Mathematics for the IB Diploma: Higher Level PDF
Preview Mathematics for the IB Diploma: Higher Level
Mathematics Higher Level for the IB Diploma Paul Fannon, Vesna Kadelburg, Ben Woolley and Stephen Ward Not for printing, sharing or distribution. cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press Th e Edinburgh Building, Cambridge CB2 8RU, UK www.cambridge.org Information on this title: www.cambridge.org/9781107661738 © Cambridge University Press 2012 Th is publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2012 Printed and bound in Great Britain by the MPG Books Group A catalogue record for this publication is available from the British Library ISBN 978-1-107-66173-8 Paperback with CD-ROM for Windows® and Mac® Cover image: Craig Jewell/Shutterstock Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work is correct at the time of fi rst printing but Cambridge University Press does not guarantee the accuracy of such information thereaft er. notice to teachers Worksheets andcopies of them remain in the copyright of Cambridge University Press and such copies may not be distributed or used in any way outside the purchasing institution. Not for printing, sharing or distribution. Contents iii Contents Introduction vii Algebra (Topic 1: Algebra) 1 Counting principles 1 1A The product principle and the addition principle 1 1B Counting arrangements 6 1C Algebra of factorials 9 1D Counting selections 11 1E Exclusion principle 15 1F Counting ordered selections 18 1G Keeping objects together or separated 21 Algebra, functions and equations (Topics 1: Algebra & 2: Functions & equations) 2 Exponents and logarithms 28 2A Laws of exponents 28 2B Exponential functions 33 2C The value e 39 2D Introducing logarithms 40 2E Laws of logarithms 44 2F Graphs of logarithms 49 2G Solving exponential equations 50 3 Polynomials 58 3A Working with polynomials 58 3B Remainder and factor theorems 63 3C Sketching polynomial functions 68 3D The quadratic formula and the discriminant 76 4 Algebraic structures 87 4A Solving equations by factorising 87 4B Solving equations by substitution 89 4C Features of graphs 91 4D Using a graphical calculator to solve equations 94 4E Solving simultaneous equations by substitution 97 4F Systems of linear equations 101 4G Solving inequalities 110 4H Working with identities 113 5 The theory of functions 118 5A Relations, functions and graphs 118 5B Function notation 123 5C Domain and range 125 5D Composite functions 131 5E Inverse functions 133 5F Rational functions 141 6 Transformations of graphs 154 6A Translations 155 6B Stretches 157 6C Refl ections 161 6D Modulus transformations 163 6E Consecutive transformations 167 6F Reciprocal transformations 174 6G Symmetries of graphs and functions 178 7 Sequences and series 190 7A General sequences 190 7B General series and sigma notation 193 7C Arithmetic sequences 195 7D Arithmetic series 199 7E Geometric sequences 201 © Cambridge University Press 2012 Not for printing, sharing or distribution. iv Contents 7F Geometric series 205 7G Infi nite geometric series 207 7H Mixed questions 211 8 Binomial expansion 217 8A Introducing the binomial theorem 217 8B Applying the binomial theorem 219 8C Products of binomial expansions 223 8D Binomial expansions as approximations 226 Geometry (Topics 3: Circular functions & trigonometry & 4: Vectors) 9 Circular measure and trigonometric functions 232 9A Radian measure 232 9B Defi nitions and graphs of sine and cosine functions 239 9C Defi nition and graph of the tangent function 248 9D Exact values of trigonometric functions 251 9E Transformations of trigonometric graphs 254 9F Modelling using trigonometric functions 261 9G Inverse trigonometric functions 264 10 Trigonometric equations and identities 273 10A Introducing trigonometric equations 273 10B Harder trigonometric equations 281 10C Trigonometric identities 291 10D Using identities to solve equations 297 11 Geometry of triangles and circles 304 11A Right-angled triangles 305 11B The sine rule 307 11C The cosine rule 312 11D Area of a triangle 318 11E Trigonometry in three dimensions 321 11F Length of an arc 326 11G Area of a sector 330 11H Triangles and circles 333 12 Further trigonometry 346 12A Double angle identities 347 12B Compound angle identities 354 12C Functions of the form a sin x + b cos x 359 12D Reciprocal trigonometric functions 364 13 Vectors 375 13A Positions and displacements 375 13B Vector algebra 384 13C Distances 389 13D Angles 393 13E Properties of the scalar product 397 13F Areas 402 13G Properties of the vector product 404 14 Lines and planes in space 413 14A Vector equation of a line 413 14B Solving problems with lines 420 14C Other forms of equation of a line 432 14D Equation of a plane 439 14E Angles and intersections between lines and planes 449 14F Intersection of three planes 457 14G Strategies for solving problems with lines and planes 463 © Cambridge University Press 2012 Not for printing, sharing or distribution. Contents v Calculus (Topic 6: Calculus) Algebra (Topic 1: Algebra) 16 Basic