SULLIVAN

M I R A N DA

SECOND EDITION

CALCULUSE A R LY T R A N S C E N D E N T A L S

WE DO MORE SO YOU CANACHIEVE MORE.WE DO MORE SO YOU CAN ACHIEVE MORE.

…THROUGH CLEARLY WRITTEN CONTENT

…IN UNDERSTANDING CONCEPTS AND PROBLEM SOLVING

…IN PREPARATION FOR EDUCATION AND CAREER SUCCESS

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For more information, contact your local Macmillan Learning representative or visit macmillanlearning.com/exploresullivan2e

EXAMPLES are given titles that clearly spell out their purpose, and offer detailed and annotated step-by-step solutions.

reach every student through clearly written content

CALCULUS, SECOND EDITION, was written so that students can read it. The engaging style, clear presentation, detailed examples, and just-in-time margin notes will help students learn and retain concepts.

Clear ContentOBJECTIVES focus students on the concepts they are responsible for learning.

“It’s unique to use this narrative style and makes the reader to feel that someone is talking to him/her.”–Li Zhang, The Citadel

Balanced RigorTHEOREMS are clearly presented and often accompanied by proofs. More involved proofs are available for students in Appendix B.

“Good, clear progression on proof.”

–Marion Foster, Houston Community

College

“The level of rigor is just right.”

–Tilak DeAlwis, Southeastern

Louisiana University

CALC CLIPS are short whiteboard videos that walk the student through examples illustrating key concepts from the text.

A NOW WORK callout in most examples directs students to specific exercises in the section’s problems set, reinforcing learning by facilitating immediate practice.

“The progressively more difficult examples demonstrating

the aspects of Part 1 of the Fundamental Theorem of Calculus [chapter 5] are excellent. Much better than my current book.” –Nicholas Belloit, Florida State

College at Jacksonville

NEED TO REVIEW? Just in time review for students who need remediation.

“Remediation tools—this is exactly what students need.”

–Val Mohanakumar, Hillsborough Community College

IN WORDS restates complex formulas, theorems, proofs, rules and definitions using language that students might more easily understand.

Understand COnceptsThe authors promote a rigorous conceptual understanding of calculus while also emphasizing computational and problem-solving skills. Methodically structured exercise sets move the student from basic comprehension to skill building to application and extension exercises and, finally, more difficult challenge problems.

reach every student inunderstanding Concepts and Problem Solving

Visualize CalculusVIVID ILLUSTRATIONS AND GRAPHS make concepts easier to understand.

“I think the problems offer a nice variety of not only type,

but also difficulty.” –Patrick Taylor,

Shelton State College

“I like the delineation of the problems in relation to their difficulties. It makes it much easier to determine the homework assignments with a balanced approach.”

–Shawn Chiappetta, University of Sioux Falls

Concepts and Vocabulary

Applications and extensions

Skill Building

Challenge problems

Sullivan/Miranda, Calculus, Early Transcendentals, 2E

DEVELOP PROBLEM SOLVING SKILLSA variety of problem types, graduated in difficulty, help students develop a richer understanding. Application problems draw from a variety of disciplines.

DYNAMIC FIGURES allow students to manipulate figures from the text to enhance their understanding of core concepts, such as finding the volume of a solid using the Shell Method.

reach every student in preparation for education and career success

CH. 4 Applications of the Derivative

CH. 14 Mulitple Integrals

CH. 13 Directional Derivatives, Gradients, and Extrema

CH. 1 Limits and Continuitypreparation for successStudents come to this course with a diversity of educational and

career goals. The authors have filled their book with an array of

applied examples and exercises—mined from biology, chemistry,

physics, environmental studies, astronomy, engineering, sports

science, technology, geography, epidemiology, meteorology,

business and finance, and more—designed to appeal to your

students’ varied interests and reveal the richness of this course.

CH. 15 Vector Calculus

CH. 7 Techniques of Integration

Sullivan/Miranda, Calculus, Early Transcendentals, 2E

Each chapter opens with an engaging case study

that demonstrates how the concepts within

apply to situations across biology, engineering,

astronomy, economics, and other fields.

Students revisit these case studies at the end of

the chapter with an extended project.

PREPARING FOR CALCULUS P.1 Functions and Their GraphsP.2 Library of Functions; Mathematical ModelingP.3 Operations on Functions; Graphing TechniquesP.4 Inverse FunctionsP.5 Exponential and Logarithmic FunctionsP.6 Trigonometric FunctionsP.7 Inverse Trigonometric FunctionsP.8 Sequences; Summation Notation; the Binomial Theorem

1 LIMITS AND CONTINUITY1.1 Limits of Functions Using Numerical andGraphical Techniques1.2 Limits of Functions Using Properties of Limits1.3 Continuity1.4 Limits and Continuity of Trigonometric, Exponential, and Logarithmic Functions1.5 Infinite Limits; Limits at Infinity; Asymptotes1.6 The e–d Definition of a LimitChapter ReviewChapter Project: Pollution in Clear Lake

