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Calculus
Concepts and Contexts | 4e
James Stewart
McMaster University
and
University of Toronto
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Calculus: Concepts and Contexts, Fourth Edition
James Stewart
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Contents
Preface
xiii
To the Student
xxiii
Diagnostic Tests
thomasmayerarchive.com
1
xxiv
A Preview of Calculus
3
Functions and Models
11
1.1
Four Ways to Represent a Function
1.2
Mathematical Models: A Catalog of Essential Functions
12
1.3
New Functions from Old Functions
37
1.4
Graphing Calculators and Computers
46
1.5
Exponential Functions
1.6
Inverse Functions and Logarithms
1.7
Parametric Curves
52
61
71
Laboratory Project ■ Running Circles Around Circles
Review
Limits and Derivatives
83
89
2.1
The Tangent and Velocity Problems
2.2
The Limit of a Function
2.3
Calculating Limits Using the Limit Laws
2.4
Continuity
2.5
Limits Involving Infinity
2.6
Derivatives and Rates of Change
90
95
104
113
124
135
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Writing Project ■ Early Methods for Finding Tangents
2.7
The Derivative as a Function
146
2.8
What Does f ⬘ Say about f ?
158
Review
79
80
Principles of Problem Solving
2
25
145
164
Focus on Problem Solving
169
vii
viii
CONTENTS
3
Differentiation Rules
3.1
173
Derivatives of Polynomials and Exponential Functions
Applied Project ■ Building a Better Roller Coaster
3.2
The Product and Quotient Rules
3.3
Derivatives of Trigonometric Functions
3.4
The Chain Rule
183
183
190
197
Laboratory Project ■ Bézier Curves
208
Applied Project ■ Where Should a Pilot Start Descent?
3.5
Implicit Differentiation
3.6
Inverse Trigonometric Functions and Their Derivatives
3.7
Derivatives of Logarithmic Functions
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Discovery Project
■
227
Rates of Change in the Natural and Social Sciences
3.9
Linear Approximations and Differentials
240
Laboratory Project ■ Taylor Polynomials
4
247
248
251
Applications of Differentiation
4.1
Related Rates
4.2
Maximum and Minimum Values
255
256
262
Applied Project ■ The Calculus of Rainbows
4.3
Derivatives and the Shapes of Curves
4.4
Graphing with Calculus and Calculators
4.5
Indeterminate Forms and l’Hospital’s Rule
270
271
282
290
Writing Project ■ The Origins of l’Hospital’s Rule
4.6
Optimization Problems
299
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Applied Project ■ The Shape of a Can
4.7
Newton’s Method
4.8
Antiderivatives
Review
312
317
323
Focus on Problem Solving
327
216
221
Hyperbolic Functions
Focus on Problem Solving
209
209
3.8
Review
174
311
299
228
CONTENTS
5
Integrals
331
5.1
Areas and Distances
332
5.2
The Definite Integral
343
5.3
Evaluating Definite Integrals
Discovery Project
5.4
■
356
Area Functions
366
The Fundamental Theorem of Calculus
367
Writing Project ■ Newton, Leibniz, and the Invention of Calculus
5.5
The Substitution Rule
5.6
Integration by Parts
5.7
Additional Techniques of Integration
5.8
Integration Using Tables and Computer Algebra Systems
375
383
389
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Discovery Project ■ Patterns in Integrals
5.9
Approximate Integration
5.10
Improper Integrals
Review
401
423
428
Applications of Integration
6.1
More About Areas
6.2
Volumes
431
432
438
Discovery Project ■ Rotating on a Slant
6.3
Volumes by Cylindrical Shells
6.4
Arc Length
455
Average Value of a Function
Applied Project
6.6
448
449
Discovery Project ■ Arc Length Contest
6.5
400
413
Focus on Problem Solving
6
■
460
460
Where To Sit at the Movies
Applications to Physics and Engineering
464
464
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Discovery Project ■ Complementary Coffee Cups
6.7
Applications to Economics and Biology
6.8
Probability
Review
374
480
487
Focus on Problem Solving
491
476
475
394
ix
x
CONTENTS
7
Differential Equations
493
7.1
Modeling with Differential Equations
494
7.2
Direction Fields and Euler’s Method
Separable Equations 508
499
7.3
Applied Project ■ How Fast Does a Tank Drain?
Applied Project
7.4
■
Which Is Faster, Going Up or Coming Down?
Exponential Growth and Decay
Courtesy of Frank O. Gehry
7.6
8
The Logistic Equation
Predator-Prey Systems
Review
547
530
540
Focus on Problem Solving
551
Infinite Sequences and Series
8.1
Sequences
529
553
554
Laboratory Project ■ Logistic Sequences
8.2
8.3
8.4
8.5
8.6
8.7
575
618
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Writing Project ■ How Newton Discovered the Binomial Series
Applications of Taylor Polynomials
Applied Project
Review
■
619
Radiation from the Stars
631
Vectors and the Geometry of Space
9.1
9.2
9.3
9.4
627
628
Focus on Problem Solving
9
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564
Series 565
The Integral and Comparison Tests; Estimating Sums
Other Convergence Tests 585
Power Series
592
Representations of Functions as Power Series 598
Taylor and Maclaurin Series 604
Laboratory Project ■ An Elusive Limit
8.8
Three-Dimensional Coordinate Systems
Vectors
639
The Dot Product 648
The Cross Product 654
633
634
Discovery Project ■ The Geometry of a Tetrahedron
9.5
Equations of Lines and Planes
663
Laboratory Project ■ Putting 3D in Perspective
9.6
518
519
Applied Project ■ Calculus and Baseball
7.5
517
Functions and Surfaces
673
672
662
618
CONTENTS
9.7
Cylindrical and Spherical Coordinates
Laboratory Project ■ Families of Surfaces
Review
10 Vector Functions
10.2
Courtesy of Frank O. Gehry
10.3
10.4
691
693
Vector Functions and Space Curves 694
Derivatives and Integrals of Vector Functions 701
Arc Length and Curvature 707
Motion in Space: Velocity and Acceleration 716
Applied Project ■ Kepler’s Laws
10.5
Parametric Surfaces
Review
733
11 Partial Derivatives
11.2
11.3
11.4
11.5
11.6
11.7
726
727
Focus on Problem Solving
11.1
687
688
Focus on Problem Solving
10.1
682
735
737
Functions of Several Variables 738
Limits and Continuity 749
Partial Derivatives 756
Tangent Planes and Linear Approximations 770
The Chain Rule 780
Directional Derivatives and the Gradient Vector 789
Maximum and Minimum Values 802
Applied Project ■ Designing a Dumpster
811
Courtesy of Frank O. Gehry
Discovery Project ■ Quadratic Approximations and Critical Points
11.8
Lagrange Multipliers
Applied Project
813
Rocket Science
820
Applied Project ■ Hydro-Turbine Optimization
Review
822
Focus on Problem Solving
12 Multiple Integrals
12.1
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■
12.2
12.3
12.4
12.5
12.6
827
829
Double Integrals over Rectangles 830
Iterated Integrals 838
Double Integrals over General Regions 844
Double Integrals in Polar Coordinates 853
Applications of Double Integrals 858
Surface Area 868
821
812
xi
xii
CONTENTS
12.7
Triple Integrals
873
Discovery Project ■ Volumes of Hyperspheres
12.8
Triple Integrals in Cylindrical and Spherical Coordinates
Applied Project
■
Discovery Project
12.9
Roller Derby
■
13 Vector Calculus
13.2
13.3
13.4
13.5
13.6
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13.7
The Intersection of Three Cylinders
13.9
903
905
Vector Fields 906
Line Integrals 913
The Fundamental Theorem for Line Integrals
Green’s Theorem 934
Curl and Divergence 941
Surface Integrals 949
Stokes’ Theorem
960
The Divergence Theorem
Summary
973
Review
974
967
Focus on Problem Solving
977
Appendixes
A
B
C
D
E
F
G
H
I
J
925
966
A1
Intervals, Inequalities, and Absolute Values A2
Coordinate Geometry A7
Trigonometry
A17
Precise Definitions of Limits A26
A Few Proofs A36
Sigma Notation A41
Integration of Rational Functions by Partial Fractions
Polar Coordinates A55
Complex Numbers
A71
Answers to Odd-Numbered Exercises A80
Index A135
890
891
Writing Project ■ Three Men and Two Theorems
13.8
883
889
Change of Variables in Multiple Integrals
Review
899
Focus on Problem Solving
13.1
883
A47
Preface
When the first edition of this book appeared twelve years ago, a heated debate about calculus 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 traditionalists 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 textbooks is that it is more streamlined. For instance, there is no complete chapter on techniques 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 interwoven 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 guidance 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.
