Introduction

Since 1970, the content, teaching, and learning of calculus have been affected by the increasing importance of mathematics in the sciences; the availability of powerful, inexpensive computing technologies; and the needs of the more diverse students entering calculus. These developments make teaching calculus an exciting challenge. This book, written for students majoring in science, engineering, or the mathematical sciences, is our response to this challenge.

With the help of grants from the National Science Foundation, we have written a text intended to help these students learn calculus more effectively so that they are better prepared for subsequent technical courses and possess more experience in solving realistic, multistep problems. Our intent is to do this within a fairly traditional syllabus.

We have provided for a more effective learning and problem-solving experience in several ways. Among the skills of good problem solvers is flexibility, including the use of numerical, graphical, and symbolic representations of a problem or its solution. We often use such representations and ask students to model our practice in their work.

In the first three chapters we discuss approximation and error, trigonometric and exponential functions, and vectors and parametric equations, all of which are important in the engineering, mathematical, and physical sciences. We return to these ideas in subsequent chapters, thus giving students repeated practice with important concepts. We have included as many realistic problems as possible, both in section exercises and in the student projects at the end of each chapter.

Approach

Our text was written for a three-semester sequence, with a provision for "early vectors."
Why Early Vectors?
We have provided an option for early study of vectors and parametric equations in the plane for several reasons. Students taking engineering physics during their first two semesters often experience positive feedback from topics common to their calculus and physics courses. In addition, parametric equations provide a uniform framework for the study of many important topics in calculus including polar coordinates, arc length, surface area, and graphing. Throughout our text we often motivate new topics by using the intuitive content of motion, which leads naturally to parametric equations and vectors. By considering only planar vectors in Calculus I, students get a chance to understand vectors, parametric equations, and motion in the more familiar two-dimensional setting before they must deal with the additional complications of working in three dimensions. In addition, by introducing vectors early, students have more time to become familiar with vectors, parametric equations, and parametric surfaces before studying the Divergence Theorem and Stokes' Theorem towards the end of Calculus III. Finally, students in some engineering curricula require only one year of calculus. In recent years some engineering faculty have expressed a desire that these students get some exposure to multivariable calculus during this year. Exposure to vectors in the first semester sets the stage for some multivariable calculus during the second semester.

Early Transcendentals. We have followed the trend toward the early introduction of the transcendental functions and their derivatives. This makes possible more practice with these functions and more realistic examples and exercises. Along with many other contemporary calculus books, in this text we now complete the study (basic properties and differentiation) of trigonometric functions, exponentials, logarithms, and inverse trigonometric functions in the first semester.

Linear Approximation. We have used the idea of linear approximation as a unifying theme throughout the book. The idea of linear approximation lies behind many concepts in calculus and can be part of a framework on which a student can build conceptual understanding of calculus. We introduce the idea of linear approximation when we zoom-in on the graphs of functions to find the rate of change at a point. The linear approximation theme continues in various applications including Newton's method and Euler's method. In later chapters we generalize linear approximation to higher-order approximations through Taylor polynomials and lay the groundwork for multivariable approximations by studying linear functions in two and three dimensions. We use linear approximations for functions of several variables to develop many of the ideas of multivariable calculus, including differentiability, the chain rule, and the change-of-variables formula for multiple integrals.

Flexibility for a Variety of Courses

The following flowcharts outline several ways of using our text. The sequence labeled "EV" addresses a growing demand by engineering and physical science programs for exposure to vectors and some topics from multivariable calculus during the first year. Even more multivariable material can be included in the second semester by moving Chapter 7 (Infinite Series, Sequences, and Approximations) to the end of the third semester and including Chapter 9 (Functions of Several Variables) in the second semester.

