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INSTRUCTOR’S RESOURCE GUIDE

AND TEST BANK

BERNARD GILLETT University of Colorado Boulder

CALCULUS FOR SCIENTISTS AND ENGINEERS

EARLY TRANSCENDENTALS

William Briggs University of Colorado Denver

Lyle Cochran Whitworth University

Bernard Gillett University of Colorado Boulder

with the assistance of

Eric Schulz Walla Walla Community College


www.pearsonhighered.com

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 from electronic files supplied by the author. Copyright © 2013 Pearson Education, Inc. Publishing as Pearson, 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: 978-0-321-78538-1 ISBN-10: 0-321-78538-X 1 2 3 4 5 6 OPM 16 15 14 13 12


Copyright © 2013 Pearson Education, Inc.

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Table of Contents Preface viii Learning Objectives xii Index of Applications xxxi Chapter 1 Functions 1 1.1 Review of Functions 1 1.2 Representing Functions 4 1.3 Inverse, Exponential, and Logarithmic Functions 8 1.4 Trigonometric Functions and Their Inverses 11 Chapter 1 Key Terms and Concepts 14 Chapter 1 Review Questions 15 Chapter 1 Test Bank Exercises 16

Chapter 2 Limits 19 2.1 The Idea of Limits 19 2.2 Definitions of Limits 22 2.3 Techniques for Computing Limits 25 2.4 Infinite Limits 28 2.5 Limits at Infinity 31 2.6 Continuity 34 2.7 Precise Definitions of Limits 37 Chapter 2 Key Terms and Concepts 39 Chapter 2 Review Questions 40 Chapter 2 Test Bank Exercises 41

Chapter 3 Derivatives 43 3.1 Introducing the Derivative 43 3.2 Rules of Differentiation 46 3.3 The Product and Quotient Rules 49 3.4 Derivatives of Trigonometric Functions 52 3.5 Derivatives as Rates of Change 54 3.6 The Chain Rule 56 3.7 Implicit Differentiation 59 3.8 Derivatives of Logarithmic and Exponential Functions 62 3.9 Derivatives of Inverse Trigonometric Functions 64 3.10 Related Rates 67 Chapter 3 Key Terms and Concepts 69 Chapter 3 Review Questions 70 Chapter 3 Test Bank Exercises 72



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Chapter 4 Applications of the Derivative 75 4.1 Maxima and Minima 75 4.2 What Derivatives Tell Us 78 4.3 Graphing Functions 81 4.4 Optimization Problems 84 4.5 Linear Approximation and Differentials 86 4.6 Mean Value Theorem 89 4.7 L’Hôpital’s Rule 92 4.8 Newton’s Method 95 4.9 Antiderivatives 99 Chapter 4 Key Terms and Concepts 101 Chapter 4 Review Questions 102 Chapter 4 Test Bank Exercises 103

Chapter 5 Integration 105 5.1 Approximating Areas under Curves 105 5.2 Definite Integrals 108 5.3 Fundamental Theorem of Calculus 111 5.4 Working with Integrals 114 5.5 Substitution Rule 116 Chapter 5 Key Terms and Concepts 118 Chapter 5 Review Questions 119 Chapter 5 Test Bank Exercises 120

Chapter 6 Applications of Integration 123 6.1 Velocity and Net Change 123 6.2 Regions Between Curves 128 6.3 Volume by Slicing 131 6.4 Volume by Shells 134 6.5 Length of Curves 136 6.6 Surface Area 138 6.7 Physical Applications 142 6.8 Logarithmic and Exponential Functions Revisited 145 6.9 Exponential Models 147 6.10 Hyperbolic Functions 149 Chapter 6 Key Terms and Concepts 153 Chapter 6 Review Questions 154 Chapter 6 Test Bank Exercises 156

Chapter 7 Integration Techniques 161 7.1 Basic Approaches 161 7.2 Integration by Parts 165 7.3 Trigonometric Integrals 169 7.4 Trigonometric Substitutions 172 7.5 Partial Fractions 175 7.6 Other Integration Strategies 177 7.7 Numerical Integration 179 7.8 Improper Integrals 181 Chapter 7 Key Terms and Concepts 183 Chapter 7 Review Questions 184 Chapter 7 Test Bank Exercises 185



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Chapter 8 Differential Equations 187 8.1 Basic Ideas 187 8.2 Direction Fields and Euler’s Method 191 8.3 Separable Differential Equations 194 8.4 Special First-Order Linear Differential Equations 196 8.5 Modeling with Differential Equations 198 Chapter 8 Key Terms and Concepts 200 Chapter 8 Review Questions 201 Chapter 8 Test Bank Exercises 203

Chapter 9 Sequences and Infinite Series 205 9.1 An Overview 205 9.2 Sequences 208 9.3 Infinite Series 210 9.4 The Divergence and Integral Tests 212 9.5 The Ratio, Root, and Comparison Tests 214 9.6 Alternating Series 217 Chapter 9 Key Terms and Concepts 219 Chapter 9 Review Questions 220 Chapter 9 Test Bank Exercises 222

Chapter 10 Power Series 223 10.1 Approximating Functions with Polynomials 223 10.2 Properties of Power Series 226 10.3 Taylor Series 228 10.4 Working with Taylor Series 230 Chapter 10 Key Terms and Concepts 232 Chapter 10 Review Questions 233 Chapter 10 Test Bank Exercises 234

Chapter 11 Parametric and Polar Curves 237 11.1 Parametric Equations 237 11.2 Polar Coordinates 240 11.3 Calculus in Polar Coordinates 243 11.4 Conic Sections 246 Chapter 11 Key Terms and Concepts 249 Chapter 11 Review Questions 250 Chapter 11 Test Bank Exercises 251

Chapter 12 Vectors and Vector-Valued Functions 253 12.1 Vectors in the Plane 254 12.2 Vectors in Three Dimensions 257 12.3 Dot Products 260 12.4 Cross Products 262 12.5 Lines and Curves in Space 264 12.6 Calculus of Vector-Valued Functions 267 12.7 Motion in Space 269 12.8 Length of Curves 271 12.9 Curvature and Normal Vectors 273 Chapter 12 Key Terms and Concepts 276 Chapter 12 Review Questions 277 Chapter 12 Test Bank Exercises 279



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Chapter 13 Functions of Several Variables 281 13.1 Planes and Surfaces 281 13.2 Graphs and Level Curves 284 13.3 Limits and Continuity 287 13.4 Partial Derivatives 290 13.5 The Chain Rule 292 13.6 Directional Derivatives and the Gradient 294 13.7 Tangent Planes and Linear Approximation 297 13.8 Maximum/Minimum Problems 299 13.9 Lagrange Multipliers 301 Chapter 13 Key Terms and Concepts 304 Chapter 13 Review Questions 305 Chapter 13 Test Bank Exercises 307

Chapter 14 Multiple Integration 311 14.1 Double Integrals over Rectangular Regions 311 14.2 Double Integrals over General Regions 314 14.3 Double Integrals in Polar Coordinates 317 14.4 Triple Integrals 319 14.5 Triple Integrals in Cylindrical and Spherical Coordinates 322 14.6 Integrals for Mass Calculations 326 14.7 Change of Variables in Multiple Integrals 328 Chapter 14 Key Terms and Concepts 330 Chapter 14 Review Questions 331 Chapter 14 Test Bank Exercises 333

Chapter 15 Vector Calculus 337 15.1 Vector Fields 337 15.2 Line Integrals 341 15.3 Conservative Vector Fields 344 15.4 Green’s Theorem 346 15.5. Divergence and Curl 348 15.6 Surface Integrals 350 15.7 Stokes’ Theorem 353 15.8 Divergence Theorem 355 Chapter 15 Key Terms and Concepts 357 Chapter 15 Review Questions 358 Chapter 15 Test Bank Exercises 359



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Chapter 1 Guided Projects 1. Problem-solving skills 361 2. Constant-rate problems 364 3. Functions in action I 366 4. Functions in action II 368 5. Supply and demand 370 6. Phase and amplitude 373 7. Atmospheric CO2

375 8. Acid, noise, and earthquakes 376

Chapter 2 Guided Projects 9. Fixed point iteration 378 10. Local linearity 380

Chapter 3 Guided Projects 11. Numerical differentiation 382 12. Enzyme kinetics 384 13. Elasticity in economics 386 14. Pharmacokinetics—drug metabolism 388

Chapter 4 Guided Projects 15. Oscillators 389 16. Ice cream, geometry and calculus 391 17. Newton’s method 393

Chapter 5 Guided Projects 18. Limits of sums 396 19. Distribution of wealth 397 20. Symmetry in integrals 399

Chapter 6 Guided Projects 21. Means and tangent lines 400 22. Landing an airliner 402 23. Geometric probability 404 24. Mathematics of the CD player 407 25. Designing a water clock 409 26. Buoyancy and Archimedes’ principle 411 27. Dipstick problems 413 28. Hyperbolic functions 416 29. Optimizing fuel use 418 30. Inverse sine from geometry 420

Chapter 7 Guided Projects 31. Simpson’s rule 422 32. How long will your iPod last? 425 33. Mercator projections 427

