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100+1 Problems in Advanced Calculus

A Creative Journey through the Fjords of Mathematical Analysis for Beginners

  • Convenes problems in Advanced Calculus with solutions that favor creativity
  • Covers all the classical topics seen in first-year Calculus and Mathematical Analysis courses
  • Offers a theoretical background in every chapter, with relevant definitions and theorems for self-study

Part of the book series: Problem Books in Mathematics (PBM)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xvii
  2. Summary of basic theory of inequalities

    • Paolo Toni, Pier Domenico Lamberti, Giacomo Drago
    Pages 1-3
  3. Sets, sequences, functions

    • Paolo Toni, Pier Domenico Lamberti, Giacomo Drago
    Pages 5-31
  4. Limits of functions, continuity

    • Paolo Toni, Pier Domenico Lamberti, Giacomo Drago
    Pages 33-93
  5. Differentiation

    • Paolo Toni, Pier Domenico Lamberti, Giacomo Drago
    Pages 95-114
  6. Classical Theorems of differential calculus

    • Paolo Toni, Pier Domenico Lamberti, Giacomo Drago
    Pages 115-128
  7. Monotonicity, concavity, minima, maxima, inflection points

    • Paolo Toni, Pier Domenico Lamberti, Giacomo Drago
    Pages 129-145
  8. Graphs of functions

    • Paolo Toni, Pier Domenico Lamberti, Giacomo Drago
    Pages 147-202
  9. Integrals

    • Paolo Toni, Pier Domenico Lamberti, Giacomo Drago
    Pages 203-220

About this book

This book convenes a collection of carefully selected problems in mathematical analysis, crafted to achieve maximum synergy between analytic geometry and algebra and favoring mathematical creativity in contrast to mere repetitive techniques. With eight chapters, this work guides the student through the basic principles of the subject, with a level of complexity that requires good use of imagination.

In this work, all the fundamental concepts seen in a first-year Calculus course are covered. Problems touch on topics like inequalities, elementary point-set topology, limits of real-valued functions, differentiation, classical theorems of differential calculus (Rolle, Lagrange, Cauchy, and l’Hospital), graphs of functions, and Riemann integrals and antiderivatives. Every chapter starts with a theoretical background, in which relevant definitions and theorems are provided; then, related problems are presented. Formalism is kept at a minimum, and solutions can be found atthe end of each chapter.


Instructors and students of Mathematical Analysis, Calculus and Advanced Calculus aimed at first-year undergraduates in Mathematics, Physics and Engineering courses can greatly benefit from this book, which can also serve as a rich supplement to any traditional textbook on these subjects as well.


Reviews

“This reviewer warmly welcomes the publication of this nice selection of problems and would recommend its use as a complimentary reading for analysis and calculus courses. … The book under review nicely complements existing literature on the subject; it is a stimulating reading requiring, as suggested by the authors, a bit of critical reflection.” (Svitlana P. Rogovchenko, zbMATH 1494.00003, 2022)

“A sequence of a traditional textbook for an undergraduate calculus course. From this standpoint, this book can be used as a supplement to any conventional calculus text. … It is well known that students struggle with inducing abstract math concepts to gain insight into behavior in real-world phenomena to understand better those behaviors; including such examples is seen as a potential asset.” (Andrzej Sokolowski, MAA Reviews, August 1, 2022)

Authors and Affiliations

  • Liceo Scientifico Statale Enrico Fermi, Padova, Italy

    Paolo Toni

  • Dipartimento di Tecnica e Gestione dei Sistemi Industriali (DTG), University of Padova, Vicenza, Italy

    Pier Domenico Lamberti

  • Istituto Tecnico Economico Calvi, Padova, Italy

    Giacomo Drago

About the authors

Paolo Toni is a Teacher of Mathematics and Physics, retired from Liceo Scientifico E. Fermi in Padua, Italy, with prolific research activities on didactics of mathematics. He obtained a Mathematics degree (1972) from the University of Firenze and authored a few books, beginning with recreational mathematics. For many years, Prof. Toni took an active role in mathematical competitions in his home country, having co-created, directed, and promoted the “Città di Padova” Mathematical Competition, aimed at students of secondary level.

Pier Domenico Lamberti is a Full Professor in Mathematical Analysis at the University of Padua, Italy. He got a PhD in Mathematics from the University of Padua (2003), and has been visiting researcher at the University of Cardiff, UK; University of Athens, Greece; and Complutense University of Madrid, Spain. His research interests lie in mathematical analysis, with focus on partial differential equations, spectral theory, theory of functions spaces, functional analysis, calculus of variations, and homogenization theory. 


Giacomo Drago is a Teacher of Mathematics and Physics at Istituto Tecnico Economico Calvi in Padua, Italy. He obtained a Master degree in Mathematics (2011) from the University of Padua, Italy. As a student, he distinguished himself in the national finals of three consecutive editions (2004-2006) of the Kangourou Mathematics Competitions. 

Bibliographic Information

  • Book Title: 100+1 Problems in Advanced Calculus

  • Book Subtitle: A Creative Journey through the Fjords of Mathematical Analysis for Beginners

  • Authors: Paolo Toni, Pier Domenico Lamberti, Giacomo Drago

  • Series Title: Problem Books in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-91863-7

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

  • Hardcover ISBN: 978-3-030-91862-0Published: 09 March 2022

  • Softcover ISBN: 978-3-030-91865-1Published: 10 March 2023

  • eBook ISBN: 978-3-030-91863-7Published: 08 March 2022

  • Series ISSN: 0941-3502

  • Series E-ISSN: 2197-8506

  • Edition Number: 1

  • Number of Pages: XVII, 220

  • Number of Illustrations: 171 b/w illustrations

  • Topics: Real Functions, Analysis

Buy it now

Buying options

eBook USD 29.99 USD 49.99
40% discount Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 39.99 USD 64.99
38% discount Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 39.99 USD 64.99
38% discount Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access