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12MATHS Quest
121212MAMATTHS HS QuestSpecialist Mathematics
VCE MATHEMATICS UNITS 3 & 4
prelims SM Page i Thursday, October 12, 2000 2:47 PM
prelims SM Page ii Thursday, October 12, 2000 2:47 PM
121212MAMATTHS HS QuestSpecialist Mathematics
Support materialVal Augustin • John Dowsey • Dennis Fitzgerald
David Phillips • David Tynan
Contributing authorsRobert Cahn • George Dimitriadis • Don Morelli
VCE MATHEMATICS UNITS 3 & 4
JENNIFER NOLAN GEOFF PHILLIPS
ROSS ALLEN MURRAY ANDERSON
MQ Spec Maths Prelims SE Page iii Tuesday, July 22, 2003 2:08 PM
First published 2000 byJohn Wiley & Sons Australia, Ltd33 Park Road, Milton, Qld 4064
Offices also in Sydney and Melbourne
Typeset in 10.5/12.5pt Times
© John Wiley & Sons Australia, Ltd 2000
National Library of AustraliaCataloguing-in-Publication data
Maths Quest 12: Specialist Mathematics VCE MathematicsUnits 3 and 4
ISBN 0 7016 3400 6
1. Mathematics. 2. Victorian Certificate of Educationexamination — Study guides. I. Nolan, Jennifer.
510
All rights reserved. No part of this publication may bereproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical,photocopying, recording, or otherwise, without the priorpermission of the publisher.
Edited by Merv Littmann, Margaret Falk, Joy Window,Jennifer Wright and Shukla Chakraborty
Illustrated by the Wiley Art Department
Cover photograph and CD face: © 2000 PhotoDisc
Printed in Singapore byCraft Print International Ltd
10 9 8 7 6 5 4 3 2
MQ Spec Maths Prelims SE Page iv Tuesday, July 22, 2003 1:43 PM
ContentsCHAPTER 1� Coordinate geometry 1Coordinate geometry 2
Sketch graphs of y = axm + bx−n + c where m = 1 or 2 and n = 1 or 2 3
Exercise 1A 11Reciprocal graphs 13
Exercise 1B 18Graphs of ellipses 21
Exercise 1C 24Graphs of hyperbolas 26
Exercise 1D 29Partial fractions 31
Exercise 1E 37Sketch graphs using partial fractions 39
Exercise 1F 45Further graphing and partial fractions 46Summary 47Chapter review 49
CHAPTER 2� Circular (trigonometric) functions 55Reciprocal trigonometric functions 56
Exercise 2A 59Graphs of reciprocal trigonometric functions 61Graphs of reciprocal trigonometric
functions 62Exercise 2B 67
Trigonometric identities 69Exercise 2C 72
Compound and double angle formulas 73Exercise 2D 78
More identities 80Inverse circular functions and their graphs 81
Exercise 2E 86How long is a piece of rope? 88Summary 89Chapter review 91
CHAPTER 3� Complex numbers 95Introduction to complex numbers 96
Exercise 3A 99
Basic operations using complex numbers 100Exercise 3B 104
Conjugates and division of complex numbers 105
Exercise 3C 108Complex numbers in polar form 110History of mathematics: Abraham de Moivre 114
Exercise 3D 115Basic operations on complex numbers in polar
form 117Exercise 3E 123
Factorisation of polynomials in C 125Exercise 3F 129
Solving equations in C 131
The exact values of cos and sin 136
Exercise 3G 137Summary 138Chapter review 140
CHAPTER 4� Representation of relations and regions in the complex plane 143Introduction 144Rays and lines 144
Exercise 4A 148Circles and ellipses 150
Exercise 4B 152Combination graphs and regions 154
Exercise 4C 157Graphs of other simple curves 160
Exercise 4D 164Summary 166Chapter review 167
CHAPTER 5� Differential calculus 171Introduction 172The derivative of tan kx 172
Exercise 5A 175Second derivatives 177
Exercise 5B 179
2π5
------2π5
------
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viAnalysing the behaviour of functions using the
second derivative 181Exercise 5C 188
Derivatives of inverse circular functions 190Exercise 5D 195
Antidifferentiation involving inverse circular functions 198
