4th版单变量微积分概念与应用：融合传统与改革

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"Single Variable Calculus Concepts and Contexts 4th Edition 是一本专门针对美国大学一元微积分入门的英文教材，旨在融合改革与传统教学方法，强调概念理解。该书通过视觉、口头、数值和代数手段帮助学生掌握微积分的实用性和内在美感。不同于作者以往较为传统的教材，这本第四版更加精简，如没有完整的积分技术章节，证明的定理较少，并且将超越函数和参数方程的内容分散在全书中，而非独立章节。本书的特点包括： 1. 注重概念: 教材通过直观的方式传达微积分的基本原理，重视让学生理解和应用数学工具，而不只是机械地练习。 2. 简化结构: 缩减了传统章节，如没有专门讨论积分技巧，减少了定理证明，使内容更为紧凑，适合那些希望专注于核心概念的学生。 3. 整合内容: 特殊多项式的因式分解、二项式定理、圆锥曲线公式等几何和代数知识与微积分主题紧密结合，强化实际问题解决能力。 4. 应用实例: 提供了丰富的例题和练习，让学生能够在解决实际问题的过程中深化对微积分的理解，同时体验其在生活和科学中的实际应用价值。 5. 几何公式: 包括三角形、圆、扇形、球体、圆柱、圆锥的面积、周长和体积计算公式，以及距离和中点的几何关系。 6. 线性代数基础: 包含直线的斜率和方程、点斜式和斜截式方程的介绍，以及圆的方程表达形式。 7. 精确度与教学策略: 作者在保持教学内容精炼的同时，也提及了关于严谨性（rigor）的讨论，提醒读者教学过程中需平衡理论证明与实践操作。这本教材以现代教育理念为导向，提供了适合不同教学风格和学习需求的一元微积分学习材料，旨在帮助学生在理解的基础上欣赏到微积分的魅力。对于希望寻求全面传统教学覆盖的教师，作者还有其他更全面的教材可供选择。"

When the ﬁrst edition of this book appeared twelve years ago, a heated debate about cal-

culus reform was taking place. Such issues as the use of technology, the relevance of rigor,

and the role of discovery versus that of drill were causing deep splits in mathematics

departments. Since then the rhetoric has calmed down somewhat as reformers and tradi-

tionalists have realized that they have a common goal: to enable students to understand and

appreciate calculus.

The ﬁrst three editions were intended to be a synthesis of reform and traditional

approaches to calculus instruction. In this fourth edition I continue to follow that path by

emphasizing conceptual understanding through visual, verbal, numerical, and algebraic

approaches. I aim to convey to the student both the practical power of calculus and the

intrinsic beauty of the subject.

The principal way in which this book differs from my more traditional calculus text-

books is that it is more streamlined. For instance, there is no complete chapter on tech-

niques of integration; I don’t prove as many theorems (see the discussion on rigor on page

xv); and the material on transcendental functions and on parametric equations is inter-

woven throughout the book instead of being treated in separate chapters. Instructors who

prefer fuller coverage of traditional calculus topics should look at my books Calculus,

Sixth Edition, and Calculus: Early Transcendentals, Sixth Edition.

What’s New in the Fourth Edition?

The changes have resulted from talking with my colleagues and students at the University

of Toronto and from reading journals, as well as suggestions from users and reviewers.

Here are some of the many improvements that I've incorporated into this edition:

■

At the beginning of the book there are four diagnostic tests, in Basic Algebra,

Analytic Geometry, Functions, and Trigonometry. Answers are given and students

who don’t do well are referred to where they should seek help (Appendixes, review

sections of Chapter 1, and the website stewartcalculus.com).

■

The majority of examples now have titles.

■

Some material has been rewritten for greater clarity or for better motivation. See,

for instance, the introduction to maximum and minimum values on pages 262–63

and the introduction to series on page 565.

■

New examples have been added and the solutions to some of the existing examples

have been ampliﬁed. For instance, I added details to the solution of Example 2.3.10

because when I taught Section 2.3 last year I realized that students need more guid-

ance when setting up inequalities for the Squeeze Theorem.

■

A number of pieces of art have been redrawn.

■

The data in examples and exercises have been updated to be more timely.

■

In response to requests of several users, the material motivating the derivative is

briefer: The former sections 2.6 and 2.7 have been combined into a single section

called Derivatives and Rates of Change.

