Multivariable Calculus, 7th Edition Solution Manual

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’s GuidePrepared byDouglas ShawUniversity of Northern IowaCalculusSEVENTH EDITIONJames StewartMcMaster University and University of Toronto

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PrefaceI once facilitated a mandatory workshop on the teaching of calculus that was attended by a diverse mixtureof professors and teaching assistants. Before the seminar began, I asked for written answers to the followingquestion: “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 tellmewhatto teach, but don’t tell mehow.” 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.” Andmany wanted specifics: “How can you teach the Chain Rule ‘reform-style’?” “How much homework shouldI assign?”This’s Guide tries to address the issues brought out by the above comments. The overall goal wasnot to write a reference book for a shelf, but to provide a user-friendly source of suggestions and activities forany teacher of calculus within a typical calculus curriculum. Instructors that have used previous editions ofthis book have said that it saved them a great deal of time, and helped them to teach a more student-orientedcourse. They have also reported that their classes have become more “fun,” but agreed that this unfortunateby-product of an engaged student population can’t always be avoided.This guide should be used together withCalculus, Seventh Editionas a source of both supplementary and com-plementary material. Depending on individual preference, instructors can choose from occasionally glancingthrough the Guide for content ideas and alternate approaches, or using the material from the’s Guideas 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 everysection.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 ofapproaches. This guide includes some conceptual and geometric problems in topics as mundane as rules fordifferentiation, and as traditional asε-δlimits. Whether a class consists of a straight lecture or an hour ofgroup 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 errorsand to suggest additional material for future editions. I am especially interested in which particular parts ofthe guide are the most and the least useful. Please email any feedback tocalculus@dougshaw.com.iii

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PREFACEThis guide could not have been completed without the help of many people. I especially want to thank JimStewart 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 guidehave benefitted from the input of Virge Cornelius, Tom Hull, Joe Mercer, Melissa Pfohl, Michael Prophet, andSuzanne 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, JohnHall, Robert Hesse, Harvey Keynes, Michael Lawler, and Dan O’Loughlin for their contributions to earlierincarnations of this guide. The book’s typesetter and proofreader, Andy Bulman-Fleming, again went aboveand beyond the call of duty, both in his work on the book, and in keeping me humble by regularly trouncingme at online Scrabble as we produced it. The talents of these people and others at Cengage have truly helpedto make writing this guide a learning experience.This guide is dedicated to Russ Campbell.Doug Shawiv

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ContentsHow to Use the’s GuidexiiiHow to Implement the ProjectsxvHow to Use the Review SectionsxviiHow to Use the Problem-Solving SectionsxviiiTips on In-Class Group Workxix1Functions and Limits11.1Four Ways to Represent a Function11.2Mathematical Models: A Catalog of Essential Functions61.3New Functions from Old Functions141.4The Tangent and Velocity Problems211.5The Limit of a Function271.6Calculating Limits Using the Limit Laws371.7The Precise Definition of a Limit431.8Continuity54Chapter 1 Sample Exam64Chapter 1 Sample Exam Solutions702Derivatives732.1Derivatives and Rates of Change73Writing Project¦Early Methods for Finding Tangents812.2The Derivative as a Function822.3Differentiation Formulas105Applied Project¦Building a Better Roller Coaster1152.4Derivatives of Trigonometric Functions1162.5The Chain Rule122Special Section¦Derivative Hangman129Applied Project¦Where Should a Pilot Start Descent?1292.6Implicit Differentiation130Laboratory Project¦Families of Implicit Curves1392.7Rates of Change in the Natural and Social Sciences140v

