### Transcription

''JUST THE MATHS''byA.J. HobsonTEACHING UNITS - TABLE OF CONTENTS(Average number of pages 1038 140 7.4 per unit)All units are in presented as .PDF files[Home] [Foreword] [About the Author]UNIT 1.1 - ALGEBRA 1 - INTRODUCTION TO ALGEBRA1.1.1 The Language of Algebra1.1.2 The Laws of Algebra1.1.3 Priorities in Calculations1.1.4 Factors1.1.5 Exercises1.1.6 Answers to exercises (6 pages)UNIT 1.2 - ALGEBRA 2 - NUMBERWORK1.2.1 Types of number1.2.2 Decimal numbers1.2.3 Use of electronic calculators1.2.4 Scientific notation1.2.5 Percentages1.2.6 Ratio1.2.7 Exercises1.2.8 Answers to exercises (8 pages)UNIT 1.3 - ALGEBRA 3 - INDICES AND RADICALS (OR SURDS)1.3.1 Indices1.3.2 Radicals (or Surds)1.3.3 Exercises1.3.4 Answers to exercises (8 pages)UNIT 1.4 - ALGEBRA 4 - LOGARITHMS1.4.1 Common logarithms1.4.2 Logarithms in general1.4.3 Useful Results1.4.4 Properties of logarithms1.4.5 Natural logarithms1.4.6 Graphs of logarithmic and exponential functions1.4.7 Logarithmic scales1.4.8 Exercises1.4.9 Answers to exercises (10 pages)UNIT 1.5 - ALGEBRA 5 - MANIPULATION OF ALGEBRAIC EXPRESSIONS1.5.1 Simplification of expressions1.5.2 Factorisation1 of 20

1.5.3 Completing the square in a quadratic expression1.5.4 Algebraic Fractions1.5.5 Exercises1.5.6 Answers to exercises (9 pages)UNIT 1.6 - ALGEBRA 6 - FORMULAE AND ALGEBRAIC EQUATIONS1.6.1 Transposition of formulae1.6.2 Solution of linear equations1.6.3 Solution of quadratic equations1.6.4 Exercises1.6.5 Answers to exercises (7 pages)UNIT 1.7 - ALGEBRA 7 - SIMULTANEOUS LINEAR EQUATIONS1.7.1 Two simultaneous linear equations in two unknowns1.7.2 Three simultaneous linear equations in three unknowns1.7.3 Ill-conditioned equations1.7.4 Exercises1.7.5 Answers to exercises (6 pages)UNIT 1.8 - ALGEBRA 8 - POLYNOMIALS1.8.1 The factor theorem1.8.2 Application to quadratic and cubic expressions1.8.3 Cubic equations1.8.4 Long division of polynomials1.8.5 Exercises1.8.6 Answers to exercises (8 pages)UNIT 1.9 - ALGEBRA 9 - THE THEORY OF PARTIAL FRACTIONS1.9.1 Introduction1.9.2 Standard types of partial fraction problem1.9.3 Exercises1.9.4 Answers to exercises (7 pages)UNIT 1.10 - ALGEBRA 10 - INEQUALITIES 11.10.1 Introduction1.10.2 Algebraic rules for inequalities1.10.3 Intervals1.10.4 Exercises1.10.5 Answers to exercises (5 pages)UNIT 1.11 - ALGEBRA 11 - INEQUALITIES 21.11.1 Recap on modulus, absolute value or numerical value1.11.2 Interval inequalities1.11.3 Exercises1.11.4 Answers to exercises (5 pages)UNIT 2.1 - SERIES 1 - ELEMENTARY PROGRESSIONS AND SERIES2.1.1 Arithmetic progressions2.1.2 Arithmetic series2.1.3 Geometric progressions2.1.4 Geometric series2.1.5 More general progressions and series2.1.6 Exercises2 of 20

