Cambridge Mathematics 2 Unit Advanced - Year 11 - Enhanced 2nd Edition

430 Pages·2016·4.63 MB·English

by  Bill Pender

YEAR
11
CAMBRIDGE
Mathematics
2 Unit
Second Edition
Enhanced
BILL PENDER
DAVID SADLER
JULIA SHEA
DEREK WARD
ISBN: 9781107679573 © Bill Pender, David Sadler, Julia Shea, Derek Ward 2012 Cambridge University Press
Photocopying is restricted under law and this material must not be transferred to another party
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ISBN: 9781107679573 © Bill Pender, David Sadler, Julia Shea, Derek Ward 2012 Cambridge University Press
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
HowtoUseThisBook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
AbouttheAuthors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
ChapterOne—MethodsinAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1A Arithmetic with Pronumerals . . . . . . . . . . . . . . . . . . . . . . . 1
1B Expanding Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1C Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1D Algebraic Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1E Factoring the Sum and Difference of Cubes . . . . . . . . . . . . . . . 13
1F Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1G Solving Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . 17
1H Solving Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . 20
1I Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1J Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 25
ChapterTwo—NumbersandSurds . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2A Integers and Rational Numbers . . . . . . . . . . . . . . . . . . . . . . 27
2B Terminating and Recurring Decimals . . . . . . . . . . . . . . . . . . . 30
2C Real Numbers and Approximations . . . . . . . . . . . . . . . . . . . . 33
2D Surds and their Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . 36
2E Further Simplification of Surds . . . . . . . . . . . . . . . . . . . . . . 38
2F Rationalising the Denominator . . . . . . . . . . . . . . . . . . . . . . 40
2G Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 42
ChapterThree—FunctionsandtheirGraphs . . . . . . . . . . . . . . . . . . . . . 45
3A Functions and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3B Review of Linear Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3C Review of Quadratic Graphs . . . . . . . . . . . . . . . . . . . . . . . . 52
3D Higher Powers of x and Circles . . . . . . . . . . . . . . . . . . . . . . 55
3E Two Graphs that have Asymptotes . . . . . . . . . . . . . . . . . . . . 57
3F Transformations of Known Graphs . . . . . . . . . . . . . . . . . . . . 59
3G Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 63
ChapterFour—GraphsandInequations . . . . . . . . . . . . . . . . . . . . . . . . 66
4A Inequations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 66
4B Solving Quadratic Inequations . . . . . . . . . . . . . . . . . . . . . . . 69
4C Intercepts and Sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4D Odd and Even Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 74
4E The Absolute Value Function . . . . . . . . . . . . . . . . . . . . . . . 77
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iv Contents
4F Using Graphs to Solve Equations and Inequations . . . . . . . . . . . . 81
4G Regions in the Number Plane . . . . . . . . . . . . . . . . . . . . . . . 85
4H Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . 89
ChapterFive—Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5A Trigonometry with Right-Angled Triangles . . . . . . . . . . . . . . . . 92
5B Problems Involving Right-Angled Triangles . . . . . . . . . . . . . . . 97
5C Trigonometric Functions of a General Angle . . . . . . . . . . . . . . .101
5D The Quadrant, the Related Angle and the Sign . . . . . . . . . . . . .105
5E Given One Trigonometric Function, Find Another . . . . . . . . . . . .111
5F Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . .113
5G Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . .117
5H The Sine Rule and the Area Formula . . . . . . . . . . . . . . . . . . .123
5I The Cosine Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129
5J Problems Involving General Triangles . . . . . . . . . . . . . . . . . . .133
5K Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . .138
Appendix — Proving the Sine, Cosine and Area Rules . . . . . . . . .141
ChapterSix—CoordinateGeometry . . . . . . . . . . . . . . . . . . . . . . . . . .143
6A Lengths and Midpoints of Intervals . . . . . . . . . . . . . . . . . . . .143
6B Gradients of Intervals and Lines . . . . . . . . . . . . . . . . . . . . . .147
6C Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153
6D Further Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . .156
6E Perpendicular Distance . . . . . . . . . . . . . . . . . . . . . . . . . . .161
6F Lines Through the Intersection of Two Given Lines . . . . . . . . . . .164
6G Coordinate Methods in Geometry . . . . . . . . . . . . . . . . . . . . .167
6H Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . .170
Appendix — The Proofs of Two Results . . . . . . . . . . . . . . . . .172
ChapterSeven—IndicesandLogarithms . . . . . . . . . . . . . . . . . . . . . . .174
7A Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174
