Thomas' Calculus, 14th edition

Clarity and precision

Thomas' Calculus helps students reach the level of mathematical proficiency and maturity you require, but with support for students who need it through its balance of clear and intuitive explanations, current applications, and generalized concepts. In the 14th Edition, new co-author Christopher Heil (Georgia Institute of Technology) partners with author Joel Hass to preserve what is best about Thomas' time-tested text while reconsidering every word and every piece of art with today's students in mind. The result is a text that goes beyond memorizing formulas and routine procedures to help students generalize key concepts and develop deeper understanding. 

MyLab™ Math is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. Within its structured environment, students practice what they learn, test their understanding, and pursue a personalized study plan that helps them absorb course material and understand difficult concepts. A full suite of Interactive Figures have been added to the accompanying MyLab Math course to further support teaching and learning. Enhanced Sample Assignments include just-in-time prerequisite review, help keep skills fresh with distributed practice of key concepts, and provide opportunities to work exercises without learning aids to help students develop confidence in their ability to solve problems independently.

About the Book

Teach calculus the way you want to teach it, and at a level that prepares students for their STEM majors

• New co-author Chris Heil (Georgia Institute of Technology) and co-author Joel Hass continue Thomas’ tradition of developing students’ mathematical maturity and proficiency, going beyond memorizing formulas and routine procedures, and showing students how to generalize key concepts once they are introduced.

• The authors are careful to present key topics, such as the definition of the derivative, both informally and formally. The distinction between the two is clearly stated as each is developed, including an explanation as to why a formal definition is needed. Ideas are introduced with examples and intuitive explanations that are then generalized so that students are not overwhelmed by abstraction.
• Results are both carefully stated and proved throughout the book, and proofs are clearly explained and motivated. Students and instructors who proceed through the formal material will find it as carefully presented and explained as the informal development. If the instructor decides to downplay formality at any stage, it will not cause problems with later developments in the text. 
• A flexible table of contents divides topics into manageable sections, allowing instructors to tailor their course to meet the specific needs of their students.
• Complete and precise multivariable coverage enhances the connections of multivariable ideas with their single-variable analogues studied earlier in the book.

Assess student understanding of key concepts and skills through a wide range of time-tested exercises

• Strong exercise sets feature a great breadth of problems—progressing from skills problems to applied and theoretical problems—to encourage students to think about and practice the concepts until they achieve mastery. In the 14th Edition, the authors added new exercises throughout, many geometric in nature.
• Writing exercises placed throughout the text ask students to explore and explain a variety of calculus concepts and applications. In addition, the end of each chapter contains a list of questions for students to review and summarize what they have learned. Many of these exercises make good writing assignments.
• Technology exercises (marked with a T) are included in each section, asking students to use the calculator or computer when solving the problems. In addition, Computer Explorations give the option of assigning exercises that require a computer algebra system (CAS, such as Maple or Mathematica).

Support a complete understanding of calculus for students at varying levels

• Each major topic is developed with both simple and more advanced examples to give the basic ideas and illustrate deeper concepts.
• UPDATED! Figures are conceived and rendered to provide insight for students and support conceptual reasoning. In the 14th Edition, new figures are added to enhance understanding and graphics are revised throughout to emphasize clear visualization.
• ENHANCED! Annotations within examples (shown in blue type)  guide students through the problem solution and emphasize that each step in a mathematical argument is rigorously justified. For the 14th Edition, many more annotations were added.
• End-of-chapter materials include review questions, practice exercises covering the entire chapter, and a series of Additional and Advanced Exercises with more challenging or synthesizing problems. 

New to the Book

• Updated graphics emphasize clear visualization and mathematical correctness.
• New examples and figures have been added throughout all chapters, many based on user feedback. See, for instance, Example 3 in Section 9.1, which helps students overcome a conceptual obstacle.
• New types of homework exercises, including many geometric in nature, have been added. The new exercises provide different perspectives and approaches to each topic.
• Short URLs have been added to the historical marginnotes, allowing students to navigate directly to online information.
• New annotations within examples (in blue type) guide the student through the problem solution and emphasize that each step in a mathematical argument is rigorously justified.
• All chapters have been revised for clarity, consistency, conciseness, and comprehension.

MyLab™ Math is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. Within its structured environment, students practice what they learn, test their understanding, and pursue a personalized study plan that helps them absorb course material and understand difficult concepts. A full suite of Interactive Figures have been added to the accompanying MyLab Math course to further support teaching and learning. Enhanced Sample Assignments include just-in-time prerequisite review, help keep skills fresh with distributed practice of key concepts, and provide opportunities to work exercises without learning aids to help students develop confidence in their ability to solve problems independently.

