# ‘JUST THE MATHS”

### by

## A.J. Hobson

## TEACHING UNITS – TABLE OF CONTENTS

### (Average number of pages = 1038 �140 = 7.4 per unit)

### All units are in presented as .PDF files

## [About the Author]

**UNIT 1.1 – ALGEBRA 1 – INTRODUCTION TO ALGEBRA**

1.1.1 The Language of Algebra

1.1.2 The Laws of Algebra

1.1.3 Priorities in Calculations

1.1.4 Factors

1.1.5 Exercises

1.1.6 Answers to exercises (6 pages)

**UNIT 1.2 – ALGEBRA 2 – NUMBERWORK**

1.2.1 Types of number

1.2.2 Decimal numbers

1.2.3 Use of electronic calculators

1.2.4 Scientific notation

1.2.5 Percentages

1.2.6 Ratio

1.2.7 Exercises

1.2.8 Answers to exercises (8 pages)

**UNIT 1.3 – ALGEBRA 3 – INDICES AND RADICALS (OR SURDS)**

1.3.1 Indices

1.3.2 Radicals (or Surds)

1.3.3 Exercises

1.3.4 Answers to exercises (8 pages)

**UNIT 1.4 – ALGEBRA 4 – LOGARITHMS**

1.4.1 Common logarithms

1.4.2 Logarithms in general

1.4.3 Useful Results

1.4.4 Properties of logarithms

1.4.5 Natural logarithms

1.4.6 Graphs of logarithmic and exponential functions

1.4.7 Logarithmic scales

1.4.8 Exercises

1.4.9 Answers to exercises (10 pages)

**UNIT 1.5 – ALGEBRA 5 – MANIPULATION OF ALGEBRAIC EXPRESSIONS**

1.5.1 Simplification of expressions

1.5.2 Factorisation

1.5.3 Completing the square in a quadratic expression

1.5.4 Algebraic Fractions

1.5.5 Exercises

1.5.6 Answers to exercises (9 pages)

**UNIT 1.6 – ALGEBRA 6 – FORMULAE AND ALGEBRAIC EQUATIONS**

1.6.1 Transposition of formulae

1.6.2 Solution of linear equations

1.6.3 Solution of quadratic equations

1.6.4 Exercises

1.6.5 Answers to exercises (7 pages)

**UNIT 1.7 – ALGEBRA 7 – SIMULTANEOUS LINEAR EQUATIONS**

1.7.1 Two simultaneous linear equations in two unknowns

1.7.2 Three simultaneous linear equations in three unknowns

1.7.3 Ill-conditioned equations

1.7.4 Exercises

1.7.5 Answers to exercises (6 pages)

**UNIT 1.8 – ALGEBRA 8 – POLYNOMIALS**

1.8.1 The factor theorem

1.8.2 Application to quadratic and cubic expressions

1.8.3 Cubic equations

1.8.4 Long division of polynomials

1.8.5 Exercises

1.8.6 Answers to exercises (8 pages)

**UNIT 1.9 – ALGEBRA 9 – THE THEORY OF PARTIAL FRACTIONS**

1.9.1 Introduction

1.9.2 Standard types of partial fraction problem

1.9.3 Exercises

1.9.4 Answers to exercises (7 pages)

**UNIT 1.10 – ALGEBRA 10 – INEQUALITIES 1**

1.10.1 Introduction

1.10.2 Algebraic rules for inequalities

1.10.3 Intervals

1.10.4 Exercises

1.10.5 Answers to exercises (5 pages)

**UNIT 1.11 – ALGEBRA 11 – INEQUALITIES 2**

1.11.1 Recap on modulus, absolute value or numerical value

1.11.2 Interval inequalities

1.11.3 Exercises

1.11.4 Answers to exercises (5 pages)

**UNIT 2.1 – SERIES 1 – ELEMENTARY PROGRESSIONS AND SERIES**

2.1.1 Arithmetic progressions

2.1.2 Arithmetic series

2.1.3 Geometric progressions

2.1.4 Geometric series

2.1.5 More general progressions and series

2.1.6 Exercises

2.1.7 Answers to exercises (12 pages)

**UNIT 2.2 – SERIES 2 – BINOMIAL SERIES**

2.2.1 Pascal’s Triangle

2.2.2 Binomial Formulae

2.2.3 Exercises

2.2.4 Answers to exercises (9 pages)

**UNIT 2.3 – SERIES 3 – ELEMENTARY CONVERGENCE AND DIVERGENCE**

2.3.1 The definitions of convergence and divergence

2.3.2 Tests for convergence and divergence (positive terms)

2.3.3 Exercises

2.3.4 Answers to exercises (13 pages)

**UNIT 2.4 – SERIES 4 – FURTHER CONVERGENCE AND DIVERGENCE**

2.4.1 Series of positive and negative terms

2.4.2 Absolute and conditional convergence

2.4.3 Tests for absolute convergence

2.4.4 Power series

2.4.5 Exercises

2.4.6 Answers to exercises (9 pages)

**UNIT 3.1 – TRIGONOMETRY 1 – ANGLES AND TRIGONOMETRIC FUNCTIONS**

3.1.1 Introduction

3.1.2 Angular measure

3.1.3 Trigonometric functions

3.1.4 Exercises

3.1.5 Answers to exercises (6 pages)

**UNIT 3.2 – TRIGONOMETRY 2 – GRAPHS OF TRIGONOMETRIC FUNCTIONS**

3.2.1 Graphs of elementary trigonometric functions

3.2.2 Graphs of more general trigonometric functions

3.2.3 Exercises

3.2.4 Answers to exercises (7 pages)

**UNIT 3.3 – TRIGONOMETRY 3 – APPROXIMATIONS AND INVERSE**

FUNCTIONS

3.3.1 Approximations for trigonometric functions

