Calculus: Early Transcendentals

Third Edition ©2015 Jon Rogawski; Colin Adams

The most successful calculus book of its generation, Jon Rogawski’s Calculus offers an ideal balance of formal precision and dedicated conceptual focus, helping students build strong computational skills while continually reinforcing the relevance of calculus to their future s...

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Guided by new author Colin Adams, the new edition stays true to the late Jon Rogawski’s refreshing and highly effective approach, while drawing on extensive instructor and student feedback, and Adams’ three decades as a calculus teacher and author of math books for general audiences.

W. H. Freeman/Macmillan and WebAssign have partnered to deliver WebAssign Premium – a comprehensive and flexible suite of resources for your calculus course. Combining the most widely used online homework platform with the authoritative and interactive content from the textbook, WebAssign Premium extends and enhances the classroom experience for instructors and students.

Features

Conceptual Insights encourage the student to develop a conceptual understanding of calculus by explaining important ideas clearly but informally.

Graphical Insights enhance the students' visual understanding by making the crucial connection between graphical properties and the underlying concept.

Reminders in the margins link back to important concepts discussed earlier in the text.

Caution notes warn students of common pitfalls they can encounter in understanding the material.

Historical Perspectives are brief vignettes that place key conceptual discoveries and advancements in their historical settings. They give students a glimpse into past accomplishments of great mathematicians and an appreciation for their significance.

Assumptions Matter uses short explanations and well-chosen counterexamples to help students appreciate why hypotheses are needed in theorems.

Section Summaries summarize a section’s key points in a concise and useful way to emphasize for students what is most important in the section.

Section Exercise Sets offer a comprehensive set of exercises closely coordinated with the text. These exercises vary in levels of difficulty from routine, to moderate, to more challenging. Also included are questions appropriate for written response or use of technology:
• Preliminary Exercises begin each exercise set and need little or no computation. They can be used to check understanding of key concepts of a section before problems from the exercise set are assigned.
• Exercises offers numerous problems from the routine drill problems to moderately challenging problems. These are carefully graded and include many innovative and interesting geometric and real world applications.
• Further Insights and Challenges are more challenging problems that help to extend a section’s material.
• End of Chapter Review Exercises offer a comprehensive set of exercises closely coordinated with the chapter material to provide additional problems for self study or assignments.

New to This Edition

New author, Colin Adams
Colin Adams is an award-winning teacher, widely read author, and distinguished researcher. A user of Jon Rogawski’s textbook, he brings his own classroom experience to the project, as well as a well-regarded ability to make calculus more engaging and meaningful to students without sacrificing its precision and rigor.

Refined Exercises
The exercise sets were reviewed extensively by longtime users to ensure the utmost accuracy, clarity, and complete content coverage. Exercise sets were also modified to improve upon the grading by level of difficulty and to ensure even/odd pairing.

In addition, numerous new exercises have been added throughout the text, particularly where new applications are available or to enhance conceptual development.

New Examples, including

Example 7 in Section 4.3 where the local minimum occurs at critical point but derivative does not exist

Example 5 in Section 6.1 that computes the areas between two curves in two ways

Several new examples in Chapter 10 including Example 1 (Fibonacci sequence) and Example 4 (bounded sequence) in Section 10.1 and Example 5 (repeated decimal expansion) in Section 10.2

Several examples in Chapter 14 including Example 7 (a limit that does not exist because different paths product different limits) in Section 14.2, Example 9 (gradient vectors perpendicular to level curves) in Section 14.5 and Example 5 (Second Derivative Tests fails) in Section 14.7

New Content Based on User and Reviewer Feedback

Strategies of Integration (new section in Ch. 7), incorporates many new examples to guide students on how to tackle integration problems

Determining Which Convergence Test To Apply (new in Ch. 10, sec. 5) reviews each test and provides strategies on when to apply them

Coverage of these concepts now focuses more on concepts and methods, rather than formulas and memorization:

Derivatives of inverse trigonometric functions (Ch. 3)

Trigonometric integrals (Ch. 7)

Quadric surfaces (Ch. 12)

New Illustrations
This edition includes a number of new figures that help students visualize concepts, including illustrations that explain:

What a limit means (Fig. 5 in Section 2.1, Fig. 3 in Section 2.2)

The Intermediate Value Theorem (Fig. 2 in Section 2.8)

Vertically and horizontally simple regions (Figs. 2 and 9 in Section 6.1)

Standardized Notation
Notational changes bring this edition in line with standard notation usage in mathematics and other fields that use mathematics, presenting a consistent message to students. Other notational changes make it easier for students to comprehend the concepts.

For example, in multivariable chapters, notation for vector-valued functions is now written r(t) = <x(t), y(t)> instead of c(t) = (x(t), y(t)) and the standard notation V is used for potential functions.

LearningCurve
In a game-like format, LearningCurve adaptive and formative quizzing provides an effective way to get students involved in the coursework. It offers:

A unique learning path for each student, with quizzes shaped by each individual’s correct and incorrect answers.

A Personalized Study Plan, to guide students’ preparation for class and for exams.