Chapter 8 Guided Projects 34. Cooling coffee 430 35. Euler’s method for differential equations 432 36. Predator-prey models 435 37. Period of the pendulum 439 38. Terminal velocity 442 39. Logistic growth 445 40. A pursuit problem 448

Chapter 9 Guided Projects 41. Chaos! 450 42. Financial matters 452 43. Periodic drug dosing 455 44. Economic stimulus packages 457 45. The mathematics of loans 459 46. Archimedes’ approximation to π 460 47. Exact values of infinite series 462 48. Conditional convergence in a crystal

lattice 464

Chapter 10 Guided Projects 49. Series approximations to π 466 50. Euler’s formula (Taylor series with complex

numbers) 468 51. Stirling’s formula and n! 469 52. Three-sigma quality control 471 53. Fourier series 474

Chapter 11 Guided Projects 54. The amazing cycloid 480 55. Parametric art 483 56. Polar art 488 57. Grazing goat problems 491 58. Translations and rotations of axes 494 59. Celestial orbits 498 60. Properties of conic sections 500

Chapter 12 Guided Projects 61. Designing a trajectory 504 62. Intercepting a UFO 507 63. CORDIC algorithms: How your calculator

works 509 64. Bezier curves for graphic design 514 65. Kepler’s laws 516

Chapter 13 Guided Projects 66. Traveling waves 519 67. Ecological diversity 522 68. Economic production functions 524

Chapter 14 Guided Projects 69. How big are n-balls? 527 70. Electric field integrals 529 71. The tilted cylinder problem 533 72. The exponential Eiffel Tower 535 73. Moments of inertia 537 74. Gravitational fields 540

Chapter 15 Guided Projects 75. Ideal fluid flow 543 76. Maxwell’s equations 546 77. Planimeters and vector fields 552 78. Vector calculus in other coordinate

systems 555 Answers to Chapter-Level Content A-1 Solutions to Guided Projects A-68 Single Variable Student Study Cards SC-1 Multivariable Student Study Cards SC-8



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Preface This guide accompanies Calculus for Scientists and Engineers: Early Transcendentals by Briggs, Cochran, Gillett, and Schulz. Think of it as a roadmap to the textbook and a collection of resources for use in your course. Though one might identify the main audience of this book as graduate students or instructors early in their careers, my hope is that seasoned professors will also benefit from the material offered here. Bernard Gillett

Features of the Text The essential features of Calculus for Scientists and Engineers: Early Transcendentals are spelled out in its preface. We encourage you to become familiar with all aspects of the text, including its online components, so that you can make informed decisions about what to incorporate into your courses. The most important features are highlighted here.

• Make students aware of the explicit connection between the worked examples in the text and the Basic Skills exercises. Each block of Basic Skills exercises is directly linked to an example in the narrative so that students can refer to the example in question while solving exercises of a similar nature. The decision to do this was very deliberate: We want students to read the text. Our hope is that this decision will increase their chances of understanding the material.

• The static figures, tables, key concepts and definitions found in the text are available within MyMathLab® as PowerPoint® slides. Use them in conjunction with your own prepared slides or as a supplement to what you present at the chalkboard. The authors went to great lengths to provide figures that “speak” to students in order to facilitate geometric intuition and a deeper understanding of calculus. We feel students will benefit from viewing professionally rendered figures in those instances where even the best chalkboard artist among us falls short of the mark. Our advice: invest some time in becoming familiar with these resources, and integrate them into your lectures.

• The interactive figures in the electronic version of the text are also available within MyMathLab. These figures bring alive the concepts of calculus, generate exciting classroom discussions, and provide students with laboratories for further exploration. In short, don’t miss out on them; they will revolutionize the way we teach and the way students learn calculus.

Features of this Guide The first third of this guide consists of fifteen chapters that correspond to the chapters in the textbook. Each chapter begins with a brief overview of the material covered in the corresponding chapter of the text, sometimes sprinkled with our reasoning for structuring the text as we have. Following this introduction, we provide teaching strategies and classroom activities for each section of the text, as detailed below. Overview A quick summary of the section’s content is given to get your bearings. Lecture Support Notes Lecture Support Notes are teaching tips linked to each section of the text. We suggest strategies for covering the contents of the section, clarify technical points and terminology, recommend key examples and figures, and on occasion, provide the logic behind the choices we made when writing the book. Everything in this guide is as advertised: a guide to teaching your course. It is inevitable that the advice we give is colored by our own preferences and teaching styles. We encourage you to use this guide to complement—but not replace—your unique style of teaching.

It is also important to recognize that the Lecture Support Notes were written with the assumption that each section of the book will be covered in full. We acknowledge that this is an unrealistic expectation (see Optional Sections, p. vii); our aim is to provide guidance for every section without passing judgment on the relative importance of a particular section’s contents within the calculus curriculum.



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Interactive Figures In this section we list and briefly describe the interactive figures of the electronic book, created by Eric Schulz of Walla Walla Community College, that correspond to the text section at hand. The intent is to let you know what figures are available and to encourage you to integrate them into your classroom presentations. All the interactive figures are accessible through MyMathLab. Connections Teaching calculus is a daunting task, especially in the early stages of an instructor’s career. It’s easy to fall into the trap of compartmentalizing the knowledge you are communicating into disjoint pieces, viewing each section as a separate entity. With Connections, we attempt to point out some of the many threads that tie the ideas of calculus together. This component is aimed primarily at instructors (for planning purposes and to see the big picture), but it is also intended for your students. For example, in Section 6.3 of this guide (p. 131) we point out that the general slicing method is used in Chapter 14 to explain the inner workings of an iterated integral and that solids of revolution and their bounding surfaces are featured prominently in multivariable calculus. The first of these facts informs instructors that teaching the general slicing method now will ease the job of explaining iterated integrals in Chapter 14. The second observation may be something you’d like to share with your students. Calling attention to the links between various ideas in calculus will help students view calculus as a unified whole rather than a disparate collection of mathematical facts. Additional Activities The entries found under the heading of Additional Activities range from five-minute hands-on experiments to detailed guided projects (see Guided Projects, pp. v, ix). Each activity is linked to a particular section, though some activities can be applied to other sections to suit your preferences. A handful of activities that require the use of technology were written with the expectation that Excel will be used. We made this choice not out of preference for Excel over other available applications (e.g. Mathematica, Maple, or graphing calculators), but rather because students are likely to have access to Excel. These activities are offered for instructors who want a hassle-free means of exposing their students to a taste of technology. Quick Quizzes A Quick Quiz appears at the end of each section. These quizzes have been carefully written to test the basic facts of the section and are ideal as handouts for your students (in which case you can provide them with answers for self-assessment), in-class quizzes, or Active Learning Questions. At the conclusion of each chapter, you will find additional support material intended for both students and instructors.

• Key Terms and Concepts lists all the major ideas and theorems encountered in the chapter, with page references included. Use this list to construct an exam review sheet, or photocopy it for your students as a quick chapter summary.

• Review Questions are designed to probe your students’ understanding of the concepts introduced in the chapter rather than test problem solving skills. Though some are certainly appropriate as exam questions, you may find them most useful to generate classroom discussion during an exam review session or as a handout for your students to help them prepare for an exam.

• Test Bank Exercises provide material for exams: Cut and paste them into your exam documents or use them as inspiration for your own questions. They can also be used as review material for your students. Understand that the test bank exercises are not meant to be sample exams.

Optional Sections As noted previously, this guide was written with the assumption that every section of the text will be covered, which is unrealistic given the time constraints in most calculus courses. Following is a list of sections that can be excluded from your syllabus or covered quickly without disrupting the flow of your course. Note that we are not advocating the exclusion of any of this material, nor passing judgment on the importance of these topics in the calculus curriculum—only you can make those decisions for your course. Rather, we are simply identifying those sections that are (largely) unnecessary for moving forward with new material.


Copyright © 2013 Pearson Education, Inc. x

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• Chapter 1—This chapter covers prerequisite material for a calculus course, and it can be skipped in its entirety if you’d like to get down to the business of teaching calculus immediately. Our experience is that most students benefit from a review of the algebra and trigonometry skills necessary to survive a calculus course. Therefore, if you choose to skip Chapter 1, it’s a good idea to encourage your students to read it and work through its exercises in a self-directed study. Consider supplying a list of exercises that you feel are most important for your course. Sections 3.8 (Derivatives of Inverse Trigonometric Functions) and 7.4 (Trigonometric Substitutions) rely on an understanding of the inverse trigonometric functions. In the event that you omit Chapter 1, it may be wise to cover the second half of Section 1.4 prior to Section 3.8.

• Section 2.7—Precise Definition of Limits The remainder of the text does not rely upon an understanding of the ε-δ definition of a limit, and this section may be omitted. That said, if you intend to teach the formal definition of the limit of a sequence (Section 9.2) or limits for multivariable functions (Section 13.3), it’s wise to devote some class time to the precise definition of a limit.

• Section 3.5—Derivatives as Rates of Change This section fleshes out ideas that are present in other portions of the book, so it can be covered selectively or omitted in its entirety if need be.

