Mathematics - Analysis and Approaches SL 2

2019·80.94 MB·english

by  Michael Haese

% HAESE MATHEMATICS
Analysis and
approaches SL

for use with

IB Diploma Programme
% HAESE MATHEMATICS

Specialists in mathematics education
Mathematics
o
Amalysis and|
Approaches SL

e Michael Haese
4 \ Mark Humphries
‘ 3 Chris Sangwin
Ngoc Vo

for use with
IB Diploma
Programme
MATHEMATICS: ANALYSIS AND APPROACHES SL
Michael Haese B.Sc.(Hons.), Ph.D.
Mark Humphries B.Sc.(Hons.)
Chris Sangwin M.A., M.Sc., Ph.D.
Ngoc Vo B.Ma.Sc.
Published by Haese Mathematics
152 Richmond Road, Marleston, SA 5033, AUSTRALIA
Telephone: [phone], Fax: [phone]
Email: [email protected]
Web: www.haesemathematics.com
National Library of Australia Card Number & ISBN 978-1-925489-56-9
© Haese & Harris Publications 2019
First Edition 2019
Editorial review by Denes Tilistyak (Western International School of Shanghai).
Cartoon artwork by John Martin.
Artwork by Brian Houston, Charlotte Frost, Yi-Tung Huang, and Nicholas Kellett-Southby.
Typeset by Deanne Gallasch and Charlotte Frost. Typeset in Times Roman 10.
Computer software by Yi-Tung Huang, Huda Kharrufa, Brett Laishley, Bronson Mathews, Linden May,
Joshua Douglass-Molloy, Jonathan Petrinolis, and Nicole Szymanczyk.
Production work by Sandra Haese, Bradley Steventon, Nicholas Kellett-Southby, Cashmere Collins-McBride,
and Joseph Small.
We acknowledge the contribution of Marjut Mdenpaé, Mal Coad, and Glen Whiffen, for material from previous
courses which now appears in this book. The publishers wish to make it clear that acknowledging these
individuals does not imply any endorsement of this book by any of them, and all responsibility for the content
rests with the authors and publishers.
Printed in China by Prolong Press Limited.
This book has been developed independently from and is not endorsed by the International Baccalaureate
Organization. International Baccalaureate, Baccalauréat International, Bachillerato Internacional, and IB are
registered trademarks owned by the International Baccalaureate Organization.
This book is copyright. Except as permitted by the Copyright Act (any fair dealing for the purposes of private
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Acknowledgements: While every attempt has been made to trace and acknowledge copyright, the authors and
publishers apologise for any accidental infringement where copyright has proved untraceable. They would be
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Disclaimer: All the internet addresses (URLs) given in this book were valid at the time of printing. While the
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any such changes can be accepted by either the authors or the publisher.
FOREWORD
This book has been written for the International Baccalaureate Diploma Programme course Mathematics:
Analysis and Approaches SL, for first teaching in August 2019, and first assessment in May 2021.
This book is designed to complete the course in conjunction with the SL Mathematics
Mathematics: Core Topics SL textbook. It is expected that students will 4 WALSE wATHEMATCS
start using this book approximately 6-7 months into the two-year course, !
upon the completion of the Mathematics: Core Topics SL textbook.
The Mathematics: Analysis and Approaches courses have a focus on
algebraic rigour, and the book has been written with this focus in mind.
The material is presented in a clear, easy-to-follow style, free from
unnecessary distractions, while effort has been made to contextualise
questions so that students can relate concepts to everyday use.
Each chapter begins with an Opening Problem, offering an insight into the
application of the mathematics that will be studied in the chapter.
Important information and key notes are highlighted, while worked
examples provide step-by-step instructions with concise and relevant
explanations. Discussions, Activities, and Investigations are used
throughout the chapters to develop understanding, problem solving, and
reasoning.
In this changing world of mathematics education, we believe that the contextual approach shown in this
book, with the associated use of technology, will enhance the students’ understanding, knowledge and
appreciation of mathematics, and its universal application.
We welcome your feedback.
Email: [email protected] Web: www.haesemathematics.com
PMH, MAH, CS, NV
ACKNOWLEDGEMENTS
The photo of Dr Jonathon Hare and Dr Ellen McCallie on page 56 was reproduced from
www.creative-science.org.uk/parabola.html with permission.
4
ONLINE FEATURES
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browser or may be installed to your tablet or computer.
Students can revisit concepts taught in class and undertake their own revision and practice online.
COMPATIBILITY
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SELF TUTOR
Simply ‘click’ on the (or anywhere in the example box) to access the worked
example, with a teacher’s voice explaining each step necessary to reach the answer.
Play any line as often as you like. See how the basic processes come alive using
movement and colour on the screen.
Example 3 o) Self Tutor
Solve for z on the domain 0 < z < 27
a cosz:—@ b 2sinz—1=0
a coszzffizé b 2sinz—1=0 ¢ tanz+v3=0
i tanz = 7\/?_)
2m
3
5T
3

See Chapter 9, Trigonometric equations and identities, p. 228
INTERACTIVE LINKS
Interactive links to in-browser tools which complement the text are included to
ICON
assist teaching and learning.
Icons like this will direct you to:
interactive demonstrations to illustrate and animate concepts
games and other tools for practising your skills
graphing and statistics packages which are fast, powerful alternatives to using a graphics calculator
printable pages to save class time.
Save time, and
= make learning easier!