2.1.7 Answers to exercises (12 pages)UNIT 2.2 - SERIES 2 - BINOMIAL SERIES2.2.1 Pascal's Triangle2.2.2 Binomial Formulae2.2.3 Exercises2.2.4 Answers to exercises (9 pages)UNIT 2.3 - SERIES 3 - ELEMENTARY CONVERGENCE AND DIVERGENCE2.3.1 The definitions of convergence and divergence2.3.2 Tests for convergence and divergence (positive terms)2.3.3 Exercises2.3.4 Answers to exercises (13 pages)UNIT 2.4 - SERIES 4 - FURTHER CONVERGENCE AND DIVERGENCE2.4.1 Series of positive and negative terms2.4.2 Absolute and conditional convergence2.4.3 Tests for absolute convergence2.4.4 Power series2.4.5 Exercises2.4.6 Answers to exercises (9 pages)UNIT 3.1 - TRIGONOMETRY 1 - ANGLES AND TRIGONOMETRIC FUNCTIONS3.1.1 Introduction3.1.2 Angular measure3.1.3 Trigonometric functions3.1.4 Exercises3.1.5 Answers to exercises (6 pages)UNIT 3.2 - TRIGONOMETRY 2 - GRAPHS OF TRIGONOMETRIC FUNCTIONS3.2.1 Graphs of elementary trigonometric functions3.2.2 Graphs of more general trigonometric functions3.2.3 Exercises3.2.4 Answers to exercises (7 pages)UNIT 3.3 - TRIGONOMETRY 3 - APPROXIMATIONS AND INVERSEFUNCTIONS3.3.1 Approximations for trigonometric functions3.3.2 Inverse trigonometric functions3.3.3 Exercises3.3.4 Answers to exercises (6 pages)UNIT 3.4 - TRIGONOMETRY 4 - SOLUTION OF TRIANGLES3.4.1 Introduction3.4.2 Right-angled triangles3.4.3 The sine and cosine rules3.4.4 Exercises3.4.5 Answers to exercises (5 pages)UNIT 3.5 - TRIGONOMETRY 5 - TRIGONOMETRIC IDENTITIES AND WAVE-FORMS3.5.1 Trigonometric identities3.5.2 Amplitude, wave-length, frequency and phase-angle3.5.3 Exercises3 of 20

3.5.4 Answers to exercises (8 pages)UNIT 4.1 - HYPERBOLIC FUNCTIONS 1 - DEFINITIONS, GRAPHS AND IDENTITIES4.1.1 Introduction4.1.2 Definitions4.1.3 Graphs of hyperbolic functions4.1.4 Hyperbolic identities4.1.5 Osborn's rule4.1.6 Exercises4.1.7 Answers to exercises (7 pages)UNIT 4.2 - HYPERBOLIC FUNCTIONS 2 - INVERSE HYPERBOLIC FUNCTIONS4.2.1 Introduction4.2.2 The proofs of the standard formulae4.2.3 Exercises4.2.4 Answers to exercises (6 pages)UNIT 5.1 - GEOMETRY 1 - CO-ORDINATES, DISTANCE AND GRADIENT5.1.1 Co-ordinates5.1.2 Relationship between polar & cartesian co-ordinates5.1.3 The distance between two points5.1.4 Gradient5.1.5 Exercises5.1.6 Answers to exercises (5 pages)UNIT 5.2 - GEOMETRY 2 - THE STRAIGHT LINE5.2.1 Preamble5.2.2 Standard equations of a straight line5.2.3 Perpendicular straight lines5.2.4 Change of origin5.2.5 Exercises5.2.6 Answers to exercises (8 pages)UNIT 5.3 - GEOMETRY 3 - STRAIGHT LINE LAWS5.3.1 Introduction5.3.2 Laws reducible to linear form5.3.3 The use of logarithmic graph paper5.3.4 Exercises5.3.5 Answers to exercises (7 pages)UNIT 5.4 - GEOMETRY 4 - ELEMENTARY LINEAR PROGRAMMING5.4.1 Feasible Regions5.4.2 Objective functions5.4.3 Exercises5.4.4 Answers to exercises (9 pages)UNIT 5.5 - GEOMETRY 5 - CONIC SECTIONS (THE CIRCLE)5.5.1 Introduction5.5.2 Standard equations for a circle5.5.3 Exercises5.5.4 Answers to exercises (5 pages)UNIT 5.6 - GEOMETRY 6 - CONIC SECTIONS (THE PARABOLA)4 of 20