7B Fractional Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .179
7C Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182
7D The Laws for Logarithms . . . . . . . . . . . . . . . . . . . . . . . . .185
7E Equations Involving Logarithms and Indices . . . . . . . . . . . . . . .188
7F Graphs of Exponential and Logarithmic Functions . . . . . . . . . . .190
7G Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . .193
ChapterEight—SequencesandSeries . . . . . . . . . . . . . . . . . . . . . . . .195
8A Sequences and How to Specify Them . . . . . . . . . . . . . . . . . . .195
8B Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . .199
8C Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . .203
8D Solving Problems about APs and GPs . . . . . . . . . . . . . . . . . .207
8E Adding Up the Terms of a Sequence . . . . . . . . . . . . . . . . . . .211
8F Summing an Arithmetic Series . . . . . . . . . . . . . . . . . . . . . .214
8G Summing a Geometric Series . . . . . . . . . . . . . . . . . . . . . . .218
8H The Limiting Sum of a Geometric Series . . . . . . . . . . . . . . . . .222
8I Recurring Decimals and Geometric Series . . . . . . . . . . . . . . . .227
8J Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . .228
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Contents v
ChapterNine—TheDerivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . .230
9A The Derivative — Geometric Definition . . . . . . . . . . . . . . . . .230
9B The Derivative as a Limit . . . . . . . . . . . . . . . . . . . . . . . . .233
9C A Rule for Differentiating Powers of x . . . . . . . . . . . . . . . . . .237
dy
9D Tangents and Normals — The Notation . . . . . . . . . . . . . . .241
dx
9E Differentiating Powers with Negative Indices . . . . . . . . . . . . . . .246
9F Differentiating Powers with Fractional Indices . . . . . . . . . . . . . .249
9G The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251
9H The Product Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254
9I The Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257
9J Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . .259
9K Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .263
9L Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . .265
Appendix — Proving Some Rules for Differentiation . . . . . . . . . .267
ChapterTen—TheQuadraticFunction . . . . . . . . . . . . . . . . . . . . . . . .269
10A Factoring and the Graph . . . . . . . . . . . . . . . . . . . . . . . . . .269
10B Completing the Square and the Graph . . . . . . . . . . . . . . . . . .274
10C The Quadratic Formulae and the Graph . . . . . . . . . . . . . . . . .278
10D Equations Reducible to Quadratics . . . . . . . . . . . . . . . . . . . .280
10E Problems on Maximisation and Minimisation . . . . . . . . . . . . . .281
10F The Theory of the Discriminant . . . . . . . . . . . . . . . . . . . . . .285
10G Definite and Indefinite Quadratics . . . . . . . . . . . . . . . . . . . .289
10H Sum and Product of Roots . . . . . . . . . . . . . . . . . . . . . . . . .292
10I Quadratic Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . .295
10J Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . .298
Appendix — Identically Equal Quadratics . . . . . . . . . . . . . . . .300
ChapterEleven—LocusandtheParabola . . . . . . . . . . . . . . . . . . . . . . .301
11A A Locus and its Equation . . . . . . . . . . . . . . . . . . . . . . . . .301
11B The Geometric Definition of the Parabola . . . . . . . . . . . . . . . .305
11C Translations of the Parabola . . . . . . . . . . . . . . . . . . . . . . . .309
11D Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . .312
ChapterTwelve—TheGeometryoftheDerivative . . . . . . . . . . . . . . . . . . .313
12A Increasing, Decreasing and Stationary at a Point . . . . . . . . . . . .313
12B Stationary Points and Turning Points . . . . . . . . . . . . . . . . . . .318
12C Second and Higher Derivatives . . . . . . . . . . . . . . . . . . . . . .322
12D Concavity and Points of Inflexion . . . . . . . . . . . . . . . . . . . . .324
12E A Review of Curve Sketching . . . . . . . . . . . . . . . . . . . . . . .329
12F Global Maximum and Minimum . . . . . . . . . . . . . . . . . . . . . .333
12G Applications of Maximisation and Minimisation . . . . . . . . . . . . .335
12H Primitive Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .341
12I Chapter Review Exercise . . . . . . . . . . . . . . . . . . . . . . . . . .346
AnswerstoExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .348
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .415
ISBN: 9781107679573 © Bill Pender, David Sadler, Julia Shea, Derek Ward 2012 Cambridge University Press
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Preface
This textbook has been written for students in Years 11 and 12 taking the 2 Unit
calculuscourse‘Mathematics’for theNSW HSC. Thebookcoversallthecontent
of the course at the level required for the HSC examination. There are two
volumes — the present volume is roughly intended for Year 11, and the next
volume for Year 12. Schools will, however, differ in their choices of order of
topics and in their rates of progress.
Although the Syllabus has not been rewritten for the new HSC, there has been
a gradual shift of emphasis in recent examination papers.
• The interdependence of the course content has been emphasised.
