Specialists in mathematics publishing Mathematicsnharris17.wikispaces.com/file/view/IB Math SL Book...

for the international studentMathematics

Specialists in mathematics publishing

HAESE HARRIS PUBLICATIONS&

Mathematics SL

Marjut Menp

John Owen

Michael Haese

Robert Haese

Sandra Haese

Mark Humphries

for use withIB DiplomaProgramme

second edition

IB_SL-2edmagentacyan yellow black

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Y:\HAESE\IB_SL-2ed\IB_SL-2ed_00\001IB_SL-2_00.CDR Tuesday, 10 March 2009 9:53:51 AM PETER

MATHEMATICS FOR THE INTERNATIONAL STUDENTMathematics SL second edition

This book is copyright

Copying for educational purposes

Acknowledgements

Disclaimer

Marjut Menp B.Sc., Dip.Ed.John Owen B.Sc., Dip.T.

Robert Haese B.Sc.Sandra Haese B.Sc.Mark Humphries B.Sc.(Hons.)

Haese & Harris Publications3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIATelephone: +61 8 8355 9444, Fax: + 61 8 8355 9471Email:

National Library of Australia Card Number & ISBN 978-1-921500-09-1

Haese & Harris Publications 2009

Published by Raksar Nominees Pty Ltd.3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA

First Edition 20042005 three times , 2006, 2007, 2008 twice

Second Edition 2009

Cartoon artwork by John Martin. Artwork by Piotr Poturaj and David Purton.Cover design by Piotr Poturaj.Computer software by David Purton, Thomas Jansson and Troy Cruickshank.

Typeset in Australia by Susan Haese and Charlotte Sabel (Raksar Nominees).

Typeset in Times Roman 10 /11

The textbook and its accompanying CD have been developed independently of the InternationalBaccalaureate Organization (IBO). The textbook and CD are in no way connected with, orendorsed by, the IBO.

. Except as permitted by the Copyright Act (any fair dealing for thepurposes of private study, research, criticism or review), no part of this publication may bereproduced, 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.Enquiries to be made to Haese & Harris Publications.

: Where copies of part or the whole of the book are madeunder Part VB of the Copyright Act, the law requires that the educational institution or the bodythat administers it has given a remuneration notice to Copyright Agency Limited (CAL). Forinformation, contact the Copyright Agency Limited.

: While every attempt has been made to trace and acknowledge copyright, theauthors and publishers apologise for any accidental infringement where copyright has proveduntraceable. They would be pleased to come to a suitable agreement with the rightful owner.

: All the internet addresses (URLs) given in this book were valid at the time of printing.While the authors and publisher regret any inconvenience that changes of address may causereaders, no responsibility for any such changes can be accepted by either the authors or thepublisher.

Michael Haese B.Sc.(Hons.), Ph.D.

Reprinted (with minor corrections)

\Qw_ \Qw_\\"

[email protected]:

Reprinted (with minor corrections)2010


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Mathematics for the International Student: Mathematics SL has been written to embracethe syllabus for the two-year Mathematics SL Course, which is one of the courses of study inthe IB Diploma Programme. It is not our intention to define the course. Teachers areencouraged to use other resources. We have developed this book independently of theInternational Baccalaureate Organization (IBO) in consultation with many experiencedteachers of IB Mathematics. The text is not endorsed by the IBO.

The second edition builds on the strength of the first edition. Chapters are arranged to followthe same order as the chapters in our , making it easierfor teachers who have combined classes of SL and HL students.

Syllabus references are given at the beginning of each chapter. The new edition reflects theMathematics SL syllabus more closely, with several sections from the first edition beingconsolidated in this second edition for greater teaching efficiency. Topics such as Pythagorastheorem, coordinate geometry, and right angled triangle trigonometry, which appeared inChapters 7 and 10 in the first edition, are now in the Background Knowledge at thebeginning of the book and accessible as printable pages on the CD.

Changes have been made in response to the introduction of a calculator-free examinationpaper. A large number of questions have been added and categorised as calculator ornon calculator. In particular, the final chapter contains over 150 examination-stylequestions.

Comprehensive graphics calculator instructions are given for Casio fx-9860G, TI-84 Plus andTI- spire in an introductory chapter (see p. 17) and, occasionally, where additional help maybe needed, more detailed instructions are available as printable pages on the CD. Theextensive use of graphics calculators and computer packages throughout the book enablesstudents to realise the importance, application, and appropriate use of technology. No singleaspect of technology has been favoured. It is as important that students work with a pen andpaper as it is that they use their calculator or graphics calculator, or use a spreadsheet orgraphing package on computer.

This package is language rich and technology rich. The combination of textbook andinteractive Student CD will foster the mathematical development of students in a stimulatingway. Frequent use of the interactive features on the CD is certain to nurture a much deeperunderstanding and appreciation of mathematical concepts. The CD also offers forevery worked example. is accessed via the CD click anywhere on any workedexample to hear a teachers voice explain each step in that worked example. This is ideal forcatch-up and revision, or for motivated students who want to do some independent studyoutside school hours.

For students who may not have a good understanding of the necessary background knowledgefor this course, we have provided printable pages of information, examples, exercises, andanswers on the Student CD see Background knowledge (p. 12). To access these pages,click on the Background knowledge icon when running the CD.

The interactive features of the CD allow immediate access to our own specially designedgeometry software, graphing software and more. Teachers are provided with a quick and easyway to demonstrate concepts, and students can discover for themselves and re-visit whennecessary.

Mathematics HL (Core) second edition

n

Self Tutor

Self Tutor

FOREWORD

continued next page


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It is not our intention that each chapter be worked through in full. Time constraints may notallow for this. Teachers must select exercises carefully, according to the abilities and priorknowledge of their students, to make the most efficient use of time and give as thoroughcoverage of work as possible. Investigations throughout the book will add to the discoveryaspect of the course and enhance student understanding and learning. Many investigations aresuitable for portfolio assignments.

In this changing world of mathematics education, we believe that the contextual approachshown in this book, with the associated use of technology, will enhance the studentsunderstanding, knowledge and appreciation of mathematics, and its universal application.

We welcome your feedback.

Email:

Web:

EMM JTO PMH

RCH SHH MAH

[email protected]

www.haeseandharris.com.au

ACKNOWLEDGEMENTS

The authors and publishers would like to thank all those teachers who offered advice andencouragement on both the first and second editions of this book. Many of them read the pageproofs and offered constructive comments and suggestions. These teachers include: CameronHall, Paul Urban, Fran OConnor, Glenn Smith, Anne Walker, Malcolm Coad, Ian Hilditch,Phil Moore, Julie Wilson, David Martin, Kerrie Clements, Margie Karbassioun, BrianJohnson, Carolyn Farr, Rupert de Smidt, Terry Swain, Marie-Therese Filippi, Nigel Wheeler,Sarah Locke, Rema George, Mike Wakeford, Eddie Kemp, Pamela Vollmar, Mark Willis,Peter Hamer-Hodges, Sandra Moore, Robby Colaiacovo. To anyone we may have missed, weoffer our apologies.

The publishers wish to make it clear that acknowledging these individuals does not imply anyendorsement of this book by any of them, and all responsibility for the content rests with theauthors and publishers.


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USING THE INTERACTIVE STUDENT CD

The interactive CD is ideal for independent study.

Students can revisit concepts taught in class and undertake their ownrevision and practice. The CD also has the text of the book, allowingstudents to leave the textbook at school and keep the CD at home.

By clicking on the relevant icon, a range of interactive features can beaccessed:

: where additional help may be needed,

detailed instructions are available on the CD, as printable pages. Click on

the relevant icon for TI- spire, TI-84 Plus or Casio fx-9860G.

Graphics calculator instructions

Background knowledge (as printable pages)

Interactive links to spreadsheets, graphing and geometry software,computer demonstrations and simulations

Graphics calculator instructions

n

Self Tutor

INTERACTIVE

LINK

for the international studentMathematicsMathematics

Mathematics HLsecond edition

for use with IB Diploma Programme

Mathematics HLsecond edition

for use with IB Diploma Programme

2009

Haese & Harris Publicati

onsHaese & Harris Publi

cation

s

STUDENT CD-ROMSTUDENT CD-ROM

SLSL second editionsecond edition

TI- spiren

TI-84

Casio

Simply click on the (or anywhere in the example box) to access the workedexample, with a teachers voice explaining each step necessary to reach the answer.

