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ESSENTIALMathematical
Methods 1 & 2 CASMICHAEL EVANS
KAY LIPSONDOUG WALLACE
TI-Nspire and Casio ClassPad materialprepared in collaboration with
Jan HonnensDavid Hibbard
Cambridge University Press • Uncorrected Sample Pages • 2008 © Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
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C A M B R I D G E U N I V E R S I T Y P R E S S
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press477 Williamstown Road, Port Melbourne, VIC 3207, Australia
www.cambridge.edu.auInformation on this title: www.cambridge.org/9780521740524
C© Michael Evans, Kay Lipson & Douglas Wallace, 2008
First published 2008
Designed, Typeset & Illustrated by AptaraPrinted in China by Printplus
For cataloguing data please visit the Libraries Australia website:http://librariesaustralia.nla.gov.au
Essential Mathematical Methods CAS 1&2 with Student CD-ROM TIN/CP VersionISBN 978-0-521-74052-4 paperback
Reproduction and Communication for educational purposesThe Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of thispublication, whichever is the greater, to be reproduced and/or communicated by any educational institution forits educational purposes provided that the educational institution (or the body that administers it) has given aremuneration notice to Copyright Agency Limited (CAL) under the Act.
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Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external orthird-party internet websites referred to in this publication and does not guarantee that any content on suchwebsites is, or will remain, accurate or appropriate.
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Contents
Introduction xi
CHAPTER 1 — Reviewing Linear Equations 1
1.1 Linear equations 11.2 Constructing linear equations 71.3 Simultaneous equations 101.4 Constructing and solving simultaneous linear
equations 141.5 Solving linear inequations 171.6 Using and transposing formulae 19
Chapter summary 25Multiple-choice questions 25Short-answer questions (technology-free) 26Extended-response questions 27
CHAPTER 2 — Linear Relations 29
2.1 The gradient of a straight line 292.2 The general equation of a straight line 322.3 Finding the equation of a straight line 362.4 Equation of a straight line in intercept form and
sketching graphs 412.5 Linear models 432.6 Problems involving simultaneous linear
models 462.7 The tangent of the angle of slope and
perpendicular lines 492.8 The distance between two points 532.9 Midpoint of a line segment 542.10 Angle between intersecting lines 55
Chapter summary 57Multiple-choice questions 58Short-answer questions (technology-free) 58Extended-response questions 59
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iv Contents
CHAPTER 3 — Matrices 66
3.1 Introduction to matrices 663.2 Addition, subtraction and multiplication
by a scalar 713.3 Multiplication of matrices 763.4 Identities, inverses and determinants for 2 × 2
matrices 813.5 Solution of simultaneous equations using
matrices 85Chapter summary 89Multiple-choice questions 90Short-answer questions (technology-free) 91Extended-response questions 92
CHAPTER 4 — Quadratics 93
4.1 Expanding and collecting like terms 934.2 Factorising 984.3 Quadratic equations 1014.4 Graphing quadratics 1044.5 Completing the square 1084.6 Sketching quadratics in polynomial form 1114.7 The general quadratic formula 1154.8 Iteration 1194.9 The discriminant 1224.10 Solving quadratic inequations 1254.11 Solving simultaneous linear and quadratic
equations 1274.12 Determining quadratic rules 1304.13 Quadratic models 135
Chapter summary 139Multiple-choice questions 141Short-answer questions (technology-free) 142Extended-response questions 144
CHAPTER 5 — A Gallery of Graphs 148
5.1 Rectangular hyperbolas 1485.2 The truncus 1515.3 y = √
x = x1/2 152
5.4 Circles 154Chapter summary 158Multiple-choice questions 159Short-answer questions (technology-free) 161Extended-response questions 161
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Contents v
CHAPTER 6 — Functions, Relations andTransformations 163
6.1 Set notation and sets of numbers 1636.2 Relations, domain and range 1666.3 Functions 1716.4 Special types of functions and implied