The majority of courses will likely follow the standard sequence, designated "S" in the chart. In this sequence we defer work with vectors until Calculus II. To do this we have written Chapters 4 and 5 so that they are independent of vector material. Section 3.1 (Motion along a Line) is included in Calculus I. The remaining sections of Chapter 3, indicated by 3* in the chart, can be covered in Calculus II after Chapter 7 (Infinite Series, Sequences, and Approximations). We have written Chapters 4 (Applications of the Derivative) and 5 (The Integral) so that they are independent of the vector material in Chapter 3. In Chapter 6 (Applications of the Integral), Sections 6.1 (Volumes by Cross Section), 6.2 (Volumes by Shells), 6.3 (Polar Coordinates and Parametric Equations), 6.4 (Arc Length and Unit Tangent Vectors), 6.5 (Areas of Regions Described by Polar Equations), and 6.9 (Improper Integrals) can be covered before the vector material in Chapter 3. In Section 6.4 we use parametric representations of planar curves; we briefly review this topic in Section 6.3. Some applications in Section 6.6 (Work), 6.7 (Center of Mass), and 6.8 (Curvature, Acceleration, and Kepler's Second Law) depend upon material in Sections 3.2, 3.3, 3.4, and 3.5.

For transferring or entering students with credit for a full year of calculus, we have sequence "M" (for multivariable). Such a class can be taught from Calculus, Part II (ISBN 0-321-02046-4), a version of this text that contains the multivariable material of Chapters 8 - 12 and Chapter R, which contains some review of polar coordinates and most of the material on vectors from Calculus I and II. Depending on the backgrounds and needs of the students, the instructor can pick and choose as needed from Chapters R, 8, and 9 to fill in the material that is needed to study differentiability, multiple integrals, and vector function in Chapters 10, 11, and 12.

We also offer a version of the text for those students taking only one year of calculus, starting with Calculus I. It contains Chapters 1 - 8 of the full text.

Features

To better prepare for advanced courses in mathematics or physics or courses in their majors, students need to work actively at solving problems, trying examples, and investigating troublesome points through graphical, numerical, and analytic representations. Toward these ends we have provided a number of features to help students explore, learn, and understand calculus.

Technology
We assume that each student has convenient access to at least a graphics calculator. We feel that such calculators, as well as computer algebra systems, can be a valuable teaching and learning tool. One goal of the text is for students to become proficient at investigating problems using an appropriate combination of graphical, numerical, or symbolic modes. We have tried to model calculator and computer usage in the text and to provide students with opportunities to practice solving problems with the aid of a calculator or computer. Throughout the text, exercises requiring substantial calculator or computer algebra system (CAS) use are identified by the icon . We comment further on exercises with and without this icon in the section labeled "Exercises", which follows shortly.

Precalculus Review
Students enter calculus with a wide variety of backgrounds in precalculus, and even well-prepared students may find their skills a little rusty after a few months without practice. Thus many students may need, and even welcome, some review of precalculus material. On the other hand, calculus is a new and exciting subject, so it is important to use and enhance this excitement from the first day of class. We have tried to provide for needed review while still bringing out new ideas or looking at old material in fresh ways. In Sections 1.1 and 1.2 we review the definition of function and operations on functions. In these sections we stress ways of representing functions (numerically, graphically, and symbolically) and, for each representation, how operations on functions can be interpreted. In studying composition of functions we work with graphics calculators and see that such calculators do not always perform functions composition correctly. This illustrates early the need to look at a problem in several ways and to think critically before trusting output from a calculator or CAS. Other review is built into the development of the derivative. In Chapters 1 and 2 many examples and exercises are designed not only to illustrate new concepts, but also to give students practice with algebraic and trigonometric skills. Finally, we have provided for more review in the Appendix. Here we have summarized and, in some cases derived results from, algebra, geometry, and trigonometry that are useful in calculus.

Graphs and Tables
When every student has access to a graphics calculator or computer algebra system, the figures and tables in a text can play a larger role. To encourage more active reading, we often urge students to verify table entries or to reproduce graphs that accompany investigations and examples. In most cases, students can obtain qualitatively similar graphs with a graphics calculator.