■ Coverage of inverse trigonometric functions has been consolidated in a single dedicated section (3.6).
xiii
xiv
PREFACE
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 featured 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, 8.2.35,
9.1.42, and 11.8.20–21.
■ 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, Problem 15 on page 492, Problem 13 on page 632, and Problem 9 on page 736.
■ 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 convex 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, 8.2, 11.2, and 11.3. 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, 8.7.2, 10.2.1–2, 10.3.33–37, 11.1.1–2, 11.1.9–18, 11.3.3–10, 11.6.1–2,
11.7.3–4, 12.1.5–10, 13.1.11–18, 13.2.15–16, and 13.3.1–2).
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 combine 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 companies 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), Exercise 5.1.14 (velocity of the space shuttle Endeavour), Figure 5 in Section 5.3 (San Francisco power consumption), Example 5 in Section 5.9 (data traffic on Internet links), and
Example 3 in Section 9.6 (wave heights).
Functions of two variables are illustrated by a table of values of the wind-chill index as
a function of air temperature and wind speed (Example 1 in Section 11.1). Partial derivatives are introduced in Section 11.3 by examining a column in a table of values of the heat
index (perceived air temperature) as a function of the actual temperature and the relative
humidity. This example is pursued further in connection with linear approximations
(Example 3 in Section 11.4). Directional derivatives are introduced in Section 11.6 by
using a temperature contour map to estimate the rate of change of temperature at Reno
in the direction of Las Vegas. Double integrals are used to estimate the average snowfall
in Colorado on December 20–21, 2006 (Example 4 in Section 12.1). Vector fields are
PREFACE
xv
introduced in Section 13.1 by depictions of actual velocity vector fields showing San Francisco Bay wind patterns.
Projects
One way of involving students and making them active learners is to have them work (perhaps 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. The project after Section 11.8 uses Lagrange multipliers to determine the masses of the three stages of a rocket so as to minimize the total
mass while enabling the rocket to reach a desired velocity. Laboratory Projects involve
technology; the project following Section 3.4 shows how to use Bézier curves to design
shapes that represent letters 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. Suggested 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). Others
explore aspects of geometry: tetrahedra (after Section 9.4), hyperspheres (after Section 12.7), and intersections of three cylinders (after Section 12.8). 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 Fundamental 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 PS emphasizes some of the explicit occurrences.) 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 solution and for recognizing which problem-solving principles are relevant.
Technology
The availability of technology makes it not less important but more important to understand 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 CAS 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 instructors and students need to develop the ability to decide where the hand or the machine is
appropriate.
xvi
PREFACE
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 figures in the text; Modules are more elaborate activities and include exercises. Instructors
can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent exploration, to assigning specific 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-numbered) 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, particularly 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 working 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, Functions, 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.
1 ■ 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-
PREFACE
xvii
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.
2 ■ Limits and Derivatives
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. (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.
3 ■ 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 functions, an early introduction to Taylor polynomials) are explored in Discovery and Laboratory Projects. (A full treatment of hyperbolic functions is available to instructors on the
website.)
4 ■ 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 interaction between calculus and calculators and the analysis of families of curves. Some substantial optimization problems are provided, including an explanation of why you need to
raise your head 42° to see the top of a rainbow.
5 ■ 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 contexts 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 computer algebra systems is discussed in Section 5.8.
6 ■ 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).
7 ■ 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 consideration. These methods are applied to the exponential, logistic, and other models for population growth. Predator-prey models are used to illustrate systems of differential
equations.
8 ■ 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 applications to physics. Error estimates include those from graphing devices.
xviii
PREFACE
9 ■ Vectors and
The Geometry of Space
The dot product and cross product of vectors are given geometric definitions, motivated by
work and torque, before the algebraic expressions are deduced. To facilitate the discussion
of surfaces, functions of two variables and their graphs are introduced here.
10 ■ Vector Functions
The calculus of vector functions is used to prove Kepler’s First Law of planetary motion,
with the proofs of the other laws left as a project. In keeping with the introduction of parametric curves in Chapter 1, parametric surfaces are introduced as soon as possible, namely,
in this chapter. I think an early familiarity with such surfaces is desirable, especially with
the capability of computers to produce their graphs. Then tangent planes and areas of parametric surfaces can be discussed in Sections 11.4 and 12.6.
11 ■ Partial Derivatives
Functions of two or more variables are studied from verbal, numerical, visual, and algebraic 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. Directional derivatives are estimated from
contour maps of temperature, pressure, and snowfall.
12 ■ Multiple Integrals
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, areas of parametric surfaces, volumes of hyperspheres, and the volume of intersection
of three cylinders.
13 ■ Vector Fields
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.
Ancillaries
Calculus: Concepts and Contexts, Fourth 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 table on pages xxi and xxii lists
ancillaries available for instructors and students.
Acknowledgments
I am grateful to the following reviewers for sharing their knowledge and judgment with
me. I have learned something from each of them.
