Further SAMPLEKevin McMenamin Mathematics · P1: FXS/ABE P2: FXS 9780521740517agg.xml CUAT013-EVANS September 7, 2008 11:27 i ESSENTIAL Further Mathematics Third edition PETER JONES
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
P1: FXS/ABE P2: FXS9780521740517agg.xml CUAT013-EVANS September 7, 2008 11:27
i
ESSENTIALFurther
MathematicsThird edition
PETER JONESMICHAEL EVANS
KAY LIPSON
TI-Nspire and Casio ClassPad materialprepared in collaboration with
First published 1998Reprinted 1998Second edition 1999Reprinted 2000, 2001, 2002, 2003, 2005Third edition 2005Reprinted 2006
Cover designed by Modern Art Production GroupText designed by Sylvia WitteTypeset in India by TechbooksPrinted in China through Everbest Printing Company Pty Ltd
National Library of Australia Cataloguing in Publication dataJones, Peter, 1943-.Essential further mathematics.3rd ed.ISBN-13 978-0-521-74051-7 paperbackISBN-10 0-521-61328-0 paperback1. Mathematics – Problems, exercises, etc. I. Evans,Michael (Michael Wyndham). II. Lipson, Kay. III. Title
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:
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.
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.
All Victorian Curriculum and Assessment Authority material copyright VCAA. Reproduced by permission ofthe Victorian Curriculum and Assessment Authority Victoria, Australia.Disclaimer: This publication is independently produced for use by teachers and students. Although referenceshave been reproduced with permission of the VCAA the publication is in no way connected with or endorsedby the VCAA.
P1: FXS/ABE P2: FXS9780521740517agg.xml CUAT013-EVANS September 7, 2008 11:27
Contents
Acknowledgements xiv
CORE
CHAPTER 1 — Organising and displaying data 1
1.1 Classifying data 11.2 Organising and displaying categorical data 31.3 Organising and displaying numerical data 81.4 What to look for in a histogram 201.5 Stem-and-leaf plots and dot plots 26
P1: FXS/ABE P2: FXS9780521740517agg.xml CUAT013-EVANS September 7, 2008 11:27
Contents vii
MODULE 2 — Geometry and trigonometry
CHAPTER 12 — Geometry 360
12.1 Properties of parallel lines – a review 36012.2 Properties of triangles – a review 36212.3 Properties of regular polygons – a review 36412.4 Pythagoras’ theorem 36712.5 Similar figures 37112.6 Volumes and surface areas 37512.7 Areas, volumes and similarity 382
Key ideas and chapter summary 387Skills check 389Multiple-choice questions 390
CHAPTER 13 — Trigonometry 392
13.1 Defining sine, cosine and tangent 39213.2 The sine rule 39613.3 The cosine rule 40113.4 Area of a triangle 404
Key ideas and chapter summary 406Skills check 407Multiple-choice questions 408
CHAPTER 14 — Applications of geometry andtrigonometry 410
14.1 Angles of elevation and depression, bearings, andtriangulation 410
14.2 Problems in three dimensions 41714.3 Contour maps 421
P1: FXS/ABE P2: FXS9780521740517agg.xml CUAT013-EVANS September 7, 2008 11:27
viii Contents
MODULE 3 — Graphs and relations
CHAPTER 16 — Constructing and interpreting lineargraphs 441
16.1 The gradient of a straight line 44116.2 The general equation of a straight line 44316.3 Finding the equation of a straight line 44516.4 Equation of a straight line in intercept form 44916.5 Linear models 45016.6 Simultaneous equations 45216.7 Problems involving simultaneous linear
equations 45616.8 Break-even analysis 458
Key ideas and chapter summary 460Skills check 461Multiple-choice questions 461
CHAPTER 17 — Graphs 465
17.1 Line segment graphs 46517.2 Step graphs 46817.3 Non-linear graphs 47017.4 Relations of the form y = kxn for
n = 1, 2, 3, −1, −2 47217.5 Linear representation of non-linear relations 475
23.1 Introduction and definitions 61423.2 Planar graphs and Euler’s formula 61923.3 Complete graphs 62223.4 Euler and Hamilton paths 62323.5 Weighted graphs 626
What is the difference between categorical and numerical data?
What is a frequency table, how is it constructed and when is it used?
What is the mode and how do we determine its value?
What are bar charts, histograms, stem plots and dot plots? How are they
constructed and when are they used?
How do you describe the features of bar charts, histograms and stem plots when
writing a statistical report?
1.1 Classifying dataStatistics is a science concerned with understanding the world through data. The first step in
this process is to put the data into a form that makes it easier to see patterns or trends.
Some dataThe data contained in Table 1.1 is part of a larger set of data collected from a group of
university students.
Table 1.1 Student data
Height Weight Age Sex Plays sport Pulse rate(cm) (kg) (years) M male 1 regularly (beats/min)
F female 2 sometimes3 rarely
173 57 18 M 2 86179 58 19 M 2 82167 62 18 M 1 96195 84 18 F 1 71173 64 18 M 3 90184 74 22 F 3 78175 60 19 F 3 88140 50 34 M 3 70
Source: www.statsci.org/data/oz/ms212.html. Used with permission.