Angle relationships ‘O
The 2-coordinate of P, is the same as the y-coordinate of Py. The y-coordinate
prev next
of P, is the is the same as the z-coordinate of P,. These relationships are the
complementary angle formulae.
Ly
—
sin(§ — 6) = cos6 Normal probabilty distribution ©
cos(j —0) =sinf v
1
Py(cosb,sin ) 08
0.6
0.4
0.2
# / 6 5 4 3 2 1 1 2 3 4 5 64

See Chapter 9, R
Trigonometric equations and identities, p. 234 v
5 A 4 0 1 2 3
o 0.75
05 075 1 125 15 175 2

See Chapter 21, The normal distribution, p. 509
Graphics calculator instruction booklets are available for the Casio fx-CG50, TI-84 Plus CE,
TI-nspire, and the HP Prime. Click on the relevant icon below.
CASIO
#x-CG50 T1-84 Plus CE Tl-nspire HP Prime
When additional calculator help may be needed, specific instructions are

available from icons within the text.
GRAPHICS
CALCULATOR
INSTRUCTIONS
6
TABLE OF CONTENTS
TABLE OF CONTENTS
15 Natural logarithms 154
THE BINOMIAL THEOREM
- g
Logarithmic equations 157
Factorial notation 16 y
W
m
p
The change of base rule 159
Binomial expansions 17 m
Q T
The binomial theorem 21 Solving exponential equations
Q
using logarithms 160
Review set 1A 26
o
Logarithmic functions 164
Review set 1B 27
o
Review set 6A 168
Review set 6B 170
29
QUADRATIC FUNCTIONS

N
Quadratic functions 31
E THE UNIT CIRCLE AND

Graphs of quadratic functions 33
~
W
RADIAN MEASURE 173
Using the discriminant 40
w
Radian measure 174
Q Finding a quadratic from its graph 43
O » Arc length and sector area 177
The intersection of graphs 47
w
m
The unit circle 181
Problem solving with quadratics 50
O
T
Multiples of % and F 187
Optimisation with quadratics 53 o
O
m
The Pythagorean identity 190
T Quadratic inequalities 57
@
Finding angles 192
Review set 2A 61
T
Q The equation of a straight line 194
Review set 2B 62
Review set 7A 195
Review set 7B 197
FUNCTIONS 65

w
Relations and functions 66
»
199
TRIGONOMETRIC FUNCTIONS

Function notation 69
w
>
O Periodic behaviour 200
Domain and range 72
w
g
The sine and cosine functions 204
Rational functions 78 Q
m
a General sine and cosine functions 206
E Composite functions 83
o
O
Modelling periodic behaviour 211
Inverse functions 86
g
T
m The tangent function 216
Q Absolute value functions 91
Review set 8A 219
Review set 3A 93
Review set 8B 221
Review set 3B 96
TRIGONOMETRIC EQUATIONS
TRANSFORMATIONS 0o
»
223
AND IDENTITIES
99
OF FUNCTIONS
Trigonometric equations 224

Translations 100

>
> Using trigonometric models 232
Stretches 103

w
w
Q g Trigonometric identities 234
Reflections 109
a
c Double angle identities 237
Miscellaneous transformations 112
o
Review set 9A 241
Review set 4A 115
Review set 9B 243
Review set 4B 116
245
REASONING AND PROOF
EXPONENTIAL FUNCTIONS 119
=
n
u
Logical connectives 248
Rational exponents 120

>
r
Proof by deduction 249
Algebraic expansion and factorisation 122
w
W O Proof by equivalence 253
Exponential equations 125
Q
Q
Definitions 256
Exponential functions 127 o

O Review set 10A 259
Growth and decay 132
m
Review set 10B 259
The natural exponential 138
M
Review set SA 141
Review set 5B 143 INTRODUCTION TO
DIFFERENTIAL CALCULUS 261
LOGARITHMS 145 Rates of change 263

o
>
Instantaneous rates of change 266
Logarithms in base 10 146
w
>
Q Limits 269
Logarithms in base a 149

a
w The gradient of a tangent 274
Laws of logarithms 151 c
Q
7
TABLE OF CONTENTS

The derivative function 276 399
DEFINITE INTEGRALS

m
=
Differentiation from first principles 278 » Definite integrals 400
-
p
Review set 11A 281
The area under a curve 404
W
Review set 11B 283
The area above a curve 409
Q
The area between two functions 411
o
RULES OF DIFFERENTIATION 285 m Problem solving by integration 416