Fourth Edition Reviewers
Jennifer Bailey, Colorado School of Mines
Lewis Blake, Duke University
James Cook, North Carolina State University
Costel Ionita, Dixie State College
Lawrence Levine, Stevens Institute of Technology
Scott Mortensen, Dixie State College
Drew Pasteur, North Carolina State University
Jeffrey Powell, Samford University
Barbara Tozzi, Brookdale Community College
Kathryn Turner, Utah State University
Cathy Zucco-Tevelof, Arcadia University
Previous Edition Reviewers
Irfan Altas, Charles Sturt University
William Ardis, Collin County Community College
Barbara Bath, Colorado School of Mines
Neil Berger, University of Illinois at Chicago
Jean H. Bevis, Georgia State University
Martina Bode, Northwestern University
Jay Bourland, Colorado State University
Paul Wayne Britt, Louisiana State University
Judith Broadwin, Jericho High School (retired)
Charles Bu, Wellesley University
Meghan Anne Burke, Kennesaw State University
Robert Burton, Oregon State University
PREFACE
Roxanne M. Byrne, University of Colorado at Denver
Maria E. Calzada, Loyola University–New Orleans
Larry Cannon, Utah State University
Deborah Troutman Cantrell,
Chattanooga State Technical Community College
Bem Cayco, San Jose State University
John Chadam, University of Pittsburgh
Robert A. Chaffer, Central Michigan University
Dan Clegg, Palomar College
Camille P. Cochrane, Shelton State Community College
James Daly, University of Colorado
Richard Davis, Edmonds Community College
Susan Dean, DeAnza College
Richard DiDio, LaSalle University
Robert Dieffenbach, Miami University–Middletown
Fred Dodd, University of South Alabama
Helmut Doll, Bloomsburg University
William Dunham, Muhlenberg College
David A. Edwards, The University of Georgia
John Ellison, Grove City College
Joseph R. Fiedler, California State University–Bakersfield
Barbara R. Fink, DeAnza College
James P. Fink, Gettysburg College
Joe W. Fisher, University of Cincinnati
Robert Fontenot, Whitman College
Richard L. Ford, California State University Chico
Laurette Foster, Prairie View A & M University
Ronald C. Freiwald, Washington University in St. Louis
Frederick Gass, Miami University
Gregory Goodhart, Columbus State Community College
John Gosselin, University of Georgia
Daniel Grayson,
University of Illinois at Urbana–Champaign
Raymond Greenwell, Hofstra University
Gerrald Gustave Greivel, Colorado School of Mines
John R. Griggs, North Carolina State University
Barbara Bell Grover, Salt Lake Community College
Murli Gupta, The George Washington University
John William Hagood, Northern Arizona University
Kathy Hann, California State University at Hayward
Richard Hitt, University of South Alabama
Judy Holdener, United States Air Force Academy
Randall R. Holmes, Auburn University
Barry D. Hughes, University of Melbourne
Mike Hurley, Case Western Reserve University
Gary Steven Itzkowitz, Rowan University
xix
Helmer Junghans, Montgomery College
Victor Kaftal, University of Cincinnati
Steve Kahn, Anne Arundel Community College
Mohammad A. Kazemi,
University of North Carolina, Charlotte
Harvey Keynes, University of Minnesota
Kandace Alyson Kling, Portland Community College
Ronald Knill, Tulane University
Stephen Kokoska, Bloomsburg University
Kevin Kreider, University of Akron
Doug Kuhlmann, Phillips Academy
David E. Kullman, Miami University
Carrie L. Kyser, Clackamas Community College
Prem K. Kythe, University of New Orleans
James Lang, Valencia Community College–East Campus
Carl Leinbach, Gettysburg College
William L. Lepowsky, Laney College
Kathryn Lesh, University of Toledo
Estela Llinas, University of Pittsburgh at Greensburg
Beth Turner Long,
Pellissippi State Technical Community College
Miroslav Lovrić, McMaster University
Lou Ann Mahaney, Tarrant County Junior College–Northeast
John R. Martin, Tarrant County Junior College
Andre Mathurin, Bellarmine College Prep
R. J. McKellar, University of New Brunswick
Jim McKinney,
California State Polytechnic University–Pomona
Richard Eugene Mercer, Wright State University
David Minda, University of Cincinnati
Rennie Mirollo, Boston College
Laura J. Moore-Mueller, Green River Community College
Scott L. Mortensen, Dixie State College
Brian Mortimer, Carleton University
Bill Moss, Clemson University
Tejinder Singh Neelon,
California State University San Marcos
Phil Novinger, Florida State University
Richard Nowakowski, Dalhousie University
Stephen Ott, Lexington Community College
Grace Orzech, Queen’s University
Jeanette R. Palmiter, Portland State University
Bill Paschke, University of Kansas
David Patocka, Tulsa Community College–Southeast Campus
Paul Patten, North Georgia College
Leslie Peek, Mercer University
xx
PREFACE
Mike Pepe, Seattle Central Community College
Dan Pritikin, Miami University
Fred Prydz, Shoreline Community College
Denise Taunton Reid, Valdosta State University
James Reynolds, Clarion University
Hernan Rivera, Texas Lutheran University
Richard Rochberg, Washington University
Gil Rodriguez, Los Medanos College
David C. Royster, University of North Carolina–Charlotte
Daniel Russow, Arizona Western College
Dusty Edward Sabo, Southern Oregon University
Daniel S. Sage, Louisiana State University
N. Paul Schembari, East Stroudsburg University
Dr. John Schmeelk, Virginia Commonwealth University
Bettina Schmidt, Auburn University at Montgomery
Bernd S.W. Schroeder, Louisiana Tech University
Jeffrey Scott Scroggs, North Carolina State University
James F. Selgrade, North Carolina State University
Brad Shelton, University of Oregon
Don Small, United States Military Academy–West Point
Linda E. Sundbye, The Metropolitan State College of Denver
Richard B. Thompson,The University of Arizona
William K. Tomhave, Concordia College
Lorenzo Traldi, Lafayette College
Alan Tucker, State University of New York at Stony Brook
Tom Tucker, Colgate University
George Van Zwalenberg, Calvin College
Dennis Watson, Clark College
Paul R. Wenston, The University of Georgia
Ruth Williams, University of California–San Diego
Clifton Wingard, Delta State University
Jianzhong Wang, Sam Houston State University
JingLing Wang, Lansing Community College
Michael B. Ward, Western Oregon University
Stanley Wayment, Southwest Texas State University
Barak Weiss, Ben Gurion University–Be’er Sheva, Israel
Teri E. Woodington, Colorado School of Mines
James Wright, Keuka College
In addition, I would like to thank Ari Brodsky, David Cusick, Alfonso Gracia-Saz,
Emile LeBlanc, Tanya Leise, Joe May, Romaric Pujol, Norton Starr, Lou Talman, and Gail
Wolkowicz for their advice and suggestions; Al Shenk and Dennis Zill for permission to
use exercises from their calculus texts; COMAP for permission to use project material;
Alfonso Gracia-Saz, B. Hovinen, Y. Kim, Anthony Lam, Romaric Pujol, Felix Recio, and
Paul Sally for ideas for exercises; Dan Drucker for the roller derby project; and Tom
Farmer, Fred Gass, John Ramsay, Larry Riddle, V. K. Srinivasan, and Philip Straffin for
ideas for projects. I’m grateful to Dan Clegg, Jeff Cole, and Tim Flaherty for preparing the
answer manuscript and suggesting ways to improve the exercises.