■

The section on Rates of Change in the Natural and Social Sciences has been moved

later in Chapter 3 (Section 3.8) in order to incorporate more differentiation rules.

Preface

xiii

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xiv

PREFACE

■

Coverage of inverse trigonometric functions has been consolidated in a single dedi-

cated section (3.6).

■

The former sections 4.6 and 4.7 have been merged into a single section, with a

briefer treatment of optimization problems in business and economics.

■

There is now a full section on volumes by cylindrical shells (6.3).

■

Sections 8.7 and 8.8 have been merged into a single section. I had previously fea-

tured the binomial series in its own section to emphasize its importance. But I

learned that some instructors were omitting that section, so I decided to incorporate

binomial series into 8.7.

■

More than 25% of the exercises in each chapter are new. Here are a few of my

favorites: 2.5.46, 2.5.49, 2.8.6–7, 3.3.50, 3.5.45– 48, 4.3.49–50, 5.2.47– 49, and

8.2.35.

■

There are also some good new problems in the Focus on Problem Solving sections.

See, for instance, Problem 5 on page 252, Problems 17 and 18 on page 429, Prob-

lem 15 on page 492, and Problem 13 on page 632.

■

The new project on page 475, Complementary Coffee Cups, comes from an article

by Thomas Banchoff in which he wondered which of two coffee cups, whose con-

vex and concave proﬁles ﬁt together snugly, would hold more coffee.

Features

Conceptual Exercises

The most important way to foster conceptual understanding is through the problems that

we assign. To that end I have devised various types of problems. Some exercise sets begin

with requests to explain the meanings of the basic concepts of the section. (See, for

instance, the ﬁrst couple of exercises in Sections 2.2. 2.4, 2.5, 5.3, and 8.2. I often use them

as a basis for classroom discussions.) Similarly, review sections begin with a Concept

Check and a True-False Quiz. Other exercises test conceptual understanding through

graphs or tables (see Exercises 1.7.22–25, 2.6.17, 2.7.33–44, 3.8.5–6, 5.2.47–49,

7.1.11–13, and 8.7.2).

Another type of exercise uses verbal description to test conceptual understanding (see

Exercises 2.4.10, 2.7.54, 2.8.9, 2.8.13–14, and 5.10.55). I particularly value problems that com-

bine and compare graphical, numerical, and algebraic approaches (see Exercises 2.5.38,

2.5.43– 44, 3.8.25, and 7.5.2).

Graded Exercise Sets

Each exercise set is carefully graded, progressing from basic conceptual exercises and

skill-development problems to more challenging problems involving applications and

proofs.

Real-World Data

My assistants and I have spent a great deal of time looking in libraries, contacting compa-

nies and government agencies, and searching the Internet for interesting real-world data

to introduce, motivate, and illustrate the concepts of calculus. As a result, many of the

examples and exercises deal with functions deﬁned by such numerical data or graphs. See,

for instance, Figure 1 in Section 1.1 (seismograms from the Northridge earthquake), Exer-

cise 5.1.14 (velocity of the space shuttle Endeavour), Figure 5 in Section 5.3 (San Fran-

cisco power consumption), and Example 5 in Section 5.9 (data trafﬁc on Internet links).

Projects

One way of involving students and making them active learners is to have them work (per-

haps in groups) on extended projects that give a feeling of substantial accomplishment

when completed. Applied Projects involve applications that are designed to appeal to the

imagination of students. The project after Section 3.1 asks students to design the ﬁrst

ascent and drop for a roller coaster. Laboratory Projects involve technology; the project

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PREFACE xv

following Section 3.4 shows how to use Bézier curves to design shapes that represent let-

ters for a laser printer. Writing Projects ask students to compare present-day methods with

those of the founders of calculus—Fermat’s method for ﬁnding tangents, for instance. Sug-

gested references are supplied. Discovery Projects anticipate results to be discussed later

or cover optional topics (hyperbolic functions) or encourage discovery through pattern

recognition (see the project following Section 5.8). Additional projects can be found in the

Instructor’s Guide (see, for instance, Group Exercise 5.1: Position from Samples) and also

in the CalcLabs supplements.

