Preface
A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.
The art of teaching, Mark Van Doren said, is the art of assisting discovery. I have tried to write a book that assists students in discovering calculus—both for its practical power and its surprising beauty. In this edition, as in the first seven editions, I aim to convey to the student a sense of the utility of calculus and develop technical competence, but I also strive to give some appreciation for the intrinsic beauty of the subject. Newton undoubtedly experienced a sense of triumph when he made his great discoveries. I want students to share some of that excitement.
The emphasis is on understanding concepts. I think that nearly everybody agrees that this should be the primary goal of calculus instruction. In fact, the impetus for the current calculus reform movement came from the Tulane Conference in 1986, which formulated as their first recommendation:
Focus on conceptual understanding.
I have tried to implement this goal through the Rule of Three: “Topics should be presented geometrically, numerically, and algebraically.” Visualization, numerical and graphical experimentation, and other approaches have changed how we teach conceptual reasoning in fundamental ways. More recently, the Rule of Three has been expanded to become the Rule of Four by emphasizing the verbal, or descriptive, point of view as well.
In writing the eighth edition my premise has been that it is possible to achieve conceptual understanding and still retain the best traditions of traditional calculus. The book contains elements of reform, but within the context of a traditional curriculum.
I have written several other calculus textbooks that might be preferable for some instructors. Most of them also come in single variable and multivariable versions.
● Calculus: Early Transcendentals, Eighth Edition, is similar to the present textbook except that the exponential, logarithmic, and inverse trigonometric functions are covered in the first semester.
● Essential Calculus, Second Edition, is a much briefer book (840 pages), though it contains almost all of the topics in Calculus, Eighth Edition. The relative brevity is achieved through briefer exposition of some topics and putting some features on the website.
● Essential Calculus: Early Transcendentals, Second Edition, resembles Essential Calculus, but the exponential, logarithmic, and inverse trigonometric functions are covered in Chapter 3.
george polya
● Calculus: Concepts and Contexts, Fourth Edition, emphasizes conceptual understanding even more strongly than this book. The coverage of topics is not encyclopedic and the material on transcendental functions and on parametric equations is woven throughout the book instead of being treated in separate chapters.
● Calculus: Early Vectors introduces vectors and vector functions in the first semester and integrates them throughout the book. It is suitable for students taking engineering and physics courses concurrently with calculus.
● Brief Applied Calculus is intended for students in business, the social sciences, and the life sciences.
● Biocalculus: Calculus for the Life Sciences is intended to show students in the life sciences how calculus relates to biology.
● Biocalculus: Calculus, Probability, and Statistics for the Life Sciences contains all the content of Biocalculus: Calculus for the Life Sciences as well as three additional chapters covering probability and statistics.
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:
● The data in examples and exercises have been updated to be more timely.
● New examples have been added (see Examples 5.1.5 and 11.2.5, for instance). And the solutions to some of the existing examples have been amplified.
● Two new projects have been added: The project Planes and Birds: Minimizing Energy (page 271) asks how birds can minimize power and energy by flapping their wings versus gliding. The project Controlling Red Blood Cell Loss During Surgery (page 473) describes the ANH procedure, in which blood is extracted from the patient before an operation and is replaced by saline solution. This dilutes the patient’s blood so that fewer red blood cells are lost during bleeding and the extracted blood is returned to the patient after surgery.
● More than 20% of the exercises in each chapter are new. Here are some of my favorites: 2.1.61, 2.2.34–36, 3.3.30, 3.3.54, 3.7.39, 3.7.67, 4.1.19–20, 4.2.67–68, 4.4.63, 5.1.51, 6.2.79, 6.7.54, 6.8.90, 8.1.39. In addition, there are some good new Problems Plus. (See Problems 10–12 on page 201, Problem 10 on page 290, and Problems 14–15 on pages 353–54.)
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 few exercises in Sections 1.5, 1.8, and 11.2.) Similarly, all the review
sections begin with a Concept Check and a True-False Quiz. Other exercises test conceptual understanding through graphs or tables (see Exercises 2.1.17, 2.2.33–36, 2.2.45–50, 9.1.11–13, 10.1.24–27, and 11.10.2).
Another type of exercise uses verbal description to test conceptual understanding (see Exercises 1.8.10, 2.2.64, 3.3.57–58, and 7.8.67). I particularly value problems that combine and compare graphical, numerical, and algebraic approaches (see Exercises 2.7.25, 3.4.33–34, and 9.4.4).
graded e xercise 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 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 2.2.33 (unemployment rates), Exercise 4.1.16 (velocity of the space shuttle Endeavour), and Figure 4 in Section 4.4 (San Francisco power consumption).
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. I have included four kinds of projects: Applied Projects involve applications that are designed to appeal to the imagination of students. The project after Section 9.3 asks whether a ball thrown upward takes longer to reach its maximum height or to fall back to its original height. (The answer might surprise you.) 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. 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 encourage discovery through pattern recognition (see the one following Section 7.6). 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 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 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 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.
dual treatment of e xponential 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 finds advocates of both approaches teaching the same course, I include full treatments of both methods. In Sections 6.2, 6.3, and 6.4 the exponential function is defined first, followed by the logarithmic 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 defined 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 first approach is taken, then much of Chapter 6 can be covered before Chapters 4 and 5, if desired. To accommodate this choice of presentation there are specially identified 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 definite integral in the first 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 first 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 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 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 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-bystep tutorials through text examples, with links to the textbook and to video solutions.
1 functions and limits
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)
2 derivatives
3 applications of differentiation 4 integrals
The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Functions, and Trigonometry.
This is an overview of the subject and includes a list of questions to motivate the study of calculus.
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 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 introduced 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 optimization 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 contexts and on estimating their values from graphs and tables. diagnostic tests a Preview of calculus
5 applications of integration
6 inverse functions:
e xponential, logarithmic, and inverse trigonometric functions
7 techniques of integration
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.
8 further applications of integration
9 differential equations
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 biology, 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 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 predatorprey models to illustrate systems of differential equations.
10 Parametric equations and Polar coordinates
11 infinite Sequences and Series
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.
Single Variable 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 xx–xxi describe each of these ancillaries.