1
Chapter 9 – Arithmetic and geometric sequences 237
How to use a graphics calculator to generate the terms of an arithmetic sequence on theHome screen
Generate the first five terms of the arithmetic sequence: 2, 7, 12, 17, 22, . . .
Steps1 Start on the Home screen. Clear. Enter the value of the
first term 2. Press Í.
2 The common difference for this sequence is 5. So, type
in + 5. Press Í. The second term in the sequence,
7, is generated.
3 Pressing Í again generates the next term, 12.
4 Pressing Í again generates the next term, 17.
Keep pressing Í until the required number
of terms is generated.
Being able to recognise an arithmetic sequence is another skill that you need to develop. The
key idea here is that the successive terms in an arithmetic sequence differ by a constant amount
(the common difference).
Example 1 Testing for an arithmetic sequence
a Is the sequence 20, 17, 14, 11, 8, . . . arithmetic?
Solution
Strategy: Subtract successive terms in the sequence to see whether they differ by a constant
amount. If they do, the sequence is arithmetic.
1 Write down the terms of the sequence.
2 Subtract successive terms.
20, 17, 14, 11, 8, . . .
17− 20= −314− 17 = −311− 14= −3 and so on
3 Write down your conclusion Sequence is arithmetic as terms differ by aconstant amount.
Chapter 4 — Displaying and describing relationships between two variables 107
Clearly, traffic volume is a very good predictor of carbon monoxide levels in the air. Knowing
the traffic volume will enable us to predict carbon monoxide levels with a high degree of
accuracy. This contrasts with the next example, which concerns the ability to predict
mathematical ability from verbal ability.
Example 3 Calculating and interpreting the coefficient of determination
Scores on tests of verbal and mathematical ability are linearly related with:
rmathematical, verbal = +0.275
Determine the value of the coefficient of determination, write it in percentage terms, and
interpret. In this relationship, mathematical ability is the DV.
Solution
The coefficient of determination is:
r 2 = (0.275)2 = 0.0756 . . . or 0.076× 100= 7.6%
Therefore, only 7.6% of the variation observed in scores on the test of mathematical ability canbe explained by the variation in scores obtained on the test of verbal ability.
Clearly, scores on the verbal ability test are not good predictors of the scores on the
mathematical ability test; 92.4% of the variation in mathematical ability is explained by other
factors.
Exercise 4G
1 For each of the following values of r, calculate the value of the coefficient of determination
and convert to a percentage (correct to one decimal place).
a r = 0.675 b r = 0.345 c r = −0.567 d r = −0.673 e r = 0.124
2 a For the relationship described by the scatterplot
shown opposite, the coefficient of determination = 0.8215.
Determine the value of the correlation coefficient r
(correct to three decimal places).
b For the relationship described by the scatterplot shown
opposite, the coefficient of determination = 0.1243.
Determine the value of the correlation coefficient r
(correct to three decimal places).
a vibrant full colour text with aclear layout that makes mathsmore accessible for students
‘Using a graphics calculator’boxes within chapters explainhow to do problems using the TI-83/Plus and TI-84 graphicscalculators, and include screenshots to further assist students
a wealth of worked examplesthat support theory explanationswithin chapters
carefully graduated exercisesthat include a number of easierlead-in questions to providestudents with a greateropportunity for immediatesuccess
chapter reviews that include keyideas and chapter summary andskills check lists, and multiple-choice and extended-responsequestions
chapter summaries at the end ofeach chapter provide studentswith a coherent overview
Appendices that include a TI-83/84 Plus help guide and step-by-step worked examples usingTI-89 Graphics Calculators
a comprehensive glossary ofmathematical terms with pagereferences to assist in the ‘openbook’ exam
revision chapters to helpconsolidate student knowledge
Review
Chapter 6 — Data transformation 173
Key ideas and chapter summary
Data transformation This means changing the scale on either the x or y axis. It is
performed when a residual plot shows that the underlying
relationship in a set of bivariate data is clearly non-linear.
x2 or y2 transformation The square transformation stretches out the upper end of
the scale on an axis.
log x or log y transformation The log transformation compresses the upper end of the
scale on an axis.1
xor
1
ytransformation The reciprocal transformation compresses the upper end
of the scale on an axis to a greater extent than the log
transformation.
Residual plots Residual plots are used to assess the effectiveness of each
data transformation.
Coefficient of determination (r2) The transformation which results in a linear relationship
and which has the highest value of the coefficient of
determination is considered to be the best transformation.
The circle of transformations The circle of transformations provides guidance in
choosing the transformations that can be used to linearise
various types of scatterplots. See page 166.
