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- Mathematics: Calculus
- Calculus

# Calculus

## 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 studies and theirThe most successful calculus book of its generation, Jon Rogawski’s

Read more
*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.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**extends and enhances the classroom experience for instructors and students.

*Premium***Maximize Teaching and Learning with WebAssign Premium**

Macmillan Learning 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 authoritative textbook content and Macmillan’s esteemed Calctools, WebAssign

*Premium*extends and enhances the classroom experience for instructors and students. Preview course content and sample assignments at www.webassign.net/whfreeman.

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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.

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**extends and enhances the classroom experience for instructors and students.

*Premium***Maximize Teaching and Learning with WebAssign Premium**

Macmillan Learning 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 authoritative textbook content and Macmillan’s esteemed Calctools, WebAssign

*Premium*extends and enhances the classroom experience for instructors and students. Preview course content and sample assignments at www.webassign.net/whfreeman.

Features

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.

**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.**

Refined Exercises

Refined Exercises

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

**, including**

New Examples

New Examples

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 11 including Example 1 (Fibonacci sequence) and Example 4 (bounded sequence) in Section 11.1 and Example 5 (repeated decimal expansion) in Section 11.2

Several examples in Chapter 15 including Example 7 (a limit that does not exist because different paths product different limits) in Section 15.2, Example 9 (gradient vectors perpendicular to level curves) in Section 15.5 and Example 5 (Second Derivative Tests fails) in Section 15.7

New Content Based on User and Reviewer Feedback

New Content Based on User and Reviewer Feedback

**Strategies of Integration**(new section in Ch. 8), incorporates many new examples to guide students on how to tackle integration problems

**Determining Which Convergence Test To Apply**(new in Ch. 11, sec. 5) reviews each test and provides strategies on when to apply them.

**More Focus on Concepts**

Derivatives of inverse trigonometric functions (Ch. 7)

Trigonometric integrals (Ch. 8)

Quadric surfaces (Ch. 13)

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

New Illustrations

New Illustrations

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.

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

ONLINE HOMEWORK OPTION

ONLINE HOMEWORK OPTION

**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

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.

**
Calculus**

Third Edition| ©2015

Jon Rogawski; Colin Adams

# Digital Options

## E-book

Read online (or offline) with all the highlighting and notetaking tools you need to be successful in this course.

**Calculus**

Third Edition| 2015

Jon Rogawski; Colin Adams

## Table of Contents

**Rogawski/Adams: Calculus 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 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 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 Graph Sketching and Asymptotes

4.6 Applied Optimizations

4.7 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

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: Exponential Functions**7.1 Derivative of f(x)=bx and the Number

*e*

7.2 Inverse Functions

7.3 Logarithms and their Derivatives

7.4 Exponential Growth and Decay

7.5 Compound Interest and Present Value

7.6 Models Involving y’= k(y-b)

7.7 L’Hôpital’s Rule

7.8 Inverse Trigonometric Functions

7.9 Hyperbolic Functions

Chapter Review Exercises

**Chapter 8: Techniques of Integration**8.1 Integration by Parts

8.2 Trigonometric Integrals

8.3 Trigonometric Substitution

8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions

8.5 The Method of Partial Fractions

8.6 Strategies for Integration

8.7 Improper Integrals

8.8 Probability and Integration

8.9 Numerical Integration

Chapter Review Exercises

**Chapter 9: Further Applications of the Integral and Taylor Polynomials **9.1 Arc Length and Surface Area

9.2 Fluid Pressure and Force

9.3 Center of Mass

9.4 Taylor Polynomials

Chapter Review Exercises

**Chapter 10: Introduction to Differential Equations**10.1 Solving Differential Equations

10.2 Graphical and Numerical Methods

10.3 The Logistic Equation

10.4 First-Order Linear Equations

Chapter Review Exercises

**Chapter 11: Infinite Series**11.1 Sequences

11.2 Summing an Infinite Series

11.3 Convergence of Series with Positive Terms

11.4 Absolute and Conditional Convergence

11.5 The Ratio and Root Tests

11.6 Power Series

11.7 Taylor Series

Chapter Review Exercises

**Chapter 12: Parametric Equations, Polar Coordinates, and Conic Sections **12.1 Parametric Equations

12.2 Arc Length and Speed

12.3 Polar Coordinates

12.4 Area and Arc Length in Polar Coordinates

12.5 Conic Sections

Chapter Review Exercises

**Chapter 13: Vector Geometry**13.1 Vectors in the Plane

13.2 Vectors in Three Dimensions

13.3 Dot Product and the Angle Between Two Vectors

13.4 The Cross Product

13.5 Planes in Three-Space

13.6 A Survey of Quadric Surfaces

13.7 Cylindrical and Spherical Coordinates

Chapter Review Exercises

**Chapter 14: Calculus of Vector-Valued Functions **14.1 Vector-Valued Functions

14.2 Calculus of Vector-Valued Functions

14.3 Arc Length and Speed

14.4 Curvature

14.5 Motion in Three-Space

14.6 Planetary Motion According to Kepler and Newton

Chapter Review Exercises

**Chapter 15: Differentiation in Several Variables**15.1 Functions of Two or More Variables

15.2 Limits and Continuity in Several Variables

15.3 Partial Derivatives

15.4 Differentiability and Tangent Planes

15.5 The Gradient and Directional Derivatives

15.6 The Chain Rule

15.7 Optimization in Several Variables

15.8 Lagrange Multipliers: Optimizing with a Constraint

Chapter Review Exercises

**Chapter 16: Multiple Integration**16.1 Integration in Variables

16.2 Double Integrals over More General Regions

16.3 Triple Integrals

16.4 Integration in Polar, Cylindrical, and Spherical Coordinates

16.5 Applications of Multiplying Integrals

16.6 Change of Variables

Chapter Review Exercises

**Chapter 17: Line and Surface Integrals**17.1 Vector Fields

17.2 Line Integrals

17.3 Conservative Vector Fields

17.4 Parametrized Surfaces and Surface Integrals

17.5 Surface Integrals of Vector Fields

Chapter Review Exercises

**Chapter 18: Fundamental Theorems of Vector Analysis**18.1 Green’s Theorem

18.2 Stokes’ Theorem

18.3 Divergence Theorem

**Appendices**A. The Language of Mathematics

B. Properties of Real Numbers

C. Mathematical Induction and the Binomial Theorem

D. Additional Proofs of Theorems

Answers to Odd-Numbered Exercises

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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**Calculus**

Third Edition| 2015

Jon Rogawski; Colin Adams

## Related Titles

**Calculus**

Third Edition| 2015

Jon Rogawski; Colin Adams

## Videos

Colin Adams' Calculus 3e Co-authorship Video

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

Author Talk

Colin Adams' Various Calculus Books Video

Colin Adams describes his supplemental texts and new novel, Zombies & Calculus.

Author Talk

Colin Adams' knot theory Video

Colin Adams describes how he began working on Knot Theory.

Author Talk

Transitioning to Homework Video

Colin Adams describes how Calculus 3e helps students transition from class to homework.

Author Talk

Notation Video

Colin Adams explains important updates to the notation in Calculus 3e.

Author Talk

Minimizing Memorization Video

Colin Adams discusses his focus on concepts and minimizing memorization in Calculus 3e.

Author Talk

Understanding Formulas Video

Colin Adams talks about how the new edition helps students understand formulas.

Author Talk

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