M
ATHS
B
M
ATHS
B
11
11
Y
YEAR
EAR
NICK SIMPSON ROBERT ROWLAND
FOR QUEENSLAND
MATHS
Quest
MATHS
Quest
Quest
First published 2001 by John Wiley & Sons Australia, Ltd 33 Park Road, Milton, Qld 4064
Offices also in Sydney and Melbourne
Typeset in 10.5/12.5pt Times
© John Wiley & Sons Australia, Ltd
National Library of Australia Cataloguing-in-Publication data
Simpson, Nick.
Maths quest: Mathematics B, Year 11 for Queensland.
ISBN 0 7016 3626 2.
1. Mathematics. 2. Mathematics — Problems, exercises, etc. I. Rowland, Robert, 1963–. II. Title. (Series: Maths quest series).
510
All rights reserved. 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 permission of the publisher.
Edited by Sharon Nevile, Merv Littmann and Jennifer Wright
Illustrated by the Wiley Art Department
Cover photograph, internal design images and CD-ROM label: © 2001 PhotoDisc Inc.
Printed in Singapore by Craft Print International Ltd
Contents
Introduction vii Acknowledgements ix
CHAPTER 1
Modelling using linear
functions 1
Introduction 2
Solving linear equations 3
Exercise 1A 6
Rearrangement and substitution 7
Exercise 1B 10
Career profile: Rick Morris — Vigneron 13
Gradient of a straight line 14
Exercise 1C 17
Equations of the form y=mx +c 22
Exercise 1D 22
Sketching linear graphs using intercepts 25
Exercise 1E 27
Simultaneous equations 28
Exercise 1F 33
Formula for finding the equation of a straight line 34
Exercise 1G 36
Linear modelling 38
Exercise 1H 41
Summary 43
Chapter review 44
CHAPTER 2
Relations and functions 49
Introduction 50 Set notation 52
Exercise 2A 54
Relations and graphs 55
Exercise 2B 58
Domain and range 61
Exercise 2C 65
Investigation — Interesting relations 67
Types of relations (including functions) 67
Exercise 2D 70
Function notation 72 Special types of function 73
Exercise 2E 75
Investigation — A special relation 78
Circles 79
Exercise 2F 81
Functions and modelling 83
Exercise 2G 84
Summary 86
Chapter review 88
CHAPTER 3
Other graphs and
modelling 95
Introduction 96
The parabola (turning point form) 99
Exercise 3A 103
Investigation — Plotting quadratic equations 105
The cubic function of the form
y=a(x−h)3+k 106
Exercise 3B 110
Investigation — Plotting cubic equations 112
The hyperbola 113
Exercise 3C 117
The square root function 120
Exercise 3D 123
The absolute value function 125
Career profile: Janet Williams — Studio Manager/Graphic Designer 127
Exercise 3E 128
Addition of ordinates 129
Exercise 3F 132
Modelling 133
Exercise 3G 138
Investigation — Goal accuracy 142
Summary 143
Chapter review 145
CHAPTER 4
Triangle trigonometry 149
Introduction 150
Calculating trigonometric ratios 150
Investigation — Looking at the tangent ratio 150
Investigation — Looking at the sine ratio 152 Investigation — Looking at the cosine
ratio 153
Exercise 4A 155
Finding an unknown side 155
Exercise 4B 159
Finding angles 162
Exercise 4C 164
iv
Applications of right-angled triangles 167
Exercise 4D 171
Investigation — Fly like a bird 172
The sine rule 173
Investigation — Derivation of the sine rule 173
Exercise 4E 177
Investigation — Bearing east and west 179
Exercise 4F 181
The cosine rule 183
Exercise 4G 185
Career profile: Richard Kennedy — Site Engineer 187
Summary 188
Chapter review 189
CHAPTER 5
Graphing periodic
functions 193
Introduction 194 Period and
amplitude of a periodic function 194
Investigation — Temperature and tide 195
Exercise 5A 198
Investigation — Rhythm of life 200 Investigation — Ferris wheeling 200
Radian measure 200
Investigation — Finding a radian 200
Exercise 5B 202
Exact values 203
Exercise 5C 210
Symmetry 211
Exercise 5D 215
Trigonometric graphs 216
Investigation — The effect of 2 219
Exercise 5E 223
Investigation — How high? 226
Applications 227
Exercise 5F 229
Summary 232
Chapter review 234
CHAPTER 6
Trigonometric equations 237
Introduction 238
Simple trigonometric equations 238
Exercise 6A 241
Equations using radians 242
Exercise 6B 244
Further trigonometric equations 245
Exercise 6C 247
Investigation — Fishing 247
Identities 247
