Second Edition Single Variable SISSON · SZARVAS with Early Transcendentals CALCULUSLead Editor: Danielle C. Bess Editors: Allison Conger, Ian Craig, Marvin Glover, James Daniel Madden, Claudia Vance Creative Services Manager: Trudy Tronco Designers: Lizbeth Mendoza, Patrick Thompson, Joel Travis Cover Design: Joel Travis Design and Layout Assistance: U. Nagesh, E. Jeevan Kumar, D. Kanthi, K.V.S. Anil Cover Sculpture: Arabesque XXIX 12˝ H × 10½˝ W × 9½˝ D Bubinga Wood by Robert Longhurst A division of Quant Systems, Inc. 546 Long Point Road Mount Pleasant, SC 29464 Copyright © 2024, 2016 by Hawkes Learning / Quant Systems, Inc. All rights reserved. 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, or otherwise, without the prior written consent of the publisher. Mathematica is a registered trademark of Wolfram Research, Inc. Maple is a registered trademark of Waterloo Maple Inc. Library of Congress Control Number: 2023938712 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN: 978-1-64277-596-9 AIE ISBN: 978-1-64277-597-6In Memoriam, Carol Ann Sisson and Drs. István Tibor Szarvas and Katalin Ilona WeiszivTable of Contents TABLE OF CONTENTS Preface From the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix About the Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .x Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Chapter 1 A Review of Functions Chapter 2 Limits and the Derivative vTable of Contents Chapter 3 Differentiation Chapter 4 Applications of Differentiation Chapter 5 Integration viTable of Contents Chapter 6 Applications of the Definite Integral Chapter 7 Techniques of Integration Chapter 8 Differential Equations viiTable of Contents Chapter 9 Parametric Equations and Polar Coordinates Chapter 10 Sequences and Series Appendices A Fundamentals of Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A-2 B Properties of Exponents and Logarithms, Graphs of Exponential and Logarithmic Functions . . . . . . . . . . . .A-7 C Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A-8 D Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A-13 E Proofs of Selected Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A-19 Table of Integrals Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-1 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AK-1viiiFrom the Authors FROM THE AUTHORS This book arises from our deeply held belief that teaching and learning calculus should be a fascinating and rewarding experience for student and professor alike, playing a major role in a student’s overall academic growth. This is true not only because calculus ranks among the monumental achievements of the human intellect, thus providing an excellent introduction to higher-order thinking, but because of its wide-ranging applications in mathematics, the sciences, the business world, and the social sciences. Our goal was to produce a text that builds on the natural intuition and curiosity of the reader, blending a student-friendly style of exposition with precision and depth. We believe that if done well, the calculus sequence should be a highly enjoyable journey of discovery and growth for the student, reflecting the journeys of discovery experienced by those who originally developed calculus centuries ago. In other words, we strived to produce a book that is not only instructive, but also enjoyable to read—one that takes its readers from intuitive problem introductions to the rigor of concepts, definitions, and proofs in a natural manner. Some of the distinctive features of our text include the following: • A large number of examples and exercises in each section that demonstrate problem-solving techniques, reinforce conceptual understanding, and stimulate interest in the subject • Carefully selected exercises that gradually increase in level of difficulty, ranging from skill-building “drill-and-practice” problems to less routine, more challenging, and occasionally deep theoretical questions • Multistep, guided exploratory exercises that allow students to discover certain principles and connections on their own • A rich variety of application problems from within and outside of mathematics and the sciences • A constant emphasis on modern technology and its potential to enhance teaching and problem solving, as well as being a tool for investigation, reinforcement, and illustration This second edition incorporates suggestions and requests from users of the first edition, additional historical context and notes, updated exercise sets, and new applied chapter projects. In summary, we have aimed for a comprehensive, mathematically rigorous exposition that not only uncovers the inherent beauty and depth of calculus, but also provides insight into the many applications of the subject. We hope you enjoy the journey to the fullest. Let us know how we did and where we can improve! Paul Sisson and Tibor SzarvasixAcknowledgements ACKNOWLEDGEMENTS We are grateful to all who have guided and assisted us throughout the long, satisfying process of writing this text. Thank you to Dr. James Hawkes, Marcel Prevuznak, Emily Cook, Kim Cumbie, and all the people at Hawkes Learning for their dedication to this project. In particular, we would like to thank our editor, Claudia Vance, and her editorial team members for their hard work and commitment: Robert Alexander, Danielle Bess, Doug Chappell, Robin Hendrix, Barbara Miller, Nina Waldron, and Tee Jay Zajac. We are deeply appreciative of the efforts of our reviewers for their many insightful comments and reviews: Carryn Bellomo-Warren University of Nevada, Las Vegas John F. Beyers University of Maryland–University College Mariah Birgen Wartburg College Douglas K. Brown Catawba College Julian M. Buck Francis Marion University Teena Carroll Emory & Henry College Brian Dalpiaz Spoon River College–Canton David Dixon Mountainair High School Sarah Duffin Southern Utah University Vincent Ferlini Keene State College Thomas L. Fitzkee Francis Marion University Roy Harris Stephen F. Austin State University Pramod Kanwar Ohio University–Zanesville Noureen Khan University of North Texas at Dallas Ravinder Kumar Tougaloo College Zsolt Lengvárszky Louisiana State University–Shreveport Richard Mabry Louisiana State University–Shreveport Rita Marie O’Brien Navarro College–Corsicana Campus Jennie Pegg Holmes Community College–Grenada Stanley Perrine Georgia Gwinnett College Paul Rokicky Cuyahoga Community College–Western Campus Melinda Rudibaugh Chandler-Gilbert Community College–Pecos Jacob Siehler Washington and Lee University John Taylor University of North Carolina–Charlotte Ruth Trubnik Delaware Valley College Robert P. Vilardi Troy University–Montgomery Campus We are indebted to uncounted family members, teachers, colleagues, and students for helping us along the path that led to the writing of Calculus with Early Transcendentals; the list is far too long for us to acknowledge them individually. But we are particularly grateful to our parents, William Reid Sisson and Carol Ann Sisson and Dr. István Tibor Szarvas and Dr. Katalin Ilona Weisz, for setting us on this path initially. Finally, Paul thanks his wife Cindy and Tibor thanks his wife Anita and sons David, Daniel, and Gergely for their unstinting support and understanding at all times, especially those many early-morning and weekend hours spent writing!Next >