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CONTENTS2.8Related Rates1432.9Linear Approximations and Differentials149Laboratory Project¦Taylor Polynomials154Chapter 2 Sample Exam155Chapter 2 Sample Exam Solutions1593Applications of Differentiation1613.1Maximum and Minimum Values161Applied Project¦The Calculus of Rainbows1673.2The Mean Value Theorem1683.3How Derivatives Affect the Shape of a Graph1743.4Limits at Infinity; Horizontal Asymptotes1893.5Summary of Curve Sketching1973.6Graphing with CalculusandCalculators2063.7Optimization Problems211Applied Project¦The Shape of a Can2183.8Newton’s Method2193.9Antiderivatives225Chapter 3 Sample Exam234Chapter 3 Sample Exam Solutions2404Integrals2434.1Areas and Distances2434.2The Definite Integral254Discovery Project¦Area Functions2644.3The Fundamental Theorem of Calculus2654.4Indefinite Integrals and the Net Change Theorem274Writing Project¦Newton, Leibniz, and the Invention of Calculus2814.5The Substitution Rule282Chapter 4 Sample Exam289Chapter 4 Sample Exam Solutions295vi

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CONTENTS5Applications of Integration2995.1Areas Between Curves299Applied Project¦The Gini Index3045.2Volume3055.3Volumes by Cylindrical Shells3165.4Work3205.5Average Value of a Function326Applied Project¦Calculus and Baseball332Chapter 5 Sample Exam333Chapter 5 Sample Exam Solutions3366Inverse Functions:Exponential, Logarithmic, and Inverse Trigonometric Functions3396.1Inverse Functions3396.2Exponential Functions andTheir Derivatives3466.2*The Natural LogarithmicFunction3686.3LogarithmicFunctions3526.3*The Natural ExponentialFunction3786.4Derivatives ofLogarithmic Functions3596.4*General Logarithmic andExponential Functions3816.5Exponential Growth and Decay3886.6Inverse Trigonometric Functions395Applied Project¦Where to Sit at the Movies4006.7Hyperbolic Functions4016.8Indeterminate Forms and l’Hospital’s Rule405Writing Project¦The Origins of l’Hospital’s Rule410Chapter 6 Sample Exam411Chapter 6 Sample Exam Solutions4157Techniques of Integration4177.1Integration by Parts4177.2Trigonometric Integrals4237.3Trigonometric Substitution4287.4Integration of Rational Functions by Partial Fractions433vii

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CONTENTS7.5Strategy for Integration4407.6Integration Using Tables and Computer Algebra Systems450Discovery Project¦Patterns in Integrals4537.7Approximate Integration4547.8Improper Integrals458Chapter 7 Sample Exam468Chapter 7 Sample Exam Solutions4738Further Applications of Integration4778.1Arc Length477Discovery Project¦Arc Length Contest4898.2Area of a Surface of Revolution490Discovery Project¦Rotating on a Slant4958.3Applications to Physics and Engineering496Discovery Project¦Complementary Coffee Cups5018.4Applications to Economics and Biology5028.5Probability505Chapter 8 Sample Exam512Chapter 8 Sample Exam Solutions5149Differential Equations5179.1Modeling with Differential Equations5179.2Direction Fields and Euler’s Method5269.3Separable Equations548Applied Project¦How Fast Does a Tank Drain?559Applied Project¦Which is Faster, Going Up or Coming Down?5599.4Models for Population Growth5609.5Linear Equations5659.6Predator-Prey Systems570Chapter 9 Sample Exam576Chapter 9 Sample Exam Solutions581viii

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CONTENTS10Parametric Equations and Polar Coordinates58510.1Curves Defined by Parametric Equations585Laboratory Project¦Running Circles Around Circles59710.2Calculus with Parametric Curves598Laboratory Project¦Be´zier Curves60410.3Polar Coordinates605Laboratory Project¦Families of Polar Curves61110.4Areas and Lengths in Polar Coordinates61210.5Conic Sections61810.6Conic Sections in Polar Coordinates621Chapter 10 Sample Exam624Chapter 10 Sample Exam Solutions62711Infinite Sequences and Series62911.1Sequences629Laboratory Project¦Logistic Sequences63811.2Series63911.3The Integral Test and Estimates of Sums65011.4The Comparison Tests65511.5Alternating Series66211.6Absolute Convergence and the Ratio and Root Tests66711.7Strategy for Testing Series67211.8Power Series67411.9Representation of Functions as Power Series68011.10Taylor and Maclaurin Series686Laboratory Project¦An Elusive Limit695Writing Project¦How Newton Discovered the Binomial Series69511.11Applications of Taylor Polynomials696Applied Project¦Radiation from the Stars700Chapter 11 Sample Exam701Chapter 11 Sample Exam Solutions704ix

