詹姆斯·斯图尔特的《微积分》第八版

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"《微积分》，第八版，詹姆斯·斯图尔特，2015年出版" 本书《微积分》是詹姆斯·斯图尔特所著的第八版，他是麦克马斯特大学和多伦多大学的教授。该书在全球范围内广受欢迎，包括澳大利亚、巴西、墨西哥、新加坡、英国和美国等多个国家的读者都在使用。书中详细阐述了微积分的基本概念、理论和应用，是高等教育数学课程的经典教材。微积分是数学的一个核心分支，它研究的是连续变化的理论，包括极限、导数、积分和无穷级数等概念。斯图尔特的这部作品深入浅出地介绍了这些概念，旨在帮助学生理解并掌握微积分的基础知识。书中的内容涵盖了单变量微积分和多元微积分，以及一些更高级的主题，如曲线的弧长、曲面的面积、物理问题的应用等。第八版的更新可能包括了对原有材料的修订，以反映最新的教学方法和研究进展，同时也可能增加了新的例子和习题，以增强学生的实践能力。此外，这本书特别强调了实际问题中的应用，将抽象的数学理论与现实世界的问题相结合，帮助学生看到微积分在工程、物理、经济学和其他科学领域的应用。书中的索引部分详细列出了所有关键术语和概念，方便读者查找和复习。尽管由于电子版权的限制，部分第三方内容可能在电子版中被省略，但这些被抑制的内容被认为不会实质性地影响整体的学习体验。出版社保留根据后续的权利限制进一步删除内容的权利。《微积分》第八版是学习微积分的权威参考资料，它提供了丰富的理论和实践练习，是大学数学课程的理想教材，不仅适合初学者，也适合希望深入理解微积分概念的进阶读者。

xiv Preface

velocity. Laboratory Projects involve technology; the one following Section 10.2 shows

how to use Bézier curves to design shapes that represent letters for a laser printer. Writ-

ing Projects ask students to compare present-day methods with those of the founders of

calculus—Fermat’s method for nding tangents, for instance. Suggested references are

supplied. Discovery Projects anticipate results to be discussed later or encourage dis-

covery through pattern recognition (see the one following Section 7.6). Others explore

aspects of geometry: tetrahedra (after Section 12.4), hyperspheres (after Section 15.6),

and intersections of three cylinders (after Section 15.7). Additional projects can be found

in the Instructor’s Guide (see, for instance, Group Exercise 4.1: Position from Samples).

Problem Solving

Students usually have difculties with problems for which there is no single well-dened

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 following Chapter 1. They are applied, both explicitly

and implicitly, throughout the book. After the other chapters I have placed sections called

Problems Plus, 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 difcult 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 signi-

cantly for ideas toward a solution and for recognizing which problem-solving principles

are relevant.

Dual Treatment of Exponential and Logarithmic Functions

There are two possible ways of treating the exponential and logarithmic functions and

each method has its passionate advocates. Because one often nds advocates of both

approaches teaching the same course, I include full treatments of both methods. In Sec-

tions 6.2, 6.3, and 6.4 the exponential function is dened rst, followed by the logarith-

mic function as its inverse. (Students have seen these functions introduced this way since

high school.) In the alternative approach, presented in Sections 6.2*, 6.3*, and 6.4*, the

logarithm is dened as an integral and the exponential function is its inverse. This latter

method is, of course, less intuitive but more elegant. You can use whichever treatment

you prefer.

If the rst approach is taken, then much of Chapter 6 can be covered before Chap-

ters 4 and 5, if desired. To accommodate this choice of presentation there are specially

identied problems involving integrals of exponential and logarithmic functions at the

end of the appropriate sections of Chapters 4 and 5. This order of presentation allows a

faster-paced course to teach the transcendental functions and the denite integral in the

rst semester of the course.

For instructors who would like to go even further in this direction I have prepared an

alternate edition of this book, called Calculus: Early Transcendentals, Eighth Edition,

in which the exponential and logarithmic functions are introduced in the rst chapter.

Their limits and derivatives are found in the second and third chapters at the same time

as polynomials and the other elementary functions.

Tools for Enriching Calculus

TEC is a companion to the text and is intended to enrich and complement its contents.

(It is now accessible in the eBook via CourseMate and Enhanced WebAssign. Selected

Visuals and Modules are available at www.stewartcalculus.com.) Developed by Harvey

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

40621_FM_p01_hr_i-xxx.indd 14 4/19/15 4:05 PM

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Preface xv

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 anima-

tions of gures in text; Modules are more elaborate activities and include exercises.