3.3.2 Inverse trigonometric functions

3.3.3 Exercises

3.3.4 Answers to exercises (6 pages)

**UNIT 3.4 – TRIGONOMETRY 4 – SOLUTION OF TRIANGLES**

3.4.1 Introduction

3.4.2 Right-angled triangles

3.4.3 The sine and cosine rules

3.4.4 Exercises

3.4.5 Answers to exercises (5 pages)

**UNIT 3.5 – TRIGONOMETRY 5 – TRIGONOMETRIC IDENTITIES AND WAVE-FORMS**

3.5.1 Trigonometric identities

3.5.2 Amplitude, wave-length, frequency and phase-angle

3.5.3 Exercises

3.5.4 Answers to exercises (8 pages)

**UNIT 4.1 – HYPERBOLIC FUNCTIONS 1 – DEFINITIONS, GRAPHS AND IDENTITIES**

4.1.1 Introduction

4.1.2 Definitions

4.1.3 Graphs of hyperbolic functions

4.1.4 Hyperbolic identities

4.1.5 Osborn’s rule

4.1.6 Exercises

4.1.7 Answers to exercises (7 pages)

**UNIT 4.2 – HYPERBOLIC FUNCTIONS 2 – INVERSE HYPERBOLIC FUNCTIONS**

4.2.1 Introduction

4.2.2 The proofs of the standard formulae

4.2.3 Exercises

4.2.4 Answers to exercises (6 pages)

**UNIT 5.1 – GEOMETRY 1 – CO-ORDINATES, DISTANCE AND GRADIENT**

5.1.1 Co-ordinates

5.1.2 Relationship between polar & cartesian co-ordinates

5.1.3 The distance between two points

5.1.4 Gradient

5.1.5 Exercises

5.1.6 Answers to exercises (5 pages)

**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 exercises (7 pages)

**UNIT 5.4 – GEOMETRY 4 – ELEMENTARY LINEAR PROGRAMMING**

5.4.1 Feasible Regions

5.4.2 Objective functions

5.4.3 Exercises

5.4.4 Answers to exercises (9 pages)

**UNIT 5.5 – GEOMETRY 5 – CONIC SECTIONS (THE CIRCLE)**

5.5.1 Introduction

5.5.2 Standard equations for a circle

5.5.3 Exercises

5.5.4 Answers to exercises (5 pages)

**UNIT 5.6 – GEOMETRY 6 – CONIC SECTIONS (THE PARABOLA)**

5.6.1 Introduction (the standard parabola)

5.6.2 Other forms of the equation of a parabola

5.6.3 Exercises

5.6.4 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 ellipse

5.7.2 Exercises

5.7.3 Answers to exercises (4 pages)

**UNIT 5.8 – GEOMETRY 8 – CONIC SECTIONS (THE HYPERBOLA)**

5.8.1 Introduction (the standard hyperbola)

5.8.2 Asymptotes

5.8.3 More general forms for the equation of a hyperbola

5.8.4 The rectangular hyperbola

5.8.5 Exercises

5.8.6 Answers to exercises (8 pages)

**UNIT 5.9 – GEOMETRY 9 – CURVE SKETCHING IN GENERAL**

5.9.1 Symmetry

5.9.2 Intersections with the co-ordinate axes

5.9.3 Restrictions on the range of either variable

5.9.4 The form of the curve near the origin

5.9.5 Asymptotes

5.9.6 Exercises

5.9.7 Answers to exercises (10 pages)

**UNIT 5.10 – GEOMETRY 10 – GRAPHICAL SOLUTIONS**

5.10.1 The graphical solution of linear equations

5.10.2 The graphical solution of quadratic equations

5.10.3 The graphical solution of simultaneous equations

5.10.4 Exercises

5.10.5 Answers to exercises (7 pages)

**UNIT 5.11 – GEOMETRY 11 – POLAR CURVES**

5.11.1 Introduction

5.11.2 The use of polar graph paper

5.11.3 Exercises

5.11.4 Answers to exercises (10 pages)

**UNIT 6.1 – COMPLEX NUMBERS 1 – DEFINITIONS AND ALGEBRA **

6.1.1 The definition of a complex number

6.1.2 The algebra of complex numbers

6.1.3 Exercises

6.1.4 Answers to exercises (8 pages)

**UNIT 6.2 – COMPLEX NUMBERS 2 – THE ARGAND DIAGRAM**

6.2.1 Introduction

6.2.2 Graphical addition and subtraction

6.2.3 Multiplication by j

6.2.4 Modulus and argument

6.2.5 Exercises

6.2.6 Answers to exercises (7 pages)

**UNIT 6.3 – COMPLEX NUMBERS 3 – THE POLAR AND EXPONENTIAL FORMS**

6.3.1 The polar form

6.3.2 The exponential form

6.3.3 Products and quotients in polar form

6.3.4 Exercises

6.3.5 Answers to exercises (8 pages)

**UNIT 6.4 – COMPLEX NUMBERS 4 – POWERS OF COMPLEX NUMBERS**

6.4.1 Positive whole number powers

6.4.2 Negative whole number powers

6.4.3 Fractional powers & De Moivre’s Theorem

6.4.4 Exercises

6.4.5 Answers to exercises (5 pages)

**UNIT 6.5 – COMPLEX NUMBERS 5 – APPLICATIONS TO TRIGONOMETRIC IDENTITIES**

6.5.1 Introduction

6.5.2 Expressions for cosn q, sinn q in terms of cosq, sinq

6.5.3 Expressions for cos^{n}q and sin^{n}q in terms of sines and cosines of whole multiples of x

6.5.4 Exercises

6.5.5 Answers to exercises (5 pages)

**UNIT 6.6 – COMPLEX NUMBERS 6 – COMPLEX LOCI**

6.6.1 Introduction

