Making Everything Easier!™ 2nd Edition Get the confidence and skills u s you need to get started with calculus 2nd Edition C a l c u l P P r e - Are you confused by calculus? This hands-on workbook helps you master r e the skills needed to excel in pre-calculus and conquer the next contender — calculus. You'll get hundreds of valuable exercises, problem-solving - k shortcuts, plenty of workspace, and step-by-step solutions to every problem. C b o o k o r You'll also memorize the most frequently used equations, see how to a W 100s of avoid common mistakes, understand tricky trig proofs, and much more. l c Problems! u l Detailed, fully worked-out u solutions to problems s The inside scoop on analyzing and W interpreting data o A wealth of tips, tricks, and warnings r k How to guesstimate with confidence b o Yang Kuang, PhD, is a professor of mathematics o at Arizona State University. k Michelle Rose Gilman is nn−−11 the co-author of Pre-Calculus 66 ⎛⎛ 11 ⎞⎞ For Dummies. ∑∑ 44 −− == Plain-English explanations foGr void teoohso , Dwst-eutopm -abrytmi-cslteiesep, oserx. tcaomo sphmloep®s!, Swciothr et ryuosuterd h sigtrhaetsetg iine spre-calculus nn==33 ⎜⎝⎜⎝ 22 ⎟⎠⎟⎠ Step-by-step procedures Master all the essential topics Hands-on practice exercises Test your skills with practical 44⎜⎛⎛⎜−−11⎟⎞⎞⎟33−−11++44⎜⎛⎛⎜−−11⎟⎞⎞⎟44−−11++44⎜⎛⎛⎜−−11⎟⎞⎞⎟55−−11++44⎜⎛⎛⎜−−11⎟⎞⎞⎟66−−11== Ample workspace to work out problems exercises ⎝⎝ 22⎠⎠ ⎝⎝ 22⎠⎠ ⎝⎝ 22⎠⎠ ⎝⎝ 22⎠⎠ A dash of humor and fun ⎛⎛ 11 ⎞⎞ 11 ⎛⎛ 11⎞⎞ Yang Kuang, PhD 11++⎜⎜−− ⎟⎟++ ++⎜⎜−− ⎟⎟ == ⎝⎝ 22⎠⎠ 44 ⎝⎝ 88⎠⎠ School of Mathematical and Statistical Sciences ISBN: 978-0-470-92322-1 Arizona State University 55 Michelle Rose Gilman $16.99 US Kuang 88 President and Founder, Fusion Academy and $19.99 CN Gilman Learning Center £13.99 UK Mathematics/Calculus Get More and Do More at Dummies.com® Start with FREE Cheat Sheets Cheat Sheets include (cid:129) Checklists (cid:129) Charts Apps (cid:129) Common Instructions Mobile (cid:129) And Other Good Stuff! 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Find out “HOW” at Dummies.com *Sweepstakes not currently available in all countries; visit Dummies.com for offi cial rules. u s u l a l c C P r e - o k o b o r k W FOR S E m I ‰ M U D 2ND EDITION by Dr. Yang Kuang and Michelle Rose Gilman, with Elleyne Kase 0011__99778800447700992233222211--ffffiirrss..iinndddd ii 33//22//1111 33::4455 PPMM Pre-Calculus Workbook For Dummies®, 2nd Edition Published by Wiley Publishing, Inc. 111 River St. Hoboken, NJ 07030-5774 www.wiley.com Copyright © 2011 by Wiley Publishing, Inc., Indianapolis, Indiana Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permit- ted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Reference for the Rest of Us!, The Dummies Way, Dummies Daily, The Fun and Easy Way, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc. and/ or its affi liates in the United States and other countries, and may not be used without written permission. All other trademarks are the property of their respective owners. Wiley Publishing, Inc., is not associated with any product or vendor mentioned in this book. LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITH- OUT LIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS. THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION. THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OR OTHER PROFESSIONAL SERVICES. IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF A COMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT. NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM. THE FACT THAT AN ORGANIZA- TION OR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY MAKE. FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ. For general information on our other products and services, please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002. For technical support, please visit www.wiley.com/techsupport. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Control Number: 2011921773 ISBN: 978-0-470-92322-1 Manufactured in the United States of America 10 9 8 7 6 5 4 3 2 1 0011__99778800447700992233222211--ffffiirrss..iinndddd iiii 33//22//1111 33::4466 PPMM About the Authors Dr. Yang Kuang is a proud father of three: Youny, Foris, and Belany. He has been a professor of mathematics at Arizona State University (ASU) since 1988. He entered the University of Science and Technology of China in 1980, went to the University of Oxford in 1984 for a Master’s degree and completed a PhD in mathematics at University of Alberta (Canada) in 1988, becoming a mathematics professor at ASU in the same year. Yang Kuang has published over 120 refereed publications and authored or edited many books. He has directed a dozen PhD dissertations and many funded research projects in mathematical biology and medicine. He founded and continues to edit the journal Mathematical Biosciences and Engineering. He has given mathematical research talks all over the world and has organized many international mathematical biology and medi- cine meetings. He is well-known for his pioneer work in the application of delay differen- tial equations in models of biology and medicine and his ongoing work in establishing a solid foundation and framework for building accurate, dynamic, rich population models that explicitly include resource quality dynamics. His more recent interest is to formulate scientifi cally well-grounded and computation- ally tractable mathematical models to describe the rich and intriguing dynamics of various within-host diseases and their treatments, including various types of cancer, diabetic diseases, and virus-induced diseases such as infl uenza and hepatitis B virus (HBV) infection. He hopes that these models will help speed up the much-needed devel- opment of personalized medicine. Yang Kuang teaches and