Fundamentals of University Mathematics

Fundamentals of University Mathematics

3rd Edition - October 20, 2010

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  • Authors: Colin McGregor, Jonathan Nimmo, Wilson Stothers
  • eBook ISBN: 9780857092243
  • Paperback ISBN: 9780857092236

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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

  • 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: Differentiation—Fundamentals

    8.1 First Principles

    8.2 Properties of Derivatives

    8.3 Some Standard Derivatives

    8.4 Higher Derivatives

    8.X Exercises

    Chapter 9: Differentiation—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: Integration—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

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

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