2 THE DERIVATIVE2.1 Rates of Change and the Derivative2.2 The Derivative as a Function2.3 The Derivative of a Polynomial Function;The Derivative of y = ex2.4 Differentiating the Product and the Quotient of Two Functions; Higher-Order Derivatives2.5 The Derivative of the Trigonometric FunctionsChapter ReviewChapter Project: The Lunar Module

3 MORE ABOUT DERIVATIVES3.1 The Chain Rule3.2 Implicit Differentiation3.3 Derivatives of the Inverse Trigonometric Functions

of a Solid of Revolution6.6 Work6.7 Hydrostatic Pressure and Force6.8 Center of Mass; Centroid; The Pappus TheoremChapter ReviewChapter Project: Determining the Amount of ConcreteNeeded for a Cooling Tower

7 TECHNIQUES OF INTEGRATION7.1 Integration by Parts7.2 Integrals Containing Trigonometric Functions7.3 Integration Using Trigonometric Substitution:Integrands Containing a2 − x2, x2 + a2, or x2 − a2, a > 07.4 Integrands Containing ax2 + bx + c, a ≠ 07.5 Integration of Rational Functions Using Partial Fractions; the Logistic Model7.6 Approximating Integrals: The Trapezoidal Rule, the Midpoint Rule, Simpson’s Rule7.7 Improper Integrals7.8 Integration Using Tables and ComputerAlgebra Systems7.9 Mixed PracticeChapter ReviewChapter Project: The Birds of Rugen Island

8 INFINITE SERIES8.1 Sequences8.2 Infinite Series8.3 Properties of Series; Series with Positive Terms; the Integral Test8.4 Comparison Tests8.5 Alternating Series; Absolute Convergence8.6 Ratio Test; Root Test8.7 Summary of Tests8.8 Power Series8.9 Taylor Series; Maclaurin Series8.10 Approximations Using Taylor/Maclaurin ExpansionsChapter ReviewChapter Project: How Calculators Calculate

3.4 Derivatives of Logarithmic Functions3.5 Differentials; Linear Approximations; Newton’s Method3.6 Hyperbolic FunctionsChapter ReviewChapter Project: World Population

4 APPLICATIONS OF THE DERIVATIVE4.1 Related Rates4.2 Maximum and Minimum Values; Critical Numbers4.3 The Mean Value Theorem4.4 Local Extrema and Concavity4.5 Indeterminate Forms and L’Hôpital’s Rule4.6 Using Calculus to Graph Functions4.7 Optimization4.8 Antiderivatives; Differential EquationsChapter ReviewChapter Project: The U.S. Economy

5 THE INTEGRAL5.1 Area5.2 The Definite Integral5.3 The Fundamental Theorem of Calculus5.4 Properties of the Definite Integral5.5 The Indefinite Integral; Method of Substitution5.6 Separable First-Order Differential Equations;Uninhibited and Inhibited Growth and Decay ModelsChapter ReviewChapter Project: Managing the Klamath River

6 APPLICATIONS OF THE INTEGRAL6.1 Area Between Graphs6.2 Volume of a Solid of Revolution: Disks and Washers6.3 Volume of a Solid of Revolution: Cylindrical Shells6.4 Volume of a Solid: Slicing6.5 Arc Length; Surface Area

CHAPTER P AND APPENDIX A: This material provides more depth than competitors on foundational precalculus topics important for calculus. Complete exercise sets are included.

CHAPTERS 2 AND 3: The development of the derivative and derivative formulas is presented in two chapters. Spreading this information out is less overwhelming for students and provides ample opportunity for practice. For example, the Chain Rule is presented at the beginning of Chapter 3, allowing more time for the student to master this topic.

CHAPTER 4: This chapter covers applications of the derivative. The related rates section gives a very detailed step-by-step approach. L’Hopital’s Rule is presented in this chapter to make the chapter complete in its coverage and to allow for its use in graphing functions. Antiderivatives and an introduction to differential equations are also noteworthy topics covered here.

CHAPTER 5: Area is covered just before the definite integral, and first-order differential equations are introduced here (compared to later introduction in some of the competitors). Emphasis is placed on providing a conceptual understanding of the definite integral. See, for example, the Objectives for Sections 5.2 and 5.3.

CHAPTER 6: The geometric applications of the integral begin with area. Two sections follow for finding volumes of surfaces of revolution: the disk method and the shell method. This is followed by the more general, and more difficult, slicing method. A robust collection of physical applications follows.

CHAPTER 7: Section 7.9 Mixed Practice summarizes strategies and provides opportunities for students to practice deciding what technique of integration to use.