Exercise 5E 202History of mathematics: Sofia Kovalevskaya 204Rate of change of angle 205Summary 206Chapter review 207
CHAPTER 6� Integral calculus 211Integration techniques and applications 212Technique 1: Substitution where the
derivative is present in the integrand 212Exercise 6A 218
Technique 2: Linear substitution 221Exercise 6B 224
Technique 3: Antiderivatives involving trigonometric identities 226
Exercise 6C 231The graph of a function and the graphs of its antiderivatives 233Technique 4: Antidifferentiation using partial
fractions 234Exercise 6D 238
Definite integrals 240Exercise 6E 245
Applications of integration 248Exercise 6F 255
Volumes of solids of revolution 258Exercise 6G 261
Approximate evaluation of definite integrals and areas 264
Exercise 6H 268Is there a connection between area and volume of revolution? 269Summary 270Chapter review 273
CHAPTER 7� Differential equations 279Differential equations: related rates 280
Exercise 7A 284
Verifying solutions 287Exercise 7B 289
Differential equations of the form
= f(x) 290
Exercise 7C 293Differential equations of the form
= f(x) 295
History of mathematics: Maria Agnesi 297
Exercise 7D 298Differential equations of the form
= g(y) 300
Exercise 7E 304Setting up and solving differential
equations 306Exercise 7F 311
Input/output of mixing problems 313Exercise 7G 317
Solutions to differential equations 319
Numerical solutions by Euler’s method 319Exercise 7H 323
Improved Euler’s method 324
Summary 325Chapter review 327
CHAPTER 8� Kinematics 331Differentiation and displacement, velocity and
acceleration 332Exercise 8A 338
Using antidifferentiation 341Exercise 8B 345
Motion under constant acceleration 348Exercise 8C 353
Velocity–time graphs 356Exercise 8D 361
Applying differential equations to rectilinear motion 365
Exercise 8E 370The falling ball bearing 373
Summary 374Chapter review 375
dydx------
d2 y
dx2---------
dydx------
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vii
CHAPTER 9� Vectors in two and three dimensions 379Vector calculus 428Vectors and scalars 380
Exercise 9A 385Position vectors in two and three
dimensions 389Exercise 9B 395
Multiplying two vectors — the dot product 400
Exercise 9C 404Using vectors in geometry 406Three-dimensional non-zero vectors 408
Exercise 9D 409History of mathematics: Charles Lutwidge Dodgson 410Resolving vectors — scalar and vector
resolutes 411Exercise 9E 414
Time-varying vectors 415Exercise 9F 419
Summary 421Chapter review 423
CHAPTER 10� Vector calculus 427Vector calculus 428Position, velocity and acceleration 428
Exercise 10A 435Cartesian equations and antidifferentiation of
vectors 438Exercise 10B 443
Applications of vector calculus 446Exercise 10C 453
Projectile motion 456Distance travelled along a curve 466
Exercise 10D 467Summary 470Chapter review 472
CHAPTER 11� Newton’s laws of motion 475Force diagrams and the triangle of
forces 476Exercise 11A 483
History of mathematics: Sir Isaac Newton 485Newton’s First Law of Motion 486
Exercise 11B 492Newton’s Second Law of Motion 495
Exercise 11C 502Applications of Newton’s First and Second
Laws of Motion 506Exercise 11D 511
Applications of Newton’s first and second laws to connected masses 514
Exercise 11E 518Variable forces 520Simple harmonic motion 523
Exercise 11F 524Momentum and Newton’s Third Law of
Motion 526Exercise 11G 531
Conservation of momentum for an isolated system 533
Exercise 11H 537Summary 539Chapter review 541
Answers 545
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Introduction
Maths Quest 12 Specialist Mathematics
is one of the exciting new
MathsQuest
resources specifically designed for the new VCE (2000–2003) Math-ematics course. It breaks new ground in Mathematics textbook publishing.