• Section 3.10—Related Rates As noted in Section 3.10 of this guide (p. 67, Additional Activities), most related rates problems can be solved without resorting to the technique normally taught in the classroom. The topic of related rates does not appear again in the text, so this section may be omitted.

• Section 4.3—Graphing Functions This section assimilates ideas already presented earlier in the text (in particular, Sections 4.1 and 4.2, but also Sections 1.1, 2.4, 2.5 and 3.1). If you want to teach students how to sketch the graph of a function without devoting an entire section to the topic, extend the examples in Sections 4.1 and 4.2 and skip this section.

• Section 4.6—The Mean Value Theorem Though the Mean Value Theorem is a fundamental building block in the theoretical framework of calculus, it can be covered quickly by focusing on what the theorem asserts, its immediate theoretical consequences, and its applicability to problems in the real world. Example 2 and Theorems 4.9 and 4.11 are sufficient for these purposes.

• Section 4.8—Newton’s Method This section is optional; Newton’s method does not appear again in the text, save for a few scattered exercises.

• Chapter 6—Applications of Integration This chapter is devoted to applications of integration, an important component in the calculus curriculum. However, most instructors do not cover all the bases due to time considerations. Rather than listing those topics that may be omitted, here we point out material that is essential for future work. An understanding of how to compute the area of a general region in the plane (Section 6.2) is needed for multiple integrals; we also appeal to the general slicing method from Section 6.3 to explain the mechanics of iterated integrals. Arc length (Section 6.5) should be covered so that arc length parameterizations (Section 12.8) and line integrals (Chapter 15) can be understood. In Section 6.7, mass as the integral of a density function and the concept of work are important ideas for multivariable calculus. The remainder of the material in this chapter can be incorporated into your course as you wish.

• Section 7.6—Other Integration Strategies None of the material in this section is required for future work.

• Section 7.7—Numerical Integration This section may also be omitted. If you want to be sure your students hear the message that many integrals require numerical methods, recognize that Section 10.4 provides another opportunity to approximate the value of a definite integral by employing power series

solutions for integrals such as 21

0

xe dx−∫ .

• Chapter 8—Differential Equations As long as you avoid exercises in future chapters that include a differential equation component (they are few in number), this chapter may be omitted.

• Section 10.4—Working with Taylor Series This section provides a potpourri of applications of power series, none of which appears again later in the text.

• Section 11.4—Conic Sections Most students have encountered conic sections in prior courses. If that describes the population you work with, this section may be quickly reviewed (an understanding of conic sections in Cartesian coordinates is necessary for working with quadric surfaces in Section 13.1).

• Section 12.7—Motion in Space Vector-valued functions are used to describe motion in space, and the relationships between position, velocity, and acceleration are useful for interpreting vector fields in



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Chapter 15. However, much of the remaining material in this section may be treated quickly or omitted.

• Section 12.9—Curvature and Normal Vectors The material in this section does not appear again in the text.

• Section 13.3—Limits and Continuity A brief discussion of the concepts of limits and continuity for multivariable functions is sufficient for future work.

• Section 13.9—Lagrange Multipliers Lagrange multipliers provide another approach to solving optimization problems, but they do not appear again in the text.

• Section 14.7—Change of Variables in Multiple Integrals None of this material is required for future work.

• Chapter 15—Vector Calculus Our experience is that reaching the end of Chapter 15 before the term is over requires a minor miracle. The topics in Sections 15.1–15.4 are rather sequential, and therefore it is necessary to cover just about everything in these sections to make sense of Green’s Theorem in both its forms. Once beyond Green’s Theorem, make an assessment of how much additional information you can fit into the end of your course, and plan accordingly. Strategies for trimming material from the final three sections are discussed in Section 15.6 of this guide (p. 350). If by Section 15.5 it is already apparent you will not make it to the end of the text, consider covering only one of the three-dimensional versions of the divergence and curl, whichever suits your purposes best for your end game.

Guided Projects The Guided Projects section of this guide is a collection of 78 projects that cover a wide range of applications, calculations, and theoretical topics. They are designed to be worked on independently by students or small groups of students in a step-by-step fashion. The projects allow students to step outside the bounds of a typical calculus course and explore related topics. They also provide instructors with an excellent alternative form of assessment.

Answers and Solutions Answers for all chapter-level content and solutions for the guided projects occupy the last third of this guide. Here you will find answers to the multiple-choice Quick Quiz questions, the Chapter Review Questions, and the Test Bank Exercises from each chapter. Full solutions are provided for each of the guided projects.

Study Cards The final pages of this guide contain study cards that accompany the textbook, split into single variable and multivariable cards. Post them on your class website or make copies and distribute them to your students.

Acknowledgments We would like to thank the following professors, mathematicians, and artists for their contributions to this manual. Carrie Green Stan Perrine Charleston Southern University Jim Hagler University of Denver Sandra Scholten Karen Hartpence IllustraTech Anthony Tongen James Madison University Mitch Keller Georgia Institute of Technology Marie Vanisko Carroll College Shawna Mahan Pikes Peak Community College Tom Wegleitner Mark Naber Monroe Community College David Zeigler California State University–Sacramento Patricia Nelson

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Learning Objectives CHAPTER 1 Functions

Section 1.1 Review of Functions

1. Understand function notation and recognize functions.

2. Find the domain and range of a function.

3. Compose and decompose functions.

4. Evaluate and simplify difference quotients.

5. Identify symmetry in graphs and functions.

Section 1.2 Representing Functions

1. Recognize the standard catalog of functions used in calculus (polynomial, rational, algebraic, exponential and logarithmic, and trigonometric and inverse trigonometric functions).

2. Use transformations to sketch graphs.

3. Graph and write formulas for piecewise functions.

4. Analyze graphs and tables.

Section 1.3 Inverse, Exponential, and Logarithmic Functions

1. Understand one-to-one functions and their inverses.

2. Find and graph inverse functions.

3. Use properties of exponentials and logs to solve and simplify.

4. Graph exponential and log functions.

Section 1.4 Trigonometric Functions and Their Inverses

1. Evaluate trigonometric expressions.

2. Prove trigonometric identities.

3. Solve trigonometric equations.

4. Evaluate inverse trigonometric expressions.

5. Use a right triangle to simplify compositions.

6. Graph trigonometric functions.

CHAPTER 2 Limits

Section 2.1 The Idea of Limits

1. Calculate average and instantaneous velocity.

2. Calculate slopes of secant and tangent lines.

3. Understand the equivalence between the limits used to compute instantaneous velocity and the slope of a tangent line.

Section 2.2 Definitions of Limits

1. Understand limit definitions.

2. Find limits from a graph.

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3. Estimate limits from a table.

4. Analyze limits.

Section 2.3 Techniques for Computing Limits

1. Evaluate limits of linear functions.

2. Apply limit laws.

3. Evaluate limits using algebraic manipulation.

4. Evaluate one-sided limits.

5. Understand the relationship between one-sided and two-sided limits.

6. Find limits using the Squeeze Theorem.

Section 2.4 Infinite Limits

1. Understand infinite limits.

2. Find infinite limits numerically or graphically.

3. Evaluate limits analytically.

4. Find vertical asymptotes.

Section 2.5 Limits at Infinity

1. Understand end behavior and its connection to horizontal asymptotes.

2. Evaluate limits at infinity.

3. Determine end behavior and sketch graphs.

4. Use limits to find steady states in applications.

5. Find horizontal, vertical, and slant asymptotes and sketch graphs.

Section 2.6 Continuity

1. Understand continuity.

2. Find points of discontinuity or intervals of continuity.

3. Evaluate limits using principles of continuity.

4. Use the Intermediate Value Theorem.

Section 2.7 Precise Definitions of Limits

1. Understand the precise definition of a limit.

2. Determine a value for δ from a graph.

3. Find a symmetric interval.

4. Write a proof for a given finite limit.

5. Write a proof for a given infinite limit.

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CHAPTER 3 Derivatives

Section 3.1 Introducing the Derivative

1. Evaluate derivatives and work with equations of tangent lines.

2. Understand derivatives graphically.

3. Understand differentiability and relate it to continuity.

Section 3.2 Rules of Differentiation

1. Take derivatives of constant multiples of powers and sums of functions.

2. Simplify expressions and take their derivatives.

3. Know the derivative of the natural exponential function .xe

4. Use derivatives to find slopes, tangent lines, and higher-order derivatives.

5. Find limits related to derivatives.

Section 3.3 The Product and Quotient Rules

1. Understand the product and quotient rules.

2. Take derivatives of products and quotients.

3. Compare two ways of taking the derivative.

4. Find derivatives using the extended power rule.

5. Find derivatives of functions that involve exponentials or a combination of rules.

6. Use derivatives to solve problems.

Section 3.4 Derivatives of Trigonometric Functions

1. Find limits involving trigonometric functions.

2. Calculate derivatives involving trigonometric functions.

3. Use derivatives of trigonometric functions to solve problems.

Section 3.5 Derivatives as Rates of Change

1. Understand definitions of average and instantaneous velocity.

2. Relate position, velocity and acceleration.

3. Solve business applications (e.g. marginal cost).

4. Solve other applications.

Section 3.6 The Chain Rule

1. Use Leibniz notation to take the derivative of a composition.

2. Use function notation to take the derivative of a composition.

3. Find the derivative using the chain rule in combination with other rules.

4. Solve applications involving the chain rule.

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Section 3.7 Implicit Differentiation

