Contents

• Foreword
  • What the Course Tries to Do
    • Aims
    • Learning Outcomes
  • These Notes
    • The Web Version
    • The EG1006 Mailing List
    • Books
    • Managing Your Learning
    • Acknowledgements
  
  
• List of Figures
• Revision
  • Powers
  • Algebraic Manipulation
  • Summations
  • The Binomial Theorem
    • Pascal's Triangle
    • Using the Binomial Theorem
  • Cartesian Coordinates
    • Distance between points
    • Straight Lines
    • Finding the intersection of two lines
  • Angles and Trigonometry
    • Trig Functions
  • Circles
    • Tangents
  • Parameters
  • Polar Co-ordinates
  
  
• The Derivative
  • Introduction
  • The Derivative
  • The Derivative as a Rate of Change
  • Three Standard Derivatives
    • The derivative of xⁿ
    • The derivatives of sine and cosine
  • Rules for Differentiation
    • SUMS
    • PRODUCTS
    • QUOTIENTS
    • THE CHAIN RULE
  
  
• Rates of Change etc.
  • Introduction
  • Higher Derivatives
    • The meaning of the second derivative
  • Parametric Differentiation
  
  
• More Functions
  • Introduction
    • The inverse trig functions
    • Differentiating the inverse trig functions
    • Log and Exp
    • The derivative of log
    • The Exponential Function
    • The derivative
    • The scientific importance of exp
    • The hyperbolic trig functions
  
  
• Maxima and Minima
  • Introduction
    • Finding critical points and determining their nature
    • Global Maxima and Minima
  
  
• Integration
  • Introduction
    • Indefinite Integrals
    • The Constant of Integration
    • How to Integrate
    • The Definite Integral
  
  
• Techniques of Integration
  • Introduction
  • Integration by Substitution
    • The technique
    • How to handle definite integrals
  • Integration by Parts
    • Guidelines on the choice of u and v
  • Partial Fractions
    • The Partial Fractions Routine
    • The Integration Stage
  
  
• Applications of Integration
  • Introduction
  • Volumes of Revolution
    • The Basic Case
    • Curves given parametrically
  • Lengths of Curves
  • Centroids
    • Calculating M_x, the moment with respect to the x-axis.
    • Calculating M_y, the moment with respect to the y-axis.
    • Special Case
    • The Centroid
  • Surfaces of Revolution
  
  
• Reduction Formulas
  • Reduction Formulas
  
  
• Complex Numbers
  • Introduction
  • The Arithmetic of Complex Numbers
    • Square Roots
    • Complex Conjugates
  • The Argand Diagram
  • Modulus and Argument
  • Products
  • De Moivre's Theorem
  • The Roots of Unity
  • Polynomials
  • Questions 1 (Hints and solutions start on page .)
  
  
• Matrices
  • Introduction
  • Terminology
  • Matrix Algebra
    • Addition of Matrices
    • Subtraction of Matrices
    • Multiplication by a Number
    • Multiplication of Matrices
    • Origins of the Definition
  • Properties of Matrix Algebra
    • The Identity and Zero Matrices
    • Relating Scalar and Matrix Multiplication
    • Transpose of a Product
    • Examples
  • Inverses of Matrices
    • Complex Numbers and Matrices
  • Determinants
    • Properties of Determinants
  • The Fourier Matrix
    • Rapid convolution and the FFT
  • Linear Systems of Equations
  • Geometrical Interpretation
  • Gaussian Reduction
    • The Simple Algorithm
    • Complications
    • Solving Systems in Practice
  • Calculating Inverses
  • Questions 2 (Hints and solutions start on page .)
  
  
• Approximation and Taylor Series
  • Introduction
  • Accuracy
  • Linear Approximation
    • Small Changes
  • Solving Equations--Approximately
    • Newton's Method
  • Higher Approximations
    • Second Approximation
    • Higher Approximations
  • Taylor Series
    • Infinite Series
    • Geometric Series
    • Taylor Series
    • The Binomial Series
  
  
• Differential Equations
  • Introduction
  • Separable Equations
    • The Malthus Equation
  • Generalities
  • Linear First-Order Equations
  • Linear Differential Equations
    • The Inhomogeneous Case
  
  
• Numbers
  • Measures of Error
  • Decimal Places
  • Scientific Notation
    • Accuracy of Calculations
  
  
• Solutions to Exercises
  • Solutions for Questions 1 (page ).
  • Solutions for Questions 2 (page ).
  
  
• Bibliography
• Index
• About this document ...