Thomas’ calculus

1. Functions1.1 Functions and Their Graphs1.2 Combining Functions
 Shifting and Scaling Graphs1.3 Trigonometric Functions1.4 Graphing with Calculators and Computers 2. Limits and Continuity2.1 Rates of Change and Tangents to Curves2.2 Limit of a Function and Limit Laws2.3 The Precise Definition of a Limit2.4 One-Sided Limits2.5 Continuity2.6 Limits Involving Infinity
 Asymptotes of Graphs 3. Differentiation3.1 Tangents and the Derivative at a Point3.2 The Derivative as a Function3.3 Differentiation Rules3.4 The Derivative as a Rate of Change3.5 Derivatives of Trigonometric Functions3.6 The Chain Rule3.7 Implicit Differentiation3.8 Related Rates3.9 Linearization and Differentials 4. Applications of Derivatives4.1 Extreme Values of Functions4.2 The Mean Value Theorem4.3 Monotonic Functions and the First Derivative Test4.4 Concavity and Curve Sketching4.5 Applied Optimization4.6 Newton's Method4.7 Antiderivatives 5. Integration5.1 Area and Estimating with Finite Sums5.2 Sigma Notation and Limits of Finite Sums5.3 The Definite Integral5.4 The Fundamental Theorem of Calculus5.5 Indefinite Integrals and the Substitution Method5.6 Substitution and Area Between Curves 6. Applications of Definite Integrals6.1 Volumes Using Cross-Sections6.2 Volumes Using Cylindrical Shells6.3 Arc Length6.4 Areas of Surfaces of Revolution6.5 Work and Fluid Forces6.6 Moments and Centers of Mass 7. Transcendental Functions7.1 Inverse Functions and Their Derivatives7.2 Natural Logarithms7.3 Exponential Functions7.4 Exponential Change and Separable Differential Equations7.5 Indeterminate Forms and L'Hopital's Rule7.6 Inverse Trigonometric Functions7.7 Hyperbolic Functions7.8 Relative Rates of Growth 8. Techniques of Integration8.1 Integration by Parts8.2 Trigonometric Integrals8.3 Trigonometric Substitutions8.4 Integration of Rational Functions by Partial Fractions8.5 Integral Tables and Computer Algebra Systems8.6 Numerical Integration8.7 Improper Integrals 9. First-Order Differential Equations9.1 Solutions, Slope Fields, and Euler's Method9.2 First-Order Linear Equations9.3 Applications9.4 Graphical Solutions of Autonomous Equations9.5 Systems of Equations and Phase Planes 10. Infinite Sequences and Series10.1 Sequences10.2 Infinite Series10.3 The Integral Test10.4 Comparison Tests10.5 The Ratio and Root Tests10.6 Alternating Series, Absolute and Conditional Convergence10.7 Power Series10.8 Taylor and Maclaurin Series10.9 Convergence of Taylor Series10.10 The Binomial Series and Applications of Taylor Series 11. Parametric Equations and Polar Coordinates11.1 Parametrizations of Plane Curves11.2 Calculus with Parametric Curves11.3 Polar Coordinates11.4 Graphing in Polar Coordinates11.5 Areas and Lengths in Polar Coordinates11.6 Conic Sections11.7 Conics in Polar Coordinates 12. Vectors and the Geometry of Space12.1 Three-Dimensional Coordinate Systems12.2 Vectors12.3 The Dot Product12.4 The Cross Product12.5 Lines and Planes in Space12.6 Cylinders and Quadric Surfaces 13. Vector-Valued Functions and Motion in Space13.1 Curves in Space and Their Tangents13.2 Integrals of Vector Functions
 Projectile Motion13.3 Arc Length in Space13.4 Curvature and Normal Vectors of a Curve13.5 Tangential and Normal Components of Acceleration13.6 Velocity and Acceleration in Polar Coordinates 14. Partial Derivatives14.1 Functions of Several Variables14.2 Limits and Continuity in Higher Dimensions14.3 Partial Derivatives14.4 The Chain Rule14.5 Directional Derivatives and Gradient Vectors14.6 Tangent Planes and Differentials14.7 Extreme Values and Saddle Points14.8 Lagrange Multipliers14.9 Taylor's Formula for Two Variables14.10 Partial Derivatives with Constrained Variables 15. Multiple Integrals15.1 Double and Iterated Integrals over Rectangles15.2 Double Integrals over General Regions15.3 Area by Double Integration15.4 Double Integrals in Polar Form15.5 Triple Integrals in Rectangular Coordinates15.6 Moments and Centers of Mass15.7 Triple Integrals in Cylindrical and Spherical Coordinates15.8 Substitutions in Multiple Integrals 16. Integration in Vector Fields16.1 Line Integrals16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux16.3 Path Independence, Conservative Fields, and Potential Functions16.4 Green's Theorem in the Plane16.5 Surfaces and Area16.6 Surface Integrals16.7 Stokes' Theorem16.8 The Divergence Theorem and a Unified Theory 17. Second-Order Differential Equations (online)17.1 Second-Order Linear Equations17.2 Nonhomogeneous Linear Equations17.3 Applications17.4 Euler Equations17.5 Power Series Solutions Appendices1. Real Numbers and the Real Line2. Mathematical Induction3. Lines, Circles, and Parabolas4. Proofs of Limit Theorems5. Commonly Occurring Limits6. Theory of the Real Numbers7. Complex Numbers8. The Distributive Law for Vector Cross Products9. The Mixed Derivative Theorem and the Increment Theorem