6.6.2 The circle

6.6.3 The half-straight-line

6.6.4 More general loci

6.6.5 Exercises

6.6.6 Answers to exercises (8 pages)

**UNIT 7.1 – DETERMINANTS 1 – SECOND ORDER DETERMINANTS**

7.1.1 Pairs of simultaneous linear equations

7.1.2 The definition of a second order determinant

7.1.3 Cramer’s Rule for two simultaneous linear equations

7.1.4 Exercises

7.1.5 Answers to exercises (7 pages)

**UNIT 7.2 – DETERMINANTS 2 – CONSISTENCY AND THIRD ORDER DETERMINANTS**

7.2.1 Consistency for three simultaneous linear equations in two unknowns

7.2.2 The definition of a third order determinant

7.2.3 The rule of Sarrus

7.2.4 Cramer’s rule for three simultaneous linear equations in three unknowns

7.2.5 Exercises

7.2.6 Answers to exercises (10 pages)

**UNIT 7.3 – DETERMINANTS 3 – FURTHER EVALUATION OF 3 X 3 DETERMINANTS**

7.3.1 Expansion by any row or column

7.3.2 Row and column operations on determinants

7.3.3 Exercises

7.3.4 Answers to exercises (10 pages)

**UNIT 7.4 – DETERMINANTS 4 – HOMOGENEOUS LINEAR EQUATIONS**

7.4.1 Trivial and non-trivial solutions

7.4.2 Exercises

7.4.3 Answers to exercises (7 pages)

**UNIT 8.1 – VECTORS 1 – INTRODUCTION TO VECTOR ALGEBRA**

8.1.1 Definitions

8.1.2 Addition and subtraction of vectors

8.1.3 Multiplication of a vector by a scalar

8.1.4 Laws of algebra obeyed by vectors

8.1.5 Vector proofs of geometrical results

8.1.6 Exercises

8.1.7 Answers to exercises (7 pages)

**UNIT 8.2 – VECTORS 2 – VECTORS IN COMPONENT FORM**

8.2.1 The components of a vector

8.2.2 The magnitude of a vector in component form

8.2.3 The sum and difference of vectors in component form

8.2.4 The direction cosines of a vector

8.2.5 Exercises

8.2.6 Answers to exercises (6 pages)

**UNIT 8.3 – VECTORS 3 – MULTIPLICATION OF ONE VECTOR BY ANOTHER**

8.3.1 The scalar product (or ‘dot’ product)

8.3.2 Deductions from the definition of dot product

8.3.3 The standard formula for dot product

8.3.4 The vector product (or ‘cross’ product)

8.3.5 Deductions from the definition of cross product

8.3.6 The standard formula for cross product

8.3.7 Exercises

8.3.8 Answers to exercises (8 pages)

**UNIT 8.4 – VECTORS 4 – TRIPLE PRODUCTS**

8.4.1 The triple scalar product

8.4.2 The triple vector product

8.4.3 Exercises

8.4.4 Answers to exercises (7 pages)

**UNIT 8.5 – VECTORS 5 – VECTOR EQUATIONS OF STRAIGHT LINES**

8.5.1 Introduction

8.5.2 The straight line passing through a given point and parallel to a given vector

8.5.3 The straight line passing through two given points

8.5.4 The perpendicular distance of a point from a straight line

8.5.5 The shortest distance between two parallel straight lines

8.5.6 The shortest distance between two skew straight lines

8.5.7 Exercises

8.5.8 Answers to exercises (14 pages)

**UNIT 8.6 – VECTORS 6 – VECTOR EQUATIONS OF PLANES**

8.6.1 The plane passing through a given point and perpendicular to a given vector

8.6.2 The plane passing through three given points

8.6.3 The point of intersection of a straight line and a plane

8.6.4 The line of intersection of two planes

8.6.5 The perpendicular distance of a point from a plane

8.6.6 Exercises

8.6.7 Answers to exercises (9 pages)

**UNIT 9.1 – MATRICES 1 – DEFINITIONS AND ELEMENTARY MATRIX ALGEBRA**

9.1.1 Introduction

9.1.2 Definitions

9.1.3 The algebra of matrices (part one)

9.1.4 Exercises

9.1.5 Answers to exercises (8 pages)

**UNIT 9.2 – MATRICES 2 – FURTHER MATRIX ALGEBRA**

9.2.1 Multiplication by a single number

9.2.2 The product of two matrices

9.2.3 The non-commutativity of matrix products

9.2.4 Multiplicative identity matrices

9.2.5 Exercises

9.2.6 Answers to exercises (6 pages)

**UNIT 9.3 – MATRICES 3 – MATRIX INVERSION AND SIMULTANEOUS EQUATIONS**

9.3.1 Introduction

9.3.2 Matrix representation of simultaneous linear equations

9.3.3 The definition of a multiplicative inverse

9.3.4 The formula for a multiplicative inverse

9.3.5 Exercises

9.3.6 Answers to exercises (11 pages)

**UNIT 9.4 – MATRICES 4 – ROW OPERATIONS**

9.4.1 Matrix inverses by row operations

9.4.2 Gaussian elimination (the elementary version)