1. Find the derivative using implicit differentiation.

2. Find equations of tangent lines using implicit differentiation.

3. Find derivatives of functions with rational exponents.

4. Solve applications using implicit differentiation.

Section 3.8 Derivatives of Logarithmic and Exponential Functions

1. Use definitions to find expressions or verify identities.

2. Find derivatives involving logarithms or general exponential functions.

3. Solve applications involving exponential models.

4. Find tangent lines using logarithmic differentiation.

5. Find derivatives using logarithmic differentiation.

Section 3.9 Derivatives of Inverse Trigonometric Functions

1. Find the derivative of functions involving inverse trigonometric functions.

2. Solve applications involving the rate of change of an angle with respect to a side.

3. Find derivatives of general inverse functions at a given point.

Section 3.10 Related Rates

1. Solve related rates applications for the rate of change of distance, area or volume.

2. Solve related rates applications for the rate of change of an angle.

CHAPTER 4 Applications of the Derivative

Section 4.1 Maxima and Minima

1. Use graphs to illustrate or identify extreme points.

2. Find critical points and extreme points.

3. Solve applications involving extreme points.

Section 4.2 What Derivatives Tell Us

1. Sketch functions from properties.

2. Determine intervals of increase and decrease.

3. Use the first derivative test to classify critical points.

4. Determine the concavity on intervals and find inflection points.

5. Use the second derivative test to classify critical points.

6. Compare , and .f f ' f ''

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Section 4.3 Graphing Functions

1. Sketch curves with given properties.

2. Sketch functions using analytic methods.

3. Sketch functions by combining analytic methods with the use of graphing calculators.

4. Sketch special curves or curves used in applications.

5. Identify properties of the graphs of functions.

Section 4.4 Optimization Problems

1. Solve applications by maximizing or minimizing functions.

2. Explain optimization problems and objective functions.

Section 4.5 Linear Approximation and Differentials

1. Write, graph, and use the linear approximation equation.

2. Use linear approximations to estimate a quantity.

3. Solve applications by estimating the change in a given variable.

4. Express the derivative in differential form (e.g. ( )dy f ' x dx= ).

Section 4.6 Mean Value Theorem

1. Find points guaranteed to exist by Rolle's Theorem.

2. Find points guaranteed to exist by the Mean Value Theorem.

3. Solve applications using the Mean Value Theorem.

4. Know that functions with equal derivatives differ by a constant.

Section 4.7 L’Hôpital’s Rule

1. Evaluate limits of the form 0/0.

2. Evaluate limits of the form / , 0 ,∞ ∞ ⋅∞ and .∞ −∞

3. Evaluate limits of the form 01 ,0 ,∞ and 0.∞

4. Compare growth rates.

5. Evaluate limits using the appropriate method.

6. Identify indeterminate forms and convert limits to other indeterminate forms.

Section 4.8 Newton’s Method

1. Understand how Newton’s method uses tangent lines to approximate roots.

2. Use the method’s iterative formula to approximate roots.

3. Compute and interpret residuals.

4. Be aware that Newton’s method can fail to locate a root for various reasons.

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Section 4.9 Antiderivatives

1. Find general antiderivatives and indefinite integrals.

2. Find particular antiderivatives and solve initial value problems.

3. Relate solutions of initial value problems to their graphs.

4. Solve applications involving antiderivatives.

5. Use identities to find antiderivatives.

6. Verify indefinite integrals by differentiation.

CHAPTER 5 Integration

Section 5.1 Approximating Areas under Curves

1. Understand how Riemann sums approximate area under a curve.

2. Approximate displacement using left, right, or midpoint sums.

3. Calculate left, right, and midpoint sums, and draw their corresponding rectangles.

4. Calculate Riemann sums from tables and graphs.

5. For larger values of n, use sigma notation and approximate sums using a calculator.

6. Use geometry to find the area under a curve.

Section 5.2 Definite Integrals

1. Understand the properties of the definite integral.

2. Approximate net area and definite integrals.

3. Relate definite integrals and Riemann sums.

4. Relate net area and definite integrals.

5. Use properties of the definite integral to evaluate integrals.

6. Find the exact value of a definite integral using its limit definition.

Section 5.3 Fundamental Theorem of Calculus

1. Understand the relationship between the antiderivative F, the function f, and the area function A.

2. Work with area functions.

3. Evaluate definite integrals using the Fundamental Theorem.

4. Find the area of regions bounded by the graph of f.

5. Take the derivative of functions expressed as integrals.

Section 5.4 Working with Integrals

1. Understand symmetry and average value.

2. Use symmetry to evaluate integrals.

3. Find the average value of a function.

4. Find the point at which a function equals its average value.

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Section 5.5 Substitution Rule

1. Understand the substitution rule.

2. Find indefinite integrals by trial and error.

3. Use the substitution rule to find indefinite integrals.

4. Use the substitution rule to evaluate definite integrals.

5. Find integrals involving 2sin x and 2cos .x

CHAPTER 6 Applications of Integration

Section 6.1 Velocity and Net Change

1. Find the displacement over a given interval.

2. Determine position functions.

3. Determine distance traveled.

4. Find the position and velocity of an object given its acceleration.

5. Find applied quantities given an equation or graph of the derivative.

6. Find the future value of a quantity given its derivative.

Section 6.2 Regions Between Curves

1. Find the area of regions bounded by curves.

2. Find the area of compound regions.

3. Express area as an integral with respect to x and to y.

Section 6.3 Volume by Slicing

1. Use the general slicing method to find volumes.

2. Use the disk method to find volumes.

3. Use the washer method to find volumes.

4. Use the method of your choice to find volumes.

5. Find volumes of regions revolved around different axes.

Section 6.4 Volume by Shells

1. Find the volume of a solid of revolution around the y-axis.

2. Find the volume of a solid of revolution around the x-axis.

3. Find the volume of a solid of revolution around a given line.

4. Find the volume by the disk/washer and shell methods.

Section 6.5 Length of Curves

1. Find arc length by integrating with respect to x.

2. Find arc length using a calculator.

3. Find arc length by integrating with respect to y.

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Section 6.6 Surface Area

1. Compute surface area by integrating with respect to x.

2. Compute surface area by integrating with respect to y.

3. Evaluate the surface area integral with a calculator.

Section 6.7 Physical Applications

1. Find the mass of bars with given density functions.

2. Find the amount of work done.

3. Find the force on a given surface.

Section 6.8 Logarithmic and Exponential Functions Revisited

1. Evaluate derivatives with ln .x

2. Evaluate integrals with ln .x

3. Evaluate integrals with .xe

4. Evaluate derivatives of tower functions.

5. Evaluate integrals of exponential functions with general bases.

Section 6.9 Exponential Models

1. Compare the growth rates of linear and exponential functions.

2. Find and use exponential growth functions for given data.

3. Find and use exponential decay functions for given data.

4. Answer questions based on exponential growth or decay.

Section 6.10 Hyperbolic Functions

1. Verify hyperbolic identities.

2. Find derivatives and integrals of hyperbolic functions.

3. Evaluate inverse hyperbolic functions.

4. Compute derivatives and integrals involving inverse hyperbolic functions.

5. Solve application problems involving hyperbolic functions.

CHAPTER 7 Integration Techniques

Section 7.1 Basic Approaches

1. Review basic integral formulas.

2. Review the substitution rule for integrals.

3. Evaluate integrals by manipulating the integrand with various techniques.

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Section 7.2 Integration by Parts

1. Evaluate indefinite integrals by applying integration by parts once.

2. Evaluate indefinite integrals by applying integration by parts more than once.

3. Evaluate definite integrals using integration by parts.

4. Prove or apply reduction formulas.

5. Combine integration methods.

Section 7.3 Trigonometric Integrals

1. Evaluate integrals involving powers of sine and cosine.

2. Evaluate integrals involving powers of the other four trigonometric functions.

3. Combine techniques to evaluate integrals involving trigonometric functions.

4. Exploit trigonometric identities and formulas to solve integrals.

Section 7.4 Trigonometric Substitutions

1. Evaluate integrals containing 2 2.a x−

2. Evaluate integrals containing 2 2.x a+

3. Evaluate integrals containing 2 2.x a−

4. Complete the square to prepare for trigonometric substitution.

Section 7.5 Partial Fractions

1. Set up partial fraction decompositions.

2. Evaluate integrals involving only linear factors.

3. Evaluate integrals involving irreducible quadratic factors.

4. Evaluate integrals that require a preliminary step.

Section 7.6 Other Integration Strategies

1. Use a table of integrals to evaluate integrals.

2. Use a table of integrals to evaluate integrals that require a preliminary step.

3. Use a computer algebra system to evaluate integrals.

Section 7.7 Numerical Integration

1. Find an approximation for an integral using the Midpoint, Trapezoid, or Simpson's Rule.

2. Determine the error for a Midpoint, Trapezoid, and/or Simpson's approximation.

Section 7.8 Improper Integrals

1. Evaluate improper integrals with an infinite limit of integration.

2. Find area and volume using improper integrals.

3. Evaluate improper integrals with an unbounded integrand.

4. Evaluate improper integrals using symmetry or integration by parts.

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CHAPTER 8 Differential Equations

