Thomas' Calculus: Early Transcendentals, 15th Edition © 2023
Joel R. Hass | Christopher E. Heil | Przemyslaw Bogacki | Maurice D. Weir
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 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 eTextbook and MyLab® Math course.
- 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 eTextbook.
- All Interactive Figures have been updated for accessibility to meet WCAG standards. The 180 figures can be used in lectures 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.
- 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. 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
Joel R. Hass
University of California, Davis
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 E. Heil
Georgia Institute of Technology
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.
Przemyslaw Bogacki
Old Dominion University
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.
Maurice D. Weir
Naval Postgraduate School
The late Maurice D. Weir of 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.
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