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thomas calculus 12th solutions.pdf

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    cover (ISM)
    title page
    copyright (12th)
    preface
    contents
    ch01 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
    Practice Exercises
    Additional and Advanced Exercises
    ch02 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
    Practice Exercises
    Additional and Advanced Exercises
    ch03 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 Related Rates
    3.9 Linearizations and Differentials
    Practice Exercises
    Additional and Advanced Exercises
    ch04 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 Applied Optimization
    4.6 Newton's Method
    4.7 Antiderivatives
    Practice Exercises
    Additional and Advanced Exercises
    ch05 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 Rule
    5.6 Substitution and Area Between Curves
    Practice Exercises
    Additional and Advanced Exercises
    ch06 Applications of Definite Integrals
    6.1 Volumes Using Cross-Sections
    6.2 Volumes Using Cylindrical Shells
    6.3 Arc Lengths
    6.4 Areas of Surfaces of Revolution
    6.5 Work and Fluid Forces
    6.6 Moments and Centers of Mass
    Practice Exercises
    Additional and Advanced Exercises
    ch07 Transcendental Functions
    7.1 Inverse Functions and Their Derivatives
    7.2 Natural Logarithms
    7.3 Exponential Functions
    7.4 Exponential Change and Separable Differential Equations
    7.5 Indeterminate Forms and L'Hopital's Rule
    7.6 Inverse Trigonometric Functions
    7.7 Hyperbolic Functions
    7.8 Relative Rates of Growth
    Practice Exercises
    Additional and Advanced Exercises
    ch08 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
    Practice Exercises
    Additional and Advanced Exercises
    ch09 First-Order Differential Equations
    9.1 Solutions, Slope Fields and Euler's Method
    9.2 First-Order Linear Equations
    9.3 Applications
    9.4 Graphical Solutions of Autonomous Equations
    9.5 Systems of Equations and Phase Planes
    Practice Exercises
    Additional and Advanced Exercises
    ch10 Infinite Sequences and Series
    10.1 Sequences
    10.2 Infinite Series
    10.3 The Integral Test
    10.4 Comparison Tests
    10.5 The Ratio and Root Tests
    10.6 Alternating Series, Absolute and Conditional Convergence
    10.7 Power Series
    10.8 Taylor and Maclaurin Series
    10.9 Convergence of Taylor Series
    10.10 The Binomial Series and Applications of Taylor Series
    Practice Exercises
    Additional and Advanced Exercises
    ch11 Parametric Equations and Polar Coordinates
    11.1 Parametrizations of Plane Curves
    11.2 Calculus with Parametric Curves
    11.3 Polar Coordinates
    11.4 Graphing in Polar Coordinates
    11.5 Areas and Lengths in Polar Coordinates
    11.6 Conic Sections
    11.7 Conics in Polar Coordinates
    Practice Exercises
    Additional and Advanced Exercises
    ch12 Vectors and the Geometry of Space
    12.1 Three-Dimensional Coordinate Systems
    12.2 Vectors
    12.3 The Dot Product
    12.4 The Cross Product
    12.5 Lines and Planes in Space
    12.6 Cylinders and Quadric Surfaces
    Practice Exercises
    Additional Exercises
    ch13 Vector-Valued Functions and Motion in Space
    13.1 Curves in Space and Their Tangents
    13.2 Integrals of Vector Functions; Projectile Motion
    13.3 Arc Length in Space
    13.4 Curvature and Normal Vectors of a Curve
    13.5 Tangential and Normal Components of Acceleration
    13.6 Velocity and Acceleration in Polar Coordinates
    Practice Exercises
    Additional Exercises
    ch14 Partial Derivatives
    14.1 Functions of Several Variables
    14.2 Limits and Continuity in Higher Dimensions
    14.3 Partial Derivatives
    14.4 The Chain Rule
    14.5 Directional Derivatives and Gradient Vectors
    14.6 Tangent Planes and Differentials
    14.7 Extreme Values and Saddle Points
    14.8 Lagrange Multipliers
    14.9 Taylor's Formula for Two Variables
    14.10 Partial Derivatives with Constrained Variables
    Practice Exercises
    Additional Exercises
    ch15 Multiple Integrals
    15.1 Double and Iterated Integrals over Rectangles
    15.2 Double Integrals over General Regions
    15.3 Area by Double Integration
    15.4 Double Integrals in Polar Form
    15.5 Triple Integrals in Rectangular Coordinates
    15.6 Moments and Centers of Mass
    15.7 Triple Integrals in Cylindrical and Spherical Coordinates
    15.8 Substitutions in Multiple Integrals
    Practice Exercises
    Additional Exercises
    ch16 Integration in Vector Fields
    16.1 Line Integrals
    16.2 Vector Fields and Line Integrals; Work, Circulation, and Flux
    16.3 Path Independence, Potential Functions, and Conservative Fields
