Thomas' Calculus, 15th edition

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
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
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
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
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
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
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
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
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
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
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

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

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

Answers to Odd-Numbered Exercises

Applications Index

Subject Index

A Brief Table of Integrals

Credits