As well, I thank those who have contributed to past editions: Ed Barbeau, George
Bergman, David Bleecker, Fred Brauer, Andy Bulman-Fleming, Tom DiCiccio, Martin
Erickson, Garret Etgen, Chris Fisher, Stuart Goldenberg, Arnold Good, John Hagood,
Gene Hecht, Victor Kaftal, Harvey Keynes, E. L. Koh, Zdislav Kovarik, Kevin Kreider,
Jamie Lawson, David Leep, Gerald Leibowitz, Larry Peterson, Lothar Redlin, Peter Rosenthal, Carl Riehm, Ira Rosenholtz, Doug Shaw, Dan Silver, Lowell Smylie, Larry Wallen,
Saleem Watson, and Alan Weinstein.
I also thank Stephanie Kuhns, Rebekah Million, Brian Betsill, and Kathi Townes of
TECH-arts for their production services; Marv Riedesel and Mary Johnson for their careful proofing of the pages; Thomas Mayer for the cover image; and the following Brooks/
Cole staff: Cheryll Linthicum, editorial production project manager; Jennifer Jones,
Angela Kim, and Mary Anne Payumo, marketing team; Peter Galuardi, media editor; Jay
Campbell, senior developmental editor; Jeannine Lawless, associate editor; Elizabeth
Neustaetter, editorial assistant; Bob Kauser, permissions editor; Becky Cross, print/media
buyer; Vernon Boes, art director; Rob Hugel, creative director; and Irene Morris, cover
designer. They have all done an outstanding job.
I have been very fortunate to have worked with some of the best mathematics editors in the
business over the past three decades: Ron Munro, Harry Campbell, Craig Barth, Jeremy Hayhurst, Gary Ostedt, Bob Pirtle, and now Richard Stratton. Special thanks go to all of them.
JAMES STEWART
Ancillaries for Instructors
PowerLecture CD-ROM with JoinIn and ExamView
ISBN 0-495-56049-9
Contains all art from the text in both jpeg and PowerPoint
formats, key equations and tables from the text, complete
pre-built PowerPoint lectures, and an electronic version of
the Instructor’s Guide. Also contains JoinIn on TurningPoint
personal response system questions and ExamView algorithmic
test generation. See below for complete descriptions.
TEC Tools for Enriching™ Calculus
by James Stewart, Harvey Keynes, Dan Clegg,
and developer Hu Hohn
TEC provides a laboratory environment in which students
can explore selected topics. TEC also includes homework
hints for representative exercises. Available online at
www.stewartcalculus.com.
Instructor’s Guide
by Douglas Shaw and James Stewart
ISBN 0-495-56047-2
Each section of the main text is discussed from several viewpoints and contains suggested time to allot, points to stress,
text discussion topics, core materials for lecture, workshop/
discussion suggestions, group work exercises in a form suitable
for handout, and suggested homework problems. An electronic
version is available on the PowerLecture CD-ROM.
Instructor’s Guide for AP ® Calculus
by Douglas Shaw
ExamView
Create, deliver, and customize tests and study guides (both print
and online) in minutes with this easy-to-use assessment and
tutorial software on CD. Includes full algorithmic generation of
problems and complete questions from the Printed Test Bank.
JoinIn on TurningPoint
Enhance how your students interact with you, your lecture, and
each other. Brooks/Cole, Cengage Learning is now pleased to
offer you book-specific content for Response Systems tailored to
Stewart’s Calculus, allowing you to transform your classroom
and assess your students’ progress with instant in-class quizzes
and polls. Contact your local Cengage representative to learn
more about JoinIn on TurningPoint and our exclusive infrared
and radio-frequency hardware solutions.
Text-Specific DVDs
ISBN 0-495-56050-2
Text-specific DVD set, available at no charge to adopters. Each
disk features a 10- to 20-minute problem-solving lesson for
each section of the chapter. Covers both single- and multivariable calculus.
Solution Builder
www.cengage.com/solutionbuilder
The online Solution Builder lets instructors easily build and
save personal solution sets either for printing or posting on
password-protected class websites. Contact your local sales
representative for more information on obtaining an account
for this instructor-only resource.
Ancillaries for Instructors and Students
ISBN 0-495-56059-6
Taking the perspective of optimizing preparation for the AP
exam, each section of the main text is discussed from several
viewpoints and contains suggested time to allot, points to
stress, daily quizzes, core materials for lecture, workshop/
discussion suggestions, group work exercises in a form suitable
for handout, tips for the AP exam, and suggested homework
problems.
Complete Solutions Manual
Single Variable
by Jeffery A. Cole and Timothy J. Flaherty
ISBN 0-495-56060-X
ISBN 0-495-56121-5
Whether you prefer a basic downloadable eBook or a premium
multimedia eBook with search, highlighting, and note taking
capabilities as well as links to videos and simulations, this new
edition offers a range of eBook options to fit how you want to
read and interact with the content.
Stewart Specialty Website
www.stewartcalculus.com
Contents: Algebra Review Additional Topics Drill
Web Links History of
exercises Challenge Problems
Mathematics Tools for Enriching Calculus (TEC)
N
N
Multivariable
N
N
N
N
by Dan Clegg
ISBN 0-495-56056-1
Includes worked-out solutions to all exercises in the text.
Printed Test Bank
by William Tomhave and Xuequi Zeng
ISBN 0-495-56123-1
Contains multiple-choice and short-answer test items that key
directly to the text.
|||| Electronic items
eBook Option
|||| Printed items
Enhanced WebAssign
Instant feedback, grading precision, and ease of use are just
three reasons why WebAssign is the most widely used homework system in higher education. WebAssign’s homework delivery system lets instructors deliver, collect, grade and record
assignments via the web. And now, this proven system has been
enhanced to include end-of-section problems from Stewart’s
Calculus: Concepts and Contexts—incorporating exercises,
(Table continues on page xxii.)
xxi
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into the boilers, and accumulated there. It saponified and formed a
foam which filled the whole boiler and caused the water to be worked
over with the steam as fast as it could be fed in. I have always
wondered why the engine, being vertical, should not have exhibited
any sign of the water working through it at the upper end of the
cylinder. The explanation after all appears simple. The water on
entering the steam chest mostly fell to the bottom and little passed
through the upper ports. The trouble from oil was not felt at all in the
Lancashire boiler. This, I suppose, was due to three causes. The
latter held a far greater body of water, had a much larger extent of
evaporating surface, and far greater steam capacity. I was always
sorry that I did not give the Harrison boiler the better chance it would
have had with a jet condenser.