5.6.1 Introduction (the standard parabola)5.6.2 Other forms of the equation of a parabola5.6.3 Exercises5.6.4 Answers to exercises (6 pages)UNIT 5.7 - GEOMETRY 7 - CONIC SECTIONS (THE ELLIPSE)5.7.1 Introduction (the standard ellipse)5.7.2 A more general form for the equation of an ellipse5.7.2 Exercises5.7.3 Answers to exercises (4 pages)UNIT 5.8 - GEOMETRY 8 - CONIC SECTIONS (THE HYPERBOLA)5.8.1 Introduction (the standard hyperbola)5.8.2 Asymptotes5.8.3 More general forms for the equation of a hyperbola5.8.4 The rectangular hyperbola5.8.5 Exercises5.8.6 Answers to exercises (8 pages)UNIT 5.9 - GEOMETRY 9 - CURVE SKETCHING IN GENERAL5.9.1 Symmetry5.9.2 Intersections with the co-ordinate axes5.9.3 Restrictions on the range of either variable5.9.4 The form of the curve near the origin5.9.5 Asymptotes5.9.6 Exercises5.9.7 Answers to exercises (10 pages)UNIT 5.10 - GEOMETRY 10 - GRAPHICAL SOLUTIONS5.10.1 The graphical solution of linear equations5.10.2 The graphical solution of quadratic equations5.10.3 The graphical solution of simultaneous equations5.10.4 Exercises5.10.5 Answers to exercises (7 pages)UNIT 5.11 - GEOMETRY 11 - POLAR CURVES5.11.1 Introduction5.11.2 The use of polar graph paper5.11.3 Exercises5.11.4 Answers to exercises (10 pages)UNIT 6.1 - COMPLEX NUMBERS 1 - DEFINITIONS AND ALGEBRA6.1.1 The definition of a complex number6.1.2 The algebra of complex numbers6.1.3 Exercises6.1.4 Answers to exercises (8 pages)UNIT 6.2 - COMPLEX NUMBERS 2 - THE ARGAND DIAGRAM6.2.1 Introduction6.2.2 Graphical addition and subtraction6.2.3 Multiplication by j6.2.4 Modulus and argument6.2.5 Exercises5 of 20

6.2.6 Answers to exercises (7 pages)UNIT 6.3 - COMPLEX NUMBERS 3 - THE POLAR AND EXPONENTIAL FORMS6.3.1 The polar form6.3.2 The exponential form6.3.3 Products and quotients in polar form6.3.4 Exercises6.3.5 Answers to exercises (8 pages)UNIT 6.4 - COMPLEX NUMBERS 4 - POWERS OF COMPLEX NUMBERS6.4.1 Positive whole number powers6.4.2 Negative whole number powers6.4.3 Fractional powers & De Moivre's Theorem6.4.4 Exercises6.4.5 Answers to exercises (5 pages)UNIT 6.5 - COMPLEX NUMBERS 5 - APPLICATIONS TO TRIGONOMETRIC IDENTITIES6.5.1 Introduction6.5.2 Expressions for cosn q, sinn q in terms of cosq, sinqnn6.5.3 Expressions for cos q and sin q in terms of sines and cosines of whole multiples ofx6.5.4 Exercises6.5.5 Answers to exercises (5 pages)UNIT 6.6 - COMPLEX NUMBERS 6 - COMPLEX LOCI6.6.1 Introduction6.6.2 The circle6.6.3 The half-straight-line6.6.4 More general loci6.6.5 Exercises6.6.6 Answers to exercises (6 pages)UNIT 7.1 - DETERMINANTS 1 - SECOND ORDER DETERMINANTS7.1.1 Pairs of simultaneous linear equations7.1.2 The definition of a second order determinant7.1.3 Cramer's Rule for two simultaneous linear equations7.1.4 Exercises7.1.5 Answers to exercises (7 pages)UNIT 7.2 - DETERMINANTS 2 - CONSISTENCY AND THIRD ORDER DETERMINANTS7.2.1 Consistency for three simultaneous linear equations in two unknowns7.2.2 The definition of a third order determinant7.2.3 The rule of Sarrus7.2.4 Cramer's rule for three simultaneous linear equations in three unknowns7.2.5 Exercises7.2.6 Answers to exercises (10 pages)UNIT 7.3 - DETERMINANTS 3 - FURTHER EVALUATION OF 3 X 3 DETERMINANTS7.3.1 Expansion by any row or column7.3.2 Row and column operations on determinants7.3.3 Exercises7.3.4 Answers to exercises (10 pages)6 of 20

