Customer Reviews

Calculus Without Limits: Almost

5 star:		(6)
4 star:		(2)
3 star:		(0)
2 star:		(0)
1 star:		(2)

 

 

This book is a very atypical introduction to calculus, which some people may like and others will hate. It clearly shows the orientation of its author, who is an engineer and not a mathematician.

Calculus can be presented in two approaches, one using infinitesimals (the way Leibniz thought of it) and one using limits (closer to Newton's picture, though he too would use infinitesimals when it suited him). Originally, either mode of thinking was applied intuitively, but mathematicians could not define the infinitesimal approach in a rigorous manner, and in the 19th century the subject was rigorously developed using limits. Eventually a rigorous approach was developed using infinitesimals by Robinson in the 20th century, but most mathematicians came to prefer the limit approach, because it was the first one developed with a degree of rigor to suit the mathematical community. But scientists and engineers developed an intuitive way of thinking about calculus, based primarily on the infinitesimal approach. So a typical scientist would come to calculus by means of a freshman course that presented the subject using limits, with only a slight reduction of the rigor to a level appropriate to college freshmen, and only in his physics and chemistry courses be presented with calculus as a problem-solving tool using a more intuitive approach based on infinitesimals. The author of this book seems to feel that it is better to present an intuitive approach to calculus, forgetting about rigor (he was once, he states, exposed to calculus in a course that was overly rigorous, even beyond the level usually met in freshman calculus classes!) and it is this which he does in this book.

The author starts off with an intuitive proof of the Pythagorean theorem, which works quite well. He does follow this with an equally intuitive proof of a geometric fallacy, which should clearly demonstrate the limits of intuitive thinking! But it seems that despite this, he intends to go on purely intuitively. The resulting book is one that mathematicians will find infuriating, but scientists and engineers will see as reflecting the way calculus is actually used in their disciplines. And so, I think, what you will think of this book will depend on your own orientation.

Before concluding I should say that one serious defect of this book, I would say, is the total absence of an index. In addition, I think that the title is somewhat misleading; the author uses the limit concept more than he implies. A better title would be "Calculus by an intuitive approach." It is the intuitive approach, even more than the fact that he uses infinitesimals rather than limits for many of his arguments, that defines the uniqueness of this book.

I have a PhD in Materials Science and have been "practicing" scientific and engineering R&D for about 25 years. As a consequence, I'm fairly astute regarding calculus matters.

I was intrigued about this book because of its' approach to calculus with minimal emphasis on the limit theorem and because my son is entering high school mathematics.

I found the book to be very well laid out and the progression of subject matter to be very logical. Most importantly, the book provided real world examples for "using" the mathematical concepts introduced. One of my favorite was the example of transporting a ladder around a corner. While one would not use calculus in such a situation, these sort of examples, really analogies, provide an excellent perspective for the reader.

Finally, the book read almost like a novel. It was enjoyable and left one to anticipate the next chapter.

I think this book is an enjoyable read and suggest it for 1) someone like myself with the challenge of helping bring relevance to their young students, 2) a student currently being introduced to calculus and 3) a former student seeking a refresher.

Only someone who has fallen in love with calculus can know its poetry. Just as any
musician can tell you how music is far deeper than mere spots on a score sheet,
John C. Sparks leads you to understand how calculus is far deeper than
mere cryptic symbols on a blackboard.

The sad truth is that too many calculus classes and texts teach students only
the rules by which you manipulate the cryptic symbols. But those cryptic symbols,
like all symbols, stand for something greater. Spots on a score sheet stand for
melodic sounds, and mathematical symbols stand for beautiful ideas. It is these
ideas that Calculus Without Limits teaches.

Without grasping the ideas and concepts of calculus, a student is left only with
the grind of applying pencil lead to paper in this or that prescribed manner and
has thoroughly missed the point. So it's hardly a surprise that so many calculus
students are frustrated. But for a student who reads John C. Sparks' explanations
of how the symbols assemble themselves into something meaningful, the symbology
becomes just a tool -- as it should have been all along. The real knowledge and
beauty behind the symbols glimmer through. And for a student who makes the special
effort (and all mathematics learning requires special effort, even from those who
love it) to follow that light where it leads -- that student too might very well
fall in love and know the poetry.