    16.4 Green's Theorem in the Plane
    16.5 Surfaces and Area
    16.6 Surface Integrals
    16.7 Stokes's Theorem
    16.8 The Divergence Theorem and a Unified Theory
    Practice Exercises
    Additional Exercises
    ch17 Second-Order Differential Equations
    17.1 Second-Order Linear Equations
    17.2 Nonhomogeneous Linear Equations
    17.3 Applications
    17.4 Euler Equations
    17.5 Power-Series Solutions
    608070 _ISM_ThomasCalc_WeirHass_ttl.qxd:harsh_569709_ttl 9/3/09 3:11 PM Page 1 INSTRUCTOR’S SOLUTIONS MANUAL SINGLE VARIABLE WILLIAM ARDIS Collin County Community College THOMAS’ CALCULUS TWELFTH EDITION BASED ON THE ORIGINAL WORK BY George B. Thomas, Jr. Massachusetts Institute of Technology AS REVISED BY Maurice D. Weir Naval Postgraduate School Joel Hass University of California, Davis
    608070 _ISM_ThomasCalc_WeirHass_ttl.qxd:harsh_569709_ttl 9/3/09 3:11 PM Page 2 This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permit- ted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials. The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs. Reproduced by Pearson Addison-Wesley from electronic files supplied by the author. Copyright © 2010, 2005, 2001 Pearson Education, Inc. Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116. 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. ISBN-13: 978-0-321-60807-9 ISBN-10: 0-321-60807-0 1 2 3 4 5 6 BB 12 11 10 09
    PREFACE TO THE INSTRUCTOR This Instructor's Solutions Manual contains the solutions to every exercise in the 12th Edition of THOMAS' CALCULUS by Maurice Weir and Joel Hass, including the Computer Algebra System (CAS) exercises. The corresponding Student's Solutions Manual omits the solutions to the even-numbered exercises as well as the solutions to the CAS exercises (because the CAS command templates would give them all away). ì ì conforms exactly to the methods, procedures and steps presented in the text is mathematically correct includes all of the steps necessary so a typical calculus student can follow the logical argument and algebra includes a graph or figure whenever called for by the exercise, or if needed to help with the explanation is formatted in an appropriate style to aid in its understanding In addition to including the solutions to all of the new exercises in this edition of Thomas, we have carefully revised or rewritten every solution which appeared in previous solutions manuals to ensure that each solution Every CAS exercise is solved in both the MAPLE and an example of the CAS commands needed to execute the solution is provided for each exercise type. Similar exercises within the text grouping require a change only in the input function or other numerical input parameters associated with the problem (such as the interval endpoints or the number of iterations). MATHEMATICA computer algebra systems. A template showing ì ì ì For more information about other resources available with Thomas' Calculus, visit http://pearsonhighered.com.
    TABLE OF CONTENTS 1 Functions 1 1.1 Functions and Their Graphs 1 1.2 Combining Functions; Shifting and Scaling Graphs 8 1.3 Trigonometric Functions 19 1.4 Graphing with Calculators and Computers 26 Practice Exercises 30 Additional and Advanced Exercises 38 2 Limits and Continuity 43 2.1 Rates of Change and Tangents to Curves 43 2.2 Limit of a Function and Limit Laws 46 2.3 The Precise Definition of a Limit 55 2.4 One-Sided Limits 63 2.5 Continuity 67 2.6 Limits Involving Infinity; Asymptotes of Graphs 73 Practice Exercises 82 Additional and Advanced Exercises 86 3 Differentiation 93 3.1 Tangents and the Derivative at a Point 93 3.2 The Derivative as a Function 99 3.3 Differentiation Rules 109 3.4 The Derivative as a Rate of Change 114 3.5 Derivatives of Trigonometric Functions 120 3.6 The Chain Rule 127 3.7 3.8 Related Rates 142 3.9 Linearizations and Differentials 146 Practice Exercises 151 Additional and Advanced Exercises 162 Implicit Differentiation 135 4 Applications of Derivatives 167 4.1 Extreme Values of Functions 167 4.2 The Mean Value Theorem 179 4.3 Monotonic Functions and the First Derivative Test 188 4.4 Concavity and Curve Sketching 196 4.5 Applied Optimization 216 4.6 Newton's Method 229 4.7 Antiderivatives 233 Practice Exercises 239 Additional and Advanced Exercises 251 5 Integration 257 5.1 Area and Estimating with Finite Sums 257 5.2 Sigma Notation and Limits of Finite Sums 262 5.3 The Definite Integral 268 5.4 The Fundamental Theorem of Calculus 282 5.5 5.6 Substitution and Area Between Curves 296 Indefinite Integrals and the Substitution Rule 290 Practice Exercises 310 Additional and Advanced Exercises 320