Exercise 6D 251
Investigation — Further trigonometric identities 252
Using the Pythagorean identity 253
Exercise 6E 254
Summary 255
Chapter review 256
CHAPTER 7
Exponential and logarithmic
functions 257
Introduction 258 Index laws 258
Exercise 7A 262
Negative and rational powers 264
Exercise 7B 267
Indicial equations 268
Exercise 7C 271
Investigation — Simulating radioactivity 272
Graphs of exponential functions 274
Exercise 7D 276
Investigation — A world population model 277
Investigation — Bode’s Law 278
Logarithms 279
Exercise 7E 281
Solving logarithmic equations 283
Exercise 7F 285
Investigation — Logarithmic graphs 286 Investigation — The slide rule 287
Applications of exponential and logarithmic functions 289
Exercise 7G 291
Investigation — The decibel 293 Investigation — The Richter scale 294
Summary 295
Chapter review 297
CHAPTER 8
Applications of exponential
and logarithmic functions in
financial mathematics 301
Introduction 302
Geometric sequences 302
Exercise 8A 306
Geometric series 308
v
Investigation — Crossing the road 313
Introduction to growth and decay 314 Growth and decay functions 314
Exercise 8C 323
Compound interest formula 329
Exercise 8D 334
Reducing balance loans 339 Loan schedules 340
Exercise 8E 345
Investigation — Loan schedules using spreadsheets 350
Investigation — Spreadsheets and investing for the future 351
The annuities formula 352
Exercise 8F 358
Investigation — Buying a home 363
Summary 364
Chapter review 366
CHAPTER 9
Presentation of data 371
Introduction 372
Types of variables and data 372
Investigation — Types of data 374
Exercise 9A 375
Collection of data 376
Investigation — Gallup poll 376 Investigation — Identifying the target population 377
Exercise 9B 381
Investigation — Census or sample? 383
Bias 383
Investigation — Biased sampling 384 Investigation — Women and work 384 Investigation — Cost of a house 384
Displaying data 385 Stem plots 385
Exercise 9C 388
Frequency histograms and bar charts 390
Exercise 9D 394
Investigation — Segmented bar chart 394 Investigation — Looking at cost 395 Investigation — Using a database 395
Describing the shape of stem plots and histograms 396
Exercise 9E 398
Cumulative data 401
Investigation — A different display 404
Exercise 9F 405
Summary 409
Chapter review 411
CHAPTER 10
Summary statistics 413
Introduction 414
Measures of central tendency 415
Exercise 10A 421
Investigation — Mean and median amount of soft drink 425
Range and interquartile range 426
Exercise 10B 430
Investigation — Range of soft drink amounts 432
The standard deviation 432
Exercise 10C 435
Investigation — Standard deviation of soft drink amounts 437
Boxplots 438
Exercise 10D 442
Back-to-back stem plots 445
Exercise 10E 448
Parallel boxplots 450
Exercise 10F 452
Summary 454
Chapter review 456
CHAPTER 11
Introduction to
probability 461
Introduction 462
Informal description of chance 462
Exercise 11A 465
Investigation — What will the weather be? 467
Single event probability 467
Exercise 11B 470
Investigation — Comparing theoretical probabilities with experimental results 473 Investigation — Experimental or
theoretical? 474
Relative frequency 474
Exercise 11C 476
Investigation — Researching relative frequencies 479
Investigation — Applying relative frequency 479
Modelling probability 479
Exercise 11D 482
Investigation — Random choice 483
Long-run proportion 483
Exercise 11E 488
Summary 490
Chapter review 491
vi
CHAPTER 12
Rates of change 493
Introduction 494 Constant rates 494
Exercise 12A 496
Variable rates 499
Exercise 12B 500
Average rates of change 502
Exercise 12C 504
Instantaneous rates 508
Exercise 12D 510
Motion graphs 513
Exercise 12E 516
Relating the gradient function to the original function 521
Exercise 12F 522
Relating velocity–time graphs to position– time graphs 523
Exercise 12G 525
Rates of change of polynomials 528
Exercise 12H 531
Career Profile: Roger Farrer — Data Manager 533
Summary 534
Chapter review 536
CHAPTER 13
Differentiation and
applications 541
Introduction 542 The limit 543