9.4.3 Exercises

9.4.4 Answers to exercises (10 pages)

**UNIT 9.5 – MATRICES 5 – CONSISTENCY AND RANK**

9.5.1 The consistency of simultaneous linear equations

9.5.2 The row-echelon form of a matrix

9.5.3 The rank of a matrix

9.5.4 Exercises

9.5.5 Answers to exercises (9 pages)

**UNIT 9.6 – MATRICES 6 – EIGENVALUES AND EIGENVECTORS**

9.6.1 The statement of the problem

9.6.2 The solution of the problem

9.6.3 Exercises

9.6.4 Answers to exercises (9 pages)

**UNIT 9.7 – MATRICES 7 – LINEARLY INDEPENDENT AND NORMALISED EIGENVECTORS**

9.7.1 Linearly independent eigenvectors

9.7.2 Normalised eigenvectors

9.7.3 Exercises

9.7.4 Answers to exercises (5 pages)

**UNIT 9.8 – MATRICES 8 – CHARACTERISTIC PROPERTIES AND SIMILARITY TRANSFORMATIONS**

9.8.1 Properties of eigenvalues and eigenvectors

9.8.2 Similar matrices

9.8.3 Exercises

9.7.4 Answers to exercises (9 pages)

**UNIT 9.9 – MATRICES 9 – MODAL AND SPECTRAL MATRICES**

9.9.1 Assumptions and definitions

9.9.2 Diagonalisation of a matrix

9.9.3 Exercises

9.9.4 Answers to exercises (9 pages)

**UNIT 9.10 – MATRICES 10 – SYMMETRIC MATRICES AND QUADRATIC FORMS**

9.10.1 Symmetric matrices

9.10.2 Quadratic forms

9.10.3 Exercises

9.10.4 Answers to exercises (7 pages)

**UNIT 10.1 – DIFFERENTIATION 1 – FUNTIONS AND LIMITS**

10.1.1 Functional notation

10.1.2 Numerical evaluation of functions

10.1.3 Functions of a linear function

10.1.4 Composite functions

10.1.5 Indeterminate forms

10.1.6 Even and odd functions

10.1.7 Exercises

10.1.8 Answers to exercises (12 pages)

**UNIT 10.2 – DIFFERENTIATION 2 – RATES OF CHANGE**

10.2.1 Introduction

10.2.2 Average rates of change

10.2.3 Instantaneous rates of change

10.2.4 Derivatives

10.2.5 Exercises

10.2.6 Answers to exercises (7 pages)

**UNIT 10.3 – DIFFERENTIATION 3 – ELEMENTARY TECHNIQUES OF DIFFERENTIATION**

10.3.1 Standard derivatives

10.3.2 Rules of differentiation

10.3.3 Exercises

10.3.4 Answers to exercises (9 pages)

**UNIT 10.4 – DIFFERENTIATION 4 – PRODUCTS, QUOTIENTS AND LOGARITHMIC DIFFERENTIATION**

10.4.1 Products

10.4.2 Quotients

10.4.3 Logarithmic differentiation

10.4.4 Exercises

10.4.5 Answers to exercises (10 pages)

**UNIT 10.5 – DIFFERENTIATION 5 – IMPLICIT AND PARAMETRIC FUNCTIONS**

10.5.1 Implicit functions

10.5.2 Parametric functions

10.5.3 Exercises

10.5.4 Answers to exercises (5 pages)

**UNIT 10.6 – DIFFERENTIATION 6 – DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS**

10.6.1 Summary of results

10.6.2 The derivative of an inverse sine

10.6.3 The derivative of an inverse cosine

10.6.4 The derivative of an inverse tangent

10.6.5 Exercises

10.6.6 Answers to exercises (7 pages)

**UNIT 10.7 – DIFFERENTIATION 7 – DERIVATIVES OF INVERSE HYPERBOLIC FUNCTIONS**

10.7.1 Summary of results

10.7.2 The derivative of an inverse hyperbolic sine

10.7.3 The derivative of an inverse hyperbolic cosine

10.7.4 The derivative of an inverse hyperbolic tangent

10.7.5 Exercises

10.7.6 Answers to exercises (7 pages)

**UNIT 10.8 – DIFFERENTIATION 8 – HIGHER DERVIVATIVES**

10.8.1 The theory

10.8.2 Exercises

10.8.3 Answers to exercises (4 pages)

**UNIT 11.1 – DIFFERENTIATION APPLICATIONS 1 – TANGENTS AND NORMALS**

11.1.1 Tangents

11.1.2 Normals

11.1.3 Exercises

11.1.4 Answers to exercises (5 pages)

**UNIT 11.2 – DIFFERENTIATION APPLICATIONS 2 – LOCAL MAXIMA, LOCAL MINIMA AND POINTS OF INFLEXION**

11.2.1 Introduction

11.2.2 Local maxima

11.2.3 Local minima

11.2.4 Points of inflexion

11.2.5 The location of stationary points and their nature

11.2.6 Exercises

11.2.7 Answers to exercises (14 pages)

**UNIT 11.3 – DIFFERENTIATION APPLICATIONS 3 – CURVATURE**

11.3.1 Introduction

11.3.2 Curvature in cartesian co-ordinates

11.3.3 Exercises

11.3.4 Answers to exercises (6 pages)

**UNIT 11.4 – DIFFERENTIATION APPLICATIONS 4 – CIRCLE, RADIUS AND CENTRE OF CURVATURE**

11.4.1 Introduction

11.4.2 Radius of curvature

11.4.3 Centre of curvature

11.4.4 Exercises

11.4.5 Answers to exercises (5 pages)

**UNIT 11.5 – DIFFERENTIATION APPLICATIONS 5 – MACLAURIN’S AND TAYLOR’S SERIES**

11.5.1 Maclaurin’s series