    6 Applications of Definite Integrals 327 6.1 Volumes Using Cross-Sections 327 6.2 Volumes Using Cylindrical Shells 337 6.3 Arc Lengths 347 6.4 Areas of Surfaces of Revolution 353 6.5 Work and Fluid Forces 358 6.6 Moments and Centers of Mass 365 Practice Exercises 376 Additional and Advanced Exercises 384 Inverse Functions and Their Derivatives 389 ^ 7 Transcendental Functions 389 7.1 7.2 Natural Logarithms 396 7.3 Exponential Functions 403 7.4 Exponential Change and Separable Differential Equations 414 7.5 7.6 7.7 Hyperbolic Functions 436 7.8 Relative Rates of Growth 443 Indeterminate Forms and L'Hopital's Rule 418 Inverse Trigonometric Functions 425 Practice Exercises 447 Additional and Advanced Exercises 458 Integration by Parts 461 8 Techniques of Integration 461 8.1 8.2 Trigonometric Integrals 471 8.3 Trigonometric Substitutions 478 8.4 8.5 8.6 Numerical Integration 502 8.7 Improper Integrals 510 Practice Exercises 518 Additional and Advanced Exercises 528 Integration of Rational Functions by Partial Fractions 484 Integral Tables and Computer Algebra Systems 491 9 First-Order Differential Equations 537 9.1 Solutions, Slope Fields and Euler's Method 537 9.2 First-Order Linear Equations 543 9.3 Applications 546 9.4 Graphical Solutions of Autonomous Equations 549 9.5 Systems of Equations and Phase Planes 557 Practice Exercises 562 Additional and Advanced Exercises 567 10 Infinite Sequences and Series 569 10.1 Sequences 569 10.2 Infinite Series 577 10.3 The Integral Test 583 10.4 Comparison Tests 590 10.5 The Ratio and Root Tests 597 10.6 Alternating Series, Absolute and Conditional Convergence 602 10.7 Power Series 608 10.8 Taylor and Maclaurin Series 617 10.9 Convergence of Taylor Series 621 10.10 The Binomial Series and Applications of Taylor Series 627 Practice Exercises 634 Additional and Advanced Exercises 642
    TABLE OF CONTENTS 10 Infinite Sequences and Series 569 10.1 Sequences 569 10.2 Infinite Series 577 10.3 The Integral Test 583 10.4 Comparison Tests 590 10.5 The Ratio and Root Tests 597 10.6 Alternating Series, Absolute and Conditional Convergence 602 10.7 Power Series 608 10.8 Taylor and Maclaurin Series 617 10.9 Convergence of Taylor Series 621 10.10 The Binomial Series and Applications of Taylor Series 627 Practice Exercises 634 Additional and Advanced Exercises 642 11 Parametric Equations and Polar Coordinates 647 11.1 Parametrizations of Plane Curves 647 11.2 Calculus with Parametric Curves 654 11.3 Polar Coordinates 662 11.4 Graphing in Polar Coordinates 667 11.5 Areas and Lengths in Polar Coordinates 674 11.6 Conic Sections 679 11.7 Conics in Polar Coordinates 689 Practice Exercises 699 Additional and Advanced Exercises 709 12 Vectors and the Geometry of Space 715 12.1 Three-Dimensional Coordinate Systems 715 12.2 Vectors 718 12.3 The Dot Product 723 12.4 The Cross Product 728 12.5 Lines and Planes in Space 734 12.6 Cylinders and Quadric Surfaces 741 Practice Exercises 746 Additional Exercises 754 Integrals of Vector Functions; Projectile Motion 764 13 Vector-Valued Functions and Motion in Space 759 13.1 Curves in Space and Their Tangents 759 13.2 13.3 Arc Length in Space 770 13.4 Curvature and Normal Vectors of a Curve 773 13.5 Tangential and Normal Components of Acceleration 778 13.6 Velocity and Acceleration in Polar Coordinates 784 Practice Exercises 785 Additional Exercises 791 Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
    14 Partial Derivatives 795 14.1 Functions of Several Variables 795 14.2 Limits and Continuity in Higher Dimensions 804 14.3 Partial Derivatives 810 14.4 The Chain Rule 816 14.5 Directional Derivatives and Gradient Vectors 824 14.6 Tangent Planes and Differentials 829 14.7 Extreme Values and Saddle Points 836 14.8 Lagrange Multipliers 849 14.9 Taylor's Formula for Two Variables 857 14.10 Partial Derivatives with Constrained Variables 859 Practice Exercises 862 Additional Exercises 876 15 Multiple Integrals 881 15.1 Double and Iterated Integrals over Rectangles 881 15.2 Double Integrals over General Regions 882 15.3 Area by Double Integration 896 15.4 Double Integrals in Polar Form 900 15.5 Triple Integrals in Rectangular Coordinates 904 15.6 Moments and Centers of Mass 909 15.7 Triple Integrals in Cylindrical and Spherical Coordinates 914 15.8 Substitutions in Multiple Integrals 922 Practice Exercises 927 Additional Exercises 933 16 Integration in Vector Fields 939 16.1 Line Integrals 939 16.2 Vector Fields and Line Integrals; Work, Circulation, and Flux 944 16.3 Path Independence, Potential Functions, and Conservative Fields 952 16.4 Green's Theorem in the Plane 957 16.5 Surfaces and Area 963 16.6 Surface Integrals 972 16.7 Stokes's Theorem 980 16.8 The Divergence Theorem and a Unified Theory 984 Practice Exercises 989 Additional Exercises 997 Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
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