domains 1776.5 Hybrid functions 1806.6 Miscellaneous exercises 1816.7 Inverse functions 1846.8 Translations of functions 1876.9 Dilations and reflections 1906.10 Combinations of transformations 1946.11 Functions and modelling exercises 197
Chapter summary 200Multiple-choice questions 202Short-answer questions (technology-free) 203Extended-response questions 204
CHAPTER 7 — Cubic and Quartic Functions 207
7.1 Functions of the form f: R → R,f (x) = a(x − h)n + k 208
7.2 Division of polynomials 2127.3 Factorisation of polynomials 2157.4 Factor theorem 2177.5 Solving cubic equations 2217.6 Graphs of cubic functions 2257.7 Solving cubic inequations 2297.8 Finding equations for given cubic graphs 2297.9 Graphs of quartic functions 2337.10 Finite differences for sequences generated by
polynomials 2347.11 Applications of polynomial functions 240
Chapter summary 244Multiple-choice questions 246Short-answer questions (technology-free) 247Extended-response questions 249
CHAPTER 8 — Applications of matrices and usingparameters 252
8.1 Systems of equations and using parameters 2528.2 Using matrices with transformations 261
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vi Contents
8.3 Using parameters to describe familiesof curves 266
8.4 Transformation of graphs of functions withmatrices 269Chapter summary 272Multiple-choice questions 274Short-answer questions (technology-free) 275Extended-response questions 276
CHAPTER 9 — Revision of Chapters 2–8 278
9.1 Multiple-choice questions 2789.2 Extended-response questions 283
CHAPTER 10 — Probability 287
10.1 Random experiments and events 28810.2 Determining empirical probabilities 29210.3 Determining probabilities by symmetry 29710.4 The addition rule 30210.5 Probability tables and Karnaugh maps 305
Chapter summary 312Multiple-choice questions 312Short-answer questions (technology-free) 314Extended-response questions 315
CHAPTER 11 — Conditional probability andMarkov chains 317
11.1 Conditional probability and themultiplication rule 317
11.2 Independent events 32411.3 Displaying conditional probabilities with
matrices 32911.4 Transition matrices and Markov chains 334
Chapter summary 342Multiple-choice questions 343Short-answer questions (technology-free) 344Extended-response questions 345
CHAPTER 12 — Counting Methods 347
12.1 Addition and multiplication principles 34712.2 Arrangements 35112.3 Selections 35512.4 Applications to probability 360
Chapter summary 363
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Contents vii
Multiple-choice questions 363Short-answer questions (technology-free) 364Extended-response questions 364
CHAPTER 13 — Discrete Probability Distributions andSimulation 366
13.1 Discrete random variables 36613.2 Sampling without replacement 37213.3 Sampling with replacement: the binomial
distribution 37513.4 Solving probability problems using
simulation 38413.5 Random number tables 387
Chapter summary 391Multiple-choice questions 391Short-answer questions (technology-free) 393Extended-response questions 394
CHAPTER 14 — Revision of Chapters 10–13 397
14.1 Multiple-choice questions 39714.2 Extended-response questions 400
CHAPTER 15 — Exponential Functions andLogarithms 406
15.1 Graphs of exponential functions 40715.2 Reviewing rules for exponents (indices) 41415.3 Rational exponents 41815.4 Solving exponential equations and
inequations 42015.5 Logarithms 42315.6 Using logarithms to solve exponential equations
and inequations 42715.7 Graph of y = loga x, where a > 1 43015.8 Exponential models and applications 433
Chapter summary 439Multiple-choice questions 439Short-answer questions (technology-free) 441Extended-response questions 442
CHAPTER 16 — Circular Functions 445
16.1 Measuring angles in degrees and radians 44516.2 Defining circular functions: sine and cosine 448
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viii Contents
16.3 Another circular function: tangent 45016.4 Reviewing trigonometric ratios 45116.5 Symmetry properties of circular functions 45216.6 Exact values of circular functions 45516.7 Graphs of sine and cosine 45816.8 Sketch graphs of y = a sin n(t ± ε) and
y = a cos n(t ± ε) 46416.9 Solution of trigonometric equations 46616.10 Sketch graphs of y = a sin n(t ± ε) ± b and
y = a cos n(t ± ε) ± b 47116.11 Further symmetry properties and the
