UNITEXT - La Matematica per il 3+2

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1 UNITEXT - La Matematica per il 3+2 Volume 101 Editor-in-chief A. Quarteroni Series editors L. Ambrosio P. Biscari C. Ciliberto M. Ledoux W.J. Runggaldier

2 More information about this series at

3 Marco Baronti Filippo De Mari Robertus van der Putten Irene Venturi Calculus Problems 123

4 Marco Baronti Dipartimento di Matematica Università di Genova Genova Italy Filippo De Mari Dipartimento di Matematica Università di Genova Genova Italy Robertus van der Putten DIME Università di Genova Genova Italy Irene Venturi Dipartimento di Matematica Università di Genova Genova Italy ISSN ISSN (electronic) UNITEXT - La Matematica per il 3+2 ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

5 Die mathematische Analysis gewissermassen eine einzige Symphonie des Unendlichen ist (D. Hilbert, Über das Unendliche, 1927) In a certain sense, mathematical analysis is a symphony of the infinite (D. Hilbert, On the Infinite, 1927)

6 Preface The aim of this book is to provide a practical working tool for students in Engineering, Mathematics, and Physics, or in any other field where rigorous Calculus is needed. The emphasis is thus on problems that enhance students skill in solving standard exercises with a careful attitude, encouraging them to devote an attentive eye to what may or may not be done in manipulating formulae or deriving correct conclusions, while maintaining, whenever possible, a fresh approach, that is, seeking guiding ideas. Every chapter starts with a summary of the main results that should be kept in mind and used for the exercises of that chapter; this is followed by a selection of guided exercises. The theoretical preamble is meant to recapitulate the main definitions and results and should also offer a bird s-eye view on the topic treated in the chapter. Hence, the student can quickly review the main theoretical facts and then, most importantly, learn by examples, becoming acquainted with the specific techniques by seeing them applied directly to the problems. Each exercise ends with a short comment which underlines the main issues of that specific exercise, the leading ideas, and the main techniques. A selection of problems closes each chapter, the answers to which are all listed in Solutions. The reader is urged to try to solve some of these problems, which are similar, but not always trivially analogous, to those that have been presented in detail. Mathematics is never just an application of rules, but requires understanding, clear thought, and a bit of imagination. A different problem, a new question, a slightly skew formulation: this is where one really begins to master a technique and to consolidate it. So, rather than feeling discouraged, the student should develop curiosity and be aware that any significant progress does require some effort, and a little sweat is really part of the game. Perhaps the most distinctive feature of this book is that our approach is very direct and refers to a concrete experience. The material is in fact mostly taken from actual written tests that have been delivered in the years at the Engineering School of the University of Genova. Literally, thousands of students have worked on these problems, so our first and foremost acknowledgment goes to them, because they have helped us greatly over the years, tuning our views and vii

7 viii Preface letting us see where the main difficulties really are, those that need both clear statements and specifically designed exercises. Their fellow colleagues, the present and future students, are of course our public and our intended readers. Some complementary standard material has also been added, especially where the main thrust is in the direction of unraveling the details of basic techniques and achieving a reasonably complete panorama of possible scenarios. The book ends with a chapter of problems that are not designed with a single issue in mind but rather require a variety of techniques, and should perhaps be addressed as a final check on the global preparation. Indeed, they have all been assigned in written tests and have all been worked on by large numbers of students. Intentionally, no solutions are given or even hinted at, and they are not ordered according to increasing difficulty. The student who wants to challenge him- or herself with questions that many other students have faced as a final exam should look with particular interest at Chap. 16. The topics covered are those that are typically taught in a first-year engineering undergraduate Calculus course in Italy, with possible variants. The basic focus is on functions of one real variable. As for basic ordinary differential equations, separation of variables, linear first-order, and constant coefficients ODEs are discussed. We believe that anyone who can solve the suggested problems with a reasonable degree of accuracy is in a safe position to achieve a positive result in most Italian universities. Our international experience also tells us that the same may be claimed for most universities around the world, for undergraduate Calculus or Advanced Calculus. Genova, Italy May 2016 Marco Baronti Filippo De Mari Robertus van der Putten Irene Venturi

