Subject: Calculus
Dear Melvin,
I am touched to know that you have found my web site useful and inspiring.   I have a few responses from graduate students from California who have said how much it has helped them in Mathematical Analysis.   My modest aim is to share what I have found useful, what I have found it difficult to understand and how I have strived to understand, and from quite a different perspective to provide a selected window, probably with a somewhat biased emphasis on the mathematics that I have tried to present as clearly and as logically and not always in a very polished elegant way (when most of the detail would have been left out).  I have also been in correspondence with a retired gentleman from Belgian who (since 2000) started to learn calculus upon the retirement at the age of 75!  Good for him!
Yes, mathematics is about finding truth and sometimes truth is not what it seems just like real life events.

Calculus is particularly difficult because it deals with the real numbers and the the set of real numbers or its many manifestations is not easy to understand.  Lots of questions can be asked and many can be directed towards a logical foundation of mathematics.  Many structures are built up from the real numbers, first complex numbers, then the quaternions, the diviison algebras, vectors spaces, abstract algebras, and other structures that are built upon theses structures and further structures on these structures.  It is not surprising when we have time to ponder about the real numbers and its immediate structured domain, calculus that we find there are still many questions one can ask and try to understand, without ever finding them stale and that each time the question is refreshing even though it may have been known and one's own inner working in arriving at the understanding is as invigorating as life itself. 

As now is the Christmas Season, I would like to share some of my philosophy or what guides me in my teaching and learning and in life itself.  It is to share ones  knowledge for the betterment of human kind and NEVER for the harm or destruction of life in whatever form.   I believe in sharing knowledge for the good of the community to spread the understanding as wide as possible, to solve problems, resolve conflicts, eradicate misery, sorrow, hunger, diseases, wanton hopelessness, right injustices and boldly to reach out to mend the very fabric of peace and to bring love to this very very troubled world, where human conflicts abound and meaningless death are very much a human weakness and more so of world leaders.   
And I would also like to share this.  Very often acquiring knowledge seems to be an end in itself and now it has become a competitive tools for advancement and climb up the academic ladder. (Like the Babel Tower, heaps and heaps of research papers are written every year, at a modest count at least 30,000 a year.)  We have forgotten the reasons or purposes for acquiring knowledge or have replaced them by self-absorbed, self-serving ones  propelled by careerism and academic pecking order.   It is not how much mathematics we can create but how well we use the mathematics and it is not always the mathematics that is on the boundary of what mathematicians consider as frontier; it is about how well and how we can impart the use of mathematics towards a better understanding of our world for us to share and to live in.   Mathematics understanding can have its simplest and humblest beginning in the smallest of  steps one takes.  The very first steps one takes to understanding how diseases spread, how to control the spread of cancer cells by building a mathematical model, the mathematics will present itself in ones search for solution.  The very first steps one thinks of how to bring employment to a work deprived neighbourhood will bring together the knowledge of local conditions, existing infra structures, natural resources, geography, historical, social and cultural background and knowledge of the people of the neighbourhood and the beginning of a sophisticated economic/industrial model built from those input and with a very different kind of mathematics that finds logics, combinatorics, discrete mathematics, statistics, operations research, business mathematics and hence calculus and even geometry together in a myriad and probably unstructured form.   Mathematics should beckon you and ask you questions and along the way the training and the care with which one answers those questions are solid foundation for the use of mathematics effectively in solving problems (not necessarily mathematical in nature) and also in mathematics itself.   Mathematics is reflective in nature and undergoes a lot of self correction very much like human thoughts ( a la Morris Kline).   My advice to students is always this: Be true to yourself; if you do not understand, seek understanding; ask your teacher, you will probably find that many fellow students do not understand too ; work in a group if you can, share your thoughts or understanding or what you find conflicting, collectively you will probably be able to come to a better understanding or to a better position to judge how far you have succeeded in understanding; try to solve the problem or answer the questions yourself;  ask questions and answer your questions yourselves, very often your own questions are the landmarks of your own understanding; if all else fails, ask your teachers politely or someone whom you think can explain to you in detail.  
Warmest Regards,
Merry Christmas
God Bless
Tze Beng
Greetings from a fellow mathematics teacher in Billings, Mt.  I teach AP Calculus at Billings Central Catholic High School in Billings, Montana
and I am always looking for sites for my students to check out.  I found yours last year and was impressed with your philosophy statement, which
read to my class.   I just finished your Definite Integral quiz and am proud to say that I scored 5/5!!  Not bad for an old calc teacher.  It
is amazing how much you continue to learn about mathematics, years after having taken the course work.  I just figured out the formula for
finding F'(x) last year and gave it to my students. I used this formula to solve
the last integral on the quiz. 

        F'(x) = f(g(x))*g'(x)    where g(x) is the upper limit. 

I'll have to admit that most of your problems are far above my high school students capabilities at this time.  I do have some great raw
talent but it will take some time to develop it.  I will give them your web address so they can see what a tough calc course would look like.
Keep up the good work.  Any ideas you might have for my students please feel free to share.


Mel Wahl
BCCHS Mathematics Dept.