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This book is for instructors who think that most calculus textbooks are too long. In writing the book, James Stewart asked himself: What is essential for a three-semester calculus course for scientists and engineers? ESSENTIAL CALCULUS: EARLY TRANSCENDENTALS, Second Edition, offers a concise approach to teaching calculus that focuses on major concepts, and supports those concepts with precise definitions, patient explanations, and carefully graded problems. The book is only 900 pages--two-thirds the size of Stewart's other calculus texts, and yet it contains almost all of the same topics. The author achieved this relative brevity primarily by condensing the exposition and by putting some of the features on the book's website, www.StewartCalculus.com. Despite the more compact size, the book has a modern flavor, covering technology and incorporating material to promote conceptual understanding, though not as prominently as in Stewart's other books. ESSENTIAL CALCULUS: EARLY TRANSCENDENTALS features the same attention to detail, eye for innovation, and meticulous accuracy that have made Stewart's textbooks the best-selling calculus texts in the world.

1. FUNCTIONS AND LIMITS.

Functions and Their Representations. A Catalog of Essential Functions. The Limit of a Function. Calculating Limits. Continuity. Limits Involving Infinity.

2. DERIVATIVES.

Derivatives and Rates of Change. The Derivative as a Function. Basic Differentiation Formulas. The Product and Quotient Rules. The Chain Rule. Implicit Differentiation. Related Rates. Linear Approximations and Differentials.

3. INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS.

Exponential Functions. Inverse Functions and Logarithms. Derivatives of Logarithmic and Exponential Functions. Exponential Growth and Decay. Inverse Trigonometric Functions. Hyperbolic Functions. Indeterminate Forms and l'Hospital's Rule.

4. APPLICATIONS OF DIFFERENTIATION.

Maximum and Minimum Values. The Mean Value Theorem. Derivatives and the Shapes of Graphs. Curve Sketching. Optimization Problems. Newton's Method. Antiderivatives.

5. INTEGRALS.

Areas and Distances. The Definite Integral. Evaluating Definite Integrals. The Fundamental Theorem of Calculus. The Substitution Rule.

6. TECHNIQUES OF INTEGRATION.

Integration by Parts. Trigonometric Integrals and Substitutions. Partial Fractions. Integration with Tables and Computer Algebra Systems. Approximate Integration. Improper Integrals.

7. APPLICATIONS OF INTEGRATION.

Areas between Curves. Volumes. Volumes by Cylindrical Shells. Arc Length. Area of a Surface of Revolution. Applications to Physics and Engineering. Differential Equations.

8. SERIES.

Sequences. Series. The Integral and Comparison Tests. Other Convergence Tests. Power Series. Representing Functions as Power Series. Taylor and Maclaurin Series. Applications of Taylor Polynomials.

9. PARAMETRIC EQUATIONS AND POLAR COORDINATES.

Parametric Curves. Calculus with Parametric Curves. Polar Coordinates. Areas and Lengths in Polar Coordinates. Conic Sections in Polar Coordinates.

10. VECTORS AND THE GEOMETRY OF SPACE.

Three-Dimensional Coordinate Systems. Vectors. The Dot Product. The Cross Product. Equations of Lines and Planes. Cylinders and Quadric Surfaces. Vector Functions and Space Curves. Arc Length and Curvature. Motion in Space: Velocity and Acceleration.

11. PARTIAL DERIVATIVES.

Functions of Several Variables. Limits and Continuity. Partial Derivatives. Tangent Planes and Linear Approximations. The Chain Rule. Directional Derivatives and the Gradient Vector. Maximum and Minimum Values. Lagrange Multipliers.

12. MULTIPLE INTEGRALS.

Double Integrals over Rectangles. Double Integrals over General Regions. Double Integrals in Polar Coordinates. Applications of Double Integrals. Triple Integrals. Triple Integrals in Cylindrical Coordinates. Triple Integrals in Spherical Coordinates. Change of Variables in Multiple Integrals.

