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    Cover
    Title Page
    Copyright Page
    ISBN-13: 9780321717399
    Contents
    Preface
    Acknowledgments
    1 Functions
    1.1 Functions and Their Graphs
    1.2 Combining Functions; Shifting and Scaling Graphs
    1.3 Trigonometric Functions
    1.4 Graphing with Calculators and Computers
    1.5 Exponential Functions
    1.6 Inverse Functions and Logarithms
    2 Limits and Continuity
    2.1 Rates of Change and Tangents to Curves
    2.2 Limit of a Function and Limit Laws
    2.3 The Precise Definition of a Limit
    2.4 One-Sided Limits
    2.5 Continuity
    2.6 Limits Involving Infinity; Asymptotes of Graphs
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    3 Differentiation
    3.1 Tangents and the Derivative at a Point
    3.2 The Derivative as a Function
    3.3 Differentiation Rules
    3.4 The Derivative as a Rate of Change
    3.5 Derivatives of Trigonometric Functions
    3.6 The Chain Rule
    3.7 Implicit Differentiation
    3.8 Derivatives of Inverse Functions and Logarithms
    3.9 Inverse Trigonometric Functions
    3.10 Related Rates
    3.11 Linearization and Differentials
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    4 Applications of Derivatives
    4.1 Extreme Values of Functions
    4.2 The Mean Value Theorem
    4.3 Monotonic Functions and the First Derivative Test
    4.4 Concavity and Curve Sketching
    4.5 Indeterminate Forms and L’Hôpital’s Rule
    4.6 Applied Optimization
    4.7 Newton’s Method
    4.8 Antiderivatives
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    5 Integration
    5.1 Area and Estimating with Finite Sums
    5.2 Sigma Notation and Limits of Finite Sums
    5.3 The Definite Integral
    5.4 The Fundamental Theorem of Calculus
    5.5 Indefinite Integrals and the Substitution Method
    5.6 Substitution and Area Between Curves
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    6 Applications of Definite Integrals
    6.1 Volumes Using Cross-Sections
    6.2 Volumes Using Cylindrical Shells
    6.3 Arc Length
    6.4 Areas of Surfaces of Revolution
    6.5 Work
    6.6 Moments and Centers of Mass
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    7 Integrals and Transcendental Functions
    7.1 The Logarithm Defined as an Integral
    7.2 Exponential Change and Separable Differential Equations
    7.3 Hyperbolic Functions
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    8 Techniques of Integration
    8.1 Integration by Parts
    8.2 Trigonometric Integrals
    8.3 Trigonometric Substitutions
    8.4 Integration of Rational Functions by Partial Fractions
    8.5 Integral Tables and Computer Algebra Systems
    8.6 Numerical Integration
    8.7 Improper Integrals
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    9 Infinite Sequences and Series
    9.1 Sequences
    9.2 Infinite Series
    9.3 The Integral Test
    9.4 Comparison Tests
    9.5 The Ratio and Root Tests
    9.6 Alternating Series, Absolute and Conditional Convergence
    9.7 Power Series
    9.8 Taylor and Maclaurin Series
    9.9 Convergence of Taylor Series
    9.10 The Binomial Series and Applications of Taylor Series
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    10 Parametric Equations and Polar Coordinates
    10.1 Parametrizations of Plane Curves
    10.2 Calculus with Parametric Curves
    10.3 Polar Coordinates
    10.4 Graphing in Polar Coordinates
    10.5 Areas and Lengths in Polar Coordinates
    10.6 Conics in Polar Coordinates
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    11 Vectors and the Geometry of Space
    11.1 Three-Dimensional Coordinate Systems
    11.2 Vectors
    11.3 The Dot Product
    11.4 The Cross Product
    11.5 Lines and Planes in Space
    11.6 Cylinders and Quadric Surfaces
