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Multivariable Calculus, 7th Edition Solution Manual

Multivariable Calculus, 7th Edition Solution Manual makes tackling textbook exercises a breeze, with clear and concise answers to every problem.

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    ’s Guide Prepared by Douglas Shaw University of Northern Iowa Calculus SEVENTH EDITION James Stewart McMaster University and University of Toronto

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    Preface I once facilitated a mandatory workshop on the teaching of calculus that was attended by a diverse mixture of professors and teaching assistants. Before the seminar began, I asked for written answers to the following question: “Ideally, what would you like to get out of the next two days’ activities?” Their responses formed a collection of contradictory expectations. Some were cautionary: “You can tell me what to teach, but don’t tell me how .” Some wanted help with group work and cooperative learning, while others just wanted a general idea of what it meant to teach calculus “with modern pedagogy.” And many wanted speci fi cs: “How can you teach the Chain Rule ‘reform-style’?” “How much homework should I assign?” This ’s Guide tries to address the issues brought out by the above comments. The overall goal was not to write a reference book for a shelf, but to provide a user-friendly source of suggestions and activities for any teacher of calculus within a typical calculus curriculum. Instructors that have used previous editions of this book have said that it saved them a great deal of time, and helped them to teach a more student-oriented course. They have also reported that their classes have become more “fun,” but agreed that this unfortunate by-product of an engaged student population can’t always be avoided. This guide should be used together with Calculus, Seventh Edition as a source of both supplementary and com- plementary material. Depending on individual preference, instructors can choose from occasionally glancing through the Guide for content ideas and alternate approaches, or using the material from the ’s Guide as a major component in planning their day-to-day classes as well as to set homework assignments and read- ing quizzes. There are student activities and worksheets, sample exam questions, and examples for every section. Some of the continuing debates about changes in calculus content and pedagogy are rendered moot by adopt- ing the principle that the instruction of any topic in calculus can be enhanced by using a wider range of approaches. This guide includes some conceptual and geometric problems in topics as mundane as rules for differentiation, and as traditional as ε - δ limits. Whether a class consists of a straight lecture or an hour of group work, the materials provided are meant to help. I value reactions from all my colleagues who are teaching calculus from this guide, both to correct any errors and to suggest additional material for future editions. I am especially interested in which particular parts of the guide are the most and the least useful. Please email any feedback to calculus@dougshaw.com . iii

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    PREFACE This guide could not have been completed without the help of many people. I especially want to thank Jim Stewart for his continuing belief in this project and trust in me. My editor, Liza Neustaetter, has been won- derful. John Samons has been a big help on short notice, and I appreciate it. Previous versions of this guide have bene fi tted from the input of Virge Cornelius, Tom Hull, Joe Mercer, Melissa Pfohl, Michael Prophet, and Suzanne Riehl. Over the years, I’ve had students read through the guide and offer suggestions from their per- spective. Thanks go to Kate Degner, Ken Doss, Job Evers, Slade Hovick, Patricia Kloeckner, Jordan Meyer, Ben Nicholson, Paul Schou, Laura Waechter, and Cody Wilson. Further thanks go to James Stewart, John Hall, Robert Hesse, Harvey Keynes, Michael Lawler, and Dan O’Loughlin for their contributions to earlier incarnations of this guide. The book’s typesetter and proofreader, Andy Bulman-Fleming, again went above and beyond the call of duty, both in his work on the book, and in keeping me humble by regularly trouncing me at online Scrabble as we produced it. The talents of these people and others at Cengage have truly helped to make writing this guide a learning experience. This guide is dedicated to Russ Campbell. Doug Shaw iv

