Understanding Calculus

Understanding Calculus is a complete online introductory book that focuses on concepts. Integrated throughout the e-book are many engineering applications aimed at developing the student's scientific approach towards problem solving.

The book has as much to do with calculus as with philosophy. My motivation in writing it was to prove to myself that I could understand a complex subject like calculus by applying simple rules of logic and reason. As Henry Ford said, " Nothing is particularly hard if you divide it into small jobs ".

I have now forgotten a lot of the calculus, but what remains is that sense that anything can be understood. Too much of the world is complicated by layers of evolution. If you understand how each layer is put down then you can begin to understand the complex systems that govern our world. Charles Darwin wrote in 1859 in his On The Origin of Species,

" When we no longer look at an organic being as a savage looks at a ship, as at something wholly beyond his comprehension; when we regard every production of nature as one which had a history; when we contemplate every complex structure and instinct as the summing up of many contrivances, each useful to the possessor, nearly in the same as when we look at any great mechanical invention as the summing of the labour, the experience, the reason, and even the blunders of numerous workmen; when we thus view each organic being, how far more interesting, I speak from experience, will the study of natural history become! "

Any system is a product of countless years of evolution. Merely changing names, rhetoric or ideology does not undo all the history that led us to where we are. Darwin's words are equally applicable to art as well as science. Every artist's performance requires a great deal of craft. Even stand-up comedy is not a question of going on stage and saying funny things. There is an enormous amount of work, practice and thought that would have to go into it.

My advice to students is to find something that fascinates you. As Physics Nobel Laureate Richard Feynman said, " Study hard what interests you the most in the most undisciplined and original manner possible." The price for seeking understanding is that one must be willing to tolerate ambiguities and accept one's own ignorance. Feynman said, " I can live with doubt and uncertainty. I think it is much more interesting to live not knowing than to have answers which might be wrong."

I hope you find the book enlightening as the goal is to learn by understanding, not by memorizing. I am currently writing another book called, Understanding Physics. Please do not hesitate to contact me with your feedback and questions. Let me know what complexities of life you are trying to solve!

Faraz Hussain - Author

Testimonials on Understanding Calculus

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I graduated from BGSU in 1974 where I took a year in Calculus. It always bothered me that I did not fully understand the Fundamental Theorem of Calculus or more specifically the connection of the derivative and anti derivative to the theorem. As in reference to your preface, I memorized the procedures without understanding what I was doing. After 35 years it makes sense to me now. Thank you! Steve Wright

Great text. Best and clearest resource I've found - especially for topics requiring geometrical interpretation. Consulted others, Stewart, Thomas, Keisler, etc. Exposition crystal clear. Others not. Many thanks.

Mr Faraz, I must say that your site on understanding calculus is the best resource I have seen on Calculus. I am doing a masters course in mechanical engineering in New york, and I have been gone from college for a while before continuing to my masters. I forgot a lot, but when I found your site and started reading, I understood things i never saw reason with.

I understand why calculus is used in almost all engineering courses, and I havent even gotten to half of your book. Thanks a lot.

I am a high school physics teacher and am trying to teach the course by having the students use as much logic as they can. I try to include as much calculus as I can without getting bogged down in the equations. I like your approach to the subject of calculus and I am looking for ways to incorporate more calculus into the curriculum.

I really greatly appreciate your effort to make learning calculus easier than ever. I used to do very excellent in mathematics when I was in High School. Unfortunetly I get divorced with mathematics when I joined Agricultural University where a freshmen calculus instructor used to mock us by saying mathematics has little to do with farmers, this much is enough for you (for excuses not get in to detail). That is almost 20 years ago. After that some times I feel very guilty of abandoning Mathematics and Physics which I used to like very much and which I can understand the logic behind of most seemingly complex formulas. I went to library and tried to get the simplest introductory book on calculus. But all the books I tried to read were to complicated to me and I dropped the idea of doing mathematics again. So I start to browse the internet to get some tips on learning Mathematics specially calculus. So I soon came across of this electronic book which I found very help full to regain my mathematics skill and knowledge. Thank you so much for writing this book and posting it on the internet so that every one can have access free of charge.

Problem with many math text I have read is they are far too concerned with correctness and proofs. This is fine once the subject has been understood on a basic level, but to first learn Calculus you need a more example driven approach like you do. I can only say I thank you for making this available.

I am from Greece, specifically Chania - Crete, just finished architecture in Athens, and now I started a masters in the department of Mechanics and Science of Materials in the Technical University of Crete. That includes lessons that require deep understanding of mathematics, and therefore I find your site very helpful.

I am a software engineer working for an investment bank. In a pursuit to understand all the risk positions and corresponding calculations, I started reading some stuff and soon came to know that I need a deep conceptual understanding of calculus and linear algebra to know the business. Actually I had learned the calculus when I was doing my bachelors in India but it was blind learning without necessarily understanding the applications in real life. I don't know whether it was my inclination to study blindly or there was something lacking in educational system which never made the subjects interesting. The way you have explained Maths as a language of science to describe the changes in situation seems so true to me after I started relating real life problems to mathematical calculations. As of now I am done with reading of first 12 chapters and already started understanding many financial risk analysis computations.

I think that you must assume that your visitor actually has the passion for math "or why he visits your site at all". So you should decrease the long intro. which turned into pure philosophical essay instead of an intro to the main subject. Finally , I must admit it's really a good essay about philosophy "in the wrong place" which i have really enjoyed.

I am a student of PUNJAB AGRICULTURAL UNIVERSITY (INDIA) and pursuing B.Tech in Agricultural Engineering. I am very much thankful to you for providing such a good material for Calculus.

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