Exercise 13A 546
Investigation — Sneaking up on a limit 548
Limits of discontinuous, rational and hybrid functions 548
Investigation — Dirichlet’s function 550
Exercise 13B 551
Differentiation using first principles 553
Investigation — Secants and tangents 554
Exercise 13C 556
Finding derivatives by rule 557
Exercise 13D 560
Investigation — Graphs of derivatives 562
Rates of change 563
Exercise 13E 566
Solving maximum and minimum problems 570
Exercise 13F 573
Investigation — When is a maximum not a maximum? 575
Summary 576
Chapter review 577
Answers 581
Introduction
Maths Quest Maths B Year 11 for Queensland is one of the exciting new
Maths Quest resources specifically designed for the Queensland senior mathematics syllabuses beginning in 2002. Written and compiled by practising Maths B teachers, Maths Quest breaks new ground in mathematics textbook publishing.
This resource contains: • a student textbook • a student CD-ROM.
Student textbook
Full colour is used throughout to produce clearer graphs and headings, to provide bright, stimulating photographs and to make navigation through the text easier.
Clear, concise theory sections contain worked examples, graphics calculator tips, highlighted important text and remember boxes.
Worked examples in a Think/Write format provide a clear explanation of key steps and suggest how solutions can be presented.
Exercises contain many carefully graded skills and application problems, including multiple-choice questions. Cross-references to relevant worked examples appear beside the first ‘matching’ question throughout the exercises.
Investigations, often suggesting the use of technology, provide further learning opportunities.
Each chapter concludes with asummary and chapter review exercise contain-ing questions that help consolidate students’ learncontain-ing of new concepts.
Technology is fully integrated within the resource. As well as use of graphics calculators, the Maths Quest for Queensland seriesfeatures computer algebra systems, spreadsheets, dynamic geometry software and several graphing packages. Not only does the text promote these technologies as learning tools, but demonstration versions of the programs (with the exception of Microsoft Excel) are also included, as well as hundreds of supporting files on the free accompanying CD-ROM.
Student CD-ROM
The accompanying CD-ROM contains the entire student textbook plus additional exercises. Students may work from the CD on laptops, school or home computers, and cut and paste material for revision or assignments. Clearly labelled icons within the electronic version of the text hyperlink to hundreds of technology files for programs such as Mathcad, Excel and Cabri Geometry to allow further exploration of ‘what if’ scenarios.
TI-83 graphics calculator programs can be downloaded to students’ calculators using the Graphlink software provided.
F O RQ U E E N
S L A N D
F O RQ U E E N
S L A N D MATHS Quest
MATHS QuestQuest YYEAREAR
11
11
M
ATHS
B
M
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B
NICK SIMPSON
ROB ROWLAND
IN
TERACTIVE
C
D- ROM
EXCE
L Spreadsheet
Mathc
ad
Grap
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Cabr
i Geometry
viii
Casio graphics calculator tips are also included on the CD-ROM.
WorkSHEET and Test yourself icons link to editable Word 97 documents that may be completed on screen, or printed and completed later.
SkillSHEET icons link to printable pages that contain additional examples and problems designed to help students revise required concepts.
Minimum system requirements
Windows 95/98 or NT Macintosh
Processor: Pentium Macintosh OS 7.6 CD-ROM drive speed: 4x Processor: PowerPC 16 MB RAM CD-ROM drive speed: 4x speakers 16 MB RAM
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 for
the 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
Trouble-shooting
If you have problems with the operation of this CD-ROM:
• Check that you have the right equipment (see Minimum system requirements).
