Thomas' Calculus

Joel Hass received his PhD from the University of California—Berkeley. He is currently a professor of mathematics at the University of California—Davis. He has coauthored six widely used calculus texts as well as two calculus study guides. He is currently on the editorial board of Geometriae Dedicata and Media-Enhanced Mathematics. He has been a member of the Institute for Advanced Study at Princeton University and of the Mathematical Sciences Research Institute, and he was a Sloan Research Fellow. Hass’s current areas of research include the geometry of proteins, three dimensional manifolds, applied math, and computational complexity. In his free time, Hass enjoys kayaking.  Maurice D. Weir holds a DA and MS from Carnegie-Mellon University and received his BS at Whitman College. He is a Professor Emeritus of the Department of Applied Mathematics at the Naval Postgraduate School in Monterey, California. Weir enjoys teaching Mathematical Modeling and Differential Equations. His current areas of research include modeling and simulation as well as mathematics education. Weir has been awarded the Outstanding Civilian Service Medal, the Superior Civilian Service Award, and the Schieffelin Award for Excellence in Teaching. He has coauthored eight books, including the University Calculus series and the twelfth edition of Thomas’ Calculus.  George B. Thomas, Jr. (late) of the Massachusetts Institute of Technology, was a professor of mathematics for thirty-eight years; he served as the executive officer of the department for ten years and as graduate registration officer for five years. Thomas held a spot on the board of governors of the Mathematical Association of America and on the executive committee of the mathematics division of the American Society for Engineering Education. His book, Calculus and Analytic Geometry, was first published in 1951 and has since gone through multiple revisions. The text is now in its twelfth edition and continues to guide students through their calculus courses. He also co-authored monographs on mathematics, including the text Probability and Statistics.

• 1. Functions1.1 Functions and Their Graphs1.2 Combining Functions; Shifting and Scaling Graphs1.3 Trigonometric Functions1.4 Graphing with Software1.5 Exponential Functions1.6 Inverse Functions and Logarithms  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 Derivatives of Inverse Functions and Logarithms3.9 Inverse Trigonometric Functions3.10 Related Rates3.11 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 Indeterminate Forms and L’Hôpital’s Rule4.6 Applied Optimization4.7 Newton's Method4.8 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. Integrals and Transcendental Functions7.1 The Logarithm Defined as an Integral7.2 Exponential Change and Separable Differential Equations7.3 Hyperbolic Functions7.4 Relative Rates of Growth  8. Techniques of Integration8.1 Using Basic Integration Formulas8.2 Integration by Parts8.3 Trigonometric Integrals8.4 Trigonometric Substitutions8.5 Integration of Rational Functions by Partial Fractions8.6 Integral Tables and Computer Algebra Systems8.7 Numerical Integration8.8 Improper Integrals8.9 Probability  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 Absolute Convergence; The Ratio and Root Tests10.6 Alternating Series 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 Polar Coordinate Equations11.5 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. Integrals and 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