• Graphs have been used much more freely in argument.
• Structured problem solving has been expanded.
• There has been more stress on explanation and proof.
This text addresses these new emphases, and the exercises contain a wide variety
of different types of questions.
There is an abundance of questions and problems in each exercise — too many
for any one student —carefullygrouped inthree gradedsets, so that with proper
selection, the book can be used at all levels of ability in the 2 Unit course.
This new second editionhasbeenthoroughlyrewrittentomakeitmoreacces-
sibletoallstudents. Theexercisesnowhavemoreearlydrillquestionstoreinforce
each new skill, there are more worked exercises on each new algorithm, and some
chapters and sections have been split into two so that ideas can be introduced
more gradually. We have also added a review exercise to each chapter.
WewouldliketothankourcolleaguesatSydneyGrammarSchoolandNewington
College for their invaluable help in advising us and commenting on the successive
drafts. We would also like to thank the Headmasters of our two schools for their
encouragement of this project, and Peter Cribb, Chris Gray and the team at
CambridgeUniversityPress,Melbourne,fortheirsupportandhelpindiscussions.
Finally, our thanks go to our families for encouraging us, despite the distractions
that the project has caused to family life.
Preface to the enhanced version
Toprovidestudentswithpracticeforthenewobjectiveresponse(multiplechoice)
questions to be included in HSC examinations, online self-marking quizzes have
been provided for each chapter, on Cambridge GO (access details can be found
in the following pages). In addition, an interactive textbook version is available
through the same website.
Dr Bill Pender Julia Shea
Subject Master in Mathematics Director of Curriculum
Sydney Grammar School Newington College
College Street 200 Stanmore Road
Darlinghurst NSW 2010 Stanmore NSW 2048
David Sadler Derek Ward
Mathematics Mathematics
Sydney Grammar School Sydney Grammar School
ISBN: 9781107679573 © Bill Pender, David Sadler, Julia Shea, Derek Ward 2012 Cambridge University Press
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How to Use This Book
This book has been written so that it is suitable for the full range of 2 Unit
students, whatever their abilities and ambitions.
TheExercises: No-one should try to do all the questions! We have written long
exercises so that everyone will find enough questions of a suitable standard —
each student will need to select from them, and there should be plenty left for
revision. The book provides a great variety of questions, and representatives of
all types should be attempted.
Each chapter is divided into a number of sections. Each of these sections has its
own substantial exercise, subdivided into three groups of questions:
Foundation: These questions are intended to drill the new content of the sec-
tion at a reasonably straightforward level. There is little point in proceeding
without mastery of this group.
Development: This group is usually the longest. It contains more substantial
questions, questions requiring proof or explanation, problems where the new
content can be applied, and problems involving content from other sections
and chapters to put the new ideas in a wider context.
Challenge: Many questions in recent 2 Unit HSC examinations have been
very demanding, and this section is intended to match the standard of those
recent examinations. Some questions are algebraically challenging, some re-
quire more sophistication in logic, some establish more difficult connections
between topics, and some complete proofs or give an alternative approach.
TheTheoryandtheWorkedExercises: All the theory in the course has been properly
developed, but students and their teachers should feel free to choose how thor-
oughly the theory is presented in any particular class. It can often be helpful to
learn a method first and then return to the details of the proof and explanation
when the point of it all has become clear.
The main formulae, methods, definitions and results have been boxed and num-
bered consecutively through each chapter. They provide a bare summary only,
and students are advised to make their own short summary of each chapter using
the numbered boxes as a basis.
The worked examples have been chosen to illustrate the new methods introduced
in the section. They should provide sufficient preparation for the questions in the
following exercise, but they cannot possibly cover the variety of questions that
can be asked.
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viii HowtoUseThisBook
The Chapter Review Exercises: A Chapter Review Exercise has been added to each
chapter of the second edition. These exercises are intended only as a basic review
ofthechapter—forharderquestions, studentsareadvisedtoworkthroughmore
of the later questions in the exercises.
The Order of the Topics: We have presented the topics in the order that we have
found most satisfactory in our own teaching. There are, however, many effective
orderings of the topics, and apart from questions that provide links between
topics, the book allows all the flexibility needed in the many different situations
that apply in different schools.
We have left Euclidean geometry and probability until Chapter Seven of the
Year 12 volume for two reasons. First, we believe that functions and calculus
should be developed as early as possible because these are the fundamental ideas
in the course. Secondly, the courses in Years 9 and 10 already develop most of
the work in Euclidean geometry and probability, at least in an intuitive fashion,
so that revisiting them in Year 12, with a greater emphasis now on proof in
geometry, seems an ideal arrangement.
