Calculus: Mathematics and Modeling

The mathematical study of change pioneered by Newton and Leibniz was free of the artificial separation of topics found in today's curriculum. In the 300 years following their work, the explosion of ideas and techniques which they inspired has resulted in an enormous body of mathematical knowledge. The sheer volume of this knowledge has resulted in a separation of originally unified concepts into distinct topics studied in a variety of courses such as Discrete Dynamical Modeling, Calculus, and Differential Equations. As a result of this separation, many students of mathematics never obtain a global understanding of the material necessary for creative and effective application in new situations. We propose a text which reunites these concepts and tools into a single study.
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Background

The first generation of calculus reformers exploited emerging technologies and the theme of multiple representations of functions to unify the study of Calculus. These pioneers also demonstrated to the mathematical community the efficacy of a number of innovative pedagogical techniques. Students' interest, engagement, and performance profited from the use of collaborative learning, writing to clarify ideas, discovery activities, and extended problem solving that requires reinterpretation of mathematical results in terms of the original problem. In addition, these reformers changed the number, order, and emphasis of topics and, in the process, highlighted the importance of differential equations.

Simultaneously, new technology-based developments have made visual the qualitative studies of differential equations developed over the last century. The latest undergraduate texts emphasize such qualitative studies and visualization.
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Implementation

In this second generation of calculus reform, we combine the lessons of the first generation along with the advances in differential equations through the use of discrete dynamical systems. An understanding of such systems provides a solid foundation for thinking about applications in science, engineering, economics, and mathematics. In addition, the study of these applications naturally lends itself to the new pedagogy.

The implementation of our philosophy will retain the intellectual core of the traditional calculus course. This is made possible through a whole-hearted adoption of symbolic manipulation software as the computational platform. Students will use technology to perform the lion's share of symbolic computations previously done with paper-and-pencil manipulation.
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Role of Technology

Implementation of our philosophy requires a computational environment in which the student can move smoothly between symbolic, numeric, graphic, and textual contexts. With the introduction of the TI-92, such an environment is now affordable for a wide audience. The preliminary addition of the text will therefore be written to make consistent and essential use of the TI-92.
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Outcomes

Students will come away form this course with a conceptual and practical knowledge of the derivative and the integral. They will also have experience in applying calculus concepts in the context of science, engineering, economics, and mathematics.
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Distinctive Features

Calculus textbooks are expected to have a number of features, even in preliminary editions. This calculus book will include the following in a fully integrated fashion.
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Reading

To encourage active reading by the students, ÒReflectionsÓ are included in the body of the text. In a Reflection, the students are asked to briefly consider a question about what has just been read or connect one idea with another. Reflections appear frequently at the beginning of the text. As students become more familiar with the types of questions they might ask themselves when reading mathematics, Reflections appear less and less often. Students, through these Reflections, can improve their ability to read mathematics. This is an additional feature that most books do not have, do not even suspect they should have.
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Writing

Perhaps 25% of all the exercises in the text will require writing. Special writing exercises will not be created to insert in the text to satisfy this need. Rather, the authors believe that the interpretation of results are an important and integral part of doing mathematics. Students will be asked to interpret the results of computations in practical situations and to describe the meaning of the mathematics in those contexts that are primarily mathematical in nature. In particular, in the early chapters, students will be asked to describe the behavior of functions relating to mathematical models of situations and of abstract functions represented as graphs. In the second and third parts of the text, the student will be frequently asked to interpret tables, graphs, and symbols involved in investigating the solution to differential equations as mathematical entities and models of real world phenomena.
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Applied Problems

Throughout the book mathematical ideas and concepts are introduced and developed in the context of significant practical applications from chemistry, biology, medical sciences, decision sciences, and mathematics. This basic approach to the book will carry over into the exercises where applied problems will appear prominently, but not to the exclusion of purely mathematical problems.
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Group Problems

The authors believe that the communication of mathematics both in writing and verbally improves some students' understanding of mathematics and provides students with the ability to transfer their use of mathematics from mathematics courses to physics, or chemistry, or economics courses, or to the engineering workbench. Group work will be incorporated into the examples in the text so that mathematics professors will feel comfortable doing ÒThink and ShareÓ or group work in class. Students will learn from such in-class activities how to participate effectively in group work outside class. Group activities make (for many students) an enormous difference in how much mathematics they can learn, master, and apply in other courses. There also will be group activities in the exercises to take advantage of what students learn about group work in class.
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Discovery Activities

Once again, these activities will be an integral part of the text. For example, there will be a trial and error aspect to most of the modeling activity based on difference equations in the first several chapters. Similarly, discovery approaches will be used in investigating differential equations by conjecturing the features of the solution from tables, graphs, and symbols. There will not be special discovery activities, but rather standard exercises that frequently require the students to discover results. Each activity will be designed to provide the student with closure on its key aspects. We will not depend upon the student to create knowledge, but rather we will capitalize on the ideas that occur when students try to come up with their own ways of working through a mathematical concept.
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Projects

The authors believe that extended projects that allow students to see the mathematics that they have learned in action and to reflect upon how the mathematics works are an important part of student mathematical development. Our text will include 1 or 2 such extended activities at the end of every chapter and several project activities at the end of the book from a variety of different areas (including mathematics) that involve many, if not all, of the ideas that the student has mastered in the course.
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Technology Activities

The TI-92 computer symbolic algebra graphing calculator will be used in the expository part of the text throughout the book. A major tenet of this book is that computer symbolic algebra technology in conjunction with the technology of the graphing calculator allows the first course in college or university mathematics to change significantly to the benefit of the mathematical understanding of students. We expect the professor who uses this book to frequently use the TI-92 ViewScreen in class. The text will allow the professor to take advantage of this calculator in the exercises throughout the book.
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Interludes

Where appropriate, chapters will be followed by Theoretical, Technical, and Historical Interludes. Interludes will include material that is not central to the development of the text, but can be used by students to further expand their understanding.
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Summary Remarks

Just as the authors believe in the importance of lectures by the professor, so too, they believe in the usefulness of summary remarks in the text. These remarks will not be a rehash of what happened in each section, but a statement of what was learned and why. It will also place the chapter in the context of the book and the intellectual history of the modern world.
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