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CONTENTS12Vectors and the Geometry of Space70912.1Three-Dimensional Coordinate Systems70912.2Vectors71812.3The Dot Product72712.4The Cross Product736Discovery Project¦The Geometry of a Tetrahedron74212.5Equations of Lines and Planes743Laboratory Project¦Putting 3D in Perspective75212.6Cylinders and Quadric Surfaces753Chapter 12 Sample Exam756Chapter 12 Sample Exam Solutions75813Vector Functions76113.1Vector Functions and Space Curves76113.2Derivatives and Integrals of Vector Functions76713.3Arc Length and Curvature77513.4Motion in Space: Velocity and Acceleration784Applied Project¦Kepler’s Laws790Chapter 13 Sample Exam791Chapter 13 Sample Exam Solutions79514Partial Derivatives79914.1Functions of Several Variables79914.2Limits and Continuity81014.3Partial Derivatives81614.4Tangent Planes and Linear Approximations82314.5The Chain Rule82714.6Directional Derivatives and the Gradient Vector83314.7Maximum and Minimum Values840Applied Project¦Designing a Dumpster846Discovery Project¦Quadratic Approximations and Critical Points84614.8Lagrange Multipliers847x

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CONTENTSApplied Project¦Rocket Science853Applied Project¦Hydro-Turbine Optimization853Chapter 14 Sample Exam854Chapter 14 Sample Exam Solutions85715Multiple Integrals86115.1Double Integrals over Rectangles86115.2Iterated Integrals86915.3Double Integrals over General Regions87415.4Double Integrals in Polar Coordinates88015.5Applications of Double Integrals88715.6Surface Area89215.7Triple Integrals897Discovery Project¦Volumes of Hyperspheres90415.8Triple Integrals in Cylindrical Coordinates905Discovery Project¦The Intersection of Three Cylinders90915.9Triple Integrals in Spherical Coordinates910Applied Project¦Roller Derby91815.10Change of Variables in Multiple Integrals919Chapter 15 Sample Exam925Chapter 15 Sample Exam Solutions92916Vector Calculus93316.1Vector Fields93316.2Line Integrals94216.3The Fundamental Theorem for Line Integrals95016.4Green’s Theorem96016.5Curl and Divergence96616.6Parametric Surfaces and Their Areas97316.7Surface Integrals98516.8Stokes’ Theorem991Writing Project¦Three Men and Two Theorems997xi

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CONTENTS16.9The Divergence Theorem998Chapter 16 Sample Exam1005Chapter 16 Sample Exam Solutions101017Second-Order Differential Equations101317.1Second-Order Linear Equations101317.2Nonhomogeneous Linear Equations101817.3Applications of Second-Order Differential Equations102217.4Series Solutions1025Chapter 17 Sample Exam1029Chapter 17 Sample Exam Solutions1030xii