Section 8.1 Basic Ideas

1. Become familiar with differential equation terminology.

2. Verify general solutions.

3. Verify solutions to initial value problems.

4. Solve differential equations of the form ( ) ( ).y' t f t=

5. Work with differential equations in applications.

Section 8.2 Direction Fields and Euler’s Method

1. Sketch and analyze direction fields.

2. Sketch solution curves on a direction field.

3. Identify equilibrium solutions.

4. Find approximate solutions using Euler’s method.

Section 8.3 Separable Differential Equations

1. Identify separable differential equations.

2. Solve separable differential equations.

3. Express a solution to a differential equation in implicit form.

Section 8.4 Special First-Order Differential Equations

1. Understand the meaning of each term in the equation ( ) ( ) .y' t ky t b= +

2. Solve equations of the form ( ) ( ) .y' t ky t b= +

3. Understand the difference between stable and unstable equilibrium solutions.

4. Be familiar with Newton’s Law of Cooling.

Section 8.5 Modeling with Differential Equations

1. Find the carrying capacity for a population model.

2. Work with logistic models and Gompertz models.

3. Solve stirred-tank reaction problems.

4. Work with predator-prey models.

CHAPTER 9 Sequences and Infinite Series

Section 9.1 An Overview

1. Understand the distinction between sequences and series.

2. Write terms of explicit and recursive sequences.

3. Find an explicit or recursive formula for a sequence.

4. Determine the limit of explicit or recursive sequences.

5. Work with partial sums of infinite series.

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Section 9.2 Sequences

1. Use properties and theorems to determine the limit of a sequence.

2. Determine the limit of geometric sequences.

3. Use the Squeeze Theorem to find a limit.

4. Solve applications involving sequences.

5. Compare growth rates of sequences.

6. Prove limits using the formal definition.

Section 9.3 Infinite Series

1. Evaluate finite geometric sums.

2. Evaluate infinite geometric series.

3. Evaluate telescoping series.

4. Determine the number of terms needed to make the error less than a specified tolerance.

5. Understand functions defined as series.

Section 9.4 The Divergence and Integral Tests

1. Use the divergence test.

2. Know that the harmonic series diverges.

3. Use the integral test and knowledge of the p-series to determine the convergence of a series.

4. Find remainders and estimates for series.

5. Use properties to find the sum of the series.

6. Determine convergence using any test.

Section 9.5 The Ratio, Root, and Comparison Tests

1. Use the ratio or root test to determine whether a series converges.

2. Use the comparison or limit comparison test to determine whether a series converges.

3. Find the values of p or x for which a series converges.

Section 9.6 Alternating Series

1. Use the alternating series test to determine convergence.

2. Find the number of terms needed for an error less than a specified tolerance.

3. Approximate an infinite sum with an error less than a specified tolerance.

4. Determine whether a series converges absolutely or conditionally.

CHAPTER 10 Power Series

Section 10.1 Approximating Functions with Polynomials

1. Make linear and quadratic approximations for functions.

2. Find Taylor polynomials centered at zero.

3. Make approximations using Taylor polynomials and estimate error.

4. Find Taylor polynomials centered at nonzero values of a.

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5. Find the remainder term in Taylor polynomials.

6. Find the number of terms in a Taylor polynomial to approximate within a given error.

Section 10.2 Properties of Power Series

1. Find the interval and radius of convergence of a power series.

2. Combine power series.

3. Differentiate and integrate power series.

4. Find power series representations of functions.

5. Write power series using summation notation.

6. Find functions represented by power series.

Section 10.3 Taylor Series

1. Find Maclaurin series.

2. Find Taylor series.

3. Find terms of binomial series and make approximations.

4. Find remainders in Taylor series.

5. Use Taylor series to approximate numbers.

6. Show that a Taylor series converges to the function that generated it.

Section 10.4 Working with Taylor Series

1. Evaluate limits using Taylor series.

2. Differentiate Taylor series.

3. Find power series solutions to differential equations.

4. Approximate definite integrals using Taylor series.

5. Approximate real numbers using Taylor series.

6. Evaluate infinite series.

7. Identify functions represented by power series.

CHAPTER 11 Parametric and Polar Curves

Section 11.1 Parametric Equations

1. Graph parametric equations and eliminate the parameter.

2. Work with parametric equations of circles, arcs, and ellipses.

3. Solve applications of circular motion.

4. Find a parametric description of line segments or curves.

5. Identify the orientation of a parametrically defined curve.

6. Find the slope of a parametric curve at a point.

Section 11.2 Polar Coordinates

1. Plot polar coordinates and find alternative representations.

2. Convert between polar and Cartesian coordinates.

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3. Graph polar curves by using the Cartesian-to-polar method.

4. Convert polar equations to Cartesian form.

5. Graph polar curves using a graphing utility.

6. Find intersection points.

Section 11.3 Calculus in Polar Coordinates

1. Find slopes and equations of lines tangent to polar curves.

2. Find the area of a region bounded by polar curves.

Section 11.4 Conic Sections

1. Graph parabolas.

2. Find the equation of a parabola given a graph or conditions.

3. Graph ellipses.

4. Find the equation of an ellipse given a graph or conditions.

5. Graph hyperbolas.

6. Find the equation of a hyperbola given a graph or conditions.

7. Find an equation and graph of a conic given eccentricity.

8. Graph conic sections given in polar form.

9. Find equations of tangent lines.

CHAPTER 12 Vectors and Vector-Valued Functions

Section 12.1 Vectors in the Plane

1. Perform vector operations.

2. Find the magnitudes of vectors.

3. Identify equal vectors.

4. Find unit vectors.

5. Find position vectors.

6. Solve vector equations.

7. Express a vector as the linear combination of other vectors.

8. Use vectors in an applied setting.

Section 12.2 Vectors in Three Dimensions

1. Find and plot points in xyz-space.

2. Sketch planes parallel to coordinate planes.

3. Use the distance formula in xyz-space.

4. Work with the equation of a sphere.

5. Perform vector operations.

6. Find unit vectors and magnitude.

7. Find parallel vectors.

8. Determine if points are collinear.

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Section 12.3 Dot Products

1. Compute dot products using the definition.

2. Compute dot products and find angles between the vectors.

3. Sketch and calculate orthogonal projections.

4. Calculate work done in given situations.

5. Find the parallel and normal components of a vector.

6. Find orthogonal vectors.

7. Find the distance between a point and a line.

Section 12.4 Cross Products

1. Work with properties of the cross product.

2. Express vectors as linear combinations of the unit coordinate vectors i, j, and k.

3. Compute cross products.

4. Find the areas of parallelograms and triangles.

5. Find normal vectors.

6. Compute torque.

7. Solve vector equations.

Section 12.5 Lines and Curves in Space

1. Find equations of lines and line segments.

2. Graph curves in space.

3. Identify the orientation of a parametrically defined curve.

4. Evaluate limits.

5. Find domains of vector-valued functions.

6. Find points of intersection.

Section 12.6 Calculus of Vector-Valued Functions

1. Find derivatives of vector-valued functions.

2. Find tangent vectors and unit tangent vectors.

3. Evaluate indefinite and definite integrals of vector-valued functions.

4. Be familiar with derivative rules for vector-valued functions.

Section 12.7 Motion in Space

1. Find the velocity and acceleration of objects.

2. Compare trajectories.

3. Solve trajectory problems.

4. Solve problems involving two-dimensional motion.

5. Solve problems involving three-dimensional motion.

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Section 12.8 Length of Curves

1. Compute arc length.

2. Find speed on and length of trajectories.

3. Approximate arc length using a calculator.

4. Find arc lengths of polar curves.

5. Understand arc length parameterization.

Section 12.9 Curvature and Normal Vectors

1. Compute curvature at a point.

2. Find curvature using the alternative formula.

3. Find the principal unit normal vector.

4. Find tangential and normal components of acceleration.

5. Find and analyze curvature functions.

6. Find the unit binormal vector and compute torsion.

CHAPTER 13 Functions of Several Variables

Section 13.1 Planes and Surfaces

1. Find equations of planes.

2. Find the intersections of lines and planes.

3. Determine whether planes are parallel, orthogonal, identical, or none of these.

4. Sketch graphs of cylinders and quadric surfaces.

5. Identify cylinders and quadric surfaces.

6. Find angles between planes.

7. Identify properties of lines, planes, and surfaces.

Section 13.2 Graphs and Level Curves

1. Find domains of functions.

2. Graph surfaces.

3. Graph level curves of functions.

4. Work with level surfaces.

Section 13.3 Limits and Continuity

1. Evaluate limits.

2. Prove limits exist or do not exist.

3. Determine where functions are continuous.

4. Understand the definitions of interior and boundary points.

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Section 13.4 Partial Derivatives

