thomas calculus twelfth edition by george b thomas revised pdf

www.EngineeringBooksPDF.com 608070 _ISM_ThomasCalc_WeirHass_ttl.qxd:harsh_569709_ttl 9/3/09 3:11 PM Page INSTRUCTOR’S SOLUTIONS MANUAL SINGLE VARIABLE Collin County Community College WILLIAM ARDIS 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 www.EngineeringBooksPDF.com 608070 _ISM_ThomasCalc_WeirHass_ttl.qxd:harsh_569709_ttl 9/3/09 3:11 PM Page 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 permitted 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 BB 12 11 10 09 www.EngineeringBooksPDF.com 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) 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 ì conforms exactly to the methods, procedures and steps presented in the text ì is mathematically correct ì includes all of the steps necessary so a typical calculus student can follow the logical argument and algebra ì includes a graph or figure whenever called for by the exercise, or if needed to help with the explanation ì is formatted in an appropriate style to aid in its understanding Every CAS exercise is solved in both the MAPLE and MATHEMATICA computer algebra systems A template showing 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) For more information about other resources available with Thomas' Calculus, visit http://pearsonhighered.com www.EngineeringBooksPDF.com TABLE OF CONTENTS Functions 1.1 1.2 1.3 1.4 Functions and Their Graphs Combining Functions; Shifting and Scaling Graphs Trigonometric Functions 19 Graphing with Calculators and Computers 26 Practice Exercises 30 Additional and Advanced Exercises 38 Limits and Continuity 43 2.1 2.2 2.3 2.4 2.5 2.6 Rates of Change and Tangents to Curves 43 Limit of a Function and Limit Laws 46 The Precise Definition of a Limit 55 One-Sided Limits 63 Continuity 67 Limits Involving Infinity; Asymptotes of Graphs 73 Practice Exercises 82 Additional and Advanced Exercises 86 Differentiation 93 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Tangents and the Derivative at a Point 93 The Derivative as a Function 99 Differentiation Rules 109 The Derivative as a Rate of Change 114 Derivatives of Trigonometric Functions 120 The Chain Rule 127 Implicit Differentiation 135 Related Rates 142 Linearizations and Differentials 146 Practice Exercises 151 Additional and Advanced Exercises 162 Applications of Derivatives 167 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Extreme Values of Functions 167 The Mean Value Theorem 179 Monotonic Functions and the First Derivative Test 188 Concavity and Curve Sketching 196 Applied Optimization 216 Newton's Method 229 Antiderivatives 233 Practice Exercises 239 Additional and Advanced Exercises 251 Integration 257 5.1 5.2 5.3 5.4 5.5 5.6 Area and Estimating with Finite Sums 257 Sigma Notation and Limits of Finite Sums 262 The Definite Integral 268 The Fundamental Theorem of Calculus 282 Indefinite Integrals and the Substitution Rule 290 Substitution and Area Between Curves 296 Practice Exercises 310 Additional and Advanced Exercises 320 www.EngineeringBooksPDF.com Applications of Definite Integrals 327 6.1 6.2 6.3 6.4 6.5 6.6 Volumes Using Cross-Sections 327 Volumes Using Cylindrical Shells 337 Arc Lengths 347 Areas of Surfaces of Revolution 353 Work and Fluid Forces 358 Moments and Centers of Mass 365 Practice Exercises 376 Additional and Advanced Exercises 384 Transcendental Functions 389 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Inverse Functions and Their Derivatives 389 Natural Logarithms 396 Exponential Functions 403 Exponential Change and Separable Differential Equations 414 ^ Indeterminate Forms and L'Hopital's Rule 418 Inverse Trigonometric Functions 425 Hyperbolic Functions 436 Relative Rates of Growth 443 Practice Exercises 447 Additional and