CALCULUS Paul SissonTibor Szarvas with Early Transcendentals Executive Editor: Claudia Vance Executive Project Manager: Kimberly Cumbie Vice President, Research and Development: Marcel Prevuznak Editorial Assistants: DanielleC.Bess, DougChappell, SusanFuller, MargaretGibbs, RobinHendrix, BarbaraMiller, NinaWaldron, BarryWright,III Copy Editors: PhillipBushkar, TaylorHamrick, MaryKatherineHuffman, RebeccaJohnson, JustinLamothe, SojwalPohekar, EricPowers, KaraRoché, JosephTracy Answer Key Editors: Taylor Jones, Jason Ling, Jake Stauch Review Coordinator: Lisa Young Senior Designer: Tee Jay Zajac Layout & Original Graphics: Tee Jay Zajac Graphics: RobertAlexander, MargaretGibbs, JenniferMoran, TeeJayZajac Quant Systems India: E. Jeevan Kumar, D. Kanthi, U. Nagesh, B. Syam Prasad Cover Design: Tee Jay Zajac Cover Sculpture: Arabesque XXIX 12˝ H × 10½˝ W × 9½˝ D Bubinga Wood by Robert Longhurst www.robertlonghurst.com Chapter Opening Artwork: Calculus Series 4.625˝ H × 8.315˝ W Digital Painting by Jameson Deichman www.jamesondeichman.com A division of Quant Systems, Inc. 546 Long Point Road, Mount Pleasant, SC 29464 Copyright © 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. Printed in the United States of America Mathematica is a registered trademark of Wolfram Research, Inc. Maple is a registered trademark of Waterloo Maple Inc. Library of Congress Control Number: 2014957010 ISBN: 978-1-935782-21-6In Memoriam, Carol Ann Sisson and Dr. István Tibor SzarvasTABLE OF CONTENTS Preface From the Authors .........................................................................................................................xi Acknowledgements .................................................................................................................... xii About the Cover ........................................................................................................................ xiii Features .....................................................................................................................................xv Chapter 1 A Function Primer 1.1 Functions and How We Represent Them ..................................................................................... 3 1.2 A Function Repertory ................................................................................................................. 21 1.3 Transforming and Combining Functions ..................................................................................... 41 1.4 Inverse Functions ....................................................................................................................... 58 1.5 Calculus, Calculators, and Computer Algebra Systems ............................................................. 80 Review Exercises ....................................................................................................................... 96 Project ....................................................................................................................................... 99 Chapter 2 Limits and the Derivative 2.1 Rates of Change and Tangents ................................................................................................ 103 2.2 Limits All Around the Plane ...................................................................................................... 117 2.3 The Mathematical Definition of Limit ......................................................................................... 134 2.4 Determining Limits of Functions ............................................................................................... 146 2.5 Continuity ................................................................................................................................ 159 2.6 Rate of Change Revisited: TheDerivative ................................................................................. 179 Review Exercises ..................................................................................................................... 192 Project ..................................................................................................................................... 196Chapter 3 Differentiation 3.1 Differentiation Notation and Consequences ............................................................................. 199 3.2 Derivatives of Polynomials, Exponentials, Products, and Quotients .......................................... 216 3.3 Derivatives of Trigonometric Functions ..................................................................................... 239 3.4 The Chain Rule ........................................................................................................................ 253 3.5 Implicit Differentiation ............................................................................................................... 266 3.6 Derivatives of Inverse Functions ............................................................................................... 277 3.7 Rates of Change in Use ........................................................................................................... 293 3.8 Related Rates .......................................................................................................................... 304 3.9 Linearization and Differentials ................................................................................................... 315 Review Exercises ..................................................................................................................... 324 Project ..................................................................................................................................... 328 Chapter 4 Applications of Differentiation 4.1 Extreme Values of Functions .................................................................................................... 331 4.2 The Mean Value Theorem ........................................................................................................ 344 4.3 The First and Second Derivative Tests ..................................................................................... 354 4.4 L’Hôpital’s Rule ........................................................................................................................ 370 4.5 Calculus and Curve Sketching ................................................................................................. 381 4.6 Optimization Problems ............................................................................................................. 394 4.7 Antiderivatives ......................................................................................................................... 407 Review Exercises ..................................................................................................................... 417 Project ..................................................................................................................................... 421 Chapter 5 Integration 5.1 Area, Distance, and Riemann Sums ......................................................................................... 425 5.2 The Definite Integral ................................................................................................................. 437 5.3 The Fundamental Theorem of Calculus .................................................................................... 451 5.4 Indefinite Integrals and the Substitution Rule ............................................................................ 465 5.5 The Substitution Rule and Definite Integration .......................................................................... 475 Review Exercises ..................................................................................................................... 485 Project ..................................................................................................................................... 488Chapter 6 Applications of the Definite Integral 6.1 Finding Volumes Using Slices .................................................................................................. 491 6.2 Finding Volumes Using Cylindrical Shells .................................................................................. 505 6.3 Arc Length and Surface Area ................................................................................................... 515 6.4 Moments and Centers of Mass ................................................................................................ 525 6.5 Force, Work, and Pressure ...................................................................................................... 539 6.6 Hyperbolic Functions ............................................................................................................... 552 Review Exercises ..................................................................................................................... 564 Project ..................................................................................................................................... 569 Chapter 7 Techniques of Integration 7.1 Integration by Parts ................................................................................................................. 