Mathematics - Applications and Interpretation SL 2

2019·73.3 MB·english

by  Michael Haese

Applicationsfand:
% HAESE MATHEMATICS

Specialists in mathematics education
Mathematics
Applications andl
Imterpretation SIL

Michael Haese
Mark Humphries
Chris Sangwin
Ngoc Vo

for use with
IB Diploma
Programme
MATHEMATICS: APPLICATIONS AND INTERPRETATION 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 978-1-925489-57-6
© 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.
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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
pleased to come to a suitable agreement with the rightful owner.
Disclaimer: 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 cause readers, no responsibility for
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:
Applications and Interpretation 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
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: Applications and Interpretation courses have a focus on
technology, and the book has been written with this focus in mind. Where
appropriate, graphics calculator screenshots and instructions have been
provided to help students use technology to solve problems. An algebraic
approach to solving the problem may be included for completeness, and to
help students enhance their understanding. 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, Investigations, and Research
exercises 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

Maps that have been provided by OpenStreetMap are available freely at www.openstreetmap.org.
Licensing terms can be viewed at www.openstreetmap.org/copyright.
The cartoon on page 392 is by Randall Munroe (xkcd), and is available at https:/xkcd.com/882/.
Licensing terms can be viewed at https:/xkcd.com/license.html.
4
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Access is granted through SNOWFLAKE, our book viewing software that can be used in your web
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.
€7, TICY) LR R (T
Sketch the graph of y = 3cos2z for 0° < z < 360°.

a =3, so the amplitude is |3|=3. Y=3 c os 2
b =2, so the period is
360° _ 360° _ o0
O o
b 2
We stretch y = cosx vertically with scale
factor 3 to give y = 3cosz, then stretch
y = 3cosx horizontally with scale factor
1 to give y = 3cos2z.
See Chapter 9, Trigonometric functions, p. 229
INTERACTIVE LINKS
Interactive links to in-browser tools which complement the text are included to
assist teaching and learning. ICON
Icons like this will direct you to:
e interactive demonstrations to illustrate and animate concepts
e games and other tools for practising your skills
e graphing and statistics packages which are fast, powerful alternatives to using a graphics calculator
e printable pages to save class time.

Save time, and
Filling Containers
make learning easier!

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-3 -2 -1 0 1 2 3
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See Chapter 15, The normal distribution, p. 365
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 TI-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
APPROXIMATIONS AND ERROR QUADRATIC FUNCTIONS 133

= o>
»
Rounding numbers 16 Quadratic functions 35
w
w
O
Approximations 20 Graphs from tables of values 137
Q o
Errors in measurement 22 g Axes intercepts 39
a
m
g Absolute and percentage error 25 Graphs of the form y = ax? 141
m
T
Review set 1A 29 Graphs of quadratic functions 43
o
Review set 1B 30 a Axis of symmetry 144
m
I Vertex 47
T
Finding a quadratic from its graph 149
LOANS AND ANNUITIES 31 I

N
32 Intersection of graphs 52
Loans
>
-
38 Quadratic models 153
Annuities

w
43 Review set 6A 59
Review set 2A
44 Review set 6B 161
Review set 2B
DIRECT AND INVERSE VARIATION 163
45
FUNCTIONS

N
w
W >
Direct variation 64
» Relations and functions 46
w
r Powers in direct variation 168
Function notation 49 O
w
o Inverse variation 70
Domain and range 53
O m
Powers in inverse variation 172
Graphs of functions 57
g H
Determining the variation model 73
Sign diagrams 60 T
m
m Using technology to find variation models 175
m Transformations of graphs 63
Review set 7A 78
O
Inverse functions 69
Review set 7B 80
Review set 3A 73
Review set 3B 76
EXPONENTIALS AND
LOGARITHMS 183
79
MODELLING

s
Exponential functions 85

The modelling cycle 80
>
» Graphing exponential functions from
Linear models 86
w @
a table of values 86
Piecewise linear models 89

Graphs of exponential functions 87
0
Systems of equations 94
g
Q Exponential equations 91
Review set 4A 96
o
Growth and decay 92
Review set 4B 98 m
The natural exponential 99
m
O Logarithms in base 10 204
BIVARIATE STATISTICS 101
nu Natural logarithms 208
T
Association between numerical variables 102

Review set 8A 211
»
Pearson’s product-moment

Review set 8B 213
w
correlation coefficient 107
Line of best fit by eye 112

0 TRIGONOMETRIC FUNCTIONS 217

The least squares regression line 116
o o
>
The unit circle 218
Spearman’s rank correlation coefficient 123
»
m
Periodic behaviour 221
m
Review set S5A 128
®
The sine and cosine functions 224
Review set 5B 130
O
General sine and cosine functions 226
o
m Modelling periodic behaviour 231
Review set 9A 236
Review set 9B 239
7
TABLE OF CONTENTS

241 14 DISCRETE RANDOM VARIABLES 335
DIFFERENTIATION

o
=
»
Rates of change 243 Random variables 336

w >
Instantaneous rates of change 247 Discrete probability distributions 338
O
W
a
Limits 251 Expectation 342
Q
o
The gradient of a tangent 252 The binomial distribution 347
m
o
@ The derivative function 254 Using technology to find
m
m
binomial probabilities 352
Differentiation 256
T
The mean and standard deviation of
Q Rules for differentiation 259
e
a binomial distribution 355
Review set 10A 265
Review set 14A 357
Review set 10B 267
Review set 14B 358
269
PROPERTIES OF CURVES

