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Dedication

To Jo Dee and Piki. These terrific people manage to get along with all three of us. Imagine that!

Acknowledgements

The authors, Bill Davis, Horacio Porta , and Jerry Uhl wholeheartedly thank the persons whose ideas and actions have been essential to Calculus&Mathematica.

Calculus&Mathematica students at the University of Illinois at Urbana - Champaign and The Ohio State University for leading the way on the first calculus adventure in three hundred years. Over a five year period, these students played an essential role in helping us to develop Calculus&Mathematica. They taught us that the modern student does have math passions and can really turn on to a calculus course. They taught us what will work and what won't work. They taught us how to write this course. The course benefitted greatly from their ideas about how it should be run. In fact, the Literacy Sheet component of Calculus&Mathematica owes its existence to student input. At the times when our calculus batteries needed to be recharged, all we had to do was to look at student work or to sit down and talk to students about their work while they were doing it. The way Calculus&Mathematica students throw themselves into their work is our continual inspiration.

Don Brown, coauthor of the preliminary version of Calculus&Mathematica, for giving us the confidence we needed in the initial phase of the project. As an Illinois undergraduate physics major, he joined the project at the very beginning to provide the computer know-how needed to transform the project from vaporware to reality. The original versions of the fonts and the Mathematica-to-TeX printing programs are all results of his powerfully fertile mind. The project could not have gotten off the ground without the help of this friend.

Alan DeGuzman , Justin Gallivan and David Wiltz for forming the original Calculus&Mathematica Development Team. All three of these fellows were Calculus&Mathematica students in the first full year of the project who stepped forward to share their expertise with us. They taught us how to set up and run a lab and they have run the technical phase of the project for three years. Their contribution of their intellectual energy and creativity was essential to the project. They are all fellows who will go very far in their own endeavors. When they leave we'll miss them.

Gary Binyamin , Martha Grover , David Kirzner , Corey Mutter , David Taubenheim , Jenny Welch , Thea Colwell , Ben Halperin , Lorien Ryan , and Chris Zeller for joining the Development Team. Operating under the tutilage of the original development team, they are all making their own fundamental contributions to Calculus&Mathematica and will continue to do so until they graduate.

Chintan Amin, Mikal Arneborn, Lorien Ryan, Dave Banyots, Steve Casburn, Brigitte Blaney, Karl Berghauer, and Neil Mickelson for plenty of good advice and for being exemplary Lab Instructors.

Glenn Scholebo, master typesetter, for his judgement, artistry and energy in converting the electronic course into print in both the preliminary edition and in this edition. Glenn is the main reason the printed supplement to the electronic course is so readable. It's a real pleasure to work with an authentic expert and gentleman.

Beate Zimmer for overall consulting and extraordinary proofreading. When one of us was forced to the sidelines by bad health, she stepped in to keep the project on schedule. She is our friend.

Stephen Wolfram , principal author of Mathematica , for calling us over to his office in 1988 and giving us his ideas for what he called "a calculus book in the form of Mathematica notebooks ." Although we didn't know it at the time, this was the beginning of the Calculus&Mathematica project. We thank him also for his encouragement and support in every phase of the whole project. He understood what we were trying to do before we did. We think Stephen is one heck of a guy.

Francis Sullivan , of the Institute for Defense Analysis, for his advice and direction both mathematical and non-mathematical. He encouraged us to start the project and has been a continual source of wisdom for the project. Among many other things, he helped us to understand the difference between traditional classroom calculus and honest calculus as practiced outside the traditional classroom. We think Francis is also one heck of a guy.

Bruce Carpenter for understanding Calculus&Mathematica better than the authors understand it. He is a man of incredible originality, insight, and drive; we are happy to have him on our team.

Theodore Gray for inventing the notebook front end of Mathematica , to which this course is a response.

Dan Grayson , coauthor of Mathematica , for introducing us to Mathematica and for plenty of pithy comments.

Allan Wylde, our original Addison-Wesley, Inc. publisher, editor, and visionary who went so far as to back our project with badly needed financial and personal support at a time when most everyone was skeptical about the project. The day he left Addison-Wesley, Inc. was a sorry day for us.

Allan Wylde for telling us that if it can be said on the Johnny Carson show, it can be said in Calculus&Mathematica, and Prunella Goodbody of Addison-Wesley, Inc. , for telling us that Allan was wrong, and for teaching us a completely new meaning of the word "sucker."

Jerome Grant, our current Addison-Wesley, Inc. editor, who surprised us by proving to be a math editor with brains, vision and drive. He gave us the freedom we never expected to get. We like this guy.