Feedback for each question with live links to relevant e-book pages, guiding students to the reading they need to do to improve their areas of weakness.

ONLINE HOMEWORK OPTION
In addition to the robust online homework system in LaunchPad, instructors can take advantage of the following W. H. Freeman partnerships:

WeBWorK
webwork.maa.org
W. H. Freeman offers approximately 2,500 algorithmically generated questions (with full solutions) through this free open source online homework system developed at the University of Rochester. Adopters also have access to a shared national library test bank with thousands of additional questions, including 1,500 problem sets correlated to the Third Edition.

WebAssign Premium
www.webassign.net/whfreeman
Premium for Calculus, Third Edition integrates the book’s exercises into the world’s most popular and trusted online homework system, making it easy to assign algorithmically generated homework and quizzes. WebAssign Premium also offers access to all of the book’s digital resources, with the option of including the complete e-Book.

"The clarity of examples, as well as their interconnectedness, remains a strong point. This fact alone goes a long way toward helping students better learn the concepts."
--Erik Tou, instructor, Carthage College

“It strikes the right balance between readability for the student and rigor for the instructor.”
--Debra Carney, instructor, Colorado School of Mines

"It is refreshing to see that even in the chapter on limits, there are application problems from a variety of disciplines, including engineering, physics and biology. The applications feel realistic and relevant as opposed to constructed and stilted."
--Maria Siopsis, instructor, Maryville College

"The strenghts are in the Conceptual and Graphical Insights. These are the kinds of comments that can make things 'click' and fall into place for the students."
--Berit Givens, instructor, California State Polytechnic University, Pomona

"The notation is a strong point of this book. It is used consistently and the authors do not shy away from using math instead of excess prose. Those two features alone put these chapters well ahead of my present text."
--Jonathan Pearsall, instructor, College of Southern Nevada

"It's an invitation to learn calculus the right way."
--Nadjib Bouzar, instructor, University of Indianapolis

Calculus: Early Transcendentals

Jon Rogawski; Colin Adams

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Calculus: Early Transcendentals

Third Edition 2015

Jon Rogawski; Colin Adams

Rogawski/Adams: Calculus Early Transcendentals 3e Table of Contents

Chapter 1: Precalculus Review
1.1 Real Numbers, Functions, and Graphs
1.2 Linear and Quadratic Functions
1.3 The Basic Classes of Functions
1.4 Trigonometric Functions
1.5 Inverse Functions
1.6 Exponential and Logarithmic Functions
1.7 Technology: Calculators and Computers
Chapter Review Exercises

Chapter 2: Limits
2.1 Limits, Rates of Change, and Tangent Lines
2.2 Limits: A Numerical and Graphical Approach
2.3 Basic Limit Laws
2.4 Limits and Continuity
2.5 Evaluating Limits Algebraically
2.6 Trigonometric Limits
2.7 Limits at Infinity
2.8 Intermediate Value Theorem
2.9 The Formal Definition of a Limit
Chapter Review Exercises

Chapter 3: Differentiation
3.1 Definition of the Derivative
3.2 The Derivative as a Function
3.3 Product and Quotient Rules
3.4 Rates of Change
3.5 Higher Derivatives
3.6 Trigonometric Functions
3.7 The Chain Rule
3.8 Implicit Differentiation
3.9 Derivatives of General Exponential and Logarithmic Functions
3.10 Related Rates
Chapter Review Exercises

Chapter 4: Applications of the Derivative
4.1 Linear Approximation and Applications
4.2 Extreme Values
4.3 The Mean Value Theorem and Monotonicity
4.4 The Shape of a Graph
4.5 L’Hopital’s Rule
4.6 Graph Sketching and Asymptotes
4.7 Applied Optimization
4.8 Newton’s Method
Chapter Review Exercises

Chapter 5: The Integral
5.1 Approximating and Computing Area
5.2 The Definite Integral
5.3 The Indefinite Integral
5.4 The Fundamental Theorem of Calculus, Part I
5.5 The Fundamental Theorem of Calculus, Part II
5.6 Net Change as the Integral of a Rate
5.7 Substitution Method
5.8 Further Transcendental Functions
5.9 Exponential Growth and Decay
Chapter Review Exercises

Chapter 6: Applications of the Integral
6.1 Area Between Two Curves
6.2 Setting Up Integrals: Volume, Density, Average Value
6.3 Volumes of Revolution
6.4 The Method of Cylindrical Shells
6.5 Work and Energy
Chapter Review Exercises

Chapter 7: Techniques of Integration
7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitution
7.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions
7.5 The Method of Partial Fractions
7.6 Strategies for Integration
7.7 Improper Integrals
7.8 Probability and Integration
7.9 Numerical Integration
Chapter Review Exercises

Chapter 8: Further Applications of the Integral and Taylor Polynomials
8.1 Arc Length and Surface Area
8.2 Fluid Pressure and Force
8.3 Center of Mass
8.4 Taylor Polynomials
Chapter Review Exercises

Chapter 9: Introduction to Differential Equations
9.1 Solving Differential Equations
9.2 Models Involving y^'=k(y-b)
9.3 Graphical and Numerical Methods
9.4 The Logistic Equation
9.5 First-Order Linear Equations
Chapter Review Exercises