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    Contents How to Use the ’s Guide xiii How to Implement the Projects xv How to Use the Review Sections xvii How to Use the Problem-Solving Sections xviii Tips on In-Class Group Work xix 1 Functions and Limits 1 1.1 Four Ways to Represent a Function 1 1.2 Mathematical Models: A Catalog of Essential Functions 6 1.3 New Functions from Old Functions 14 1.4 The Tangent and Velocity Problems 21 1.5 The Limit of a Function 27 1.6 Calculating Limits Using the Limit Laws 37 1.7 The Precise De fi nition of a Limit 43 1.8 Continuity 54 Chapter 1 Sample Exam 64 Chapter 1 Sample Exam Solutions 70 2 Derivatives 73 2.1 Derivatives and Rates of Change 73 Writing Project ¦ Early Methods for Finding Tangents 81 2.2 The Derivative as a Function 82 2.3 Differentiation Formulas 105 Applied Project ¦ Building a Better Roller Coaster 115 2.4 Derivatives of Trigonometric Functions 116 2.5 The Chain Rule 122 Special Section ¦ Derivative Hangman 129 Applied Project ¦ Where Should a Pilot Start Descent? 129 2.6 Implicit Differentiation 130 Laboratory Project ¦ Families of Implicit Curves 139 2.7 Rates of Change in the Natural and Social Sciences 140 v

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    CONTENTS 2.8 Related Rates 143 2.9 Linear Approximations and Differentials 149 Laboratory Project ¦ Taylor Polynomials 154 Chapter 2 Sample Exam 155 Chapter 2 Sample Exam Solutions 159 3 Applications of Differentiation 161 3.1 Maximum and Minimum Values 161 Applied Project ¦ The Calculus of Rainbows 167 3.2 The Mean Value Theorem 168 3.3 How Derivatives Affect the Shape of a Graph 174 3.4 Limits at In fi nity; Horizontal Asymptotes 189 3.5 Summary of Curve Sketching 197 3.6 Graphing with Calculus and Calculators 206 3.7 Optimization Problems 211 Applied Project ¦ The Shape of a Can 218 3.8 Newton’s Method 219 3.9 Antiderivatives 225 Chapter 3 Sample Exam 234 Chapter 3 Sample Exam Solutions 240 4 Integrals 243 4.1 Areas and Distances 243 4.2 The De fi nite Integral 254 Discovery Project ¦ Area Functions 264 4.3 The Fundamental Theorem of Calculus 265 4.4 Inde fi nite Integrals and the Net Change Theorem 274 Writing Project ¦ Newton, Leibniz, and the Invention of Calculus 281 4.5 The Substitution Rule 282 Chapter 4 Sample Exam 289 Chapter 4 Sample Exam Solutions 295 vi

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    CONTENTS 5 Applications of Integration 299 5.1 Areas Between Curves 299 Applied Project ¦ The Gini Index 304 5.2 Volume 305 5.3 Volumes by Cylindrical Shells 316 5.4 Work 320 5.5 Average Value of a Function 326 Applied Project ¦ Calculus and Baseball 332 Chapter 5 Sample Exam 333 Chapter 5 Sample Exam Solutions 336 6 Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions 339 6.1 Inverse Functions 339 6.2 Exponential Functions and Their Derivatives 346 6.2* The Natural Logarithmic Function 368 6.3 Logarithmic Functions 352 6.3* The Natural Exponential Function 378 6.4 Derivatives of Logarithmic Functions 359 6.4* General Logarithmic and Exponential Functions 381 6.5 Exponential Growth and Decay 388 6.6 Inverse Trigonometric Functions 395 Applied Project ¦ Where to Sit at the Movies 400 6.7 Hyperbolic Functions 401 6.8 Indeterminate Forms and l’Hospital’s Rule 405 Writing Project ¦ The Origins of l’Hospital’s Rule 410 Chapter 6 Sample Exam 411 Chapter 6 Sample Exam Solutions 415 7 Techniques of Integration 417 7.1 Integration by Parts 417 7.2 Trigonometric Integrals 423 7.3 Trigonometric Substitution 428 7.4 Integration of Rational Functions by Partial Fractions 433 vii