Richard Wilson , Associate Chancellor of the University of Illinois at Urbana - Champaign , for his unflagging support from the very beginning. Without this friend, we would have been dead in the water.

Ward Henson of the University of Illinois at Urbana - Champaign , for saving this project from an early death during his term as chair of the mathematics department at Illinois .

Joan Leitzel who, during her term as Associate Provost at The Ohio State University , gave the support necessary for establishing the early Calculus&Mathematica labs there.

The University of Illinois at Urbana - Champaign and The Ohio State University for giving us the freedom to develop, revise, and teach Calculus&&Mathematica in their classrooms.

Amy Young, member of the first-ever Calculus&Mathematica class, for her uniquely incisive advice and support from the very beginning. She is a very special friend.

All the other folks at Wolfram Research, Inc. for unwavering support of the project. We thank especially Shawn Sheridan, Glenn Scholebo, Prem Chawla, Scott May, Tom Wickham-Jones, Paul Katula, Kevin MacIsaac, Eric Blankenburg, Paul Abbott, Dara Pond, Zoe Midler, Lisa Shipley-Jones, Jane Rich, Laurie Gilmore, Jerry Keiper, Dan Lichtblau, Ben Friedman, Brad Horn, Tom Sherlock, Bo Liu, Joe Grohens, Jamie Peterson, John Cochran, Melissa Idleman, Christy Uden, Joe Kaiping, David Withoff and John Bonadies, who all did things for us that they didn't really have to do.

Andy Fisher, Karen Wernholm, Peter Blaiwas, and Sean Angus of Addison-Wesley, Inc. for going about their business with cheer and aplomb.

Tony Peressini , at Illinois , for lots of advice and support. Every project like this needs a steady source of practical wisdom; he is ours. His conversion from the traditional course to Calculus&Mathematica was a significant event for us. He is a very special friend.

Alayne Parson and Tom Ralley at The Ohio State University for their enthusiastic support for the project. They have, at times, been more optimistic than we have about the future of the course, and, in particular, in the pedagogical tenets upon which it is based.

Paul McCreary for sharing his ideas on group learning and for proving that Calculus&Mathematica can be a beneficial experience for some students who would normally be considered to be at risk in a calculus course.

Dana Scott for his belief in the project and his efforts on our behalf.

Jonathan Manton, Eric Hjelmfeldt and Soren Lundsgaard for help in the very early stages of Calculus&Mathematica at Illinois .

The National Science Foundation for grants for development.

Apple Computer, Inc. for machine support and for making two videos about the project. John Noon for convincing Apple Computer, Inc. that this project should be supported.

Ron Weissman and David Spitzler of NeXT Computer, Inc. for loaner computers and for their positive attitude about the project.

Bernard Madison for his early recognition of the merits of the project and for writing the first accurate, succinct description of Calculus&Mathematica.

Emily Peck , Associate Dean of LAS at Illinois , for her support.

George Badger , director of the University of Illinois Computer Services Office, for cooperation far beyond what we expected or deserved.

Charlie Bender , Director of the Ohio Supercomputer Center, and Director of Academic Computing at The Ohio State University , for getting the project off the ground there by suggesting the construction of a computer lab usable for Calculus&Mathematica, and for letting Holly Hirst, then his assistant, teach in the first year of the project.

Bob Dixon , Steve Gordon, Randy Jackson, Terry Reeves , and Jeff Schluep of Academic Computing Services at The Ohio State University for support, installation and systems support for the various labs which have been homes to Calculus&Mathematica.

Juan Manfredi of the University of Pittsburgh for his continuing support. When he told us he liked the project enough to teach it, we began to feel that it could become real.

Elias Saab , chair of the mathematics department at the University of Missouri - Columbia , for proving that a large-scale implementation of Calculus&Mathematica can be successful.

Russell Howell, Dennis Schneider, Juan Manfredi , Lew Lefton, Enid Steinbart, Ken Holladay, Elias Saab Dana Weston, Carruth McGehee, Neil Stoltzfus, Paul Wellin, Bill Emerson, Louis Talman, Dan Yates, Dennis Schneider, Tony Peressini , Don Sherbert, Peter Loeb , Elliott Weinberg , Bob Muncaster , Alice Iverson, Barbara Beechler, Mel Henrickson, Rob Beezer, Barry Turett, Jack Nachman, Elizabeth Covington, Robin Sanders, Jerry Rubin, Fred Andrew, and Tom Morley for giving Calculus&Mathematica an early try in their university and community college calculus courses. They proved that Calculus&Mathematica can ignite students' mathematical passions in a variety of settings.