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
technology reviewers
Maria Andersen, Muskegon Community College
Eric Aurand, Eastfield 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
Previous edition 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
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
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
Jack Ceder, University of California, Santa Barbara
Scott Chapman, Trinity University
Zhen-Qing Chen, University of Washington—Seattle
James Choike, Oklahoma State University
Barbara Cortzen, DePaul University
Carl Cowen, Purdue University
Philip S. Crooke, Vanderbilt University
Charles N. Curtis, Missouri Southern State College
Daniel Cyphert, Armstrong State College
Robert Dahlin
M. Hilary Davies, University of Alaska Anchorage
Gregory J. Davis, University of Wisconsin–Green Bay
Elias Deeba, University of Houston–Downtown
Daniel DiMaria, Suffolk Community College
Seymour Ditor, University of Western Ontario
Greg Dresden, Washington and Lee University
Daniel Drucker, Wayne State University
kenn Dunn, Dalhousie University
Dennis Dunninger, Michigan State University
Bruce Edwards, University of Florida
David Ellis, San Francisco State University
John Ellison, Grove City College
Martin Erickson, Truman State University
Garret Etgen, University of Houston
Theodore G. Faticoni, Fordham University
Laurene V. Fausett, Georgia Southern University
Norman Feldman, Sonoma State University
Le Baron O. Ferguson, University of California—Riverside
Newman Fisher, San Francisco State University
José D. Flores, The University of South Dakota
William Francis, Michigan Technological University
James T. Franklin, Valencia Community College, East
Stanley Friedlander, Bronx Community College
Patrick Gallagher, Columbia University–New York
Paul Garrett, University of Minnesota–Minneapolis
Frederick Gass, Miami University of Ohio
Bruce Gilligan, University of Regina
Matthias k. Gobbert, University of Maryland, Baltimore County
Gerald Goff, Oklahoma State University
Stuart Goldenberg, California Polytechnic State University
John A. Graham, Buckingham Browne & Nichols School
Richard Grassl, University of New Mexico
Michael Gregory, University of North Dakota
Charles Groetsch, University of Cincinnati
Paul Triantafilos Hadavas, Armstrong Atlantic State University
Salim M. Haïdar, Grand Valley State University
D. W. Hall, Michigan State University
Robert L. Hall, University of Wisconsin–Milwaukee
Howard B. Hamilton, California State University, Sacramento
Darel Hardy, Colorado State University
Shari Harris, John Wood Community College
Gary W. Harrison, College of Charleston
Melvin Hausner, New York University/Courant Institute
Curtis Herink, Mercer University
Russell Herman, University of North Carolina at Wilmington
Allen Hesse, Rochester Community College
Randall R. Holmes, Auburn University
James F. Hurley, University of Connecticut
Amer Iqbal, University of Washington—Seattle
Matthew A. Isom, Arizona State University
Gerald Janusz, University of Illinois at Urbana-Champaign
John H. Jenkins, Embry-Riddle Aeronautical University, Prescott Campus
Clement Jeske, University of Wisconsin, Platteville
Carl Jockusch, University of Illinois at Urbana-Champaign
Jan E. H. Johansson, University of Vermont
Jerry Johnson, Oklahoma State University
Zsuzsanna M. kadas, St. Michael’s College
Nets katz, Indiana University Bloomington
Matt kaufman
Matthias kawski, Arizona State University
Frederick W. keene, Pasadena City College
Robert L. kelley, University of Miami
Akhtar khan, Rochester Institute of Technology
Marianne korten, Kansas State University
Virgil kowalik, Texas A&I University
kevin kreider, University of Akron
Leonard krop, DePaul University
Mark krusemeyer, Carleton College
John C. Lawlor, University of Vermont
Christopher C. Leary, State University of New York at Geneseo
David Leeming, University of Victoria
Sam Lesseig, Northeast Missouri State University
Phil Locke, University of Maine
Joyce Longman, Villanova University
Joan McCarter, Arizona State University
Phil McCartney, Northern Kentucky University
Igor Malyshev, San Jose State University
Larry Mansfield, Queens College
Mary Martin, Colgate University
Nathaniel F. G. Martin, University of Virginia
Gerald Y. Matsumoto, American River College
James Mckinney, California State Polytechnic University, Pomona
Tom Metzger, University of Pittsburgh
Richard Millspaugh, University of North Dakota
Lon H. Mitchell, Virginia Commonwealth University
Michael Montaño, Riverside Community College
Teri Jo Murphy, University of Oklahoma
Martin Nakashima, California State Polytechnic University, Pomona
Ho kuen Ng, San Jose State University
Richard Nowakowski, Dalhousie University
Hussain S. Nur, California State University, Fresno
Norma Ortiz-Robinson, Virginia Commonwealth University
Wayne N. Palmer, Utica College
Vincent Panico, University of the Pacific
F. J. Papp, University of Michigan–Dearborn
Mike Penna, Indiana University–Purdue University Indianapolis
Mark Pinsky, Northwestern University
Lothar Redlin, The Pennsylvania State University
Joel W. Robbin, University of Wisconsin–Madison
Lila Roberts, Georgia College and State University
E. Arthur Robinson, Jr., The George Washington University
Richard Rockwell, Pacific Union College
Rob Root, Lafayette College
Richard Ruedemann, Arizona State University
David Ryeburn, Simon Fraser University
Richard St. Andre, Central Michigan University
Ricardo Salinas, San Antonio College
Robert Schmidt, South Dakota State University
Eric Schreiner, Western Michigan University
Mihr J. Shah, Kent State University–Trumbull
Qin Sheng, Baylor University
Theodore Shifrin, University of Georgia
Wayne Skrapek, University of Saskatchewan
Larry Small, Los Angeles Pierce College
Teresa Morgan Smith, Blinn College
William Smith, University of North Carolina
Donald W. Solomon, University of Wisconsin–Milwaukee
Edward Spitznagel, Washington University
Joseph Stampfli, Indiana University
kristin Stoley, Blinn College
M. B. Tavakoli, Chaffey College
Magdalena Toda, Texas Tech University
Ruth Trygstad, Salt Lake Community College
Paul Xavier Uhlig, St. Mary’s University, San Antonio
Stan Ver Nooy, University of Oregon
Andrei Verona, California State University–Los Angeles
klaus Volpert, Villanova University
Russell C. Walker, Carnegie Mellon University
William L. Walton, McCallie School
Peiyong Wang, Wayne State University
Jack Weiner, University of Guelph
Alan Weinstein, University of California, Berkeley
Theodore W. Wilcox, Rochester Institute of Technology
Steven Willard, University of Alberta
Robert Wilson, University of Wisconsin–Madison
Jerome Wolbert, University of Michigan–Ann Arbor
Dennis H. Wortman, University of Massachusetts, Boston
Mary Wright, Southern Illinois University–Carbondale
Paul M. Wright, Austin Community College
Xian Wu, University of South Carolina
In addition, I would like to thank R. B. Burckel, Bruce Colletti, David Behrman, John Dersch, Gove Effinger, Bill Emerson, Dan kalman, Quyan khan, Alfonso Gracia-Saz, Allan MacIsaac, Tami Martin, Monica Nitsche, Lamia Raffo, Norton Starr, and Jim Trefzger for their suggestions; Al Shenk and Dennis Zill for permission to use exercises from their calculus texts; COMAP for permission to use project material; George Bergman, David Bleecker, Dan Clegg, Victor kaftal, Anthony Lam, Jamie Lawson, Ira Rosenholtz, Paul Sally, Lowell Smylie, and Larry Wallen for ideas for exercises; Dan Drucker for the roller derby project; Thomas Banchoff, Tom Farmer, Fred Gass, John Ramsay, Larry Riddle, Philip Straffin, and klaus Volpert for ideas for projects; Dan Anderson, Dan Clegg, Jeff Cole, Dan Drucker, and Barbara Frank for solving the new exercises and suggesting ways to improve them; Marv Riedesel and Mary Johnson for accuracy in proofreading; Andy Bulman-Fleming, Lothar Redlin, Gina Sanders, and Saleem Watson for additional proofreading; and Jeff Cole and Dan Clegg for their careful preparation and proofreading of the answer manuscript.