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    12 Vector-Valued Functions and Motion in Space
    12.1 Curves in Space and Their Tangents
    12.2 Integrals of Vector Functions; Projectile Motion
    12.3 Arc Length in Space
    12.4 Curvature and Normal Vectors of a Curve
    12.5 Tangential and Normal Components of Acceleration
    12.6 Velocity and Acceleration in Polar Coordinates
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    13 Partial Derivatives
    13.1 Functions of Several Variables
    13.2 Limits and Continuity in Higher Dimensions
    13.3 Partial Derivatives
    13.4 The Chain Rule
    13.5 Directional Derivatives and Gradient Vectors
    13.6 Tangent Planes and Differentials
    13.7 Extreme Values and Saddle Points
    13.8 Lagrange Multipliers
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    14 Multiple Integrals
    14.1 Double and Iterated Integrals over Rectangles
    14.2 Double Integrals over General Regions
    14.3 Area by Double Integration
    14.4 Double Integrals in Polar Form
    14.5 Triple Integrals in Rectangular Coordinates
    14.6 Moments and Centers of Mass
    14.7 Triple Integrals in Cylindrical and Spherical Coordinates
    14.8 Substitutions in Multiple Integrals
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    15 Integration in Vector Fields
    15.1 Line Integrals
    15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux
    15.3 Path Independence, Conservative Fields, and Potential Functions
    15.4 Green’s Theorem in the Plane
    15.5 Surfaces and Area
    15.6 Surface Integrals
    15.7 Stokes’ Theorem
    15.8 The Divergence Theorem and a Unified Theory
    QUESTIONS TO GUIDE YOUR REVIEW
    PRACTICE EXERCISES
    ADDITIONAL AND ADVANCED EXERCISES
    Appendices
    A.1 Real Numbers and the Real Line
    A.2 Mathematical Induction
    A.3 Lines, Circles, and Parabolas
    A.4 Conic Sections
    A.5 Proofs of Limit Theorems
    A.6 Commonly Occurring Limits
    A.7 Theory of the Real Numbers
    A.8 Complex Numbers
    A.9 The Distributive Law for Vector Cross Products
    A.10 The Mixed Derivative Theorem and the Increment Theorem
    A.11 Taylor’s Formula for Two Variables
    Answers to Odd-Numbered Exercises
    Index
    A
    B
    C
    D
    E
    F
    G
    H
    I
    J
    K
    L
    M
    N
    O
    P
    Q
    R
    S
    T
    U
    V
    W
    X
    Y
    Z
    Credits
    A Brief Table of Integrals
    UNIVERSITY CALCULUS EARLY TRANSCENDENTALS Second Edition Joel Hass University of California, Davis Maurice D. Weir Naval Postgraduate School George B. Thomas, Jr. Massachusetts Institute of Technology
    Editor in Chief: Deirdre Lynch Senior Acquisitions Editor: William Hoffman Sponsoring Editor: Caroline Celano Senior Content Editor: Elizabeth Bernardi Editorial Assistant: Brandon Rawnsley Senior Managing Editor: Karen Wernholm Associate Managing Editor: Tamela Ambush Senior Production Project Manager: Sheila Spinney Digital Assets Manager: Marianne Groth Supplements Production Coordinator: Kerri McQueen Associate Media Producer: Stephanie Green Software Development: Kristina Evans and Marty Wright Executive Marketing Manager: Jeff Weidenaar Marketing Coordinator: Kendra Bassi Senior Author Support/Technology Specialist: Joe Vetere Rights and Permissions Advisor: Michael Joyce Image Manager: Rachel Youdelman Manufacturing Manager: Evelyn Beaton Senior Manufacturing Buyer: Carol Melville Senior Media Buyer: Ginny Michaud Design Manager: Andrea Nix Production Coordination, Composition, and Illustrations: Nesbitt Graphics, Inc. Cover Design: Andrea Nix Cover Image: Black Shore III—Iceland, 2007. All content copyright © 2009 Josef Hoflehner For permission to use copyrighted material, grateful acknowledgment is made to the copyright holders on page C-1, which is hereby made part of this copyright page. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Pearson Education was aware of a trademark claim, the designations have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data Hass, Joel. University calculus: early transcendentals/Joel Hass, Maurice D. Weir, George B. Thomas, Jr.—2nd ed. p. cm. Rev. ed. of: University calculus. c2007. ISBN 978-0-321-71739-9 (alk. paper) 1. Calculus—Textbooks. I. Weir, Maurice D. III. Title. QA303.2.H373 2011 515—dc22 II. Thomas, George B. (George Brinton), 1914–2006. 2010035141 Copyright © 2012, 2007, Pearson Education, 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 permission of the publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm. 1 2 3 4 5 6—CRK—14 13 12 11 www.pearsonhighered.com ISBN 13: 978-0-321-71739-9 ISBN 10: 0-321-71739-2
    CONTENTS 1 2 3 Preface Functions 1.1 1.2 1.3 1.4 1.5 1.6 Functions and Their Graphs Combining Functions; Shifting and Scaling Graphs Trigonometric Functions Graphing with Calculators and Computers Exponential Functions Inverse Functions and Logarithms 21 29 33 1 39 Limits and Continuity 2.1 2.2 2.3 2.4 2.5 2.6 Differentiation 59 70 52 Rates of Change and Tangents to Curves Limit of a Function and Limit Laws The Precise Definition of a Limit One-Sided Limits Continuity 86 Limits Involving Infinity; Asymptotes of Graphs QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 110 111 113 79 116 120 Tangents and the Derivative at a Point The Derivative as a Function Differentiation Rules 129 The Derivative as a Rate of Change Derivatives of Trigonometric Functions The Chain Rule 156 Implicit Differentiation Derivatives of Inverse Functions and Logarithms Inverse Trigonometric Functions 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 Related Rates 3.11 Linearization and Differentials 149 180 186 195 139 164 QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 206 206 211 14 97 170 ix 1 52 116 iii
    iv Contents 4 5 6 7 Applications of Derivatives 214 Extreme Values of Functions The Mean Value Theorem 222 4.1 4.2 4.3 Monotonic Functions and the First Derivative Test 4.4 4.5 4.6 4.7 4.8 Concavity and Curve Sketching Indeterminate Forms and L’Hôpital’s Rule Applied Optimization Newton’s Method Antiderivatives QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 246 281 281 285 255 266 235 271 Integration 5.1 5.2 5.3 5.4 5.5 5.6 299 305 289 Area and Estimating with Finite Sums Sigma Notation and Limits of Finite Sums The Definite Integral The Fundamental Theorem of Calculus Indefinite Integrals and the Substitution Method Substitution and Area Between Curves QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 345 345 349 335 317 Applications of Definite Integrals Volumes Using Cross-Sections Volumes Using Cylindrical Shells Arc Length Areas of Surfaces of Revolution 6.1 6.2 6.3 6.4 6.5 Work 6.6 Moments and Centers of Mass 383 372 353 364 378 389 230 328 214 289 353 QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 397 397 399 Integrals and Transcendental Functions 401 7.1 7.2 7.3 401 The Logarithm Defined as an Integral Exponential Change and Separable Differential Equations Hyperbolic Functions QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 429 428 428 420 411
