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# Calculus by Rogawski

*Calculus, Early Transcendentals, Second Edition*

## Contents

- 1 Chapter 1: Precalculus Review
- 2 Chapter 2: Limits
- 3 Chapter 3: Differentiation
- 4 Chapter 4: Applications of the Derivative
- 5 Chapter 5: The Integral
- 6 Chapter 6: Applications of the Integral
- 7 Chapter 7: Techniques of Integration
- 8 Chapter 8: Further Applications of the Integral and Taylor Polynomials
- 9 Chapter 9: Introduction to Differential Equations
- 10 Chapter 10: Infinite Series
- 11 Chapter 11: Parametric Equations, Polar Coordinates, and Conic Sections
- 12 Chapter 12: Vector Geometry
- 13 Chapter 13: Calculus of Vector-Valued Functions
- 14 Chapter 14: Differentiation in Several Variables
- 15 Chapter 15: Multiple Integration
- 16 Chapter 16: Line and Surface Integrals
- 17 Chapter 17: Fundamental Theorems of Vector Analysis
- 18 Appendices

## 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 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 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

## 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 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 15: Multiple Integration

15.1 Integration in 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 Multiplying Integrals 15.6 Change of Variables

## 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 17: Fundamental Theorems of Vector Analysis

17.1 Green’s Theorem 17.2 Stokes’ Theorem 17.3 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