1. Find first partial derivatives.

2. Find higher-order partial derivatives.

3. Determine whether functions are continuous and differentiable.

Section 13.5 The Chain Rule

1. Use the chain rule to find derivatives.

2. Use tree diagrams to write chain rule formulas for derivatives.

3. Differentiate implicitly.

4. Understand intermediate variables and implicitly defined functions.

Section 13.6 Directional Derivatives and the Gradient

1. Compute directional derivatives.

2. Compute gradients.

3. Interpret the gradient in various contexts.

4. Compute slopes of lines tangent to level curves.

5. Work with the gradient in three dimensions.

Section 13.7 Tangent Planes and Linear Approximation

1. Find equations of planes tangent to surfaces.

2. Find linear approximations.

3. Use differentials to approximate changes in functions.

4. Find points at which surfaces have horizontal tangent planes.

Section 13.8 Maximum/Minimum Problems

1. Find and classify critical points using the Second Derivative Test.

2. Find local and absolute extrema of functions on closed and bounded sets.

3. Find absolute extrema of functions on open sets when possible.

Section 13.9 Lagrange Multipliers

1. Use Lagrange multipliers to find extreme values of functions of two variables.

2. Use Lagrange multipliers to find extreme values of functions of three variables.

CHAPTER 14 Multiple Integration

Section 14.1 Double Integrals over Rectangular Regions

1. Evaluate double integrals over rectangular regions.

2. Compute average values of functions over plane regions.

3. Find volumes of solids.

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Section 14.2 Double Integrals over General Regions

1. Write iterated integrals of continuous functions over general regions.

2. Evaluate double integrals over general regions.

3. Find volumes of solid regions.

4. Change orders of integration.

5. Find areas of plane regions.

6. Find average values over general regions.

Section 14.3 Double Integrals in Polar Coordinates

1. Find volumes of solids described in polar coordinates.

2. Evaluate double integrals in polar coordinates.

3. Convert double integrals in Cartesian coordinates to polar coordinates.

4. Find areas of polar regions and solve applications involving area.

5. Find average values over planar polar regions.

Section 14.4 Triple Integrals

1. Evaluate triple integrals.

2. Find volumes of solids using triple integrals.


4. Find average values of functions of three variables.

Section 14.5 Triple Integrals in Cylindrical and Spherical Coordinates

1. Sketch sets in cylindrical and spherical coordinates.

2. Evaluate triple integrals in cylindrical and spherical coordinates.

3. Find masses of objects with given density functions.

4. Use cylindrical and spherical coordinates to find volumes of solid regions.

5. Convert between cylindrical, spherical and rectangular coordinates.


Section 14.6 Integrals for Mass Calculations

1. Compute centers of mass of one-dimensional objects.

2. Compute centers of mass of two-dimensional objects.

3. Compute centers of mass of three-dimensional objects.

Section 14.7 Change of Variables in Multiple Integrals

1. Find the image of a transformation.

2. Compute Jacobian in two and three variables.

3. Evaluate double and triple integrals with a change of variables.

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CHAPTER 15 Vector Calculus

Section 15.1 Vector Fields

1. Graph two- and three-dimensional vector fields.

2. Identify rotation and radial fields.

3. Find normal and tangential vector fields.

4. Find gradient fields for potential functions.

Section 15.2 Line Integrals

1. Evaluate line integrals in two- and three-dimensions.

2. Find average values of functions on curves.

3. Find lengths of curves using scalar line integrals.

4. Evaluate line integrals of vector fields.

5. Find the work required to move objects on curves.

6. Compute circulation.

7. Compute flux for vector fields and curves.

Section 15.3 Conservative Vector Fields

1. Work with test for conservative vector fields.

2. Find potential functions.

3. Evaluate line integrals.

4. Work with the Fundamental Theorem for line integrals.

5. Find the work required to move objects in force fields.

Section 15.4 Green's Theorem

1. Use Green's theorem to evaluate line integrals.

2. Calculate areas of plane regions using Green's theorem.

3. Compute circulation and flux on general regions.

4. Find the two-dimensional curl and divergence of a field.

5. Find stream functions for fields.

Section 15.5 Divergence and Curl

1. Find and interpret the divergence of vector fields.

2. Find and interpret the curl of vector fields.

3. Prove properties and identities of the divergence and curl.

4. Know the properties of conservative vector fields.

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Section 15.6 Surface Integrals

1. Work with parametric surfaces.

2. Find surface areas.

3. Find surface integrals.

4. Find average values.

Section 15.7 Stokes' Theorem

1. Verify that the two integrals in Stokes' Theorem are equal.

2. Use Stokes' Theorem to evaluate integrals.

3. Interpret and graph the curl.

4. Use Stokes' Theorem to find the circulation.

Section 15.8 Divergence Theorem

1. Verify the Divergence Theorem by integrating both integrals.

2. Compute flux using the Divergence Theorem.

APPENDICES

Appendix A

1. Perform operations on polynomials and rational expressions.

2. Solve inequalities.

3. Graph linear equations and use the distance formula.

4. Use equations of circles.

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Index of Applications

Engineering and Physical Sciences Air drop, 739 Airline regulation, 267 Airline travel, 177 Airplane in the wind, 788, 797,

799, 877, 1096 Altitude of a jet, 223 Angle of elevation, 217 Angle to a particle, 218 Angular size, 217 Ant on a page, 789 Approximating displacement,

327–328, 336 Arch height, 374, 375 Arm torque, 819 Arrow velocity, 224 Artillery shell velocity, 396 Atmospheric pressure, 282, 479 Avalanche forecasting, 289 Balancing forces, 786–787 Balloon volume, 10 Ballpark shadows, 214 Baseball height, 38, 388 Baseball pitch, 850–851 Bend in the road, 869 Bicycle brakes, 819 Biking, 401, 499, 736 Billboard viewing angle, 230 Block on inclined plane, 800, 808 Block position, 393–394 Boat travel, 223, 226, 227, 230 Boiling point, 51 Box surface area, 239 Box volume, 25, 270, 926, 969,

970, 978 Bug on a parabola, 223, 224 Building a tunnel, 681 Bullet velocity, 848 Bungee jumper, 181 Buoyancy, 462 Calorie-free milkshake, 462 Carbon dating, 480 Car deceleration, 400, 499 Car travel, 326–327, 338, 572 Catenary, 492, 495 Channel flow, 752, 1119, 1168 Chemical compound reaction,

582, 590, 597 Circular path of turtle, 731 Clock hands, 225, 736 Clock vectors, 789 Coiling a rope, 461 Compressing and stretching a

spring, 453, 459, 461, 501 Conservation of energy, 936, 1106 Converging airplanes, 220–221 Cooling time, 602, 603, 604, 616 Corral construction, 266–267

Covering a marble, 270 Crankshaft, 272 Crease-length problem, 271 Cutting a wire, 270 Cyclist, 391–392, 399 Cylinder radius, 79 Cylindrical tank, 80, 181, 224 Dancing on a parabola, 239 Day hike, 399 Depletion of natural resources,

401, 568 Diluting a solution, 638 Displacement by geometry, 387 Displacement from velocity, 339,

340, 386, 388, 499 Dog retrieving a ball, 274 Double glass, 646 Drag racer acceleration, 288, 400 Draining a trough, 227 Drinking a soda, 224 Drinking juice, 461 Earth circumference, 49 Earth-Mars system, 751 Earthquake, 207 Eiffel Tower, 569 Electric field, 80, 901–902, 905,

927, 949, 982, 1129, 1162–1163, 1165–1166

Electrical charge, 37 Electrical resistors, 960 Electromagnetic waves, 1130 Electron speed, 819 Electronic chips, 569 Electrostatic force, 159–160, 282 Elevation change, 323 Elevator angle of elevation, 225–

226 Energy change, 324 Energy consumption, 476, 479 Estimating car speed, 277, 282,

283 Estimating travel time, 282, 283 Falling probe, 877 Ferris wheel, 226, 273, 736 Field goal angle, 49 Filling a reservoir, 402 Filming a race, 225 Firework, 878 Flight of a baseball, 843–844, 878 Flight of a golf ball, 845, 848, 850 Flight of an eagle, 855 Flipping a switch, 68 Flow in an ocean basin, 1118–

1119 Flow of heat, 928 Flow rate, 340, 499 Fluid flow, 933–934, 935, 960,

1130 Fly rod box, 800 Flying a kite, 224

Flying into a headwind, 399 Folded boxes, 271 Force on a building, 460 Force on a camel, 790 Force on a dam, 501 Force on a moving charge, 818 Force on a proton, 816–817 Force on a window, 460–461 Four-cable load, 800 Free fall, 480, 497, 582, 590, 597 Fuel consumption, 499 Gateway Arch, 441 Go-cart, 736 Golden earring, 1053 Golden Gate cables, 441, 772 Golf slice, 828 Gravitational force, 160, 1106,