I recently finished reading "Calculus Without Limits-Almost", by John C. Sparks and found it to be a fascinating book. I thoroughly enjoyed the historical perspectives, the diversion of the poetry and the interesting examples...especially the geometric proof of the Pythagorean Theorem, and Curry's paradox. I (allegedly) learned calculus 20 some years ago and was taught using both limits and differentials. Either way works pretty well. The geometric proofs used throughout this book provide an easily grasped visual component to the differential approach and I learned things about calculus from them that I had taken on faith for many years. I believe the differential approach is an easier concept to grasp than the limit approach. Lets face it, the subject of calculus is difficult enough; anything that can be done to simplify it, at least initially, is a positive development. Going from the frustration of "I don't get it" to the power, beauty and empowerment of "Wow, I get it" has got to be a prime objective of both student and teacher. Once achieved, a chain reaction occurs where confidence and curiosity displace anxiety and intimidation. At that point studying and homework go from "have to" to "want to". This is when the love of learning and specifically of mathematics sets in. John discusses this passion beginning with the first sentence of the book. If you've been there, you understand; if not, maybe this book can help. The book covers a lot of material and a lot of practical applications including optimization, surfaces and volumes of revolution, heat transfer, linear motion and financial applications. At times the algebra gets a little tedious but nothing worthwhile was ever accomplished without some degree of struggle. It may have been intended as a primer (and works quite well in that application) but it also makes a good reference book for those of us who have been out of the classroom for a while. By any measure I found it well worth the price of admission.

With a Ph.D. in applied mathematics (emphasis on solid mechanics) and more than 30 years experience in solving complex mechanics problems using numerical, closed form solutions and experimental techniques, I have a thorough understanding and working knowledge of the subject of calculus. Thus I feel well qualified to evaluate this newly released calculus book entitled "Calculus Without Limits --- Almost" by John C. Sparks. The foreword of the book speaks about the outstanding work the author has done, not only in mathematics, but in the fields of management, teaching and poetry as well. The book is a unique blend of calculus principles, simplified and understandable illustrations, and motivational descriptions of significant contributions by various world-renowned mathematicians. Living up to the author's objective, the book makes the complex subject of calculus simple to understand and easy to apply. Thus it takes the fear out of a subject that so many young scholars find intimidating. The author has creatively used his own diverse background to apply calculus to the solution of different problems. The book is laid out in a very orderly, systematic, easy to follow manner, and very nicely addresses all the basics of calculus. The book is divided into 9 chapters. Each chapter has a well thought out objective that is accomplished through examples and diagrams. The book has more than 330 pages, including 83 figures and 4 tables.

The book begins with a quote by Isaac Newton, emphasizing the importance of the company one keeps in one's life. It is well known that the caliber of a person is evaluated by the caliber of people in his company.

Within each chapter, the author provides numerous, fully worked out, problems as examples, and then provides practice problems and answers. The book very nicely addresses the complex subject of calculus in a manner such that a new student to this area of learning more easily understands it. On the other hand, it is an excellent book for practicing engineers to use as a means to refresh their math skills. Clearly the author has done an excellent job of meeting these objectives in an interesting and creative way. Congratulations John on a job well done.

My primary objective is to transition high school and first-year college students from algebra/trigonometry/analytical geometry to calculus. My students are pretty sharp and don't appear to be troubled by limits if the concepts are explained and implemented with simple, cogent examples. I enjoyed "Calculus without Limits - almost" but was intrigued when none (zero) of my high school or first-year college calculus tutorial students found the book helpful. On the other hand if I hand them a copy of "Calculus Made Easy" by Silvanus P. Thompson or "Teach Yourself Calculus" by Hugh Neil they zoom right along with limits, differentials, derivatives, and so on. My one-star recommendation is therefore based upon its poor performance as an educational tool for my advanced high school and first-year college calculus students.

This is a very interesting approach to understanding Calculus. John is an excellent writer and not only has strong background in Mathematics but also in Science & Engineering. He is well-versed in Computational Science and has many years of experience in applying mathematics to the solving of real world problems. I would highly recommend this book to anyone who is learning or teaching Calculus.

The 3rd Edition of Calculus without Limits has a full-subject index and expanded material on applied optimization problems, which includes a calculus-based demonstration of the Pythagorean Theorem. One of the unique features of Calculus without Limits is the smooth transition into elementary differential equations and associated physics. This is made possible by the almost exclusive use of differentials in order to develop the subject manner. The book has four intended uses: 1) as a supplement to a standard college calculus course, 2) as a self-help study guide, 3) as a resource for bright high school students about to tackle the AP calculus exam, 4) and as a primary high-school text for a single-term introduction to single-variable calculus (a suggested course outline is included in the introduction)--especially good for home-schoolers.

As I started reading this book I noticed not only math but peppered throughout is a bit of poetry, well I thought that this concept was refreshing. The integration of two subjects in one book is a novel idea.
Calculus is taxing enough on its own, however when it is broken up with any other discipline I think that it is a good thing, this tends to break up the monotonie. Great job Mr. Sparks and best of luck on the others because I know that where will be other.

While Sparks has tried to explain basic calculus without limits, he really adds nothing to existing books. Calculus can truly be used without limits. In fact, limits did not exist when calculus was invented.

Sparks' book is wishy, lacks rigour and is somewhat disappointing. Poetry has no place in mathematics, only cold, hard facts. Unfortunately, Sparks' book falls short in many places because it indirectly uses the exact methods used in the process of finding limits. It is still 'Calculus with limits', not 'calculus without limits'.

For a new and refreshing calculus, I invite you to visit:
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