    8 9 10 11 Contents v 431 448 Techniques of Integration 8.1 8.2 8.3 8.4 8.5 8.6 8.7 444 439 432 Integration by Parts Trigonometric Integrals Trigonometric Substitutions Integration of Rational Functions by Partial Fractions Integral Tables and Computer Algebra Systems 456 Numerical Integration Improper Integrals QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 481 481 483 461 471 Infinite Sequences and Series 486 486 498 Sequences Infinite Series The Integral Test 507 Comparison Tests 512 The Ratio and Root Tests Alternating Series, Absolute and Conditional Convergence Power Series Taylor and Maclaurin Series Convergence of Taylor Series 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 The Binomial Series and Applications of Taylor Series 538 543 529 517 550 522 QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 559 559 561 Parametric Equations and Polar Coordinates 563 563 570 579 Polar Coordinates 10.1 Parametrizations of Plane Curves 10.2 Calculus with Parametric Curves 10.3 10.4 Graphing in Polar Coordinates 10.5 Areas and Lengths in Polar Coordinates 10.6 Conics in Polar Coordinates 591 QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 599 583 587 598 600 Vectors and the Geometry of Space 602 607 11.1 Three-Dimensional Coordinate Systems 11.2 Vectors 11.3 The Dot Product 11.4 The Cross Product 624 11.5 Lines and Planes in Space 11.6 Cylinders and Quadric Surfaces 616 630 638 602 QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 644 643 646
    vi Contents 12 Vector-Valued Functions and Motion in Space 649 Integrals of Vector Functions; Projectile Motion 12.1 Curves in Space and Their Tangents 12.2 12.3 Arc Length in Space 12.4 Curvature and Normal Vectors of a Curve 12.5 Tangential and Normal Components of Acceleration 12.6 Velocity and Acceleration in Polar Coordinates 664 668 649 657 674 679 QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 683 682 685 13 Partial Derivatives 686 Partial Derivatives Functions of Several Variables 13.1 13.2 Limits and Continuity in Higher Dimensions 13.3 13.4 The Chain Rule 13.5 Directional Derivatives and Gradient Vectors 13.6 Tangent Planes and Differentials 13.7 Extreme Values and Saddle Points 13.8 Lagrange Multipliers 740 703 730 748 714 694 723 QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 758 757 761 14 Multiple Integrals 763 14.1 Double and Iterated Integrals over Rectangles 14.2 Double Integrals over General Regions 768 14.3 Area by Double Integration 14.4 Double Integrals in Polar Form 780 14.5 Triple Integrals in Rectangular Coordinates 14.6 Moments and Centers of Mass 14.7 Triple Integrals in Cylindrical and Spherical Coordinates 14.8 795 777 786 Substitutions in Multiple Integrals QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 823 814 823 825 686 763 802
    15 Integration in Vector Fields Contents vii 828 828 Path Independence, Conservative Fields, and Potential Functions 15.1 Line Integrals 15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 15.3 15.4 Green’s Theorem in the Plane 15.5 15.6 15.7 15.8 The Divergence Theorem and a Unified Theory 870 Surfaces and Area Surface Integrals 880 Stokes’ Theorem 889 900 858 834 847 QUESTIONS TO GUIDE YOUR REVIEW PRACTICE EXERCISES ADDITIONAL AND ADVANCED EXERCISES 911 911 914 First-Order Differential Equations Online Solutions, Slope Fields, and Euler’s Method First-Order Linear Equations 16.1 16.2 16.3 Applications 16.4 Graphical Solutions of Autonomous Equations 16.5 Systems of Equations and Phase Planes 16-10 16-16 16-29 16-2 16-22 Second-Order Differential Equations Online 16 17 Second-Order Linear Equations 17.1 17.2 Nonhomogeneous Linear Equations 17.3 Applications 17.4 Euler Equations 17.5 Power Series Solutions 17-17 17-23 17-26 17-1 17-8 Appendices AP-1 AP-10 Real Numbers and the Real Line AP-6 A.1 A.2 Mathematical Induction A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10 The Mixed Derivative Theorem and the Increment Theorem AP-44 A.11 Taylor’s Formula for Two Variables Lines, Circles, and Parabolas AP-18 Conic Sections Proofs of Limit Theorems Commonly Occurring Limits Theory of the Real Numbers Complex Numbers AP-33 The Distributive Law for Vector Cross Products AP-29 AP-31 AP-43 AP-26 AP-48 Answers to Odd-Numbered Exercises Index Credits A Brief Table of Integrals AP-1 A-1 I-1 C-1 T-1
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