Prelims

Mathematics Extension 1 HSC Course

maths

Mathematics Extension 1 HSC Course

maths Margaret Grove

Text © 2010 Grove and Associates Pty Ltd Illustrations and design © 2010 McGraw-Hill Australia Pty Ltd Additional owners o copyright are acknowledged in on-page credits Every eort has been made to trace and acknowledge copyrighted material. The authors and publishers tender their apologies should any inringement have occurred. Reproduction and communication or educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum o one chapter or 10% o the pages o this work, whichever is the greater, to be reproduced and/or communicated by any educational institution or its educational purposes provided that the institution (or the body that administers it) has sent a Statutory Educational notice to Copyright Agency Limited (CAL) and been granted a licence. For details o statutory educational and other copyright licences contact: Copyright Agency Limited, Level 15, 233 Castlereagh Street, Sydney NSW 2000. Telephone: (02) 9394 7600. Website: www.copyright.com.au Reproduction and communication or other purposes Apart rom any air dealing or the purposes o study, research, criticism or review, as permitted under the Act, no part o this publication may be reproduced, distributed or transmitted in any orm or by any means, or stored in a database or retrieval system, without the written permission o McGraw-Hill Australia including, but not limited to, any network or other electronic storage. Enquiries should be made to the publisher via www.mcgraw-hill.com.au National Library o Australia Cataloguing-in-Publication Data Author: Grove, Margaret. Title: Maths in ocus: mathematics extension 1 HSC course/Margaret Grove. Edition: 2nd ed. ISBN: 9780070278592 (pbk.) Target Audience: For secondary school age. Subjects: Mathematics. Mathematics–Problems, exercises, etc. Dewey Number: 510.76 Published in Australia by McGraw-Hill Australia Pty Ltd Level 2, 82 Waterloo Road, North Ryde NSW 2113 Publisher: Eiko Bron Managing Editor: Kathryn Fairax Production Editor: Natalie Crouch Editorial Assistant: Ivy Chung Art Director: Astred Hicks Cover and Internal Design: Simon Rattray, Squirt Creative Cover Image: Corbis Prooreaders: Terence Townsend and Ron Buck CD-ROM Preparation: Nicole McKenzie Typeset in ITC Stone seri, 10/14 by diacriTech Printed in China on 80 gsm matt art by iBook

