Precalculus - University of Washington

[Pages:335]Precalculus

Precalculus

David H. Collingwood

Department of Mathematics University of Washington

K. David Prince

Minority Science and Engineering Program College of Engineering

University of Washington

Matthew M. Conroy

Department of Mathematics University of Washington

September 2, 2011

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Copyright c 2003 David H. Collingwood and K. David Prince. Copyright c 2011 David H. Collingwood, K. David Prince, and Matthew M. Conroy. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover, and with no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

Author Note

For most of you, this course will be unlike any mathematics course you have previously encountered. Why is this?

Learning a new language

Colleges and universities have been designed to help us discover, share and apply knowledge. As a student, the preparation required to carry out this three part mission varies widely, depending upon the chosen field of study. One fundamental prerequisite is fluency in a "basic language"; this provides a common framework in which to exchange ideas, carefully formulate problems and actively work toward their solutions. In modern science and engineering, college mathematics has become this "basic language", beginning with precalculus, moving into calculus and progressing into more advanced courses. The difficulty is that college mathematics will involve genuinely new ideas and the mystery of this unknown can be sort of intimidating. However, everyone in this course has the intelligence to succeed!

Is this course the same as high school Precalculus?

There are key differences between the way teaching and learning takes place in high schools and universities. Our goal is much more than just getting you to reproduce what was done in the classroom. Here are some key points to keep in mind:

? The pace of this course will be faster than a high school class in precalculus. Above that, we aim for greater command of the material, especially the ability to extend what we have learned to new situations.

? This course aims to help you build the stamina required to solve challenging and lengthy multi-step problems.

? As a rule of thumb, this course should on average take 15 hours of effort per week. That means that in addition to the 5 classroom hours per week, you would spend 10 hours extra on the class. This is only an average and my experience has shown that 12?15 hours iii

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of study per week (outside class) is a more typical estimate. In other words, for many students, this course is the equivalent of a halftime job!

? Because the course material is developed in a highly cumulative manner, we recommend that your study time be spread out evenly over the week, rather than in huge isolated blocks. An analogy with athletics is useful: If you are preparing to run a marathon, you must train daily; if you want to improve your time, you must continually push your comfort zone.

Prerequisites

This course assumes prior exposure to the "mathematics" in Chapters 1-12; these chapters cover functions, their graphs and some basic examples. This material is fully developed, in case you need to brush up on a particular topic. If you have never encountered the concept of a function, graphs of functions, linear functions or quadratic functions, this course will probably seem too advanced. You are not assumed to have taken a course which focuses on mathematical problem solving or multi-step problem solving; that is the purpose of this course.

Internet

There is a great deal of archived information specific to this course that can be accessed via the World Wide Web at the URL address



Why are we using this text?

Prior to 1990, the performance of a student in precalculus at the University of Washington was not a predictor of success in calculus. For this reason, the mathematics department set out to create a new course with a specific set of goals in mind:

? A review of the essential mathematics needed to succeed in calculus.

? An emphasis on problem solving, the idea being to gain both experience and confidence in working with a particular set of mathematical tools.

This text was created to achieve these goals and the 2004-05 academic year marks the eleventh year in which it has been used. Several thousand students have successfully passed through the course.

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Notation, Answers, etc.

This book is full of worked out examples. We use the the notation "Solution." to indicate where the reasoning for a problem begins; the symbol is used to indicate the end of the solution to a problem. There is a Table of Contents that is useful in helping you find a topic treated earlier in the course. It is also a good rough outline when it comes time to study for the final examination. The book also includes an index at the end. Finally, there is an appendix at the end of the text with "answers" to most of the problems in the text. It should be emphasized these are "answers" as opposed to "solutions". Any homework problems you may be asked to turn in will require you include all your work; in other words, a detailed solution. Simply writing down the answer from the back of the text would never be sufficient; the answers are intended to be a guide to help insure you are on the right track.

How to succeed in Math 120.

Most people learn mathematics by doing mathematics. That is, you learn it by active participation; it is very unusual for someone to learn the material by simply watching their instructor perform on Monday, Wednesday, and Friday. For this reason, the homework is THE heart of the course and more than anything else, study time is the key to success in Math 120. We advise 15 hours of study per week, outside class. Also, during the first week, the number of study hours will probably be even higher as you adjust to the viewpoint of the course and brush up on algebra skills.

Here are some suggestions: Prior to a given class, make sure you have looked over the reading assigned. If you can't finish it, at least look it over and get some idea of the topic to be discussed. Having looked over the material ahead of time, you will get FAR MORE out of the lecture. Then, after lecture, you will be ready to launch into the homework. If you follow this model, it will minimize the number of times you leave class in a daze. In addition, spread your study time out evenly over the week, rather than waiting until the day before an assignment is due.

Acknowledgments

The efforts of numerous people have led to many changes, corrections and improvements. We want to specifically thank Laura Acun~ a, Patrick Averbeck, Jim Baxter, Sandi Bennett, Daniel Bjorkegren, Cindy Burton, Michael D. Calac, Roll Jean Cheng, Jerry Folland, Dan Fox, Grant Galbraith, Peter Garfield, Richard J. Golob, Joel Grus, Fred Kuczmarski, Julie Harris, Michael Harrison, Teri Hughes, Ron Irving, Ian Jannetty, Mark Johnson, Michael Keynes, Andrew Loveless, Don Marshall, Linda

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Martin, Alexandra Nichifor, Patrick Perkins, Lisa Peterson, Ken Plochinski, Eric Rimbey, Tim Roberts, Aaron Schlafly, David Schneider, Marilyn Stor, Lukas Svec, Sarah Swearinger, Jennifer Taggart, Steve Tanner, Paul Tseng, and Rebecca Tyson. I am grateful to everyone for their hard work and dedication to making this a better product for our students.

The Minority Science and Engineering Program (MSEP) of the College of Engineering supports the development of this textbook. It is also authoring additional materials, namely, a student study guide and an instructor guide. MSEP actively uses these all of these materials in its summer mathematics program for freshman pre-engineers. We want to thank MSEP for its contributions to this textbook.

We want to thank Intel Corporation for their grant giving us an "Innovation in Education" server donation. This computer hardware was used to maintain and develop this textbook.

Comments

Send comments, corrections, and ideas to colling@math.washington.edu or kdp@engr.washington.edu.

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