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PREFACE TO THE INSTRUCTOR
Hell's library
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and to spend as little time as possible reviewing precalculus material. However,
college freshmen constitute a very diverse group, with widely different levels of
mathematical preparation. For this reason I have included a first chapter on pre-
calculus material, which I urge teachers to skim over as lightly as they think ad-
visable for their particular students. This chapter is written in enough detail so
that individual students who need to spend more time on the preliminaries should
be able to absorb most of it on their own with a little extra effort.*
Problems For students, the most important parts of their calculus book may well
be the problem sets, because this is where they spend most of their time and en-
ergy. There are more than 7,000 problems in this book, including many old stand-
bys familiar to all calculus teachers and dating back to the time of Euler and even
earlier. I have tried to repay my debt to the past by inventing new problems when-
ever possible. The problem sets are carefully constructed, beginning with routine
drill exercises and building up to more complex problems requiring higher lev-
els of thought and skill. The most challenging problems are marked with an as-
terisk (*). In general, each set contains approximately twice as many problems
as most teachers will want to assign for homework, so that a large number will
be left over for students to use as review material.
Most of the chapters conclude with long lists of additional problems. Many of
these are intended only to provide further scope and variety to the problems sets
at the ends of the sections. However, teachers and students alike should treat these
additional problems with special care, because a few are quite subtle and diffi-
cult and should be attacked only by students with ample reserves of drive and
tenacity.
I should also mention that there are several sections scattered throughout the
book with no corresponding problems at all. Sometimes these sections occur in
small groups and are merely convenient subdivisions of what I consider a single
topic and intend as a single assignment, as with Sections 6.1, 6.2, 6.3, and 6.4,
6.5. In other cases (e.g., Sections 15.5 and 19.4), the absence of problems is a
tacit suggestion that the subject matter of these sections should be touched upon
only lightly and briefly.
There are a great many so-called "story problems" spread through the entire
book. All teachers know that students shudder at these problems, because they
usually require nonroutine thinking. However, the usefulness of mathematics in
the various sciences demands that we try to teach our students how to penetrate
into the meaning of a story problem, how to judge what is relevant to it, and how
to translate it from words into sketches and equations. Without these skills —
which are equally valuable for students who will become doctors, lawyers, fi-
nancial analysts, or thinkers of any kind—there is no mathematics education
worthy of the name.
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*A more complete exposition of high school mathematics that is still respectably concise can be found
in my little book. Precalculus Mathematics in a Nutshell (Janson Publications, Dedham, MA, 1981),
119 pages.
+
I cannot let the opportunity pass without quoting a classic story problem that appeared in The New
Yorker magazine many years ago. "You know those terrible arithmetic problems about how many
peaches some people buy, and so forth? Well, here's one we like, made up by a third-grader who was
asked to think up a problem similar to the ones in his book: "My father is forty-four years old. My
dog is eight. If my dog was a human being, he would be fifty-six years old. How old would my fa-
ther plus my dog be if they were both human beings?"
By GARY LARSON