N
=
»
Simple rules of differentiation 286 Review set 17A 419
w
The chain rule 291 Review set 17B 422
Q
a
The product rule 294
o
The quotient rule 297
g KINEMATICS 425

m =
427
Derivatives of exponential functions 299 e
Displacement
m
w 429
Derivatives of logarithmic functions 303
T Velocity
O
O
436
Derivatives of trigonometric functions 306
Acceleration
o a
439
T Second derivatives 308 g Speed
444
Review set 12A 310
Review set 18A
446
Review set 12B 311
Review set 18B
313
PROPERTIES OF CURVES 19 BIVARIATE STATISTICS 449
-w
314
Tangents Association between numerical variables 450
>
319
»
Normals
Pearson’s product-moment

w w
321 455
Increasing and decreasing correlation coefficient
O
a 326 460
Stationary points Line of best fit by eye

o
331 464
Shape O The least squares regression line
m
o
m 333 471
Inflection points The regression line of x against y
m
T
474
Understanding functions Review set 19A
Q
476
and their derivatives 338
Review set 19B
Review set 13A 340
Review set 13B 342
0
DISCRETE RANDOM VARIABLES 479

N
Random variables 480

14 >
APPLICATIONS OF w
Discrete probability distributions 482
Q
345
DIFFERENTIATION
Expectation 486
O
346
A Rates of change o The binomial distribution 492
g
352
B Optimisation
Using technology to find
m
362
Review set 14A binomial probabilities 496
363
Review set 14B The mean and standard deviation of

1
a binomial distribution 498
Review set 20A 500
15 INTRODUCTION TO
Review set 20B 502
INTEGRATION 365
Approximating the area under a curve 366

» 505
The Riemann integral 369 THE NORMAL DISTRIBUTION
w
N 507
Antidifferentiation 372 Introduction to the normal distribution

>
510
g The Fundamental Theorem of Calculus 374 e Calculating probabilities
w 518
Review set 15A 379 The standard normal distribution
Q
522
Review set 15B 380 g Quantiles
528
Review set 21A
529
Review set 21B
TECHNIQUES FOR INTEGRATION 381

=
» Discovering integrals 382
ANSWERS 531
w Rules for integration 384
Q Particular values 388
o Integrating f(ax + b) 390 INDEX 611
m
Integration by substitution 393
Review set 16A 396
Review set 16B 397
8
SYMBOLS AND NOTATION USED IN THIS COURSE
the set of positive integers and zero, t he common difference of an
{0,1,2,3,...} arithmetic sequence
e set of integers, {0, 1, £2, £3, .... t he common ratio of a geometric
1l

sequence
he set of positive integers, {1, 2, 3, ...}
1l
t he sum of the first n terms of a
the set of rational numbers
sequence, uy + uz + ... + Uy,
he set of irrational numbers
1l
the sum to infinity of a sequence,
the set of real numbers
Uy T+ U2

{z1, T2, ... he set with elements z1, zo, ....
tl

Uy + Uz + o Uy
the number of elements in set A
the set of all = such that X (n—1)x(n..—. x23)x2xx1
T
is an element of
he r binomial coefficient,
t
is not an element of r=0,1,2, ... in the expansion of
the empty (null) set (a+0b)"
the universal set
[ is a function which maps « onto y
union
t he image of under the function f
intersection
t e inverse function of the function f
is a proper subset of
he composite function of f and g
t
is a subset of
he limit of f(z) as tends to a

the complement of the set A =
e derivative of y with respect to
a to the power of l, nth root of a
n
e derivative of f(x) with respect
(if a>0 then /a>0)
=
t
o
a to the power %, square root of a
he second derivative of y with
1l
(if a>0 then a>0)
respect to =
he second derivative of f(z) with
the modulus or absolute value of x t
espect to =
|$|:{ rforr >0 =xzeR T
—zforr <0 zeR he indefinite integral of y with
respect to =
identity or is equivalent to

l
is approximately equal to
o
e definite integral of y with respect
is greater than
t o « between the limits =a and
v
is greater than or equal to r=>b

v 2 V
exponential function of x
is less than

A
is less than or equal to log, = t he logarithm in base a of

N N
I 2 I
is not greater than Inz t he natural logarithm of x, log,

W
is not less than sin, cos, tan t he circular functions
A
the nth term of a sequence or series sin™!, he inverse circular functions
cos™!, tan™!
the point A in the plane with
Cartesian coordinates = and y
the line segment with end points
A and B
AB the length of [AB]
(AB)
the line containing points A and B
the angle at A
CAB
the angle between [CA] and [AB]
AABC
the triangle whose vertices are
A, B,and C
is parallel to
is perpendicular to
P(4)
probability of event A
P(4’)
probability of the event ‘not A’
P(A| B)
probability of the event A given B
T1, T2y e observations of a variable
1, fas o frequencies with which the
observations i, T3, T3, ..... occur
probabilities with which the
observations 1, xa, T3, ..... oceur
the probability distribution function
of the discrete random variable X
the probability mass function of a
discrete random variable X
the expected value of the random
variable X'
population mean
population standard deviation
population variance
sample mean
sample variance
standard deviation of the sample
binomial distribution with parameters
n and p
normal distribution with mean p and
variance o>
is distributed as
T—p
standardised normal z-score, z =

o
Pearson’s product-moment
correlation coefficient