13. VECTOR CALCULUS.

Vector Fields. Line Integrals. The Fundamental Theorem for Line Integrals. Green's Theorem. Curl and Divergence. Parametric Surfaces and Their Areas. Surface Integrals. Stokes' Theorem. The Divergence Theorem.

Appendix A. Trigonometry.

Appendix B. Proofs.

Appendix C. Sigma Notation.

Appendix D. The Logarithm Defined as an Integral

- The book begins with four diagnostic tests in Basic Algebra, Analytic Geometry, Functions, and Trigonometry. These tests address the gaps in many students' prerequisite skills--and help them start the course with confidence--by giving them opportunities to get up to speed or brush up.
- Some material has been rewritten for greater clarity or for better motivation.
- Based on reviewers' suggestions, a new section on surface area has been added to Chapter 7, "Applications of Integration."
- Updated data in examples and exercises assures that the book maintains its currency and relevance.
- More than 35% of the exercises are new, providing instructors with new assignment options, and students with an abundance of practice opportunities.
- Enhanced WebAssign® for Stewart's ESSENTIAL CALCULUS: EARLY TRANSCENDENTALS, Second Edition--which combines the mathematics content in this text with the most powerful online homework solution, WebAssign--is enhanced with new features and functions. These include Enhanced Feedback for incorrect problems, Visualizing Calculus modules and animations, Lecture Video Problems, and a Show Your Work feature that gives you the option of seeing students' detailed solutions. Plus, Cengage YouBook, a Flash-based eBook version of the text that is interactive and customizable, is fully integrated into Enhanced WebAssign.

- The text presents a concise approach to calculus for instructors who want to focus on essential principles and who feel no need for frills.
- Brevity is achieved through condensed exposition, fewer examples in some sections, fewer technology and conceptual problems, and fewer appendixes. Problems Plus and Projects have been moved to the author's website at www.stewartcalculus.com.
- Certain topics, for example, the treatment of the integral and the remainder term in Taylor Series, are presented in a manner that is more traditional than in Stewart's other books.
- For the convenience of instructors who wish to cover additional material, the www.stewartcalculus.com website contains: Review of Algebra, Trigonometry, Analytic Geometry, and Conic Sections; Additional Examples; Projects; Archived Problems (drill exercises from Stewart's other books) with solutions; Challenge Problems; Lies My Calculator and Computer Told Me, and more.
- This version of the text presents exponential, logarithmic, and inverse trigonometric functions in Chapter Three. Those who wish to cover such functions later, with the logarithm defined as an integral, should consider the book entitled simply, ESSENTIAL CALCULUS.
- Each chapter ends with a comprehensive review that includes a true/false quiz, Concept Check Questions, and exercises for every topic covered in the chapter.
- Tools for Enriching Calculus--a free, online, interactive resource that allows Calculus students to work with animations that deepen their understanding by helping them visualize the concepts they are learning--has been updated with new problems and a new Flash design that is more visually appealing and engaging. Written and narrated by James Stewart, these assignable animations are also integrated into Cengage YouBook, and many are available directly to students at www.stewartcalculus.com.
- Additional topics (with exercises) available at www.stewartcalculus.com include Principles of Problem Solving, Strategy for Integration, Strategy for Testing Series, Fourier Series, Area of a Surface of Revolution, Linear Differential Equations, Second Order Linear Differential Equations, Nonhomogeneous Linear Equations, Applications of Second Order Differential Equations, Using Series to Solve Differential Equations, Complex Numbers, and Rotation of Axes. Links to outside web resources and the History of Mathematics, with links to historical websites, are also available at the site.

**James Stewart**

McMaster University

The late James Stewart received his M.S. from Stanford University and his Ph.D. from the University of Toronto. He did research at the University of London and was influenced by the famous mathematician George Polya at Stanford University. Stewart was most recently Professor of Mathematics at McMaster University, and his research field was harmonic analysis. Stewart was the author of a best-selling calculus textbook series published by Cengage Learning, including CALCULUS, CALCULUS: EARLY TRANSCENDENTALS, and CALCULUS: CONCEPTS AND CONTEXTS, as well as a series of precalculus texts.

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