Chapter 10: Infinite Series
10.1 Sequences
10.2 Summing an Infinite Series
10.3 Convergence of Series with Positive Terms
10.4 Absolute and Conditional Convergence
10.5 The Ratio and Root Tests
10.6 Power Series
10.7 Taylor Series
Chapter Review Exercises

Chapter 11: Parametric Equations, Polar Coordinates, and Conic Sections
11.1 Parametric Equations
11.2 Arc Length and Speed
11.3 Polar Coordinates
11.4 Area and Arc Length in Polar Coordinates
11.5 Conic Sections
Chapter Review Exercises

Chapter 12: Vector Geometry
12.1 Vectors in the Plane
12.2 Vectors in Three Dimensions
12.3 Dot Product and the Angle Between Two Vectors
12.4 The Cross Product
12.5 Planes in Three-Space
12.6 A Survey of Quadric Surfaces
12.7 Cylindrical and Spherical Coordinates
Chapter Review Exercises

Chapter 13: Calculus of Vector-Valued Functions
13.1 Vector-Valued Functions
13.2 Calculus of Vector-Valued Functions
13.3 Arc Length and Speed
13.4 Curvature
13.5 Motion in Three-Space
13.6 Planetary Motion According to Kepler and Newton
Chapter Review Exercises

Chapter 14: Differentiation in Several Variables
14.1 Functions of Two or More Variables
14.2 Limits and Continuity in Several Variables
14.3 Partial Derivatives
14.4 Differentiability and Tangent Planes
14.5 The Gradient and Directional Derivatives
14.6 The Chain Rule
14.7 Optimization in Several Variables
14.8 Lagrange Multipliers: Optimizing with a Constraint
Chapter Review Exercises

Chapter 15: Multiple Integration
15.1 Integration in Two Variables
15.2 Double Integrals over More General Regions
15.3 Triple Integrals
15.4 Integration in Polar, Cylindrical, and Spherical Coordinates
15.5 Applications of Multiple Integrals
15.6 Change of Variables
Chapter Review Exercises

Chapter 16: Line and Surface Integrals
16.1 Vector Fields
16.2 Line Integrals
16.3 Conservative Vector Fields
16.4 Parametrized Surfaces and Surface Integrals
16.5 Surface Integrals of Vector Fields
Chapter Review Exercises

Chapter 17: Fundamental Theorems of Vector Analysis
17.1 Green’s Theorem
17.2 Stokes’ Theorem
17.3 Divergence Theorem
Chapter Review Exercises

Appendices
A. The Language of Mathematics
B. Properties of Real Numbers
C. Induction and the Binomial Theorem
D. Additional Proofs

Answers to Odd-Numbered Exercises
References
Index

Calculus: Early Transcendentals

Third Edition 2015

Jon Rogawski; Colin Adams

Find Your Rep

Jon Rogawski

Jon Rogawski received his undergraduate and master’s degrees in mathematics simultaneously from Yale University, and he earned his PhD in mathematics from Princeton University, where he studied under Robert Langlands. Before joining the Department of Mathematics at UCLA in 1986, where he was a full professor, he held teaching and visiting positions at the Institute for Advanced Study, the University of Bonn, and the University of Paris at Jussieu and Orsay. Jon’s areas of interest were number theory, automorphic forms, and harmonic analysis on semisimple groups. He published numerous research articles in leading mathematics journals, including the research monograph Automorphic Representations of Unitary Groups in Three Variables (Princeton University Press). He was the recipient of a Sloan Fellowship and an editor of the Pacific Journal of Mathematics and the Transactions of the AMS. As a successful teacher for more than 30 years, Jon Rogawski listened and learned much from his own students. These valuable lessons made an impact on his thinking, his writing, and his shaping of a calculus text. Sadly, Jon Rogawski passed away in September 2011. Jon’s commitment to presenting the beauty of calculus and the important role it plays in students’ understanding of the wider world is the legacy that lives on in each new edition of Calculus.

Colin Adams

Colin Adams is the Thomas T. Read professor of Mathematics at Williams College, where he has taught since 1985. Colin received his undergraduate degree from MIT and his PhD from the University of Wisconsin. His research is in the area of knot theory and low-dimensional topology. He has held various grants to support his research, and written numerous research articles. Colin is the author or co-author of The Knot Book, How to Ace Calculus: The Streetwise Guide, How to Ace the Rest of Calculus: The Streetwise Guide, Riot at the Calc Exam and Other Mathematically Bent Stories, Why Knot?, Introduction to Topology: Pure and Applied, and Zombies & Calculus. He co-wrote and appears in the videos “The Great Pi vs. E Debate” and “Derivative vs. Integral: the Final Smackdown.” He is a recipient of the Haimo National Distinguished Teaching Award from the Mathematical Association of America (MAA) in 1998, an MAA Polya Lecturer for 1998-2000, a Sigma Xi Distinguished Lecturer for 2000-2002, and the recipient of the Robert Foster Cherry Teaching Award in 2003. Colin has two children and one slightly crazy dog, who is great at providing the entertainment.