New to MyLab Math: 

• The new edition continues to expand the comprehensive auto-graded exercise options. The pre-existing exercises were carefully reviewed, vetted, and improved using aggregated student usage and performance data over time. 
• A full suite of Interactive Figures has been added to support teaching and learning. The figures illustrate key concepts and allow manipulation. They have been designed to be used in lecture as well as by students independently. Videos that use the Interactive Figures to explain key concepts are included. The figures are editable using the freely available GeoGebra software. The figures were created by Marc Renault (Shippensburg University), Steve Phelps (University of Cincinnati), Kevin Hopkins (Southwest Baptist University), and Tim Brzezinski (Berlin High School, CT).
• Setup & Solve Exercises require students to first set up, then solve a problem. This better matches what they are asked to do on tests and promotes long-term retention of the skill. 
• Additional Conceptual Questions augment the text exercises to focus on deeper, theoretical understanding of the key concepts in calculus. These questions were written by faculty at Cornell University under an NSF grant and are also assignable through Learning Catalytics.
• Enhanced Sample Assignments are crafted to maximize student performance in the course. These section-level assignments include: (a) personalized, just-in-time prerequisite review exercises; (b) systematic distributed practice of key concepts (such as the Chain Rule) in order to help keep skills fresh, and (c) periodic removal of learning aids to help students develop confidence in their ability to solve problems independently.
• Integrated Review MyLab Math courses provide a full suite of supporting resources for the main course content plus additional assignments and study aids for students who will benefit from remediation. Assignments for the Integrated Review content are preassigned in MyLab Math, making it easier than ever to create your course.
• More assignable exercises -- Instructors now have more exercises than ever to choose from in assigning homework. There are approximately 8080 assignable exercises in MyLab Math.
• More instructional videos -- Over 200 new instructional videos, featuring Greg Wisloski and Dan Radelet (both of Indiana University of PA), augment the already robust collection within the course. These videos support the overall approach of the text--specifically, they go beyond routine procedures to show students how to generalize and connect key concepts.

Content Updates:

Chapter 1

• Shortened 1.4 to focus on issues arising in use of mathematical software and potential pitfalls. Removed peripheral material on regression, along with associated exercises.

• Added new Exercises: 1.1: 59–62, 1.2: 21–22; 1.3: 64–65, PE: 29–32.

Chapter 2

• Added definition of average speed in 2.1.

• Clarified definition of limits to allow for arbitrary domains. The definition of limits is now consistent with the definition in multivariable domains later in the text and with more general mathematical usage.

• Reworded limit and continuity definitions to remove implication symbols and improve comprehension.

• Added new Example 7 in 2.4to illustrate limits of ratios of trig functions.

• Rewrote 2.5 Example 11 to solve the equation by finding a zero, consistent with previous discussion.

• Added new Exercises: 2.1: 15–18; 2.2: 3h–k, 4f–i; 2.4: 19–20, 45–46; 2.6: 69–72; PE: 49–50; AAE: 33.

Chapter 3

• Clarified relation of slope and rate of change.

• Added new Figure 3.9 using the square root function to illustrate vertical tangent lines.

• Added figure of x sin (1>x) in 3.2 to illustrate how oscillation can lead to nonexistence of a derivative of a continuous function.

• Revised product rule to make order of factors consistent throughout text, including later dot product and cross product formulas.

• Added new Exercises: 3.2: 36, 43–44; 3.3: 51–52; 3.5: 43–44, 61bc; 3.6: 65–66, 97–99; 3.7: 25–26; 3.8: 47; AAE: 24–25.

Chapter 4

• Added summary to 4.1.

• Added new Example 3 with new Figure 4.27 to give basic and advanced examples of concavity.

• Added new Exercises: 4.1: 61–62; 4.3: 61–62; 4.4: 49–50, 99–104; 4.5: 37–40; 4.6: 7–8; 4.7: 93–96; PE: 1–10; AAE: 19–20, 33. Moved Exercises 4.1: 53–68 to PE.

Chapter 5

• Improved discussion in 5.4 and added new Figure 5.18 to illustrate the Mean Value Theorem.

• Added new Exercises: 5.2: 33–36; PE: 45–46.

Chapter 6

• Clarified cylindrical shell method.

• Converted 6.5 Example 4 to metric units.

• Added introductory discussion of mass distribution along a line, with figure, in 6.6.

• Added new Exercises: 6.1: 15–16; 6.2: 45–46; 6.5: 1–2; 6.6: 1–6, 19–20; PE: 17–18, 35–36.

Chapter 7

• Added explanation for the terminology “indeterminate form.”

• Clarified discussion of separable differential equations in 7.4.

• Replaced sin-1 notation for the inverse sine function with arcs in as default notation in 7.6, and similarly for other trig functions.

• Added new Exercises: 7.2: 5–6, 75–76; 7.3: 5–6, 31–32, 123–128, 149–150; 7.6: 43–46, 95–96; AAE: 9–10, 23.

Chapter 8

• Updated 8.2 Integration by Parts discussion to emphasize u(x) y(x) dx form rather than u dy. Rewrote Examples 1–3 accordingly.

