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Section 3 About Calculus Volume 1

Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 1 covers functions, limits, derivatives, and integration.

Subsection 3.1 Coverage and scope

Our Calculus Volume 1 textbook adheres to the scope and sequence of most general calculus courses nationwide. We have worked to make calculus interesting and accessible to students while maintaining the mathematical rigor inherent in the subject. With this objective in mind, the content of the three volumes of Calculus have been developed and arranged to provide a logical progression from fundamental to more advanced concepts, building upon what students have already learned and emphasizing connections between topics and between theory and applications. The goal of each section is to enable students not just to recognize concepts, but work with them in ways that will be useful in later courses and future careers. The organization and pedagogical features were developed and vetted with feedback from mathematics educators dedicated to the project.

  • Volume 1

    • Chapter 1: Functions and Graphs

    • Chapter 2: Limits

    • Chapter 3: Derivatives

    • Chapter 4: Applications of Derivatives

    • Chapter 5: Integration

    • Chapter 6: Applications of Integration

  • Volume 2

    • Chapter 1: Integration

    • Chapter 2: Applications of Integration

    • Chapter 3: Techniques of Integration

    • Chapter 4: Introduction to Differential Equations

    • Chapter 5: Sequences and Series

    • Chapter 6: Power Series

    • Chapter 7: Parametric Equations and Polar Coordinates

  • Volume 3

    • Chapter 1: Parametric Equations and Polar Coordinates

    • Chapter 2: Vectors in Space

    • Chapter 3: Vector-Valued Functions

    • Chapter 4: Differentiation of Functions of Several Variables

    • Chapter 5: Multiple Integration

    • Chapter 6: Vector Calculus

    • Chapter 7: Second-Order Differential Equations

Subsection 3.2 Pedagogical foundation

Throughout Calculus Volume 1 you will find examples and exercises that present classical ideas and techniques as well as modern applications and methods. Derivations and explanations are based on years of classroom experience on the part of long-time calculus professors, striving for a balance of clarity and rigor that has proven successful with their students. Motivational applications cover important topics in probability, biology, ecology, business, and economics, as well as areas of physics, chemistry, engineering, and computer science. Student Projects in each chapter give students opportunities to explore interesting sidelights in pure and applied mathematics, from determining a safe distance between the grandstand and the track at a Formula One racetrack, to calculating the center of mass of the Grand Canyon Skywalk or the terminal speed of a skydiver. Chapter Opening Applications pose problems that are solved later in the chapter, using the ideas covered in that chapter. Problems include the hydraulic force against the Hoover Dam, and the comparison of relative intensity of two earthquakes. Definitions, Rules, and Theorems are highlighted throughout the text, including over 60 Proofs of theorems.

Subsection 3.3 Assessments that reinforce key concepts

In-chapter Examples walk students through problems by posing a question, stepping out a solution, and then asking students to practice the skill with a “Checkpoint” question. The book also includes assessments at the end of each chapter so students can apply what they’ve learned through practice problems. Many exercises are marked with a [T] to indicate they are suitable for solution by technology, including calculators or Computer Algebra Systems (CAS). Answers for selected exercises are available in the Answer Key at the back of the book. The book also includes assessments at the end of each chapter so students can apply what they’ve learned through practice problems.

Subsection 3.4 Early or late transcendentals

Calculus Volume 1 is designed to accommodate both Early and Late Transcendental approaches to calculus. Exponential and logarithmic functions are introduced informally in Chapter 1 and presented in more rigorous terms in Chapter 6. Differentiation and integration of these functions is covered in Chapters 3–5 for instructors who want to include them with other types of functions. These discussions, however, are in separate sections that can be skipped for instructors who prefer to wait until the integral definitions are given before teaching the calculus derivations of exponentials and logarithms.

Subsection 3.5 Comprehensive art program

Our art program is designed to enhance students’ understanding of concepts through clear and effective illustrations, diagrams, and photographs.