In this pair of diagrams, which are copied from the catalogue of
Ormerod, Grierson & Co., the low steam pressure, 29 pounds above
the atmosphere, will be observed. This was about the pressure
commonly carried. The pressure in the exhibition boilers, 75 pounds,
was exhibited by Mr. John Hick, of Bolton, as a marked advance on
the existing practice.
In preparing for the governor manufacture I had my first revelation
of the utter emptiness of the Whitworth Works. Iron gear patterns
were required, duplicates of those which had been cut for me at
home by Mr. Pratt. The blanks for these gears were turned as soon
as possible after I reached Manchester, and sent to the Whitworth
Works to be cut. It seemed as though we should never get them.
Finally, after repeated urging, the patterns came. I was sent for to
come into the shop and see them. They were in the hands of the
best fitter we had, who, by the way, was a Swedenborgian preacher
and preached every Sunday. The foreman told me he had given
them to this man to see if it was possible to do anything with them,
and he thought I ought to see them before he set about it. I could
hardly believe my eyes. There was no truth about them. The spaces
and the teeth differed so much that the same tooth would be too
small for some spaces and could not be wedged into others; some
would be too thick or too thin at one end. They were all alike bad,
and presented all kinds of badness. It was finally concluded to make
the best of them, and this careful man worked on them more than
two days to make them passable.
The first governor order that was booked was the only case that
ever beat me. I went to see the engine. It was a condensing beamengine of good size, made by Ormerod, Grierson & Co. to maintain
the vacuum in a tube connecting two telegraph offices in
Manchester, and had been built to the plans and specifications of the
telegraph company’s engineer. The engine had literally nothing to
do. A little steam air-pump that two men could have lifted and set on
a bench would have been just suitable for the work. They could not
carry low enough pressure nor run slowly enough. On inspection I
reported that we should have nothing to do with it.
The custom of making whatever customers order and taking no
responsibility was first illustrated to me in this curious way. I saw a
queer-looking boiler being finished in the boiler-shop. In reply to my
question the foreman told me they were making it for a cottonspinner, according to a plan of his own. It consisted of two boilers,
one within the other. The owner’s purpose was to carry the ordinary
steam pressure in the outer boiler, and a pressure twice as great in
the inner one, when the inner boiler would have to suffer the stress
of only one half the pressure it was carrying.
I asked the superintendent afterwards why they did not tell that
man that he could not maintain steam at two different temperatures
on the opposite sides of the same sheets. He replied: “Because we
do not find it profitable to quarrel with our customers. That is his
idea. If we had told him there was nothing in it, he would not have
believed us, but would have got his boiler made somewhere else.”
Perhaps the most curious experience I ever had was that of
getting the governor into cotton-mills. There was a vast field all
around us, and we looked for plenty of orders. This was the
reception I met with every time. After listening to the winning story I
had to tell, the cotton lord would wind up with this question: “Well, sir,
have you got a governor in a large cotton-mill?” After my answer in
the negative I was bowed out. I early got an order from Titus Salt &
Son, of Saltaire, for two large governors but these did not weigh at
all with a cotton-spinner; they made alpaca goods.
The way the governor was finally got into cotton-mills, where
afterwards its use became general, was the most curious part. A mill
in the city of Manchester was troubled by having its governor fly in
pieces once in a while. After one of these experiences the owners
thought that they might cure the difficulty by getting one of my
governors. That flew in pieces in a week. I went to see the engine.
The cause of all the trouble appeared at a glance. The fly-wheel was
on the second-motion shaft which ran at twice the speed of the main
shaft, and the gearing between them was roaring away enough to
deafen one. The governor was driven by gearing. The vibrations
transmitted to the governor soon tired the arms out. I saw the son of
the principal owner, and explained the cause of the failure of every
governor they had tried, and told him the only remedy, which would
be a complete one, would be to drive the governor by a belt. That, he
replied, was not to be thought of for an instant. I told him he knew
himself that a governor could not endure if driven in any other way,
and that I had hundreds of governors driven by belts, which were
entirely reliable in all cases. “But,” said he, “supposing the belt runs
off the pulley.” “The consequence,” I replied, “cannot be worse than
when the governor flies in pieces.” After wasting considerable time in
talk, he said, “Well, leave it till my father comes home; he is absent
for a few days.” “No,” said I, “if I can’t convince a young man, I shall
not try to convince an old man.” Finally, with every possible
stipulation to make it impossible for the belt to come off, he yielded
his assent, and I had the governor on in short order, lacing the belt
myself, to make sure that it was butt-jointed and laced in the
American fashion.
More than three years afterwards, two days before I was to sail for
home, I met this man on High Street, in Manchester. It was during
the Whitsuntide holidays, and the street was almost deserted. He
came up to me, holding out both hands and grasping mine most
cordially. “Do you know,” said he, “that we have increased our
product 10 per cent., and don’t have half as many broken threads as
we had before, and it’s all that belt.”
Condenser and Air-pump designed by Mr. Porter. (Cross-section)
The tendency towards the horizontal type of engine, in place of the
beam-engine, began to be quite marked in England about that time.
This was favorable to the use of the Allen engine. The only thing that
seemed wanting to its success was a directly connected jet
condenser. No one believed that an air-pump could be made to run
successfully at the speed of 150 double strokes per minute. Yet this
had to be done, or I could not look for any considerable adoption of
the high-speed engine. This subject occupied my mind continually.
When I returned from Oporto, I had thought out the plan of this
condenser, and at once set about the drawings for it. No alteration
was ever made from the first design of the condenser, which I
intended to show with the engine at the coming Paris Exposition in
1867, and which I finally did succeed in showing there, but under
very different and unexpected relations.
The philosophy of this condenser is sufficiently shown in the
accompanying vertical cross-section. A hollow ram, only equal in
weight to the water which it displaced, ran through a stuffing-box at
the front end of the chamber, and was connected with an extension
of the piston-rod of the engine. So the center line of the engine
extended through this single-acting ram, which had the full motion of
the piston. It ran through the middle of a body of water, the surface of
which fell as the ram was withdrawn, and rose as it returned. A quiet
movement of the water was assured by three means: First, the
motion of the ram was controlled by the crank of the engine, and so
began and ceased insensibly. Second, the motion of the ram, of two
feet, produced a rise or fall of the surface of the water of only about
one inch. Third, the end of the ram was pointed, a construction which
does not appear in this sectional view, permitting it to enter and
leave the water at every point gradually. Both the condenser and the
hot-well were located above the chamber in which the ram worked.
The problem was to obtain complete displacement by means of
solid water without any admixture of free air, the expansion of which
as the plunger was withdrawn would reduce the efficiency of the airpump. To effect this object the air must be prevented from mingling
with the water, and must be delivered into the hot-well first. This was
accomplished by two means: First, placing the condenser as well as
the hot-well above the air-pump chamber, as already stated, and
secondly, inclining the bottom of the condenser, so that the water
would pass through the inlet valves at the side farthest from, and the
air at the side nearest to, the hot-well. Thus the air remained above
the water, and as the latter rose it sent the air before it quite to the
delivery valves. Pains were taken to avoid any place where air could
be trapped, so it was certain that on every stroke the air would be
sent through the delivery valves first, mingled air and water, if there
were any, next, and the solid water last, insuring perfect
displacement.