Play any line as often as you like. See how the basic processes come alive usingmovement and colour on the screen.

Ideal for students who have missed lessons or need extra help.

Self Tutor

SELF TUTOR is an exciting feature of this book.

The icon on each worked example denotes an active link on the CD.Self Tutor

See , , p. 356Chapter 13 Lines and planes in space

Find a the vector b the parametric c the Cartesian equation of the line passing

through the point A(1, 5) with direction

32

.

a a =!OA =

15

and b =

32

But r = a + tb )

x

y

=

15

+ t

32

, t 2 R

b From a, x = 1 + 3t and y = 5 + 2t, t 2 R

c Now t =x 13

=y 52

) 2x 2 = 3y 15) 2x 3y = 13 fgeneral formg

Example 1 Self Tutor

A

R

a

r

O

& *


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6 TABLE OF CONTENTS

SYMBOLS AND NOTATION

USED IN THIS BOOK 10

BACKGROUND KNOWLEDGE 12

SUMMARY OF

CIRCLE PROPERTIES 12

SUMMARY OF

MEASUREMENT FACTS 14

GRAPHICS CALCULATOR

INSTRUCTIONS 17

1 FUNCTIONS 45

2 SEQUENCES AND SERIES 69

3 EXPONENTIALS 93

4 LOGARITHMS 119

5 GRAPHING AND

TRANSFORMING FUNCTIONS 143

A Surds and radicals CD

B Scientific notation (standard form) CD

C Number systems and set notation CD

D Algebraic simplification CD

E Linear equations and inequalities CD

F Modulus or absolute value CD

G Product expansion CD

H Factorisation CD

Another factorisation

technique CD

I Formula rearrangement CD

J Adding and subtracting algebraic fractions CD

K Congruence and similarity CD

L Pythagoras theorem CD

M Coordinate geometry CD

N Right angled triangle trigonometry CD

A Casio fx-9860G 18

B Texas Instruments TI-84 Plus 26

C Texas Instruments TI- Spire 35

A Relations and functions 46

B Function notation 49

C Domain and range 51

: Fluid filling functions 54

D Composite functions 55

E Sign diagrams 56

F The reciprocal function 60

G Asymptotes of other rational functions 61

: Finding asymptotes 61

H Inverse functions 62

Review set 1A 65

Review set 1B 66

Review set 1C 68

A Number patterns 70

B Sequences of numbers 71

C Arithmetic sequences 72

D Geometric sequences 76

E Series 82

: Von Kochs snowflake curve 89

Review set 2A 90

Review set 2B 90

Review set 2C 91

A Index notation 94

B Evaluating powers 95

C Index laws 96

D Rational indices 99

E Algebraic expansion and factorisation 101

F Exponential equations 104

G Graphs of exponential functions 105

: Exponential graphs 106

H Growth and decay 109

I The natural exponential 113

: Continuous compound

interest 113

Review set 3A 116

Review set 3B 117

Review set 3C 118

A Logarithms 120

B Logarithms in base 10 122

C Laws of logarithms 125

: Discovering the laws

of logarithms 125

D Natural logarithms 128

E Exponential equations using logarithms 131

F The change of base rule 133

G Graphs of logarithmic functions 134

H Growth and decay 137

Review set 4A 140

Review set 4B 140

Review set 4C 141

A Families of functions 144

: Function families 144

B Transformation of graphs 146

Review set 5A 151

Review set 5B 152

Review set 5C 153

Investigation:

Investigation 1

Investigation 2

Investigation

Investigation 1

Investigation 2

Investigation

Investigation

n

e

TABLE OF CONTENTS


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TABLE OF CONTENTS 7

6 QUADRATIC EQUATIONS AND

FUNCTIONS 155

7 THE BINOMIAL EXPANSION 189

8 THE UNIT CIRCLE AND RADIAN

MEASURE 197

9 NON-RIGHT ANGLED TRIANGLE

TRIGONOMETRY 217

10 ADVANCED TRIGONOMETRY 235

11 MATRICES 273

12 VECTORS IN 2 AND

3 DIMENSIONS 309

13 LINES AND PLANES IN SPACE 353

A Quadratic equations 157

B The discriminant of a quadratic 162

C Graphing quadratic functions 164

:

Graphing 164

:

Graphing 165

D Finding a quadratic from its graph 173

: Finding quadratic

functions 176

E Where functions meet 177

F Problem solving with quadratics 179

G Quadratic optimisation 182

: Sum and product of roots 185

Review set 6A 185

Review set 6B 186

Review set 6C 187

A Binomial expansions 190

: The binomial

expansion of , 191

B The binomial theorem 193

: The binomial coefficient 193

Review set 7 196

A Radian measure 198

B Arc length and sector area 200

C The unit circle and the basic

trigonometric ratios 203

: Parametric equations 209

D The equation of a straight line 213

Review set 8A 214

Review set 8B 215

Review set 8C 216

A Areas of triangles 218

B The cosine rule 221

C The sine rule 224

: The ambiguous case 225

D Using the sine and cosine rules 229

Review set 9A 232

Review set 9B 233

Review set 9C 234

A Observing periodic behaviour 237

B The sine function 240

: The family 241

: The family

, 242

: The families

and 244

C Modelling using sine functions 246

D The cosine function 249

E The tangent function 251

F General trigonometric functions 254

G Trigonometric equations 255

H Using trigonometric models 261

I Trigonometric relationships 263

J Double angle formulae 266

: Double angle formulae 266

K Trigonometric equations in quadratic form 269

Review set 10A 269

Review set 10B 270

Review set 10C 271

A Matrix structure 274

B Matrix operations and definitions 276

C The inverse of a matrix 291

D matrices 297

E Solving systems of linear equations 299

: Using matrices

in cryptography 302

Review set 11A 304

Review set 11B 305

Review set 11C 307

A Introduction 310

B Geometric operations with vectors 314

C 2-D vectors in component form 322

D 3-D coordinate geometry 327

E 3-D vectors in component form 330

F Algebraic operations with vectors 333

G Parallelism 337

H Unit vectors 338

I The scalar product of two vectors 341

Review set 12A 347

Review set 12B 349

Review set 12C 351

A Lines in 2-D and 3-D 355

B Applications of a line in a plane 360

C Relationships between lines 368

Investigation 1

Investigation 2

Investigation 3

Investigation 4

Investigation 1

Investigation 2

Investigation

Investigation

Investigation 1

Investigation 2

Investigation 3

Investigation 4

Investigation

y a x h k

a b n

y a x

y bx b >

y x c y x d

= +

+ 4

= sin

=sin 0

=sin =sin +

2 2

( )

( )

( )

n

3 3

y a x p x q = ( )( )

X

>


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8 TABLE OF CONTENTS

Review set 13A 371

Review set 13B 371

Review set 13C 372

A Key statistical concepts 376

B Measuring the centre of data 381

: Merits of the mean

and median 384

C Measuring the spread of data 394

D Cumulative frequency graphs 399

E Statistics using technology 404

F Variance and standard deviation 406

G The significance of standard deviation 412

Review set 14A 414

Review set 14B 415

Review set 14C 416

A Experimental probability 422

: Tossing drawing pins 422

: Coin tossing experiments 423

: Dice rolling experiments 424

B Sample space 426

C Theoretical probability 427

D Tables of outcomes 431

E Compound events 433

: Probabilities

of compound events 433

: Revisiting drawing pins 434

F Using tree diagrams 438

G Sampling with and without replacement 440

: Sampling simulation 442

H Binomial probabilities 444

I Sets and Venn diagrams 446

J Laws of probability 452

K Independent events 456

Review set 15A 457

Review set 15B 458

Review set 15C 459

: How many should I plant? 460

A Limits 462

B Finding asymptotes using limits 466

: Limits in number

sequences 467

C Rates of change 468

: Instantaneous speed 468

: The gradient of a tangent 470

D Calculation of areas under curves 471

: Estimating

476

Review set 16 477

A The derivative function 480

: Finding gradients

of functions 482

B Derivatives at a given -value 483

C Simple rules of differentiation 485

: Simple rules

of differentiation 485

D The chain rule 489

:

Differentiating composites 490

E The product rule 493

F The quotient rule 495

G Tangents and normals 497

H The second derivative 501

Review set 17A 503

Review set 17B 504

Review set 17C 505

A Time rate of change 508

B General rates of change 509

C Motion in a straight line 513

: Displacement, velocity

and acceleration graphs 517

D Some curve properties 520

E Rational functions 528

F Inflections and shape 533

G Optimisation 538

Review set 18A 547

Review set 18B 548

Review set 18C 549

A Exponential 552

: The derivative of 552

: Finding when

and 553

B Natural logarithms 557

C Derivatives of logarithmic functions 560

: The derivative of 560

D Applications 563

Review set 19A 566

Review set 19B 567

Review set 19C 568

Investigation

Investigation 1

Investigation 2

Investigation 3

Investigation 4

Investigation 5

Investigation 6

Investigation 7

Investigation 1

Investigation 2

Investigation 3

Investigation 4

Investigation 1

Investigation 2

Investigation 3

Investigation

Investigation 1

Investigation 2

Investigation 3

x

e

y a

a

y a

x

=

=

ln

x

14 DESCRIPTIVE STATISTICS 375

15 PROBABILITY 419

16 INTRODUCTION TO CALCULUS 461

17 DIFFERENTIAL CALCULUS 479

18 APPLICATIONS OF

DIFFERENTIAL CALCULUS 507

19 DERIVATIVES OF

EXPONENTIAL AND

LOGARITHMIC FUNCTIONS 551

R 33 e

(x22 )dx

dy

dx=a


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TABLE OF CONTENTS 9

R baf (x) dx

20 DERIVATIVES OF

TRIGONOMETRIC FUNCTIONS 569

21 INTEGRATION 579

22 APPLICATIONS OF

INTEGRATION 605

23 STATISTICAL DISTRIBUTIONS

OF DISCRETE RANDOM

VARIABLES 629

24 STATISTICAL DISTRIBUTIONS

OF CONTINUOUS RANDOM

VARIABLES 645

25 MISCELLANEOUS QUESTIONS 667

ANSWERS 695

INDEX 763

A Derivatives of trigonometric functions 570

: Derivatives of

and 570

B Optimisation with trigonometry 575

Review set 20 577

A Antidifferentiation 580

B The fundamental theorem of calculus 582

: The area function 582

C Integration 587

D Integrating 594

E Definite integrals 598

Review set 21A 602

Review set 21B 602

Review set 21C 603

: and areas 606

A Finding areas between curves 606

B Motion problems 612

C Problem solving by integration 617

D Solids of revolution 619

Review set 22A 625

Review set 22B 626

Review set 22C 627

A Discrete random variables 630

B Discrete probability distributions 632

C Expectation 635

D The binomial distribution 639

Review set 23A 643

Review set 23B 643

Review set 23C 644

A Continuous probability density functions 646

B Normal distributions 648

: Standard deviation

significance 650

C The standard normal distribution

( -distribution) 653

: Properties of 653

D Quantiles or -values 659

E Applications of the normal distribution 661

Review set 24A 664

Review set 24B 665

Review set 24C 666

A Non-calculator questions 668

B Calculator questions 681

Investigation

Investigation

Investigation

Investigation 1

Investigation 2

sin cos

+

t

f ax b

Z

k

t

( )

z =x


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10

N the set of positive integers and zero,

f0, 1, 2, 3, ......gZ the set of integers, f0, 1, 2, 3, ......gZ + the set of positive integers, f1, 2, 3, ......gQ the set of rational numbers

Q + the set of positive rational numbers,

fx jx > 0 , x 2 Q gR the set of real numbers

R + the set of positive real numbers,

fx jx > 0 , x 2 R g

a j b a divides bun the nth term of a sequence or series

d the common difference of an arithmeticsequence

r the common ratio of a geometric sequence

Sn the sum of the first n terms of a sequence,u1 + u2 + :::::+ un

S1 or S the sum to infinity of a sequence,u1 + u2 + :::::

nXi=1

ui u1 + u2 + :::::+ unnr

the rth binomial coefficient, r = 0, 1, 2, ....

in the expansion of (a+ b)n

f : A ! B f is a function under which each elementof set A has an image in set B

f : x 7! y f is a function which maps x onto yf(x) the image of x under the function f

f1 the inverse function of the function f

f g the composite function of f and glimx!a

f(x) the limit of f(x) as x tends to a

dy

dxthe derivative of y with respect to x

f 0(x) the derivative of f(x) with respect to x

d2y

dx2the second derivative of y with respect to x

f 00(x) the second derivative of f(x) with respectto x

SYMBOLS AND NOTATION USED IN THIS BOOK

fx1, x2, ....g the set with elements x1, x2, .....n(A) the number of elements in set A

fx j .... the set of all x such that2 is an element of=2 is not an element of? the empty (null) set

U the universal set

[ union\ intersection is a proper subset of is a subset ofA0 the complement of the set A

a1n , n

pa a to the power of 1

n, nth root of a

(if a > 0 then npa > 0)

a12 ,

pa a to the power 1

2, square root of a

(if a > 0 thenpa > 0)

jxj the modulus or absolute value of xjxj =

nx for x > 0 x 2 Rx for x < 0 x 2 R

identity or is equivalent to is approximately equal to> is greater than

or > is greater than or equal to< is less than

or 6 is less than or equal tois not greater than

is not less than

Ry dx the indefinite integral of y with respect to xZ b

a

y dx the definite integral of y with respect to x

between the limits x = a and x = b

ex exponential function of x

loga x logarithm to the base a of x

lnx the natural logarithm of x, loge x

sin, cos, tan the circular functions


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11

A(x, y) the point A in the plane with Cartesian

coordinates x and y

[AB] the line segment with end points A and B

AB the length of [AB]

(AB) the line containing points A and BbA the angle at A[CAB or CbAB the angle between [CA] and [AB]

ABC the triangle whose vertices are A, B and C

k is parallel to? is perpendicular tov the vector v!AB the vector represented in magnitude and

direction by the directed line segment

from A to B

a the position vector!OA

i, j, k unit vectors in the directions of the

Cartesian coordinate axes

jaj the magnitude of vector a!AB

the magnitude of!AB

v w the scalar product of v and wA1 the inverse of the non-singular matrix A

detA or jAj the determinant of the square matrix AI the identity matrix

P(A) probability of event A

P(A0) probability of the event not A

P(A j B) probability of the event A given B

x1, x2, .... observations of a variable

f1, f2, .... frequencies with which the observationsx1, x2, x3, ..... occur

P(X = x) the probability distribution function of

the discrete random variable X

E(X) the expected value of the random

variable X

population mean

population standard deviation

2 population variance

x sample mean

s 2n sample variance

sn standard deviation of the sample

s 2n1 unbiased estimate of the populationvariance

B(n, p) binomial distribution with parameters

n and p

N(, 2) normal distribution with mean and

variance 2

X B(n, p) the random variable X has a binomialdistribution with parameters n and p

X N(, 2) the random variable X has a normaldistribution with mean and variance 2

the cumulative distribution functionof the standardised normal variable

with distribution N(0, 1)


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12 BACKGROUND KNOWLEDGE AND GEOMETRIC FACTS

A circle is a set of points which are equidistant froma fixed point, which is called its centre.

The circumference is the distance around the entirecircle boundary.

An arc of a circle is any continuous part of thecircle.

A chord of a circle is a line segment joining anytwo points of a circle.

A semi-circle is a half of a circle.

A diameter of a circle is any chord passing throughits centre.

A radius of a circle is any line segment joining itscentre to any point on the circle.

A tangent to a circle is any line which touches thecircle in exactly one point.

Before starting this course you can make sure that you have a good

understanding of the necessary background knowledge. Click on the icon

alongside to obtain a printable set of exercises and answers on this

background knowledge.

Click on the icon to access printable facts about number sets.