Judy Holdener, Janice Malouf, Bruce Carpenter , Bill Hammack , Eric Hjelmfeldt, David Ose, Judy Walker, Todd Will, Heather Hulett, Tammy Hummel, Myung Chung, Gregory Michalopoulos and all the other Illinois teaching assistants and professors who were game enough to run early sections of Calculus&Mathematica as it was being developed.

Holly Hirst, Tom Ralley, Alayne Parson, Amy Lee, Cheryl Stitt, Ed Overman, Steve Giust, Jeremy Green, Marc McClure, Jim Brazelton and all the other Ohio State teaching assistants and professors who were also game enough to run early sections of Calculus&Mathematica as it was being developed.

Jeff Hirst for sharing his perspective on the course, and his support for Holly while we struggled through the first year of learning about the course.

David Appleyard for letting his ideas rub off on us during the semester he spent in Calculus&Mathematica lab at Illinois.

Francis Sullivan, Karen Uhlenbeck, Luis Caffarelli, Albert Fassler, "Spud" Bradley, Martin Flashman , Deborah Hughes Hallett, Bernard Madison, Cleve Moler, Peter Lax, Peter Ponzo, Dana Scott, Gil Strang, David Tall, Edward Tufte , Mac Van Valkenberg, Ken Wilson, Jerry Johnson and Stephen Wolfram for taking time out of their own schedules to visit the lab as Calculus&Mathematica was being developed.

Shirley Treadway of Robinson High School for suggesting that Calculus&Mathematica could be used by high school students in distance education (with the result being C&M NetMath and a few years later, CROSU).

Shirley Treadway, Kay Hall, Deana Brashear, Jackie Wood, Mark Catt, and Mark Calvert for sponsoring Calculus&Mathematica at their high schools and for helping to prove that Calculus&Mathematica can be used successfully in distance education (with the result being C&M NetMath and a few years later, CROSU).

Nora Sabelli, Lisa Bienvue , and John Duban of the National Center for Supercomputer Applications for help in networking for distance education.

Deborah Crocker for the first systematic study of the learning, learning styles and characteristics of Calculus&Mathematica students in her doctoral dissertation at The Ohio State University . We were very pleased with her conclusion that one of the striking differences between Calculus&Mathematica students and traditional students is that C&M students believe they can attack and solve problems that require the use of calculus. We were glad to have her further evidence that Calculus&Mathematica students develop a stronger conceptual understanding of calculus than their peers in traditional courses.

Kyung Mee Park for the first systematic comparison of Calculus&Mathematica students against traditionally trained students in her doctoral dissertation at the University of Illinois at Urbana - Champaign. We were very gratified by her conclusions that Calculus&Mathematica students are considerably more likely to try multiple approaches to a problem than students from the traditional course, that Calculus&Mathematica students cannot be distinguished from students in the traditional course on the basis of their ability at hand calculation, and that Calculus&Mathematica students demonstrate considerably richer ability to identify relations among calculus ideas than students in the traditional course.

Kenneth Travers for being the director of Kyung Mee Park's disseration research and for being a steady source of calm wisdom. His continued interest in the course has been much appreciated.

Albert Fassler for his work in transforming Calculus&Mathematica for the Swiss classroom and for ringing the Calculus&Mathematica bell in Europe.

William Graves, Ladnor Geissinger, Lester Senechal, and Gerald Porter for selecting Calculus&Mathematica as the protype interactive text for the national MAA- IBM- NSF Interactive Mathematics Text Project .

Peter Lax for issuing the first call for calulus revolution by proclaiming that

Calculus as currently taught is full of inert topics.

Ralph Abraham for his interest in the project and for his view that

Mathematical concepts may be communicated easily in a format which combines visual, verbal and symbolic representations in tight coordination.

He has been at this business a lot longer than we have. We hope he likes what he sees.

Lynn Arthur Steen for smelling a course like Calculus&Mathematica in the wind even before we began to work on it.

The authors of the reports Everybody Counts and Moving BeyondMyths, by the National Research Council, for issuing a mandate for a course like Calculus&Mathematica. In a very real way, Calculus&Mathematica is a reaction to these reports. In fact we believe Calculus&Mathematica to be in line with every recommendation in these reports - from the electronic text to mathematics as it really is and to classes without lectures.

David Tall for papers on using computers for visualization in the mathematics classroom. His ideas opened our eyes to what it is possible to do. His visit to our lab gave us a jump start.