In addition, I thank those who have contributed to past editions: Ed Barbeau, Jordan Bell, George Bergman, Fred Brauer, Andy Bulman-Fleming, Bob Burton, David Cusick, Tom DiCiccio, Garret Etgen, Chris Fisher, Leon Gerber, Stuart Goldenberg, Arnold Good, Gene Hecht, Harvey keynes, E. L. koh, Zdislav kovarik, kevin kreider, Emile LeBlanc, David Leep, Gerald Leibowitz, Larry Peterson, Mary Pugh, Lothar Redlin, Carl Riehm, John Ringland, Peter Rosenthal, Dusty Sabo, Doug Shaw, Dan Silver, Simon Smith, Norton Starr, Saleem Watson, Alan Weinstein, and Gail Wolkowicz.
I also thank kathi Townes, Stephanie kuhns, kristina Elliott, and kira Abdallah of TECHarts for their production services and the following Cengage Learning staff: Cheryll Linthicum, content project manager; Stacy Green, senior content developer; Samantha Lugtu, associate content developer; Stephanie kreuz, product assistant; Lynh Pham, media developer; Ryan Ahern, marketing manager; and Vernon Boes, art director. 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, Richard Stratton, Liz Covello, and now Neha Taleja. All of them have contributed greatly to the success of this book.
james stewart
Instructor’s Guide
by Douglas Shaw
ISBN 978-1-305-27178-4
Each section of the text is discussed from several viewpoints. The Instructor’s Guide 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 assignments.
Complete Solutions Manual
Single Variable
By Daniel Anderson, Jeffery A. Cole, and Daniel Drucker
ISBN 978-1-305-27610-9
Includes worked-out solutions to all exercises in the text.
Printed Test Bank
By William Steven Harmon
ISBN 978-1-305-27180-7
Contains text-specific multiple-choice and free response test items.
Cengage Learning Testing Powered by Cognero (login.cengage.com)
This flexible online system allows you to author, edit, and manage test bank content from multiple Cengage Learning solutions; create multiple test versions in an instant; and deliver tests from your LMS, your classroom, or wherever you want.
Stewart Website
www.stewartcalculus.com
Contents: Homework Hints n Algebra Review n Additional Topics n Drill exercises n Challenge Problems n Web Links n History of Mathematics n Tools for Enriching Calculus (TEC)
TEC TOOLS FOR ENRICHING™ CALCULUS
By
James Stewart, Harvey keynes, Dan Clegg, and developer Hubert Hohn
Tools for Enriching Calculus (TEC) functions as both a powerful tool for instructors and as a tutorial environment in which students can explore and review selected topics. The Flash simulation modules in TEC include instructions, written and audio explanations of the concepts, and exercises. TEC is accessible in the eBook via CourseMate and Enhanced WebAssign. Selected Visuals and Modules are available at www.stewartcalculus.com.
Enhanced WebAssign®
www.webassign.net
Printed Access Code: ISBN 978-1-285-85826-5
Instant Access Code ISBN: 978-1-285-85825-8
Exclusively from Cengage Learning, Enhanced WebAssign offers an extensive online program for Stewart’s Calculus to encourage the practice that is so critical for concept mastery. The meticulously crafted pedagogy and exercises in our proven texts become even more effective in Enhanced WebAssign, supplemented by multimedia tutorial support and immediate feedback as students complete their assignments. Key features include:
n Thousands of homework problems that match your textbook’s end-of-section exercises
n Opportunities for students to review prerequisite skills and content both at the start of the course and at the beginning of each section
n Read It eBook pages, Watch It videos, Master It tutorials, and Chat About It links
n A customizable Cengage YouBook with highlighting, notetaking, and search features, as well as links to multimedia resources
n Personal Study Plans (based on diagnostic quizzing) that identify chapter topics that students will need to master
n A WebAssign Answer Evaluator that recognizes and accepts equivalent mathematical responses in the same way an instructor grades
n A Show My Work feature that gives instructors the option of seeing students’ detailed solutions
n Visualizing Calculus Animations, Lecture Videos, and more
Cengage Customizable YouBook
YouBook is an eBook that is both interactive and customizable. Containing all the content from Stewart’s Calculus, YouBook features a text edit tool that allows instructors to modify the textbook narrative as needed. With YouBook, instructors can quickly reorder entire sections and chapters or hide any content they don’t teach to create an eBook that perfectly matches their syllabus. Instructors can further customize the text by adding instructor-created or YouTube video links. Additional media assets include animated figures, video clips, highlighting and note-taking features, and more. YouBook is available within Enhanced WebAssign.
CourseMate
CourseMate is a perfect self-study tool for students, and requires no set up from instructors. CourseMate brings course concepts to life with interactive learning, study, and exam preparation tools that support the printed textbook. CourseMate for Stewart’s Calculus includes an interactive eBook, Tools for Enriching Calculus, videos, quizzes, flashcards, and more. For instructors, CourseMate includes Engagement Tracker, a first-of-its-kind tool that monitors student engagement.
CengageBrain.com
To access additional course materials, please visit www.cengagebrain.com . At the CengageBrain.com home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page where these resources can be found.
Student Solutions Manual
Single Variable
By Daniel Anderson, Jeffery A. Cole, and Daniel Drucker
ISBN 978-1-305-27181-4
Provides completely worked-out solutions to all oddnumbered exercises in the text, giving students a chance to
■ Electronic items ■ Printed items
check their answer and ensure they took the correct steps to arrive at the answer. The Student Solutions Manual can be ordered or accessed online as an eBook at www.cengagebrain.com by searching the ISBN.