This resource contains:• a student textbook with accompanying CD-ROM and • a teacher CD-ROM.
Textbook
Full colour
is used throughout to produce clearer graphs and headings, toprovide bright, stimulating photos and to make navigation through the texteasier.
Clear, concise
theory sections
contain
worked examples
,
graphics calculatortips
and
highlighted important text
and
remember boxes
.
Worked examples
in a Think/Write format provide clear explanation of keysteps and suggest presentation of solutions.
Exercises
contain many carefully graded skills and application problems,including multiple-choice questions. Cross references to relevant workedexamples appear beside the first ‘matching’ question throughout theexercises.
History of mathematics
profiles of mathematicians place mathematicalconcepts in context.
Investigations
, often suggesting the use of technology, provide furtherdiscovery learning opportunities.
Each chapter concludes with a
summary
and
chapter review
exercise contain-ing examination style questions (multiple choice, short answer and analysis)which help consolidate students’ learning of new concepts.
Technology
is fully integrated (in line with VCE 2000 recommendations). Aswell as featuring graphics calculators,
Maths Quest
employs computeralgebra systems, spreadsheets, dynamic geometry software and severalgraphing packages. Not only does the text promote these technologies aslearning tools, but demonstration versions of the programs (with theexception of Microsoft Excel) are also included, as well as hundreds ofsupporting files on the
bonus
accompanying CD-ROM.
Student CD-ROM
The accompanying CD-ROM contains the entire student textbook plus addi-tional exercises. Students may work from the CD on laptops, school or homecomputers, and cut and paste material for revision or assignments.
Clearly labelled icons within the electronic version of the text
hyperlink
tohundreds of technology files for programs such as Mathcad, Excel and CabriGeometry to allow further exploration of ‘what if’ scenarios.
TI-83 graphics calculator programs can be downloaded to students’ calcu-lators using the Graphlink software provided.
121212MATHS QuestQuest
JAC
ARANDA
MA
THSQUES
T
VCE MATHEMATICS UNITS 3 & 4
JENNIFER NOLAN GEOFF PHILLIPS ROSS ALLEN MURRAY ANDERSONJENNIFER NOLAN GEOFF PHILLIPS ROSS JENNIFER NOLAN GEOFF PHILLIPS ROSS ALLEN MURRAALLEN MURRAY Y ANDERSONANDERSON
Specialist MathematicsSpecialist Mathematics
INTE
RACTIVE
C
D- ROM
Cabri
Geometry
EXCEL
Spreadsheet
Mathca
d
GCpro
gram
MQ Spec Maths Prelims SE Page viii Sunday, July 13, 2003 11:25 AM
ix
WorkSHEET
and
Test yourself
icons link to editable Word 97 documents, andmay be completed on screen or printed and completed.
Programs included
Mathcad Explorer
: a computer algebra system and graphing program
Graphmatica
: an excellent graphing utility
Equation grapher and regression analyser
: like a graphics calculator forthe PC
GrafEq
: graphs any relation, including complicated inequalities
Poly
: for visualising 3D polyhedra and their nets
TI Graphlink 83 and 89
: calculator screen capture and program transfer
Cabri Geometry II
: dynamic geometry program
Adobe
®
Acrobat
®
Reader 4.0
Teacher CD-ROM
The teacher CD-ROM contains everything in the student CD-ROM and more.A test for each chapter, fully worked solutions to
WorkSHEETs
, the workprogram and other curriculum advice in editable Word 97 format areprovided.
Web site:
www.jaconline.com.au/maths
The
Maths Quest
Web site will provide support in the form of additionaltechnology files, worksheets, links to other sites, assessment materialsincluding practice examinations and more.
Maths Quest
is a rich collection of teaching and learning resources withinone package.
Maths Quest 12 Specialist Mathematics
provides ample material, such asexercises, analysis questions, investigations, worksheets and technology files,from which teachers may set school assessed coursework (SAC).