-
361
THE NORMAL DISTRIBUTION

Tangents 270 =

p Introduction to the normal distribution 363

Normals 273
w
»
Calculating probabilities 366
w
Increasing and decreasing 276

O Quantiles 373

Stationary points 280
O O
Review set 15A 377
Review set 11A 284
Review set 15B 378
Review set 11B 285
381
HYPOTHESIS TESTING
12 APPLICATIONS OF
=
382
P Statistical hypotheses
DIFFERENTIATION 287
384
W
Student’s ¢-test
Rates of change 288

>
Q
The two-sample ¢-test for
Optimisation 293

w comparing population means 393
Modelling with calculus 301
a
The x? goodness of fit test 395

Review set 12A 303 g
The x? test for independence 405

Review set 12B 304 m
Review set 16A 413
Review set 16B 415
307
INTEGRATION

—w
308
Approximating the area under a curve

> VORONOI DIAGRAMS 417
313
The Riemann integral
—
w
Voronoi diagrams 418
W 317
The Fundamental Theorem of Calculus
Q W>
Constructing Voronoi diagrams 422
320
Antidifferentiation and indefinite integrals
w

Adding a site to a Voronoi diagram 427
o 322
Q
Rules for integration
m
Nearest neighbour interpolation 431
324 o
M Particular values
T m The Largest Empty Circle problem 433
325
Definite integrals
O
Review set 17A 437
328
The area under a curve

T
Review set 17B 439
331
Review set 13A
333
Review set 13B
ANSWERS 441
INDEX 503
8
SYMBOLS AND NOTATION USED IN THIS COURSE
N the set of positive integers and zero, # is not greater than
{0,1,2,3, ..}
e is not less than
Z the set of integers, {0, £1, +2, £3, ...} Up the nth term of a sequence or series

7+ the set of positive integers, {1, 2, 3, ....} d the common difference of an
Q the set of rational numbers arithmetic sequence
Q the set of irrational numbers T the common ratio of a geometric
sequence
R the set of real numbers qu
. Sn the sum of the first n terms of a

{1, z2, ...} the set with elements z1, za, .... sequence -
uy 4 ug + ..+
n(A) the number of elements in set A o "
Soc or S the sum to infinity of a sequence,
{z]. the set of all = such that up + Us A+ ...

€ is an element of n

¢ is not an element of 1:21 vi YLz T T
@or{ } theempty (null) set n! nx(n—1)x(n—2)x..x3x2x1
v t1§ universal set (") or"C, the r™ binomial coefficient,
u union r=0,1, 2, .... in the expansion of
n intersection (a+b)"
c fs a proper subset of f(z) the image of z under the function f
C is a subset of 1 i i i
o th 1 { of the set A the inverse function of the function f
e complement of the se . .
. lim f(z) the limit of f(z) as z tends to a
L 1 z—a
a", {/a a to the power of —, nth root ofa d
" d_y the derivative of y with respect to x
(if a>0 then /a>0) v
1 f(z) the derivative of f(z) with respect
a?, \a a to the power %, square root of a tox
(if >0 then a>0) fy dx th e indefinite integral ofy with
respect to =
|| the modulus or absolute value of . P
|z| = zforz>0 zeR / y dz the definite integral of y with respect
—zforz <0 zekR a to between the limits = = a and
= identity or is equivalent to T="b
~ is approximately equal to e* exponential function of x
> is greater than log the logarithm in base 10 of=
>or > is greater than or equal to Inz the natural logarithm of x, log, z
< is less than sin, cos, tan the circular functions
<or< is less than or equal to sin~?t, the inverse circular functions
cos™!, tan™!
E(X)
the point A in the plane with the expected value of the random
Cartesian coordinates = and y variable X
population mean
the line segment with end points
A and B population standard deviation
AB
the length of [AB]
population variance
(AB) the line containing points A and B
sample mean
PB(A, B)
the perpendicular bisector of [AB]
sample variance
the angle at A standard deviation of the sample
CAB
binomial distribution with
the angle between [CA] and [AB]
AABC parameters n and p
the triangle whose vertices are
A, B,and C normal distribution with mean s
and variance o2
is parallel to
is distributed as
is perpendicular to
Pearson’s product-moment
P(4) probability of event A
correlation coefficient
P(A)
probability of the event ‘not A’
the null hypothesis
P(A| B)
probability of the event A given B
the alternative hypothesis
T1, T2y e observations of a variable
the random variable 7" has the
frequencies with which the Student’s ¢ distribution with
f1s fay e
observations 1, T2, T3, ..... occur n —1 degrees of freedom
P1s P2, -t probabilities with which the chi-squared
observations i, xa, T3, ..... oceur
calculated chi-squared value
P(X =2x) the probability distribution function
critical value of the chi-squared
of the discrete random variable X
distribution
P(z) the probability mass function of a
observed frequency
fobs
discrete random variable X
Jexp expected frequency