Study Guide
Single Variable
By Richard St. Andre
ISBN 978-1-305-27913-1
For each section of the text, the Study Guide provides students with a brief introduction, a short list of concepts to master, and summary and focus questions with explained answers. The Study Guide also contains self-tests with exam-style questions. The Study Guide can be ordered or accessed online as an eBook at www.cengagebrain.com by searching the ISBN.
A Companion to Calculus
By
Dennis Ebersole, Doris Schattschneider, Alicia Sevilla, and Kay Somers
ISBN 978-0-495-01124-8
Written to improve algebra and problem-solving skills of students taking a calculus course, every chapter in this companion is keyed to a calculus topic, providing conceptual background and specific algebra techniques needed to understand and solve calculus problems related to that topic. It is designed for calculus courses that integrate the review of precalculus concepts or for individual use. Order a copy of the text or access the eBook online at www.cengagebrain.com by searching the ISBN.
Linear Algebra for Calculus
by Konrad J. Heuvers, William P. Francis, John H. Kuisti, Deborah F. Lockhart, Daniel S. Moak, and Gene M. Ortner
ISBN 978-0-534-25248-9
This comprehensive book, designed to supplement the calculus course, provides an introduction to and review of the basic ideas of linear algebra. Order a copy of the text or access the eBook online at www.cengagebrain.com by searching the ISBN.
Diagnostic Tests
Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry. The following tests are intended to diagnose weaknesses that you might have in these areas. After taking each test you can check your answers against the given answers and, if necessary, refresh your skills by referring to the review materials that are provided.
A1. Evaluate each expression without using a calculator.
2. Simplify each expression. Write your answer without negative exponents. (a) s200 s32
3. Expand and simplify.
4. Factor each expression.
5. Simplify the rational expression.
6. Rationalize the expression and simplify. (a) s10 s5 2 (b) s4 1 h 2 h
7. Rewrite by completing the square. (a) x 2 1 x 1 1 (b) 2 x 2 12 x 1 11
8. Solve the equation. (Find only the real solutions.) (a) x 1 5 − 14 1 2 x
(c) x 2 x 12 − 0
x 4 3x 2 1 2 −
− 0
9. Solve each inequality. Write your answer using interval notation. (a) 4 , 5 3x < 17 (b) x 2 , 2 x 1 8 (c) x s x 1ds x 1 2d . 0 (d) | x 4 | , 3 (e) 2 x 3 x 1 1 < 1
10. State whether each equation is true or false. (a) s p 1 qd2 − p 2 1 q 2 (b) sab − sa sb (c) sa 2 1 b 2 − a 1 b (d) 1 1 TC C − 1 1 T (e) 1 x y − 1 x 1 y
answers TO D ia G n O s T ic T es T a: al G ebra
1. (a) 81 (b) 81 (c) 1 81 (d) 25 (e) 9 4 (f) 1 8
2. (a) 6 s2 (b) 48 a 5b7 (c) x 9y7
3. (a) 11 x 2 (b) 4 x 2 1 7x 15
(c) a b (d) 4 x 2 1 12 x 1 9
(e) x 3 1 6 x 2 1 12 x 1 8
4. (a) s2 x 5ds2 x 1 5d (b) s2 x 3ds x 1 4d
(c) s x 3ds x 2ds x 1 2d (d) x s x 1 3ds x 2 3x 1 9d (e) 3x 1y2s x 1ds x 2d (f) x y s x 2ds x 1 2d
5. (a) x 1 2 x 2 (b) x 1 x 3 (c) 1 x 2 (d) s x 1 yd
6. (a) 5 s2 1 2 s10 (b) 1 s4 1 h 1 2
7. (a) s x 1 1 2 d 2 1 3 4 (b) 2s x 3d2 7
8. (a) 6 (b) 1 (c) 3, 4 (d) 1 6 1 2 s2 (e) 61, 6s2 (f) 2 3 , 22 3 (g) 12 5
9. (a) f 4, 3d (b) s 2, 4d (c) s 2, 0d ø s1, `d (d) s1, 7d (e) s 1, 4g
10. (a) False (b) True (c) False (d) False (e) False (f) True
If you had difficulty with these problems, you may wish to consult the Review of Algebra on the website www.stewartcalculus.com.
1. Find an equation for the line that passes through the point s2, 5d and (a) has slope 3
(b) is parallel to the x-axis
(c) is parallel to the y-axis
(d) is parallel to the line 2 x 4 y − 3
2. Find an equation for the circle that has center s 1, 4d and passes through the point s3, 2d
3. Find the center and radius of the circle with equation x 2 1 y 2
4. Let As 7, 4d and Bs5, 12d be points in the plane.
1 10 y 1 9 − 0.
(a) Find the slope of the line that contains A and B. (b) Find an equation of the line that passes through A and B. What are the intercepts?
(c) Find the midpoint of the segment AB.
(d) Find the length of the segment AB.
(e) Find an equation of the perpendicular bisector of AB. (f) Find an equation of the circle for which AB is a diameter.
5. Sketch the region in the x y-plane defined by the equation or inequalities. (a) 1 < y < 3 (b) | x | , 4 and | y | , 2 (c) y , 1 1 2 x (d) y > x 2 1
(e) x 2 1 y 2 , 4
answers TO D ia G n O s T ic T es T b: analy T ic G e O me T ry
1. (a) y − 3x 1 1 (b) y − 5
(c) x − 2 (d) y − 1 2 x 6
2. s x 1 1d2 1 s y 4d2 − 52
3. Center s3, 5d, radius 5
4. (a) 4 3
(b) 4 x 1 3y 1 16 − 0; x-intercept 4, y-intercept 16 3
(c) s 1, 4d
(d) 20
(e) 3x 4 y − 13
(f) s x 1 1d2 1 s y 1 4d2 − 100
(f) 9x 2 1 16y 2 − 144
If you had difficulty with these problems, you may wish to consult the review of analytic geometry in Appendixes B and C.
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of odds cannot score till he has holed two balls, each of which has to be put back on the table (see Rule 18).
The rules for foul strokes are the same as at billiards, but if a player wilfully touch a moving ball, he loses the game;[21] to do so accidentally makes that stroke foul (see also Rule 23).
I once saw a pretty commentary on Rule 18, which directs that a ball which has to be put up be placed on the pyramid spot. The player, who owed four, made a hazard and got exact position behind the pyramid spot in a line with the corner pocket, and, screwing back each time, holed each ball as it was put up.
Balls forced off the table are put up again, but the striker’s break is at an end unless he also holes a ball (see Rules 18 and 19).
Nothing can be said on the question of handicapping players. Of course their relative skill at billiards affords no criterion of their relative powers at pyramids. A few games will best decide the question; but it may be remarked that to give a ball is a far higher handicap than to owe a ball, as the adversary starts with a point to the good, and there are only fourteen coloured balls on the table; whereas the player who owes a ball only forfeits the first hazard he makes.