I have a friend who has often asked me, with a manner showing
his conviction that the question could not be answered, “How can
you know that anything will work until you have tried it?” In this case I
did know that this condenser would work at rapid speed before I tried
it. The event proved it, and any engineer could have seen that it
must have worked. The only question in my mind was as to the
necessity of the springs behind the delivery valves. Experiment was
needed to settle that question, which it did in short order. At the
speed at which the engine ran, the light springs improved the
vacuum a full pound, showing that without them these valves did not
close promptly.
The following important detail must not be overlooked. The rubber
disk valves were backed by cast-iron plates, which effectually
preserved them from being cut or even marked by the brass
gratings. These plates were made with tubes standing in the middle
of them, as shown. These tubes afforded long guides on the stems,
and a projection of them on the under side held the valves in place
without any wear. They also determined the rise of the valves. The
chambers, being long and narrow, accommodated three inlet and
three outlet valves. The jet of water struck the opposite wall with
sufficient force to fill the chamber with spray.
When the plans for this condenser were completed, and the Evan
Leigh engine had been vindicated, I felt that the success of the highspeed system was assured, and looked forward to a rapidly growing
demand for the engines. We got out an illustrated catalogue of sizes,
in which I would have put the condenser, but the firm decided that it
would be better to wait for that until it should be on the same footing
with the engine, as an accomplished fact.
Suddenly, like thunder from a clear sky, I received notice that
Ormerod, Grierson & Co. were in difficulties, had stopped payment,
placed their books in the hands of a firm of accountants, and called a
meeting of their creditors, and the works were closed. Some of their
enormous contracts had proved losing ones. I had made such
provision in my contract with them that on their failure my license to
them became void. Otherwise it would have been classed among
their assets.
CHAPTER XII
Introduction to the Whitworth Works. Sketch of Mr. Whitworth. Experience in the
Whitworth Works. Our Agreement which was never Executed. First Engine in
England Transmitting Power by a Belt.
was still debating with myself what course to take,
when I received a note from Mr. W. J. Hoyle, secretary
of the Whitworth Company, inquiring if I were free from
any entanglement with the affairs of Ormerod,
Grierson & Co., to which I was able to make a
satisfactory reply. Mr. Hoyle was then a stranger to
me. It appeared that he was an accomplished steam engineer, and
had been employed as an expert to test one of my engines in
operation, an engine which we had made for a mill-owner in
Bradford. He had been very favorably impressed by the engine, so
much so as to form this scheme. He had been with the Whitworth
Company only a short time, and was struck with the small amount of
work they were doing in their tool department; and after his
observation of the engine at Bradford, learning of the stoppage of
Ormerod, Grierson & Co., it occurred to him that it would be a good
thing for his company to undertake the manufacture of these
engines. After receiving my answer to his preliminary inquiry, having
Mr. Whitworth, as he afterwards told me, where he could not get
away, on a trip from London to Manchester, he laid the plan before
him and talked him into it. I directly after received an invitation to
meet Mr. Whitworth at his office, and here commenced what I verily
believed was one of the most remarkable experiences that any man
ever had.
William J. Hoyle
In the course of our pretty long interview, which terminated with
the conclusion of a verbal agreement, Mr. Whitworth talked with me
quite freely, and told me several things that surprised me. One was
the frank statement that he divided all other toolmakers in the world
into two classes, one class who copied him without giving him any
credit, and the other class who had the presumption to imagine that
they could improve on him. His feelings towards both these classes
evidently did not tend to make him happy. Another thing, which I
heard without any sign of my amazement, was that he had long
entertained the purpose of giving to the world the perfect steamengine. “That is,” he explained, “an engine embodying all those
essential principles to which steam-engine builders must sooner or
later come.” This, he stated, had been necessarily postponed while
he was engaged in developing his system of artillery, but he was
nearing the completion of that work and should then be able to
devote himself to it.
I cannot perhaps do better than stop here and give my
impressions of Mr. Whitworth. He was in all respects a phenomenal
man. As an engineer, or rather a toolmaker, he addressed himself to
all fundamental constructive requirements and problems, and
comprehended everything in his range and grasp of thought,
continually seeking new fields to conquer. Long after the period here
referred to he closed his long and wonderful career by giving to the
world the hollow engine shaft and the system of hydraulic forging. At
that time he was confidently anticipating the adoption by all nations
of his system of artillery. He had made an immense advance, from
spherical shot, incapable of accurate aim and having a high
trajectory, to elongated shot, swiftly rotating in its flight and having a
comparatively flat trajectory, and which could hit the mark and
penetrate with destructive effect at distances of several miles. These
fundamental features of modern artillery thus originated with Mr.
Whitworth. All his other features have been superseded, but his
elongated pointed rotating projectile will remain until nations shall
learn war no more; a time which in the gradual development of
humanity cannot be far away. Before I left England, however, he had
abandoned his artillery plans in most bitter disappointment. He had
met the English official mind. By the authorities of the war and navy
departments it had been unanimously decided that what England
wanted was, not accuracy of aim and penetration at long range, but
smashing effects at close quarters. The record of that is to be found
in the proceedings of the House of Commons in 1868, only thirtynine years ago. Think of that!
Mr. Whitworth was not only the most original engineering genius
that ever lived. He was also a monumental egotist. His fundamental
idea was always prominent, that he had taught the world not only all
that it knew mechanically, but all it ever could know. His fury against
tool-builders who improved on his plans was most ludicrous. He
drew no distinction between principles and details. He must not be
departed from even in a single line. No one in his works dared to
think. This disposition had a striking illustration only a short time—
less than a year—before I went there. He had no children. His
nearest relatives were two nephews, W. W. and J. E. Hulse. The
latter was a tool-manufacturer in Salford. W. W. Hulse was Mr.
Whitworth’s superintendent, and had been associated with him for
twenty-four years, for a long time as his partner, the firm being
Joseph Whitworth & Company. Lately the business had been taken
over by a corporation formed under the style of the Whitworth
Company, and Mr. Hulse became the general superintendent.
Mr. Whitworth was taken sick, and for a while was not expected to
live, and no one thought, even if he did get better, that he would ever
be able to visit his works again. Mr. Hulse had been chafing under
his restraint, and during Mr. Whitworth’s absence proceeded to make
a few obvious improvements in their tools, such, for example, as
supporting the table of their shaper, so that it would not yield under
the cut. To the surprise of every one, Mr. Whitworth got well, and
after more than six months’ absence, he appeared again at the
works. Walking through, he noted the changes that had been made,
sent for Mr. Hulse, discharged him on the spot, and ordered
everything restored to its original form.