Thomas West for sharing his ideas on visual learning in a long conversation and for writing a manifesto for visual learning in his book In the Mind's Eye: Visual Thinkers, Gifted People with Learning Disabilities, Computer Images, and the Ironies of Creativity. His ideas had a profound effect on the revision of Calculus&Mathematica from the preliminary to the present edition.

Andrew Gleason for reminding us over and over that common sense is the great engine for driving mathematics.

Henry Edwards for his interest in this project and for writing his book The Historical Development of Calculus which gave us many provocative morsels to savor when we were thinking about what calculus is.

Deborah Hughes Hallett, honcho of the Harvard Calculus Consortium, for her encouragement and for hour after hour of sagacious advice on what should be in, what should be out, what will work and what will not work. She is one of the rare individuals who understands that mathematics itself must be rethought before better calculus courses can begin. She has been a behind the scene advisor to the project almost from the beginning. Our special thanks go to her.

Jim Callahan, Ken Hoffman, Donal O'Shea and Lester Senechal of the Five Colleges Calculus project, for showing us what calculus reform really means. Many of their ideas are reflected in Calculus&Mathematica. They have rethought the mathematics of calculus; their view of calculus remains a beacon for those who question the traditional course.

Martin Flashman for sharing his ideas for a sensible calculus with us in a visit to our lab. His ideas have influenced this course even more than even he might believe.

Ron Douglas, Al Tucker and Tom Tucker for some profound comments about the role of algebra, trigonometry, and technology as delivery vehicles in modern calculus.

James Glimm for expressing his view that the derivative should be introduced as a measurement the growth rate rather than the slope of a tangent line. His views confirmed our belief in calculus as a course in measurements. We also thank him for his view that computers, not calculators, are the correct technology for the modern calculus classroom.

Stephen Jay Gould for reminding us, through his writings, that a course like this cannot ignite students' mathematical passions if any part of it is copied or adapted from an existing text.

Edward Tufte for his books The Visual Display of Qualitative Information and Envisioning Information .

Igor Kluvánek, our old and dear late friend, for his papers, What's Wrong with Calculus? and Archimedes Was Right. It's hard to accept that we will never again have opportunity to sit down with him with a good drink and talk with him about the central issues of mathematics and of life.

Gil Strang for his book, Introduction to Applied Mathematics. This book had a decisive effect on our ideas about vector calculus as a course about measurements rather than a course about formulas.

J.D. Murray for his book, Mathematical Biology. This book helped us to make the decision to use biological models to illustrate the meaning of the derivative in early calculus.

Jacob T. Scwartz for his view that

Courseware can concentrate on one skill at a time, in a manner impossible for a textbook and hardly available to the classroom teacher, namely by asking the student to handle only that part of a procedure on which pedagogical stress is to be laid, while other aspects of the same procedure are handled automatically by the computer.

Emil Artin for his Princeton calculus lectures. His course proved that a strong calculus course need be neither formal nor laden down with heavy terminolgy or algebra. We also thank him for his position that when a mathematical idea has a geometric interpretation, then that idea should be introduced via geometry instead of algebra.

Blaise Pascal , Gottfried Wilhelm von Leibniz , and Augusta Ada Byron for calling early attention, in their writings, to the fact that machine calculations have an essential role in mathematics.

Leonhard Euler , for his book, Introduction to the Analysis of the Infinite, which showed us a style of writing appropriate for Mathematica notebooks.

Henri Poincaré for arguing, in his writings, in favor of mathematics courses that emphasize insight over rigor. In doing the mathematical archeology required for a new calculus course, we came to the view that our visual approach was first advanced by Poincaré. In fact we go so far as to say that the traditional calculus and traditional differential equations courses are courses in pre-Poincaré mathematics. We offer Calculus&Mathematica as our best attempt at a post-Poincaré calculus course.

Alfred Whitehead for announcing, in his writings, a general principle for determining what parts of a mathematics course are inert and should be dropped. We also thank him for his view that

Civilization advances by extending the number of operations we can do without thinking about them.

Henri Lebesgue for his writings on the pedagogy of the integral. He shared Emil Artin 's idea that instead of defining area by an integral, the integral should be introduced as a device to measure area. It was his idea to deemphasize the indefinite integral (the integral without limits) in calculus. It was also his idea to emphasize the decimal form of a number.

Richard Feynman for his view that the most mathematics textbook authors are wrong when they confuse clarity with precision.

Many other persons, living or dead, for contributing to Calculus&Mathematica by sharing wisdom and problems through direct suggestion or in their writings. Their specific contributions are mentioned as appropriate in the main body of the course. We thank them.