IBHM Prelim .pdf



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About the Authors
Bill Roberts is an experienced educator, having taught mathematics in a number of
international schools for the past twenty years. He also works for the International
Baccalaureate as an examiner and trainer. Bill is currently based at the University of
Newcastle in England.
Sandy MacKenzie is an experienced teacher of mathematics in both the International
Baccalaureate and the Scottish systems and works for the International Baccalaureate as
an examiner. Sandy is currently Assistant Rector at Morrison’s Academy in Scotland.

i

Dedications
To my father, George Roberts, who encouraged and nurtured my love and appreciation
for mathematics.
To Nancy MacKenzie, and the Cairns sisters, Alexia and Rosemary, my three mothers,
without whom none of this would have been possible.

ii

Preface
This book is intended primarily for use by students and teachers of Higher Level
Mathematics in the International Baccalaureate Diploma Programme, but will also be of
use to students on other courses.
Detailed coverage is provided for the core part of this syllabus and provides excellent
preparation for the final examination. The book provides guidance on areas of the
syllabus that may be examined differently depending on whether a graphical calculator
is used or not.
Points of theory are presented and explained concisely and are illustrated by worked
examples which identify the key skills and techniques. Where appropriate, information
and methods are highlighted and margin notes provide further tips and important
reminders. These are supported by varied and graded exercises, which consolidate the
theory, thus enabling the reader to practise basic skills and challenging exam-style
questions. Each chapter concludes with a review exercise that covers all of the skills
within the chapter, with a clear distinction between questions where a calculator is
M

allowed and ones where it is not. The icon

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may be used, and
indicates where it may not. Many of these questions are from
past IB papers and we would like to thank the International Baccalaureate for permission
to reproduce these questions.
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Throughout the text, we have aimed to produce chapters that have been sequenced in
a logical teaching order with major topics grouped together. However, we have also
built in flexibility and in some cases the order in which chapters are used can be
changed. For example we have split differential and integral calculus, but these can be
taught as one section. Although many students will use this book in a teacher-led
environment, it has also been designed to be accessible to students for self-study.
The book is accompanied by a CD. As well as containing an electronic version of the
entire book, there is a presumed knowledge chapter covering basic skills, revision
exercises of the whole syllabus grouped into six sections, and twenty extended
exam-style questions.
We would like extend our thanks to family, friends, colleagues and students who
supported and encouraged us throughout the process of writing this book and
especially to those closest to each of us, for their patience and understanding.
Bill Roberts
Sandy MacKenzie
2007

iii

Contents
1

Trigonometry 1

1.1
1.2
1.3
1.4
1.5
1.6
1.7

Circle problems
Trigonometric ratios
Solving triangles
Trigonometric functions and graphs
Related angles
Trigonometric equations
Inverse trigonometric functions

2

Quadratic Equations, Functions and Inequalities

2.1
2.2
2.3
2.4
2.5

Introduction to quadratic functions
Solving quadratic equations
Quadratic functions
Linear and quadratic inequalities
Nature of roots of quadratic equations

3

Functions

3.1
3.2
3.3
3.4
3.5
3.6
3.7

Functions
Composite functions
Inverse functions
Graphs of inverse functions
Special functions
Drawing a graph
Transformations of functions

4

Polynomials

4.1
4.2
4.3
4.4
4.5
4.6
4.7

Polynomial functions
Factor and remainder theorems
Finding a polynomial’s coefficients
Solving polynomial equations
Finding a function from its graph
Algebraic long division
Using a calculator with polynomials

5

Exponential and Logarithmic Functions

5.1
5.2
5.3
5.4
5.5
5.6

Exponential functions
Logarithmic graphs
Rules of logarithms
Logarithms on a calculator
Exponential equations
Related graphs

6

Sequences, Series and Binomial Theorem

6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8

Arithmetic sequences
Sum of the first n terms of an arithmetic sequence
Geometric sequences and series
Sum of an infinite series
Applications of sequences and series
Sigma notation
Factorial notation
Binomial theorem

iv

1–34
1
7
9
17
24
27
32
35–57
35
39
41
47
51
58–85
58
62
64
67
69
73
76
86–107
87
91
93
95
98
100
103
108–129
109
112
114
117
120
125
130–158
131
133
136
138
142
143
146
152

