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Zusatztext "MacCluer's book . . . is calculus 'done right.' And in a mere 162 pages and 1.1 lb! . . . The approach is axiomatic! but the writing style is breezy and inviting. The student will see right away that this is not simply a rehash of material seen in high school--there are some genuinely new ideas here. These include topology! compactness! quantum mechanics! differential equations! and a host of other fascinating topics. . . . This could be an exciting book from which to teach. This is a book that will allow the instructor to build a fascinating course in a variety of different ways. Teaching from this book should be a joy for all." ---Steven G. Krantz! UMAP Journal Informationen zum Autor Charles R. MacCluer is Professor of Mathematics and director of the industrial mathematics program at Michigan State University. His first interest was algebraic number theory but later turned to the more practical disciplines of control theory, signal processing, building science, and industrial problems. He is the author of Industrial Mathematics, Boundary Value Problems and Fourier Expansions , and Calculus of Variations . Klappentext This is the first modern calculus book to be organized axiomatically and to survey the subject's applicability to science and engineering. A challenging exposition of calculus in the European style, it is an excellent text for a first-year university honors course or for a third-year analysis course. The calculus is built carefully from the axioms with all the standard results deduced from these axioms. The concise construction, by design, provides maximal flexibility for the instructor and allows the student to see the overall flow of the development. At the same time, the book reveals the origins of the calculus in celestial mechanics and number theory. The book introduces many topics often left to the appendixes in standard calculus textbooks and develops their connections with physics, engineering, and statistics. The author uses applications of derivatives and integrals to show how calculus is applied in these disciplines. Solutions to all exercises (even those involving proofs) are available to instructors upon request, making this book unique among texts in the field. Focuses on single variable calculus Provides a balance of precision and intuition Offers both routine and demanding exercises Zusammenfassung Featuring a challenging exposition of calculus in the European style, this text is suitable for a first-year university honors course or for a third-year analysis course. It uses applications of derivatives and integrals to show how calculus is applied in these disciplines. It also offers both routine and demanding exercises. Inhaltsverzeichnis Preface xi Acknowledgments xiii Chapter 1: Functions on Sets 1.1 Sets 1 1.2 Functions 2 1.3 Cardinality 5 Exercises 6 Chapter 2: The Real Numbers 2.1 The Axioms 12 2.2 Implications 14 2.3 Latter-Day Axioms 16 Exercises 16 Chapter 3: Metric Properties 3.1 The Real Line 19 3.2 Distance 20 3.3 Topology 21 3.4 Connectedness 22 3.5 Compactness 23 Exercises 27 Chapter 4: Continuity 4.1 The Definition 30 4.2 Consequences 31 4.3 Combinations of Continuous Functions 33 4.4 Bisection 36 4.5 Subspace Topology 37 Exercises 38 Chapter 5: Limits and Derivatives 5.1 Limits 41 5.2 The Derivative 43 5.3 Mean Value Theorem 46 5.4 Derivatives of Inverse Functions 48 5.5 Derivatives of Trigonometric Functions 50 Exercises 53 Chapter 6: Applications of the Derivative 6.1 Tangents 60 6.2 Newton's Method 63 6.3 Linear Approximation and Sensitivity 65 6.4 Optimization 66 6.5 Rate of Change 67 6.6 Related Rates 68 6.7 Ordinary Differential Equations 69 6.8 Kepler's Laws 71 6.9 Universal Gravitation 73 6.10 Concavity 76 6.11 Differentials 79 Exercises 80 Chapter 7: The Riemann Integral 7.1 Dar...