Rigor I include fewer proofs than in my more traditional books, but I think it is still worthwhile

to expose students to the idea of proof and to make a clear distinction between a proof and

a plausibility argument. The important thing, I think, is to show how to deduce something

that seems less obvious from something that seems more obvious. A good example is the

use of the Mean Value Theorem to prove the Evaluation Theorem (Part 2 of the Fundamen-

tal Theorem of Calculus). I have chosen, on the other hand, not to prove the convergence

tests but rather to argue intuitively that they are true.

Problem Solving Students usually have difﬁculties with problems for which there is no single well-deﬁned

procedure for obtaining the answer. I think nobody has improved very much on George

Polya’s four-stage problem-solving strategy and, accordingly, I have included a version of

his problem-solving principles at the end of Chapter 1. They are applied, both explicitly

and implicitly, throughout the book. (The logo emphasizes some of the explicit occur-

rences.) After the other chapters I have placed sections called Focus on Problem Solving,

which feature examples of how to tackle challenging calculus problems. In selecting the

varied problems for these sections I kept in mind the following advice from David Hilbert:

“A mathematical problem should be difﬁcult in order to entice us, yet not inaccessible lest

it mock our efforts.” When I put these challenging problems on assignments and tests I

grade them in a different way. Here I reward a student signiﬁcantly for ideas toward a solu-

tion and for recognizing which problem-solving principles are relevant.

Technology The availability of technology makes it not less important but more important to under-

stand clearly the concepts that underlie the images on the screen. But, when properly used,

graphing calculators and computers are powerful tools for discovering and understanding

those concepts. I assume that the student has access to either a graphing calculator or a

computer algebra system. The icon

;

indicates an exercise that deﬁnitely requires the use

of such technology, but that is not to say that a graphing device can’t be used on the other

exercises as well. The symbol is reserved for problems in which the full resources of a

computer algebra system (like Derive, Maple, Mathematica, or the TI-89/92) are required.

But technology doesn’t make pencil and paper obsolete. Hand calculation and sketches are

often preferable to technology for illustrating and reinforcing some concepts. Both instruc-

tors and students need to develop the ability to decide where the hand or the machine is

appropriate.

Tools for Enriching™ Calculus TEC is a companion to the text and is intended to enrich and complement its contents. (It

is now accessible from the Internet at www.stewartcalculus.com.) Developed by Harvey

Keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory

approach. In sections of the book where technology is particularly appropriate, marginal

icons direct students to TEC modules that provide a laboratory environment in which they

can explore the topic in different ways and at different levels. Visuals are animations of ﬁg-

ures in the text; Modules are more elaborate activities and include exercises. Instructors

can choose to become involved at several different levels, ranging from simply encourag-

ing students to use the Visuals and Modules for independent exploration, to assigning spe-

ciﬁc exercises from those included with each Module, or to creating additional exercises,

labs, and projects that make use of the Visuals and Modules.

CAS

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xvi

PREFACE

TEC also includes Homework Hints for representative exercises (usually odd-num-

bered) in every section of the text, indicated by printing the exercise number in red. These

hints are usually presented in the form of questions and try to imitate an effective teaching

assistant by functioning as a silent tutor. They are constructed so as not to reveal any more

of the actual solution than is minimally necessary to make further progress.

Enhanced WebAssign

Technology is having an impact on the way homework is assigned to students, particu-

larly in large classes. The use of online homework is growing and its appeal depends on

ease of use, grading precision, and reliability. With the fourth edition we have been work-

ing with the calculus community and WebAssign to develop an online homework system.

Many of the exercises in each section are assignable as online homework, including free

response, multiple choice, and multi-part formats. The system also includes Active Examples,

in which students are guided in step-by-step tutorials through text examples, with links to

the textbook and to video solutions.

Website: www.stewartcalculus.com

This website includes the following.

■

Algebra Review

■

Lies My Calculator and Computer Told Me

■

History of Mathematics, with links to the better historical websites

■

Additional Topics (complete with exercise sets):

Trigonometric Integrals, Trigonometric Substitution, Strategy for Integration,

Strategy for Testing Series, Fourier Series, Formulas for the Remainder Term in

Taylor Series, Linear Differential Equations, Second-Order Linear Differential

Equations, Nonhomogeneous Linear Equations, Applications of Second-Order

Differential Equations, Using Series to Solve Differential Equations, Rotation of

Axes, and (for instructors only) Hyperbolic Functions

■

Links, for each chapter, to outside Web resources

■

Archived Problems (drill exercises that appeared in previous editions, together

with their solutions)

■

Challenge Problems (some from the Focus on Problem Solving sections of prior

editions)

Content

Diagnostic Tests

The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Func-

tions, and Trigonometry.