Before the game commences the first step is to set up the balls properly. In theory each ball should touch its immediate neighbours, but in practice this is of course impossible. They should, however, be collected in the triangle, and then rolled smartly up and down parallel to the sides of the table, the apex-ball never going beyond the pyramid spot. After this has been done two or three times the motion should be sharply stopped when the apex-ball is on the spot, and the pyramid will then be fairly correct. There are three ways of playing the first stroke, two of them unsound. The first wrong way is to smash the pyramid by a vicious hit from baulk, for which Captain Crawley, in ‘The Billiard Book,’ recommends a mysterious ‘underhanded stroke;’ but in whatever way the stroke is played it is unsound, as there are only two pockets behind the pyramid into which to drive a ball. The second wrong method is to play slowly up the table with a little side, missing the pyramid on the upward journey, and just dropping on to it from the top cushion. Oldfashioned players are fond of this opening, but it is not sound, as the adversary can easily get safe, or, if he likes, he can smash with four
pockets open to him. Whether he be wise to do so is another question; with a weak adversary, to whom he is giving odds, it may be advisable. I have seen no less than five balls disappear after such a stroke; but if a winning hazard is not made, the break of course goes to the other side: it is a matter of speculation, the chances being naturally in favour of the stronger player. At no time should a smash be tried except when four pockets are open—i.e. from the top of the table; a stab or screw back should be used, so as to avoid as far as possible the mob of flying balls, which may kiss the white into a pocket. The third and orthodox opening is to play at the end ball of the row next the base of the pyramid with strength sufficient to leave the white ball as near to the bottom cushion as possible. If the pyramid is properly set up, the opponent has no easy stroke left, though occasionally a ball is malignant enough to detach itself and come down the table. This is generally the result of either careless setting up or of striking too hard, but if this opening stroke is properly played, the second player will have nothing better to play for than a more or less difficult stroke for safety, and so the game will proceed till the pyramid is gradually shaken and finally broken up; but in playing for safety it is sometimes advisable to disturb the pyramid, if possible, it being difficult for one’s opponent to steer a safe course when there are rocks ahead in the shape of balls.
It is often safe to leave the white ball near the pyramid, provided that it has not been greatly disturbed; for, if the players are equal, neither should risk a smash, for it is, after all, even betting which player profits. The best safety of all is to leave the adversary far away from a ball and as near to a cushion as possible; but if he can retaliate in kind, not much good will have been done. Watch the score, and play to the score. The leader should play a cautious rather than a dashing game, as a losing hazard not only diminishes his score by a point, but also gives his adversary the advantage of playing from baulk with an extra ball on the table. ‘When in doubt play for safety,’ is a golden rule, but a doubtful hazard may often be tried when one can get safety as well. Beginners should be cautioned to watch carefully for foul strokes, especially when the rest or spider is being used. A knowledge of the spot stroke and its variations is invaluable, involving, as this stroke does, every form of screw, stab, and following stroke; while the stop stroke is also most useful—i.e. a sort of stab that leaves the white ball on the place just vacated by the red.
When a player is familiar with ordinary winning hazards, and can make them with some facility, he should devote himself to the making of breaks—i.e. a series of hazards. Diagrams I. and II. may serve as examples, showing how position should be got so that one hazard may lead on to another.
Diagram I.
Nor should the famous dictum about the spot stroke be forgotten— viz. that the first and most important point is to make sure of putting in the red. Supposing, then, that the balls are left as in Diagram I., the problem is to take them all in a break, which may be done as follows, the figures representing the successive positions of the white ball:—From position 1 there is an easy hazard, and position 2 may be easily gained. The same remarks apply to the next stroke, but from 3 a ‘run through’ with right-hand side is required so as to get to 4. Here there is the option of dropping ball X quietly into the middle pocket, leaving an easy shot on Y, or of stabbing Y and getting position 5. If
the latter stroke is successfully played, the rule of ‘never play for a middle pocket at single pool’ should decide the striker to drive X gently down to the left-hand bottom pocket, leaving his ball safe under the side cushion.
The break shown in Diagram II. is by no means so easy. It may be played as follows:—From position 1 a gentle stab, screwing back a little, should be played; from 2 is required a semi-follow with lefthand side. For the third stroke strong right-hand side is used, the top cushion being utilised or not according to fancy; and the fourth also requires some right-hand side, but the proper play is to try to get such position as will leave a shot for a corner and not a middle pocket. It may also be noted that by playing on ball Y first there will be but a poor chance of getting ball X in the course of the break, as ball Z will clearly be the next one to play at. These two breaks are only suggested for useful practice, and to show the beginner some of the devices necessary for success.
Diagram II.
I will now discuss certain strokes of frequent occurrence, for which special hints are necessary, plants and doubles being among the most important. They have been to some extent dealt with already,[22] but are more common at winning-hazard games than at billiards, and consequently not only do they demand careful attention, but also verification by practice, the relative positions of the balls being frequently altered, and the varieties in the results noted and studied. Another very important class of strokes occurs when the object ball is under the cushion, a common situation in all games of pool. Diagram III. shows two examples, though stroke B is really only a modification of stroke A; still, it deserves separate consideration, as the hazard is very difficult, and the position of the striker’s ball after the stroke has been played is most important. Example A may be considered typical; the player’s ball is on the centre of the Ｄ and the object ball half way up cushion 2. Play slowly, about No. 1 strength, so as to hit ball and cushion simultaneously. Ball 2 will drop into the pocket, and ball 1 will travel towards the spot. If position is desired to the right of the spot, a little left-hand side should be used, and it even seems to make the hazard easier; a sharper stroke with right-hand side will bring ball 1 towards the middle of the table. This stroke should be practised with ball 2 at such positions as P, Q, R, and S, and the resting place of ball 1 should be carefully observed. It is clearly not a good stroke for single pool, as the balls are left too close together. Stroke B is not at all easy, but it is worth playing for, as it cannot leave much. Ball 2 must be cut very finely—in fact, play just not to miss it. If it is missed on the upward journey, left-hand side, which is almost essential to the stroke, will cause ball 2 to be hit from the cushion, X and Y show the direction of ball 1 according as no side or left-hand side is used.
There are no strokes more common and none which require more care than those in which the object ball is close to a cushion, nearly at right angles to the path of ball 1 and a long distance from it. The paths of both seem, from the mere proximity of the cushion, to be regulated by an entirely new code of dynamic laws, the fact being that the whole conditions of the case are not correctly realised. It is here that the inestimable qualities of side, as an agent productive of pace, are called in to assist; for by playing with direct side and cutting ball 2 very fine, its course will be restricted, and the side will cause ball 1 to travel freely down the table; but here, as in all things,
an ounce of practice is worth a ton of theory, and more can be learned by an hour’s practice than a week’s reading.
