0521609976agg.xml cuau021-evans january 1, 1904 7:26 i ... · 7.7 solving cubic inequations 229 7.8...

P1: FXS/ABE P2: FXS

0521609976agg.xml CUAU021-EVANS January 1, 1904 7:26

i

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

SAMPLE

P1: FXS/ABE P2: FXS


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.

For details of the CAL licence for educational institutions contact:

Copyright Agency LimitedLevel 15, 233 Castlereagh StreetSydney NSW 2000Telephone: (02) 9394 7600Facsimile: (02) 9394 7601Email: [email protected]

Reproduction and Communication for other purposesExcept as permitted under the Act (for example a fair dealing for the purposes of study, research, criticism orreview) no part of this publication may be reproduced, stored in a retrieval system, communicated ortransmitted in any form or by any means without prior written permission. All inquiries should be made to thepublisher at the address above.

Student CD-ROM licencePlease see the file ‘licence.txt’ on the Student CD-ROM that is packed with this book.

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.


SAMPLE

P1: FXS/ABE P2: FXS


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


iiiCambridge University Press • Uncorrected Sample Pages • 2008 © Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

SAMPLE

P1: FXS/ABE P2: FXS


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 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


SAMPLE

P1: FXS/ABE P2: FXS


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 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 8 — Applications of matrices and usingparameters 252

8.1 Systems of equations and using parameters 2528.2 Using matrices with transformations 261


SAMPLE

P1: FXS/ABE P2: FXS


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 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 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


SAMPLE

P1: FXS/ABE P2: FXS


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 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 16 — Circular Functions 445

16.1 Measuring angles in degrees and radians 44516.2 Defining circular functions: sine and cosine 448


SAMPLE

P1: FXS/ABE P2: FXS


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 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 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


SAMPLE

P1: FXS/ABE P2: FXS


Contents ix

19.4 Limits and continuity 55719.5 When is a function differentiable? 563


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 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 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


SAMPLE

P1: FXS/ABE P2: FXS


x Contents

23.5 Numerical methods for finding areas 658Chapter summary 663Multiple-choice questions 664Short-answer questions (technology-free) 665Extended-response questions 666


24.1 Multiple-choice questions 670

Glossary 673

Appendix A: Computer Algebra System (TI-Nspire) 680

Appendix B: Computer Algebra System 693

Answers 704


SAMPLE

P1: FXS/ABE P2: FXS


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.

xiCambridge University Press • Uncorrected Sample Pages • 2008 © Evans, Lipson, Wallace TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

SAMPLE

P1: FXS/ABE P2: FXS



SAMPLE

0521609976agg.xml cuau021-evans january 1, 1904 7:26 i ... · 7.7 solving cubic inequations 229 7.8...

Documents