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# Calculus: Early Transcendentals, Single Variable

## Chapters 1-11Second Edition| ©2012 Jon Rogawski

What’s the ideal balance? How can you make sure students get both the computational skills they need and a deep understanding of the significance of what they are learning? With your teaching—supported by Rogawski’s

*Calculus*Second Edition—the mo...What’s the ideal balance? How can you make sure students get both the computational skills they need and a deep understanding of the significance of what they are learning? With your teaching—supported by Rogawski’s

Widely adopted in its first edition, Rogawski’s

Now Rogawski’s

*Calculus*Second Edition—the most successful new calculus text in 25 years!Widely adopted in its first edition, Rogawski’s

*Calculus*worked for instructors and students by balancing formal precision with a guiding conceptual focus. Rogawski engages students while reinforcing the relevance of calculus to their lives and future studies. Precise mathematics, vivid examples, colorful graphics, intuitive explanations, and extraordinary problem sets all work together to help students grasp a deeper understanding of calculus.Now Rogawski’s

*Calculus*success continues in a meticulously updated new edition. Revised in response to user feedback and classroom experiences, the new edition provides an even smoother teaching and learning experience.

*Calculus*Second Edition—the most successful new calculus text in 25 years!

Widely adopted in its first edition, Rogawski’s

*Calculus*worked for instructors and students by balancing formal precision with a guiding conceptual focus. Rogawski engages students while reinforcing the relevance of calculus to their lives and future studies. Precise mathematics, vivid examples, colorful graphics, intuitive explanations, and extraordinary problem sets all work together to help students grasp a deeper understanding of calculus.

Now Rogawski’s

*Calculus*success continues in a meticulously updated new edition. Revised in response to user feedback and classroom experiences, the new edition provides an even smoother teaching and learning experience.

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 i**n 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

Jon Rogawski’s new edition of*Calculus*builds on the strengths of the bestselling First Edition by incorporating his own classroom experience, as well as extensive feedback from many in the mathematics community, including adopters, nonusers, reviewers, and students. Every section has been carefully revised in order to further polish a text that has been enthusiastically recognized for its meticulous pedagogy and its careful balance among the fundamental pillars of calculus instruction: conceptual understanding, skill development, problem solving, and innovative real world applications.

**Enhanced Exercise Sets—with Approximately 25% New and Revised Problems**

The Second Edition features thousands of new and updated problems. Exercise sets were meticulously reviewed by users and nonusers to assist the author as he revised this cornerstone feature of the text. Rogawski carefully evaluated and rewrote exercise sets as needed to further refine quality, pacing, coverage and quantity.

N

**ew and Larger Variety of Applications**

To show students how calculus directly relates to the real world, the Second Edition features many fresh and creative examples and exercises centered on innovative, contemporary applications from engineering, the life sciences, physical sciences, business, economics, medicine, and the social sciences.

**Updated Art Program— with Approximately 15% New Figures**

Throughout the Second Edition, there are numerous new and updated figures with refined labeling to increase student understanding. The author takes special care to position the art with the related exposition and provide multiple figures rather than a single one for increased visual support of the concepts.

**Key Content Changes**

Rogawski’s Second Edition includes several content changes in response to feedback from users and reviewers. The key revisions include:

- The topic “Limits at Infinity” has been moved forward from Chapter 4 to Section 2.7 so all types of limits are introduced together (Chapter 2 Limits).

- Coverage of “Differentials” has been expanded in ET Section 4.1 and ET Section 14.4 for those who wish to emphasize differentials in their approach to Linear Approximation.

- “L'Hôpital’s Rule” has been moved up in the ET version so this topic can support the section on graph sketching (Chapter 4 Applications of the Derivative in ET).

- The section on “Numerical Integration” has been moved to the end of the chapter after the techniques of integration are presented (Techniques of Integration chapter).

- A section on “Probability and Integration” was added to allow students to explore new applications of integration important in the physical sciences, as well as in business and the social sciences (Techniques of Integration chapter).

- Multivariable Calculus: Recognized as especially strong in Rogawski’s Calculus, the multivariable chapters have been meticulously refined to enhance pedagogy and conceptual clarity. Exercise sets have been improved and rebalanced to fully support basic skill development, as well as conceptual and visual understanding.

- A new section “Applications of Multiple Integrals” has been added to ET Chapter 15 (LT Chapter 16) to unify the approach to applications of multiple integration and give students an even broader selection of applied problems from the physical and social sciences.

**
Calculus: Early Transcendentals, Single Variable**

Second Edition| ©2012

Jon Rogawski

# Digital Options

**Calculus: Early Transcendentals, Single Variable**

Second Edition| 2012

Jon Rogawski

## 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 2: Limits**2.1 Limits, Rates of Change, and Tangent Lines

2.2 Limits: A Numerical and Graphical Approach2.3 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 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 Derivatives of Inverse Functions

3.9 Derivatives of General Exponential and Logarithmic Functions

3.10 Implicit Differentiation

3.11 Related Rates

**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

4.9 Antiderivatives

**Chapter 5: The Integral**

5.1 Approximating and Computing Area

5.2 The Definite Integral

5.3 The Fundamental Theorem of Calculus, Part I

5.4 The Fundamental Theorem of Calculus, Part II

5.5 Net Change as the Integral of a Rate

5.6 Substitution Method

5.7 Further Transcendental Functions

5.8 Exponential Growth and Decay

**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 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 Improper Integrals

7.7 Probability and Integration

7.8 Numerical Integration

**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 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 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 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 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

## 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.

**Calculus: Early Transcendentals, Single Variable**

Second Edition| 2012

Jon Rogawski

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