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Thomas' Calculus: Early Transcendentals, 15th edition

Published by Pearson (February 8, 2022) © 2023

  • Joel R. Hass University of California, Davis
  • Christopher E. Heil Georgia Institute of Technology
  • Maurice D. Weir Naval Postgraduate School
  • Przemyslaw Bogacki Old Dominion University

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For 3-semester or 4-quarter courses in Calculus for students majoring in mathematics, engineering or science.

Clarity and precision

Thomas' Calculus: Early Transcendentals goes beyond memorizing formulas and routine procedures to help students develop deeper understanding. It guides students to a level of mathematical proficiency and maturity needed for the course, with support for those who require it through its balance of clear and intuitive explanations, current applications and generalized concepts. The 15th Edition meets the needs of students with increasingly varied levels of readiness for the calculus sequence. This revision also adds exercises, revises figures and narrative for clarity, and updates many applications with modern topics.

Hallmark features of this title

  • Key topics are presented both informally and formally.
  • Results are carefully stated and proved throughout, and proofs are clearly explained and motivated.
  • Strong exercise sets feature a wide range from skills problems to applied and theoretical problems.
  • Writing exercises ask students to explore and explain various concepts and applications. A list of questions at the end of each chapter asks them to review and summarize what they have learned.
  • Technology exercises in each section ask students to use the calculator or computer when solving the problems. Computer Explorations offer exercises requiring a computer algebra system such as Maple or Mathematica.
  • Annotations within examples guide students through the problem solution and emphasize that each step in a mathematical argument is justified.

New and updated features of this title

  • Many narrative clarifications and revisions have been made throughout the text.
  • A new appendix on Determinants and Gradient Descent has been added, covering many topics relevant to students interested in Machine Learning and Neural Networks.
  • Many updated graphics and figures have been enhanced to bring out clear visualization and mathematical correctness.
  • Many exercise instructions have been clarified, such as suggesting where the use of a calculator may be needed.
  • Notation of inverse trig functions has been changed throughout the text to favor arcsin notation over sin^{-1}, etc.
  • New advanced online chapters and sections are offered on Complex Functions, Fourier Series and Wavelets in the eText and MyLab Math course.

Features of MyLab Math for the 15th Edition

  • 100 additional Setup & Solve exercises have been selected by author Przemyslaw Bogacki. These exercises focus students on the process of problem solving by requiring them to set up their equations before moving on to the solution.
  • Integrated Review quizzes and personalized homework are now built into all MyLab Math courses. No separate Integrated Review course is required.
  • New online chapters and sections on Complex Functions, Fourier Series and Wavelets offer exercises, as requested by many users. These are also available in the standalone eText.
  • All Interactive Figures have been updated for accessibility to meet WCAG standards. The 180 figures can be used in lecture and by students independently.
    • Figures are editable using the free GeoGebra software; they were created by Marc Renault (Shippensburg University), Kevin Hopkins (Southwest Baptist University), Steve Phelps (University of Cincinnati), and Tim Brzezinzki (Southington High School, CT).
  • GeoGebra Exercises are gradable graphing and computational exercises that help students demonstrate their understanding, enabling them to interact directly with the graph in a manner that reflects how they would graph on paper.

Features of Pearson eText for the 15th Edition

  • New advanced online chapters and sections are offered on Complex Functions, Fourier Series and Wavelets.
  • A new appendix on Determinants and Gradient Descent covers many topics relevant to students interested in Machine Learning and Neural Networks.
  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
    • 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'Hôpital'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
    • 8.9 Probability
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
    • Technology Application Projects
  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
    • Questions to Guide Your Review
    • Practice Exercises
    • Technology Application Projects
  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
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
    • Technology Application Projects
  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
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
    • Technology Application Projects
  12. Vectors and the Geometry of Space
    • 12.1 Three-Dimensional Coordinate Systems
    • 12.2 Vectors
    • 12.3 The Dot Product
    • 12.4 The Cross Product
    • 12.5 Lines and Planes in Space
    • 12.6 Cylinders and Quadric Surfaces
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
    • Technology Application Projects
  13. Vector-Valued Functions and Motion in Space
    • 13.1 Curves in Space and Their Tangents
    • 13.2 Integrals of Vector Functions; Projectile Motion
    • 13.3 Arc Length in Space
    • 13.4 Curvature and Normal Vectors of a Curve
    • 13.5 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
  14. Partial Derivatives
    • 14.1 Functions of Several Variables
    • 14.2 Limits and Continuity in Higher Dimensions
    • 14.3 Partial Derivatives
    • 14.4 The Chain Rule
    • 14.5 Directional Derivatives and Gradient Vectors
    • 14.6 Tangent Planes and Differentials
    • 14.7 Extreme Values and Saddle Points
    • 14.8 Lagrange Multipliers
    • 14.9 Taylor's Formula for Two Variables
    • 14.10 Partial Derivatives with Constrained Variables
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
    • Technology Application Projects
  15. Multiple Integrals
    • 15.1 Double and Iterated Integrals over Rectangles
    • 15.2 Double Integrals over General Regions
    • 15.3 Area by Double Integration
    • 15.4 Double Integrals in Polar Form
    • 15.5 Triple Integrals in Rectangular Coordinates
    • 15.6 Applications
    • 15.7 Triple Integrals in Cylindrical and Spherical Coordinates
    • 15.8 Substitutions in Multiple Integrals
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced Exercises
    • Technology Application Projects
  16. Integrals and Vector Fields
    • 16.1 Line Integrals of Scalar Functions
    • 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux
    • 16.3 Path Independence, Conservative Fields, and Potential Functions
    • 16.4 Green's Theorem in the Plane
    • 16.5 Surfaces and Area
    • 16.6 Surface Integrals
    • 16.7 Stokes' Theorem
    • 16.8 The Divergence Theorem and a Unified Theory
    • Questions to Guide Your Review
    • Practice Exercises
    • Additional and Advanced 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 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

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.

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MyLab

Customize your course to teach your way. MyLab® is a flexible platform merging world-class content with dynamic study tools. It takes a personalized approach designed to ignite each student's unique potential. And, with the freedom it affords to adapt your pedagogy, you can reinforce select concepts and guide students to real results.

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