Mathematical Methods Units 3 & 4

Series editor: John Dowsey

Philip Swedosh Natalie Caruso John Dowsey

Peter Flynn Philip Wayth David Tynan

Contributing authors:

Laure Berman John Burrow

Peter Galbraith Bruce Henry Helen Keating Gurcharn Rehill

Mathematical

Methods

Units

3 & 4

CAS active

MathsW

First published 2006 by

MACMILLANEDUCATIONAUSTRALIAPTYLTD

627 Chapel Street, South Yarra 3141

Visit our website at www.macmillan.com.au

Associated companies and representatives throughout the world.

Copyright © J. Dowsey, P. Swedosh, N. Caruso, P. Flynn, P. Wayth 2006

All rights reserved.

Except under the conditions described in the

Copyright Act 1968 of Australia (the Act) and subsequent amendments, no part of this publication may be reproduced,

stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the copyright owner.

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Licence restrictions must be adhered to. For details of the CAL licence contact: Copyright Agency Limited, Level 12, 233 Castlereagh Street, Sydney, NSW 2000. Telephone: (02) 9394 7600. Facsimile: (02) 9394 7601. Email: [email protected]

National Library of Australia cataloguing in publication data

MathsWorld: mathematical methods. Units 3 and 4.

For VCE year 12 students. ISBN 978 0 7329 9227 9 ISBN 0 7329 9227 3.

1. Mathematics—Problems, exercises, etc. 2. Mathematics— Textbooks. 3. Mathematics—Study and teaching

(Secondary). I. Dowsey, John.

510.76

Publisher: David Tynan Project editor: Colin McNeil Editor: Marta Veroni

Cover and text designer: Anne Stanhope Photo research: Kath Selkirk and Naomi Parker Permissions clearance: Naomi Parker

Typeset in 10 on 13 point Slimbach Book and 10 on 12 point Avenir Book by Sun Photoset Pty Ltd Cover image: Getty Images

Printed by McPherson’s Printing Group, Australia

Internet addresses

At the time of printing, the internet addresses appearing in this book were correct. Owing to the dynamic nature of the Internet, however, we cannot guarantee that all these addresses will remain correct.