To return now to my own experience. Since Mr. Whitworth had
been absorbed in his artillery development he had given only a
cursory oversight to the tool manufacture. Mr. Hulse had been
succeeded as superintendent by a man named Widdowson, whose
only qualification for his position was entire subserviency to Mr.
Whitworth.
Sir Joseph Whitworth
My drawings and patterns were purchased by the Whitworth
Company, and I was installed with one draftsman in a separate
office, and prepared to put the work in hand at once for a 12×24-inch
engine for the Paris Exposition, where Ormerod, Grierson & Co. had
secured the space, and the drawings for which I had completed. If I
remember rightly, the patterns were finished also. While I was getting
things in order, Mr. Widdowson came into my office, and in a very
important manner said to me: “You must understand, sir, that we
work here to the decimal system and all drawings must be
conformed to it.” I received this order meekly, and we went to work to
make our drawings all over, for the single purpose of changing their
dimensions from binary to decimal divisions of the inch. There was of
course quite a body of detail drawings, and to make these over, with
the pains required to make these changes to an unaccustomed
system, and make and mount the tracings, took us nearly three
weeks. When finished I took the roll of tracings to Mr. Widdowson’s
office. He was not in, and I left them for him. An hour or so later he
came puffing and blowing into my office with the drawings. He was a
heavy man, and climbing upstairs exhausted him. When he got his
breath, he broke out: “We can’t do anything with these. Haven’t got a
decimal gauge in the shop.” “You gave me express orders to make
my drawings to the decimal system.” “Damn it, I meant in halves and
quarters and all that, and write them decimals.” So all that work and
time were thrown away, and we had to make a new set of tracings
from the drawings I had brought, in order to figure the dimensions in
decimals. He told me afterwards that when Mr. Whitworth
commenced the manufacture of cylindrical gauges he made them to
the decimal divisions of the inch, imagining that was a better mode of
division than that by continual bisection, and supposing that he had
influence enough to effect the change. But nobody would buy his
gauges. He had to call them in and make what people wanted. “And
now,” said Mr. Widdowson, “there is not a decimal gauge in the
world.” He knew, too, for up to that time they made them all. So Mr.
Whitworth could make a mistake, and I found that this was not the
worst one that he had made.
While time was being wasted in this manner, the subject of
manufacturing the governors came up. Mr. Whitworth concluded that
he would first try one on his own shop engine, so one was bought
from Ormerod, Grierson & Co. I had a message from Mr. Widdowson
to come to the shop and see my governor. It was acting in a manner
that I had seen before, the counterpoise rising and dropping to its
seat twice every time the belt lap came around. “Total failure, you
see,” said Mr. Widdowson, “and I got a new belt for it, too.” I saw a
chance to make an interesting observation, and asked him if he
would get an old belt and try that. This he did, lapping the ends as
before about 18 inches, according to the universal English custom,
which I had long before found it necessary carefully to avoid. As I
knew would be the case, the action was not improved at all. I then
cut off the lap, butted the ends of the belt, and laced them in the
American style, and lo! the trouble vanished. The governor stood
motionless, only floating up and down slightly with the more
important changes of load. Mr. Whitworth was greatly pleased, and
at once set about their manufacture, in a full line of sizes.
He made the change, to which I have referred already, from the
urn shape to the semi-spherical form of the counterpoise. In this
connection he laid the law down to me in this dogmatic fashion: “Let
no man show me a mechanical form for which he cannot give me a
mechanical reason.” But Jove sometimes nods. They were to exhibit
in Paris a large slotting-machine. The form of the upright did not suit
Mr. Whitworth exactly. He had the pattern set up in the erectingshop, and a board tacked on the side, cut to an outline that he
directed. He came to look at it every day for a week, and ordered
some change or other. Finally it was gotten to his mind, the pattern
was altered accordingly, and a new casting made. This was set up in
the shop, and I happened to be present when he came to see it.
“Looks like a horse that has been taught to hold his head up,” said
he. “Mechanical reason,” thought I, fresh from my lesson. When
finished the slotting-machine was tried in the shop, and found to
yield in the back. The tool sprang away from its work and rounded
the corner. Mr. Whitworth had whittled the pattern away and ruined it.
Instead of being sent to Paris, it was broken up.
My experiment with the governor proved the defect in the English
system of lacing belts. Every machine in the land, of whatever kind,
tool or loom or spinning or drawing frame, or whatever it was, driven
by a belt, halted in its motion every time the lap in the belt passed
over a pulley, sufficiently to drop my governor, when the same
motion was given to it, and no one had ever observed this
irregularity.
I thought they would never be ready to set about work on the
engine. First, Mr. Widdowson ordered that every casting and forging,
large and small, must be in the shop before one of them was put in
hand. After this was done I found a number of men at work making
sheet-iron templets of everything. I saw one man filing the threads in
the edges of a templet for a ³⁄₈-inch bolt. When these were all
finished and stamped, an operation that took quite a week, a great
fuss was made about commencing work on everything
simultaneously.
I went into the shop to see what was going on. The first thing to
attract my attention was the steam-chest, then made separate from
the cylinder. A workman—their best fitter, as I afterwards learned—
was engaged in planing out the cavities in which the exhaust valves
worked. I saw no center line, and asked him where it was. He had
never heard of such a thing. “What do you measure from?” “From
the side of the casting.” I called his attention to the center line on the
drawing, from which all the measurements were taken, and told him
all about it. He seemed very intelligent, and under my direction set
the chest up on a plane table and made a center line around it and
another across it, and set out everything from these lines, and I left
him going on finely. An hour later I looked in again. He was about his
job in the old way. To my question he explained that his foreman had
come around and told him I had no business in the shop, that he
gave him his directions, and he must finish his job just as he began
it.
I made no reply but went to Mr. Hoyle’s office, and asked him if he
knew what they were doing in the shop. He smiled and said, “I
suppose they are finally making an engine for you.” “No, they are
not.” “What are they doing?” “Making scrap iron.” “What do you
mean?” I told him the situation. He took his hat and went out, saying,
“I must see this myself.”
A couple of hours later he sent for me, and told me this. “I have
been all around the works and seen all that is doing. It is all of the
same piece. I have had a long interview with Mr. Widdowson, and
am sorry to tell you that we can’t make your engine; we don’t know
how. It seems to be entirely out of our line. The intelligence does not
exist in these works to make a steam-engine. Nobody knows how to
set about anything. I have stopped the work, and want to know what
you think had better be done about it?” I asked him to let me think
the matter over till the next morning. I then went to him and
suggested to him to let me find a skilled locomotive-erecter who was
also a trained draftsman, and to organize a separate department for
the engine and governor manufacture, and put this man at the head
of it, to direct it without interference. This was gladly agreed to. I
found a young man, Mr. John Watts, who proved to be the very man
for the place. In a week we were running under Mr. Watts’ direction,
and the engine was saved. But what a time the poor man had!
Everything seemed to be done wrong. It is hardly to be believed. He
could not get a rod turned round, or a hole bored round.