Diagram III.
To the serious student the ‘R.-W. Billiard Diagram Notebook’[23] is recommended, in order that the results of practice and observation may be recorded, for, as Captain Cuttle might have said, ‘These things, when found, should be made a note of.’
Diagram IV. shows a useful double in stroke A, ball 2 being some distance below the middle pocket, and two or three inches from the cushion. Ball 1 should be placed approximately as shown in the diagram; but practice and experience can alone show the exact place, depending as it does on the position of ball 2. To make the double, play full on ball 2. Ball 1 can be made to reach the top of the table, if desired, by the use of strong left-hand side and follow. For single pool a stab is of course the right stroke. If the top pocket is blocked, or if for any other reason (e.g. for the sake of position) this stroke is
undesirable, there is a good chance of a double as shown in stroke B, ball 2 being struck half-ball, and ball 1 following approximately the lines terminating in X. For the sake of practice the position of ball 2 should be shifted towards the pocket, and also further down the table. The further it lies from baulk, the finer must be the cut, and the harder the stroke. The position of ball 1 should be carefully noted each time, and also the point at which the object ball having been gradually moved down the table, a kiss occurs and prevents the double into the middle pocket.
Diagram IV.
The stroke shown in Diagram V. is of common occurrence in single pool, and may appropriately be here explained, as every game of pyramids eventually becomes one of single pool. The type of stroke is so important that it should be practised from various positions—first without side, and then with side both right and left, the ultimate position of the striker’s ball being the main feature of the stroke. The direct hazard is of course on, but only special circumstances would
justify any but a first-rate hazard striker in trying for it. In the diagram P Z shows the course of ball 1 when no side is used, Q Y when played with strong left-hand side, and R X when strong right-hand side is employed. Strength is most important, and observation alone will show when a kiss occurs as the balls cross each other’s track; but the chances of this are much diminished when reverse side (in this case left side) is used. If ball 2 is near the cushion, a sharp stroke is necessary, but the double shown in stroke A, Diagram VI., is the better game. All the doubles shown in this diagram are useful, especially for single pool. Stroke A is played with a stab, stroke B with follow, so as to leave ball 1 under the top cushion. Stroke C also requires a stab, strength being judged so as to leave ball 2 close to the pocket, if it is not holed. In both B and C ball 2 might possibly be cut in, but the double is, for pool at least, the safer stroke.
The question of plants has been already alluded to in Chapter VII., which should be carefully studied, as such shots are infinitely more common with the fifteen-pyramid balls than at billiards. In the plant, pure and simple, the balls are touching or practically touching; but if what may be called the second object ball is fairly near to the pocket, a plant is often worth trying, though some caution is necessary, as a leave is very likely to result if the stroke fails. The principle may be described as playing a ball on to a certain point in a third ball, this point being on the line leading to the centre of the pocket. Thus, by means of ball 2, ball 3 may be holed, though, with a view to a possible leave for the adversary, the stroke is too risky to be recommended for general use.
Diagram V.
Fig. 1
It should be noticed that a pocket is considerably enlarged so to speak, when there is a ball in the position shown by fig. 2, as from almost anywhere to the right of the diagonal drawn through that pocket a ball may be holed off ball 3, either directly or off cushion 2, or it may be put in without touching ball 3, which will then be left for the next stroke.
Fig. 3
2
Fig. 3 shows a neat stroke. Balls 2 and 3 are touching. The line passing through their centres is at right angles to the line drawn from ball 2 to the centre of the pocket. Then from any point below P Q, and even from some distance above it, a winning hazard on ball 2 is with ordinary care a certainty.
Fig.
In fig. 4 a useful but rare stroke is shown. The two balls are touching or nearly touching, but are not aligned on the pocket. By playing a push shot, quite quietly, the point of the cue, never quitting ball 1, gradually directs ball 2 towards the pocket. The cue should be directed as much above P as the pocket is below it. A stroke with left-hand side will have the same effect, but to enter into reasons would be to open up the whole question of push strokes.
Diagram VI.
4
In Diagram VII. stroke A suggests a method of making a winning hazard which, though in itself easy, may be dangerous when the player is in a cramped position. Ball 2 is close to the pocket, and ball 1 is in a straight line with it, but so hampered by the cushion that a stab shot is out of the question. The hazard may easily be made by playing off cushion 6, as shown, and ball 1 may be left in the direction of the spot. This type of stroke, by the way, is capable of much development and should be studied. The strokes marked B1, B2, and B3 suggest three methods of play in case ball 1 should be angled, ball 2 being in the jaws of the middle pocket. Fortunately such an occurrence is rare; but I once saw it happen at pool, and the player— a very good one—played the stroke marked B2 and brought it off.
Experiment gives the best results with B1 and the worst with B3; but which of the three should be essayed depends on the exact position of ball 2 and the chances of making a loser. The point Q is about six inches below the baulk-line, but a few trials will show the exact place. B2 is of the nature of a massé, and even if ball 1 strikes the cushion above the middle pocket, there is still a chance of success. Should ball 1 be angled for ball 2, the latter being in a corner pocket, the massé stroke is the only chance; in fact, there is a very old trick stroke, made when balls 1 and 3 are in the jaws of the corner pockets and ball 2 in the jaws of the middle, all on the same side; by a similar
Fig.
species of massé ball 1 curves round and outside ball 2 and holes ball 3.
Diagram VII.
A propos of stroke A, there is a useful method of getting position at the top of the table, if ball 1 can be struck freely. If plenty of follow is used, and ball 2 is struck nearly full, the striker’s ball will rebound towards the middle of the table and then spring forward towards the top cushion again. The stroke requires much freedom, and the explanation of it is to be found in Chapter VI., On Rotation.
Stroke C may be found useful at some time or another. The object ball is resting against the upper jaw of the middle pocket, in such a way that it is impossible to cut it in from baulk; but with a kiss the stroke is absurdly easy. By playing from the end of the Ｄ with No. 1 strength, and hitting the red about three-quarters right, the kiss will send it into the pocket and leave ball 1 in the middle of the table.
The question of occasionally giving a miss may deserve a word, but, as a matter of fact, the opportunities of playing such a stroke with profit are very rare. To begin with, the penalty is a very heavy one, and can only be afforded by a player who has a most commanding lead and whose adversary cannot dare to follow suit. With a score of, say, nine to one, when the leading player has the game in hand, he may, if he please, sacrifice a ball in the hopes of getting a break afterwards; but when the scores are nearly equal, it is clear that if it is worth A.’s while to give a miss, B. can hardly do better than follow his example.
