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# Calculus: Early Transcendentals

## Third Edition| ©2015New Edition Available 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 studies and

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 studies and their lives.

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.

**Achieve for Calculus** redefines homework by offering guidance for every student and support for every instructor. Homework is designed to teach by correcting students' misconceptions through targeted feedback, meaningful hints, and full solutions, helping teach students conceptual understanding and critical thinking in real-world contexts.

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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 studies and their lives.

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.

**Achieve for Calculus** redefines homework by offering guidance for every student and support for every instructor. Homework is designed to teach by correcting students' misconceptions through targeted feedback, meaningful hints, and full solutions, helping teach students conceptual understanding and critical thinking in real-world contexts.

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

"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.""It's an invitation to learn calculus the right way."

--Jonathan Pearsall, instructor,College of Southern Nevada

--Nadjib Bouzar, instructor,University of Indianapolis

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## Table of Contents

**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

References

Index

## Authors

### 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.

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## Related Titles

## Videos

Colin Adams' Calculus 3e Co-authorship Video

Colin Adams discusses how he became involved with co-authoring Calculus 3e.

Colin Adams' Various Calculus Books Video

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Transitioning to Homework Video

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Notation Video

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Minimizing Memorization Video

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Understanding Formulas Video

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