Instructors can choose to become involved at several different levels, ranging from sim-

ply encouraging students to use the Visuals and Modules for independent exploration,

to assigning specic exercises from those included with each Module, or to creating

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

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 Eighth Edition we have been

working with the calculus community and WebAssign to develop an online homework

system. Up to 70% 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

Visit CengageBrain.com or stewartcalculus.com for these additional materials:

●

Homework Hints

●

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): Fourier Series, Formulas for the

Remainder Term in Taylor Series, Rotation of Axes

●

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

their solutions)

●

Challenge Problems (some from the Problems Plus sections from prior editions)

●

Links, for particular topics, to outside Web resources

●

Selected Visuals and Modules from Tools for Enriching Calculus (TEC)

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

tions, and Trigonometry.

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

of calculus.

Diagnostic Tests

A Preview of Calculus

40621_FM_p01_hr_i-xxx.indd 15 4/19/15 4:05 PM

xvi Preface

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

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

standard functions from these four points of view. The material on limits is motivated

by a prior discussion of the tangent and velocity problems. Limits are treated from

descriptive, graphical, numerical, and algebraic points of view. Section 1.7, on the precise

epsilon-delta deﬁntion of a limit, is an optional section.

The material on derivatives is covered in two sections in order to give students more

time to get used to the idea of a derivative as a function. The examples and exercises

explore the meanings of derivatives in various contexts. Higher derivatives are intro-

duced in Section 2.2.

The basic facts concerning extreme values and shapes of curves are deduced from the

Mean Value Theorem. Graphing with technology emphasizes the interaction between

calculus and calculators and the analysis of families of curves. Some substantial optimi-

zation problems are provided, including an explanation of why you need to raise your

head 42° to see the top of a rainbow.

The area problem and the distance problem serve to motivate the deﬁnite integral, with

sigma notation introduced as needed. (Full coverage of sigma notation is provided in

Appendix E.) Emphasis is placed on explaining the meanings of integrals in various con-

texts and on estimating their values from graphs and tables.

Here I present the applications of integration—area, volume, work, average value—that

can reasonably be done without specialized techniques of integration. General methods

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.

As discussed more fully on page xiv, only one of the two treatments of these functions

need be covered. Exponential growth and decay are covered in this chapter.

All the standard methods are covered but, of course, the real challenge is to be able to

recognize which technique is best used in a given situation. Accordingly, in Section 7.5,

I present a strategy for integration. The use of computer algebra systems is discussed in

Section 7.6.

Here are the applications of integration—arc length and surface area—for which it is

useful to have available all the techniques of integration, as well as applications to biol-

ogy, economics, and physics (hydrostatic force and centers of mass). I have also

included a section on probability. 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.

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

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

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

1 Functions and Limits

2 Derivatives

3 Applications of Dierentiation

4 Integrals

5 Applications of Integration

6 Inverse Functions:

Exponential, Logarithmic, and

Inverse Trigonometric Functions

7 Techniques of Integration

8 Further Applications

of Integration

9 Dierential Equations

40621_FM_p01_hr_i-xxx.indd 16 4/19/15 4:05 PM

Preface xvii

consideration. These methods are applied to the exponential, logistic, and other models

for population growth. The ﬁrst four or ﬁve sections of this chapter serve as a good

introduction to ﬁrst-order differential equations. An optional ﬁnal section uses predator-

prey models to illustrate systems of differential equations.

This chapter introduces parametric and polar curves and applies the methods of calculus

to them. Parametric curves are well suited to laboratory projects; the two presented here

involve families of curves and Bézier curves. A brief treatment of conic sections in polar

coordinates prepares the way for Kepler’s Laws in Chapter 13.

The convergence tests have intuitive justiﬁcations (see page 759) as well as formal

proofs. 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 applications

to physics. Error estimates include those from graphing devices.

The material on three-dimensional analytic geometry and vectors is divided into two

chapters. Chapter 12 deals with vectors, the dot and cross products, lines, planes, and

surfaces.

This chapter covers vector-valued functions, their derivatives and integrals, the length

and curvature of space curves, and velocity and acceleration along space curves, culmi-

nating in Kepler’s laws.

Functions of two or more variables are studied from verbal, numerical, visual, and alge-

braic points of view. In particular, I introduce partial derivatives by looking at a speciﬁc

column in a table of values of the heat index (perceived air temperature) as a function

of the actual temperature and the relative humidity.

Contour maps and the Midpoint Rule are used to estimate the average snowfall and

average temperature in given regions. Double and triple integrals are used to compute

probabilities, surface areas, and (in projects) volumes of hyperspheres and volumes of

intersections of three cylinders. Cylindrical and spherical coordinates are introduced in

the context of evaluating triple integrals.

