Thomas' Calculus

Thomas' Calculus goes beyond memorizing formulas and routine procedures to help you develop deeper understanding. It guides you to a level of mathematical proficiency, with additional support if needed through its clear and intuitive explanations, current applications and generalized concepts. Technology exercises in every section use the calculator or computer for solving problems, and Computer Explorations offer exercises requiring a computer algebra system like Maple or Mathematica. The 15th Edition adds exercises, revises figures and language for clarity, and updates many applications; new online chapters cover Complex Functions, Fourier Series and Wavelets.

About our authors

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 widely used calculus texts as well as calculus study guides. He is currently on the editorial board of several publications, including the Notices of the American Mathematical Society. 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.

Christopher Heil received his PhD from the University of Maryland. He is currently a professor of mathematics at the Georgia Institute of Technology. He is the author of a graduate text on analysis and a number of highly cited research survey articles. He serves on the editorial boards of Applied and Computational Harmonic Analysis and The Journal of Fourier Analysis and Its Applications. Heil's current areas of research include redundant representations, operator theory, and applied harmonic analysis. In his spare time, Heil pursues his hobby of astronomy.

The late Maurice D. Weir of the the Naval Postgraduate School in Monterey, California was Professor Emeritus as a member of the Department of Applied Mathematics. He held a DA and MS from Carnegie-Mellon University and received his BS at Whitman College. Weir was awarded the Outstanding Civilian Service Medal, the Superior Civilian Service Award, and the Schieffelin Award for Excellence in Teaching. He co-authored eight books, including University Calculus and Thomas' Calculus.

Przemyslaw Bogacki is an Associate Professor of Mathematics and Statistics and a University Professor at Old Dominion University. He received his PhD in 1990 from Southern Methodist University. He is also the author of a text on linear algebra, which appeared in 2019. He is actively involved in applications of technology in collegiate mathematics. His areas of research include computer aided geometric design and numerical solution of initial value problems for ordinary differential equations.

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

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