Thomas' Calculus: Early Transcendentals

Chapter 1: Functions

• • Functions and Their Graphs
• • Combining Functions
• • Trigonometric Functions
• • Graphing with Software

Chapter 2: Limits and Continuity

• • Rates of Change and Tangent Lines
• • Limit of a Function and Limit Laws
• • The Precise Definition of a Limit
• • One-Sided Limits
• • Continuity
• • Limits Involving Infinity

Chapter 3: Derivatives

• • Tangents and the Derivative at a Point
• • The Derivative as a Function
• • Differentiation Rules
• • The Derivative as a Rate of Change
• • Derivatives of Trigonometric Functions
• • The Chain Rule
• • Implicit Differentiation
• • Related Rates
• • Linearization and Differentials

Chapter 4: Applications of Derivatives

• • Extreme Values of Functions
• • The Mean Value Theorem
• • Monotonic Functions and the First Derivative Test
• • Concavity and Curve Sketching
• • Applied Optimization
• • Newton's Method
• • Antiderivatives

Chapter 5: Integrals

• • Area and Estimating with Finite Sums
• • Sigma Notation and Limits of Finite Sums
• • The Definite Integral
• • The Fundamental Theorem of Calculus
• • Indefinite Integrals and the Substitution Method
• • Definite Integral Substitutions and the Area Between Curves

Chapter 6: Applications of Definite Integrals

• • Volumes Using Cross-Sections
• • Volumes Using Cylindrical Shells
• • Arc Length
• • Areas of Surfaces of Revolution
• • Work and Fluid Forces
• • Moments and Centers of Mass

Chapter 7: Integrals and Transcendental Functions

• • The Logarithm Defined as an Integral
• • Exponential Functions
• • Hyperbolic Functions
• • Relative Rates of Growth

Chapter 8: Techniques of Integration

• • Using Basic Integration Formulas
• • Integration by Parts
• • Trigonometric Integrals
• • Trigonometric Substitutions
• • Integration of Rational Functions by Partial Fractions
• • Integral Tables and Computer Algebra Systems
• • Numerical Integration
• • Improper Integrals

Chapter 9: First-Order Differential Equations

• • Solutions, Slope Fields, and Euler's Method
• • First-Order Linear Equations
• • Applications
• • Graphical Solutions of Autonomous Equations
• • Systems of Equations and Phase Planes

Chapter 10: Infinite Sequences and Series

• • Sequences
• • Infinite Series
• • The Integral Test
• • Comparison Tests
• • The Ratio and Root Tests
• • Alternating Series, Absolute and Conditional Convergence
• • Power Series
• • Taylor and Maclaurin Series
• • Convergence of Taylor Series
• • Applications of Taylor Series

Chapter 11: Parametric Equations and Polar Coordinates

• • Parametric Equations for Plane Curves
• • Calculus with Parametric Curves
• • Polar Coordinates
• • Graphing in Polar Coordinates
• • Areas and Lengths in Polar Coordinates
• • Conic Sections
• • Conics in Polar Coordinates

Chapter 12: Vectors and the Geometry of Space

• • Three-Dimensional Coordinate Systems
• • Vectors
• • The Dot Product
• • The Cross Product
• • Lines and Planes in Space
• • Cylinders and Quadric Surfaces

Chapter 13: Vector-Valued Functions and Motion in Space

• • Curves in Space and Their Tangents
• • Integrals of Vector Functions
• • Arc Length in Space
• • Curvature and Normal Vectors of a Curve
• • Tangential and Normal Components of Acceleration
• • Velocity and Acceleration in Polar Coordinates

Chapter 14: Partial Derivatives

• • Functions of Several Variables
• • Limits and Continuity in Higher Dimensions
• • Partial Derivatives
• • The Chain Rule
• • Directional Derivatives and Gradient Vectors
• • Tangent Planes and Differentials
• • Extreme Values and Saddle Points
• • Lagrange Multipliers

Chapter 15: Multiple Integrals

• • Double and Iterated Integrals over Rectangles
• • Double Integrals over General Regions
• • Area by Double Integration
• • Double Integrals in Polar Form
• • Triple Integrals in Rectangular Coordinates
• • Applications
• • Triple Integrals in Cylindrical and Spherical Coordinates
• • Substitutions in Multiple Integrals