v

Contents PREFACE

iii

ACKNOWLEDGEMENTS

iii

CREDITS

iii

FEATURES OF THIS BOOK

iii

SYLLABUS MATRIX

ix

STUDY SKILLS

x

Chapter 1: Geometry 2

2

INTRODUCTION

3

PlaNe FIgURe geOmeTRy

3

SURFaCe aReaS aND VOlUmeS

16

COORDINaTe meThODS IN geOmeTRy

21

CIRCLE PROPERTIES

25

TeST yOURSelF 1

43

ChalleNge exeRCISe 1

45

Chapter 2: Geometrical Applications of Calculus

50

INTRODUCTION

51

gRaDIeNT OF a CURVe

51

TyPeS OF STaTIONaRy POINTS

57

hIgheR DeRIVaTIVeS

61

SIgN OF The SeCOND DeRIVaTIVe

62

DeTeRmININg TyPeS OF STaTIONaRy POINTS

70

CURVe SkeTChINg

73

FURTHER CURvE SKETCHING

77

maxImUm aND mINImUm ValUeS

79

PROblem S INVOlVINg maxIma aND mINIma

83

PRImITIVe FUNCTIONS

95

TeST yOURSelF 2

100


102

Chapter 3: Integration

104

INTRODUCTION

105

aPPROxImaTION meThODS

105

INTegRaTION aND The PRImITIVe FUNCTION

117

DeFINITe INTegRalS

120

INDeFINITe INTegRalS

123

aReaS eNClOSeD by The x -axIS

128

aReaS eNClOSeD by The y -axIS

133

SUmS aND DIFFeReNCeS OF aReaS

136

VOlUmeS

138

INTEGRATION USING SUBSTITUTION

145

TeST yOURSelF 3

150


151

Practice Assessment Task Set 1

153

vi

Chapter 4: Exponential and Logarithmic Functions

160

INTRODUCTION

161

DIFFERENTIATION OF EXPONENTIAL FUNCTIONS

161

INTEGRATION OF EXPONENT IAL FUNCTIONS

169

LOGARITHMS

172

DERIVATIVE OF THE LOGARITHMI C FUNCTION

183

INTEGRATION AND THE LOGARITHMIC FUNCTION

187

TEST YOURSELF 4

190

CHALLENGE EXERCISE 4

191

Chapter 5: Trigonometric Functions INTRODUCTION

194 195

CIRCULAR MEASURE

195

TRIGONOMETRIC RESULTS

199

FURTHER TRIGONOMETRIC EQUATIONS

204

CIRCLE RESULTS

209

SMALL ANGLES

218

TRIGONOMETRIC GRAPHS

222

DIFFERENTIATION OF TRIGONOMETRIC FUNCTIONS

236

INTEGRATION OF TRIGONOMETRIC FUNCTIONS

240

2

244

INTEGRATION OF SIN

X

AND COS

2

X

TEST YOURSELF 5

247


248

Chapter 6: Applications of Calculus to the Physical World

250

INTRODUCTION

251

RATES OF CHANGE

251

RATES INVOLVING TWO OR MORE VARIABLES EXPONENT IAL GROWTH AND DECAY

A MORE COMPLEX FORMULA FOR GROWTH AND DECAY

255 260

269

MOTION OF A PARTICLE IN A STRAIGHT LINE

275

MOTION AND DIFFERENTIATION

283

MOTION AND INTEGRATION

290

VELOCITY AND ACCELERATION IN TERMS OF X

294

SIMPLE HARMONIC MOTION

302

PROJECTILES

313

TEST YOURSELF 6

324


326


Chapter 7: Inverse Functions

329

334

INTRODUCTION

335

INVERSE FUNCTIONS

335

GRAPH OF INVERSE FUNCTIONS

339

INVERSE TRIGONOMETRIC FUNCTIONS

352

DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS

363

INTEGRATION OF INVERSE TRIGONOMETRIC FUNCTIONS

370

TEST YOURSELF 7

373


374

vii

Chapter 8: Series INTRODUCTION

376 377

geNeRal SeRIeS

377

SIgma NOTaTION

383

aRIThmeTIC SeRIeS

385

geOmeTRIC SeRIeS

394

aPPlICaTIONS OF SeRIeS

410

PROOF BY MATHEMATICAL INDUCTION

431

TeST yOURSelF 8

436


438


Chapter 9: Polynomials 2

440

446

INTRODUCTION

447

ESTIMATION OF ROOTS

447

TEST YOURSELF 9

464


465

Chapter 10: The Binomial Theorem

466

INTRODUCTION

467

COMBINATIONS

467

BINOMIAL THEOREM

474

FURTHER WORK WITH COEFFICIENTS

483

TEST YOURSELF 10

494


495

Chapter 11: Probability

496

INTRODUCTION

497

SImPle PRObabIlITy

497

mUlTI-STage eVeNTS

509

COUNTING TECHNIqUES

521

BINOMIAL PROBABILITY DISTRIBUTION

533

TeST yOURSelF 11

543


545


Sample Examination Papers Answers

548

552 562

viii

PREFACE This book covers the HSC syllabus or Mathematics and Extension 1. It ollows the same style as the Year 11 Preliminary course, and provides a thorough coverage o the HSC syllabus. The extension material is easy to see as it has purple headings and there is purple shading next to all extension questions and answers. The syllabus is available through the NSW Board o Studies website at www.boardostudies .nsw.edu.au. You can also access resources, study techniques, examination technique, sample and past examination papers through other websites such as www.math.nsw.edu.au and www.csu.edu .au. Searching the Internet generally will pick up many websites supporting the work in this course. Each chapter has comprehensive ully worked examples and explanations as well as ample sets o graded exercises. The theory ollows a logical order, although some topics may be learned in any order. Each chapter contains Test Yoursel and Challenge exercises, and there are several practice assessment tasks throughout the book. I you have trouble doing the Test Yoursel exercises at the end o a chapter, you will need to go back into the chapter and revise it beore trying them again. Don’t attempt to do the Challenge exercises until you are condent that you can do the Test Yoursel exercises, as these are more dicult and are designed to test the more able students who understand the topic really well.

ACKNOWLEDGEMENTS Thanks go to my family, especially my husband Geoff, for supporting me in writing this book.