differentiation and its applications 527 16A Sketching derivatives 527 16B Differentiation from fi rst principles 535 16C Rules of differentiation 538 16D Interpreting derivatives and second derivatives 541 16E Trigonometric functions 547 16F The exponential and natural logarithm functions 549 16G Tangents and normals 550 16H Stationary points 554 16I General points of infl exion 560 16J Optimisation 562 17 Basic integration and its applications 569 17A Reversing differentiation 569 17B Constant of integration 571 17C Rules of integration 572 17D Integrating x–1 and ex 575 17E Integrating trigonometric functions 576 17F Finding the equation of a curve 577 17G Defi nite integration 579 17H Geometrical signifi cance of defi nite integration 581 17I The area between a curve and the y-axis 588 17J The area between two curves 591 18 Further differentiation methods 599 18A Differentiating composite functions using the chain rule 599 18B Differentiating products using the product rule 604 18C Differentiating quotients using the quotient rule 607 18D Implicit differentiation 611 18E Differentiating inverse trigonometric functions 616 19 Further integration methods 622 19A Reversing standard derivatives 622 19B Integration by substitution 626 19C Using trigonometric identities in integration 634 19D Integration using inverse trigonometric functions 641 19E Other strategies for integrating quotients 644 19F Integration by parts 649 20 Further applications of calculus 659 20A Related rates of change 659 20B Kinematics 663 20C Volumes of revolution 668 20D Optimisation with constraints 674 15 Complex numbers 476 15A Defi nition and basic arithmetic of i 476 15B Geometric interpretation 482 15C Properties of complex conjugates 487 15D Complex solutions to polynomial equations 493 15E Sums and products of roots of polynomials 498 15F Operations in polar form 504 15G Complex exponents 510 15H Roots of complex numbers 512 15I Using complex numbers to derive trigonometric identities 517 © Cambridge University Press 2012 Not for printing, sharing or distribution. vi Contents Probability and statistics (Topic 5: Statistics & probability) Algebra (Topic 1: Algebra) 21 Summarising data 689 21A Some important concepts in statistics 689 21B Measures of spread 691 21C Frequency tables and grouped data 695 22 Probability 705 22A Introduction to probability 705 22B Combined events and Venn diagrams 711 22C Tree diagrams and fi nding the intersection 715 22D Independent events 720 22E Counting principles in probability 722 22F Conditional probability 724 22G Further Venn diagrams 727 22H Bayes’ theorem 732 23 Discrete probability distributions 743 23A Random variables 743 23B Expectation, median and variance of a discrete random variable 747 23C The binomial distribution 751 23D The Poisson distribution 758 24 Continuous distributions 769 24A Continuous random variables 769 24B Expectation and variance of continuous random variables 773 24C The normal distribution 777 24D Inverse normal distribution 783 25 Mathematical induction 791 25A The principle of mathematical induction 791 25B Induction and series 793 25C Induction and sequences 795 25D Induction and differentiation 799 25E Induction and divisibility 801 25F Induction and inequalities 803 26 Questions crossing all chapters 809 Examination tips 820 Answers 821 Index 911 Acknowledgements 917 Terms and conditions of use for the CD-ROM 918 © Cambridge University Press 2012 Not for printing, sharing or distribution. Introduction vii Structure of the book Th e book is split roughly into four blocks: chapters 2 to 8 cover algebra and functions; chapters 9 to 14 cover geometry; chapters 16 to 20 cover calculus; and chapters 21 to 24 cover probability and statistics. Chapters 1, 15 and 25; on counting principles, complex numbers and induction (respectively), bring together several areas of the course and chapter 26 includes questions that mix diff erent parts of the course – a favourite trick in International Baccalaureate® (IB) examinations. You do not have to work through the book in the order presented, but (given how much the International Baccalaureate® likes to mix up topics) you will fi nd that several questions refer to material in previous chapters. In these cases a ‘rewind panel’ will tell you that the material has been covered previously so that you can either remind yourself or decide to move on. We have tried to include in the book only the material that will be examinable. Th ere are many proofs and ideas which are useful and interesting, and these are on the CD-ROM if you would like to explore them. Each chapter starts with a list of learning objectives to give you an idea about what the chapter contains. Th ere is also an introductory problem that illustrates what you will be able to do aft er you have completed the chapter. Some introductory problems relate to ‘real life’ situations, while others are purely mathematical. You should not expect to be able to solve the problem, but you may want to think about possible strategies and