You are previewing Fundamentals of University Mathematics, 3rd Edition.
O'Reilly logo
Fundamentals of University Mathematics, 3rd Edition

Book Description

The third edition of this popular and effective textbook provides in one volume a unified treatment of topics essential for first year university students studying for degrees in mathematics. Students of computer science, physics and statistics will also find this book a helpful guide to all the basic mathematics they require. It clearly and comprehensively covers much of the material that other textbooks tend to assume, assisting students in the transition to university-level mathematics.

Expertly revised and updated, the chapters cover topics such as number systems, set and functions, differential calculus, matrices and integral calculus. Worked examples are provided and chapters conclude with exercises to which answers are given. For students seeking further challenges, problems intersperse the text, for which complete solutions are provided. Modifications in this third edition include a more informal approach to sequence limits and an increase in the number of worked examples, exercises and problems.

The third edition of Fundamentals of university mathematics is an essential reference for first year university students in mathematics and related disciplines. It will also be of interest to professionals seeking a useful guide to mathematics at this level and capable pre-university students.

  • One volume, unified treatment of essential topics
  • Clearly and comprehensively covers material beyond standard textbooks
  • Worked examples, challenges and exercises throughout

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Preface to the Third Edition
  6. Notation
  7. Chapter 1: Preliminaries
    1. 1.1 Number Systems
    2. 1.2 Intervals
    3. 1.3 The Plane
    4. 1.4 Modulus
    5. 1.5 Rational Powers
    6. 1.6 Inequalities
    7. 1.7 Divisibility and Primes
    8. 1.8 Rationals and Irrationals
    9. 1.X Exercises
  8. Chapter 2: Functions and Inverse Functions
    1. 2.1 Functions and Composition
    2. 2.2 Real Functions
    3. 2.3 Standard Functions
    4. 2.4 Boundedness
    5. 2.5 Inverse Functions
    6. 2.6 Monotonic Functions
    7. 2.X Exercises
  9. Chapter 3: Polynomials and Rational Functions
    1. 3.1 Polynomials
    2. 3.2 Division and Factors
    3. 3.3 Quadratics
    4. 3.4 Rational Functions
    5. 3.X Exercises
  10. Chapter 4: Induction and the Binomial Theorem
    1. 4.1 The Principle of Induction
    2. 4.2 Picking and Choosing
    3. 4.3 The Binomial Theorem
    4. 4.X Exercises
  11. Chapter 5: Trigonometry
    1. 5.1 Trigonometric Functions
    2. 5.2 Identities
    3. 5.3 General Solutions of Equations
    4. 5.4 The t-formulae
    5. 5.5 Inverse Trigonometric Functions
    6. 5.X Exercises
  12. Chapter 6: Complex Numbers
    1. 6.1 The Complex Plane
    2. 6.2 Polar Form and Complex Exponentials
    3. 6.3 De Moivre’s Theorem and Trigonometry
    4. 6.4 Complex Polynomials
    5. 6.5 Roots of Unity
    6. 6.6 Rigid Transformations of the Plane
    7. 6.X Exercises
  13. Chapter 7: Limits and Continuity
    1. 7.1 Function Limits
    2. 7.2 Properties of Limits
    3. 7.3 Continuity
    4. 7.4 Approaching Infinity
    5. 7.X Exercises
  14. Chapter 8: Differentiation—Fundamentals
    1. 8.1 First Principles
    2. 8.2 Properties of Derivatives
    3. 8.3 Some Standard Derivatives
    4. 8.4 Higher Derivatives
    5. 8.X Exercises
  15. Chapter 9: Differentiation—Applications
    1. 9.1 Critical Points
    2. 9.2 Local and Global Extrema
    3. 9.3 The Mean Value Theorem
    4. 9.4 More on Monotonic Functions
    5. 9.5 Rates of Change
    6. 9.6 L’Hôpital’s Rule
    7. 9.X Exercises
  16. Chapter 10: Curve Sketching
    1. 10.1 Types of Curve
    2. 10.2 Graphs
    3. 10.3 Implicit Curves
    4. 10.4 Parametric Curves
    5. 10.5 Conic Sections
    6. 10.6 Polar Curves
    7. 10.X Exercises
  17. Chapter 11: Matrices and Linear Equations
    1. 11.1 Basic Definitions
    2. 11.2 Operations on Matrices
    3. 11.3 Matrix Multiplication
    4. 11.4 Further Properties of Multiplication
    5. 11.5 Linear Equations
    6. 11.6 Matrix Inverses
    7. 11.7 Finding Matrix Inverses
    8. 11.X Exercises
  18. Chapter 12: Vectors and Three Dimensional Geometry
    1. 12.1 Basic Properties of Vectors
    2. 12.2 Coordinates in Three Dimensions
    3. 12.3 The Component Form of a Vector
    4. 12.4 The Section Formula
    5. 12.5 Lines in Three Dimensional Space
    6. 12.X Exercises
  19. Chapter 13: Products of Vectors
    1. 13.1 Angles and the Scalar Product
    2. 13.2 Planes and the Vector Product
    3. 13.3 Spheres
    4. 13.4 The Scalar Triple Product
    5. 13.5 The Vector Triple Product
    6. 13.6 Projections
    7. 13 X Exercises
  20. Chapter 14: Integration—Fundamentals
    1. 14.1 Indefinite Integrals
    2. 14.2 Definite Integrals
    3. 14.3 The Fundamental Theorem of Calculus
    4. 14.4 Improper Integrals
    5. 14.X Exercises
  21. Chapter 15: Logarithms and Exponentials
    1. 15.1 The Logarithmic Function
    2. 15.2 The Exponential Function
    3. 15.3 Real Powers
    4. 15.4 Hyperbolic Functions
    5. 15.5 Inverse Hyperbolic Functions
    6. 15 X Exercises
  22. Chapter 16: Integration—Methods and Applications
    1. 16.1 Substitution
    2. 16.2 Rational Integrals
    3. 16.3 Trigonometric Integrals
    4. 16.4 Integration by Parts
    5. 16.5 Volumes of Revolution
    6. 16.6 Arc Lengths
    7. 16.7 Areas of Revolution
    8. 16.X Exercises
  23. Chapter 17: Ordinary Differential Equations
    1. 17.1 Introduction
    2. 17.2 First Order Separable Equations
    3. 17.3 First Order Homogeneous Equations
    4. 17.4 First Order Linear Equations
    5. 17.5 Second Order Linear Equations
    6. 17.X Exercises
  24. Chapter 18: Sequences and Series
    1. 18.1 Reed Sequences
    2. 18.2 Sequence Limits
    3. 18.3 Series
    4. 18.4 Power Series
    5. 18.5 Taylor’s Theorem
    6. 18.X Exercises
  25. Chapter 19: Numerical Methods
    1. 19.1 Errors
    2. 19.2 The Bisection Method
    3. 19.3 Newton’s Method
    4. 19.4 Definite Integrals
    5. 19.5 Euler’s Method
    6. 19.X Exercises
  26. Appendix A: Answers to Exercises
  27. Appendix B: Solutions to Problems
  28. Appendix C: Limits and Continuity—A Rigorous Approach
  29. Appendix D: Properties of Trigonometric Functions
  30. Appendix E: Table of Integrals
  31. Appendix F: Which Test for Convergence?
  32. Appendix G: Standard Maclaurin Series
  33. Index