• Removed discussion of tabular integration and associated exercises.

• Updated discussion in 8.5 on how to find constants in the method of partial fractions.

• Updated notation in 8.8 to align with standard usage in statistics.

• Added new Exercises: 8.1: 41–44; 8.2: 53–56, 72–73; 8.3: 75–76; 8.4: 49–52; 8.5: 51–66, 73–74; 8.8: 35–38, 77–78; PE: 69–88.

Chapter 9

• Added new Example 3 with Figure 9.3 to illustrate how to construct a slope field.

• Added new Exercises: 9.1: 11–14; PE:  17–22, 43–44.

Chapter 10

• Clarified the differences between a sequence and a series.

• Added new Figure 10.9 to illustrate sum of a series as area of a histogram.

• Added to 10.3 a discussion on the importance of bounding errors in approximations.

• Added new Figure 10.13 illustrating how to use integrals to bound remainder terms of partial sums.

• Rewrote Theorem 10 in 10.4 to bring out similarity to the integral comparison test.

• Added new Figure 10.16 to illustrate the differing behaviors of the harmonic and alternating harmonic series.

• Renamed the nth-Term Test the “nth-Term Test for Divergence” to emphasize that it says nothing about convergence.

• Added new Figure 10.19 to illustrate polynomials converging to ln (1 + x), which illustrates convergence on the halfopen interval (-1, 14.

• Used red dots and intervals to indicate intervals and points where divergence occurs, and blue to indicate convergence, throughout Chapter 10.

• Added new Figure 10.21 to show the six different possibilities for an interval of convergence.

• Added new Exercises: 10.1: 27–30, 72–77; 10.2: 19–22, 73–76, 105; 10.3: 11–12, 39–42; 10.4: 55–56; 10.5: 45–46, 65–66; 10.6: 57–82; 10.7: 61–65; 10.8: 23–24, 39–40; 10.9: 11–12, 37–38; PE: 41–44, 97–102.

Chapter 11

• Added new Example 1 and Figure 11.2 in 11.1 to give a straightforward first example of a parametrized curve.

• Updated area formulas for polar coordinates to include conditions for positive r and nonoverlapping u.

• Added new Example 3 and Figure 11.37 in 11.4 to illustrate intersections of polar curves.

• Added new Exercises: 11.1: 19–28; 11.2: 49–50; 11.4: 21–24.

Chapter 12

• Added new Figure 12.13(b) to show the effect of scaling a vector.

• Added new Example 7 and Figure 12.26 in 12.3 to illustrate projection of a vector.

• Added discussion on general quadric surfaces in 12.6, with new Example 4 and new Figure 12.48 illustrating the description of an ellipsoid not centered at the origin via completing the square.

• Added new Exercises: 12.1: 31–34, 59–60, 73–76; 12.2: 43–44; 12.3: 17–18; 12.4: 51–57; 12.5: 49–52.

Chapter 13

• Added sidebars on how to pronounce Greek letters such as kappa, tau, etc.

• Added new Exercises: 13.1: 1–4, 27–36; 13.2: 15–16, 19–20; 13.4: 27–28; 13.6: 1–2.

Chapter 14

• Elaborated on discussion of open and closed regions in 14.1.

• Standardized notation for evaluating partial derivatives, gradients, and directional derivatives at a point, throughout the chapter.

• Renamed “branch diagrams” as “dependency diagrams,” which clarifies that they capture dependence of variables.

• Added new Exercises: 14.2: 51–54; 14.3: 51–54, 59–60, 71–74, 103–104; 14.4: 20–30, 43–46, 57–58; 14.5: 41–44; 14.6: 9–10, 61; 14.7: 61–62.

Chapter 15

• Added new Figure 15.21b to illustrate setting up limits of a double integral.

• Added new 15.5 Example 1, modified Examples 2 and 3, and added new Figures 15.31, 15.32, and 15.33 to give basic examples of setting up limits of integration for a triple integral.

• Added new material on joint probability distributions as an application of multivariable integration.

• Added new Examples 5, 6 and 7 to Section 15.6.

• Added new Exercises: 15.1: 15–16, 27–28; 15.6: 39–44; 15.7: 1–22.

Chapter 16

• Added new Figure 16.4 to illustrate a line integral of a function.

• Added new Figure 16.17 to illustrate a gradient field.

• Added new Figure 16.18 to illustrate a line integral of a vector field.

• Clarified notation for line integrals in 16.2.

• Added discussion of the sign of potential energy in 16.3.

• Rewrote solution of Example 3 in 16.4 to clarify connection to Green’s Theorem.

• Updated discussion of surface orientation in 16.6 along with Figure 16.52.

• Added new Exercises: 16.2: 37–38, 41–46; 16.4: 1–6; 16.6: 49–50; 16.7: 1–6; 16.8: 1–4.

Appendices: Rewrote Appendix A7 on complex numbers.