Bücher der Reihe SUMS Readings

This textbook provides a gentle overview of fundamental concepts related to one-variable calculus. The original approach is a result of the author's 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 "e;side trips"e; to relevant subjects not usually seen in traditional calculus textbooks, touching on topics like curvature, the isoperimetric inequality, Riemann's rearrangement theorem, Snell's law, Buffon's needle problem, Gregory's 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.

This textbook introduces generalized trigonometric functions through the exploration of imperfect circles: curves defined by |x|p + |y|p = 1 where p ¿ 1. Grounded in visualization and computations, this accessible, modern perspective encompasses new and old results, casting a fresh light on duality, special functions, geometric curves, and differential equations. Projects and opportunities for research abound, as we explore how similar (or different) the trigonometric and squigonometric worlds might be.Comprised of many short chapters, the book begins with core definitions and techniques. Successive chapters cover inverse squigonometric functions, the many possible re-interpretations of ¿, two deeper dives into parameterizing the squigonometric functions, and integration. Applications include a celebration of Piet Hein¿s work in design. From here, more technical pathways offer further exploration. Topicsinclude infinite series; hyperbolic, exponential, and logarithmic functions; metrics and norms; and lemniscatic and elliptic functions. Illuminating illustrations accompany the text throughout, along with historical anecdotes, engaging exercises, and wry humor.Squigonometry: The Study of Imperfect Circles invites readers to extend familiar notions from trigonometry into a new setting. Ideal for an undergraduate reading course in mathematics or a senior capstone, this book offers scaffolding for active discovery. Knowledge of the trigonometric functions, single-variable calculus, and initial-value problems is assumed, while familiarity with multivariable calculus and linear algebra will allow additional insights into certain later material.

This text provides a detailed presentation of the main results for infinite products, as well as several applications.

This book takes the reader on a journey from familiar high school mathematics to undergraduate algebra and number theory.