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Unformatted text preview: Volume 1 Calculus Volume 1 SENIOR CONTRIBUTING AUTHORS EDWIN "JED" HERMAN, UNIVERSITY OF WISCONSIN-STEVENS POINT GILBERT STRANG, MASSACHUSETTS INSTITUTE OF TECHNOLOGY OpenStax Rice University 6100 Main Street MS-375 Houston, Texas 77005 To learn more about OpenStax, visit . Individual print copies and bulk orders can be purchased through our website. ©2017 Rice University. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution Non-Commercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0). Under this license, any user of this textbook or the textbook contents herein can share, remix, and build upon the content for noncommercial purposes only. Any adaptations must be shared under the same type of license. 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The Maxfield Foundation supports projects with potential for high impact in science, education, sustainability, and other areas of social importance. Our mission at The Michelson 20MM Foundation is to grow access and success by eliminating unnecessary hurdles to affordability. We support the creation, sharing, and proliferation of more effective, more affordable educational content by leveraging disruptive technologies, open educational resources, and new models for collaboration between for-profit, nonprofit, and public entities. The Bill and Stephanie Sick Fund supports innovative projects in the areas of Education, Art, Science and Engineering. I WOULDN’T THIS PENS I LOOK BETTER TUDENT E ON A BRAND MEET SC E NEW IPAD QUIREMENT I MINI? URSES. THESE AR EER-REVIEWED TEXTS WR ROFESSIONAL CONTENT EVELOPERS. ADOPT A BO ODAY FOR A TURNKEY LASSROOM SOLUTION OR TO SUIT YOUR TEACHING PPROACH. FREE ONLINE Knowing where our textbooks are used can help us provide better services to students and receive more grant support for future projects. If you’re using an OpenStax textbook, either as required for your course or just as an extra resource, send your course syllabus to [email protected] and you’ll be entered to win an iPad Mini. If you don’t win, don’t worry – we’ll be holding a new contest each semester. Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1: Functions and Graphs . . . . . . . . . . . . . . . . . 1.1 Review of Functions . . . . . . . . . . . . . . . . . . . . 1.2 Basic Classes of Functions . . . . . . . . . . . . . . . . 1.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . 1.4 Inverse Functions . . . . . . . . . . . . . . . . . . . . . 1.5 Exponential and Logarithmic Functions . . . . . . . . . . Chapter 2: Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 A Preview of Calculus . . . . . . . . . . . . . . . . . . . 2.2 The Limit of a Function . . . . . . . . . . . . . . . . . . . 2.3 The Limit Laws . . . . . . . . . . . . . . . . . . . . . . . 2.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Precise Definition of a Limit . . . . . . . . . . . . . . Chapter 3: Derivatives . . . . . . . . . . . . . . . . . . . . . . . 3.1 Defining the Derivative . . . . . . . . . . . . . . . . . . . 3.2 The Derivative as a Function . . . . . . . . . . . . . . . . 3.3 Differentiation Rules . . . . . . . . . . . . . . . . . . . . 3.4 Derivatives as Rates of Change . . . . . . . . . . . . . . 3.5 Derivatives of Trigonometric Functions . . . . . . . . . . 3.6 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . 3.7 Derivatives of Inverse Functions . . . . . . . . . . . . . . 3.8 Implicit Differentiation . . . . . . . . . . . . . . . . . . . 3.9 Derivatives of Exponential and Logarithmic Functions . . . Chapter 4: Applications of Derivatives . . . . . . . . . . . . . . 4.1 Related Rates . . . . . . . . . . . . . . . . . . . . . . . 4.2 Linear Approximations and Differentials . . . . . . . . . . 4.3 Maxima and Minima . . . . . . . . . . . . . . . . . . . . 4.4 The Mean Value Theorem . . . . . . . . . . . . . . . . . 4.5 Derivatives and the Shape of a Graph . . . . . . . . . . . 4.6 Limits at Infinity and Asymptotes . . . . . . . . . . . . . . 4.7 Applied Optimization Problems . . . . . . . . . . . . . . 4.8 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . 4.9 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . 4.10 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . Chapter 5: Integration . . . . . . . . . . . . . . . . . . . . . . . 5.1 Approximating Areas . . . . . . . . . . . . . . . . . . . . 5.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . 5.3 The Fundamental Theorem of Calculus . . . . . . . . . . 5.4 Integration Formulas and the Net Change Theorem . . . . 5.5 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Integrals Involving Exponential and Logarithmic Functions 5.7 Integrals Resulting in Inverse Trigonometric Functions . . Chapter 6: Applications of Integration . . . . . . . . . . . . . . 6.1 Areas between Curves . . . . . . . . . . . . . . . . . . . 6.2 Determining Volumes by Slicing . . . . . . . . . . . . . . 6.3 Volumes of Revolution: Cylindrical Shells . . . . . . . . . 6.4 Arc Length of a Curve and Surface Area . . . . . . . . . 6.5 Physical Applications . . . . . . . . . . . . . . . . . . . . 6.6 Moments and Centers of Mass . . . . . . . . . . . . . . 6.7 Integrals, Exponential Functions, and Logarithms . . . . . 6.8 Exponential Growth and Decay . . . . . . . . . . . . . . 6.9 Calculus of the Hyperbolic Functions . . . . . . . . . . . Appendix A: Table of Integrals . . . . . . . . . . . . . . . . . . . Appendix B: Table of Derivatives . . . . . . . . . . . . . . . . . Appendix C: Review of Pre-Calculus . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 5 . 6 34 60 76 94 123 124 135 160 179 194 213 214 232 247 266 277 287 299 309 319 341 342 354 366 379 390 407 439 454 472 485 507 508 529 549 566 584 595 608 623 624 636 656 671 685 703 721 734 745 763 769 771 861 This OpenStax book is available for free at Preface 1 PREFACE Welcome to Calculus Volume 1, an OpenStax resource. This textbook was written to increase student access to high-quality learning materials, maintaining highest standards of academic rigor at little to no cost. About OpenStax OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education. Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 20 books for college and AP courses used by hundreds of thousands of students. Our adaptive learning technology, designed to improve learning outcomes through personalized educational paths, is being piloted in college courses throughout the country. Through our partnerships with philanthropic foundations and our alliance with other educational resource organizations, OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed. About OpenStax Resources Customization Calculus Volume 1 is licensed under a C reative C ommons Attribution 4.0 International (C CBY) license, which means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its content contributors. Because our books are openly licensed, you are free to use the entire book or pick and choose the sections that are most relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and sections in your syllabus, in the order that you prefer. You can even provide a direct link in your syllabus to the sections in the web view of your book. Faculty also have the option of creating a customized version of their OpenStax book through the platform. The custom version can be made available to students in low-cost print or digital form through their campus bookstore. Visit your book page on openstax.org for a link to your book on . Errata All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors sometimes occur. Since our books are web based, we can make updates periodically when deemed pedagogically necessary. If you have a correction to suggest, submit it through the link on your book page on openstax.org. Subject matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on openstax.org. Format You can access this textbook for free in web view or PDF through openstax.org, and for a low cost in print. About Calculus Volume 1 C alculus 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. 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 2 Preface 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 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. 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. 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. Comprehensive Art Program Our art program is designed to enhance students’ understanding of concepts through clear and effective illustrations, diagrams, and photographs. This OpenStax book is available for free at Preface 3 Additional Resources Student and Instructor Resources We’ve compiled...
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