Fundamentals of university mathematics / Colin McGregor, Jonathan Nimmo and Wilson Stothers.

Includes index.

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 -- 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 -- 3. Polynomials and Rational Functions -- 3.1. Polynomials -- 3.2. Division and Factors -- 3.3. Quadratics -- 3.4. Rational Functions -- 3.X. Exercises -- 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 -- 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 -- 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 -- 7. Limits and Continuity -- 7.1. Function Limits -- 7.2. Properties of Limits -- 7.3. Continuity -- 7.4. Approaching Infinity -- 7.X. Exercises -- 8. Differentiation -- Fundamentals -- 8.1. First Principles -- 8.2. Properties of Derivatives -- 8.3. Some Standard Derivatives -- 8.4. Higher Derivatives -- 8.X. Exercises -- 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'Hopital's Rule -- 9.X. Exercises -- 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 -- 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 -- 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 -- 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 -- 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 -- 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 -- 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 -- 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 -- 18. Sequences and Series -- 18.1. Real Sequences -- 18.2. Sequence Limits -- 18.3. Series -- 18.4. Power Series -- 18.5. Taylor's Theorem -- 18.X. Exercises -- 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 -- A. Answers to Exercises -- B. Solutions to Problems -- C. Limits and Continuity -- A Rigorous Approach -- C.1. Function Limits -- C.2. Continuity -- C.3. L'Hoptal's Rule -- C.4. Sequence Limits -- D. Properties of Trigonometric Functions -- E. Table of Integrals -- F. Which Test for Convergence? -- G. Standard Maclaurin Series.

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 topicsClearly and comprehensively covers material beyond standard textbooksWorked examples, challenges and exercises throughout.