BACKGROUND

KNOWLEDGE

NUMBER

SETS

arcchord

centre

circle

diameter

radius

tangent

point of contact

SUMMARY OF CIRCLE PROPERTIES

BACKGROUND KNOWLEDGE


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BACKGROUND KNOWLEDGE AND GEOMETRIC FACTS 13

AM = BM

A

MB

O

OAT = 90o

AP = BP

AOB = 2ACB

ADB = ACB

BAS = BCA

A

B

O P

AB

DC

A

B

S

C

T

A

T

O

A B

C

O

Chords of

a circle

Radius-tangent

Angle at

the centre

Angles

subtended by

the same arc

Angle

between a

tangent and

a chord

Tangents

from an

external point

Click on the appropriate icon to revisit these well known theorems.

Name of theorem Statement Diagram

Angle in a

semi-circleABC = 90o

OA C

B

GEOMETRY

PACKAGE

The angle in asemi-circle is a rightangle.

The perpendicularfrom the centre of acircle to a chordbisects the chord.

The tangent to acircle is perpendicularto the radius at thepoint of contact.

Tangents from anexternal point areequal in length.

The angle at thecentre of a circle istwice the angle on thecircle subtended bythe same arc.

Angles subtended byan arc on the circleare equal in size.

The angle between atangent and a chordat the point of contactis equal to the anglesubtended by thechord in the alternatesegment.

b

b

b

b

b

b

b

b

GEOMETRY

PACKAGE

GEOMETRY

PACKAGE

GEOMETRY

PACKAGE

GEOMETRY

PACKAGE

GEOMETRY

PACKAGE

GEOMETRY

PACKAGE


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Y:\HAESE\IB_SL-2ed\IB_SL-2ed_00a\013IB_SL-2_01.CDR Tuesday, 27 January 2009 9:50:09 AM PETER


PERIMETER FORMULAE

The distance around a closed figure is its perimeter.

For some shapes we can derive a formula for perimeter. The

formulae for the most common shapes are given below:

The length of an arcis a fraction of thecircumference of a

circle.

P =4 l P =2(l+w) P =a+ b+ c l=( )360 2ror

C=2rC=d

triangle

b

c

a

square

l l

w

rectangle circle

r d

arc

r

AREA FORMULAE

Shape Figure Formula

Rectangle Area = length width

Triangle Area = 12base height

Parallelogram Area = base height

T

Trapezoid

orrapezium Area =(

(

a+ b

2h

Circle Area = r2

SectorArea =

360r2

length

width

base base

height

r

r

height

base

a

b

h)

)

SUMMARY OF MEASUREMENT FACTS


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Y:\HAESE\IB_SL-2ed\IB_SL-2ed_00a\014IB_SL-2_01a.CDR Wednesday, 28 January 2009 9:25:33 AM PETER

BACKGROUND KNOWLEDGE AND GEOMETRIC FACTS 15

SURFACE AREA FORMULAE

RECTANGULAR PRISM

A = 2(ab+ bc+ ac)

CYLINDER CONE

SPHERE

A = 4r2

Object Outer surface area

Hollow cylinder A=2rh

(no ends)

Open can A=2rh+r2

(one end)

Solid cylinder A=2rh+2r2

(two ends)

r

h

hollow

hollow

r

h

hollow

solid

r

h

solid

solid

Object Outer surface area

Open cone A=rs(no base)

Solid cone A=rs+r2

(solid)

r

s

r

s

r

ab

c


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VOLUME FORMULAE

Object

Volume of uniform solid

= area of end length

Volume of a pyramid

or cone

= 13

(area of base height)

Volume of a sphere

= 43r3

r

height

endheight

end

base

height

base

h

height

VolumeF eigur

Solids of

uniform

cross-section

Pyramids

and cones

Spheres


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Graphics calculatorinstructions

Contents: A

B

C

Casio fx-9860G

Texas Instruments TI-84Plus

Texas Instruments TI- spiren


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Y:\HAESE\IB_SL-2ed\IB_SL-2ed_00b\017IB_SL-2_00b.CDR Friday, 13 March 2009 4:15:44 PM PETER

18 GRAPHICS CALCULATOR INSTRUCTIONS

In this course it is assumed that you have a graphics calculator. If you learn how to operate

your calculator successfully, you should experience little difficulty with future arithmetic

calculations.

There are many different brands (and types) of calculators. Different calculators do not have

exactly the same keys. It is therefore important that you have an instruction booklet for your

calculator, and use it whenever you need to.

However, to help get you started, we have included here some basic instructions for the

Casio fx-9860G, the Texas Instruments TI-84 plus and the Texas Instruments TI-nspire

calculators. Note that instructions given may need to be modified slightly for other models.

The instructions have been divided into three sections, one for each of the calculator models.

BASIC FUNCTIONS

GROUPING SYMBOLS (BRACKETS)

The Casio has bracket keys that look like ( and ) .

Brackets are regularly used in mathematics to indicate an expression which needs to be

evaluated before other operations are carried out.

For example, to evaluate 2 (4 + 1) we type 2 ( 4 + 1 ) EXE .

We also use brackets to make sure the calculator understands the expression we are typing in.

For example, to evaluate 24+1 we type 2 ( 4 + 1 ) EXE .

If we typed 2 4 + 1 EXE the calculator would think we meant 24 + 1.

In general, it is a good idea to place brackets around any complicated expressions which need

to be evaluated separately.

POWER KEYS

The Casio has a power key that looks like ^ . We type the base first, press the power key,then enter the index or exponent.

For example, to evaluate 253 we type 25 ^ 3 EXE .

Numbers can be squared on the Casio using the special key x2 .

For example, to evaluate 252 we type 25 x2 EXE .

CASIO FX-9860GA


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Y:\HAESE\IB_SL-2ed\IB_SL-2ed_00b\018IB_SL-2_00b.CDR Friday, 6 February 2009 3:31:38 PM PETER

GRAPHICS CALCULATOR INSTRUCTIONS 19

ROOTS

To enter roots on the Casio we need to use the secondary function key SHIFT .

We enter square roots by pressing SHIFT x2 .

For example, to evaluatep36 we press SHIFT x2 36 EXE .

If there is a more complicated expression under the square root sign you should enter it in

brackets.

For example, to evaluatep18 2 we press SHIFT x2 ( 18 2 ) EXE .

Cube roots are entered by pressing SHIFT ( .

For example, to evaluate 3p8 we press SHIFT ( 8 EXE .

Higher roots are entered by pressing SHIFT ^ .

For example, to evaluate 4p81 we press 4 SHIFT ^ 81 EXE .

LOGARITHMS

We can perform operations involving logarithms in base 10 using the log button.

To evaluate log(47) press log 47 EXE .

To evaluate log3 11, press SHIFT 4 (CATALOG), and select

logab( . You can use the alpha keys to navigate the catalog, so

in this example press I to jump to L.

Press 3 , 11 ) EXE .

INVERSE TRIGONOMETRIC FUNCTIONS

The inverse trigonometric functions sin1, cos1 and tan1 are the secondary functions ofsin , cos and tan respectively. They are accessed by using the secondary function key

SHIFT .

For example, if cosx = 35 , then x = cos1 3

5

.

To calculate this, press SHIFT cos ( 3 5 ) EXE .

SCIENTIFIC NOTATION

If a number is too large or too small to be displayed neatly on the screen, it will be expressed

in scientific notation, which is the form a 10k where 1 6 a < 10 and k is an integer.To evaluate 23003, press 2300 ^ 3 EXE . The answer

displayed is 1:2167e+10, which means 1:2167 1010.To evaluate 320 000 , press 3 20 000 EXE . The answerdisplayed is 1:5e04, which means 1:5 104.


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Y:\HAESE\IB_SL-2ed\IB_SL-2ed_00b\019IB_SL-2_00b.cdr Friday, 6 February 2009 3:33:15 PM PETER


You can enter values in scientific notation using the EXP key.

For example, to evaluate 2:61014

13 , press 2:6 EXP 14 13EXE . The answer is 2 1013.

SECONDARY FUNCTION AND ALPHA KEYS

The shift function of each key is displayed in yellow above the key. It is accessed by pressing

the SHIFT key followed by the key corresponding to the desired shift function.

For example, to calculatep36, press SHIFT x2 (

p) 36 EXE .

