Second Edition

One way to find out what drives our text is to read the prefaces---one for instructors and one for students---from the first edition.

(The instructors' preface has considerable detail on philosophy, use of technology, pedagogical goals, etc.)

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Changes from the first edition. Why revise? What's changed?
**

In short, to make the text easier to use, both for teachers
and for students.
To this end, building on many suggestions from both teachers
and students, we've made various changes in narrative, exercises,
content, emphasis, order of presentation, added answers to selected
in the back, etc.---some of these changes are outlined below.

For more detailed information, click either link below:

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What has not changed?
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The basic principles and strategies underlying 1/e remain unchanged.
Conceptual understanding is still the main goal, and combining
various viewpoints is still the main strategy for achieving it.
We've retained the basic assumptions
and operating premises of the first edition: an emphasis on concepts
and sense-making; complementing symbolic with graphical and numerical
points of view; exercises of varied nature and difficulty; a narrative
aimed at student readers.

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Getting more quickly to derivatives
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We have somewhat compressed the ``pre-calculus'' material in order to
get to the derivative idea faster. Chapter~1 now includes essentially
complete coverage of the graphical point of view; derivatives now
appear first in Section~1.4. Chapter 2 introduces and interprets the
symbolic point of view, and Chapter 3 presents the combinatorial rules
for calculating derivatives (e.g., the product and quotient rules).

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New treatment of DEs
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Differential equations now appear a little earlier and more often
than in the first edition. Section 2.5 (a few sections after the
ideas of derivative and antiderivative are first met in symbolic form)
introduces the basic idea. Mentioning the DE idea early lets us say
early (and economically) that exponential functions are important
largely because they have a certain crucial growth property, thanks
to the DE * y'=ky*. In the same spirit, we can say early that
the trigonometric functions satisfy the DE *y''=-y* that free-fall
conditions correspond in a certain way to *y''=k* ---and that these
facts account largely for the importance of the functions we'll work
so hard with. DEs also provide a natural approach to scientific and
engineering applications.

Once introduced, DEs appear repeatedly in following sections and in exercises. Slope fields (aka direction fields) now appear in Chapter 4---they appeared in Chapter 12 in 1/e. Euler's method now appears in Chapter 6 (along with numerical integration methods), and separation of variables appear in Chapter~7 as an application of symbolic antidifferentiation.

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Not a crypto-DE course
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Although we stress DEs a little more in 2/e, we have no intention of
jamming a DE course into the already well-filled calculus syllabus.
We want to convey the * idea * of a DE and its solutions; we remark,
for example, that the antiderivative equation * y'=f(x) * is itself
a DE. But we make * no * effort to cover, or even catalog, the
huge variety of DEs and solution techniques. Instead, we sometimes use
DEs to motivate new techniques and concepts as they develop naturally
over the course.

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Content changes and reorganization
**

Several changes have been made to the content and the organization of
the text. The
somewhat increased treatment of DEs, for example, was discussed above.
Other ``content'' changes include a new brief chapter on function
approximation, centering on Taylor polynomial approximation but also
including basic discussion of Fourier polynomials.

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Exercises and such
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Many users found 1/e short on ``routine'' exercises; many more have been
added to 2/e. (But 2/e will, like 1/e, contain plenty of
``interesting'' problems.) We've also added more exercises that point to
specific issues and examples in the narrative.
We are also including answers (not solutions)
to odd-numbered exercises in the back of the book, again in response to
many user requests (from teachers, not just from students!).

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Interludes
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Most chapters end with one or more ``interludes''---brief
project-oriented expositions with exercises, designed for
independent student work---that address topics or questions
that are ``optional'' or out of the given chapter's main stream of
development.

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Help with reading
**

To help students read the text more successfully, we've added
more examples to many sections, rewritten many parts of the narrative,
and included more detail and brief commentary on many calculations.
We've also included more problems in the exercise sets that draw students'
attention explicitly to aspects of the narrative.

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Annotated Instructor's Edition
**

We plan a special edition of the textbook intended specifically
for instructors, with marginal annotations, hints, pointers, and
suggestions for teaching from the text. For example, the AIE
will point to topics that are ``foreshadowed,'' appearing
more than once.

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For more information
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For mathematical information, explore this Web site or email
Paul Zorn .