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How to Use the’s GuideFor each section ofCalculus, Seventh Edition, this’s Guideprovides information on the items listedbelow.1. Suggested Time and EmphasisHere 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 labeledoptional, the time range given is the amount of time for the material in the event that it is covered.2. Points to StressThis is a short summary of the main topics to be covered. The stress is on the big ideas,rather than specific details.3. Quiz QuestionsSome instructors have reported that they like to open or close class by handing out a singlequestion, either as a quiz or to start a discussion. Two types are included:•Text QuestionThis question is designed for students who have done the reading, but haven’t yet seen thematerial in class. These questions can be used to help ensure that the students are reading the textbookcarefully.•Drill QuestionThese questions are designed to be straightforward “right down the middle” questions forstudents who have tried, but not necessarily mastered, the material.4. Materials for LectureThese suggestions are meant to work along with the text to create a classroomatmosphere of experimentation and inquiry. They have a theoretical bent to help the students understandthe material at a deep conceptual level. In a course with a “lecture and discussion” format, these ideas canbe used during the lectures.5. Workshop/DiscussionThese suggestions are interesting examples and applications aimed at motivatingthe material and helping the students master it. In a course with a “lecture and discussion” format, theseideas can be used during the discussions.6. Group WorkOne of the main difficulties instructors have in presenting group work to their classes is thatof choosing an appropriate group task. Suggestions for implementation and answers to the group activitiesare providedfirst, followed by photocopy-ready handouts on separate pages. The guide’s main philosophyof group work is that there should be a solid introduction to each exercise (“What are we supposed todo?”) and good closure before class is dismissed (“Why did we just do that?”)7.TECTools for Enriching Calculusis a companion to the text, intended to enrich and complement itscontents. You canfind it in Enhanced WebAssign, CourseMate, and PowerLecture. Marginal notes in themain text direct students toTECmodules where appropriate. When aTECmodule relates to an’sGuideitem, it is referenced there as well.8. Homework ProblemsFor each section, a set of essentialCore Exercises(a bare minimum set of homeworkproblems) is provided. Using this core set as a base, aSample Assignmentis suggested, and each exercisein that assignment is classified asDescriptive,Algebraic,Numeric, and/orGraphic.•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,TECincludes “homework hints” for representative exercises (usually odd-numbered) for eachsection of the text. These hints are constructed so as not to reveal any more of the actual solution than isminimally necessary to make progress toward the solution. Also available is aStudent Solutions Manualwhich presents complete solutions to all of the odd-numbered exercises in the text.9. Sample Exam QuestionsI 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” portionof a calculus test. We do not recommend that calculus tests be composed entirely of questions from thissection. One strategy is to announce to the students that one-third of the text questions will be based onhomework, one-third will be based on in-class group work, and one-third will not be immediately familiar.10. Web ResourcesUsefulresourcescanbefoundonthewebsiteforCalculus,SeventhEdition(http://www.stewartcalculus.com).xiv

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How to Implement the ProjectsOne exciting yet intimidating aspect of teaching a calculus course is projects. An extended assignment givesstudents 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 Editionhas many possible projects throughout its chapters. Here are some tips on ensuring that yourstudents have a successful experience.TimeStudents should have two to three weeks to work on any extended out-of-class assignment. This isnot because they will need all this time to complete them!But afifteen-to-twenty-day deadline allows thestudents to beflexible in structuring their time wisely, and allows the instructors to apply fairly strict standardsin grading the work.GroupsStudents usually work in teams and are expected to have team meetings. The main problem studentshave in setting up these meetings is scheduling. Four randomly selected undergraduates will probablyfindit very hard to get together for more than a few hours, which may not be sufficient. One way to help yourstudents is to clearly specify a minimum number of meetings, and have one or all group members turn insummaries of what was accomplished at each meeting.On a commuter campus, a goodfirst grouping mightbe by location.Studies have shown that the optimal group size is three people, followed by four, then two. I advocate groupsof four whenever possible. That way, if someone doesn’t show up to a team meeting, there are still threepeople there to discuss the problems.Before thefirst project, students should discuss the different roles that are assumed in a team. Who will beresponsible for keeping people informed of where and when they meet? Who will be responsible for makingsure that thefinal copy of the report is all together when it is supposed to be? These types of jobs can beassigned 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 worktoward an audience that is bright, but not necessarily up-to-speed on this problem. For example, they canthink of themselves as professional mathematicians writing for a manager, or as research assistants writingfor 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 effortwas put into the grading. Both form and content should be commented on, and recognition of good aspects oftheir 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 onthe 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 PROJECTSProviding assistanceMake sure that the students know when you are available to help them, and what kindof help you are willing to provide. Students may be required to hand in a rough draft ten days before the duedate, to give them a little more structure and to make sure they have a solid week to write up the assignment.Individual AccountabilityIt is important that the students are individually accountable for the output of theirgroup. Giving each student a different grade is a dangerous solution, because it does not necessarily encouragethe students to discuss the material, and may actually discourage their working together. A better alternativemight be to create a feedback form. If the students are given a copy of the feedback form ahead of time, andthey know that their future group placement will be based on what they do in their present group, then theyare given an incentive to work hard. One surprising result is that when a group consists of students who werepreviously slackers, that group often does quite well. The exam question idea discussed earlier also givesindividuals an incentive to keep up with their colleagues.xvi

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