The highest possible break at pyramids (unless the striker owes one or more balls) is, of course, fifteen; this number has frequently been taken by fine players, but the chances of finding a full complement of balls on the table and of being in a position to take advantage of the opening are very small, always presupposing that the ability to clear the table exists. In the quickest game I ever played, my adversary managed to take a ball after the opening stroke, and, gradually breaking up the pyramid, secured ten, and the last five fell to me in the next turn, so that we had but three innings between us, one of them being the break. A capital performance was once done at Cambridge by an undergraduate whose adversary broke and apparently left everything safe; however, eight balls disappeared, nearly all very difficult strokes, in which the player had to consider safety as well as the hazard. In his next turn he cleared the table by a series of similar shots, all, or nearly all, so difficult that once more safety was his main object. This was a very great feat; but the reader need hardly be reminded that at pyramids, as at billiards, the art to be cultivated is the art of leaving a series of easy strokes. I once saw a man who had just made a break of 30 or 40 at billiards turn round and say, ‘What a good break! There wasn’t a single easy stroke in it!’ The real billiard-player would have described it as a series of wellmade strokes, but as a break, never.
SHELL-OUT
This game is practically the same as pyramids, but more than two take part in it. A stake, so much a ball, is agreed upon. The balls are set up as at pyramids, but under no circumstances is a ball ever put up after a miss, or when a ball has been forced off the table, or when the white has run in. If any of these things has happened, a point is added to every one’s score except the offender’s, who thus pays the stake to the other players. The score is most conveniently kept on the slate, each ball counting one, except the last, which is generally reckoned as two; all penalties incurred off it are also double, but with the exaction of the penalty the game is at an end. There is no single pool, white always playing upon red. If a player plays out of turn, he has to pay all round; he can gain nothing if he takes a ball, but the ball is not put up. If he makes a foul when only one red ball is left and takes that ball, he cannot score, but the game is at an end. The rules for foul strokes are the same as at pyramids. At the end of the game each player pays or receives the differences. Thus, if the scores stand A. 9, B. 3, C. 4, A. receives 6 from B. and 5 from C., while C. receives 1 from B.
Everything that has been said about pyramids applies equally to shell-out; but as it is a game of all against all, safety is not so much an object as hazard striking, it being clearly useless for A. to leave B. safe so that C. may profit by it. It may also be remembered that if, say, four are playing, the individual has practically a bet of three to one about each stroke. The game, in fact, is more for amusement than for scientific play, though naturally the scientific player will in the long run get more pleasure and profit out of it.
POOL
Pool, the good old-fashioned following pool, is getting out of date. The more racy games of black pool and snooker have jostled it from its place in men’s affections, so once more Cronos has been deposed, and Zeus reigns in his stead. As, however, no article on winning hazard games would be complete without a detailed reference to it, if only because of its antiquity, I will treat it as still instinct with life and energy; and, indeed, as the parent of the more modern games, it deserves our respect. The principles of the game are quite simple.
Each player receives a ball by lot—any number up to twelve can play—and starts with three lives. The order of play is decided by the sequence of the colours on the marking-board, which correspond with the balls. White is placed on the billiard spot and red plays the first stroke from the Ｄ; yellow plays on red, green on yellow and so on, the same order being kept throughout the game, unless an intermediate ball is dead (i.e. has lost all its lives), when the ball that precedes it becomes the object ball; e.g. if red is dead, yellow plays on white, or if both red and yellow are dead, green plays on white. If a player clears the table—i.e. takes all the balls on it—his ball is spotted.
The most important part of the game is to hole the object ball. When this is done, its owner loses a life, and the striker continues his break by playing at the nearest ball, and thus the game proceeds till only one player or two are left. In the first event the survivor takes the whole pool, which is called a maiden pool if he has his three lives intact; in the second the two players play on till one kills the other, or till each has an equal number of lives, in which case the pool is divided. If, however, there are three balls on the table, say red with two lives, yellow with one life, and green with two lives, then if green holes yellow, he has a shot at red, though the number of their lives is equal. This is obviously fair, as he is in the middle of his break. Green, then, has this stroke called stroke or division; if he holes red, the pool is played out to an end—i.e. till both are one-lifers or till green kills red; but if green’s stroke fails, the pool is divided at once.
Each player pays the amount of his pool to the marker, this being generally three times the value of a life, though, if lives are only sixpence, the pool should be two shillings. The table-money, generally threepence a ball, and the same for each star, is taken out of the pool before it is given to the winner. A player who is unfortunate or unskilful enough to lose his three lives early can come into the game again by paying the amount of the pool a second time over. This is called starring, as a star is put against his colour on the marking-board, but he only receives the lowest number of lives shown on the board.
As the striker is compelled to play at a particular ball, he is allowed to have any ball or balls taken up (to be replaced after the balls have ceased rolling) which are nearer than the object ball and prevent him hitting either side of it, or which in any way interfere with his stroke (see Rules 20 and 21).[24] No star is allowed when only two players are left in, but when more than eight are playing a second star is permitted. Sometimes the game is played with an unlimited number of stars, each costing the amount of the pool over and above the price of the last star—e.g. at three-shilling pool the first star costs three, the second six, and the third nine shillings, and so on (see Rules 8 and 9). The striker loses a life (Rule 13) if he holes his own ball, whether he takes the object ball or not; forces it off the table; misses; runs a coup; plays out of turn; hits a wrong ball first; or plays with the wrong ball, except when he is in hand, in which case there is no penalty. All penalties are paid to the owner of the proper object ball, however incurred. If a player wishes to have a ball up, he should not lift it himself, as such an act would be technically a foul stroke. Should a player be angled, and wish to play off the cushion, he may have any ball or balls taken up which interfere with his aim; but the old rules allowed him to move his ball just far enough to get his stroke, though he could not take a life by it.
Rule 28 says: ‘Should a player be misinformed by the marker, he may play the stroke over again, but cannot take a life.’ This seems hard, and is perhaps an instance of summum ius, summa injuria; but the moral for the player is that he must keep his attention fixed on the game and the marking-board. To attempt to replace the balls, and to allow the stroke to be played over again might give rise to much unpleasantness. By Rule 30, ‘Should the striker miss the ball
played at, no one is allowed to stop the ball, the striker having no option.’ The striker’s ball after missing the object ball may still hit others, and materially affect the subsequent progress of the game; hence a hard-and-fast law on the subject is necessary.