what sort of new facts and methods would help you. Th e solution to the introductory problem is provided at the end of the chapter, aft er the summary of the chapter contents. Key point boxes Th e most important ideas and formulae are emphasised in the ‘KEY POINT’ boxes. When the formulae are given in the Formula booklet, there will be an icon: ; if this icon is not present, then the formulae are not in the Formula booklet and you may need to learn them or at least know how to derive them. Worked examples Each worked example is split into two columns. On the right is what you should write down. Sometimes the example might include more detail then you strictly need, but it is designed to give you an idea of what is required to score full method marks in examinations. However, mathematics is about much more than examinations and remembering methods. So, on the left of the worked examples are notes that describe the thought processes and suggest which route you should use to tackle the question. We hope that these will help you with any exercise questions that diff er from the worked examples. It is very deliberate that some of the questions require you to do more than repeat the methods in the worked examples. Mathematics is about thinking! Introduction © Cambridge University Press 2012 Not for printing, sharing or distribution. viii Introduction Signposts Th ere are several boxes that appear throughout the book. Theory of knowledge issues Every lesson is a Th eory of knowledge lesson, but sometimes the links may not be obvious. Mathematics is frequently used as an example of certainty and truth, but this is oft en not the case. In these boxes we will try to highlight some of the weaknesses and ambiguities in mathematics as well as showing how mathematics links to other areas of knowledge. From another perspective Th e International Baccalaureate® encourages looking at things in diff erent ways. As well as highlighting some international diff erences between mathematicians these boxes also look at other perspectives on the mathematics we are covering: historical, pragmatic and cultural. Research explorer As part of your course, you will be asked to write a report on a mathematical topic of your choice. It is sometimes diffi cult to know which topics are suitable as a basis for such reports, and so we have tried to show where a topic can act as a jumping-off point for further work. Th is can also give you ideas for an Extended essay. Th ere is a lot of great mathematics out there! Exam hint Although we would encourage you to think of mathematics as more than just learning in order to pass an examination, there are some common errors it is useful for you to be aware of. If there is a common pitfall we will try to highlight it in these boxes. We also point out where graphical calculators can be used eff ectively to simplify a question or speed up your work, oft en referring to the relevant calculator skills sheet on the CD-ROM. Fast forward / rewind Mathematics is all about making links. You might be interested to see how something you have just learned will be used elsewhere in the course, or you may need to go back and remind yourself of a previous topic. Th ese boxes indicate connections with other sections of the book to help you fi nd your way around. How to use the questions The colour-coding Th e questions are colour-coded to distinguish between the levels. Black questions are drill questions. Th ey help you practise the methods described in the book, but they are usually not structured like the questions in the examination. Th is does not mean they are easy, some of them are quite tough. exam hint © Cambridge University Press 2012 Not for printing, sharing or distribution. Introduction ix Each diff erently numbered drill question tests a diff erent skill. Lettered subparts of a question are of increasing diffi culty. Within each lettered part there may be multiple roman-numeral parts ((i), (ii),...) , all of which are of a similar diffi culty. Unless you want to do lots of practice we would recommend that you only do one roman-numeral part and then check your answer. If you have made a mistake then you may want to think about what went wrong before you try any more. Otherwise move on to the next lettered part. Green questions are examination-style questions which should be accessible to students on the path to getting a grade 3 or 4. Blue questions are harder examination-style questions. If you are aiming for a grade 5 or 6 you should be able to make signifi cant progress through most of these. Red questions are at the very top end of diffi culty in the examinations. If you can do these then you are likely to be on course for a grade 7. Gold questions are a type that are not set in the examination, but are designed to provoke thinking and discussion in order to help you to a better understanding of a particular concept. At the end of each chapter you will see longer questions typical of the second section of International Baccalaureate® examinations. Th ese follow the