9 PARAMETRIC EQUATIONS; POLAR EQUATIONS9.1 Parametric Equations9.2 Tangent Lines9.3 Arc Length; Surface Area of a Solid of Revolution9.4 Polar Coordinates9.5 Polar Equations; Parametric Equations of Polar Equations; Arc Length of Polar Equations9.6 Area in Polar Coordinates9.7 The Polar Equation of a ConicChapter ReviewChapter Project: Polar Graphs and Microphones

10 VECTORS; LINES, PLANES, AND QUADRIC SURFACES IN SPACE10.1 Rectangular Coordinates in Space10.2 Introduction to Vectors10.3 Vectors in the Plane and in Space10.4 The Dot Product10.5 The Cross Product10.6 Equations of Lines and Planes in Space10.7 Quadric SurfacesChapter ReviewChapter Project: The Hall Effect

11 VECTOR FUNCTIONS11.1 Vector Functions and Their Derivatives11.2 Unit Tangent and Principal Unit Normal Vectors; Arc Length11.3 Arc Length as Parameter; Curvature11.4 Motion Along a Curve11.5 Integrals of Vector Functions; Projectile Motion11.6 Application: Kepler’s Laws of Planetary MotionChapter ReviewChapter Project: How to Design a Safe Road

12 FUNCTIONS OF SEVERAL VARIABLES12.1 Functions of Two or More Variables and Their Graphs12.2 Limits and Continuity

12.3 Partial Derivatives12.4 Differentiability and the Differential12.5 Chain RulesChapter ReviewChapter Project: Searching for Exoplanets

13 DIRECTIONAL DERIVATIVES, GRADIENTS, AND EXTREMA13.1 Directional Derivatives; Gradients13.2 Tangent Planes13.3 Extrema of Functions of Two Variables13.4 Lagrange MultipliersChapter ReviewChapter Project: Measuring Ice Thickness on Crystal Lake

14 MULTIPLE INTEGRALS14.1 The Double Integral over a Rectangular Region14.2 The Double Integral over Nonrectangular Regions14.3 Double Integrals Using Polar Coordinates14.4 Center of Mass; Moment of Inertia14.5 Surface Area14.6 The Triple Integral14.7 Triple Integrals Using Cylindrical Coordinates14.8 Triple Integrals Using Spherical Coordinates14.9 Change of Variables Using JacobiansChapter ReviewChapter Project: The Mass of Stars

15 VECTOR CALCULUS15.1 Vector Fields15.2 Line Integrals of Scalar Functions15.3 Line Integrals of Vector Fields; Work15.4 Fundamental Theorem of Line Integrals15.5 Green’s Theorem15.6 Parametric Surfaces15.7 Surface and Flux Integrals15.8 The Divergence Theorem15.9 Stokes’ TheoremChapter ReviewChapter Project: Modeling a Tornado

16 DIFFERENTIAL EQUATIONS16.1 Classification of Ordinary Differential Equations16.2 Separable and Homogeneous First-Order DifferentialEquations; Slope Fields; Euler’s Method16.3 Exact Differential Equations16.4 First-Order Linear Differential Equations; Bernoulli Differential Equations16.5 Power Series MethodsChapter ReviewChapter Project: The Melting Arctic Ice Cap

APPENDIX A PRECALCULUS USED IN CALCULUSA.1 Algebra Used in CalculusA.2 Geometry Used in CalculusA.3 Analytic Geometry Used in CalculusA.4 Trigonometry Used in Calculus

APPENDIX B THEOREMS AND PROOFSB.1 Limit Theorems and ProofsB.2 Theorems and Proofs Involving Inverse FunctionsB.3 Derivative Theorems and ProofsB.4 Integral Theorems and ProofsB.5 A Bounded Monotonic Sequence ConvergesB.6 Taylor’s Formula with Remainder

APPENDIX C TECHNOLOGY USED IN CALCULUSC.1 Graphing CalculatorsC.2 Computer Algebra Systems (CAS)

AnswersIndex

CHAPTER 8: Section 8.7 Summary of Tests provides strategies and exercises to give students practice in deciding what test to use to determine convergence or divergence of an infinite series.

CHAPTERS 9 AND 10: These chapters provide a detailed development of topics required for multivariable calculus, developed through Chapters 11-15.

CHAPTER 16: The study of differential equations is woven through the text. Here an overview of differential equations that requires multivariable calculus is given.

APPENDIX B: Proofs that are relatively easy are given in the text. Those that are more advanced are given here in Appendix B. Proofs beyond the scope of the text are noted as such when they appear.

table of contents

Sullivan/Miranda, Calculus 2e

SULLIVAN

M I R A N DA

SECOND EDITION

CALCULUSE A R L Y T R A N S C E N D E N T A L S

For more information, contact your local Macmillan Learning representative or visit macmillanlearning.com/exploresullivan2e

CALCULUSEARLY TRANSCENDENTALS, SECOND EDITION

Michael Sullivan, Chicago State UniversityKathleen Miranda, State University of New York, Old Westbury

@2019 Cloth • 1328 pages • 978-1-319-01835-1reach

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