As it is to the survivor or survivors that the pool eventually comes, it is of paramount importance to cling to life. Many things combine to decide the striker whether to try for safety or for a hazard, or for both together—his own temperament and skill, the state of the score, and the position and skill of the next player and his other opponents. All hands are against every man, so that general rules are impossible, but ‘When in doubt play safety’ is a capital rule. Another useful maxim is, ‘Play for safety with safety-players, play for hazards with hazard strikers,’ as the latter, if they play out boldly, are sure to sell the safety-player sooner or later—i.e. will leave his ball in a position of great danger. But if the next player is close to the cushion, one is justified in playing for a hazard when, under other conditions, safety would be the right game. Again, a bolder style is right in a big pool of, say, eight or ten players, as the chances of being sold are greater. More boldness, too, in playing out for hazards may be shown by a player who is left in with, say, three lives, while the others have only one; but even then it is better to be cautious.
It is generally good policy to star two; many things may happen before it is star’s turn to play again, and he has the great advantage of playing from the Ｄ. A glance should also be taken at the position of the other balls, as the next player may have an easy stroke to play, and every life taken is in star’s favour with a view to the pool. A deliberate miss or coup, so as to be enabled to star well, is not chivalrous, perhaps not fair, but there is nothing unsportsmanlike in playing a more open game with a view to starring. With an unlimited star the question is reduced to one of capital and temperament; but in any case starring is an expensive luxury, and the player who is not judicious may find in himself a parallel to F. C. Burnand’s heroine of suicidal tendencies, of whom he wittily writes the epitaph, ‘In memory of Itti Duffa, the ill-starred maid, who lost her one life in this pool.’
One must play for one’s own hand, regardless of the other players, and undeterred by chaff or sneer from trying for a plant or cannon; but it is generally dangerous to play for a cannon unless very easy, as
there is always a chance of the player’s ball following the other into a pocket; but it is no more bad form to try for such a stroke than to pot the white at billiards; whether it is expedient or not, is another question, which only the exigencies of the moment can decide.
The marker is the proper person to measure distances when necessary, but the beginner should learn the right way to do so. One player puts his finger firmly on the striker’s ball, while another gently slides the butt of his cue up to it, holding the other end between forefinger and thumb, thumb uppermost. The point of the cue is then lowered till the cue rests lightly on the top of the other ball, the forefinger being slid up till it just touches the ball. A similar process is gone through with the third ball, the striker’s being held steady the while, and the question of which is the nearer is solved at once. Again, it is the marker’s duty to tell the striker on which ball he has to play, and which ball plays next, the formula being, e.g. ‘Green on yellow, player brown,’ or, if brown is in hand, ‘Green on yellow, player brown in hand,’ and if the striker has to play on his player, the marker must inform him of the fact; as, however, the striker is the scapegoat even if he acts on wrong information, he should keep his attention fixed on the game.
The opening of a pool is more or less stereotyped, all the players endeavouring to lay themselves under the top cushion out of harm’s way, the player being always in hand till white’s turn comes round; thus the last player—brown, let us say, in a five-pool—has to steer himself round the other balls that are clustered at the head of the table, and find his way down to baulk, as white is nearly sure to be high up; in a big pool the last player may have some difficulty, and it is well to remember that, as he can have any ball or balls up that lie between him and the object ball, he can, by selecting a good spot from baulk, have one or two such obstacles removed. The orthodox opening shot for red, by the way, is to play full on to white from a corner of the Ｄ, just hard enough to find the cushion himself. Plenty of drag should be used and no side. Side and screw are of no value except for position, or for playing a slow stroke which is wanted to travel quickly off the cushion.
The late William Cook once made a pool record. ‘Playing in a twelve following pool at his own rooms’ (I quote the words of a fine amateur player who took part in the game), ‘in 1881, he actually
cleared the table, playing always of course on the nearest ball. He had taken 20 to 1 five or six times from spectators, and the excitement was intense when he performed this really phenomenal feat.’ As pool is limited to 12, Cook, like Alexander, had no more worlds to conquer; but his hazard striking and position must have been marvellous.
Doubles are of the utmost importance, and the strokes shown in the preceding diagrams should be noted. One may fairly play a middle-pocket double with extra strength if by so doing there is a chance of the double-double, though it is not strictly sound, and shows a certain diffidence as to one’s accuracy. Plants are rare.
A propos of doubles, the following occurrence is probably without precedent, but the story is absolutely vouched for. Three amateurs were playing three-pool. Red opened by doubling white into the right-hand bottom pocket. Yellow avenged white by doing exactly the same to red, and white made matters even by treating yellow to a precisely identical shot. Strange to say, red with his second shot holed white just as before—four consecutive doubles into the same pocket—and, though yellow spoiled the average by only doubling red into the right-hand middle pocket, white made things all square and yellow disappeared into the original pocket. Thus six consecutive doubles were made, five of them into one pocket! What are the odds against such a performance?
As even in these enlightened days the confidence trick flourishes, it may be worth while to warn beginners against innocent strangers. If these win by sheer skill, there is nothing to be said against them, and the best thing is to put down one’s cue; if they are sharps as well, they will probably hunt in couples, on the chance of one playing next to the other, when the first player, curiously enough, never quite gets safety and always leaves a ball over the pocket. I remember just such a pair, a good player and a duffer, turning up at some rooms I used to frequent, and, though none of us were innocents, they played so cleverly into each other’s hands, the apparent duffer making several slight mistakes at critical moments, that the good player had a pretty good time. Talking the matter over, we saw that we had been had, and, as we were rather a snug little coterie, arranged with the marker what was to be done if they reappeared. The pair had posed as absolute strangers and had come in separately, so we told the marker
that if the duffer came in first he was to have a ball and we would try to warm him up, but on the good player’s appearance he was to be refused a ball, while we, a fairly sturdy lot, would see the marker through any trouble. All came off splendidly. The duffer appeared first and lost two or three pools, and when the crack walked in he was at once confronted by the marker with ‘I am very sorry, sir, but I can’t give you a ball to-day.’ We expected a row, but he took it like a lamb and decamped, and the duffer, after losing another pool or two, decamped also. One of our party saw them the same evening, hobnobbing together at the Criterion, so there can be no doubt that it was a put-up job. The following occurred to a friend of mine, a good billiard-player, a particularly good pyramid-player, and well able to look after himself. After a couple of games of billiards, on both of which he won a small bet, his opponent, a stranger and apparently a Jew, suggested ‘just one game of pyramids.’ ‘What shall we play for?’ said my friend. ‘Three and one,’ said the Semitic one, which means, as usually interpreted, a shilling a ball and three shillings on the game. My friend won by thirteen balls, but his opponent, after putting up his cue, offered him just four shillings and threepence, being at the rate of three pence a ball and one shilling a game! There was nothing to be done, but I wonder what the Jew would have claimed had he won by thirteen balls.