Advanced Exercises 458 Techniques of Integration 461 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Integration by Parts 461 Trigonometric Integrals 471 Trigonometric Substitutions 478 Integration of Rational Functions by Partial Fractions 484 Integral Tables and Computer Algebra Systems 491 Numerical Integration 502 Improper Integrals 510 Practice Exercises 518 Additional and Advanced Exercises 528 First-Order Differential Equations 537 9.1 9.2 9.3 9.4 9.5 Solutions, Slope Fields and Euler's Method 537 First-Order Linear Equations 543 Applications 546 Graphical Solutions of Autonomous Equations 549 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 www.EngineeringBooksPDF.com 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 11.2 11.3 11.4 11.5 11.6 11.7 Parametrizations of Plane Curves 647 Calculus with Parametric Curves 654 Polar Coordinates 662 Graphing in Polar Coordinates 667 Areas and Lengths in Polar Coordinates 674 Conic Sections 679 Conics in Polar Coordinates 689 Practice Exercises 699 Additional and Advanced Exercises 709 12 Vectors and the Geometry of Space 715 12.1 12.2 12.3 12.4 12.5 12.6 Three-Dimensional Coordinate Systems 715 Vectors 718 The Dot Product 723 The Cross Product 728 Lines and Planes in Space 734 Cylinders and Quadric Surfaces 741 Practice Exercises 746 Additional Exercises 754 13 Vector-Valued Functions and Motion in Space 759 13.1 13.2 13.3 13.4 13.5 13.6 Curves in Space and Their Tangents 759 Integrals of Vector Functions; Projectile Motion 764 Arc Length in Space 770 Curvature and Normal Vectors of a Curve 773 Tangential and Normal Components of Acceleration 778 Velocity and Acceleration in Polar Coordinates 784 Practice Exercises 785 Additional Exercises 791 Copyright © 2010 Pearson Education Inc Publishing as Addison-Wesley www.EngineeringBooksPDF.com 14 Partial Derivatives 795 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 Functions of Several Variables 795 Limits and Continuity in Higher Dimensions 804 Partial Derivatives 810 The Chain Rule 816 Directional Derivatives and Gradient Vectors 824 Tangent Planes and Differentials 829 Extreme Values and Saddle Points 836 Lagrange Multipliers 849 Taylor's Formula for Two Variables 857 Partial Derivatives with Constrained Variables 859 Practice Exercises 862 Additional Exercises 876 15 Multiple Integrals 881 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 Double and Iterated Integrals over Rectangles 881 Double Integrals over General Regions 882 Area by Double Integration 896 Double Integrals in Polar Form 900 Triple Integrals in Rectangular Coordinates 904 Moments and Centers of Mass 909 Triple Integrals in Cylindrical and Spherical Coordinates 914 Substitutions in Multiple Integrals 922 Practice Exercises 927 Additional Exercises 933 16 Integration in Vector Fields 939 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 Line Integrals 939 Vector Fields and Line Integrals; Work, Circulation, and Flux 944 Path Independence, Potential Functions, and Conservative Fields 952 Green's Theorem in the Plane 957 Surfaces and Area 963 Surface Integrals 972 Stokes's Theorem 980 The Divergence Theorem and a Unified Theory 984 Practice Exercises 989 Additional Exercises 997 Copyright © 2010 Pearson Education Inc Publishing as Addison-Wesley www.EngineeringBooksPDF.com CHAPTER FUNCTIONS 1.1 FUNCTIONS AND THEIR GRAPHS domain œ (c_ß _); range œ [1ß _) domain œ [0ß _); range œ (c_ß 1] domain œ Ịc2ß _); y in range and y œ È5x b 10   ! Ê y can be any positive real number Ê range œ Ị!ß _) domain œ (c_ß 0Ĩ r Ò3, _); y in range and y œ Èx2 c 3x   ! Ê y can be any positive real number Ê range œ Ị!ß _) domain œ (c_ß 3Đ r Ð3, _); y in range and y œ Ê3 c t!Ê 3ct 3ct, now if t  Ê c t € ! Ê 3ct € !, or if t €  ! Ê y can be any nonzero real number Ê range œ Ðc_ß 0Đ r Ð!