# Calculus Vol 1

Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, OpenStax split the book into 3 volumes. Volume 1 covers functions, limits, derivatives, and integration. Volume 2 covers integration, differential equations, sequences and series, and parametric equations and polar coordinates. Volume 3 covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second-order differential equations.

Volume 2 can be found here or read here

Volume 3 can be found here or read here

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#### Volume 1

- Preface
- 1. Functions and Graphs
- Introduction
- 1.1. Review of Functions
- 1.2. Basic Classes of Functions
- 1.3. Trigonometric Functions
- 1.4. Inverse Functions
- 1.5. Exponential and Logarithmic Functions

- 2. Limits
- Introduction
- 2.1. A Preview of Calculus
- 2.2. The Limit of a Function
- 2.3. The Limit Laws
- 2.4. Continuity
- 2.5. The Precise Definition of a Limit

- 3. Derivatives
- Introduction
- 3.1. Defining the Derivative
- 3.2. The Derivative as a Function
- 3.3. Differentiation Rules
- 3.4. Derivatives as Rates of Change
- 3.5. Derivatives of Trigonometric Functions
- 3.6. The Chain Rule
- 3.7. Derivatives of Inverse Functions
- 3.8. Implicit Differentiation
- 3.9. Derivatives of Exponential and Logarithmic Functions

- 4. Applications of Derivatives
- Introduction
- 4.1. Related Rates
- 4.2. Linear Approximations and Differentials
- 4.3. Maxima and Minima
- 4.4. The Mean Value Theorem
- 4.5. Derivatives and the Shape of a Graph
- 4.6. Limits at Infinity and Asymptotes
- 4.7. Applied Optimization Problems
- 4.8. Lâ€™HÃ´pitalâ€™s Rule
- 4.9. Newtonâ€™s Method
- 4.10. Antiderivatives

- 5. Integration
- Introduction
- 5.1. Approximating Areas
- 5.2. The Definite Integral
- 5.3. The Fundamental Theorem of Calculus
- 5.4. Integration Formulas and the Net Change Theorem
- 5.5. Substitution
- 5.6. Integrals Involving Exponential and Logarithmic Functions
- 5.7. Integrals Resulting in Inverse Trigonometric Functions

- 6. Applications of Integration
- Introduction
- 6.1. Areas between Curves
- 6.2. Determining Volumes by Slicing
- 6.3. Volumes of Revolution: Cylindrical Shells
- 6.4. Arc Length of a Curve and Surface Area
- 6.5. Physical Applications
- 6.6. Moments and Centers of Mass
- 6.7. Integrals, Exponential Functions, and Logarithms
- 6.8. Exponential Growth and Decay
- 6.9. Calculus of the Hyperbolic Functions

- Table of Integrals
- Table of Derivatives
- Review of Pre-Calculus

#### Volume 2

- Preface
- 1. Integration
- Introduction
- 1.1. Approximating Areas
- 1.2. The Definite Integral
- 1.3. The Fundamental Theorem of Calculus
- 1.4. Integration Formulas and the Net Change Theorem
- 1.5. Substitution
- 1.6. Integrals Involving Exponential and Logarithmic Functions
- 1.7. Integrals Resulting in Inverse Trigonometric Functions

- 2. Applications of Integration
- Introduction
- 2.1. Areas between Curves
- 2.2. Determining Volumes by Slicing
- 2.3. Volumes of Revolution: Cylindrical Shells
- 2.4. Arc Length of a Curve and Surface Area
- 2.5. Physical Applications
- 2.6. Moments and Centers of Mass
- 2.7. Integrals, Exponential Functions, and Logarithms
- 2.8. Exponential Growth and Decay
- 2.9. Calculus of the Hyperbolic Functions

- 3. Techniques of Integration
- Introduction
- 3.1. Integration by Parts
- 3.2. Trigonometric Integrals
- 3.3. Trigonometric Substitution
- 3.4. Partial Fractions
- 3.5. Other Strategies for Integration
- 3.6. Numerical Integration
- 3.7. Improper Integrals

- 4. Introduction to Differential Equations
- Introduction
- 4.1. Basics of Differential Equations
- 4.2. Direction Fields and Numerical Methods
- 4.3. Separable Equations
- 4.4. The Logistic Equation
- 4.5. First-order Linear Equations

- 5. Sequences and Series
- Introduction
- 5.1. Sequences
- 5.2. Infinite Series
- 5.3. The Divergence and Integral Tests
- 5.4. Comparison Tests
- 5.5. Alternating Series
- 5.6. Ratio and Root Tests

- 6. Power Series
- Introduction
- 6.1. Power Series and Functions
- 6.2. Properties of Power Series
- 6.3. Taylor and Maclaurin Series
- 6.4. Working with Taylor Series

- 7. Parametric Equations and Polar Coordinates
- Introduction
- 7.1. Parametric Equations
- 7.2. Calculus of Parametric Curves
- 7.3. Polar Coordinates
- 7.4. Area and Arc Length in Polar Coordinates
- 7.5. Conic Sections

- Table of Integrals
- Table of Derivatives
- Review of Pre-Calculus

#### Volume 3

- Preface
- 1. Parametric Equations and Polar Coordinates
- Introduction
- 1.1. Parametric Equations
- 1.2. Calculus of Parametric Curves
- 1.3. Polar Coordinates
- 1.4. Area and Arc Length in Polar Coordinates
- 1.5. Conic Sections

- 2. Vectors in Space
- Introduction
- 2.1. Vectors in the Plane
- 2.2. Vectors in Three Dimensions
- 2.3. The Dot Product
- 2.4. The Cross Product
- 2.5. Equations of Lines and Planes in Space
- 2.6. Quadric Surfaces
- 2.7. Cylindrical and Spherical Coordinates

- 3. Vector-Valued Functions
- Introduction
- 3.1. Vector-Valued Functions and Space Curves
- 3.2. Calculus of Vector-Valued Functions
- 3.3. Arc Length and Curvature
- 3.4. Motion in Space

- 4. Differentiation of Functions of Several Variables
- Introduction
- 4.1. Functions of Several Variables
- 4.2. Limits and Continuity
- 4.3. Partial Derivatives
- 4.4. Tangent Planes and Linear Approximations
- 4.5. The Chain Rule
- 4.6. Directional Derivatives and the Gradient
- 4.7. Maxima/Minima Problems
- 4.8. Lagrange Multipliers

- 5. Multiple Integration
- Introduction
- 5.1. Double Integrals over Rectangular Regions
- 5.2. Double Integrals over General Regions
- 5.3. Double Integrals in Polar Coordinates
- 5.4. Triple Integrals
- 5.5. Triple Integrals in Cylindrical and Spherical Coordinates
- 5.6. Calculating Centers of Mass and Moments of Inertia
- 5.7. Change of Variables in Multiple Integrals

- 6. Vector Calculus
- Introduction
- 6.1. Vector Fields
- 6.2. Line Integrals
- 6.3. Conservative Vector Fields
- 6.4. Greenâ€™s Theorem
- 6.5. Divergence and Curl
- 6.6. Surface Integrals
- 6.7. Stokesâ€™ Theorem
- 6.8. The Divergence Theorem

- 7. Second-Order Differential Equations
- Introduction
- 7.1. Second-Order Linear Equations
- 7.2. Nonhomogeneous Linear Equations
- 7.3. Applications
- 7.4. Series Solutions of Differential Equations

- Table of Integrals
- Table of Derivatives
- Review of Pre-Calculus

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