11.5.2 Standard series

11.5.3 Taylor’s series

11.5.4 Exercises

11.5.5 Answers to exercises (10 pages)

**UNIT 11.6 – DIFFERENTIATION APPLICATIONS 6 – SMALL INCREMENTS AND SMALL ERRORS**

11.6.1 Small increments

11.6.2 Small errors

11.6.3 Exercises

11.6.4 Answers to exercises (8 pages)

**UNIT 12.1 – INTEGRATION 1 – ELEMENTARY INDEFINITE INTEGRALS**

12.1.1 The definition of an integral

12.1.2 Elementary techniques of integration

12.1.3 Exercises

12.1.4 Answers to exercises (11 pages)

**UNIT 12.2 – INTEGRATION 2 – INTRODUCTION TO DEFINITE INTEGRALS**

12.2.1 Definition and examples

12.2.2 Exercises

12.2.3 Answers to exercises (3 pages)

**UNIT 12.3 – INTEGRATION 3 – THE METHOD OF COMPLETING THE SQUARE**

12.3.1 Introduction and examples

12.3.2 Exercises

12.3.3 Answers to exercises (4 pages)

**UNIT 12.4 – INTEGRATION 4 – INTEGRATION BY SUBSTITUTION IN GENERAL**

12.4.1 Examples using the standard formula

12.4.2 Integrals involving a function and its derivative

12.4.3 Exercises

12.4.4 Answers to exercises (5 pages)

**UNIT 12.5 – INTEGRATION 5 – INTEGRATION BY PARTS**

12.5.1 The standard formula

12.5.2 Exercises

12.5.3 Answers to exercises (6 pages)

**UNIT 12.6 – INTEGRATION 6 – INTEGRATION BY PARTIAL FRACTIONS**

12.6.1 Introduction and illustrations

12.6.2 Exercises

12.6.3 Answers to exercises (4 pages)

**UNIT 12.7 – INTEGRATION 7 – FURTHER TRIGONOMETRIC FUNCTIONS**

12.7.1 Products of sines and cosines

12.7.2 Powers of sines and cosines

12.7.3 Exercises

12.7.4 Answers to exercises (7 pages)

**UNIT 12.8 – INTEGRATION 8 – THE TANGENT SUBSTITUTIONS**

12.8.1 The substitution t = tanx

12.8.2 The substitution t = tan(x/2)

12.8.3 Exercises

12.8.4 Answers to exercises (5 pages)

**UNIT 12.9 – INTEGRATION 9 – REDUCTION FORMULAE**

12.9.1 Indefinite integrals

12.9.2 Definite integrals

12.9.3 Exercises

12.9.4 Answers to exercises (7 pages)

**UNIT 12.10 – INTEGRATION 10 – FURTHER REDUCTION FORMULAE**

12.10.1 Integer powers of a sine

12.10.2 Integer powers of a cosine

12.10.3 Wallis’s formulae

12.10.4 Combinations of sines and cosines

12.10.5 Exercises

12.10.6 Answers to exercises (8 pages)

**UNIT 13.1 – INTEGRATION APPLICATIONS 1 – THE AREA UNDER A CURVE**

13.1.1 The elementary formula

13.1.2 Definite integration as a summation

13.1.3 Exercises

13.1.4 Answers to exercises (6 pages)

**UNIT 13.2 – INTEGRATION APPLICATIONS 2 – MEAN AND ROOT MEAN SQUARE VALUES**

13.2.1 Mean values

13.2.2 Root mean square values

13.2.3 Exercises

13.2.4 Answers to exercises (4 pages)

**UNIT 13.3 – INTEGRATION APLICATIONS 3 – VOLUMES OF REVOLUTION**

13.3.1 Volumes of revolution about the x-axis

13.3.2 Volumes of revolution about the y-axis

13.3.3 Exercises

13.3.4 Answers to exercises (7 pages)

**UNIT 13.4 – INTEGRATION APPLICATIONS 4 – LENGTHS OF CURVES**

13.4.1 The standard formulae

13.4.2 Exercises

13.4.3 Answers to exercises (5 pages)

**UNIT 13.5 – INTEGRATION APPLICATIONS 5 – SURFACES OF REVOLUTION**

13.5.1 Surfaces of revolution about the x-axis

13.5.2 Surfaces of revolution about the y-axis

13.5.3 Exercises

13.5.4 Answers to exercises (7 pages)

**UNIT 13.6 – INTEGRATION APPLICATIONS 6 – FIRST MOMENTS OF AN ARC**

13.6.1 Introduction

13.6.2 First moment of an arc about the y-axis

13.6.3 First moment of an arc about the x-axis

13.6.4 The centroid of an arc

13.6.5 Exercises

13.6.6 Answers to exercises (11 pages)

**UNIT 13.7 – INTEGRATION APPLICATIONS 7 – FIRST MOMENTS OF AN AREA**

13.7.1 Introduction

13.7.2 First moment of an area about the y-axis

13.7.3 First moment of an area about the x-axis

13.7.4 The centroid of an area

13.7.5 Exercises

13.7.6 Answers to exercises (12 pages)

**UNIT 13.8 – INTEGRATION APPLICATIONS 8 – FIRST MOMENTS OF A VOLUME**

13.8.1 Introduction

13.8.2 First moment of a volume of revolution about a plane through the origin, perpendicular to the x-axis

13.8.3 The centroid of a volume

13.8.4 Exercises

13.8.5 Answers to exercises (10 pages)

**UNIT 13.9 – INTEGRATION APPLICATIONS 9 – FIRST MOMENTS OF A SURFACE OF REVOLUTION**

13.9.1 Introduction

13.9.2 Integration formulae for first moments