8 Contents 1 Manipulation of Graphs Operations on Graphs Guided Exercises on Graphs Problems on Graphs Invertible Mappings Injective, Surjective and Bijective Mappings Inversion of a Map Monotone Functions Guided Exercises on Invertible Mappings Problems on Invertible Mapings Maximum, Minimum, Supremum, Infimum Upper and Lower Bounds, Maximum and Minimum Supremum, Infimum Guided Exercises on Maximum, Minimum, Supremum, Infimum Problems on Maximum, Minimum, Supremum, Infimum Sequences Lists of Real Numbers Convergence Notions Results on Limits Guided Exercises on Sequences Problems on Sequences Limits of Functions Convergence Notions Results on Limits Local Comparison of Functions Orders Guided Exercises on Limits of Functions Problems on Limits ix

9 x Contents 6 Continuous Functions Basic Properties of Continuous Functions Discontinuities, Continuous Extensions Global Properties of Continuous Functions Continuous Monotonic Functions Guided Exercises on Continuous Functions Problems on Continuity Differentiable Functions The Derivative of a Function Derivatives of Elementary Functions The Classical Theorems of Differential Calculus Guided Exercises on Differentiable Functions Problems on Differentiability Taylor Expansions Taylor Expansions Guided Exercises on Taylor Expansions Problems on Taylor Expansions The Geometry of Functions Asymptotes Convexity and Concavity, Inflection Points The Nature of Critical Points Guided Exercises on the Geometry of Functions Problems on the Geometry of Functions Indefinite and Definite Integrals Primitive Functions Computing Indefinite Integrals General Techniques Rational Functions Riemann Integrals The Fundamental Theorem of Calculus Guided Exercises on Integration Problems on Integration Improper Integrals and Integral Functions Improper Integrals Convergence Criteria Integral Functions Guided Exercises on Improper Integrals and Integral Functions Improper Integrals Integral Functions Problems on Improper Integrals and Integral Functions

10 Contents xi 12 Numerical Series Convergence Positive Series: Criteria Order, Series and Integrals Alternating Series Guided Exercises on Numerical Series Problems on Series Separation of Variables Differential Equations The Method of Separation of Variables Guided Exercises on Separation of Variables Problems on Separation of Variables First Order Linear Differential Equations First Order Linear Equations with Continuous Coefficients Guided Exercises on Linear First Order Equations Problems on Linear First Order Equations Constant Coefficient Linear Differential Equations Linear Equations with Constant Coefficients The Homogeneous Equation The Nonhomogeneous Equation Guided Exercises on Constant Coefficient Differential Equations Problems on Constant Coefficient Differential Equations Miscellaneous Problems Appendix A: Basic Facts and Notation Appendix B: Calculus Solutions Further Reading Index

11 About the Authors Marco Baronti was born in Genova in Since 1990, he is associate professor in Mathematical Analysis at the University of Genova. His scientific interests are mainly in Functional Analysis and in particular in Geometry of Banach Spaces. Filippo De Mari was born in Genova in In 1987 he received his Ph.D. from Washington University in St. Louis, USA. Since 1998, he is associate professor in Mathematical Analysis at the University of Genova. His scientific interests are mainly in Harmonic Analysis, Representation Theory and Lie Groups. Robertus van der Putten was born in Sanremo in In 1989 he received his Ph.D. from the University of Milan. Since 1990, he is researcher in Mathematical Analysis at the University of Genova. His scientific interests are mainly in Calculus of Variations. Irene Venturi was born in Viareggio in In 2009 she received her Ph.D. from the University of Genova and in 2011 a Master's in Security Safety and Sustainability in Transportation Systems. She is a teacher in Mathematics and has several editorial collaborations. xiii

International Series in Operations Research & Management Science

International Series in Operations Research & Management Science Volume 240 Series Editor Camille C. Price Stephen F. Austin State University, TX, USA Associate Series Editor Joe Zhu Worcester Polytechnic