MM_Prelims_2.indd ii

Contents

How to use MathsWorld

vi

Acknowledgments x

Chapter 1 Algebraic techniques

1

1.1 Algebra

of

polynomials

2

1.2 Systems of simultaneous linear equations

16

1.3 Algebra of exponential and logarithmic functions

25

1.4 Algebra of circular functions

36

1.5 Further solving techniques

48

Chapter review

52

Chapter 2 Functions and their graphs

57

2.1 Power and polynomial functions

58

2.2 Exponential and logarithmic functions

78

2.3 Circular

functions

86

2.4 Absolute value function

94

Chapter review

100

Chapter 3 Transformations of graphs

105

3.1 Refl ection

106

3.2 Dilation

113

3.3 Translation

121

3.4 Further combinations of transformations

126

3.5 Transformations with matrices

137

Chapter review

142

Chapter 4 Differentiation techniques

147

4.1 Continuity, limits and differentiability

148

4.2 Differentiation

162

4.3 Differentiation rules

171

4.4 Derivatives of other functions

179

4.5 Graphs of gradient functions

194

Chapter review

200

Chapter 5 Applications of differentiation

205

5.1 Average, instantaneous and related rate of change

206

5.2 Curve sketching

212

5.3 Tangents and normals to curves

219

5.4 Maximum and minimum problems

223

5.5 Linear approximations

230

iv

Contents

Chapter 6 Antidifferentiation and integration

237

6.1 Derivatives and antiderivatives

238

6.2 Approximations to area under a curve and defi nite integration

251

6.3 Properties of antiderivatives and defi nite integrals

265

Chapter review

271

Chapter 7 Review 1

275

Chapters 1–6 revision questions

276

Examination-style technology-free questions

280

Examination-style technology-active questions

286

Chapter 8 Applications of integration

295

8.1 Graph of an antiderivative function

296

8.2 Areas under and between curves

304

8.3 Straight-line motion

317

8.4 Finding a function given its derivative

323

8.5 Average value of a function

327

Chapter review

332

Chapter 9 The algebra of functions

337

9.1 Sum, difference and product of functions

338

9.2 Composite functions

349

9.3 Inverse functions

357

9.4 Functional equations

374

Chapter review

379

Chapter 10 Functions, graphs and modelling

383

10.1 Modelling with polynomials and rational functions

384

10.2 Modelling with exponential and logarithmic functions

390

10.3 Modelling with circular functions

395

10.4 Models for data

401

10.5 Miscellaneous modelling

408

Chapter review

414

Chapter 11 Discrete random variables

417

11.1 Discrete random variables

418

11.2 Statistics of discrete random variables

437

11.3 The binomial distribution

445

11.4 Mean and variance of a binomial random variable

455

11.5 Transition matrices

459

Chapter review

470

MM_Prelims_2.indd iv

v

Contents

Chapter 12 Continuous random variables

473

12.1 Continuous random variables

474

12.2 Statistics of continuous random variables

485

12.3 Applications of continuous random variables

494

Chapter review

500

Chapter 13 The normal distribution

503

13.1 The normal distribution

504

13.2 The standard normal distribution

513

13.3 Calculation of probabilities

517

13.4 The inverse normal distribution

526

Chapter review

533

Chapter 14 Review 2

537

Chapters 8–13 revision questions

538

Examination-style technology-free questions

542

Examination-style technology-active questions

548

vi

How to use MathsWorld

MathsWorld Mathematical Methods Units 3 & 4

is one of the

innovative

MathsWorld VCE

series of student textbooks and teacher resources

designed specifi cally for the newly accredited VCE Mathematics study design (2006–2009).

This resource comprises:

.

a student edition (book and CD)

.

a teacher edition (book and CD)

.

a solutions CD and teacher support website.

This resource covers all material from both Mathematical Methods Units 3 & 4 and

Mathematical Methods (CAS) Units 3 & 4. Where theory, worked examples and exercise

questions relate only to the Mathematical Methods (CAS) course this is clearly indicated using

icons and shading.

The student edition

Theory and worked examples in the student book are designed to

illustrate the main concepts and skills in each topic.

Handy

tips

are included to help simplify or clarify theory points,

and

warnings

about common student errors or misconceptions are

placed near the relevant text.

Sets of carefully graded

exercise questions

are placed appropriately

throughout the text. Some exercises are broken up into smaller

chunks to refl ect common teaching practice.

A navigation tab on the right of each double page clearly indicates

the section that you are working in.

Each chapter ends with a thorough

review section

, consisting of a concise

summary

and sets

of

short answer

and

extended response revision questions

.

There are two review chapters (7 and 14), one covering chapters 1 to 6 and the other covering

chapters 8 to 13. The review chapters include

examination-style technology-free

and

technology-active

questions.

A

technology toolkit

is included on the student CD for more detailed explanation of commonly

needed procedures for a graphics or CAS calculator that are covered briefl y in the text. Where

calculator images are useful they have been

included and shaded to indicate whether

they are from a TI-83/84 or TI-89 calculator,

as shown. A toolkit icon is included in

the margin, with indexed references to

procedures supplied on the CD for each

calculator.

tip

Warning

GC 1.3, 2.4 CAS 1.3, 4.7

/‡nÎÉn{ÊÃVÀii˜Ã…œÌ

/‡n™ÊÃVÀii˜Ã…œÌ

MM_Prelims_2.indd vi

vii

CD icons in the margin refer to

analysis tasks

,

modelling tasks

,

theory

extensions

,

calculator programs

,

spreadsheets

or

websites

, all of which

are available on the teacher CD. Your teacher will make these available to

you when required.

For convenience, the complete student book is available on the

student CD

in PDF format. These fi les can be easily navigated using the bookmarks and

links to all chapters and sections contained in the book.