13.9.3 The centroid of a surface of revolution

13.9.4 Exercises

13.9.5 Answers to exercises (11 pages)

**UNIT 13.10 – INTEGRATION APPLICATIONS 10 – SECOND MOMENTS OF AN ARC**

13.10.1 Introduction

13.10.2 The second moment of an arc about the y-axis

13.10.3 The second moment of an arc about the x-axis

13.10.4 The radius of gyration of an arc

13.10.5 Exercises

13.10.6 Answers to exercises (11 pages)

**UNIT 13.11 – INTEGRATION APPLICATIONS 11 – SECOND MOMENTS OF AN AREA (A)**

13.11.1 Introduction

13.11.2 The second moment of an area about the y-axis

13.11.3 The second moment of an area about the x-axis

13.11.4 Exercises

13.11.5 Answers to exercises (8 pages)

**UNIT 13.12 – INTEGRATION APPLICATIONS 12 – SECOND MOMENTS OF AN AREA (B)**

13.12.1 The parallel axis theorem

13.12.2 The perpendicular axis theorem

13.12.3 The radius of gyration of an area

13.12.4 Exercises

13.12.5 Answers to exercises (8 pages)

**UNIT 13.13 – INTEGRATION APPLICATIONS 13 – SECOND MOMENTS OF A VOLUME (A)**

13.13.1 Introduction

13.13.2 The second moment of a volume of revolution about the y-axis

13.13.3 The second moment of a volume of revolution about the x-axis

13.13.4 Exercises

13.13.5 Answers to exercises (8 pages)

**UNIT 13.14 – INTEGRATION APPLICATIONS 14 – SECOND MOMENTS OF A VOLUME (B)**

13.14.1 The parallel axis theorem

13.14.2 The radius of gyration of a volume

13.14.3 Exercises

13.14.4 Answers to exercises (6 pages)

**UNIT 13.15 – INTEGRATION APPLICATIONS 15 – SECOND MOMENTS OF A SURFACE OF REVOLUTION**

13.15.1 Introduction

13.15.2 Integration formulae for second moments

13.15.3 The radius of gyration of a surface of revolution

13.15.4 Exercises

13.15.5 Answers to exercises (9 pages)

**UNIT 13.16 – INTEGRATION APPLICATIONS 16 – CENTRES OF PRESSURE**

13.16.1 The pressure at a point in a liquid

13.16.2 The pressure on an immersed plate

13.16.3 The depth of the centre of pressure

13.16.4 Exercises

13.16.5 Answers to exercises (9 pages)

**UNIT 14.1 – PARTIAL DIFFERENTIATION 1 – PARTIAL DERIVATIVES OF THE FIRST ORDER**

14.1.1 Functions of several variables

14.1.2 The definition of a partial derivative

14.1.3 Exercises

14.1.4 Answers to exercises (7 pages)

**UNIT 14.2 – PARTIAL DIFFERENTIATION 2 – PARTIAL DERIVATIVES OF THE SECOND AND HIGHER ORDERS**

14.2.1 Standard notations and their meanings

14.2.2 Exercises

14.2.3 Answers to exercises (5 pages)

**UNIT 14.3 – PARTIAL DIFFERENTIATION 3 – SMALL INCREMENTS AND SMALL ERRORS**

14.3.1 Functions of one independent variable – a recap

14.3.2 Functions of more than one independent variable

14.3.3 The logarithmic method

14.3.4 Exercises

14.3.5 Answers to exercises (10 pages)

**UNIT 14.4 – PARTIAL DIFFERENTIATION 4 – EXACT DIFFERENTIALS**

14.4.1 Total differentials

14.4.2 Testing for exact differentials

14.4.3 Integration of exact differentials

14.4.4 Exercises

14.4.5 Answers to exercises (9 pages)

**UNIT 14.5 – PARTIAL DIFFERENTIATION 5 – PARTIAL DERIVATIVES OF COMPOSITE FUNCTIONS**

14.5.1 Single independent variables

14.5.2 Several independent variables

14.5.3 Exercises

14.5.4 Answers to exercises (8 pages)

**UNIT 14.6 – PARTIAL DIFFERENTIATION 6 – IMPLICIT FUNCTIONS**

14.6.1 Functions of two variables

14.6.2 Functions of three variables

14.6.3 Exercises

14.6.4 Answers to exercises (6 pages)

**UNIT 14.7 – PARTIAL DIFFERENTIATON 7 – CHANGE OF INDEPENDENT VARIABLE**

14.7.1 Illustrations of the method

14.7.2 Exercises

14.7.3 Answers to exercises (5 pages)

**UNIT 14.8 – PARTIAL DIFFERENTIATON 8 – DEPENDENT AND INDEPENDENT FUNCTIONS** [October 2009]

14.8.1 The Jacobian

14.8.2 Exercises

14.8.3 Answers to exercises (8 pages)

**UNIT 14.9 – PARTIAL DIFFERENTIATON 9 – TAYLOR’S SERIES FOR FUNCTIONS OF SEVERAL VARIABLES**

14.9.1 The theory and formula

14.9.2 Exercises (8 pages)

**UNIT 14.10 – PARTIAL DIFFERENTIATON 10 – STATIONARY VALUES FOR FUNCTIONS OF TWO VARIABLES**

14.10.1 Introduction

14.10.2 Sufficient conditions for maxima and minima 14.10.3 Exercises

14.10.4 Answers to exercises (9 pages)