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    CONTENTS 7.5 Strategy for Integration 440 7.6 Integration Using Tables and Computer Algebra Systems 450 Discovery Project ¦ Patterns in Integrals 453 7.7 Approximate Integration 454 7.8 Improper Integrals 458 Chapter 7 Sample Exam 468 Chapter 7 Sample Exam Solutions 473 8 Further Applications of Integration 477 8.1 Arc Length 477 Discovery Project ¦ Arc Length Contest 489 8.2 Area of a Surface of Revolution 490 Discovery Project ¦ Rotating on a Slant 495 8.3 Applications to Physics and Engineering 496 Discovery Project ¦ Complementary Coffee Cups 501 8.4 Applications to Economics and Biology 502 8.5 Probability 505 Chapter 8 Sample Exam 512 Chapter 8 Sample Exam Solutions 514 9 Differential Equations 517 9.1 Modeling with Differential Equations 517 9.2 Direction Fields and Euler’s Method 526 9.3 Separable Equations 548 Applied Project ¦ How Fast Does a Tank Drain? 559 Applied Project ¦ Which is Faster, Going Up or Coming Down? 559 9.4 Models for Population Growth 560 9.5 Linear Equations 565 9.6 Predator-Prey Systems 570 Chapter 9 Sample Exam 576 Chapter 9 Sample Exam Solutions 581 viii

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    CONTENTS 10 Parametric Equations and Polar Coordinates 585 10.1 Curves De fi ned by Parametric Equations 585 Laboratory Project ¦ Running Circles Around Circles 597 10.2 Calculus with Parametric Curves 598 Laboratory Project ¦ Be ´ zier Curves 604 10.3 Polar Coordinates 605 Laboratory Project ¦ Families of Polar Curves 611 10.4 Areas and Lengths in Polar Coordinates 612 10.5 Conic Sections 618 10.6 Conic Sections in Polar Coordinates 621 Chapter 10 Sample Exam 624 Chapter 10 Sample Exam Solutions 627 11 Infinite Sequences and Series 629 11.1 Sequences 629 Laboratory Project ¦ Logistic Sequences 638 11.2 Series 639 11.3 The Integral Test and Estimates of Sums 650 11.4 The Comparison Tests 655 11.5 Alternating Series 662 11.6 Absolute Convergence and the Ratio and Root Tests 667 11.7 Strategy for Testing Series 672 11.8 Power Series 674 11.9 Representation of Functions as Power Series 680 11.10 Taylor and Maclaurin Series 686 Laboratory Project ¦ An Elusive Limit 695 Writing Project ¦ How Newton Discovered the Binomial Series 695 11.11 Applications of Taylor Polynomials 696 Applied Project ¦ Radiation from the Stars 700 Chapter 11 Sample Exam 701 Chapter 11 Sample Exam Solutions 704 ix

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    CONTENTS 12 Vectors and the Geometry of Space 709 12.1 Three-Dimensional Coordinate Systems 709 12.2 Vectors 718 12.3 The Dot Product 727 12.4 The Cross Product 736 Discovery Project ¦ The Geometry of a Tetrahedron 742 12.5 Equations of Lines and Planes 743 Laboratory Project ¦ Putting 3D in Perspective 752 12.6 Cylinders and Quadric Surfaces 753 Chapter 12 Sample Exam 756 Chapter 12 Sample Exam Solutions 758 13 Vector Functions 761 13.1 Vector Functions and Space Curves 761 13.2 Derivatives and Integrals of Vector Functions 767 13.3 Arc Length and Curvature 775 13.4 Motion in Space: Velocity and Acceleration 784 Applied Project ¦ Kepler’s Laws 790 Chapter 13 Sample Exam 791 Chapter 13 Sample Exam Solutions 795 14 Partial Derivatives 799 14.1 Functions of Several Variables 799 14.2 Limits and Continuity 810 14.3 Partial Derivatives 816 14.4 Tangent Planes and Linear Approximations 823 14.5 The Chain Rule 827 14.6 Directional Derivatives and the Gradient Vector 833 14.7 Maximum and Minimum Values 840 Applied Project ¦ Designing a Dumpster 846 Discovery Project ¦ Quadratic Approximations and Critical Points 846 14.8 Lagrange Multipliers 847 x