The alpha function of each key is displayed in red above the key. It is accessed by pressing

the ALPHA key followed by the key corresponding to the desired letter. The main purpose

of the alpha keys is to store values which can be recalled later.

MEMORY

Utilising the memory features of your calculator allows you to recall calculations you have

performed previously. This not only saves time, but also enables you to maintain accuracy

in your calculations.

SPECIFIC STORAGE TO MEMORY

Values can be stored into the variable letters A, B, ..., Z. Storing a value in memory is useful

if you need that value multiple times.

Suppose we wish to store the number 15:4829 for use in a numberof calculations. To store this number in variable A, type in the

number then press I ALPHA X,,T (A) EXE .

We can now add 10 to this value by pressing ALPHA X,,T +

10 EXE , or cube this value by pressing ALPHA X,,T ^ 3 EXE .

ANS VARIABLE

The variable Ans holds the most recent evaluated expression,

and can be used in calculations by pressing SHIFT () . Forexample, suppose you evaluate 3 4, and then wish to subtractthis from 17. This can be done by pressing 17 SHIFT ()EXE .

If you start an expression with an operator such as + , ,etc, the previous answer Ans is automatically inserted ahead of

the operator. For example, the previous answer can be halved

simply by pressing 2 EXE .

If you wish to view the answer in fractional form, press FJID .


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RECALLING PREVIOUS EXPRESSIONS

Pressing the left cursor key allows you to edit the most recently evaluated expression, and is

useful if you wish to repeat a calculation with a minor change, or if you have made an error

in typing.

Suppose you have evaluated 100 +p132. If you now want to evaluate 100 +

p142, instead

of retyping the command, it can be recalled by pressing the left cursor key. Move the cursor

between the 3 and the 2, then press DEL 4 to remove the 3 and change it to a 4. Press EXE

to re-evaluate the expression.

LISTS

Lists enable us to store sets of data, which we can then analyse and compare.

CREATING A LIST

Selecting STAT from the Main Menu takes you to the list editor

screen.

To enter the data f2, 5, 1, 6, 0, 8g into List 1, start by movingthe cursor to the first entry of List 1. Press 2 EXE 5 EXE ......

and so on until all the data is entered.

DELETING LIST DATA

To delete a list of data from the list editor screen, move the cursor to anywhere on the list

you wish to delete, then press F6 (B) F4 (DEL-A) F1 (Yes).

REFERENCING LISTS

Lists can be referenced using the List function, which is accessed by pressing SHIFT 1.

For example, if you want to add 2 to each element of List 1 and display the results in List

2, move the cursor to the heading of List 2 and press SHIFT 1 (List) 1 + 2 EXE .

Casio models without the List function can do this by pressing OPTN F1 (LIST) F1

(List) 1 + 2 EXE .

STATISTICS

Your graphics calculator is a useful tool for analysing data and creating statistical graphs.

We will first produce descriptive statistics and graphs for the data set 5 2 3 3 6 4 5 3 7 5 71 8 9 5.

Enter the data into List 1. To obtain

the descriptive statistics, press F6 (B)

until the GRPH icon is in the bottom

left corner of the screen, then press

F2 (CALC) F1 (1 VAR).


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To obtain a boxplot of the data,

press EXIT EXIT F1 (GRPH)

F6 (SET), and set up StatGraph 1

as shown. Press EXIT F1

(GPH1) to draw the boxplot.

To obtain a vertical bar chart of the

data, press EXIT F6 (SET) F2

(GPH 2), and set up StatGraph 2

as shown. Press EXIT F2

(GPH 2) to draw the bar chart (set Start

to 0, and Width to 1).

We will now enter a second set of data,

and compare it to the first.

Enter the data set 9 6 2 3 5 5 7 5 6 76 3 4 4 5 8 4 into List 2, then press

F6 (SET) F2 (GPH2) and set up

StatGraph 2 to draw a boxplot of this

data set as shown. Press EXIT F4

(SEL), and turn on both StatGraph 1

and StatGraph 2. Press F6 (DRAW)

to draw the side-by-side boxplots.

STATISTICS FROM GROUPED DATA

To obtain descriptive statistics for the

data in the table alongside, enter the data

values into List 1, and the frequency

values into List 2.

Data Frequency

2 3

3 4

4 8

5 5

Press F2 (CALC) F6 (SET), and change the 1 Var Freq

variable to List 2. Press EXIT F1 (1Var) to view the statistics.

BINOMIAL PROBABILITIES

To find P(X = 2) for X B(10, 0:3), select STAT from theMain Menu and press F5 (DIST) F5 (BINM) F1 (Bpd).

Set up the screen as shown. Go to Execute and press EXE to

display the result, which is 0:233 .


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To find P(X 6 5) for X B(10, 0:3), select STAT fromthe Main Menu and press F5 (DIST) F5 (BINM) F2 (Bcd).

Set up the screen as shown.

Go to Execute and press EXE to display the result, which is

0:953 .

NORMAL PROBABILITIES

Suppose X is normally distributed with mean 10 and standard deviation 2.

To find P(8 6 X 6 11), select STAT from the Main Menu

and press F5 (DIST) F1 (NORM) F2 (Ncd).



To find P(X > 7), select STAT from the Main Menu and press

F5 (DIST) F1 (NORM) F2 (Ncd).



To find a such that P(X 6 a) = 0:479, select STAT from the

Main Menu and press F5 (DIST) F1 (NORM) F3 (InvN).


display the result, which is a 9:89 .

WORKING WITH FUNCTIONS

GRAPHING FUNCTIONS

Selecting GRAPH from the Main Menu takes you to the Graph

Function screen, where you can store functions to graph. Delete

any unwanted functions by scrolling down to the function and

pressing DEL F1 (Yes).

To graph the function y = x23x5, move the cursor to Y1and press X,,T x2 3 X,,T 5 EXE . This storesthe function into Y1. Press F6 (DRAW) to draw a graph of

the function.

To view a table of values for the function, press MENU and

select TABLE. The function is stored in Y1, but not selected.

Press F1 (SEL) to select the function, and F6 (TABL) to view

the table. You can adjust the table settings by pressing EXIT

and then F5 (SET) from the Table Function screen.


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ADJUSTING THE VIEWING WINDOW

When graphing functions it is important that you are able to view all the important features

of the graph. As a general rule it is best to start with a large viewing window to make sure

all the features of the graph are visible. You can then make the window smaller if necessary.

The viewing window can be adjusted by pressing SHIFT F3

(V-Window). You can manually set the minimum and maximum

values of the x and y axes, or press F3 (STD) to obtain the

standard viewing window 10 6 x 6 10, 10 6 y 6 10:

FINDING POINTS OF INTERSECTION

It is often useful to find the points of intersection of two graphs, for instance, when you are

trying to solve simultaneous equations.

We can solve y = 11 3x and y = 12 x2

simultane-

ously by finding the point of intersection of these two lines.

Select GRAPH from the Main Menu, then store 113x intoY1 and

12 x2

into Y2. Press F6 (DRAW) to draw a graph

of the functions.

To find their point of intersection, press F5 (G-Solv) F5

(ISCT). The solution x = 2, y = 5 is given.

If there is more than one point of intersection, the remaining

points of intersection can be found by pressing I .

FINDING x-INTERCEPTS

In the special case when you wish to solve an equation of the form f(x) = 0, this can be

done by graphing y = f(x) and then finding when this graph cuts the x-axis.

To solve x3 3x2 + x + 1 = 0, select GRAPH from theMain Menu and store x3 3x2 +x+1 into Y1. Press F6(DRAW) to draw the graph.

To find where this function cuts the x-axis, press F5 (G-Solv)

F1 (ROOT). The first solution x 0:414 is given.Press I to find the remaining solutions x = 1 and x 2:41 .

TURNING POINTS

To find the turning point or vertex of y = x2 + 2x+ 3, select GRAPH from the MainMenu and store x2 + 2x+ 3 into Y1. Press F6 (DRAW) to draw the graph.From the graph, it is clear that the vertex is a maximum, so to

find the vertex press F5 (G-Solv) F2 (MAX). The vertex is

(1, 4).


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FINDING THE TANGENT TO A FUNCTION

To find the equation of the tangent to y = x2 when x = 2,

we first press SHIFT MENU (SET UP), and change the

Derivative setting to On. Draw the graph of y = x2, then

press SHIFT F4 (Sketch) F2 (Tang).