Differential Equations
We tell students that calculus is the language of science and engineering. This statement is very well illustrated by the wide range of physical and biological phenomena that are modeled with differential equations. At the end of Chapter 2 we introduce differential equations. Our objective here is not to learn to solve differential equations, but rather to begin to understand what is meant by a solution to a differential equation and to get an idea of how to interpret and build equations that describe changing situations. As part of Section 3.6 on Newton's laws and as a prelude to the Fundamental Theorem of Calculus in Chapter 5, we use simple antidifferentiation to infer the position and velocity of objects in motion from Newton's second law. In an optional section of Chapter 4 we discuss Euler's method for solving differential equations. In Chapter 5 we discuss separable equations and how to solve them by integration. In Chapter 12 we touch on exact equations as a by-product of our work with potential functions and the Fundamental Theorem of Line Integrals.

Exercises The problem sets at the end of each section contain a variety of exercises. Many of the problems reflect skills or problem-solving techniques encountered in the section. To encourage close reading of the examples, a few of these problems ask students to fill in some details in an example or to rework an example with different numerical data. Every exercise set also contains problems whose solution method is not covered in an example. In these problems students may be required to work a little beyond the material discussed in the text or to think about and use the text concepts in ways not illustrated in the examples. All of these problems can be solved using skills the student should already have mastered. Students find problems of this sort challenging and sometimes frustrating. However, one of the best ways for them to develop good problem-solving skills is to practice on problems that encourage them to think in new directions. For those who use calculus, it is also important to communicate ideas and conclusions. Thus, in some problems, students are asked to write a sentence or two explaining their result in words, as in Exercise 22 on page 77 or Exercise 32 on page 201. In others, we ask students to look at several examples illustrating a situation, then write a paragraph describing a conclusion that might be conjectured based on these examples. For example, see Exercises 38 and 39 on page 19 and Exercise 31 on page 528.

Most of the exercises in our text can be done by hand. At the same time, we have assumed that each student has convenient access to at least a graphics calculator. We expect that students will sometimes use their calculator or CAS in solving exercises involving routine graphing of functions or in evaluating a function or expression. We have not marked such exercises as requiring technology. We have marked with the icon exercises requiring something beyond routine graphing or function evaluation. We have, for example, marked with exercises involving graphical determination of error for an approximation or graphs of surfaces described parametrically. As students progress through the text they should become better at recognizing when the use of technology is appropriate and when it is not. Sound judgments of this sort are an important part of problem solving in today's engineering and science fields.

Investigations
We often use the context of an "investigation" to introduce a topic using physical, numerical, graphical, or symbolic ideas. These investigations introduce students to the main ideas in a topic in advance of the formal definitions and development. Following each investigation, we use examples to show how one may solve the more or less standard kinds of problems associated with the topic. For example see the Investigations on pages 543 - 544 (Section 7.1, Taylor Polynomials) or page 445 (Section 6.1, Volumes by Cross Section).

Chapter-End Materials
Each chapter ends with review materials, beginning with a brief summary of the chapter highlights. Following these highlights is a review grid that summarizes many of these ideas with a graph, a formula, and a short example. Following the review grid are chapter review exercises that draw on material from the entire chapter.

Student Projects. This book grew from an NSF grant to write projects designed to give students significant experience in solving multistep problems, writing, using appropriate computing technology, and performing group work. These projects have been class-tested extensively over a period of years. Several of these projects are included at the end of each chapter. For example, see pages 189 and 537. Maple worksheets, Mathematica notebooks, and MATLAB M-files for these projects are available on the Web site located at www.aw.com/johnston .