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    CONTENTS Applied Project ¦ Rocket Science 853 Applied Project ¦ Hydro-Turbine Optimization 853 Chapter 14 Sample Exam 854 Chapter 14 Sample Exam Solutions 857 15 Multiple Integrals 861 15.1 Double Integrals over Rectangles 861 15.2 Iterated Integrals 869 15.3 Double Integrals over General Regions 874 15.4 Double Integrals in Polar Coordinates 880 15.5 Applications of Double Integrals 887 15.6 Surface Area 892 15.7 Triple Integrals 897 Discovery Project ¦ Volumes of Hyperspheres 904 15.8 Triple Integrals in Cylindrical Coordinates 905 Discovery Project ¦ The Intersection of Three Cylinders 909 15.9 Triple Integrals in Spherical Coordinates 910 Applied Project ¦ Roller Derby 918 15.10 Change of Variables in Multiple Integrals 919 Chapter 15 Sample Exam 925 Chapter 15 Sample Exam Solutions 929 16 Vector Calculus 933 16.1 Vector Fields 933 16.2 Line Integrals 942 16.3 The Fundamental Theorem for Line Integrals 950 16.4 Green’s Theorem 960 16.5 Curl and Divergence 966 16.6 Parametric Surfaces and Their Areas 973 16.7 Surface Integrals 985 16.8 Stokes’ Theorem 991 Writing Project ¦ Three Men and Two Theorems 997 xi

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    CONTENTS 16.9 The Divergence Theorem 998 Chapter 16 Sample Exam 1005 Chapter 16 Sample Exam Solutions 1010 17 Second-Order Differential Equations 1013 17.1 Second-Order Linear Equations 1013 17.2 Nonhomogeneous Linear Equations 1018 17.3 Applications of Second-Order Differential Equations 1022 17.4 Series Solutions 1025 Chapter 17 Sample Exam 1029 Chapter 17 Sample Exam Solutions 1030 xii

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    How to Use the ’s Guide For each section of Calculus, Seventh Edition , this ’s Guide provides information on the items listed below. 1. Suggested Time and Emphasis Here are suggestions for the amount of time to spend in a class of “average” students, and whether or not the material is essential to the rest of the course. If a section is labeled optional, the time range given is the amount of time for the material in the event that it is covered. 2. Points to Stress This is a short summary of the main topics to be covered. The stress is on the big ideas, rather than speci fi c details. 3. Quiz Questions Some instructors have reported that they like to open or close class by handing out a single question, either as a quiz or to start a discussion. Two types are included: • Text Question This question is designed for students who have done the reading, but haven’t yet seen the material in class. These questions can be used to help ensure that the students are reading the textbook carefully. • Drill Question These questions are designed to be straightforward “right down the middle” questions for students who have tried, but not necessarily mastered, the material. 4. Materials for Lecture These suggestions are meant to work along with the text to create a classroom atmosphere of experimentation and inquiry. They have a theoretical bent to help the students understand the material at a deep conceptual level. In a course with a “lecture and discussion” format, these ideas can be used during the lectures. 5. Workshop/Discussion These suggestions are interesting examples and applications aimed at motivating the material and helping the students master it. In a course with a “lecture and discussion” format, these ideas can be used during the discussions. 6. Group Work One of the main dif fi culties instructors have in presenting group work to their classes is that of choosing an appropriate group task. Suggestions for implementation and answers to the group activities are provided fi rst, followed by photocopy-ready handouts on separate pages. The guide’s main philosophy of group work is that there should be a solid introduction to each exercise (“What are we supposed to do?”) and good closure before class is dismissed (“Why did we just do that?”) 7. TE C Tools for Enriching Calculus is a companion to the text, intended to enrich and complement its contents. You can fi nd it in Enhanced WebAssign, CourseMate, and PowerLecture. Marginal notes in the main text direct students to TEC modules where appropriate. When a TEC module relates to an ’s Guide item, it is referenced there as well. 8. Homework Problems For each section, a set of essential Core Exercises (a bare minimum set of homework problems) is provided. Using this core set as a base, a Sample Assignment is suggested, and each exercise in that assignment is classi fi ed as Descriptive , Algebraic , Numeric , and/or Graphic . • Descriptive: The student is required to translate mathematical concepts into everyday terms, or vice- versa. • Algebraic: The student is required to use algebraic and/or symbolic manipulation and computation. xiii

