Calculus 8th Edition

Book Preface

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.
● 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, 11.2.5, and 14.3.3, for  instance). And the solutions to some of the existing examples have been amplified.
● Three 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. In the project The Speedo  LZR Racer (page 976) it is explained that this suit reduces drag in the water and, as  a result, many swimming records were broken. Students are asked why a small  decrease in drag can have a big effect on performance.
● I have streamlined Chapter 15 (Multiple Integrals) by combining the first two sections so that iterated integrals are treated earlier.
● 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, 12.5.81, 12.6.29–30, 14.6.65–66.
In addition, there are some good new Problems Plus. (See Problems 10–12 on  page 201, Problem 10 on page 290, Problems 14–15 on pages 353–54, and Problem 8 on page 1026.)  May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learningexperience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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, 11.2, 14.2, and 14.3.) 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, 11.10.2, 13.2.1–2, 13.3.33–39, 14.1.1–2, 14.1.32–38,  14.1.41–44, 14.3.3–10, 14.6.1–2, 14.7.3–4, 15.1.6–8, 16.1.11–18, 16.2.17–18, and  16.3.1–2). 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 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 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). 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 14.1.2). Partial derivatives are introduced  in Section 14.3 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 14.4.3).  Directional derivatives are introduced in Section 14.6 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 15.1.9). Vector fields are introduced in Section 16.1 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. 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.) The project after  Section 14.8 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  May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
velocity. Laboratory Projects involve technology; the one following Section 10.2 shows  how to use Bézier curves to design shapes that represent letters for a laser printer. 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). Others explore  aspects of geometry: tetrahedra (after Section 12.4), hyperspheres (after Section 15.6),  and intersections of three cylinders (after Section 15.7). Additional projects can be found  in the Instructor’s Guide (see, for instance, Group Exercise 4.1: Position from Samples).

Problem Solving
Students usually have 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 Exponential and Logarithmic Functions
There are two possible ways of treating the exponential and logarithmic functions and  each method has its passionate advocates. Because one often 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

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