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Mathematics - Applications and Interpretation SL 2 PDF

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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: +61 8 8210 4666, Fax: +61 8 8354 1238 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. This book is copyright. Except as permitted by the Copyright Act (any fair dealing for the purposes of private study, research, criticism or review), no part of this publication may be reproduced, 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 Mathematics. Copying for educational purposes: Where copies of part or the whole of the book are made under Part VB of the Copyright Act, the law requires that the educational institution or the body that administers it has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the Copyright Agency Limited. 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 ONLINE FEATURES With the purchase of a new textbook you will gain 24 months subscription to our online product. This subscription can be renewed for a small fee. 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 For iPads, tablets, and other mobile devices, some of the interactive features may not work. However, the digital version of the textbook can be viewed online using any of these devices. REGISTERING You will need to register to access the online features of this textbook. Visit www.haesemathematics.com/register and follow the instructions. Once registered, you can: e activate your digital textbook e use your account to make additional purchases. To activate your digital textbook, contact Haese Mathematics. On providing proof of purchase, your digital textbook will be activated. It is important that you keep your receipt as proof of purchase. For general queries about registering and subscriptions: e Visit our SNOWFLAKE help page: https://snowflake haesemathematics.com.au/help e Contact Haese Mathematics: [email protected] 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! Normal probability distribution © Y 1 0.8 0.6 G0 mI T 600 ml 00 il 100wl ] 1200 ml 50ml T 150 mi] 250wl T 550 ml 0.4 0.2 S30ml | 550 ml 200 ml 200 ml 150 ml 1150 ml 200ml | 1300 ml S0ml T 1200 ml oe -6 -5 4 -3 -2 -1 1 2 3 4 5 6 p:oo2 See Chapter 10, Differentiation, p. 242 v -3 -2 -1 0 1 2 3 o 0.75 0.5 0.75 1 1.25 15 L.75 2 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

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