x the logarithm to the base a of x
ln x the natural logarithm of x, elog x
sin, cos, tan the circular functions
arcsin, arccos,arctan
the inverse circular functions
csc, sec, cot the reciprocal circular functions
A( , )x y the point A in the plane with Cartesian coordinates x and y
AB the line segment with end points A and B
AB the length of AB
AB the line containing points A and B
 the angle at A
Further mathematics HL guide56
Notation listNotation list
Further mathematics HL guide 4
ˆCAB the angle between CA and AB
ABC the triangle whose vertices are A, B and C
v the vector v
AB
the vector represented in magnitude and direction by the directed line segment from A to B
a the position vector OA
i, j, k unit vectors in the directions of the Cartesian coordinate axes
a the magnitude of a
|AB|
the magnitude of AB
v w the scalar product of v and w
v w the vector product of v and w
1A the inverse of the non-singular matrix A
TA the transpose of the matrix A
det A the determinant of the square matrix A
I the identity matrix
P( )A the probability of event A
P( )A the probability of the event “not A ”
P( | )A B the probability of the event A given B
1 2, , ...x x observations
1 2, , ...f f frequencies with which the observations 1 2, , ...x x occur
Px the probability distribution function P( = )X x of the discrete random variable X
( )f x the probability density function of the continuous random variable X
( )F x the cumulative distribution function of the continuous random variable X
E( )X the expected value of the random variable X
Var ( )X the variance of the random variable X
population mean
2 population variance,
2
2 1( )
k
i ii
f x
n
, where 1
k
ii
n f
population standard deviation
x sample mean
Further mathematics HL guide 57
Notation listNotation list
Further mathematics HL guide 5
2ns sample variance,
2
2 1( )
k
i ii
n
f x xs
n
, where 1
k
ii
n f
ns standard deviation of the sample
21ns
unbiased estimate of the population variance,
2
2 2 11
( )
1 1
k
i ii
n n
f x xns s
n n
, where
1
k
ii
n f
B , n p binomial distribution with parameters n and p
Po m Poisson distribution with mean m
2N , normal distribution with mean and variance 2
~ B ,X n p the random variable X has a binomial distribution with parameters n and p
~ PoX m the random variable X has a Poisson distribution with mean m
2~ N ,X the random variable X has a normal distribution with mean and
variance 2
cumulative distribution function of the standardized normal variable with distribution N 0,1
number of degrees of freedom
\A B the difference of the sets A and B (that is,
\ { and }A B A B x x A x B )
A B the symmetric difference of the sets A and B (that is,
( \ ) ( \ )A B A B B A )
n a complete graph with n vertices
,n m a complete bipartite graph with one set of n vertices and another set of m vertices
p the set of equivalence classes {0,1, 2, , 1}p of integers modulo p
gcd( , )a b the greatest common divisor of integers a and b
lcm( , )a b the least common multiple of integers a and b
GA the adjacency matrix of graph G
GC the cost adjacency matrix of graph G
Recommended