## Description

**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 SI 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.

*For three-semester or four-quarter courses in Calculus for students majoring in mathematics, engineering, or science*

## Features

- 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.
- A flexible table of contents divides topics into manageable sections, allowing instructors to tailor their course to meet the specific needs of their students.
- Assess student understanding of key concepts and skills through a wide range of time-tested exercises
- 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.
- Support a complete understanding of calculus for students at varying levels
- UPDATED! Figures are conceived and rendered to provide insight for students and support conceptual reasoning. In the 14th SI Edition, new figures are added to enhance understanding and graphics are revised throughout to emphasize clear visualization.
- 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.
- Engage students with the power of calculus through a variety of multimedia resourcesUPDATED! Instructional videos: Hundreds of videos are available as learning aids within exercises and for self-study. The Guide to Video-Based Assignments makes it easy to assign videos for homework by showing which MyLab Math exercises correspond to each video.
- Assess student understanding of concepts and skills through a wide range of exercises.
- NEW! Setup & Solve exercises require students to first describe how they will set up and approach the problem. This reinforces conceptual understanding of the process applied in approaching the problem, promotes long term retention of the skill and mirrors what students will be expected to do on a test.
- NEW!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.

##### New

Co-authors Joel Hass and Chris Heil reconsidered every word, symbol, and piece of art, motivating students to consider the content from different perspectives and compelling a deeper, geometric understanding.

- 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.

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.

MyLabTM Math not included. Students, if MyLab is a recommended/mandatory component of the course, please ask your instructor for the correct ISBN and course ID. MyLab should only be purchased when required by an instructor. Instructors, contact your Pearson representative for more information.

- 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.
- More assignable exercises -- Instructors now have more exercises than ever to choose from in assigning homework.
- More instructional videos -- Numerous 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.

## Table of Contents

**1. Functions**

1.1 Functions and Their Graphs

1.2 Combining Functions; Shifting and Scaling Graphs

1.3 Trigonometric Functions

1.4 Exponential Functions

**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 Limits Involving Infinity; Asymptotes of Graphs

2.6 Continuity

**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 Related Rates

3.9 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 Indeterimnate 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

**9. Infinite Sequences and Series**

9.1 Sequences

9.2 Infinite Series

9.3 The Integral Test

9.4 Comparison Tests

9.5 Absolute Convergence; The Ratio and Root Tests

9.6 Alternating Series and Conditional Convergence

9.7 Power Series

9.8 Taylor and Maclaurin Series

9.9 Convergence of Taylor Series

9.10 Applications of Taylor Series

**10. Parametric Equations and Polar Coordinates**

10.1 Parametrizations of Plane Curves

10.2 Calculus with Parametric Curves

10.3 Polar Coordinates

10.4 Graphing Polar Coordinate Equations

10.5 Areas and Lengths in Polar Coordinates

10.6 Conic Sections

10.7 Conics in Polar Coordinates

**11. Vectors and the Geometry of Space**

11.1 Three-Dimensional Coordinate Systems

11.2 Vectors

11.3 The Dot Product

11.4 The Cross Product

11.5 Lines and Planes in Space

11.6 Cylinders and Quadric Surfaces

**12. Vector-Valued Functions and Motion in Space**

12.1 Curves in Space and Their Tangents

12.2 Integrals of Vector Functions; Projectile Motion

12.3 Arc Length in Space

12.4 Curvature and Normal Vectors of a Curve

12.5 Tangential and Normal Components of Acceleration

12.6 Velocity and Acceleration in Polar Coordinates

**13. Partial Derivatives**

13.1 Functions of Several Variables

13.2 Limits and Continuity in Higher Dimensions

13.3 Partial Derivatives

13.4 The Chain Rule

13.5 Directional Derivatives and Gradient Vectors

13.6 Tangent Planes and Differentials

13.7 Extreme Values and Saddle Points

13.8 Lagrange Multipliers

13.9 Taylor’s Formula for Two Variables

13.10 Partial Derivatives with Constrained Variables

**14. Multiple Integrals**

14.1 Double and Iterated Integrals over Rectangles

14.2 Double Integrals over General Regions

14.3 Area by Double Integration

14.4 Double Integrals in Polar Form

14.5 Triple Integrals in Rectangular Coordinates

14.6 Applications

14.7 Triple Integrals in Cylindrical and Spherical Coordinates

14.8 Substitutions in Multiple Integrals

**15. Integrals and Vector Fields**

15.1 Line Integrals of Scalar Functions

15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux

15.3 Path Independence, Conservative Fields, and Potential Functions

15.4 Green’s Theorem in the Plane

15.5 Surfaces and Area

15.6 Surface Integrals

15.7 Stokes’ Theorem

15.8 The Divergence Theorem and a Unified Theory

**16. First-Order Differential Equations**

16.1 Solutions, Slope Fields, and Euler’s Method

16.2 First-Order Linear Equations

16.3 Applications

16.4 Graphical Solutions of Autonomous Equations

16.5 Systems of Equations and Phase Planes

**17. Second-Order Differential Equations (Online)**

17.1 Second-Order Linear Equations

17.2 Nonhomogeneous Linear Equations

17.3 Applications

17.4 Euler Equations

17.5 Power-Series Solutions

**Appendices**

1. Real Numbers and the Real Line

2. Mathematical Induction

3 Lines, Circles, and Parabolas

4 Proofs of Limit Theorems

5 Commonly Occurring Limits

6 Theory of the Real Numbers

7 Complex Numbers

8. Probability

9. The Distributive Law for Vector Cross Products

10. The Mixed Derivative Theorem and the Increment Theorem

## MyLab Mathematics

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