# Calculus

## Second EditionJon Rogawski

©2012What’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 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.## 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 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 3: Differentiation**3.1 Definition of the Derivative

3.2 The Derivative as a Function

3.3 Product and Quotient Rates

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 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

4.8 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

**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: The Exponential Function**

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 8: Techniques of Integration**

8.1 Integration by Parts

8.2 Trigonometric Integral

8.3 Trigonometric Substitution

8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions

8.5 The Method of Partial Fractions

8.6 Improper Integrals

8.7 Probability and Integration

8.8 Numerical Integration

**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 10: Introduction to Differential Equations**

10.1 Solving Differential Equations

10.2 Graphical and Numerical Method

10.3 The Logistic Equation

10.4 First-Order Linear Equations

**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 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 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 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 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 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 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 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

E. Taylor Polynomials

**Answers to Odd-Numbered ExercisesReferencesPhoto CreditsIndex**

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