| Description |
1 online resource (xvi, 213 pages) : illustrations (some color) |
| Series |
Springer undergraduate mathematics series. SUMS readings |
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Springer undergraduate mathematics series. SUMS readings
|
| Contents |
Intro -- Preface -- Contents -- Introduction -- 2400 Years of Calculus -- Calculus and Education -- 1 The Real Numbers -- The Rational Line -- Density of Q -- Some Basic Notions -- Irrationality of Q -- From Eudoxus to Dedekind -- The Real Line -- Dyadic Series-A Construction of R -- The Scarcity of Q -- The Completeness of R -- Cardinality -- Exercises -- 2 Sequences and Series -- Sequences -- Limits of Sequences -- Cantor's Nested Intervals Theorem -- Subsequences -- Series -- The Harmonic Series -- Series of Positive Terms -- Series with Positive and Negative Terms |
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The Riemann Series Theorem -- Absolute and Unconditional Convergence -- Exercises -- 3 Functions -- The Elementary Functions -- Polynomials -- Circular Functions -- The Exponential Function: Bernoulli's Inequality -- Irrationality of e -- Convergence of k=1∞(1+ak) and of k=1∞ak -- Hyperbolic Functions -- Injectivity and Inverse Functions -- Curves in the Plane: Parametrized Curves -- The Cycloid -- Pythagorean Triples -- Continuity -- Bolzano and Weierstrass -- Limits -- Limits in Ancient Greece: The Area of a Circle -- Three Important Limits -- Exercises -- 4 The Derivative -- Derivative |
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Integration and Products: Integration by Parts -- Stirling's Formula -- Integration and Composition: Integration by Substitution -- A Note on Notation -- Length of Curves. The Catenary -- Area Enclosed by a Simple Closed Curve -- Exercises -- 6 More Derivatives -- Second Derivative, Best-Fitting Parabola, and Curvature -- The Taylor Polynomial of Order Two -- Curvature -- Random Walk and the Gauss Curve -- The Taylor Series -- Exercises -- 7 Convexity and the Isoperimetric Inequality -- The Arithmetic-Geometric Inequality -- Convexity -- Young, Hölder, Jensen, Cauchy-Schwarz.. |
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The Isoperimetric Inequality -- Exercises -- 8 More Integrals -- Volume -- Double Integrals -- The Basel Problem -- Solids of Revolution -- Integration of e -- Density Functions, Barycenter, and Expectation -- Center of Mass or Barycenter -- Pappus' Theorem -- The Method -- Surface Area -- Normal Distribution. Gauss, Laplace, and Stirling -- Exercises -- 9 The Gamma Function -- The Gamma Function -- Weierstrass' Formula -- Growth of the Harmonic Series, Again -- Exercises -- Bibliography -- Index |
| Summary |
This textbook provides a gentle overview of fundamental concepts related to one-variable calculus. The original approach is a result of the authors forty years of experience in teaching the subject at universities around the world. In this book, Dr. Zalduendo makes use of the history of mathematics and a friendly, conversational approach to attract the attention of the student, emphasizing what is more conceptually relevant and putting key notions in a historical perspective. Such an approach was conceived to help them to overcome potential difficulties in teaching and learning of this subject caused, in many cases, by an excess of technicalities and computations. Besides covering the core of the discipline real number, sequences and series, functions, derivatives, integrals, convexity and inequalities the book is enriched by "side trips" to relevant subjects not usually seen in traditional calculus textbooks, touching on topics like curvature, the isoperimetric inequality, Riemanns rearrangement theorem, Snells law, Buffons needle problem, Gregorys series, random walk and the Gauss curve, and more. An insightful collection of exercises and applications completes this book, making it ideal as a supplementary textbook for a calculus course or the main textbook for an honors course on the subject |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Online resource; title from PDF title page (SpringerLink, viewed November 16, 2022) |
| Subject |
Calculus.
|
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calculus.
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Cálculo
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Calculus
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| Form |
Electronic book
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| ISBN |
9783031157653 |
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3031157656 |
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9788303115768 |
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8303115766 |
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