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.
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.
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What has not changed?
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.
Getting more quickly to derivatives
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).
New treatment of DEs
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.
Not a crypto-DE course
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.
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.
Exercises and such
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!).
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.
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.
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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