1129, 1166 Headstart problem, 322 Heat flux, 810, 1096, 1129, 1145,

1166 Height vs. volume, 264 Highway travel, 177 Hiking trail, 372 Hot–air balloon, 221–222, 224,

230 Ice cream cone, 1038 Ideal gas law, 283, 907, 922–923,

926, 936 Jet angle of elevation, 230 Kiln design, 497 Knee torque, 877 Ladder, 50, 223, 268–269, 270 Lapse rate, 286 Laying cable, 271, 272 Lifting a pendulum, 461 Light sources, 271 Light transmission, 272 Light travel, 274, 275 Lighthouse, 50 Lighthouse beam, 226 Magnetic field, 531, 1154 Making silos, 271 Man on a hillside, 877 Mass of rod, 340 Mass on an inclined plane, 790 Maximum printable area, 323 Melting snowball, 223 Metal rain gutters, 272 Minimum painting surface, 323 Mixing tank, 189 Motion on the moon, 849 Motion with gravity, 319, 321,

577, 581 Mountains, 906 Moving shadow, 223 Ocean wave, 493, 496 Oil consumption, 479 Oil production, 400, 553, 558 Opening a laptop, 818

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Optimal garden, 270 Paddle wheel, 1129, 1150–1151,

1169 Parabolic dam, 460 Parachute in the wind, 788 Paths of moons, 738 Peak oil extraction, 400 Peaks and valleys, 906 Pen construction, 270, 273, 275 Pendulum, 548, 557, 724 Piston compression, 223 Piston position, 230 Planetary orbits, 375, 855–856 Poles, 50, 274, 274 Power and energy, 140, 181, 189,

402–403 Power lines, 495 Pressure and altitude, 180, 207 Pressure on a dam, 458 Probe speed, 399 Projectile launch, 739, 846–847,

849, 861 Proton and nucleus, 570 Pulling a sled, 809 Pulling a suitcase, 788, 809 Pulling a wagon, 786 Pumping gasoline, 455–456 Pumping water, 455, 460 Pursuit curve, 264 Pushing a stroller, 809 Radioactive decay, 501, 627 Rain on a roof, 1141, 1145 RC circuit, 614 Resistors in electrical circuit, 905,

927 Revolving an asteroid, 449 Road system, 273 Rocket launch, 12, 150, 324, 461,

849 Rocket velocity, 230 Roller coaster curve, 824 Rolling wheels, 733, 738 Rose petals, 751, 755–756, 759 Sandpile, 221, 224 Searchlight beam, 226, 272–273 Seesaw balance, 1044 Ship travel, 225 Shock absorber, 459 Shooting a basket, 878 Ski jump, 850 Skydiving, 38, 228, 395, 508, 542 Slinky curve, 824–825 Slowest shortcut, 273 Soap bubbles, 647 Soda can, 272, 1053 Spaceship length, 79 Speed of boat in a current, 785–

786, 788 Speeding, 289 Spherical balloon, 223, 230 Spin on a baseball, 849 Spin on a soccer ball, 849

Spreading oil, 219–220 Spring oscillations, 49–50, 180,

188, 189, 311, 322, 399, 514 Spring runoff, 181 Stirred tank, 609, 613, 614, 616 Stone trajectory, 236–237, 238 Submarine course, 799 Submarine dive rate, 223 Suspended loaf, 788 Suspension system, 271 Swimming pool, 223, 225, 459,

568, 993 Temperature, 25, 181, 286, 288,

556, 1023, 1082, 1083, 1137, 1144

Three-cable load, 800 Tightening a bolt, 815–816, 818 Time and distance, 3, 10, 111–

112, 150 Time-lagged flights, 225 Towing a boat, 218, 788 Train deceleration, 400 Travel time, 239, 789 Tree notch, 275 Tsunamis, 496 Tug-of-war, 789 Uranium dating, 480 Variable gravity, 403 Velocity, 54–56, 60, 168, 170–

171, 177, 178, 179, 180, 227, 321, 399, 569

Velocity and acceleration, 172–173, 177, 178, 321, 399, 480

Viewing angles, 50, 226, 272 Volume, 198 Volume of a drilled sphere, 428 Volume of a shed, 994 Volume of a wooden object, 423 Walking and rowing, 25, 270 Walking and swimming, 268, 270 Water flow, 810 Water in a bowl, 423, 435, 1014 Water in a gas tank, 1043 Water tank, 12, 224, 225, 230,

322, 401, 459, 460, 461, 462, 501, 569, 579, 581, 597, 959–960, 982

Water tower, 10 Water trough, 460 Water waves, 906, 949 Wave height, 374 Wave on a string, 927–928 Wedge from a tree, 435 Winding a chain, 461 Wine barrel volume, 271 Economics and Business Automobile lease vs. purchase, 25 Average and marginal costs, 175–

176, 178, 180, 230, 400 Average and marginal profit, 179

Bank account, 159 Car loan, 637, 646 CD sales, 14 Cobb-Douglas production

function, 198, 905, 927, 979 Compound inflation, 480 Compound interest, 301, 475 Consumer price index, 178, 627 Demand functions, 254 Depreciation of equipment, 480 Diminishing returns, 180 DVD sales, 21 Endowment, 604 Fuel economy, 180 Gas mileage, 150 Gasoline costs, 51 House loan, 646 House locations, 970 Interest rate, 110, 111 Law of 70, 480 Manufacturing errors, 957 Marginal production, 179–180 Maximizing profit, 238, 273 Maximizing revenue, 238, 481 Mortgage payments, 110 Multiplier effect, 646 Parking fees, 21, 51, 111 Paying off a loan, 601, 604 Phone calls, 125 Postage rates, 68 Production costs, 398 Publishing costs, 51 Revenue function, 180 Rising costs, 479 Savings account, 479, 501, 569,

906 Savings plan, 209, 637, 681, 906 Shipping regulations, 965, 969,

978 Taxicab fees, 21 Travel costs, 274, 959 T-shirt profits, 894 Utility functions, 936 Life Sciences Air flow in the lungs, 402 Antibiotic decay, 158–159 Bacteria growth rate, 228 Bacteria population, 37, 385, 400 Bald eagle population, 25 Bamboo growth rate, 324 Bioavailability of a drug, 566–

567, 568 Biology Club fundraiser, 21 Blood flow, 402, 759 Body mass index, 927, 956 Carbon emissions, 481 Cell population growth, 150, 156–

157, 181, 324, 397–398, 616 Deer population, 209 Diagnostic scanning, 207

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Drug dosing, 599–600, 604, 633–634, 637, 646

Drug infusion, 582, 590, 614 Drug metabolism, 479, 627 Endangered species population,

400 Eyesight, 24–25 Falcon dive, 218 Fish length, 179 Fishing, 208, 224 Flight of an eagle, 1085 Fox population, 400 Fruit fly population, 594–595 Gliding mammals, 275 Grazing goats, 760 Hare population, 596 Hungry hippo, 638 Pharmacokinetics, 478 Prairie dog population, 400, 606,

627 Predator-prey model, 611–612,

613, 614, 616 Radioiodine treatment, 480 Radiometric dating, 477 Species population, 254, 402 Tree growth, 180 Tumor cell growth, 479, 481, 582,

597, 607 Valium metabolism, 480 Social and Behavioral Sciences China’s one-son policy, 646 Crime rate, 479 Epidemic, 596 Internet growth, 174 Population and growth, 21, 230,

400, 472–473, 480, 501, 582 Population growth in Georgia, 178 Population growth of the U.S., 179 Population of China, 479 Population of India, 616 Population of Las Vegas, 140 Population of Michigan, 479 Population of Texas, 479 Society wealth, 411 Spread of rumors, 597 U.S. population, 227, 614 World population, 208–209, 474 General Archimedes’ calculation, 645 Bagels, 387, 531 Baseball runners, 223 Batting averages, 959 Blood test, 272 Bouncing balls, 621, 626, 627,

646 Cheese wedge, 1026 Coin toss, 724 Darts, 410–411

Daylight, 49, 50, 189 Down syndrome, 202 Earned run average, 905 Hailstone sequence, 640 Harvesting, 578, 581, 604, 638,

646 Heights of U.S. men, 558 Length of a DVD groove, 879 Mercator map projection, 522–523 Quarterback ratings, 907 Rejected quarters, 727 Running pace, 289 Running race, 180–181, 289, 318,

401, 480, 849 Running stride, 151–152 Sierpinski triangle, 681 Sleep model, 639 Snowflake island, 646 Snowplow problem, 402 Stacking dominoes, 661 Sudden death playoff, 724 Tennis probabilities, 25 Traffic flow, 572 U.S. movie lengths, 558 Zeno’s paradox, 645

download instant at http://testbankinstant.com1.1 Review of Functions 1


CHAPTER 1 FUNCTIONS The opening chapter of the text focuses on functions: their properties, their graphs, and their use in applications. It can also be viewed as an overview of the prerequisite knowledge from algebra and trigonometry that is necessary for success in a calculus course. Additional review material appears in Appendix A. Some departments skip this material and begin the calculus curriculum with Chapter 2; if that describes your situation, you may refer your students to this chapter and to Appendix A when algebra and trigonometry difficulties arise. For those instructors who provide a review of basic skills, a daunting task lies ahead of you. In one or two weeks, you must race through enough material to fill a term-long course. Plan accordingly, and recognize that you won’t likely be able to cover everything. Rely on your students to fill in the details.