**UNIT 14.11 – PARTIAL DIFFERENTIATON 11 – CONSTRAINED MAXIMA AND MINIMA**

14.11.1 The substitution method

14.11.2 The method of Lagrange multipliers 14.11.3 Exercises

14.11.4 Answers to exercises (11 pages)

**UNIT 14.12 – PARTIAL DIFFERENTIATON 12 – THE PRINCIPLE OF LEAST SQUARES**

14.12.1 The normal equations

14.11.2 Simplified calculation of regression lines 14.11.3 Exercises

14.11.4 Answers to exercises (9 pages)

**UNIT 15.1 – ORDINARY DIFFERENTIAL EQUATIONS 1 – FIRST ORDER EQUATIONS (A)**

15.1.1 Introduction and definitions

15.1.2 Exact equations

15.1.3 The method of separation of the variables

15.1.4 Exercises

15.1.5 Answers to exercises (8 pages)

**UNIT 15.2 – ORDINARY DIFFERENTIAL EQUATIONS 2 – FIRST ORDER EQUATIONS (B)**

15.2.1 Homogeneous equations

15.2.2 The standard method

15.2.3 Exercises

15.2.4 Answers to exercises (6 pages)

**UNIT 15.3 – ORDINARY DIFFERENTIAL EQUATIONS 3 – FIRST ORDER EQUATIONS (C)**

15.3.1 Linear equations

15.3.2 Bernouilli’s equation

15.3.3 Exercises

15.3.4 Answers to exercises (9 pages)

**UNIT 15.4 – ORDINARY DIFFERENTIAL EQUATIONS 4 – SECOND ORDER EQUATIONS (A)**

15.4.1 Introduction

15.4.2 Second order homogeneous equations

15.4.3 Special cases of the auxiliary equation

15.4.4 Exercises

15.4.5 Answers to exercises (9 pages)

**UNIT 15.5 – ORDINARY DIFFERENTIAL EQUATIONS 5 – SECOND ORDER EQUATIONS (B)**

15.5.1 Non-homogeneous differential equations

15.5.2 Determination of simple particular integrals

15.5.3 Exercises

15.5.4 Answers to exercises (6 pages)

**UNIT 15.6 – ORDINARY DIFFERENTIAL EQUATIONS 6 – SECOND ORDER EQUATIONS (C)**

15.6.1 Recap

15.6.2 Further types of particular integral

15.6.3 Exercises

15.6.4 Answers to exercises (7 pages)

**UNIT 15.7 – ORDINARY DIFFERENTIAL EQUATIONS 7 – SECOND ORDER EQUATIONS (D)**

15.7.1 Problematic cases of particular integrals

15.7.2 Exercises

15.7.3 Answers to exercises (6 pages)

**UNIT 15.8 – ORDINARY DIFFERENTIAL EQUATIONS 8 – SIMULTANEOUS EQUATIONS (A)**

15.8.1 The substitution method

15.8.2 Exercises

15.8.3 Answers to exercises (5 pages)

**UNIT 15.9 – ORDINARY DIFFERENTIAL EQUATIONS 9 – SIMULTANEOUS EQUATIONS (B)**

15.9.1 Introduction

15.9.2 Matrix methods for homogeneous systems

15.9.3 Exercises

15.9.4 Answers to exercises (8 pages)

**UNIT 15.10 – ORDINARY DIFFERENTIAL EQUATIONS 10 – SIMULTANEOUS EQUATIONS (C)**

15.10.1 Matrix methods for non-homogeneous systems

15.10.2 Exercises

15.10.3 Answers to exercises (10 pages)

**UNIT 16.1 – LAPLACE TRANSFORMS 1 – DEFINITIONS AND RULES**

16.1.1 Introduction

16.1.2 Laplace Transforms of simple functions

16.1.3 Elementary Laplace Transform rules

16.1.4 Further Laplace Transform rules

16.1.5 Exercises

16.1.6 Answers to exercises (10 pages)

**UNIT 16.2 – LAPLACE TRANSFORMS 2 – INVERSE LAPLACE TRANSFORMS**

16.2.1 The definition of an inverse Laplace Transform

16.2.2 Methods of determining an inverse Laplace Transform

16.2.3 Exercises

16.2.4 Answers to exercises (8 pages)

**UNIT 16.3 – LAPLACE TRANSFORMS 3 – DIFFERENTIAL EQUATIONS**

16.3.1 Examples of solving differential equations

16.3.2 The general solution of a differential equation

16.3.3 Exercises

16.3.4 Answers to exercises (7 pages)

**UNIT 16.4 – LAPLACE TRANSFORMS 4 – SIMULTANEOUS DIFFERENTIAL EQUATIONS**

16.4.1 An example of solving simultaneous linear differential equations

16.4.2 Exercises

16.4.3 Answers to exercises (5 pages)

**UNIT 16.5 – LAPLACE TRANSFORMS 5 – THE HEAVISIDE STEP FUNCTION**

16.5.1 The definition of the Heaviside step function

16.5.2 The Laplace Transform of H(t – T)

16.5.3 Pulse functions

16.5.4 The second shifting theorem

16.5.5 Exercises

16.5.6 Answers to exercises (8 pages)

**UNIT 16.6 – LAPLACE TRANSFORMS 6 – THE DIRAC UNIT IMPULSE FUNCTION**

16.6.1 The definition of the Dirac unit impulse function

16.6.2 The Laplace Transform of the Dirac unit impulse function

16.6.3 Transfer functions

16.6.4 Steady-state response to a single frequency input

16.6.5 Exercises

16.6.6 Answers to exercises (11 pages)

**UNIT 16.7 – LAPLACE TRANSFORMS 7 – (AN APPENDIX)**

One view of how Laplace Transforms might have arisen (4 pages)