same colour-coding scheme. Of course, these are just guidelines. If you are aiming for a grade 6, do not be surprised if you fi nd a green question you cannot do. People are never equally good at all areas of the syllabus. Equally, if you can do all the red questions that does not guarantee you will get a grade 7; aft er all, in the examination you have to deal with time pressure and examination stress! Th ese questions are graded relative to our experience of the fi nal examination, so when you fi rst start the course you will fi nd all the questions relatively hard, but by the end of the course they should seem more straightforward. Do not get intimidated! Calculator versus non-calculator questions In the fi nal examination there will be one paper in which calculators are not allowed. Some questions specifi cally need a calculator but most could appear in either the calculator or the non- calculator paper. Some questions are particularly common in the non-calculator paper and you must be able to know how to deal with these. Th ey are highlighted by the non-calculator symbol. Some questions can be done in a clever way using a calculator or cannot be realistically done without using a calculator. Th ese questions are marked with a calculator symbol. In the fi nal examination you will not get this indication, so you must make sure you learn which types of questions have an easy calculator method. Th e calculator skills sheets on the CD-ROM can help with this. For the questions that do not have either calculator icon, you should mix up practising with and without a calculator. Be careful not to become too reliant on your calculator. Half of the core examination is done without one! © Cambridge University Press 2012 Not for printing, sharing or distribution. x Introduction On the CD-ROM On the CD-ROM there are various materials that you might fi nd useful. Prior learning Th e International Baccalaureate® syllabus lists what candidates should know before taking the examination. Th e topics on the list are not all explicitly covered in the syllabus, but their knowledge may be required to answer examination questions. Don’t worry, you do not need to know all this before starting the course, and we have indicated in the rewind boxes where a particular concept or skill is required. On the CD-ROM you will fi nd a self-assessment test to check your knowledge, and some worksheets to help you learn any skills that you might be missing or need to revise. Option chapters Each of the four options is covered by several chapters on the CD-ROM. Coursebook support Supporting worksheets include: • calculator skills sheets that give instructions for making optimal use of some of the recomended graphical calculators • extension worksheets that go in diffi culty beyond what is required at International Baccalaureate® • fi ll-in proof sheets to allow you to re-create proofs that are not required in the examination • self-discovery sheets to encourage you to investigate new results for yourself • supplementary sheets exploring some applications, international and historical perspectives of the mathematics covered in the syllabus. e-version A fl at pdf of the whole coursebook (for days when you don’t want to carry the paperback!). We hope you fi nd Higher Level Mathematics for the IB diploma an interesting and enriching course. You might also fi nd it quite challenging, but do not get intimidated, frequently topics only make sense aft er lots of revision and practice. Persevere and you will succeed. Th e author team. © Cambridge University Press 2012 Not for printing, sharing or distribution. 1 Counting principles 1 1 Counting is one of the fi rst things we learn in mathematics and at fi rst it seems very simple. If you were asked to count how many people there are in your school, this would not be too tricky. If you were asked how many chess matches would need to be played if everyone were to play everyone else, this would be a little more complicated. If you were asked how many diff erent football teams could be chosen, you might fi nd that the numbers become far too large to count without coming up with some clever tricks. Th is chapter aims to help develop strategies for counting in such diffi cult situations. 1A The product principle and the addition principle Counting very small groups is easy. So, we need to break down more complicated problems into counting small groups. But how do we then combine these together to come up with an answer to the overall problem? Th e answer lies in using the product principle and the addition principle, which can be illustrated using the following menu. Counting sometimes gets extremely diffi cult. Are there more whole numbers or odd numbers; fractions or decimals? Have a look at the work of Georg Cantor, and the result may surprise you! Counting principles Introductory problem If a computer can print a line containing all 26 letters of the alphabet in 0.01 seconds, estimate how long it would take to print all possible permutations of the alphabet. In this chapter you will learn: how to break down • complicated questions into parts that are easier to count, and then combine them together how to count the • number of ways to arrange a set of objects the algebraic • properties of a useful new tool called the factorial function in how many ways you • can choose objects from a group strategies for applying • these tools to harder problems. © Cambridge University Press 2012 Not for printing, sharing or distribution. 