WorkS
HEETtesttestCH
APTERyyourselfourself
testyyourselfourself 3.1
1
INTE
RACTIVE
C
D- ROM
MQ Spec Maths Prelims SE Page ix Sunday, July 13, 2003 11:26 AM
Acknowledgements
The authors and publisher would like to thank the following copyright holders,organisations and individuals for their assistance and for permission toreproduce copyright material in this book.
Illustrative material
• © Australian Picture Library pp. 210/John Carnemolla, 305/GrahamWarrend, 410/Corbis, 498/West Stock, 537/Universal Pictorial Press, 543; •Courtesy RACQ CareFlight Queensland, p. 477(upper); • © Corbis 1999,p. 396; • Malcolm Cross, p. 281(left); • © 2000 Digital Stock, pp. 96, 279,285, 315, 329, 532; • © Mary Evans Picture Library, 204; • © 2000PhotodDisc, Inc., (cover), pp. 1, 54, 88, 95, 143, 171, 189, 205, 211,278(upper), 278(lower), 281(right), 311, 312, 318, 331, 347, 354, 355, 363,371, 372, 379, 386, 398, 399, 414, 425, 427, 445, 448, 453, 454(two),459, 467, 468, 469, 475, 476, 477(lower), 487, 488, 493, 504, 508, 513,523, 524, 525(two), 528, 538; • © 2000 Stockbyte, p. 263; • Peter Storer,pp. 55, 94; • © photolibrary.com/Science Photo Library, p. 485; • © JohnWiley & Sons Australia, p. 490.
Software
The authors and publisher would like to thank the following software providersfor their assistance and for permission to use their materials. However, the use ofsuch material does not imply that the providers endorse this product in any way.
Third party software — registered full version ordering information
Full versions of third party software may be obtained by contacting thecompanies listed below.
Texas Instruments 83 and 89 Graphlink software
Material reproduced with permission of the publisher. © Texas InstrumentsIncorporated.
TI 83 and 89 Graphlink Software available from Texas Instruments.
Web: http://education.ti.com/us/global/software.html#connectivity
Note
: A Graphlink cable can be purchased from educational booksellers orcalculator suppliers.
Mathcad Explorer
Reproduced with permission of Mathsoft www.mathsoft.com
Distributed in Australia by Hearne Scientific Software Pty Ltd
Level 6, 552 Lonsdale Street, Melbourne 3000
e-mail: [email protected]
Web: www.hearne.com.au/
Phone: (03) 9602 5088
MQ Spec Maths Prelims SE Page x Sunday, July 13, 2003 11:26 AM
xi
Graphmatica
Reproduced with permission of kSoft, Inc.
e-mail: [email protected]
Web: www.graphmatica.com
345 Montecillo Dr., Walnut Creek, CA 94595-2654.
Software included is for evaluation purposes only. The user is expected toregister shareware if use exceeds 30 days. Order forms are available atwww.graphmatica.com/register.txt
Cabri Geometry II
Reproduced with permission of Cabrilog.
Cabrilog
7 av. Félix Viallet
38000 Grenoble FRANCE
Web: www.cabri.com
GrafEq and Poly
Evaluation copies of GrafEq™ and Poly™ have been included with permis-sion from Pedogoguery Software.
e-mail: [email protected]
Web: www.peda.com
Equation Grapher and Regression Analyser
Reproduced with permission of MFSoft International.
e-mail: [email protected]
Web: www.mfsoft.com
Microsoft® Excel
Screen Shots reproduced by permission of Microsoft Corporation.
Note
: Microsoft Software was used only in Screen Dumps.
Microsoft Excel is a registered trademark of the Microsoft Corporation in theUnited States and/or other countries.
Every effort has been made to trace the ownership of copyright material.Information that will enable the publisher to trace the copyright holders or torectify any error or omission in subsequent reprints will be welcome. In suchcases, please contact the Permission Section of John Wiley & Sons Australia,who will arrange for the payment of the usual fee.
MQ Spec Maths Prelims SE Page xi Sunday, July 13, 2003 11:28 AM
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