Many students, however, will want to study geometry in Year 11. The publishers
have therefore made this chapter available free on their website at
www.cambridge.edu.au/education/2unitGeometry
The two geometry chapters from the 3 Unit volume are also on the website.
The Structure of the Course: Recent examination papers have made the interconnec-
tions amongst the various topics much clearer. Calculus is the backbone of the
course, and the two processes of differentiation and integration, inverses of each
other, are the basis of most of the topics. Both processes are introduced as geo-
metrical ideas — differentiation is defined using tangents, and integration using
areas — but the subsequent discussions, applications and exercises give many
other ways of understanding them.
Besides linear functions, three groups of functions dominate the course:
The Quadratic Functions: (Covered in the Year 11 volume) These func-
tions are known from earlier years. They are algebraic representations of the
parabola, and arise naturally when areas are being considered or a constant
acceleration is being applied. They can be studied without calculus, but
calculus provides an alternative and sometimes quicker approach.
The Exponential and Logarithmic Functions: (Covered in the Year 12
volume) Calculus is essential for the study of these functions. We have
begun the topic with the exponential function. This has the great advantage
of emphasising the fundamental property that the exponential function with
baseeisitsownderivative—thisisthereasonwhyitisessentialforthestudy
ofnaturalgrowthanddecay, andthereforeoccursinalmosteveryapplication
of mathematics. The logarithmic function, and its relationship with the
rectangular hyperbola y = 1/x, has been covered in a separate chapter.
The Trigonometric Functions: (CoveredintheYear12volume) Calculus
isalsoessentialforthestudyofthetrigonometricfunctions. Theirdefinitions,
like the associated definition of π, are based on the circle. The graphs of the
sine and cosine functions are waves, and they are essential for the study of
all periodic phenomena.
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HowtoUseThisBook ix
Thus the three basic functions in the course, x2, ex and sinx, and the related
numbers e and π, can all be developed from the three most basic degree-2 curves
—theparabola, therectangularhyperbolaandthecircle. Inthisway, everything
in the course, whether in calculus, geometry, trigonometry, coordinate geometry
or algebra, can easily be related to everything else.
Algebra and Graphs: One of the chief purposes of the course, stressed heavily in re-
cent examinations, is to encourage arguments that relate a curve to its equation.
Algebraic arguments are constantly used to investigate graphs of functions. Con-
versely, graphs are constantly used to solve algebraic problems. We have drawn
as many sketches in the book as space allowed, but as a matter of routine, stu-
dents should draw diagrams for most of the problems they attempt. It is because
sketches can so easily be drawn that this type of mathematics is so satisfactory
for study at school.
Theory and Applications: Although this course develops calculus in a purely mathe-
matical way, using geometry and algebra, its content is fundamental to all the
sciences. In particular, the applications of calculus to maximisation, motion,
rates of change and finance are all parts of the syllabus. The course thus allows
students to experience a double view of mathematics, as a system of pure logic
on the one hand, and an essential part of modern technology on the other.
Limits,ContinuityandtheRealNumbers: Thisisafirstcourseincalculus,andrigorous
arguments about limits, continuity or the real numbers would be quite inappro-
priate. Any such ideas required in this course are not difficult to understand
intuitively. Most arguments about limits need only the limit lim 1/x = 0 and
x→∞
occasionally the sandwich principle. Introducing the tangent as the limit of the
secant is a dramatic new idea, clearly marking the beginning of calculus, and is
quite accessible. The functions in the course are too well-behaved for continuity
to be a real issue. The real numbers are defined geometrically as points on the
number line, and any properties that are needed can be justified by appealing to
intuitive ideas about lines and curves. Everything in the course apart from these
subtle issues of ‘foundations’ can be proven completely.
Technology: There is much discussion about what role technology should play in the
mathematics classroom and which calculators or software may be effective. This
is a time for experimentation and diversity. We have therefore given only a few
specific recommendations about technology, but we encourage such investigation,
and to this new colour version we have added some optional technology resources
that can be accessed via the Cambridge GO website. The graphs of functions are
at the centre of the course, and the more experience and intuitive understanding
students have, the better able they are to interpret the mathematics correctly. A
warning here is appropriate — any machine drawing of a curve should be accom-
panied by a clear understanding of why such a curve arises from the particular
equation or situation.
ISBN: 9781107679573 © Bill Pender, David Sadler, Julia Shea, Derek Ward 2012 Cambridge University Press
Photocopying is restricted under law and this material must not be transferred to another party
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ISBN: 9781107679573 © Bill Pender, David Sadler, Julia Shea, Derek Ward 2012 Cambridge University Press
Photocopying is restricted under law and this material must not be transferred to another party