CREDITS Istockphoto: p 105 Margaret Grove: p 3, p 25, p 92, p 94, p 144, p 195, p 231, p 234, p 235, p 260, p 265, p 266, p 267, p 268, p 269, p 274, p 322, p 414, p 415, p416, p 497, p 508, p 519, p 520, p 535, p 541 Shutterstock: p 21, p 251, p 301, p 335, p 514

FEATURES OF THIS BOOK This second edition retains all the features of previous Maths in Focus books while adding in new improvements. The main feature of Maths in Focus is in its readability, its plentiful worked examples and straightforward language so that students can understand it and use it in self-paced learning. The logical progression of topics, the comprehensive fully worked examples and graded exercises are still major features. A wide variety of questions is maintained, with more comprehensive and more difficult questions included in each topic. At the end of each chapter is a consolidation set of exercises (Test Yourself) in no particular order that will test whether the student has grasped the concepts contained in the chapter. There is also a Challenge set for the more able students. The four practice assessment tasks provide a comprehensive variety of mixed questions from various chapters. These have been extended to contain questions in the form of sample examination questions, including short answer, free response and multiple choice questions that students may encounter in HSC assessments. The second edition also features a short summary of general study skills that students will find useful, both in the classroom and when doing assessment tasks and examinations. A syllabus matrix is included to show where each syllabus topic fits into the book. Topics are generally arranged in a logical order but there is room for some topics to be done in a different way. For example, probability can be done at any time.

ix

SYLLABUS MATRIX This matrix shows how the syllabus is organised in the chapters o this book.

Mathematics (2 Unit) Coordinate methods in geometry (6.8)


Applications of geometrical properties (2.5)


Geometrical applications of differentiation (10.1 – 10.8)

Chapter 2: Geometrical applications of calculus

Integration (11.1 – 11.4)


Trigonometric functions (13.1 – 13.6, 13.7)

Chapter 5: Trigonometric functions

Logarithms and exponential functions (12.1 – 12.5)

Chapter 4: Exponential and logarithmic functions

Applications of calculus to the physical world (14.1 – 14.3)

Chapter 6: Applications of calculus to the physical world

Probability (3.1 – 3.3)


Series (7.1 – 7.3) and Series applications (7.5)

Chapter 8: Series

Extension 1 Methods of integration (11.5E)


Primitive of sin 2 x and cos 2 x (13.6E)

Chapter 5: Trigonometric functions

Equation

dN dt

=

k(N

-

P )

(14.2E)

Velocity and acceleration as a function of x (14.3E)



x

Projectile motion (14.3E)


Simple harmonic motion (14.4E)


Inverse functions and inverse trigonometric functions (15.1 – 15.5E)

Chapter 7: Inverse functions

Induction (7.4E)

Chapter 8: Series

Binomial theorem (17.1 – 17.3E)

Chapter 10: The binomial theorem

Further probability (18.2E)


Iterative methods for numerical estimation of the roots of a polynomial equation (16.4E)

Chapter 9: Polynomials 2

STUDY SKILLS You may have coasted through previous stages without needing to rely on regular study, but in this course many of the topics are new and you will need to systematically revise in order to build up your skills and to remember them. The Preliminary course introduces the basics of topics such as calculus that are then applied in the HSC course. You will struggle in the HSC if you don’t set yourself up to revise the preliminary topics as you learn new HSC topics. Your teachers will be able to help you build up and manage good study habits. Here are a few hints to get you started. There is no right or wrong way to learn. Different styles of learning suit different people. There is also no magical number of hours a week that you should study, as this will be different for every student. But just listening in class and taking notes is not enough, especially when learning material that is totally new. You wouldn’t go for your driver’s licence after just one trip in the car, or enter a dance competition after learning a dance routine once. These skills take a lot of practice. Studying mathematics is just the same. If a skill is not practised within the first 24 hours, up to 50% can be forgotten. If it is not practised within 72 hours, up to 85–90% can be forgotten! So it is really important that whatever your study timetable, new work must be looked at soon after it is presented to you. With a continual succession of new work to learn and retain, this is a challenge. But the good news is that you don’t have to study for hours on end!

xi

In the classroom In order to remember, rst you need to ocus on what is being said and done. According to an ancient proverb:

‘I hear and I forget I see and I remember I do and I understand’