Contents

7

Trigonometry 2

7.1
7.2
7.3
7.4
7.5

Identities
Compound angle (addition) formulae
Double angle formulae
Using double angle formulae
Wave function

8

Differential Calculus 1– Introduction

8.1
8.2
8.3
8.4
8.5
8.6
8.7

Differentiation by first principles
Differentiation using a rule
Gradient of a tangent
Stationary points
Points of inflexion
Curve sketching
Sketching the graph of the derived function

9

Differentiation 2 – Further Techniques

9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9

Differentiating trigonometric functions
Differentiating functions of functions (chain rule)
Differentiating exponential and logarithmic functions
Product rule
Quotient rule
Implicit differentiation
Differentiating inverse trigonometric functions
Summary of standard results
Further differentiation problems

10

Differentiation 3 – Applications

10.1
10.2
10.3

Optimization problems
Rates of change of connected variables
Displacement, velocity and acceleration

11

Matrices

11.1
11.2
11.3
11.4

Introduction to matrices
Determinants and inverses of matrices
Solving simultaneous equations in two unknowns
Solving simultaneous equations in three unknowns

12

Vector Techniques

12.1
12.2
12.3

Introduction to vectors
A geometric approach to vectors
Multiplication of vectors

13

Vectors, Lines and Planes

13.1
13.2
13.3
13.4

Equation of a straight line
Parallel, intersecting and skew lines
Equation of a plane
Intersecting lines and planes

14

Integration 1

14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14.9

Undoing differentiation
Constant of integration
Initial conditions
Basic results
Anti-chain rule
Definite integration
Geometric significance of integration
Areas above and below the x-axis
Area between two curves

159–182
159
163
169
173
176
183–216
184
187
190
193
201
204
209
217–245
218
221
224
229
231
234
238
242
243
246–267
246
253
259
268–304
269
278
287
291
305–336
306
315
321
337–372
338
346
353
364
373–402
373
374
377
379
380
382
385
391
395

v

Contents

15 Integration 2 – Further Techniques
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
15.10

Integration as a process of anti-differentiation – direct reverse
Integration of functions to give inverse trigonometric functions
Integration of powers of trigonometric functions
Selecting the correct technique 1
Integration by substitution
Integration by parts
Miscellaneous techniques
Further integration practice
Selecting the correct technique 2
Finding the area under a curve

16 Integration 3 – Applications
16.1
16.2
16.3
16.4
16.5
16.6

Differential equations
Solving differential equations by direct integration
Solving differential equations by separating variables
Verifying that a particular solution fits a differential equation
Displacement, velocity and acceleration
Volumes of solids of revolution

17 Complex Numbers
17.1
17.2
17.3
17.4

Imaginary numbers
Complex numbers
Argand diagrams
de Moivre’s theorem

18

Mathematical Induction

18.1
18.2
18.3

Introduction to mathematical induction
Proving some well-known results
Forming and proving conjectures

19

Statistics

19.1
19.2
19.3
19.4

Frequency tables
Frequency diagrams
Measures of dispersion
Using a calculator to perform statistical calculations

20

Probability

20.1
20.2
20.3
20.4
20.5
20.6

Introduction to probability
Conditional probability
Independent events
Bayes’ theorem
Permutations and combinations
Probability involving permutations and combinations

21

Discrete Probability Distributions

21.1
21.2
21.3
21.4

Introduction to discrete random variables
Expectation and variance
Binomial distribution
Poisson distribution

22 Continuous Probability Distributions
22.1
22.2
22.3
22.4
22.5

Continuous random variables
Using continuous probability density functions
Normal distributions
Problems involving finding μ and s
Applications of normal distributions

Answers
Index
vi

403–445
405
411
414
421
422
428
433
436
438
441
446–472
446
448
452
457
459
463
473–508
474
477
484
497
509–527
510
516
522
528–560
529
537
548
555
561–594
561
567
569
575
582
589
595–632
596
600
609
619
633–672
634
639
652
660
662
673
721



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