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Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Australia • Brazil • Mexico • Singapore • United Kingdom • United States calculusEarly TranscEndEnTalsEighTh EdiTionJamEs sTEwarTMcMaster University and University of toronto

This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Calculus: Early Transcendentals, Eighth Edition James Stewart © 2016, 2012 Cengage Learning WCN: 02-200-203 Product Manager: Neha Taleja Senior Content Developer: Stacy Green Associate Content Developer: Samantha Lugtu Product Assistant: Stephanie Kreuz Media Developer: Lynh Pham Marketing Manager: Ryan Ahern Content Project Manager: Cheryll Linthicum Art Director: Vernon Boes Manufacturing Planner: Becky Cross Production Service: TECHarts Photo and Text Researcher: Lumina Datamatics Copy Editor: Kathi Townes, TECHarts Illustrator: TECHarts Text Designer: Diane Beasley Cover Designer: Irene Morris, Morris Design Compositor: Stephanie Kuhns, Kristina Elliott, and Kira Abdallah, TECHarts Cover Image: elisanth/123RF; tharrison/Getty Images ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706. For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be e-mailed to permissionrequest@cengage.com. Library of Congress Control Number: 2014951195 Student Edition: ISBN: 978-1-285-74155-0 Loose-leaf Edition: ISBN: 978-1-305-27235-4 Cengage Learning 20 Channel Center Street Boston, MA 02210 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at www.cengage.com/global. Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Cengage Learning Solutions, visit www.cengage.com. Purchase any of our products at your local college store or at our pre- ferred online store www.cengagebrain.com. Windows is a registered trademark of the Microsoft Corporation and used herein under license. Macintosh is a registered trademark of Apple Computer, Inc. Used herein under license. Maple is a registered trademark of Waterloo Maple, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. Tools for Enriching Calculus is a trademark used herein under license. 4 1 T 2 1 k Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Printed in the United States of AmericaPrint Number: 01 Print Year: 2014

contents PrEfacE xi To ThE sTudEnT xxiii calculaTors, comPuTErs, and oThEr graPhing dEvicEs xxiv diagnosTic TEsTs xxvi a Preview of calculus 1 Four Ways to Represent a Function 10 1.1 1.2 Mathematical Models: A Catalog of Essential Functions 23 1.3 New Functions from Old Functions 36 1.4 1.5 Exponential Functions 45 Inverse Functions and Logarithms 55 Review 68 Principles of Problem solving 71 The Tangent and Velocity Problems 78 The Limit of a Function 83 2.1 2.2 2.3 Calculating Limits Using the Limit Laws 95 2.4 2.5 Continuity 114 2.6 2.7 Derivatives and Rates of Change 140 The Precise Definition of a Limit 104 Limits at Infinity; Horizontal Asymptotes 126 Writing Project • Early Methods for Finding Tangents 152 2.8 The Derivative as a Function 152 Review 165 Problems Plus 169 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. iii 12

iv Contents 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Derivatives of Polynomials and Exponential Functions 172 Applied Project • Building a Better Roller Coaster 182 The Product and Quotient Rules 183 Derivatives of Trigonometric Functions 190 The Chain Rule 197 Applied Project • Where Should a Pilot Start Descent? 208 Implicit Differentiation 208 Laboratory Project • Families of Implicit Curves 217 Derivatives of Logarithmic Functions 218 Rates of Change in the Natural and Social Sciences 224 Exponential Growth and Decay 237 Applied Project • Controlling Red Blood Cell Loss During Surgery 244 Related Rates 245 3.9 3.10 Linear Approximations and Differentials 251 Laboratory Project • Taylor Polynomials 258 3.11 Hyperbolic Functions 259 Review 266 Problems Plus 270 4.1 Maximum and Minimum Values 276 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 Applied Project • The Calculus of Rainbows 285 The Mean Value Theorem 287 How Derivatives Affect the Shape of a Graph 293 Indeterminate Forms and l’Hospital’s Rule 304 Writing Project • The Origins of l’Hospital’s Rule 314 Summary of Curve Sketching 315 Graphing with Calculus and Calculators 323 Optimization Problems 330 Applied Project • The Shape of a Can 343 Applied Project • Planes and Birds: Minimizing Energy 344 Newton’s Method 345 Antiderivatives 350 Review 358 Problems Plus 363 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 34

Contents v 5.1 5.2 5.3 5.4 5.5 Areas and Distances 366 The Definite Integral 378 Discovery Project • Area Functions 391 The Fundamental Theorem of Calculus 392 Indefinite Integrals and the Net Change Theorem 402 Writing Project • Newton, Leibniz, and the Invention of Calculus 411 The Substitution Rule 412 Review 421 Problems Plus 425 6.1 Areas Between Curves 428 Applied Project • The Gini Index 436 Volumes 438 Volumes by Cylindrical Shells 449 6.2 6.3 6.4 Work 455 6.5 Review 466 Problems Plus 468 Average Value of a Function 461 Applied Project • Calculus and Baseball 464 Applied Project • Where to Sit at the Movies 465 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Integration by Parts 472 Trigonometric Integrals 479 Trigonometric Substitution 486 Integration of Rational Functions by Partial Fractions 493 Strategy for Integration 503 Integration Using Tables and Computer Algebra Systems 508 Discovery Project • Patterns in Integrals 513 Approximate Integration 514 Improper Integrals 527 Review 537 Problems Plus 540 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 567

vi Contents 8.1 8.2 8.3 8.4 8.5 Arc Length 544 Discovery Project • Arc Length Contest 550 Area of a Surface of Revolution 551 Discovery Project • Rotating on a Slant 557 Applications to Physics and Engineering 558 Discovery Project • Complementary Coffee Cups 568 Applications to Economics and Biology 569 Probability 573 Review 581 Problems Plus 583 9.1 Modeling with Differential Equations 586 9.2 Direction Fields and Euler’s Method 591 9.3 Separable Equations 599 Applied Project • How Fast Does a Tank Drain? 608 Applied Project • Which Is Faster, Going Up or Coming Down? 609 9.4 Models for Population Growth 610 9.5 9.6 Linear Equations 620 Predator-Prey Systems 627 Review 634 Problems Plus 637 10.1 10.2 10.3 Curves Defined by Parametric Equations 640 Laboratory Project • Running Circles Around Circles 648 Calculus with Parametric Curves 649 Laboratory Project • Bézier Curves 657 Polar Coordinates 658 Laboratory Project • Families of Polar Curves 668 10.4 Areas and Lengths in Polar Coordinates 669 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 8910

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James Stewart Calculus Early Transcendentals 8th.pdf

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