uses calculus all the time. He is keenly aware of the many chal- lenges facing students and instructors. He hopes this book will be helpful to many of those who want to confront those challenges. Michelle Rose Gilman is the founder and CEO of Fusion Learning Center and Fusion Academy, a private school and tutoring/test prep facility in Solana Beach, California. She is the author of Pre-Calculus For Dummies, The ACT For Dummies, and other books on self-esteem, writing, and motivational topics. Elleyne Kase is a multidisciplined artist and graphics professional whose creative expertise has been applied commercially in many different avenues, such as fi ne art designs for the commercial interior design industry, Wayfi nding programs for large hos- pital systems, 3D displays, Web site design, and national brand identity. A bachelor’s degree in graphic design from San Jose State University began her professional career in the San Francisco Bay Area, which continued in San Diego design agencies as her family grew. Book design, a favorite of hers, began early on in her career and has now been applied to two of her own exam prep publications, Visual Quick Notes for Life Insurance and Visual Quick Notes for Life and Health. Both use her new mind mapping system, an evolution of her study on how the mind absorbs data, to explain simple terms and con- cepts in an organized visual matrix of graphic elements and text that improves memory retention. 0011__99778800447700992233222211--ffffiirrss..iinndddd iiiiii 33//22//1111 33::4466 PPMM 0011__99778800447700992233222211--ffffiirrss..iinndddd iivv 33//22//1111 33::4466 PPMM Authors’ Acknowledgments From Yang: I would like to thank Elleyne Kase for inviting me to take part in this unusual book project and all the people involved in the process. It was a joy working with all of you. I would also like to thank Professor Jamie Whittimore McGill and Dr. Scott Parsell for providing a detailed technical review of this edition. And last, but not least, I am forever grateful to my wife, Aijun, for her beautiful creation of and the great attention she devotes to the family, so that I can have time to do my work, including carrying out this project. From Elleyne: Throughout my varied career I have had the great pleasure of working with profes- sionals of many disciplines. The For Dummies series combines the skills of many very talented people. I want to express my gratitude to Bill Gladstone who gave me this rare opportunity to participate; to the editors at Wiley — Lindsay Lefevere, Tracy Boggier, Elizabeth Rea, and Christy Pingleton — for their professionalism, understanding, and skill in bringing such a complex product to market; and to Yang Kuang who juggled his demanding teaching schedule to accommodate this project. Above all I want to thank my dear mother, who has always supported me in all my varied interests, and my three children, Seth, Shane, and Mimi, who have grown to become wonderful, nurturing, caring adults and my best friends. 0011__99778800447700992233222211--ffffiirrss..iinndddd vv 33//22//1111 33::4466 PPMM Publisher’s Acknowledgments We’re proud of this book; please send us your comments at http://dummies.custhelp.com. For other com- ments, please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317- 572-3993, or fax 317-572-4002. Some of the people who helped bring this book to market include the following: Acquisitions, Editorial, and Media Development Composition Services Project Editor: Elizabeth Rea Project Coordinators: Nikki Gee, Patrick Redmond Acquisitions Editor: Tracy Boggier Layout and Graphics: Carl Byers, Carrie A. Cesavice, Joyce Haughey, Mark Pinto, Corrie Socolovitch Copy Editors: Christine Pingleton, Jennifer Tebbe Proofreaders: Betty Kish, Jessica Kramer Assistant Editor: David Lutton Indexer: Glassman Indexing Services Technical Editors: Jamie W. McGill, Scott Parsell, Ph.D. Editorial Manager: Michelle Hacker Editorial Assistants: Rachelle Amick, Jennette ElNaggar, Alexa Koschier Cartoons: Rich Tennant (www.the5thwave.com) Publishing and Editorial for Consumer Dummies Diane Graves Steele, Vice President and Publisher, Consumer Dummies Kristin Ferguson-Wagstaffe, Product Development Director, Consumer Dummies Ensley Eikenburg, Associate Publisher, Travel Kelly Regan, Editorial Director, Travel Publishing for Technology Dummies Andy Cummings, Vice President and Publisher, Dummies Technology/General User Composition Services Debbie Stailey, Director of Composition Services 0011__99778800447700992233222211--ffffiirrss..iinndddd vvii 33//22//1111 33::4466 PPMM Contents at a Glance Introduction ............................................................................1 Part I: Setting the Foundation: The Nuts and Bolts of Pre-Calculus ...............................................................5 Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus .......................................................7 Chapter 2: Real Numbers Come Clean .............................................................................................23 Chapter 3: Controlling Functions by Knowing Their Function .....................................................37 Chapter 4: Searching Roots to Get the Degree ...............................................................................69 Chapter 5: Exponential and Logarithmic Functions .......................................................................87 Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs ................................................103 Chapter 6: Basic Trigonometry and the Unit Circle .....................................................................105 Chapter 7: Graphing and Transforming Trig Functions ...............................................................127 Part III: Digging into Advanced