Chapter Summaries

Chapter 1 Rates of Change, Limits, and the Derivative
In this chapter we lay the groundwork for our study of calculus and its applications. The ideas described in this chapter - functions, limits, and rates of change - are the foundation for the rest of the text. We begin by studying functions and how they can be presented or interpreted in symbolic, graphical, or tabular form. We then introduce the rate of change as the slope of the "line" we see when we zoom in on the graph of a function. As we modify this to get a symbolic definition of rate of change, we are led to define and investigate limits. Once we have limits, we refine our definition of rate of change and obtain the definition of the derivative.

Chapter 2 Finding the Derivative
In this chapter we approach the derivative more mechanically. Our goal is to develop quick, efficient procedures for producing the derivative of a function. Students are encouraged to recreate heuristic arguments in support of differentiation rules (product, quotient, chain rule). We also work with trigonometric functions and the unit circle and use this context to find the derivative of the sine function. We discuss the special role of the number e as the base of the exponential function and are then led to the natural logarithm. We then discuss inverse functions in more generality and investigate the notion of restriction of domain to obtain a one-to-one function for purpose of defining an inverse. With this we can then define the inverse sine, tangent, and cosine functions and discuss their derivatives. We conclude this chapter with a section on modeling, in which we give examples of some of the diverse and interesting situations that can be described by using derivatives.

Chapter 3 Motion, Vectors, and Parametric Equations
We work towards an understanding of vectors through the modeling of forces, velocities, and displacements. We model motion on lines and circles using vector/parametric equations and relate these motions to Newton's laws. The chapter includes vector arithmetic and geometric interpretations, including dot products and the projection of a vector in a given direction; the calculation of the position, velocity, and speed of an object; tangent vectors and slope for curves described by parametric equations; and elementary antidifferentiation in connection with Newton's second law.

Chapter 4 Applications of the Derivative
We study such applications of the derivative as the approximation, Newton's method for solving equations like f(x)=0, implicit differentiation, the analysis of the graph of a function by classifying it as increasing or decreasing or concave up or concave down, optimizing a process by locating the maximum or minimum of a function, the calculation of certain limits by applying l'Hôpital's Rule, and Euler's method, the simplest numerical method for solving a differential equation.

Chapter 5 The Integral
The goal of the chapter is for students to understand the integral as the limit of a Riemann sum associated with geometric or physical objects. The Fundamental Theorem of Calculus is discussed through a combination of position/velocity and area approaches. The integral is used to calculate simple areas and solve simple separable differential equations. We study the usual techniques of integration (substitution, integration by parts, and integration of rational functions through partial fractions) using a combination of symbolic calculation and tables. We discuss the trapezoid, midpoint, and Simpson's rules for numerical integration.

Chapter 6 Applications of the Integral
The integral is used to model and calculate the volumes of solids, lengths of curves, work done by force fields on objects, and masses and the centers of mass of objects. We also study curvature, the tangential and normal acceleration vectors of moving objects, and the classification and evaluation of "improper integrals." A major theme in this chapter is the use of "elements" of area, volume, arc length, work, and mass. Such elements increase understanding and decrease the need for memorized formulas by making direct use of underlying geometric concepts or physical principles. Applications involving vectors, which can be postponed or skipped if vectors are to be covered later, are as follows: Work (Section 6.6), Center of Mass (Section 6.7), and Curvature, Acceleration, and Kepler's Second Law (Section 6.8).

Chapter 7 Infinite Series, Sequences, and Approximations
In an effort to find approximations that do a better job than the tangent line approximation we are led to the Taylor polynomials. To better understand such approximations, we next study sequences and infinite series. We examine the dynamics of increasing sequences and the possible behaviors of such sequences. This in turn leads to results about the behavior of infinite series, including the comparison, integral, and ratio tests. This study of series leads us to a power series, and then back to Taylor polynomials and Taylor series.

