# Fundamentals of University Mathematics

## 3rd Edition

**Authors:**Colin McGregor Jonathan Nimmo Wilson Stothers

**eBook ISBN:**9780857092243

**Paperback ISBN:**9780857092236

**Imprint:**Woodhead Publishing

**Published Date:**20th October 2010

**Page Count:**568

## 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.

## Key Features

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

## Readership

University students

## Table of Contents

Preface to the Third Edition

Notation

Chapter 1: Preliminaries

1.1 Number Systems

1.2 Intervals

1.3 The Plane

1.4 Modulus

1.5 Rational Powers

1.6 Inequalities

1.7 Divisibility and Primes

1.8 Rationals and Irrationals

1.X Exercises

Chapter 2: Functions and Inverse Functions

2.1 Functions and Composition

2.2 Real Functions

2.3 Standard Functions

2.4 Boundedness

2.5 Inverse Functions

2.6 Monotonic Functions

2.X Exercises

Chapter 3: Polynomials and Rational Functions

3.1 Polynomials

3.2 Division and Factors

3.3 Quadratics

3.4 Rational Functions

3.X Exercises

Chapter 4: Induction and the Binomial Theorem

4.1 The Principle of Induction

4.2 Picking and Choosing

4.3 The Binomial Theorem

4.X Exercises

Chapter 5: Trigonometry

5.1 Trigonometric Functions

5.2 Identities

5.3 General Solutions of Equations

5.4 The t-formulae

5.5 Inverse Trigonometric Functions

5.X Exercises

Chapter 6: Complex Numbers

6.1 The Complex Plane

6.2 Polar Form and Complex Exponentials

6.3 De Moivre’s Theorem and Trigonometry

6.4 Complex Polynomials

6.5 Roots of Unity

6.6 Rigid Transformations of the Plane

6.X Exercises

Chapter 7: Limits and Continuity

7.1 Function Limits

7.2 Properties of Limits

7.3 Continuity

7.4 Approaching Infinity

7.X Exercises

Chapter 8: Differentiationâ€”Fundamentals

8.1 First Principles

8.2 Properties of Derivatives

8.3 Some Standard Derivatives

8.4 Higher Derivatives

8.X Exercises

Chapter 9: Differentiationâ€”Applications

9.1 Critical Points

9.2 Local and Global Extrema

9.3 The Mean Value Theorem

9.4 More on Monotonic Functions

9.5 Rates of Change

9.6 L’Hôpital’s Rule

9.X Exercises

Chapter 10: Curve Sketching

10.1 Types of Curve

10.2 Graphs

10.3 Implicit Curves

10.4 Parametric Curves

10.5 Conic Sections

10.6 Polar Curves

10.X Exercises

Chapter 11: Matrices and Linear Equations

11.1 Basic Definitions

11.2 Operations on Matrices

11.3 Matrix Multiplication

11.4 Further Properties of Multiplication

11.5 Linear Equations

11.6 Matrix Inverses

11.7 Finding Matrix Inverses

11.X Exercises

Chapter 12: Vectors and Three Dimensional Geometry

12.1 Basic Properties of Vectors

12.2 Coordinates in Three Dimensions

12.3 The Component Form of a Vector

12.4 The Section Formula

12.5 Lines in Three Dimensional Space

12.X Exercises

Chapter 13: Products of Vectors

13.1 Angles and the Scalar Product

13.2 Planes and the Vector Product

13.3 Spheres

13.4 The Scalar Triple Product

13.5 The Vector Triple Product

13.6 Projections

13 X Exercises

Chapter 14: Integrationâ€”Fundamentals

14.1 Indefinite Integrals

14.2 Definite Integrals

14.3 The Fundamental Theorem of Calculus

14.4 Improper Integrals

14.X Exercises

Chapter 15: Logarithms and Exponentials

15.1 The Logarithmic Function

15.2 The Exponential Function

15.3 Real Powers

15.4 Hyperbolic Functions

15.5 Inverse Hyperbolic Functions

15 X Exercises

Chapter 16: Integration - Methods and Applications

16.1 Substitution

16.2 Rational Integrals

16.3 Trigonometric Integrals

16.4 Integration by Parts

16.5 Volumes of Revolution

16.6 Arc Lengths

16.7 Areas of Revolution

16.X Exercises

Chapter 17: Ordinary Differential Equations

17.1 Introduction

17.2 First Order Separable Equations

17.3 First Order Homogeneous Equations

17.4 First Order Linear Equations

17.5 Second Order Linear Equations

17.X Exercises

Chapter 18: Sequences and Series

18.1 Reed Sequences

18.2 Sequence Limits

18.3 Series

18.4 Power Series

18.5 Taylor’s Theorem

18.X Exercises

Chapter 19: Numerical Methods

19.1 Errors

19.2 The Bisection Method

19.3 Newton’s Method

19.4 Definite Integrals

19.5 Euler’s Method

19.X Exercises

Appendix A: Answers to Exercises

Appendix B: Solutions to Problems

Appendix C: Limits and Continuity - A Rigorous Approach

Appendix D: Properties of Trigonometric Functions

Appendix E: Table of Integrals

Appendix F: Which Test for Convergence?

Appendix G: Standard Maclaurin Series

Index

## Details

- No. of pages:
- 568

- Language:
- English

- Copyright:
- © Woodhead Publishing 2010

- Published:
- 20th October 2010

- Imprint:
- Woodhead Publishing

- eBook ISBN:
- 9780857092243

- Paperback ISBN:
- 9780857092236

## About the Author

### Colin McGregor

*Colin McGregor* is an Honorary Research Fellow in the Department of Mathematics, University of Glasgow, UK.

### Affiliations and Expertise

Glasgow University, UK

### Jonathan Nimmo

*Jonathan Nimmo* is a Reader in Mathematics in the Department of Mathematics, University of Glasgow, UK.

### Affiliations and Expertise

Glasgow University

### Wilson Stothers

*Wilson Stothers* was formerly a member in the Department of Mathematics, University of Glasgow, UK.

### Affiliations and Expertise

formerly Glasgow University, UK

## Reviews

I found this book to be extremely helpful over a wide range of topics. I must have used this to aid my revision for at least 7 or 8 exams and would probably have struggled without it. Particularly useful sections include those on Complex Numbers, Matrices, Vectors, and Differential Equations. If you can only afford to buy one or two books for your course or study, make sure this is one of them!, Rg Ellam "rob_ellam (review on www.amazon.co.uk)

This book is excellent preparation for 2nd year undergraduate Mathematics courses. It is detailed and step-by-step, so you don't get lost on certain topics. The proofs are complex, but then that's Mathematics for you!, Jon Baldie (review on www.amazon.co.uk)

If you are looking for a first year university text you should carefully look at this, a unifier of mathematical ideas at this level. I found it most valuable., The Mathematical Gazette (review of a previous edition)