UNIT 7.4 - DETERMINANTS 4 - HOMOGENEOUS LINEAR EQUATIONS7.4.1 Trivial and non-trivial solutions7.4.2 Exercises7.4.3 Answers to exercises (7 pages)UNIT 8.1 - VECTORS 1 - INTRODUCTION TO VECTOR ALGEBRA8.1.1 Definitions8.1.2 Addition and subtraction of vectors8.1.3 Multiplication of a vector by a scalar8.1.4 Laws of algebra obeyed by vectors8.1.5 Vector proofs of geometrical results8.1.6 Exercises8.1.7 Answers to exercises (7 pages)UNIT 8.2 - VECTORS 2 - VECTORS IN COMPONENT FORM8.2.1 The components of a vector8.2.2 The magnitude of a vector in component form8.2.3 The sum and difference of vectors in component form8.2.4 The direction cosines of a vector8.2.5 Exercises8.2.6 Answers to exercises (6 pages)UNIT 8.3 - VECTORS 3 - MULTIPLICATION OF ONE VECTOR BY ANOTHER8.3.1 The scalar product (or 'dot' product)8.3.2 Deductions from the definition of dot product8.3.3 The standard formula for dot product8.3.4 The vector product (or 'cross' product)8.3.5 Deductions from the definition of cross product8.3.6 The standard formula for cross product8.3.7 Exercises8.3.8 Answers to exercises (8 pages)UNIT 8.4 - VECTORS 4 - TRIPLE PRODUCTS8.4.1 The triple scalar product8.4.2 The triple vector product8.4.3 Exercises8.4.4 Answers to exercises (7 pages)UNIT 8.5 - VECTORS 5 - VECTOR EQUATIONS OF STRAIGHT LINES8.5.1 Introduction8.5.2 The straight line passing through a given point and parallel to a given vector8.5.3 The straight line passing through two given points8.5.4 The perpendicular distance of a point from a straight line8.5.5 The shortest distance between two parallel straight lines8.5.6 The shortest distance between two skew straight lines8.5.7 Exercises8.5.8 Answers to exercises (14 pages)UNIT 8.6 - VECTORS 6 - VECTOR EQUATIONS OF PLANES8.6.1 The plane passing through a given point and perpendicular to a given vector8.6.2 The plane passing through three given points8.6.3 The point of intersection of a straight line and a plane8.6.4 The line of intersection of two planes7 of 20