Pythagorean identity 47416.12 The tangent function 47616.13 Numerical methods with a CAS calculator 48016.14 General solution of circular function
equations 48316.15 Applications of trigonometric functions 487
Chapter summary 489Multiple-choice questions 492Short-answer questions (technology-free) 493Extended-response questions 494
CHAPTER 17 — Revision of Chapters 15–16 496
17.1 Multiple-choice questions 49617.2 Extended-response questions 499
CHAPTER 18 — Rates of Change 502
18.1 Recognising relationships 50318.2 Constant rates of change 50718.3 Non-constant rate of change and
average rate of change 51018.4 Finding the gradient of a curve at a
given point 51518.5 Displacement, velocity and acceleration 522
Chapter summary 529Multiple-choice questions 529Short-answer questions (technology-free) 531Extended-response questions 532
CHAPTER 19 — Differentiation of Polynomials 536
19.1 The gradient of a curve at a point, andthe gradient function 537
19.2 The derived function 54219.3 Graphs of the derived or gradient function 552
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19.4 Limits and continuity 55719.5 When is a function differentiable? 563
Chapter summary 567Multiple-choice questions 568Short-answer questions (technology-free) 569Extended-response questions 570
CHAPTER 20 — Applications of Differentiation ofPolynomials 571
20.1 Tangents and normals 57120.2 Rates of change and kinematics 57420.3 Stationary points 58420.4 Types of stationary points 58820.5 Families of functions and transformations 59320.6 Applications to maximum and minimum and rate
problems 596Chapter summary 603Multiple-choice questions 604Short-answer questions (technology-free) 604Extended-response questions 605
CHAPTER 21 — Revision of Chapters 18–20 611
21.1 Multiple-choice questions 61121.2 Extended-response questions 616
CHAPTER 22 — Differentiation Techniques 621
22.1 Differentiating xn where n is anegative integer 621
22.2 The chain rule 62422.3 Differentiating rational powers (x
pq ) 628
22.4 The second derivative 63122.5 Sketch graphs 632
Chapter summary 636Multiple-choice questions 636Short-answer questions (technology-free) 637Extended-response questions 638
CHAPTER 23 — Integration 640
23.1 Antidifferentiation of polynomial functions 64023.2 Antidifferentiation of algebraic expressions with
rational exponents 64523.3 Applications to kinematics 64823.4 Area — the definite integral 651
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23.5 Numerical methods for finding areas 658Chapter summary 663Multiple-choice questions 664Short-answer questions (technology-free) 665Extended-response questions 666
CHAPTER 24 — Revision of Chapters 22–23 670
24.1 Multiple-choice questions 670
Glossary 673
Appendix A: Computer Algebra System (TI-Nspire) 680
Appendix B: Computer Algebra System 693
Answers 704
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Introduction
This book provides a complete course for Mathematical Methods (CAS) Units 1 and 2. It has
been written as a teaching text, with understanding as its chief aim and with ample practice
offered through the worked examples and exercises. All the work has been trialed in the
classroom, and the approaches offered are based on classroom experience.
The book contains five revision chapters. These provide multiple-choice questions and
extended-response questions. Use of a CAS calculator has been included throughout the text
and there is also an appendix which provides an introduction to the use of the calculator. The
use of matrices to describe transformations, solve systems of linear equations and in the study
of Markov sequences is fully integrated. The study of families of functions is also throughout
the text.
Extended-response questions that require a CAS calculator have been incorporated. These
questions are indicated by the use of a CAS calculator icon.
The TI-Nspire calculator instructions have been completed by Jan Honnens and the Casio
ClassPad instructions have been completed by David Hibbard.
The TI-Nspire instructions are written for operating system 1.4 but can be used with other
versions.
The Casio Classpad instructions are written for operating system 3 or above.
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