| Description |
1 online resource (vii, 188 pages) : illustrations |
| Series |
Modular mathematics series |
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Modular mathematics series.
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| Contents |
Front Cover; Analysis; Copyright Page; Table of Contents; Series Preface; Preface; Acknowledgements; Chapter 1. Introduction: Why We Study Analysis; 1.1 What the computer cannot see ... ; 1.2 From counting to complex numbers; 1.3 From infinitesimals to limits; Chapter 2. Convergent Sequences and Series; 2.1 Convergence and summation; 2.2 Algebraic and order properties of limits; Summary; Further exercises; Chapter 3. Completenessand Convergence; 3.1 Completeness and sequences; 3.2 Completeness and series; 3.3 Alternating series; 3.4 Absolute and conditional convergence of series; Summary |
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Further exercisesChapter 4. Functions Definedby Power Series; 4.1 Polynomials -- and what Euler did with them!; 4.2 Multiplying power series: Cauchy products; 4.3 The radius of convergence of a power series; 4.4 The elementary transcendental functions; Summary; Further exercises; Chapter 5. Functions and Limits; 5.1 Historical interlude: curves, graphs and functions; 5.2 The modern concept of function: ordered pairs, domainand range; 5.3 Combining real functions; 5.4 Limits of real functions -- what Cauchy meant!; Summary; Further exercises; Chapter 6. Continuous Functions; 6.1 Limits that fit |
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6.2 Limits that do not fit: types of discontinuity6.3 General power functions; 6.4 Continuity of power series; Summary; Further exercises; Chapter 7. Continuity on Intervals; 7.1 From interval to interval; 7.2 Applications: fixed points, roots and iteration; 7.3 Reaching the maximum: the Boundedness Theorem; 7.4 Uniform continuity -- what Cauchy meant?; Summary; Further exercises; Chapter 8. Differentiable Real Functions; 8.1 Tangents: prime and ultimate ratios; 8.2 The derivative as a limit; 8.3 Differentiation and continuity; 8.4 Combining derivatives; 8.5 Extreme points and curve sketching |
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11.1 The Fundamental Theorem of the Calculus11.2 Integration by parts and change of variable; 11.3 Improper integrals; 11.4 Convergent integrals and convergent series; Summary; Further exercises; Chapter 12. What Next? Extensions and Developments; 12.1 Generalizations of completeness; 12.2 Approximation of functions; 12.3 Integrals of real functions: yet more completeness; Appendix A: Program Listings; A.I Sequences program; A.2 Another sequence program; A.3 Taylor series; A.4 Newton's method in one dimension; Solutions to exercises; Index |
| Summary |
Further exercises; Chapter 9. Mean Values and Taylor Series; 9.1 The Mean Value Theorem; 9.2 Tests for extreme points; 9.3 L'Hôpital's Rules and the calculation of limits; 9.4 Differentiation of power series; 9.5 Taylor's Theorem and series expansions; Summary; Further exercises; Chapter 10. The Riemann Integral; 10.1 Primitives and the 'arbitrary constant'; 10.2 Partitions and step functions: the Riemann Integral; 10.3 Criteria for integrability; 10.4 Classes of integrable functions; 10.5 Properties of the integral; Summary; Further exercises; Chapter 11. Integration Techniques |
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Building on the basic concepts through a careful discussion of covalence, (while adhering resolutely to sequences where possible), the main part of the book concerns the central topics of continuity, differentiation and integration of real functions. Throughout, the historical context in which the subject was developed is highlighted and particular attention is paid to showing how precision allows us to refine our geometric intuition. The intention is to stimulate the reader to reflect on the underlying concepts and ideas |
| Notes |
Includes index |
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English |
| Subject |
Mathematical analysis.
|
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MATHEMATICS -- Calculus.
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MATHEMATICS -- Mathematical Analysis.
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Mathematical analysis
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| Form |
Electronic book
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| ISBN |
9780080928722 |
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0080928722 |
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9786613932068 |
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661393206X |
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