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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
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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_\\"
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
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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V:\BOOKS\IB_books\IB_SL-2ed\IB_SL-2ed_00a\011IB_SL-2_00a.CDR Monday, 30 November 2009 10:16:52 AM PETER
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
14 BACKGROUND KNOWLEDGE AND GEOMETRIC FACTS
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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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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16 BACKGROUND KNOWLEDGE AND GEOMETRIC FACTS
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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Y:\HAESE\IB_SL-2ed\IB_SL-2ed_00a\016IB_SL-2_01a.CDR Wednesday, 28 January 2009 9:27:04 AM PETER
Graphics calculatorinstructions
Contents: A
B
C
Casio fx-9860G
Texas Instruments TI-84Plus
Texas Instruments TI- spiren
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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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20 GRAPHICS CALCULATOR INSTRUCTIONS
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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GRAPHICS CALCULATOR INSTRUCTIONS 21
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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22 GRAPHICS CALCULATOR INSTRUCTIONS
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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GRAPHICS CALCULATOR INSTRUCTIONS 23
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).
Set up the screen as shown. Go to Execute and press EXE to
display the result, which is 0:533 .
To find P(X > 7), select STAT from the Main Menu and press
F5 (DIST) F1 (NORM) F2 (Ncd).
Set up the screen as shown. Go to Execute and press EXE to
display the result, which is 0:933 .
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).
Set up the screen as shown. Go to Execute and press EXE to
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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24 GRAPHICS CALCULATOR INSTRUCTIONS
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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GRAPHICS CALCULATOR INSTRUCTIONS 25
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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26 GRAPHICS CALCULATOR INSTRUCTIONS
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
GROUPING SYMBOLS (BRACKETS)
The TI-84 Plus 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 ) 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
to be evaluated separately.
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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GRAPHICS CALCULATOR INSTRUCTIONS 27
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 .
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
2nd .
For example, if cosx = 35 , then x = cos1 3
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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28 GRAPHICS CALCULATOR INSTRUCTIONS
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
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.
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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GRAPHICS CALCULATOR INSTRUCTIONS 29
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 .
RECALLING PREVIOUS EXPRESSIONS
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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30 GRAPHICS CALCULATOR INSTRUCTIONS
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
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 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.
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.
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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GRAPHICS CALCULATOR INSTRUCTIONS 31
BINOMIAL PROBABILITIES
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 .
NORMAL PROBABILITIES
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 .
WORKING WITH FUNCTIONS
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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32 GRAPHICS CALCULATOR INSTRUCTIONS
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).
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.
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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GRAPHICS CALCULATOR INSTRUCTIONS 33
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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34 GRAPHICS CALCULATOR INSTRUCTIONS
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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GRAPHICS CALCULATOR INSTRUCTIONS 35
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
GROUPING SYMBOLS (BRACKETS)
The TI-nspire 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 ) 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
to be evaluated separately.
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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36 GRAPHICS CALCULATOR INSTRUCTIONS
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 .
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
ctrl .
For example, if cosx = 35 , then x = cos1 3
5
.
To calculate this, press ctrl cos 3 5 ) enter .
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GRAPHICS CALCULATOR INSTRUCTIONS 37
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
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 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 .
RECALLING PREVIOUS EXPRESSIONS
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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38 GRAPHICS CALCULATOR INSTRUCTIONS
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
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 7 1 8 9 5:
Enter the data into and name this list .List A one
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GRAPHICS CALCULATOR INSTRUCTIONS 39
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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40 GRAPHICS CALCULATOR INSTRUCTIONS
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.
STATISTICS FROM GROUPED DATA
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 .
BINOMIAL PROBABILITIES
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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GRAPHICS CALCULATOR INSTRUCTIONS 41
NORMAL PROBABILITIES
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 .
WORKING WITH FUNCTIONS
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.
IB_SL-2edmagentacyan yellow black
0 05 5
25
25
75
75
50
50
95
95
100
100 0 05 5
25
25
75
75
50
50
95
95
100
100
Y:\HAESE\IB_SL-2ed\IB_SL-2ed_00b\041IB_SL-2_00b.cdr Friday, 6 February 2009 4:41:54 PM PETER
42 GRAPHICS CALCULATOR INSTRUCTIONS
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.
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