Thomas' Calculus

1. Functions

• 1.1 Functions and Their Graphs
• 1.2 Combining Functions; Shifting and Scaling Graphs
• 1.3 Trigonometric Functions
• 1.5 Exponential Functions
• 1.6 Inverse Functions and Logarithms
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

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
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

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 Inverse Trigonometric Functions
• 3.10 Related Rates
• 3.11 Linearization and Differentials
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

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 Indeterminate Forms and L'Hopital's Rule
• 4.6 Applied Optimization
• 4.7 Newton's Method
• 4.8 Antiderivatives
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

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
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

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
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

7. Integrals and Transcendental Functions

• 7.1 The Logarithm Defined as an Integral
• 7.2 Exponential Change and Separable Differential Equations
• 7.3 Hyperbolic Functions
• 7.4 Relative Rates of Growth
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises

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
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

9. Infinite Sequences and Series

• 9.1 Sequences
• 9.2 Infinite Series
• 9.3 The Integral Test
• 9.4 Comparison Tests
• 9.5 Absolute Convergence; The Ratio and Root Tests
• 9.6 Alternating Series and Conditional Convergence
• 9.7 Power Series
• 9.8 Taylor and Maclaurin Series
• 9.9 Convergence of Taylor Series
• 9.10 Applications of Taylor Series
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

10. Parametric Equations and Polar Coordinates

• 10.1 Parametrizations of Plane Curves
• 10.2 Calculus with Parametric Curves
• 10.3 Polar Coordinates
• 10.4 Graphing Polar Coordinate Equations
• 10.5 Areas and Lengths in Polar Coordinates
• 10.6 Conic Sections
• 10.7 Conics in Polar Coordinates
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

11. Vectors and the Geometry of Space

• 11.1 Three-Dimensional Coordinate Systems
• 11.2 Vectors
• 11.3 The Dot Product
• 11.4 The Cross Product
• 11.5 Lines and Planes in Space
• 11.6 Cylinders and Quadric Surfaces
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

12. Vector-Valued Functions and Motion in Space

• 12.1 Curves in Space and Their Tangents
• 12.2 Integrals of Vector Functions; Projectile Motion
• 12.3 Arc Length in Space
• 12.4 Curvature and Normal Vectors of a Curve
• 12.5 Tangential and Normal Components of Acceleration
• 13.6 Velocity and Acceleration in Polar Coordinates
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

13. Partial Derivatives

• 13.1 Functions of Several Variables
• 13.2 Limits and Continuity in Higher Dimensions
• 13.3 Partial Derivatives
• 13.4 The Chain Rule
• 13.5 Directional Derivatives and Gradient Vectors
• 13.6 Tangent Planes and Differentials
• 13.7 Extreme Values and Saddle Points
• 13.8 Lagrange Multipliers
• 13.9 Taylor's Formula for Two Variables
• 13.10 Partial Derivatives with Constrained Variables
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

14. Multiple Integrals

• 14.1 Double and Iterated Integrals over Rectangles
• 14.2 Double Integrals over General Regions
• 14.3 Area by Double Integration
• 14.4 Double Integrals in Polar Form
• 14.5 Triple Integrals in Rectangular Coordinates
• 14.6 Applications
• 14.7 Triple Integrals in Cylindrical and Spherical Coordinates
• 14.8 Substitutions in Multiple Integrals
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

15. Integrals and Vector Fields

• 15.1 Line Integrals of Scalar Functions
• 15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux
• 15.3 Path Independence, Conservative Fields, and Potential Functions
• 15.4 Green's Theorem in the Plane
• 15.5 Surfaces and Area
• 15.6 Surface Integrals
• 15.7 Stokes' Theorem
• 15.8 The Divergence Theorem and a Unified Theory
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

16. First-Order Differential Equations

• 16.1 Solutions, Slope Fields, and Euler's Method
• 16.2 First-Order Linear Equations
• 16.3 Applications
• 16.4 Graphical Solutions of Autonomous Equations
• 16.5 Systems of Equations and Phase Planes
• Questions to Guide Your Review
• Practice Exercises
• Technology Application Projects

17. Second-Order Differential Equations

• 17.1 Second-Order Linear Equations
• 17.2 Nonhomogeneous Linear Equations
• 17.3 Applications
• 17.4 Euler Equations
• 17.5 Power-Series Solutions
• Questions to Guide Your Review
• Practice Exercises
• Additional and Advanced Exercises
• Technology Application Projects

18. Complex Functions (online)

• 18.1 Complex Numbers
• 18.2 Limits and Continuity
• 18.3 Complex Derivatives
• 18.4 The Cauchy-Riemann Equations
• 18.5 Complex Series
• 18.6 Conformal Maps

19. Fourier Series and Wavelets (online)

• 19.1 Periodic Functions
• 19.2 Summing Sines and Cosines
• 19.3 Vectors and Approximation in Three and More Dimensions
• 19.4 Approximation of Functions
• 19.5 Advanced Topic: The Haar System and Wavelets

Appendix A

• A.1 Real Numbers and the Real Line
• A.2 Graphing with Software
• A.3 Mathematical Induction
• A.4 Lines, Circles, and Parabolas
• A.5 Proofs of Limit Theorems
• A.6 Commonly Occurring Limits
• A.7 Theory of the Real Numbers
• A.8 The Distributive Law for Vector Cross Products
• A.9 Probability
• A.10 The Mixed Derivative Theorem and the Increment Theorem

Appendix B

• 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