2 Topic 1: Algebra Mains Pizza Hamburger Paella Desserts Ice-cream Fruit salad ad MENU mburger aella essert ger a erts rger la erts Pizza Anna would like to order a main course and a dessert. She can choose one of three main courses and one of two desserts. How many diff erent choices could she make? Bob would like to order either a main course or a dessert. He can choose one of the three main courses or one of the two desserts; how many diff erent orders can he make? We can use the notation n(A) to represent the number of ways of making a choice about A. Th e product principle tells us that when we want to select one option from A and one option from B we multiply the individual possibilities together. KEY POINT 1.1 KEY POINT 1.1 Th e Product principle (AND rule) Th e number of ways of in which both choice A and choice B can be made is the product of the number of options for A and the number of options for B. n(A AND B) = n(A) × n(B) Th e addition principle tells us that when we wish to select one option from A or one option from B we add the individual possibilities together. Th e addition principle has one essential restriction. You can only use it if there is no overlap between the choices for A and the choices for B. For example, you cannot apply the addition principle to counting the number of ways of getting an odd The word analysis literally means ‘breaking up’. When a problem is analysed it is broken down into simpler parts. One of the purposes of studying mathematics is to develop an analytical mind, which is considered very useful in many different disciplines. n(A) means the size of the set of options of A. See Prior Learning Section G on the CD-ROM. © Cambridge University Press 2012 Not for printing, sharing or distribution. 1 Counting principles 3 number or a prime number on a die. If there is no overlap between the choices for A and for B, the two events are mutually exclusive. KEY POINT 1.2 KEY POINT 1.2 Th e Addition Principle (OR rule) Th e number of ways of in which either choice A or choice B can be made is the sum of the number of options for A and the number of options for B. If A and B are mutually exclusive then n(A OR B) = n(A) + n(B) Th e hardest part of applying either the addition or product principle is breaking the problem down and deciding which principle to use. You must make sure that you have included all of the cases and checked that they are mutually exclusive. It is oft en useful to rewrite questions to emphasise what is required, ‘AND’ or ‘OR’. Worked example 1.1 An examination has ten questions in section A and four questions in section B. How many diff erent ways are there to choose questions if you must: (a) choose one question from each section? (b) choose a question from either section A or section B? Describe the problem accurately (a) Choose one question from A (10 ways) AND one from B (4 ways) ‘AND’ means we should apply the product principle: n (A) × n (B) Number of ways = 10 × 4 = 40 Describe the problem accurately (b) Choose one question from A (10 ways) OR one from B (4 ways) ‘OR’ means we should apply the addition principle: n (A) + n (B) Number of ways = 10 + 4 = 14 In the example above we cannot answer a question twice so there are no repeated objects; however, this is not always the case. © Cambridge University Press 2012 Not for printing, sharing or distribution. 4 Topic 1: Algebra Th is leads us to a general idea. KEY POINT 1.3 KEY POINT 1.3 Th e number of ways of selecting something r times from n objects is nr. Exercise 1A 1. If there are 10 ways of doing A, 3 ways of doing B and 19 ways of doing C, how many ways are there of doing (a) (i) both A and B? (ii) both B and C? (b) (i) either A or B? (ii) either A or C? 2. If there are 4 ways of doing A, 7 ways of doing B and 5 ways of doing C, how many ways are there of doing (a) all of A, B and C? (b) exactly one of A, B or C? 3. How many diff erent paths are there (a) from A to C? (b) from C to E? (c) from A to E? A B C D E Worked example 1.2 In a class there are awards for best mathematician, best sportsman and nicest person. Students can receive more than one award. In how many ways can the awards be distributed if there are twelve people in the class? Describe the problem accurately Choose one of 12 people for the best mathematician (12 ways) AND one of the 12 for best sportsmen (12 ways) AND one of the 12 for nicest person. (12 ways) Apply the product principle 12 × 12 × 12 = 1728 © Cambridge University Press 2012 Not for printing, sharing or distribution.
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