Vector ﬁelds are introduced through pictures of velocity ﬁelds showing San Francisco

Bay wind patterns. The similarities among the Fundamental Theorem for line integrals,

Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized.

Since ﬁrst-order differential equations are covered in Chapter 9, this ﬁnal chapter deals

with second-order linear differential equations, their application to vibrating springs and

electric circuits, and series solutions.

Calculus, Eighth Edition, is supported by a complete set of ancillaries developed

under my direction. Each piece has been designed to enhance student understanding

and to facilitate creative instruction. The tables on pages xxi–xxii describe each of these

ancillaries.

10 Parametric Equations

and Polar Coordinates

11 Innite Sequences and Series

12 Vectors and the

Geometry of Space

13 Vector Functions

14 Partial Derivatives

15 Multiple Integrals

16 Vector Calculus

17 Second-Order

Dierential Equations

40621_FM_p01_hr_i-xxx.indd 17 4/21/15 10:27 AM

xviii Preface

The preparation of this and previous editions has involved much time spent reading the

reasoned (but sometimes contradictory) advice from a large number of astute reviewers.

I greatly appreciate the time they spent to understand my motivation for the approach

taken. I have learned something from each of them.

Eighth Edition Reviewers

Jay Abramson, Arizona State University

Adam Bowers, University of California San Diego

Neena Chopra, The Pennsylvania State University

Edward Dobson, Mississippi State University

Isaac Goldbring, University of Illinois at Chicago

Lea Jenkins, Clemson University

Rebecca Wahl, Butler University

Maria Andersen, Muskegon Community College

Eric Aurand, Easteld College

Joy Becker, University of Wisconsin–Stout

Przemyslaw Bogacki, Old Dominion University

Amy Elizabeth Bowman, University of Alabama

in Huntsville

Monica Brown, University of Missouri–St. Louis

Roxanne Byrne, University of Colorado at Denver and

Health Sciences Center

Teri Christiansen, University of Missouri–Columbia

Bobby Dale Daniel, Lamar University

Jennifer Daniel, Lamar University

Andras Domokos, California State University, Sacramento

Timothy Flaherty, Carnegie Mellon University

Lee Gibson, University of Louisville

Jane Golden, Hillsborough Community College

Semion Gutman, University of Oklahoma

Diane Hoffoss, University of San Diego

Lorraine Hughes, Mississippi State University

Jay Jahangiri, Kent State University

John Jernigan, Community College of Philadelphia

Brian Karasek, South Mountain Community College

Jason Kozinski, University of Florida

Carole Krueger, The University of Texas at Arlington

Ken Kubota, University of Kentucky

John Mitchell, Clark College

Donald Paul, Tulsa Community College

Chad Pierson, University of Minnesota, Duluth

Lanita Presson, University of Alabama in Huntsville

Karin Reinhold, State University of New York at Albany

Thomas Riedel, University of Louisville

Christopher Schroeder, Morehead State University

Angela Sharp, University of Minnesota, Duluth

Patricia Shaw, Mississippi State University

Carl Spitznagel, John Carroll University

Mohammad Tabanjeh, Virginia State University

Capt. Koichi Takagi, United States Naval Academy

Lorna TenEyck, Chemeketa Community College

Roger Werbylo, Pima Community College

David Williams, Clayton State University

Zhuan Ye, Northern Illinois University

Technology Reviewers

B. D. Aggarwala, University of Calgary

John Alberghini, Manchester Community College

Michael Albert, Carnegie-Mellon University

Daniel Anderson, University of Iowa

Amy Austin, Texas A&M University

Donna J. Bailey, Northeast Missouri State University

Wayne Barber, Chemeketa Community College

Marilyn Belkin, Villanova University

Neil Berger, University of Illinois, Chicago

David Berman, University of New Orleans

Anthony J. Bevelacqua, University of North Dakota

Richard Biggs, University of Western Ontario

Robert Blumenthal, Oglethorpe University

Martina Bode, Northwestern University

Barbara Bohannon, Hofstra University

Jay Bourland, Colorado State University

Philip L. Bowers, Florida State University

Amy Elizabeth Bowman, University of Alabama in Huntsville

Stephen W. Brady, Wichita State University

Michael Breen, Tennessee Technological University

Robert N. Bryan, University of Western Ontario

David Buchthal, University of Akron

Jenna Carpenter, Louisiana Tech University

Jorge Cassio, Miami-Dade Community College

Previous Edition Reviewers

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