I you chat to riends and just take notes without really paying attention, you aren’t giving yoursel a chance to remember anything and will have to study harder at home. I you have just had a ght with a riend, have been chatting about weekend activities or myriad o other conversations outside the classroom, it helps i you can check these at the door and don’t keep chatting about them once the lesson starts. I you are unsure o something that the teacher has said, the chances are that others are also not sure. Asking questions and clariying things will ultimately help you gain better results, especially in a subject like mathematics where much o the knowledge and skills depends on being able to understand the basics. Learning is all about knowing what you know and what you don’t know. Many students eel like they don’t know anything, but it’s surprising just how much they know already. Picking up the main concepts in class and not worrying too much about other less important parts can really help. The teacher can guide you on this. Here are some pointers to get the best out o classroom learning: Q

Take control and be responsible or your own learning

Q

Clear your head o other issues in the classroom

Q

Active, not passive learning is more memorable

Q

Ask questions i you don’t understand something

Q

Listen or cues rom the teacher

Q

Look out or what are the main concepts

Note taking varies rom class to class, but there are some general guidelines that will help when you come to read over your notes later on at home: Q

Write legibly

Q

Use dierent colours to highlight important points or ormulae

Q

Make notes in textbooks (using pencil i you don’t own the textbook)

Q

Use highlighter pens to point out important points

Q

Summarise the main points

Q

I notes are scribbled, rewrite them at home

xii

At home You are responsible or your own learning and nobody else can tell you how best to study. Some people need more revision time than others, some study better in the mornings while others do better at night, and some can work at home while others preer a library. There are some general guidelines or studying at home: Q

Revise both new and older topics regularly

Q

Have a realistic timetable and be fexible

Q

Summarise the main points

Q

Revise when you are resh and energetic

Q

Divide study time into smaller rather than longer chunks

Q

Study in a quiet environment

Q

Have a balanced lie and don’t orget to have un!

I you are given exercises out o a textbook to do or homework, consider asking the teacher i you can leave some o them until later and use these or revision. It is not necessary to do every exercise at one sitting, and you learn better i you can spread these over time. People use dierent learning styles to help them study. The more variety the better, and you will nd some that help you more than others. Some people (around 35%) learn best visually, some (25%) learn best by hearing and others (40%) learn by doing. Here are some ideas to give you a variety o ways to study: Q

Summarise on cue cards or in a small notebook

Q

Use colourul posters

Q

Use mindmaps and diagrams

Q

Discuss work with a group o riends

Q

Read notes out aloud

Q

Make up songs and rhymes

Q

Do exercises regularly

Q

Role play teaching someone else

Assessment tasks and exams Many o the assessment tasks or maths are closed book examinations. You will cope better in exams i you have practiced doing sample exams under exam conditions. Regular revision will give you condence and i you eel well prepared, this will help get rid o nerves in the exam. You will also cope better i you have had a reasonable night’s sleep beore the exam. One o the biggest problems students have with exams is in timing. Make sure you don’t spend too much time on questions you’re unsure about, but work through and nd questions you can do rst. Divide the time up into smaller chunks or each question and allow some extra time to go back to questions you couldn’t do or nish. For example, in a 2 hour exam with 6 questions, allow around 15 minutes or each question. This will give an extra hal hour at the end to tidy up and nish o questions.

xiii

Here are some general guidelines or doing exams: Q

Read through and ensure you know how many questions there are

Q

Divide your time between questions with extra time at the end

Q

Don’t spend too much time on one question

Q

Read each question careully, underlining key words

Q

Show all working out, including diagrams and ormulae

Q

Cross out mistakes with a single line so it can still be read

Q

Write legibly

And fnally… Study involves knowing what you don’t know, and putting in a lot o time into concentrating on these areas. This is a positive way to learn. Rather than just saying, ‘I can’t do this’, say instead, ‘I can’t do this yet’, and use your teachers, riends, textbooks and other ways o nding out. With the parts o the course that you do know, make sure you can remember these easily under exam pressure by putting in lots o practice. Remember to look at new work Q

today

Q

tomorrow

Q

in a week

Q

in a month

Some people hardly ever nd time to study while others give up their outside lives to devote all their time to study. The ideal situation is to balance study with other aspects o your lie, including going out with riends, working and keeping up with sport and other activities that you enjoy.

Good luck with your studies!

Prelims

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