To Students from University of Arizona Students
David Brokl, Jennifer Lowe, Ian Scott, and Michael Van Zeeland

This book is different! Most math books and math classes follow the simple formula
(1) present an abstract definition or theorem,
(2) attempt to clarify this definition or theorem with examples and dialogue, and then
(3) assign exercises which can be done by repeating the examples in the book with changed numbers.

This book often works contrary to that formula in that graphical and numerical examples are presented first, and from them some general conclusions are drawn. Sometimes sections start with a data set or real world example, often about situations we have encountered in our other science and engineering classes. This is great, as it makes the book much more interesting to read.

As you are working through these examples, keep your eyes open for connections to past results. These are there, and when you finish, there are some new conclusions which await you. Sometimes there are clues as to what to look for in the ''Where We Are Going—and Why'' at the start of each chapter.

To help understand these general conclusions, we would first read through a new section as if reading a story, looking for main ideas, but not dwelling on them. Then we would go back through the section carefully with pencil and paper (and often a computer or graphing calculator) at hand to make sure the details made sense. After each example, we stopped to examine the logic and details of the analysis. In many places of the text there were questions asking why something was true (these are contained in parentheses). We found it very worthwhile to figure out the answer before going on. The words ''Comments about'' appeared at various places in the book. These were not simply idle comments, but either emphasize an important point or made useful remarks a bout a prior result. Be sure to read and think about these "Comments''!

As you work through each section, pay close attention to every single graph. In other math books, graphs may seem to be incidental, but in this book the graphs are central to understanding the ideas. A computer, or graphing calculator, was also crucial to understanding the material. We would question what we saw on the screen and guess what would happen if we changed one of the inputs. We would also make sure that what we saw agreed with our analytical work. Computers can be misleading, so we questioned things which did not make sense. We found that answering these questions led to a much better understanding of the material. It was a great feeling to discover that we could explain why the computer results were misleading.

Most of the exercises cannot be done by simply looking at the examples in the book without reading the section. However by working through the material in the book you will be able to recognize similarities between the exercises and the examples and see the proper approach for the exercise. If part of a section is confusing, look ahead to a ''How to ... '' box—which contains concise explanations—or to related definitions, or maybe to the ''What Have We Learned?'' summary at the end of the chapter.

Finally, find a friend (or friends) to study with. These people should be able to trade ideas with you and discuss alternative approaches to solving problems. Having several points of view is often a big help in solving some of the more challenging problems. Someone else can often easily spot the place where your thinking or analysis is incorrect. Of course explaining things to each other is an effective way to really understand the material. We really enjoyed using this book, and think you will also.

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