573 7.2 The Partial Fractions Method ................................................................................................... 581 7.3 Trigonometric Integrals ............................................................................................................. 592 7.4 Trigonometric Substitutions ..................................................................................................... 599 7.5 Integration Summary and Integration Using Computer Algebra Systems .................................. 608 7.6 Numerical Integration ............................................................................................................... 616 7.7 Improper Integrals .................................................................................................................... 627 Review Exercises ..................................................................................................................... 637 Project ..................................................................................................................................... 641 Chapter 8 Differential Equations 8.1 Separable Differential Equations ............................................................................................... 645 8.2 First-Order Linear Differential Equations ................................................................................... 654 8.3 Autonomous Differential Equations and Slope Fields ................................................................ 663 8.4 Second-Order Linear Differential Equations .............................................................................. 672 Review Exercises ..................................................................................................................... 682 Project ..................................................................................................................................... 686Chapter 9 Parametric Equations and Polar Coordinates 9.1 Parametric Equations .............................................................................................................. 689 9.2 Calculus and Parametric Equations ......................................................................................... 702 9.3 Polar Coordinates .................................................................................................................... 713 9.4 Calculus in Polar Coordinates .................................................................................................. 724 9.5 Conic Sections in Cartesian Coordinates ................................................................................. 733 9.6 Conic Sections in Polar Coordinates ........................................................................................ 752 Review Exercises ..................................................................................................................... 760 Project ..................................................................................................................................... 764 Chapter 10 Sequences and Series 10.1 Sequences .............................................................................................................................. 767 10.2 Infinite Series ........................................................................................................................... 781 10.3 The Integral Test ...................................................................................................................... 795 10.4 Comparison Tests .................................................................................................................... 802 10.5 The Ratio and Root Tests ........................................................................................................ 809 10.6 Absolute and Conditional Convergence ................................................................................... 816 10.7 Power Series ........................................................................................................................... 824 10.8 Taylor and Maclaurin Series ..................................................................................................... 834 10.9 Further Applications of Series .................................................................................................. 847 Review Exercises ..................................................................................................................... 859 Project .................................................................................................................................... 864 Chapter 11 Vectors and the Geometry of Space 11.1 Three-Dimensional Cartesian Space ........................................................................................ 867 11.2 Vectors and Vector Algebra ..................................................................................................... 873 11.3 The Dot Product ...................................................................................................................... 884 11.4 The Cross Product .................................................................................................................. 895 11.5 Describing Lines and Planes .................................................................................................... 905 11.6 Cylinders and Quadric Surfaces ............................................................................................... 916 Review Exercises ..................................................................................................................... 925 Project ..................................................................................................................................... 930Chapter 12 Vector Functions 12.1 Vector-Valued Functions .......................................................................................................... 933 12.2 Arc Length and the Unit Tangent Vector ................................................................................... 945 12.3 The Unit Normal and Binormal Vectors, Curvature, and Torsion ............................................... 954 12.4 Planetary Motion and Kepler’s Laws ........................................................................................ 967 Review Exercises ..................................................................................................................... 975 Project ..................................................................................................................................... 980 Chapter 13 Partial Derivatives 13.1 Functions of Several Variables ................................................................................................. 983 13.2 Limits and Continuity of Multivariable Functions ....................................................................... 994 13.3 Partial Derivatives .................................................................................................................. 1002 13.4 The Chain Rule ...................................................................................................................... 1017 13.5 Directional Derivatives and Gradient Vectors .......................................................................... 1026 13.6 Tangent Planes and Differentials ............................................................................................ 1036 13.7 Extreme Values of Functions of Two Variables ........................................................................ 1045 13.8 Lagrange Multipliers ............................................................................................................... 1058 Review Exercises ................................................................................................................... 1066 Project ................................................................................................................................... 1072 Chapter 14 Multiple Integrals 14.1 Double Integrals ..................................................................................................................... 1075 14.2 Applications of Double Integrals ............................................................................................. 1086 14.3 Double Integrals in Polar Coordinates .................................................................................... 1097 14.4 Triple Integrals ....................................................................................................................... 1105 14.5 Triple Integrals in Cylindrical and Spherical Coordinates ......................................................... 1117 14.6 Substitutions and Multiple Integrals ........................................................................................ 1129 Review Exercises ................................................................................................................... 1138 Project ................................................................................................................................... 1143Next >