Section 1.1 Review of Functions

Overview A function is defined and its properties are developed.

Lecture Support Notes It’s important to set the right tone at the beginning of a semester. Experience shows that when you set the bar high, your class will jump higher as a result. Assign a good deal of homework from these initial sections to ensure that your students get a thorough review of the prerequisites necessary for calculus, and so that they get used to devoting plenty of time to the study of calculus outside of class.

This chapter is also an ideal time to think about an early diagnostic quiz, assuming your department does not have an entrance exam for the class already in place. Providing immediate feedback for your students helps them to determine whether they are ready for the rigors of calculus, or whether they need to drop the course before your institution’s deadline. It’s an old story, but one that calculus teachers share often: The main reason for failure in calculus usually stems from difficulties in algebra and trigonometry. Try to develop a quiz—which could be given in the first week—in order to gather data for correlations between your students’ scores on the quiz and their final grade in the course. Over time, you will be able to spot insurmountable deficiencies, or problem areas where your students need help, and you can address the challenges accordingly.

• Cover the definition of a function, its geometric interpretation (the vertical line test), and the concepts of domain and range (both the domain of definition and the domain in the context of an application).

• Review composition of functions. Composite functions are featured prominently in calculus, so cover all the bases (Examples 4–7).

• Focus on the idea of a difference quotient (examples 8 and 9), and be sure your students can simplify

expressions such as ( ) ( )f x h f x

h

+ − and

( ) ( )f x f a

x a

−−

.

• Explain the notion of symmetry in graphs, and give definitions of even and odd functions. Interactive Figures

• Figures 1.4–1.5 display the domain and range of two functions. • Figure 1.7 illustrates the distinction between the vertical trajectory of a stone thrown upward and the

graph of its height as a function of time. • Figure 1.13 illustrates the ideas of symmetry about the y-axis, x-axis, and origin. • Figures 1.14–1.16 display symmetric and non-symmetric functions.

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Connections

• Exercises 57–70 and Exercises 89–92 ask students to compute the difference quotients ( ) ( )f x f a

x a

−−

and ( ) ( )f x h f x

h

+ − for various functions; these exercises will prepare students for upcoming limit

and derivative computations. • The notion of symmetry is used when graphing functions in Cartesian coordinates (Section 4.3) and

polar coordinates (Section 11.2), and when evaluating definite integrals (Section 5.4).

Additional Activities Suggested Guided Projects: Problem-solving skills and Constant-rate problems

• Problem-solving skills is a guided project that can be used in a variety of ways. • Constant-rate problems continues the theme of problem solving. Assign a few of their brain teasers to

develop critical thinking skills, and to introduce students to Pólya’s four-step method to problem solving. The exercises in these guided projects could also be used as icebreakers in the initial days of class, as they are sure to generate discussion. Finally, you could help your students get to know one another by asking them to work on a handful of exercises in groups of 2–4 students.

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Section 1.1 Quick Quiz Answer the following multiple choice questions by circling the correct response.

1. The function 3( )f x x x= − is defined for

(a) { : 0}x x ≥ . (b) { : 0}x x < . (c) { : }x x−∞ < < ∞ .

2. The range of 3( )y f x x x= = − is

(a) { : 1}y y ≥ . (b) { : }y y−∞ < < ∞ . (c) { : 0}y y < .

3. The function 2( ) 9f x x= − is defined for

(a) { : 3}.x x > (b) { : 3}.x x < (c) { : 3}.x x ≤

4. The graph of the function ( ) 3 8f x x= − + is

(a) a line with slope 8 and y-intercept (0, −3). (b) a line with slope 3 and y-intercept (0, 8).

(c) a line with slope −3 and y-intercept (0, 8).

5. Suppose the height of a soccer ball that is kicked from the ground at time t = 0 is 2( ) 5 60h t t t= − + (in feet).

An appropriate domain for this problem is (a) { : 0 12}.t t≤ ≤ (b) { : 0 6}.t t≤ ≤ (c) { : }.t t−∞ < < ∞

6. If ( )f x x= and ( ) 1/ ( 1),g x x= + then ( ( ))f g x is

(a) 1

1x +. (b)

1

1x +. (c) 1x + .

7. If 3( )f x x x= − and 2( ) ,g x x−= then ( ( ))g f x is

(a) 6 2.x x− −− (b) 3 2( )x x −− . (c) 6 2.x x−

8. With ( )f x x= and 2( ) 4 ,g x x= − the function f g is defined for

(a) all real numbers. (b) { : 2}.x x ≥ (c) { : 2}.x x ≤

9. Suppose 2( ) 3 2= −f x x . When simplified, the difference quotient ( ) ( )+ −f x h f x

h is equal to

(a) 6 3 4+ −x h . (b) 1. (c) 6 3+x h .

10. Suppose 3

( ) = −f xx

. When simplified, the difference quotient ( ) ( )−

−f x f a

x a is equal to

(a) 0. (b) 3

ax. (c) –3.

11. The function 4( ) 3 1h x x x= − +

(a) is even. (b) is odd. (c) has no symmetry.

12. The curve described by the equation 2 4 1x y− = is symmetric about the

(a) x-axis only. (b) y-axis only. (c) origin.

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Section 1.2 Representing Functions

Overview We introduce the full catalog of functions that will be encountered in calculus, and present four ways to represent a function: through formulas, graphs, tables, and words.

Lecture Support Notes • Run through the gamut of standard functions and provide representative graphs for each family of

functions. This section covers only the graphs of polynomial, algebraic, and rational functions—logarithmic, exponential, and trigonometric functions are featured in the following sections—and we keep things simple at this stage (e.g. power functions instead of polynomial functions).

• Cover piecewise functions, which are used repeatedly in the next chapter (limits). Include a piecewise definition of the absolute value function, another fact that will be used frequently in upcoming material.

• Let your students know where you stand regarding the question of technology and graphing: To what degree will they need to be able to produce graphs by hand? Whatever your policy is, it’s probably best to ask your students to become familiar with at least the basic shapes of the standard functions.

• Reviewing transformations of graphs is important, as this topic gives students the tools needed to quickly visualize more complicated functions.

• Examples 6 and 7 provide a preview of the derivative and integral. Although the main goal of the chapter is to review algebra and trigonometry, these examples give an early introduction to slope functions and area functions.

Interactive Figures

• Figures 1.24–1.25 display ny x= for n even and n odd, respectively.

• Figures 1.26–1.27 display 1 ny x= for n even and n odd, respectively.

• Figure 1.28 zooms in and out on a particular rational function. • Figure 1.32 illustrates how the graph of g, which is a slope function for the function f, is related to the

graph of f. • Figures 1.35–1.36 illustrate how an area function A is generated by a given function f. • Figures 1.37–1.42 display shifts and scalings in the x- and y-directions. Figures 1.43 and 1.44 display

similar shifts and scalings in a parabola and absolute value function, respectively.

Connections • Examples 6 and 7, and their related Exercises 35–42, are included as prelude to big ideas on the

horizon. Exercises 78–79 and 83–85 are also designed with future chapters in mind. • The theme of investigating the properties of the standard families of functions (polynomials, rational

functions, algebraic functions, etc.) continues throughout the text. We first learn how to differentiate each family of functions (Chapter 3), and then later learn how to integrate each family (Section 4.9, Chapter 5, and Chapter 7). This program is repeated in multivariable calculus (in fact, several times over, as we learn how to differentiate and integrate vector-valued functions, followed by multivariable functions, and finally vector fields).

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Additional Activities Suggested Guided Projects: Functions in action I, Functions in action II and Supply and demand

• Functions in action I and II are guided projects that explore the behavior of functions in an applied setting. Each mini-project provides a detailed and compelling look into how functions are used to model various phenomena in a variety of disciplines (biology, meteorology, physics, economics, and general interest—see Chasing a dog—to name a few).

• Supply and demand is another guided project along these lines that is devoted to illustrating principles in economics with graphs.

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Section 1.2 Quick Quiz Answer the following multiple choice questions by circling the correct response.

1. The function 1

( )2

xf x

x

+=−

is a

(a) polynomial. (b) rational function. (c) transcendental function.

2. The function 10 2( ) 2 3f x x x= − is a

(a) polynomial. (b) trigonometric function. (c) transcendental function.

3. The function 2( ) 1 3f x x x−= − − is

(a) a polynomial. (b) an algebraic function. (c) a rational function.

4. The graph of 5( ) 2f x x=

(a) lies in the first and second quadrants.

(b) lies in the second and fourth quadrants. (c) has a point for every real number x.

5. The graph to the right best represents the function

(a) 2( ) 4f x x= − .

(b)2

1( ) .

4f x

x=

−

(c) 2( ) 4f x x= − .

6. The data in the table below is best represented by the function

(a) ( ) 1.f x x= + (b) 2( ) 3.f x x= − (c) 2( ) .f x x x= −

x −2 −1 1 3 4 6 f(x) 1 −2 −2 6 13 33

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