• Visit www.jaconline.com.au to check if the answer to your problem is provided under ‘Frequently Asked Questions’ in the ‘Contact Us’ section. • Either email or write to John Wiley & Sons Australia, Ltd explaining the
problem, and providing details of the type of computer and the amount of RAM you have, the processor type and the CD-ROM speed. If you return the disk, please package it appropriately to protect it during transit. Email: [email protected]
Address: Multimedia Assistant
John Wiley & Sons Australia, Ltd PO Box 1226
MILTON QLD 4064
Work
SHEET
CASIO
3.2
test
test
CHAPTER
y
yourselfourself
test
yyourselfourselfAcknowledgements
The authors and publisher would like to thank the following copyright holders, organisations and individuals for their assistance and for permission to reproduce copyright material in this book.
Illustrative material
• © 2001 Corbis Corporation (pp. 93, 140, 182, 190, 231[lower], 376, 451, 460[lower]); © Creative Cohesions p. 197; • © Digital Stock/Corbis Corporation (pp. 1, 14[two], 24, 52[mountains], 57, 258[upper], 349, 563[right], 573); • © 2001 Digital Vision (pp. 496, 541, 563[left]); • © Getty Images/Tony Stone (pp. 413, 414); • © 2001 Image Disk (pp. 199, 492, 507); • Malcolm Cross (pp. 187, 533, 568); • © Copyright 2001 PhotoDisc Inc. (pp. 12, 16, 21(two), 41, 42(two), 48, 49, 50[two], 51[two], 52[five], 55, 82, 84, 85[two], 94, 95, 96, 141, 142, 148, 149, 150, 161, 166[two], 172, 178, 179, 186, 191[two], 192, 193, 194, 195, 203, 216, 229, 230[two], 231[upper], 236, 237, 257, 258[lower], 272, 273, 277, 278, 287, 290, 291, 292[two], 294, 300[two], 301, 302[two], 313, 318, 339, 345, 346, 348, 359[two], 362, 363[two], 369, 370, 374, 380, 382, 383, 384, 385, 394, 397, 402, 406, 407, 412, 416, 418, 422, 425, 428, 444, 445, 449, 452, 453[two], 459, 460[upper], 462, 470, 478, 479, 493, 511, 512, 517, 521, 527, 532[two], 540, 564, 565, 566, 567, 575, 580); • © 2001 Photo Essentials (pp. 33, 228, 491); • Photograph courtesy Rick Morris p. 13; • © 2001 Stockbyte (pp. 415, 461, 471); • Reproduced with permission of Victorian Police p. 543.
Software
The authors and publisher would like to thank the following software providers for their assistance and for permission to use their materials. However, the use of such 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 the companies listed below.
Texas Instruments 83 and 89 Graphlink software
Material reproduced with permission of the publisher. © Texas Instruments Incorporated.
TI 83 and 89 Graphlink Software available from Texas Instruments Web: http://education.ti.com
Note: A Graphlink cable can be purchased from educational booksellers or cal-culator 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
x
Graphmatica*
Reproduced with permission of kSoft, Inc.
345 Montecillo Dr., Walnut Creek, CA 94595–2654 e-mail: [email protected]
Web: www.pair.com/ksoft
Software included is for evaluation purposes only. The user is expected to regis-ter shareware if use exceeds 30 days. Order forms are available at www.pair.com/ksoft/register.txt
Cabri Geometry II
Reproduced with permission of Cabri. Leibniz
Cabri-géomètre 46, avenue Félix Viallet
38031 Grenoble Cedex FRANCE Web: www.cabri.net
Distributed by AAMT (Australian Association of Mathematics Teachers) Phone: (08) 8363 0288
Fax: (08) 8362 9288
e-mail: [email protected]
Web: www.aamt.edu.au
GrafEq and Poly*
Evaluation copies of GrafEq™ and Poly™ have been included with permission from Pedogoguery Software.
e-mail: [email protected]
Web: www.peda.com
Equation Grapher with 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 the United States and/or other countries.
Every effort has been made to trace the ownership of copyright material. Infor-mation that will enable the publisher to trace the copyright holders or to rectify any error or omission in subsequent reprints will be welcome. In such cases, please contact the Permission Section of John Wiley & Sons Australia, who will arrange for the payment of the usual fee.
*GrafEq, Poly, Graphmatica, Equation Grapher with Regression Analyser and Mathcad Explorer student edition are now available in Australia from:
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Fax: (03) 9722 2545