Skills check
Having completed this chapter you should be able to:
recognise which of the x2, log x,1
x, y2, log y or
1
ytransformations might be used to
linearise a bivariate relationship
apply each of these transformations to a data set
use residual plots and the coefficient of determination r2 to decide which
transformation gives the best model for the relationship
use the transformed variable as part of a regression analysis to give a model for the
relationship
Multiple-choice questions
1 The missing data values, a and b, in the table are:
value 1 2 3 4
(value)2 a 4 9 16
log(value) 0 b 0.477 0.602
A a = 0, b = 0.5 B a = 1, b = 0.5 C a = 1, b = 0.301
D a = 1, b = 0.602 E a = 1, b = 0.693 Glossary
AActivity (CPA): [p. 595] A task to be completed aspart of a project. Activities are represented by theedges in the project diagram.
Acute angle: An angle less than 90◦.
Adjacency matrix: [pp. 561, 586] A square matrixshowing the number of edges joining each pair ofvertices in a graph.
Algorithm: A step-by-step procedure for solving aparticular problem that involves applying the sameprocess repeatedly. Examples include Prim’salgorithm and the Hungarian algorithm.
Allocation problem: [p. 602] A problem thatinvolves finding the best way to match a givennumber of objects (people, machines, etc.) to a givennumber of activities.
Alternate angles: [p. 320]
Angle of depression: [p. 371] The angle between thehorizontal and a direction below the horizontal.
Angle of elevation: [p. 371] The angle between thehorizontal and a direction above the horizontal.
Angle sum of a triangle: [p. 322] In triangleABC, <A + <B + <C = 180◦.
Annuity equation: [pp. 531, 534] The equation:
A = PRn − Q(Rn − 1)
R − 1, where R = 1 + r
100.
The annuity equation can be used to determineeither the amount owed on a reducing balance loanwith regular repayments, or the value of aninvestment (annuity) with regular payments orwithdrawals.
Area of a triangle: [p. 364] See also Heron’sformula.
Arithmetic sequence: [p. 236] A sequence whosesuccessive terms differ by a constant amount (d)called the common difference. Given the value of thefirst term in an arithmetic sequence (a), there arerules for finding the nth term and the sum of the firstn terms.
Assignment problem: [p. 602] See allocationproblem.
BBar chart: [p. 4] A statistical graph used to displaythe frequency distribution of categorical data.
Bearing: [p. 371] See true bearing.
Bipartite graph (bigraph): [p. 562] A graph whoseset of vertices can be split into two subsets, X and Y,in such a way that each edge of the graph joins avertex in X and a vertex in Y.
Bivariate data: [p. 85] Data associated with tworelated variables.
Book value: [p. 526] The value of an item afterdepreciation.
Box plot (standard): [p. 41] A graphicalrepresentation of a five number summary.
Box plot (with outliers): [p. 42] A modified form ofthe standard box plot in which possible outliers areshown. Possible outliers are defined as data valuesgreater than Q3 + 1.5 × IQR and less thanQ1 − 1.5 × IQR.
Break-even analysis: [p. 418] Finding the pointwhere the revenue of a business first equals the costsof running the business. Past this point, the businessis running at a profit: profit = revenue − costs.
CCapacity of a cut: [p. 592] The sum of thecapacities (weights) of the edges directed from X to Ythat the cut passes through.
Categorical data: [p. 2] Data obtained whenclassifying or naming some quality or attribute, forexample: place of birth, hair colour.
Causal relationship: [p. 108] When the change inone variable in a relationship can be said to be thedirect result of a change in a second variable, therelationship is said to be causal. A high correlationbetween two variables does not necessarily mean thatthe variables are causally related.
Gives an extra hint in extended-responsequestions that a numerical approach isrequired
Indicates that there is an explanation inAppendix B as to how this example may bedone using TI-89 Graphics Calculators
Explaining icons in the book ...
What teachers and students will find on the CD-ROMs ...
Calculator icons
Indicates that a skillsheet is available toprovide further practice and examples in thisarea. If students are having difficulty they canapproach their teacher who can access thismaterial on the Teacher CD-ROM.
Links to Teacher CD-ROM
Live links to interactive files on the Student CD-ROM.
Links to Student CD-ROM
The Essential Further Mathematics Teacher CD-ROMcontains a wealth of time-saving assessment andclassroom resources including:
modifiable chapter tests and answers containingmultiple-choice and short-answer questionschapter review assignments with extendedproblems that can be given to students in class orcan be completed at homeprintable versions of the multiple-choice questionsfrom the Student CD-ROMprint-ready skillsheets to revise the prerequisiteknowledge and skills required for the chaptereditable Exam Question Sets from which teacherscan create their own exams.
Teacher CD-ROM
The textbook includes a Student CD-ROM that containsa PDF of the book, interactive multiple-choice questionsand unique drag-and-drop activities. Technologyapplets such as PowerPoint and Excel activites are alsoincluded.
Student CD-ROM
Additional resources ...
The Essential Further Mathematics Third EditionSolutions Supplement book provides solutions to theextended-response questions, highlighting the processas well as the answer.
Solutions Supplements
Teacher WebsiteThis dynamic website enables teachers to interact witheach other through teacher forums, to send questionsto the authors and to obtain updates.
Student WebsiteThis free student website contains a student forumallowing keen mathematics students to interact witheach other, as well as interactive tests and links toother useful sites.