**UNIT 16.8 – Z-TRANSFORMS 1 – DEFINITION AND RULES**

16.8.1 Introduction

16.8.2 Standard Z-Transform definition and results

16.8.3 Properties of Z-Transforms

16.8.4 Exercises

16.8.5 Answers to exercises (10 pages)

**UNIT 16.9 – Z-TRANSFORMS 2 – INVERSE Z-TRANSFORMS**

16.9.1 The use of partial fractions

16.9.2 Exercises

16.9.3 Answers to exercises (6 pages)

**UNIT 16.10 – Z-TRANSFORMS 3 – SOLUTION OF LINEAR DIFFERENCE EQUATIONS**

16.10.1 First order linear difference equations

16.10.2 Second order linear difference equations

16.10.3 Exercises

16.10.4 Answers to exercises (9 pages)

**UNIT 17.1 – NUMERICAL MATHEMATICS 1 – THE APPROXIMATE SOLUTION OF ALGEBRAIC EQUATIONS**

17.1.1 Introduction

17.1.2 The Bisection method

17.1.3 The rule of false position

17.1.4 The Newton-Raphson method

17.1.5 Exercises

17.1.6 Answers to exercises (8 pages)

**UNIT 17.2 – NUMERICAL MATHEMATICS 2 – APPROXIMATE INTEGRATION (A)**

17.2.1 The trapezoidal rule

17.2.2 Exercises

17.2.3 Answers to exercises (4 pages)

**UNIT 17.3 – NUMERICAL MATHEMATICS 3 – APPROXIMATE INTEGRATION (B)**

17.3.1 Simpson’s rule

17.3.2 Exercises

17.3.3 Answers to exercises (6 pages)

**UNIT 17.4 – NUMERICAL MATHEMATICS 4 – FURTHER GAUSSIAN ELIMINATION**

17.4.1 Gaussian elimination by “partial pivoting”with a check column

17.4.2 Exercises

17.4.3 Answers to exercises (4 pages)

**UNIT 17.5 – NUMERICAL MATHEMATICS 5 – ITERATIVE METHODS FOR SOLVING SIMULTANEOUS LINEAR EQUATIONS**

17.5.1 Introduction

17.5.2 The Gauss-Jacobi iteration

17.5.3 The Gauss-Seidel iteration

17.5.4 Exercises

17.5.5 Answers to exercises (7 pages)

**UNIT 17.6 – NUMERICAL MATHEMATICS 6 – NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (A)**

17.6.1 Euler’s unmodified method

17.6.2 Euler’s modified method

17.6.3 Exercises

17.6.4 Answers to exercises (6 pages)

**UNIT 17.7 – NUMERICAL MATHEMATICS 7 – NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (B)**

17.7.1 Picard’s method

17.7.2 Exercises

17.7.3 Answers to exercises (6 pages)

**UNIT 17.8 – NUMERICAL MATHEMATICS 8 – NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS (C)**

17.8.1 Runge’s method

17.8.2 Exercises

17.8.3 Answers to exercises (5 pages)

**UNIT 18.1 – STATISTICS 1 – THE PRESENTATION OF DATA**

18.1.1 Introduction

18.1.2 The tabulation of data

18.1.3 The graphical representation of data

18.1.4 Exercises

18.1.5 Selected answers to exercises (8 pages)

**UNIT 18.2 – STATISTICS 2 – MEASURES OF CENTRAL TENDENCY**

18.2.1 Introduction

18.2.2 The arithmetic mean (by coding)

18.2.3 The median

18.2.4 The mode

18.2.5 Quantiles

18.2.6 Exercises

18.2.7 Answers to exercises (9 pages)

**UNIT 18.3 – STATISTICS 3 – MEASURES OF DISPERSION (OR SCATTER)**

18.3.1 Introduction

18.3.2 The mean deviation

18.3.3 Practical calculation of the mean deviation

18.3.4 The root mean square (or standard) deviation

18.3.5 Practical calculation of the standard deviation

18.3.6 Other measures of dispersion

18.3.7 Exercises

18.3.8 Answers to exercises (6 pages)

**UNIT 18.4 – STATISTICS 4 – THE PRINCIPLE OF LEAST SQUARES**

18.4.1 The normal equations

18.4.2 Simplified calculation of regression lines

18.4.3 Exercises

18.4.4 Answers to exercises (6 pages)

**UNIT 19.1 – PROBABILITY 1 – DEFINITIONS AND RULES**

19.1.1 Introduction

19.1.2 Application of probability to games of chance

19.1.3 Empirical probability

19.1.4 Types of event

19.1.5 Rules of probability

19.1.6 Conditional probabilities

19.1.7 Exercises

19.1.8 Answers to exercises (5 pages)

**UNIT 19.2 – PROBABILITY 2 – PERMUTATIONS AND COMBINATIONS**

19.2.1 Introduction

19.2.2 Rules of permutations and combinations

19.2.3 Permutations of sets with some objects alike

19.2.4 Exercises

19.2.5 Answers to exercises (7 pages)

**UNIT 19.3 – PROBABILITY 3 – RANDOM VARIABLES**

19.3.1 Defining random variables

19.3.2 Probability distribution and

probability density functions

19.3.3 Exercises

19.3.4 Answers to exercises (9 pages)

**UNIT 19.4 – PROBABILITY 4 – MEASURES OF LOCATION AND DISPERSION**

19.4.1 Common types of measure

19.4.2 Exercises

19.4.3 Answers to exercises (6 pages)

**UNIT 19.5 – PROBABILITY 5 – THE BINOMIAL DISTRIBUTION**

19.5.1 Introduction and theory

19.5.2 Exercises

19.5.3 Answers to exercises (5 pages)

**UNIT 19.6 – PROBABILITY 6 – STATISTICS FOR THE BINOMIAL DISTRIBUTION**

19.6.1 Construction of histograms

19.6.2 Mean and standard deviation of a binomial distribution

19.6.3 Exercises

19.6.4 Answers to exercises (10 pages)

**UNIT 19.7 – PROBABILITY 7 – THE POISSON DISTRIBUTION**

19.7.1 The theory

19.7.2 Exercises

19.7.3 Answers to exercises (5 pages)

**UNIT 19.8 – PROBABILITY 8 – THE NORMAL DISTRIBUTION**

19.8.1 Limiting position of a frequency polygon

19.8.2 Area under the normal curve

19.8.3 Normal distribution for continuous variables

19.8.4 Exercises

19.8.5 Answers to exercises (10 pages)

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