Calculus

Chapter P: Preparation for Calculus

• • P.1 Graphs and Models
• • P.2 Linear Models and Rates of Change
• • P.3 Functions and Their Graphs
• • P.4 Review of Trigonometric Functions

Chapter 1: Limits and Their Properties

• • 1.1 A Preview of Calculus
• • 1.2 Finding Limits Graphically and Numerically
• • 1.3 Evaluating Limits Analytically
• • 1.4 Continuity and One-Sided Limits
• • 1.5 Infinite Limits

Chapter 2: Differentiation

• • 2.1 The Derivative and the Tangent Line Problem
• • 2.2 Basic Differentiation Rules and Rates of Change
• • 2.3 Product and Quotient Rules and Higher-Order Derivatives
• • 2.4 The Chain Rule
• • 2.5 Implicit Differentiation
• • 2.6 Related Rates

Chapter 3: Applications of Differentiation

• • 3.1 Extrema on an Interval
• • 3.2 Rolle's Theorem and the Mean Value Theorem
• • 3.3 Increasing and Decreasing Functions and the First Derivative Test
• • 3.4 Concavity and the Second Derivative Test
• • 3.5 Limits at Infinity
• • 3.6 A Summary of Curve Sketching
• • 3.7 Optimization Problems
• • 3.8 Newton's Method
• • 3.9 Differentials

Chapter 4: Integration

• • 4.1 Antiderivatives and Indefinite Integration
• • 4.2 Area
• • 4.3 Riemann Sums and Definite Integrals
• • 4.4 The Fundamental Theorem of Calculus
• • 4.5 Integration by Substitution

Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions

• • 5.1 The Natural Logarithmic Function: Differentiation
• • 5.2 The Natural Logarithmic Function: Integration
• • 5.3 Inverse Functions
• • 5.4 Exponential Functions: Differentiation and Integration
• • 5.5 Bases Other Than e and Applications
• • 5.6 Indeterminate Forms and L'Hopital's Rule
• • 5.7 Inverse Trigonometric Functions: Differentiation
• • 5.8 Inverse Trigonometric Functions: Integration
• • 5.9 Hyperbolic Functions

Chapter 6: Differential Equations

• • 6.1 Slope Fields and Euler's Method
• • 6.2 Growth and Decay
• • 6.3 Separation of Variables and the Logistic Equation
• • 6.4 First-Order Linear Differential Equations

Chapter 7: Applications of Integration

• • 7.1 Area of a Region between Two Curves
• • 7.2 Volume: The Disk Method
• • 7.3 Volume: The Shell Method
• • 7.4 Arc Length and Surfaces of Revolution
• • 7.5 Work
• • 7.6 Moments, Centers of Mass, and Centroids
• • 7.7 Fluid Pressure and Fluid Force

Chapter 8: Integration Techniques and Improper Integrals

• • 8.1 Basic Integration Rules
• • 8.2 Integration by Parts
• • 8.3 Trigonometric Integrals
• • 8.4 Trigonometric Substitution
• • 8.5 Partial Fractions
• • 8.6 Numerical Integration
• • 8.7 Integration by Tables and Other Integration Techniques
• • 8.8 Improper Integrals

Chapter 9: Infinite Series

• • 9.1 Sequences
• • 9.2 Series and Convergence
• • 9.3 The Integral Test and p-Series
• • 9.4 Comparisons of Series
• • 9.5 Alternating Series
• • 9.6 The Ratio and Root Tests
• • 9.7 Taylor Polynomials and Approximations
• • 9.8 Power Series
• • 9.9 Representation of Functions by Power Series
• • 9.10 Taylor and Maclaurin Series

Chapter 10: Conics, Parametric Equations, and Polar Coordinates

• • 10.1 Conics and Calculus
• • 10.2 Plane Curves and Parametric Equations
• • 10.3 Parametric Equations and Calculus
• • 10.4 Polar Coordinates and Polar Graphs
• • 10.5 Area and Arc Length in Polar Coordinates
• • 10.6 Polar Equations of Conics and Kepler's Laws

Chapter 11: Vectors and the Geometry of Space

• • 11.1 Vectors in the Plane
• • 11.2 Space Coordinates and Vectors in Space
• • 11.3 The Dot Product of Two Vectors
• • 11.4 The Cross Product of Two Vectors in Space
• • 11.5 Lines and Planes in Space
• • 11.6 Surfaces in Space
• • 11.7 Cylindrical and Spherical Coordinates

Chapter 12: Vector-Valued Functions

• • 12.1 Vector-Valued Functions
• • 12.2 Differentiation and Integration of Vector-Valued Functions
• • 12.3 Velocity and Acceleration
• • 12.4 Tangent Vectors and Normal Vectors
• • 12.5 Arc Length and Curvature

Chapter 13: Functions of Several Variables

• • 13.1 Introduction to Functions of Several Variables
• • 13.2 Limits and Continuity
• • 13.3 Partial Derivatives
• • 13.4 Differentials
• • 13.5 Chain Rules for Functions of Several Variables
• • 13.6 Directional Derivatives and Gradients
• • 13.7 Tangent Planes and Normal Lines
• • 13.8 Extrema of Functions of Two Variables
• • 13.9 Applications of Extrema
• • 13.10 Lagrange Multipliers

Chapter 14: Multiple Integration

• • 14.1 Iterated Integrals and Area in the Plane
• • 14.2 Double Integrals and Volume
• • 14.3 Change of Variables: Polar Coordinates
• • 14.4 Center of Mass and Moments of Inertia
• • 14.5 Surface Area
• • 14.6 Triple Integrals and Applications
• • 14.7 Triple Integrals in Other Coordinates
• • 14.8 Change of Variables: Jacobians

Chapter 15: Vector Analysis

• • 15.1 Vector Fields
• • 15.2 Line Integrals
• • 15.3 Conservative Vector Fields and Independence of Path
• • 15.4 Green's Theorem
• • 15.5 Parametric Surfaces
• • 15.6 Surface Integrals
• • 15.7 Divergence Theorem
• • 15.8 Stokes's Theorem