# Fundamentals of University Mathematics

## Description

## Key Features

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

## Readership

## Table of Contents

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

## Product details

- No. of pages: 568
- Language: English
- Copyright: © Woodhead Publishing 2010
- Published: October 20, 2010
- Imprint: Woodhead Publishing
- eBook ISBN: 9780857092243
- Paperback ISBN: 9780857092236

## About the Authors

### Colin McGregor

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

#### Affiliations and Expertise

### Jonathan Nimmo

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

#### Affiliations and Expertise

### Wilson Stothers

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