ß _) domain œ (c_ß c%Đ r Ðc4, 4Đ r Ð4, _); y in range and y œ c%  t  Ê c16 Ÿ t c 16  ! Ê nonzero real number Ê range œ Ðc_ß # c "' c 18 Ó Ÿ t2 c 16 t2 c 16 , t2 c 16 now if t  c% Ê t2 c 16 € ! Ê  !, or if t € % Ê t c 16 € ! Ê t2 c 16 € !, or if € ! Ê y can be any r Ð!ß _) (a) Not the graph of a function of x since it fails the vertical line test (b) Is the graph of a function of x since any vertical line intersects the graph at most once (a) Not the graph of a function of x since it fails the vertical line test (b) Not the graph of a function of x since it fails the vertical line test # base œ x; (height)# b ˆ x# ‰ œ x# Ê height œ È3 # x; area is a(x) œ " # (base)(height) œ " # (x) Š È3 # x‹ œ È3 x# ; perimeter is p(x) œ x b x b x œ 3x 10 s œ side length Ê s# b s# œ d# Ê s œ d È2 ; and area is a œ s# Ê a œ " # d# 11 Let D œ diagonal length of a face of the cube and j œ the length of an edge Then j# b D# œ d# and D# œ 2j# Ê 3j# œ d# Ê j œ d È3 The surface area is 6j# œ 6d# 12 The coordinates of P are ˆxß Èx‰ so the slope of the line joining P to the origin is m œ ˆx, Èx‰ œ ˆ m"# , # œ 2d# and the volume is j$ œ Š d3 ‹ Èx x œ " Èx $Ỵ# œ (x € 0) Thus, "‰ m 13 2x b 4y œ Ê y œ c "# x b 54 ; L œ ÈÐx c 0Ñ2 b Ðy c 0Ñ2 œ Éx2 b Ðc "# x b 54 Ñ2 œ Éx2 b "4 x2 c 45 x b œ É 54 x2 c 54 x b 25 16 œ É 20x c 20x b 25 16 œ È20x2 c 20x b 25 14 y œ Èx c Ê y2 b œ x; L œ ÈÐx c 4Ñ2 b Ðy c 0Ñ2 œ ÈÐy2 b c 4Ñ2 b y2 œ ÈÐy2 c 1Ñ2 b y2 œ Èy4 c 2y2 b b y2 œ Èy4 c y2 b Copyright © 2010 Pearson Education, Inc Publishing as Addison-Wesley www.EngineeringBooksPDF.com d$ 3È 25 16 Chapter Functions 15 The domain is ac_ß _b 16 The domain is ac_ß _b 17 The domain is ac_ß _b 18 The domain is Ðc_ß !Ĩ 19 The domain is ac_ß !b r a!ß _b 20 The domain is ac_ß !b r a!ß _b 21 The domain is ac_ß c5b r Ðc5ß c3Ĩ r Ị3, 5Đ r a5, _b 22 The range is Ị2, 3Đ 23 Neither graph passes the vertical line test (a) (b) Copyright © 2010 Pearson Education, Inc Publishing as Addison-Wesley www.EngineeringBooksPDF.com 1000 Chapter 16 Integration in Vector Fields (c) The center of mass of the sheet is the point axß yß zb where z œ Mxy M with Mxy œ ' ' xyz d5 and S M œ ' ' xy d5 The work done by gravity in moving the point mass at axß yß zb to the xy-plane is S gMz œ gM Š Mxy M ‹ œ gMxy œ ' ' gxyz d5 œ S È3g 20 13 (a) Partition the sphere x# b y# b (z c 2)# œ into small pieces Let ?i be the surface area of the ith piece and let (xi ß yi ß zi ) be a point on the ith piece The force due to pressure on the ith piece is approximately w(4 c zi )?i The total force on S is approximately D w(4 c zi )?i This gives the actual force to be ' ' w(4 c z) d5 i S (b) The upward buoyant force is a result of the k-component of the force on the ball due to liquid pressure The force on the ball at (xß yß z) is w(4 c z)(cn) œ w(z c 4)n , where n is the outer unit normal at (xß yß z) Hence the k-component of this force is w(z c 4)n † k œ w(z c 4)k † n The (magnitude of the) buoyant force on the ball is obtained by adding up all these k-components to obtain ' ' w(z c 4)k † n d5 S (c) The Divergence Theorem says ' ' w(z c 4)k † n d5 œ ' ' ' div(w(z c 4)k) dV œ ' ' ' w dV, where D D S is x# b y# b (z c 2)# Ÿ Ê D ' ' w(z c 4)k † n d5 œ w ' ' ' dV œ 43 1w, the weight of the fluid if it D S were to occupy the region D 14 The surface S is z œ Èx# b y# from z œ to z œ Partition S into small pieces and let ?i be the area of the ith piece Let (xi ß yi ß zi ) be a point on the ith piece Then the magnitude of the force on the ith piece due to liquid pressure is approximately Fi œ w(2 c zi )?i Ê the total force on S is approximately D Fi œ D w(2 c zi )?i Ê the actual force is ' ' w(2 c z) d5 œ ' ' w ˆ2 c Èx# b y# ‰ É1 b i S œ ' ' È2 w ˆ2 c Èx# b y# ‰ dA œ '0 21 Rxy Rxy x# x# b y# b y# x# b y# dA '12 È2w(2 c r) r dr d) œ '021 È2w