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Excursions in Calculus: An Interplay of the Continuous and the Discrete (Dolciani Mathematical Expositions)
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Prof. Young's book is a collection of some of his favourite topics in teaching elementary calculus and analysis. Intended for both teachers and motivated students of the calculus, he takes the reader through several beautiful realms of mathematical inquiry and discovery. His topics are diverse: infinite sums and products (including a brilliant presentation of some of the work of Euler, one of his favourite mathematicians), exponential spirals, Wallis's formula for pi, chaos and fractals, Cantor functions, the Weierstrass approximation theorem, and many more with an ambitious appendix on modular arithmetic and related topics such as the celebrated Chinese Remainder Theorem.
Prof. Young treats each of his subjects with not only the highest responsibility and technical acuity of a trained professional mathematician, but also with the greatest reverence and passion for the glorious field to which he has devoted his life. The book reads not like a sterile mathematical text but as an intricately woven epic of centuries of mathematical inquiry and the rich personalities responsible. Complete with hundreds of very challenging and non-trivial exercises, this book has something for everyone, whether a motivated student of freshman calculus or a sophisticated mathematician. None will be bored, all will be mystified.
This is one of those rare books that actually teaches people how to think "outside the box"--how to come up with different ways of looking at things, creative ways of solving problems. The author places an emphasis on simplicity and elegance.
If you want a book that will be a fun, easy read, yet that you will keep coming back to over and over again--or if you want a book to help you create some new and fun problems--or if you feel like you need a little infusion of that mathematical creativity that is so critically needed in advanced mathematical work, this book will be able to help you a great deal.
I urged its adoption by the Calculus instructor at that time, and sang its praises at that time, but to no avail.
(In fairness to the fine Mathematics Department, at the time computer technology was being fully integrated into classrooms,
leaving little time for new instructional books.)
Now, I do the same---that is, I urge its perusal--- utilizing this platform.
This book bristles with excitement and charm. The exposition is lucid, the mathematics ever fresh and approached with vigor.
Most of this material should be easily accessible to students, certainly by their junior year of college (or, sooner)--however,
more often than not this material is neglected , if not downplayed, in traditional Calculus courses.
Much (if not all) of the material certainly was absent from my academic curricula. A random miscellany of its contents include:
a fine selection of fascinating problems accompanying each new section, Mathematical Induction, Binomial Theorem, Fermat, Euler, Fibonacci,Generating Functions, Averages, Approximations, Dirac Delta, Number Theory ,Infinite Sums. The scope is panoramic: Numbers, Algebra, Geometry, Calculus--espousing the interplay of the discrete and the continuous. The author presents each in masterful prose and clear mathematical detail. The problems/exercises are exploratory and fun(!) to solve. The Bibliography provides entree into much more. Herewith is presented a beautiful exploration of relevant mathematical gems.