Ebook: Polynomials, Power Series and Calculus (University Series in Undergraduate Mathematics)

This book is not intended for use as a text for the calculus course now
generally given in the United States, but rather as a text for a proposed
replacement for that course.

During the author's career as a teacher of college mathematics, he
has seen three courses disappear from the standard college sequence in
mathematics in most colleges and universities-college algebra, trigo-
nometry, and analytic geometry. This book is written with the thought
that the traditional calculus course is moving toward the same fate. Each
year many mathematics departments change their calculus textbook
because the old one proved unsatisfactory, only to find that the" new"
one leads to a similar sense of defeat. The whole situation, in the author's
opinion, suggests that the course itself is at fault and that no text can be
satisfactory for it.

The traditional calculus course seeks to impart mastery of the concepts
of limit, derivative, and integral. In principle, this is a fine program, but
in practice, it seems to be getting out of hand. This could be due to the
fact that, whether he knows it or not, the student of calculus is studying
a class of functions with some very, very pathological members, and his
methodology has been elaborated in order to cope with them (for in-
stance, Riemann introduced his integral in order to investigate highly
discontinuous functions). If these difficulties were the necessary price for
achieving the benefits of calculus, the teacher could simply apologize
and proceed.

The present author believes, on the contrary, that the actual benefit
of the calculus is not the mastery of these concepts but the acquisition
of skill in dealing with relatively well-behaved functions. He offers a
more direct way of acquiring this skill and believes that nothing essential
is lost there by.