8.6.5 The perpendicular distance of a point from a plane8.6.6 Exercises8.6.7 Answers to exercises (9 pages)UNIT 9.1 - MATRICES 1 - DEFINITIONS AND ELEMENTARY MATRIX ALGEBRA9.1.1 Introduction9.1.2 Definitions9.1.3 The algebra of matrices (part one)9.1.4 Exercises9.1.5 Answers to exercises (8 pages)UNIT 9.2 - MATRICES 2 - FURTHER MATRIX ALGEBRA9.2.1 Multiplication by a single number9.2.2 The product of two matrices9.2.3 The non-commutativity of matrix products9.2.4 Multiplicative identity matrices9.2.5 Exercises9.2.6 Answers to exercises (6 pages)UNIT 9.3 - MATRICES 3 - MATRIX INVERSION AND SIMULTANEOUS EQUATIONS9.3.1 Introduction9.3.2 Matrix representation of simultaneous linear equations9.3.3 The definition of a multiplicative inverse9.3.4 The formula for a multiplicative inverse9.3.5 Exercises9.3.6 Answers to exercises (11 pages)UNIT 9.4 - MATRICES 4 - ROW OPERATIONS9.4.1 Matrix inverses by row operations9.4.2 Gaussian elimination (the elementary version)9.4.3 Exercises9.4.4 Answers to exercises (10 pages)UNIT 9.5 - MATRICES 5 - CONSISTENCY AND RANK9.5.1 The consistency of simultaneous linear equations9.5.2 The row-echelon form of a matrix9.5.3 The rank of a matrix9.5.4 Exercises9.5.5 Answers to exercises (9 pages)UNIT 9.6 - MATRICES 6 - EIGENVALUES AND EIGENVECTORS9.6.1 The statement of the problem9.6.2 The solution of the problem9.6.3 Exercises9.6.4 Answers to exercises (9 pages)UNIT 9.7 - MATRICES 7 - LINEARLY INDEPENDENT AND NORMALISED EIGENVECTORS9.7.1 Linearly independent eigenvectors9.7.2 Normalised eigenvectors9.7.3 Exercises9.7.4 Answers to exercises (5 pages)UNIT 9.8 - MATRICES 8 - CHARACTERISTIC PROPERTIES AND SIMILARITY8 of 20

TRANSFORMATIONS9.8.1 Properties of eigenvalues and eigenvectors9.8.2 Similar matrices9.8.3 Exercises9.7.4 Answers to exercises (9 pages)UNIT 9.9 - MATRICES 9 - MODAL AND SPECTRAL MATRICES9.9.1 Assumptions and definitions9.9.2 Diagonalisation of a matrix9.9.3 Exercises9.9.4 Answers to exercises (9 pages)UNIT 9.10 - MATRICES 10 - SYMMETRIC MATRICES AND QUADRATIC FORMS9.10.1 Symmetric matrices9.10.2 Quadratic forms9.10.3 Exercises9.10.4 Answers to exercises (7 pages)UNIT 10.1 - DIFFERENTIATION 1 - FUNTIONS AND LIMITS10.1.1 Functional notation10.1.2 Numerical evaluation of functions10.1.3 Functions of a linear function10.1.4 Composite functions10.1.5 Indeterminate forms10.1.6 Even and odd functions10.1.7 Exercises10.1.8 Answers to exercises (12 pages)UNIT 10.2 - DIFFERENTIATION 2 - RATES OF CHANGE10.2.1 Introduction10.2.2 Average rates of change10.2.3 Instantaneous rates of change10.2.4 Derivatives10.2.5 Exercises10.2.6 Answers to exercises (7 pages)UNIT 10.3 - DIFFERENTIATION 3 - ELEMENTARY TECHNIQUES OF DIFFERENTIATION10.3.1 Standard derivatives10.3.2 Rules of differentiation10.3.3 Exercises10.3.4 Answers to exercises (9 pages)UNIT 10.4 - DIFFERENTIATION 4 - PRODUCTS, QUOTIENTS AND LOGARITHMICDIFF

UNIT 5.2 - GEOMETRY 2 - THE STRAIGHT LINE 5.2.1 Preamble 5.2.2 Standard equations of a straight line 5.2.3 Perpendicular straight lines 5.2.4 Change of origin 5.2.5 Exercises 5.2.6 Answers to exercises (8 pages) UNIT 5.3 - GEOMETRY 3 - STRAIGHT LINE LAWS 5.3.1 Introduction 5.3.2 Laws reducible to linear form 5.3.3 The use of logarithmic graph paper 5.3.4 Exercises 5.3.5 Answers to