In their toolmaking they relied entirely on grinding with “Turkey
dust.” I once saw a gang of a dozen laborers working a long
grinding-bar, in the bore, 10 inches diameter by 8 feet long, in the
tailstock of an enormous lathe. I peered through this hole when the
bar was withdrawn. It looked like a ploughed field. Scattered over it
here and there were projections which had been ground off by these
laborers. On the other hand, the planing done in these works was
magnificent. I never saw anything to equal it. But circular work beat
them entirely. I found that the lathe hands never thought of such a
thing as getting any truth by the sliding cut. After that they went for
the surface with coarse files, and relied for such approximate truth as
they did get upon grinding with the everlasting Turkey dust.
Mr. Whitworth invented the duplex lathe tool, but I observed that
they never used it. I asked Mr. Widdowson why this was. “Because,”
said he, “the duplex tool will not turn round.” After a while I found out
why. When our engine was finished, Mr. Widdowson set it upon two
lathe beds and ran it. Lucky that he did. The bottom of the engine
bed was planed, and it could be leveled nicely on the flat surfaces of
their lathe beds. The fly-wheel ran nearly a quarter of an inch out of
truth. He set up some tool-boxes on one of the lathe beds, and
turned the rim off in place, both sides and face being out. That, of
course, made it run perfectly true. I asked the lathe hand how he
could turn out such a job. He replied, “Come and see my lathe.” I
found the spindle quite an eighth of an inch loose in the main
bearing, the wear of twenty or thirty years. He told me all of the
lathes in the works were in a similar condition. That explained many
things. The mystery of those gear patterns was solved. Every spindle
in the gear-cutting machine was wabbling loose in its holes. I can’t
call them bearings. Now it appeared why they could not use the
duplex tools. With a tool cutting on one side, they relied on the
pressure of the cut to keep the lathe spindle in contact with the
opposite side of its main bearing, and a poor reliance that was, but
with a tool cutting on each side, fancy the situation. Then boring a
true hole was obviously impossible. The workmen became
indifferent; they had no reamers, relied entirely on grinding. I asked,
Why do you not renew these worn-out bushings? but could never get
an answer to the question. Some power evidently forbade it, and the
fact is that no man about the place dared to think of such a thing as
intimating to Mr. Whitworth that one of his lathe bearings required
any fixing up, or that it was or could be anything short of perfect. He
(Mr. Whitworth) had designed it as a perfect thing; ergo, it was
perfect, and no man dared say otherwise.
Our engine work was finally, as a last resort, done by Mr. Watts on
new lathes, made for customers and used for a month or two before
they were sent out. Not only in England, but on the Continent and in
America, the Whitworth Works were regarded as the perfect
machine-shop. I remember a visit I had at the Paris Exposition from
Mr. Elwell, of the firm of Varrell, Elwell & Poulot, proprietors of the
largest mechanical establishment in Paris. After expressing his
unbounded admiration of the running of the engine, he said, “I
warrant your fly-wheel runs true.” After observing it critically, he
exclaimed, “Ah, they do those things at Whitworth’s!”
The fact was Mr. Whitworth had cursed the British nation with the
solid conical lathe-spindle bearing, a perfect bearing for ordinary-
sized lathes and a most captivating thing—when new. These
hardened steel cones, in hardened steel seats, ran in the most
charming manner. But they wore more loose in the main bearing
every day they ran, and there were no means for taking up the wear.
It came on insensibly, and no one paid any attention to it. The cream
of the joke was that people were so fascinated with this bearing that
at that time no other could be sold in England, except for very large
lathes. All toolmakers had to make it. I remember afterwards that Mr.
Freeland, our best American toolmaker, who, as I have already
mentioned, went to England and worked for some years as a
journeyman in the Whitworth Works for the purpose of learning
everything there that he could, did not bring back to America the
conical bearing.
The firm of Smith & Coventry were the first to fit their lathes with
the means for taking up this wear, which took place only in the main
bearing, where both the force of the cut and the weight of the piece
were received. They made the conical seat for the back end of the
spindle adjustable in the headstock and secured it by a thin nut on
each end. This then could be moved backward sufficiently to let the
forward cone up to its seat. This made it possible to use the solid
bearing, but it involved this error, that after this adjustment the axis of
the spindle did not coincide with the line connecting the lathe
centers; but the two lines formed an angle with each other, which
grew more decided every time the wear was taken up. This,
however, was infinitely better than not to take up the wear at all.
At that time the Whitworth Works were divided into four
departments. These were screwing machinery, gauges, guns and
machine tools. The first three of these were locked. I never entered
either of them. The latter also, like most works in England, was
closed to outsiders. No customer could see his work in progress.
This department was without a head or a drawing-office. It seemed
to be running it on its traditions. I once said to Mr. Hoyle, “There
must at some time have been here mechanical intelligence of the
highest order, but where is it?” They had occasionally an order for
something out of their ancient styles, and their attempts to fill such
orders were always ruinous. The following is a fair illustration. They
had an order for a radial drill to be back-geared and strong enough
to bore an 8-inch hole. Mr. Widdowson had the pattern for the upright
fitted with the necessary brackets, and thought it was such a good
thing that he would make two. The first one finished was tried in the
shop, and all the gears in the arm were stripped. He woke up to the
fact that he had forgotten to strengthen the transmitting parts, and
moreover that the construction would not admit anything stronger.
There was nothing to be done but to decline the order, chip off the
brackets, and make these into single-speed drills. This I saw being
done.
Mr. Widdowson told me the following amusing story. The London
Times had heard of the wonderful performance of Mr. Hoe’s multiplecylinder press, and concluded to have one of them of the largest
size, ten cylinders. But, of course, Mr. Hoe did not know how to
make his own presses. His work would do well enough for ignorant
Americans, but not for an English Journal. The press must be made
in England in the world-renowned Whitworth Works.
Mr. Hoe sent over one of his experts to give them the information
they might need, but they would not let him in the shop. Mr. Hulse
told him they had the drawings and specifications and that was all
they needed. When the press was finished they set it up in the shop
and attempted to run it. The instant it started every tape ran off its
pulleys, and an investigation showed that not a spindle or shaft was
parallel with any other. They had no idea of the method that must be
employed to ensure this universal alignment. After enormous labor
they got these so that they were encouraged to make another trial,
when after a few revolutions every spindle stuck fast in its bearings.
Mr. Whitworth, absorbed in his artillery and spending most of his
time in London, of course had no knowledge of how things were
going on in his shop, of the utter want of ordinary intelligence.
I formed a scheme for an application of Mr. Whitworth’s system of
end measurement to the production of an ideally perfect dividingwheel. In this system Mr. Whitworth employed what he termed “the
gravity piece.” This was a small steel plate about ¹⁄₈ of an inch in
thickness, the opposite sides of which were parallel and had the
most perfectly true and smooth surfaces that could be produced by
scraping. The ends of the piece to be tested were perfectly squared,
by a method which I will not stop here to describe, and were finished