Pure Mathematics Core

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Transcript of Pure Mathematics Core

  • Pure Mathematics Core

    N. P. Strickland

    December 9, 2003

  • 2

  • Contents

    1 Introduction 51.1 Technicalities about these notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2 Algebraic manipulation 72.1 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Manipulation of algebraic fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    3 Sets 193.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Sets of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Boolean operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 De Morgans laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.6 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.7 Proving relations between sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    4 General theory of functions 314.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 The range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

    5 Special functions 435.1 The exponential and the logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.3 Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4 Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    5.4.1 Trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.5 Special values of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . 505.6 Inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.7 Advanced special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    6 Differentiation 536.1 The meaning of differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Differentiation from first principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.3 Derivatives of special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.4 Rules for differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    6.4.1 The product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.4.2 The quotient rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.4.3 The power rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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  • 4 CONTENTS

    6.4.4 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.4.5 The logarithmic rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.4.6 The inverse function rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

    6.5 Examples of differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

    7 Integration 617.1 The meaning of integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.2 Guessing and checking of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.3 Integrals of standard functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

    7.3.1 Rational functions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.3.2 Trigonometric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.3.3 Exponential oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

    7.4 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.5 Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

    8 Vectors and matrices 798.1 Vectors and dot products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.3 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    8.3.1 Matrices for linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . 848.3.2 Row operations and echelon form . . . . . . . . . . . . . . . . . . . . . . . . 868.3.3 Gaussian elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    8.4 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.5 Determinants and inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

    8.5.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938.5.2 The transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.5.3 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

    A Complex numbers 103

    B Maple 105

    C The Greek alphabet 107

    D Solutions 109

    E Additional background 137

  • Chapter 1: IntroductionThis is a core course, designed to review and reinforce some fundamental ideas that are used inall areas of mathematics. The main topics are as follows:

    Manipulation of algebraic expressions and inequalities.

    Sets of numbers, vectors and other mathematical objects; geometric figures as sets of points;sets of solutions to equations or systems of equations.

    General theory of functions between sets. Composing and inverting functions; finding rangesof functions.

    Particular special functions, such as sin, cos, exp and log.

    Special classes of functions, such as polynomials, rational functions, periodic functions, bellcurves, decaying oscillations, and so on.

    Differentiation: the meaning of derivatives, and general techniques for calculating them.Derivatives of particular special functions, and of special classes of functions.

    Integration: the meaning of integrals, and general techniques for calculating them. Integralsof particular special functions, and of special classes of functions.

    Vectors and matrices, emphasising the link with systems of linear equations.

    Complex numbers are discussed briefly in an appendix, and are used occasionally in themain body of the notes.

    Many of you will have met many of these topics at school already. However, in this course, wewill take a slightly different point of view. We will stand back a little from the detailed calculations,look for patterns and common features, and try to understand a little more deeply why thingswork the way they do.

    This course will also provide some pointers to other areas of mathematics that you may end upstudying over the next few years. All the examples in this course are carefully chosen, and mosthave some kind of story behind them. Many are related to particular applications of mathematics,or they involve functions with special properties or geometric meaning, or they will arise naturallyfrom a topic in some other module. The notes will contain brief accounts of some of these stories;it is up to you to decide how far to follow them.

    There are many exercises, some of which you will be asked to prepare for discussion in tutorials.Solutions are given at the end of the notes for about half of the exercises. Solutions for theremaining exercises will be released towards the end of the course, to help you with revision. Morechallenging exercises are marked with a star.

    Finally, the flavour of this course is influenced by the availability of computer algebra systems(although we will not be using them systematically at this stage). These systems (such as Mapleand Mathematica) can carry out many kinds of manipulations automatically, such as expandingout complicated expressions, plotting graphs, differentiating and integrating, and so on. Thismakes it more important for humans to understand the larger picture and learn how to formulateproblems and extract information.

    1.1 Technicalities about these notes

    These notes are available both on paper and as a PDF file. You will need Acrobat Reader to viewthe PDF file, but that is available on the Managed XP service on the campus network, is usuallyinstalled on any reasonably new PC, and can be downloaded freely from http://www.adobe.com/products/acrobat/readermain.html. The PDF version makes extensive use of colour, and also

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  • 6 CHAPTER 1. INTRODUCTION

    has hyperlinks both within the notes and pointing to various external websites. You may need tostart up your browser before following the external links; there is no good reason why this shouldbe necessary, but it seems to be.

    You may find it convenient to view the notes in full screen mode. For this, you should firstclick on the link to the notes with the right-hand button on your mouse, which will allow you tosave the file to your own disk space rather than just opening it in the browser. You can then openthe file in the Acrobat Reader, and type CTRL-L (or select Full Screen View from the Windowmenu in the Acrobat toolbar) to change to full screen mode. The notes will then fill your wholescreen. You can return to the normal mode by pressing the Esc key, or typing CTRL-L. While infull screen mode, you can