Thomas' Calculus

Chapter 1: Functions

• • 1.1 Functions and Their Graphs
• • 1.2 Combining Functions; Shifting and Scaling Graphs
• • 1.3 Trigonometric Functions
• • 1.4 Graphing with Software

Chapter 2: Limits and Continuity

• • 2.1 Rates of Change and Tangent Lines to Curves
• • 2.2 Limit of a Function and Limit Laws
• • 2.3 The Precise Definition of a Limit
• • 2.4 One-Sided Limits
• • 2.5 Continuity
• • 2.6 Limits Involving Infinity; Asymptotes of Graphs

Chapter 3: Derivatives

• • 3.1 Tangent Lines and the Derivative at a Point
• • 3.2 The Derivative as a Function
• • 3.3 Differentiation Rules
• • 3.4 The Derivative as a Rate of Change
• • 3.5 Derivatives of Trigonometric Functions
• • 3.6 The Chain Rule
• • 3.7 Implicit Differentiation
• • 3.8 Derivatives of Inverse Functions and Logarithms
• • 3.9 Related Rates
• • 3.10 Linearization and Differentials

Chapter 4: Applications of Derivatives

• • 4.1 Extreme Values of Functions on Closed Intervals
• • 4.2 The Mean Value Theorem
• • 4.3 Monotonic Functions and the First Derivative Test
• • 4.4 Concavity and Curve Sketching
• • 4.5 Applied Optimization
• • 4.6 Newton's Method
• • 4.7 Antiderivatives

Chapter 5: Integrals

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

Chapter 6: Applications of Definite Integrals

• • 6.1 Volumes Using Cross-Sections
• • 6.2 Volumes Using Cylindrical Shells
• • 6.3 Arc Length
• • 6.4 Areas of Surfaces of Revolution
• • 6.5 Work and Fluid Forces
• • 6.6 Moments and Centers of Mass

Chapter 7: Transcendental Functions

• • 7.1 Inverse Functions and Their Derivatives
• • 7.2 Natural Logarithms
• • 7.3 Exponential Functions
• • 7.4 Exponential Change and Separable Differential Equations
• • 7.5 Indeterminate Forms and L'Hôpital's Rule
• • 7.6 Inverse Trigonometric Functions
• • 7.7 Hyperbolic Functions
• • 7.8 Relative Rates of Growth

Chapter 8: Techniques of Integration

• • 8.1 Using Basic Integration Formulas
• • 8.2 Integration by Parts
• • 8.3 Trigonometric Integrals
• • 8.4 Trigonometric Substitutions
• • 8.5 Integration of Rational Functions by Partial Fractions
• • 8.6 Integral Tables and Computer Algebra Systems
• • 8.7 Numerical Integration
• • 8.8 Improper Integrals
• • 8.9 Probability

Chapter 9: First-Order Differential Equations

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

Chapter 10: Infinite Sequences and Series

• • 10.1 Sequences
• • 10.2 Infinite Series
• • 10.3 The Integral Test
• • 10.4 Comparison Tests
• • 10.5 Absolute Convergence; The Ratio and Root Tests
• • 10.6 Alternating Series and Conditional Convergence
• • 10.7 Power Series
• • 10.8 Taylor and Maclaurin Series
• • 10.9 Convergence of Taylor Series
• • 10.10 Applications of Taylor Series

Chapter 11: Parametric Equations and Polar Coordinates

• • 11.1 Parametrizations of Plane Curves
• • 11.2 Calculus with Parametric Curves
• • 11.3 Polar Coordinates
• • 11.4 Graphing Polar Coordinate Equations
• • 11.5 Areas and Lengths in Polar Coordinates
• • 11.6 Conic Sections
• • 11.7 Conics in Polar Coordinates

Chapter Appendix A: Appendix A

• • A.1 Real Numbers and the Real Line
• • A.2 Mathematical Induction
• • A.3 Lines, Circles, and Parabolas
• • A.4 Proofs of Limit Theorems
• • A.5 Commonly Occurring Limits
• • A.6 Theory of the Real Numbers
• • A.7 The Distributive Law for Vector Cross Products
• • A.8 The Mixed Derivative Theorem and the Increment Theorem

Chapter Appendix B (online): Appendix B (online)

• • B.1 Determinants
• • B.2 Extreme Values and Saddle Points for Functions of More than Two Variables
• • B.3 The Method of Gradient Descent