Chapter 8 Vectors and Linear Functions
Chapter 8 lays the groundwork for the study of functions of several variables. Most of the ideas in this chapter are generalizations to three dimensions of ideas developed when we studied functions of one variable and vectors and motion in two dimensions. These ideas include vectors, planes, and motion in three dimensions. We also look at several simple but important ideas concerning linear functions in two and three dimensions. These ideas allow us to continue our theme of linear approximation and are the foundation of later results about differentiability of functions of several variables, the chain rule, and change of variables in multiple integrals.

Chapter 9 Functions of Several Variables
We start with a discussion of conic sections, because a sound understanding of graphs of conics is needed to understand many surface graphs. We then develop techniques to help students understand and visualize functions of several variables. Once these techniques are in place, we define and study rates of change for functions of several variables. Partial derivatives and the gradient vector are introduced, two fundamental concepts that are used in the remainder of this text.

Chapter 10 Differentiable Functions of Several Variables
In this chapter, we extend the ideas of differentiability, linear approximation, and the chain rule to higher dimensions by using the idea of a linear function and its associated matrix (see Chapter 8). The conceptual and notational connections with single variable calculus are emphasized. The chain rule is applied to express partial differential equations in new coordinate systems. Optimization is studied with and without constraints.

Chapter 11 Multiple Integrals
The goals of the chapter are for students to understand double and triple integrals as the limit of a Riemann sum associated with geometric or physical objects and to understand the process of evaluating multiple integrals through reduction to several single-dimensional integrals. Multiple integrals are used to calculate volumes of three-dimensional regions, areas of surfaces, masses, and centers of mass. We discuss the change of variable formula for multiple integrals with an emphasis on a change from rectangular to cylindrical or spherical coordinates. A major theme in this chapter is the use of "elements" of volume, surface area, and mass. Such elements increase understanding and decrease the need for memorized formulas by making direct use of underlying geometric concepts or physical principles.

Chapter 12 Line and Surface Integrals
In this chapter we study functions defined on curves or surfaces in space. We start by extending the definition of definite integral to real- and vector-valued functions defined on curves. We use integrals to calculate the mass of a wire and the work done by a force, and to study the flow of a fluid in the plane. In the second half of the chapter, we define the integral of real- and vector-valued functions defined on surfaces. With these integrals we study such phenomena as three-dimensional fluid flow and heat flux. Many of the integrals introduced in this chapter can be more easily evaluated by converting them to single, double, or triple integrals. To do this, we apply the Fundamental Theorem of Line Integrals, Green's Theorem, the Divergence Theorem, and Stokes' Theorem.

Acknowledgments

Many friends, students, and colleagues were of great help to us in writing this textbook. We were encouraged and guided by our own calculus students at Iowa State University as we tested these materials in our classrooms. The students were always willing to tell us what was working, what was not, and how things might be improved. Many of our colleagues at Iowa State University also contributed by class-testing some materials and refining many of our student projects. They include Roger Alexander, Clifford Bergman, Brian Cain, A. M. Fink, Alan Heckenbach, Irwin Hentzel, Fritz Keinert, Justin Peters, Richard Tondra, and Bruce Wagner. In addition, we'd like to express our sincere thanks to our many other colleagues and students.

During the summers of 1992, 1993, and 1996 we offered three workshops in calculus reform. The 1992 workshop was at the University of Wisconsin at La Crosse, the 1993 workshop was given at Idaho State University in Pocatello, and the 1996 workshop was at Iowa State University in Ames, Iowa. Although we went there to present our ideas and to demonstrate some of our student projects, we learned a lot by listening to participants' ideas on the directions calculus should take. We thank all of the workshop participants for their insights and ideas.

The University of Wisconsin at La Crosse Workshop:
Carrie Ash-Mott, John Bruha, Robert Coffman, Wesley Day, Ronald Dettmers, G. S. Gill, David Hardy, Marian Harty, Linda H. Host, Erna Jensen, Clement Jeske, Charles Kolsrud, Larry Krajewski, Robert Kreczner, Don Leake, Steve Leth, Franklyn Lightfoot, Rich Maresh, Dan Nicol, Gerald W. Niedfeldt, David Oakland, Marlene Pinzka, Kay Strangman, Gordon Sundberg, Jack Unbehaun, Calvin Van Niewaal, Paul Williams, Randy Wills, J. D. Wine, and Elizabeth Wood.