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    HOW TO USE THE ’S GUIDE • Numeric: The student is required to work with numerical data, or provide a numerical approximation. • Graphic: The student is required to provide or receive information presented in the form of a graph. In addition, TEC includes “homework hints” for representative exercises (usually odd-numbered) for each section of the text. These hints are constructed so as not to reveal any more of the actual solution than is minimally necessary to make progress toward the solution. Also available is a Student Solutions Manual which presents complete solutions to all of the odd-numbered exercises in the text. 9. Sample Exam Questions I recommend that tests in a calculus course have a mix of routine and non- routine questions. The sample exam questions provided are meant to inspire the “non-routine” portion of a calculus test. We do not recommend that calculus tests be composed entirely of questions from this section. One strategy is to announce to the students that one-third of the text questions will be based on homework, one-third will be based on in-class group work, and one-third will not be immediately familiar. 10. Web Resources Useful resources can be found on the website for Calculus, Seventh Edition ( http://www.stewartcalculus.com ). xiv

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    How to Implement the Projects One exciting yet intimidating aspect of teaching a calculus course is projects. An extended assignment gives students the chance to take a focused problem or project and explore it in-depth — making conjectures, discussing them, eventually drawing conclusions and writing them up in a clear, precise format. Calculus, Seventh Edition has many possible projects throughout its chapters. Here are some tips on ensuring that your students have a successful experience. Time Students should have two to three weeks to work on any extended out-of-class assignment. This is not because they will need all this time to complete them! But a fi fteen-to-twenty-day deadline allows the students to be fl exible in structuring their time wisely, and allows the instructors to apply fairly strict standards in grading the work. Groups Students usually work in teams and are expected to have team meetings. The main problem students have in setting up these meetings is scheduling. Four randomly selected undergraduates will probably fi nd it very hard to get together for more than a few hours, which may not be suf fi cient. One way to help your students is to clearly specify a minimum number of meetings, and have one or all group members turn in summaries of what was accomplished at each meeting.On a commuter campus, a good fi rst grouping might be by location. Studies have shown that the optimal group size is three people, followed by four, then two. I advocate groups of four whenever possible. That way, if someone doesn’t show up to a team meeting, there are still three people there to discuss the problems. Before the fi rst project, students should discuss the different roles that are assumed in a team. Who will be responsible for keeping people informed of where and when they meet? Who will be responsible for making sure that the fi nal copy of the report is all together when it is supposed to be? These types of jobs can be assigned within the team, or by the teacher at the outset. Tell the students that you will be grading on both content and presentation. They should gear their work toward an audience that is bright, but not necessarily up-to-speed on this problem. For example, they can think of themselves as professional mathematicians writing for a manager, or as research assistants writing for a professor who is not necessarily a mathematician. If the students are expected to put some effort into the project, it is important to let them know that some effort was put into the grading. Both form and content should be commented on, and recognition of good aspects of their work should be included along with criticism. One way to help ensure cooperation is to let the students know that there will be an exam question based on the project. If every member of the group does well on that particular question, then they can all get a bonus, either on the exam or on the project grade. xv

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    HOW TO IMPLEMENT THE PROJECTS Providing assistance Make sure that the students know when you are available to help them, and what kind of help you are willing to provide. Students may be required to hand in a rough draft ten days before the due date, to give them a little more structure and to make sure they have a solid week to write up the assignment. Individual Accountability It is important that the students are individually accountable for the output of their group. Giving each student a different grade is a dangerous solution, because it does not necessarily encourage the students to discuss the material, and may actually discourage their working together. A better alternative might be to create a feedback form. If the students are given a copy of the feedback form ahead of time, and they know that their future group placement will be based on what they do in their present group, then they are given an incentive to work hard. One surprising result is that when a group consists of students who were previously slackers, that group often does quite well. The exam question idea discussed earlier also gives individuals an incentive to keep up with their colleagues. xvi

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