The teacher edition

The teacher book includes an annotated version of every page of the

student book. Each chapter includes

teaching notes

placed near the

relevant text on most pages, including:

.

further explanation of worked examples and theory

.

extra theory or pedagogic points

.

suggested additional examples

.

listings of the relevant content dot points from the

VCE Mathematics (2006–2009) study design

.

tips to encourage students’ refl ective use of technology.

The teacher book also includes

answers

to all exercise questions

placed as near as is practicable to each question. All answers are clearly visible and included

at a reasonable size.

The accompanying

teacher CD

contains all elements present on the student

CD, clearly indexed throughout each chapter. For example the icon shown

on the left indicates that it is the 5th on-CD element in chapter 3, and

that it is a SAC-style analysis task.

The teacher CD has an easy-to-use interface of linked PDF documents.

Each chapter includes

a trio of tests

in editable format, so that part or all of

the questions supplied can be used. Each chapter test includes multiple choice,

short answer and extended response style items, and is provided with a marking scheme.

SAC-style analysis tasks

are also included in editable format, as well as solutions.

Innovative

modelling tasks

have been included, accompanied by teacher notes and worked

solutions.

Some chapters include short, focused

technology assisted investigations

, which may be set

as class work or homework.

Calculator programs

and

spreadsheets

related to learning tasks have been included on the

teacher CD where relevant and helpful. At the teacher’s discretion, these can be placed on

the school’s intranet or shared folders for easy student access.

Curriculum planning documents including a suggested teaching program have been included

on the teacher CD.

3.1_Transform

GC pr

ogram

CD

3.5Is this normal?

SAC analysis task

viii

Further teacher support

A

solutions CD

provides worked solutions to every question in MathsWorld Mathematical

Methods Units 3 & 4.

A community website run by the authors at

www.mathsworld.com.au

provides extra assessment

tasks, an online forum and regular newsletters and updates about MathsWorld.

MM_Prelims_2.indd viii

ix

The MathsWorld author team would like to dedicate this book to all their friends and family who suffered so that this book could have life. They would also like

x

Acknowledgments

The publishers wish to thank Adam Spencer for the generous donation of his time and the use of material from

Adam Spencer’s Book of Numbers, Penguin Group Australia, 2000. Photograph, pp. 24, 106, 245, 487. Text extracts, pp. 24, 35, 51, 76, 99, 132, 153, 161, 178, 218, 256, 263, 322, 365, 370, 407, 412, 431, 436, 458, 484, 505, 525, 532, 561.