The Idaho State University at Pocatello Workshop:
Sam Berney, Janet Burgoyne, John C. Eilers, Chaitan Gupta, Joseph Hwang, Jim Brennan, Bob Davis, Bob Firman, Roger Higdem, Ken Meerdink, Tom Misseldine, Eric Rowley, Madeline Schaal, Peter Wildman, Mike Prophet, Dan Schaal, Don Shimamoto, Suzanne Wisner, and Larry Ford.
The Iowa State University at Ames Workshop:
Carrie Ash-Mott, Joel Bundt, Frank Carver, Wesley R. Day, Marlin Deweerdt, Loren Flater, Dan Fuchs, Peg Griffey, Ruth A. Hartman, Robert Hoile, Jenelle Jarnagin, Jim Jefson, Steven D. Klassen, David Kofoed, Virgil E. Larsen, Sergio Loch, Wanda Long, Herbert C. Lyon, Ronald M. Mathson, Cheryl Ooten, Constantin Pirvulescu, Jerry Schmidt, Virginia Swenson, Melvern K. Taylor, and James W. Van Ark.
We are grateful to the instructors who have class tested and reviewed drafts of this book. Comments from these reviewers were invaluable in suggesting ways to improve our manuscript. Our sincere thanks to each of them including those listed here:
Gregory T. Adams, Bucknell University
James C. Alexander, Case Western Reserve University
Johan G. F. Belinfante, Georgia Institute of Technology
Mark Bridger, Northeastern University
Donatella Danielli, Johns Hopkins University
John M. Erdman, Portland State University
Earl Hamilton, North Seattle Community College
Weimin Han, University of Iowa
Phillip Johnson, University of North Carolina, Charlotte
William J. Keane, Boston College
Joseph D. Lakey, New Mexico State University
Ronald M. Mathsen, North Dakota State University
Joe Miles, University of Illinois, Urbana-Champaign
Andrew Nestler, Santa Monica College
Scott Pauls, Dartmouth College
Roy Rakestraw, Oral Roberts University
Lynn M. Siedenstrang, Grays Harbor College
William L. Siegmann, Rensselaer Polytechnic Institute
Jennifer Slimowitz, Rice University
Harvey E. Wolff, University of Toledo

Many other people have been very helpful in finding materials on which to base problems and discussions. These include Susan North of the Ames Public Library, the reference staff at the Iowa State University Library, the reference staff at the Alaska State Library, the U.S. Geological Survey, and Michael Graff of the Iowa State University Computation Center.

Many people were involved in ensuring the accuracy of this text. We wish to express our thanks to those people who helped accuracy-check this text, which went through not one or two, but three rounds of checking: Jon Booze, Dr. Lyle Cochran, Dr. Tim Mogill, Patricia Nelson, Jeffrey D. Oldham, Daniel Pick, Jeff Suzuki, Daniel P. Thompson, Marie Vanisko, Stephen Whalen, Dr. Yuri Zhorov, Dr. Holly Zullo, Dr. Mark Parker, Becky Cointin, and Tom Wegleitner.

We also wish to express our thanks to Laurel Technical Services for writing the Instructor's and Student's Solutions Manuals. Thanks to Tom Leathrum of Jacksonville State University for his extensive contributions to our Web site.

We hope that instructors and students find this text instructive and enlightening. We invite suggestions from readers- students, professors, and others- for improvements to the text. We will do our best to incorporate appropriate suggestions in future printings. Please write to us at Addison-Wesley or contact us by electronic mail.

Elgin H. Johnston
Iowa State University
ehjohnst@iastate.edu

Jerold Mathews
Iowa State University
mathews@iastate.edu

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