AboutAbe Mantell Expertise Hello,
I am a college professor of mathematics and regularly teach all levels
from elementary mathematics through differential equations, and would
be happy to assist anyone with such questions!
Experience Over 15 years teaching at the college level.
My questions appears at the very end of this writing, but I thought I’d give you first an idea of why I am asking it, the context and my personal background.
(You will not be flogged if you skip the preamble, which, quite atypically, in this case constitutes the bulk of the writing. !).
I am 66 years old, with a strong background in the humanities, including an M.A. in Philosopy and a background in Classics, from my High-School days back in Italy ( Liceo Classico),including Latin and Greek, which I have ever since kept up, until today.. . I did Maths, though, including a few years of Algebra and Trigonometry, ( although I never learned Calculus) but I have always had a strong aversion to them and, to use a litote, I never really excelled in them
In retrospect, I guess, it is because I have always felt that the formal structure of mathematical thinking, in its seeming coldness and abstraction, does not have a concrete conceptual frame of reference within which we can plot our existential concerns and the basic questions which philosophy, instead, deals with and which ,in its own ways ,it attempts to answer, even though its questions ( and the answers ) cannot be proved ( but neither disproved ) by science, precisely because the two deal with entirely different but equally valid and complementary realities, Science being concerned with causal links ( the HOW question) while Philosophy attempts to find the existential meaning of reality by asking the ultimate WHY’s of things.
As inevitable, though, my intellectual curiosity, has recently sparked a totally new and hitherto unexpected interest in Science ,via Physics, which I can no longer now , by any stretch of imagination, consider an ABSTRACT pursuit, precisely because I realize it involves the understanding of the concrete world with which we interact at every step of our lives and which we take for granted : whether it is the sun rising every day or my watching TV or using a cellular phone, the Universe and the technology we use to harness it, exploit some basic laws of nature which are not of our making, but which , perhaps because of this.., I feel challenged to explore and understand..
Even without adequate mathematical knowledge, I thought I could and I should at first grasp the conceptual aspect of some key notions of Physics , like the Theory of Relativity and Quantum Mechanics in their relation to classic newtonian physics... I have found excellent material on the Internet , and although I am sure I am only scratching the surface, I believe I have grasped the essence of them and their seeming counter-intuitive conclusions. But here we are : however nice it is to understand the conclusions, it would be even nicer to understand the mathematical thinking leading to them, at least the main points, which is where I stumble.
So I have decided to backtrack and to review my mathematical notions , deeply buried in my subconscious, after 50 years of oblivion. As well I’d like to grasp Calculus, even more so that I am accustomed to hear it is stuff for really smart people and I want to face the challenge….
I have found an excellent site on the Internet, called Calculus on the Web ( COW, for short )
I have completed the COW pre-calculus algebra section in a few months, by solving ALL the problems and exercises given (there is no key , but for every attempted step of the solution, the COW evaluates the answer and stops you if it is wrong.)
I found it very challenging indeed to rack my brains , sometimes for hours on end, but it is also deeply satisfying to finally get the correct answer. !
I’ll move to the Calculus 1 section shortly. Again, I went to the WEB to understand , before I embark on the purely mathematical aspect, the CONCEPTS of Calculus, i.e. the Derivative, the Limit , Integration, Differentiation..
I have found and downloaded a series of lectures, in which a mathematician explains in lay terms the meaning of Calculus ,by concrete and sometimes amusing examples …So now I will undestand better, I hope, what lies underneath the elucubrations of Calculus.
And now the question :
Can you refer me to books or Web resources dealing with CONCRETE applications of Calculus ( in Physics, Financial Maths, Sports ( ?) etc ) and containing drills ( lots of them ) with answers at the end ?
Thank you so much for your answer and for bearing with my long exposé..
Franco Vivona
Montreal, Canada
Answer Hello Franco,
It is always satisfying to hear from people who develop an appreciation
for or interest in mathematics after having shunned it during the
early years! :-D
As for calculus with applications...
1. Calculus, by Deborah Hughes-Hallett, Andrew M. Gleason, et. al.
-- has many nice realistic and interesting problems. Particularly
-- sections 4.3-6, all of chapt. 8, 11.5-7
2. Just about any calculus text that is written for engineering or
-- science majors...e.g. Calculus for Engineers, by Donald Trim
-- or Applied Calculus for Scientists and Engineers, by Frank Blume