A Preview of Calculus

This is an overview of the subject and includes a list of questions to motivate the study of

calculus.

■

Functions and Models

From the beginning, multiple representations of functions are stressed: verbal, numerical,

visual, and algebraic. A discussion of mathematical models leads to a review of the stan-

dard functions, including exponential and logarithmic functions, from these four points of

view. Parametric curves are introduced in the ﬁrst chapter, partly so that curves can be

drawn easily, with technology, whenever needed throughout the text. This early placement

also enables tangents to parametric curves to be treated in Section 3.4 and graphing such

curves to be covered in Section 4.4.

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PREFACE xvii

■

Limits and Derivatives The material on limits is motivated by a prior discussion of the tangent and velocity prob-

lems. Limits are treated from descriptive, graphical, numerical, and algebraic points of

view. (The precise deﬁnition of a limit is provided in Appendix D for those who wish to

cover it.) It is important not to rush through Sections 2.6–2.8, which deal with derivatives

(especially with functions deﬁned graphically and numerically) before the differentiation

rules are covered in Chapter 3. Here the examples and exercises explore the meanings of

derivatives in various contexts. Section 2.8 foreshadows, in an intuitive way and without

differentiation formulas, the material on shapes of curves that is studied in greater depth in

Chapter 4.

■

Differentiation Rules All the basic functions are differentiated here. When derivatives are computed in applied

situations, students are asked to explain their meanings. Optional topics (hyperbolic func-

tions, an early introduction to Taylor polynomials) are explored in Discovery and Labora-

tory Projects. (A full treatment of hyperbolic functions is available to instructors on the

website.)

■

Applications of Differentiation The basic facts concerning extreme values and shapes of curves are derived using the

Mean Value Theorem as the starting point. Graphing with technology emphasizes the inter-

action between calculus and calculators and the analysis of families of curves. Some sub-

stantial optimization problems are provided, including an explanation of why you need to

raise your head 42° to see the top of a rainbow.

■

Integrals The area problem and the distance problem serve to motivate the deﬁnite integral. I have

decided to make the deﬁnition of an integral easier to understand by using subintervals of

equal width. Emphasis is placed on explaining the meanings of integrals in various con-

texts and on estimating their values from graphs and tables. There is no separate chapter

on techniques of integration, but substitution and parts are covered here and other methods

are treated brieﬂy. Partial fractions are given full treatment in Appendix G. The use of com-

puter algebra systems is discussed in Section 5.8.

■

Applications of Integration General methods, not formulas, are emphasized. The goal is for students to be able to

divide a quantity into small pieces, estimate with Riemann sums, and recognize the limit

as an integral. There are more applications here than can realistically be covered in a given

course. Instructors should select applications suitable for their students and for which they

themselves have enthusiasm. Some instructors like to cover polar coordinates (Appendix H)

here. Others prefer to defer this topic to when it is needed in third semester (with Section 9.7

or just before Section 12.4).

■

Differential Equations Modeling is the theme that uniﬁes this introductory treatment of differential equations.

Direction ﬁelds and Euler’s method are studied before separable equations are solved

explicitly, so that qualitative, numerical, and analytic approaches are given equal consid-

eration. These methods are applied to the exponential, logistic, and other models for pop-

ulation growth. Predator-prey models are used to illustrate systems of differential

equations.

■

Infinite Sequences and Series Tests for the convergence of series are considered brieﬂy, with intuitive rather than formal

justiﬁcations. Numerical estimates of sums of series are based on which test was used to

prove convergence. The emphasis is on Taylor series and polynomials and their applica-

tions to physics. Error estimates include those from graphing devices.

Ancillaries

Single Variable Calculus: Concepts and Contexts, Fourth Edition, is supported by a com-

plete set of ancillaries developed under my direction. Each piece has been designed to

enhance student understanding and to facilitate creative instruction. The table on pages xxi

and xxii lists ancillaries available for instructors and students.

59726_SV_FM_FM_pi-xxiii.qk 11/22/08 3:08 PM Page xvii

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