Press 2 EXE EXE to draw the tangent at x = 2.

The tangent has gradient 4, and equation y = 4x 4.

DEFINITE INTEGRALS

To calculateR 31 x

2 dx, we first draw the graph of y = x2. Press

F5 (G-Solv) F6 F3 (Rdx) to select the integral tool. Press

1 EXE 3 EXE to specify the lower and upper bounds of the

integral.

So,R 31 x

2 dx = 823 .

MATRICES

MATRIX OPERATIONS

To find the sum of A =

0@ 2 31 45 0

1A and B =0@ 1 62 0

3 8

1A, wefirst define matrices A and B.

To find A + B, press OPTN F2 (MAT) F1 (Mat) ALPHA

X,,T (A) to enter matrix A, then + , then F1 (Mat) ALPHA

log (B) to enter matrix B.

Matrices are easily stored in a graphics calculator. This is particularly valuable if we need to

perform a number of operations with the same matrices.

STORING MATRICES

To store the matrix

0@ 2 31 45 0

1A, select RUNMAT from theMain Menu, and press F1 (IMAT). This is where you define

matrices and enter their elements.

To define the matrix as matrix A, make sure Mat A is

highlighted, and press F3 (DIM) 3 EXE 2 EXE EXE .

This indicates that the matrix has 3 rows and 2 columns.

Enter the elements of the matrix, pressing EXE after each entry.

Press EXIT twice to return to the home screen when you are

done.


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Y:\HAESE\IB_SL-2ed\IB_SL-2ed_00b\025IB_SL-2_00b.cdr Tuesday, 24 February 2009 10:39:21 AM PETER


Press EXE to display the result.

Operations of subtraction, scalar multiplication, and matrix

multiplication can be performed in a similar manner.

INVERTING MATRICES

To find the inverse of A =

2 51 3

, we first define matrix A.

Press OPTN F2 (MAT) F1 (Mat) ALPHA X,,T (A) to

enter matrix A, then SHIFT ) (x1) EXE .

So, A1 =

3 51 2

.

BASIC FUNCTIONS


The TI-84 Plus has bracket keys that look like ( and ) .



For example, to evaluate 2 (4 + 1) we type 2 ( 4 + 1 ) ENTER .We also use brackets to make sure the calculator understands the expression we are typing in.

For example, to evaluate 24+1 we type 2 ( 4 + 1 ) ENTER .

If we typed 2 4 + 1 ENTER the calculator would think we meant 24 + 1.In general, it is a good idea to place brackets around any complicated expressions which need


POWER KEYS

The TI-84 Plus has a power key that looks like ^ . We type the base first, press the powerkey, then enter the index or exponent.

For example, to evaluate 253 we type 25 ^ 3 ENTER .

Numbers can be squared on the TI-84 Plus using the special key x2 .

For example, to evaluate 252 we type 25 x2 ENTER .

TEXAS INSTRUMENTS TI-84 PLUSB


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ROOTS

To enter roots on the TI-84 Plus we need to use the secondary function key 2nd .

We enter square roots by pressing 2nd x2 .

For example, to evaluatep36 we press 2nd x2 36 ) ENTER .

The end bracket is used to tell the calculator we have finished entering terms under the square

root sign.

Cube roots are entered by pressing MATH 4: 3p

( .

To evaluate 3p8 we press MATH 4 : 8 ) ENTER .

Higher roots are entered by pressing MATH 5: xp

.

To evaluate 4p81 we press 4 MATH 5 : 81 ENTER .

LOGARITHMS

We can perform operations involving logarithms in base 10 using the log button.

To evaluate log(47), press log 47 ) ENTER .

Since loga b =log b

log a, we can use the base 10 logarithm to

calculate logarithms in other bases.

To evaluate log3 11, we note that log3 11 =log 11

log 3, so we

press log 11 ) log 3 ) ENTER .


The inverse trigonometric functions sin1, cos1 and tan1 are the secondary functions ofSIN , COS and TAN respectively. They are accessed by using the secondary function key

2nd .


5

.

To calculate this, press 2nd COS 3 5 ) ENTER .

SCIENTIFIC NOTATION

If a number is too large or too small to be displayed neatly on the screen, it will be expressed

in scientific notation, which is the form a10k where 1 6 a < 10 and k is an integer.To evaluate 23003, press 2300 ^ 3 ENTER . The answer

displayed is 1:2167e10, which means 1:2167 1010.To evaluate 320 000 , press 3 20 000 ENTER . The answerdisplayed is 1:5e4, which means 1:5 104.


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You can enter values in scientific notation using the EE function,

which is accessed by pressing 2nd , .For example, to evaluate 2:610

14

13 , press 2:6 2nd , 14 13 ENTER . The answer is 2 1013.

SECONDARY FUNCTION AND ALPHA KEYS

The secondary function of each key is displayed in blue above the key. It is accessed by

pressing the 2nd key, followed by the key corresponding to the desired secondary function.

For example, to calculatep36, press 2nd x2 (

p) 36 ) ENTER .

The alpha function of each key is displayed in green above the key. It is accessed by pressing

the ALPHA key followed by the key corresponding to the desired letter. The main purpose

of the alpha keys is to store values into memory which can be recalled later.

MEMORY







Suppose we wish to store the number 15:4829 for use in a numberof calculations.

number, then press STOI ALPHA MATH (A) ENTER .

We can now add 10 to this value by pressing ALPHA MATH

+ 10 ENTER , or cube this value by pressing ALPHA MATH

^ 3 ENTER .

ANS VARIABLE

The variable Ans holds the most recent evaluated expression,

and can be used in calculations by pressing 2nd () .

For example, suppose you evaluate 3 4, and then wish tosubtract this from 17. This can be done by pressing 17 2nd () ENTER .

To store this number in variable A, type in the


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If you start an expression with an operator such as + , ,etc, the previous answer Ans is automatically inserted ahead of

the operator. For example, the previous answer can be halved

simply by pressing 2 ENTER .

If you wish to view the answer in fractional form, press MATH

1 ENTER .


The ENTRY function recalls previously evaluated expressions, and is used by pressing 2nd

ENTER .

This function is useful if you wish to repeat a calculation with a minor change, or if you have

made an error in typing.

Suppose you have evaluated 100 +p132. If you now want to evaluate 100 +

p142, instead

of retyping the command, it can be recalled by pressing 2nd ENTER .

The change can then be made by moving the cursor over the 3 and changing it to a 4, then

pressing ENTER .

If you have made an error in your original calculation, and intended to calculate 1500+p132,

again you can recall the previous command by pressing 2nd ENTER .

Move the cursor to the first 0.

You can insert the 5, rather than overwriting the 0, by pressing 2nd DEL (INS) 5 ENTER .

LISTS

Lists are used for a number of purposes on the calculator. They enable us to store sets of

data, which we can then analyse and compare.

CREATING A LIST

Press STAT 1 to access the list editor screen.

To enter the data f2, 5, 1, 6, 0, 8g into List 1, start by movingthe cursor to the first entry of L1. Press 2 ENTER 5 ENTER

...... and so on until all the data is entered.

DELETING LIST DATA

To delete a list of data from the list editor screen, move the cursor to the heading of the list

you want to delete then press CLEAR ENTER .


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REFERENCING LISTS

Lists can be referenced by using the secondary functions of the keypad numbers 1-6.

For example, suppose you want to add 2 to each element of List1 and display the results in

List2. To do this, move the cursor to the heading of L2 and press 2nd 1 + 2 ENTER .

STATISTICS


We will first produce descriptive statistics and graphs for the data set:

5 2 3 3 6 4 5 3 7 5 7 1 8 9 5.

Enter the data set into List 1. To obtain

descriptive statistics of the data set, press

STAT I 1:1-Var Stats 2nd 1 (L1) ENTER .

To obtain a boxplot of the data, press 2nd

Y= (STAT PLOT) 1 and set up Statplot1 as

shown. Press ZOOM 9:ZoomStat to graph the

boxplot with an appropriate window.