Trig: Identities, Theorems, and Applications ..................................................143 Chapter 8: Basic Trig Identities .......................................................................................................145 Chapter 9: Advanced Identities .......................................................................................................161 Chapter 10: Solving Oblique Triangles ...........................................................................................177 Part IV: Poles, Cones, Variables, Sequences, and Finding Your Limits .......................................................195 Chapter 11: Exploring Complex Numbers and Polar Coordinates .............................................197 Chapter 12: Conquering Conic Sections ........................................................................................213 Chapter 13: Finding Solutions for Systems of Equations .............................................................245 Chapter 14: Spotting Patterns in Sequences, Series, and Binomials .........................................277 Chapter 15: Previewing Calculus ....................................................................................................289 Part V: The Part of Tens .......................................................301 Chapter 16: Ten Uses for Parent Graphs .......................................................................................303 Chapter 17: Ten Missteps to Avoid in Pre-Calculus .....................................................................311 Index ..................................................................................315 0022__99778800447700992233222211--ffttoocc..iinndddd vviiii 33//22//1111 33::4466 PPMM Table of Contents Introduction .............................................................................1 About This Book .................................................................................................................1 Conventions Used in This Book ........................................................................................2 Foolish Assumptions ..........................................................................................................2 How This Book Is Organized .............................................................................................2 Part I: Setting the Foundation: The Nuts and Bolts of Pre-Calculus ...................2 Part II: Trig Is the Key: Basic Review, the Unit Circle, and Graphs ....................2 Part III: Digging into Advanced Trig: Identities, Theorems, and Applications ................................................................................3 Part IV: Poles, Cones, Variables, Sequences, and Finding Your Limits..............3 Part V: The Part of Tens ...........................................................................................3 Icons Used in This Book ....................................................................................................3 Where to Go from Here ......................................................................................................4 Part I: Setting the Foundation: The Nuts and Bolts of Pre-Calculus ................................................................5 Chapter 1: Beginning at the Very Beginning: Pre-Pre-Calculus. . . . . . . . . . . . . . .7 Reviewing Order of Operations: The Fun in Fundamentals ..........................................7 Keeping Your Balance While Solving Equalities ...........................................................10 When Your Image Really Counts: Graphing Equalities and Inequalities ...................12 Graphing with the plug-and-chug method ...........................................................12 Graphing by using the slope-intercept form .......................................................13 Graphing inequalities by either method ..............................................................14 Using Graphs to Find Distance, Midpoint, and Slope ..................................................15 Finding the distance ...............................................................................................15 Calculating the midpoint........................................................................................15 Discovering the slope .............................................................................................16 Answers to Problems on Fundamentals ........................................................................18 Chapter 2: Real Numbers Come Clean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 Solving Inequalities ..........................................................................................................23 Expressing Inequality Solutions in Interval Notation ..................................................26 Don’t Panic with Radicals and Exponents — Just Simplify! ........................................28 Getting Out of a Sticky Situation, or Rationalizing .......................................................31 Answers to Problems on Real Numbers ........................................................................33 Chapter 3: Controlling Functions by Knowing Their Function. . . . . . . . . . . . . . . .37 Using Both Faces of the Coin: Even and Odd ................................................................37 Leaving the Nest: Transforming Parent Graphs ...........................................................39 Quadratic functions ................................................................................................39 Square root functions .............................................................................................40 0022__99778800447700992233222211--ffttoocc..iinndddd vviiiiii 33//22//1111 33::4466 PPMM