Photographs

AAP Image/Steve Holland, p. 222. Aquarius Collection/Twentieth Century Fox, p. 132. Brand X Pictures, p. 493. Colin McNeil, p. 516. Corbis/Jim Erickson, p. 105, /Kimimasa Mayama/Reuters, pp. 205, 222. Fairfax Photos/ Vince Caligiuri, p. 467, /Sebastian Costanzo, p. 270, /Brendan Esposito, p. 387, /Craig Golding, pp. 406, 458, /Estelle Grunberg, p. 435, /Eddie Jim, p. 384, /Dallas Kilponen, p. 503, /Gary Medlicott, p. 3, /Robert Pearce, p. 77, /Darren Pateman, p. 383, /Fiona Lee Quimby, p. 529, /Michael Rayner, pp. 466, 473, /Steven Siewert, p. 161, /Rick Stevens, p. 229, /Andrew Taylor, pp. 401, 542, /The Age Archives, p. 400, /Angela Wylie, pp. 427, 540. Getty Images/Abrams/Lacagnina, p. 147, /Ian Logan, p. 85, /David Maitland, pp. 275, 537, /Stephen Marks, p. 124, /Yellow Dog Productions, p. 417. Istockphoto.com, pp. 330, 349, 389, /Judi Ashlock, p. 456, /Tamara Bauer, p. 292, /Arpad Benedek, p. 171, /Adam Booth, p. 86, /Martin Bowker, p. 82, /Christian Carroll, p. 81, /Michel de Nijs, p. 287, /David Elfstrom, pp. 262, 264, /Joy Fera, p. 293, /David Ferrell, p. 211, /Jose Gil, p. 315, /Margaret Gilchrist, p. 539, /Dave Giribet, p. 322, /Roy Greve, p. 193, /Zeliha Gurkan, p. 410, /George Ihring, p. 413, /Paul Johnson, p. 290, /Ula Kapala, p. 221, /Jeroen Kloppenburg, p. 531, /Helen Koshkina, p. 395, /Neils Laan, p. 354, /Dianne McFadden, p. 356, /James McKinlay, p. 93, /Sean McLennan, p. 407, /James McQuillan, pp. 11, 41, /Linda Macpherson, p. 497, /Mike Manzano, p. 444, /Greg Nicholas, p. 76, /Peter Nguyen, pp. 207 (all), 428, /Pekka Nikonen, p. 434, /Jane Norton, p. 525, /Lee Pettet, p. 208, /Florian Rieder, p. 169, /Amanda Rohde, pp. 91, 325, /Matthew Scherf, p. 331, /Sara Sherman, p. 227, /Andrej Simeonov, p. 324, /Ben Spinler, p. 18, /Jason Stitt, p. 422, /Yen Teoh, p. 91, /Andrei Tchernov, p. 447, /Wouter Van Caspel, p. 209, /Jeffrey Waibel, p. 411, /Stephen Walls, p. 511, /John Watson, p. 257, /Torsten Wittmann, pp. 316, 499. Lonely Planet Images/John Banagan, p. 237, /Richard Cummins, p. 295, /Richard I’Anson, pp. 106, 212, /Wayne Walton, p. 57. MEA Photos/Tania Mazzeo, p. 326. Newspix/Jane Ollerenshaw, p. 386. Photolibrary/Payne Anderson, p. 16. Photos.com, pp. 172, 391, 398 (bottom). Rubberball Productions, p. 398 (bottom). Science Photo Library/Oscar Burriel, p. 341, /Cavendish Laboratory UA5 Experiment, CERN/David Parker, p. 318, /Mehau Kulyk, pp. 337, 345, /London School of Hygiene & Tropical Medicine, p. 289, /Cordelia Molloy, p. 113 (all), /Volker Steger, p. 390. The Kobal Collection/CBS-TV, p. 1, /CBS-TV/NBC/Talent Assoc. p, 454, /Columbia Pictures Corporation, p. 355, /Roadside/Goldwyn, p. 233, /Rosenthal/Zade/Columbia/Tri-Star, p. 521.

Other material

Reproduced by permission of the Victorian Curriculum and Assessment Authority, Victoria, Australia. Although references have been reproduced with permission of the VCAA the publication is in no way connected with or endorsed by the VCAA, pp. 280 (Q. 1, 2, 3), 281 (Q. 9, 11, 12), 282 (Q. 14, 15, 18), 283 (Q. 20, 22, 23), 284 (Q. 25, 27), 285 (Q. 30, 31, 34), 286 (Q. 35, 36), 287 (Q. 2), 289 (Q. 4), 290 (Q. 6), 292 (Q. 8), 294 (Q. 10, 11), 542 (Q. 1), 543 (Q. 3, 6, 7), 544 (Q. 9, 10, 13), 545 (Q. 16, 18), 547 (Q. 26, 28, 30), 548 (Q. 32, 33, 35), 549 (Q. 2), 551 (Q. 4, 5), 553 (Q. 7, 8), 555 (Q. 10), 556 (Q. 12), 557 (Q. 13), 558 (Q. 14), 560 (Q. 16), 561 (Q. 17).

While every care has been taken to trace and acknowledge copyright, the publishers tender their apologies for any accidental infringement where copyright has proved untraceable. They would be pleased to come to a suitable arrangement with the rightful owner in each case.

MM_Prelims_2.indd x