To obtain a vertical bar chart of the data, press

2nd Y= 1, and change the type of graph to

a vertical bar chart as shown. Press ZOOM

9:ZoomStat to draw the bar chart. Press

WINDOW and set the Xscl to 1, then GRAPH

to redraw the bar chart.

We will now enter a second set of data, and

compare it to the first.

Enter the data set 9 6 2 3 5 5 7 5 6 7 6

3 4 4 5 8 4 into List 2, press 2nd Y= 1,

and change the type of graph back to a boxplot

as shown. Move the cursor to the top of the

screen and select Plot2. Set up Statplot2 in the

same manner, except set the XList to L2. Press

ZOOM 9:ZoomStat to draw the side-by-side

boxplots.


To obtain descriptive statistics for the

data in the table alongside, enter the data

values into List 1, and the frequency

values into List 2.

Press STAT I 1 : 1-Var Stats 2nd

1 (L1) , 2nd 2 (L2) ENTER .

Data Frequency

2 3

3 4

4 8

5 5


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To find P(X = 2) for X B(10, 0:3), press 2nd VARS(DISTR) and select A:binompdf( . Press 10 , 0:3 , 2

) ENTER .

So, P(X = 2) 0:233 .

To find P(X 6 5) for X B(10, 0:3), press 2nd VARS(DISTR) and select B:binomcdf( . Press 10 , 0:3 , 5

) ENTER .

So, P(X 6 5) 0:953 .


Suppose X is normally distributed with mean 10 and standard deviation 2.

To find P(8 6 X 6 11), press 2nd VARS (DISTR), and select

2:normalcdf( . Press 8 , 11 , 10 , 2 ) ENTER .So, P(8 6 X 6 11) 0:533 .

To find P(X > 7), press 2nd VARS (DISTR), and select

2:normalcdf( . Press 7 , 2nd , (EE) 99 , 10 , 2 )ENTER .

So, P(X > 7) 0:933 .

To find a such that P(X 6 a) = 0:479, press 2nd VARS

(DISTR), and select 3:invNorm( .

Press 0:479 , 10 , 2 ) ENTER .So, a 9:89 .


GRAPHING FUNCTIONS

Pressing Y= selects the Y= editor, where you can store functions

to graph. Delete any unwanted functions by scrolling down to

the function and pressing CLEAR .


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To graph the function y = x2 3x 5, move the cursor toY1, and press X,T,,n x2 3 X,T,,n 5 ENTER . Thisstores the function into Y1. Press GRAPH to draw a graph of

the function.

To view a table of values for the function, press 2nd GRAPH

(TABLE). The starting point and interval of the table values can

be adjusted by pressing 2nd WINDOW (TBLSET).





Some useful commands for adjusting the viewing window

include:

ZOOM 0:ZoomFit : This command scales the y-axis to

fit the minimum and maximum val-

ues of the displayed graph within the

current x-axis range.

ZOOM 6:ZStandard : This command returns the viewing

window to the default setting of

10 6 x 6 10, 10 6 y 6 10.If neither of these commands are helpful, the viewing window

can be adjusted manually by pressing WINDOW and setting the

minimum and maximum values for the x and y axes.

FINDING POINTS OF INTERSECTION

It is often useful to find the points of intersection of two graphs, for instance, when you are

trying to solve simultaneous equations.

We can solve y = 11 3x and y = 12 x2

simultaneously

by finding the point of intersection of these two lines.

Press Y= , then store 11 3x into Y1 and 12 x2

into

Y2. Press GRAPH to draw a graph of the functions.

To find their point of intersection, press 2nd TRACE (CALC)

5:intersect. Press ENTER twice to specify the functions Y1

and Y2 as the functions you want to find the intersection of,

then use the arrow keys to move the cursor close to the point of

intersection and press ENTER once more.

The solution x = 2, y = 5 is given.


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FINDING x-INTERCEPTS

In the special case when you wish to solve an equation of the form f(x) = 0, this can be

done by graphing y = f(x) and then finding when this graph cuts the x-axis.

For example, to solve x3 3x2 + x + 1 = 0, press Y= andstore x3 3x2 + x+ 1 into Y1. Then press GRAPH .

To find where this function first cuts the x-axis, press 2nd

TRACE (CALC) 2:zero. Move the cursor to the left of the first

zero and press ENTER , then move the cursor to the right of

the first zero and press ENTER . Finally, move the cursor close

to the first zero and press ENTER once more. The solution

x 0:414 is given.

Repeat this process to find the remaining solutions x = 1 andx 2:414 .

TURNING POINTS

To find the turning point or vertex of y = x2+2x+3, pressY= and store x2+2x+3 into Y1. Press GRAPH to draw

the graph.

From the graph, it is clear that the vertex is a maximum, so press

2nd TRACE (CALC) 4:maximum.

Move the cursor to the left of the vertex and press ENTER , then

move the cursor to the right of the vertex and press ENTER .

Finally, move the cursor close to the vertex and press ENTER

once more. The vertex is (1, 4).

FINDING THE TANGENT TO A FUNCTION

To find the equation of the tangent to y = x2 when x = 2, we

first draw the graph of y = x2. Press 2nd PRGM (DRAW)

5:Tangent, then press 2 ENTER to draw the tangent at

x = 2.

The tangent has gradient 4, and equation y = 4x 4.


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DEFINITE INTEGRALS

To calculateR 31 x

2 dx, we first draw the graph of y = x2. Press

2nd TRACE (CALC) and select 7:Rf(x) dx. Press 1

ENTER 3 ENTER to specify the lower and upper limits of

the integral.

So,R 31x2 dx = 823 .

MATRICES

Matrices are easily stored in a graphics calculator. This is particularly valuable if we need to

perform a number of operations with the same matrices.

STORING MATRICES

To store the matrix

0@ 2 31 45 0

1A, press 2nd x-1 (MATRIX) todisplay the matrices screen, and use I to select the EDIT

menu. This is where you define matrices and enter the elements.

Press 1 to select 1:[A]. Press 3 ENTER 2 ENTER to define

matrix A as a 3 2 matrix.Enter the elements of the matrix, pressing ENTER after each

entry.

Press 2nd MODE (QUIT) when you are done.

MATRIX OPERATIONS

To find the sum of A =

0@ 2 31 45 0

1A and B =0@ 1 62 0

3 8

1A, wefirst define matrices A and B.

To find A + B, press 2nd x-1 1 to enter matrix A, then + ,

then 2nd x-1 2 to enter matrix B. Press ENTER to display the

results.

INVERTING MATRICES

To find the inverse of A =

2 51 3

, we first define matrix A.

Press 2nd x-1 1 to enter matrix A, then press x-1 ENTER .

Operations of subtraction, scalar multiplication and matrix

multiplication can be performed in a similar manner.


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GETTING STARTED

Pressing takes you to the home screen, where you can choose which application you

wish to use.

The TI-nspire organises any work done into pages. Every time you start a new application,

a new page is created. You can navigate back and forth between the pages you have worked

on by pressing ctrl J or ctrl I

From the home screen, press 1 to open the Calculator application. This is where most of the

basic calculations are performed.

SECONDARY FUNCTION KEY

The secondary function of each key is displayed in grey above the primary function. It is

accessed by pressing the ctrl key followed by the key corresponding to the desired secondary

function.

BASIC FUNCTIONS


The TI-nspire has bracket keys that look like ( and ) .



For example, to evaluate 2 (4 + 1) we type 2 ( 4 + 1 ) enter .We also use brackets to make sure the calculator understands the expression we are typing in.

For example, to evaluate 24+1 we type 2 ( 4 + 1 ) enter .

If we typed 2 4 + 1 enter the calculator would think we meant 24 + 1.In general, it is a good idea to place brackets around any complicated expressions which need


POWER KEYS

The TI-nspire has a power key that looks like ^ . We type the base first, press the powerkey, then enter the index or exponent.

For example, to evaluate 253 we type 25 ^ 3 enter .

Numbers can be squared on the TI-nspire using the special key x2 .

For example, to evaluate 252 we type 25 x2 enter .

TEXAS INSTRUMENTS TI-nSPIREC


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ROOTS

To enter roots on the TI-nspire we need to use the secondary function ctrl .

We enter square roots by pressing ctrl x2 .

For example, to evaluatep36 we press ctrl x2 36 enter .

