xi
Preface
is text covers single-variable differential calculus and reviews necessary algebra skills. It was
developed for a course that arose from a perennial complaint by the physics department at the
University of Guelph that the introductory calculus courses covered topics roughly a year after
they were needed. In an attempt to address this concern, a multi-disciplinary team created a
two-semester integrated calculus and physics course. is book covers the differential calculus
topics from that course. e philosophy of the course was that the calculus be delivered before it
is needed, often just in time, and that the physics serves as a substantial collection of motivating
examples that anchor the student’s understanding of the mathematics.
e course ran three times before this text was started, and it was used in draft form
for the fourth offering of the course and then for two additional years. ere is a good deal of
classroom experience and testing behind this text. ere is also enough information to confirm
our hypothesis that the course would help students. e combined drop and flunk rate for this
course is consistently under 3%, where 20% is more typical for first-year university calculus.
Co-instruction of calculus and physics works. It is important to note that we did not achieve
these results by watering down the math. e topics covered, in two semesters, are about half
as many as are covered by a standard first-year calculus course. at’s the big surprise: covering
more topics faster increased the average grade and reduced the failure rate. Using physics as a
knowledge anchor worked even better than we had hoped.
is text and its two companion volumes, Fast Start Integral Calculus and Fast Start Ad-
vanced Calculus, multivariate calculus make a number of innovations that have caused mathe-
matical colleagues to raise objections. In mathematics it is traditional, even dogmatic, that math
be taught in an order in which nothing is presented until the concepts on which it rests are al-
ready in hand. is is correct, useful dogma for mathematics students. It also leads to teaching
difficult proofs to students who are still hungover from beginning-of-semester parties. is text
neither emphasizes nor neglects theory, but it does move theory away from the beginning of the
course in acknowledgment of the fact that this material is philosophically difficult and intellec-
tually challenging. e course presents a broad integrated picture as soon as possible. Cleverness
and computational efficiency are emphasized following the philosophy that “mathematics is the
art of avoiding calculation.”
e material on the mathematical foundations of limits was presented near the end of the
first semester in the course for which these texts were developed. It appeared about mid-way
through the original single-volume edition of the book. e partitioning of the original text
into three parts left us with the question of where to put the theoretical material on limits and
continuity. We settled on the end of this text, on the differential calculus. In the past, formal
xii PREFACE
limits and continuity have been taught at the end of what is now the second text, on the integral
calculus. Instructors can teach the material where they want—other than an appropriate level of
mathematical maturity, there is no prerequisite for this material.
It is important to state what was sacrificed to make this course and this text work the
way they do. is is not a good text for math majors, unless they get the theoretical parts of
calculus later in a real analysis course. e text is relatively informal, almost entirely example
driven, and application motivated. e author is a math professor with a CalTech Ph.D. and
three decades of experience teaching math at all levels from 7th grade (as a volunteer) to graduate
education including having supervised a dozen successful doctoral students. e author’s calculus
credits include calculus for math and engineering, calculus for biology, calculus for business, and
multivariate and vector calculus.
Daniel Ashlock
August 2019
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