Higher roots are entered by pressing ctrl ^ , which creates the p template.

Enter the root number in the first entry, and the number you wish to find the root of in the

second entry. Use the arrow keys to navigate around the template.

LOGARITHMS

We can perform operations involving logarithms using the

log function, which is accessed by pressing ctrl 10x .

Put the base number in the first entry, and the number

you wish to find the logarithm of in the second entry.

For example, to find log3 11 press ctrl 10x 3 I 11

enter .

ROUNDING OFF

You can instruct the TI-nspire to round off values to a

fixed number of decimal places.

For example, to round to 3 decimal places, from the homescreen select 8 : System Info > 2 : System Settings....

Press the tab button until the Display Digits drop-box

is highlighted. Use the H key to select Fix 3, then press

enter three times to return to the page you were working

on.

To unfix the number of decimal places, select Float from the Display Digits drop-box.

DECIMAL EXPANSION OF FRACTIONS

If you press 5 8 enter , the calculator will simply return the fraction 58 . To find thedecimal expansion of 58 , press 5 8 ctrl enter .


The inverse trigonometric functions sin1, cos1 and tan1 are the secondary functions ofsin , cos and tan respectively. They are accessed by using the secondary function key

ctrl .


5

.

To calculate this, press ctrl cos 3 5 ) enter .


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SCIENTIFIC NOTATION

You can enter very large or very small values by

expressing them in scientific notation, which is in the

form a 10k where 1 6 a < 10 and k is an integer.You can do this using the EE button.

For example, to evaluate2:6 1014

13, press 2:6 EE 14

13 enter . The answer is 2 1013.

MEMORY







Suppose we wish to store the number 15:4829 for use in anumber of calculations. To store this number in variable

A, type in the number then press ctrl var (stoI) A

enter .

We can now add 10 to this value by pressing A + 10

enter or cube this value by pressing A ^ 3 enter .

ANS VARIABLE

The variable Ans holds the most recent evaluated expression, and can be used in calculations

by pressing ctrl () .For example, suppose you evaluate 3 4, and then wish to subtract this from 17. This canbe done by pressing 17 ctrl () enter .

If you start an expression with an operator such as + , , etc, the previous answer Ansis automatically inserted ahead of the operator. For example, the previous answer can be

doubled simply by pressing 2 enter .


Pressing the N key allows you to recall and edit previously evaluated expressions.

To recall a previous expression, press the N key until the desired expression is highlighted,

and press enter . The expression then appears in the current entry line, where it can be

edited.


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LISTS

Lists are used for a number of purposes on the calculator. They enable us to store sets of

data, which we can then analyse and compare.

In order to perform many of the operations involving data lists on the TI-nspire, you need to

name your lists as you define them.

CREATING A LIST

Selecting 3 : Lists & Spreadsheet from the home screen

takes you to the list editor screen.

To enter the data f2, 5, 1, 6, 0, 8g into List A, startby moving the cursor to the first entry of List A. Press

2 enter 5 enter ...... and so on until all the data is

entered.

Move the cursor to the heading of List A and use the

green alphabet keys to enter a name for the list, for

example data.

DELETING LIST DATA

To delete a list of data from the list editor screen, move the cursor to the heading of that list,

and press N to highlight the whole column. Press J to delete the list.

REFERENCING LISTS

Once you have named a list, you can use that name to

reference the list in other operations.

Suppose you want to add 2 to each element of the data listwe created in the above example, and display the results

in List B. Move the cursor to the shaded row of List B,

and press = data + 2 enter .

STATISTICS


We will first produce descriptive statistics and graphs for the data set:

5 2 3 3 6 4 5 3 7 5 7 1 8 9 5:

Enter the data into and name this list .List A one


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We will now draw some statistical graphs for the data.

Press 5 : Data & Statistics to open the Data

and Statistics application. You will see the data points

randomly scattered on the screen. Move the cursor to the

bottom of the screen until the Click to add variable box

appears. Press enter , then select one. The points will

order themselves into a dotplot along the horizontal axis.

We will now add a second set of data, and compare it to

the first.

Use the ctrl J command to return to the page

containing the list of data.

Enter the set 9 6 2 3 5 5 7 5 6 7 6 3 4 4 5 8 4 intoList B, and name the list two.

Use the ctrl I command to navigate back to the page

containing the boxplot of one.

To view boxplots of both data sets, we need to split the

page into two sections. Press ctrl , then select

5 : Page Layout > 2 : Select Layout > 3 : Layout 3.

Press ctrl tab to navigate between the sections, then

press menu 5 to open a new Data and Statistics

application. Draw another boxplot in this section, this

time using the data in two.

To obtain the descriptive statistics, press 1 : Calculator to open the Calculator application,

press menu , then select 5 : Statistics > 1 : Statistical Calculations > 1 : One-Variable

Statistics.

Press enter to choose

1 list. Select one from

the X1 List drop-box,

and press enter .

To obtain a vertical bar

chart of the data, press

menu , then select

1 : Plot Type >...

3 : Histogram.

To obtain a boxplot of

the data, press menu ,

then select 1 : Plot Type > 2 : Box Plot.


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Y:\HAESE\IB_SL-2ed\IB_SL-2ed_00b\039IB_SL-2_00b.cdr Tuesday, 10 February 2009 4:30:01 PM PETER


In order to compare the boxplots, the horizontal scales

should be the same. Press menu , then select

5 : Window/Zoom > 1 : Window Settings.

Set the X range from 0 to 10. Press ctrl tab , and do

the same with the other boxplot.


To obtain descriptive statistics

for the data in the table along-

side, enter the data values into

List A, and label the list data.

Enter the frequency values into

List B, and label the list freq.

Data Frequency

2 3

3 4

4 8

5 5

Press 1 : Calculator to open the Calculator applica-

tion. Press menu , then select 5 : Statistics > 1 : Stat

Calculations .... > 1 : One-Variable Statistics. Press

enter to choose 1 list. Select data from the X1 List

drop-box and freq from the Frequency List drop-box,

then press enter .


To find P(X = 2) for X B(10, 0:3), press1 : Calculator to open the Calculator application. Press

menu then select

4 : Probability > 5 : Distributions ... > D : Binomial

Pdf.

Press 10 tab 0:3 tab 2 tab enter . The result is

0:233 .

To find P(X 6 5) for X B(10, 0:3), press menu

4 : Probability > 5 : Distributions ... > E : Binomial

Cdf.

Press 10 tab 0:3 tab 0 tab 5 tab enter . The result

is 0:953 .

for the Calculator application and select


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Suppose X is normally distributed with mean 10 andstandard deviation 2.

To find P(8 6 X 6 11), press menu from the

Calculator application and select

4 : Probability > 5 : Distributions .... > 2 : Normal

Cdf.

Press 8 tab 11 tab 10 tab 2 tab enter . The result

is 0:533 .

To find P(X > 7), press menu from the Calculator

application and select

4 : Probability > 5 : Distributions .... > 2 : Normal

Cdf.

Set up the screen as shown, and select OK. The result is

0:933 .

To find a such that P(X 6 a) = 0:479, press menu

from the Calculator application and select

4 : Probability > 5 : Distributions .... > 3 : Inverse

Normal.

Press 0:479 tab 10 tab 2 tab enter . The solution is

a 9:89 .


GRAPHING FUNCTIONS

Selecting 2 : Graphs & Geometry from the home screen

opens the graphing application, where you can graph

functions.

To graph the function y = x2 3x 2, press X x2 3 X 2 enter .

To view a table of values for the function, press menu ,

then select 2 : View > 9 : Add Function Table.

The page splits into 2, with the graph on one side and thetable of values on the other side.


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To adjust the viewing window, press menu then select 4 : Window . The most useful

options are:

1 : Window Settings : With this option you can set

the minimum and maximum values for the x and y axes

manually.

5 : Zoom-Standard : This option returns the viewing

window to the default setting of 10 6 x 6 10,203 6 y 6 203 .A : Zoom-Fit : This option scales the y-axis to fit the

minimum and maximum values of the displayed graph

within the c

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Transcript of Specialists in mathematics publishing Mathematicsnharris17.wikispaces.com/file/view/IB Math SL Book...