# Book: Calculus (OpenStax)

This Textmap guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them.

Thumbnail: Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot. (Public Domain; Jacj via Wikipedia)

## Book Title: Precalculus

Book Description: Note: This OpenStax book was imported into Pressbooks on July 29, 2019, to make it easier for instructors to edit, build upon, and remix the content. The OpenStax import process isn't perfect, so there are a number of formatting errors in the book that need attention. As such, we don't recommend you use this book in the classroom. This also means that, while the original version of this book is accessible, this Pressbooks copy is not.

For information about how to get your own copy of this book to work on, see the Add Content part in the Pressbooks Guide.

You can access the original version of this textbook here: Precalculus: OpenStax.

## Product description

Senior Contributing Authors:
Gilbert Strang (Massachusetts Institute of Technology)Edwin Jed Herman (University of Wisconsin-Stevens Point)

Contributing Authors:
Alfred K. Mulzet (Florida State College at Jacksonville)
Sheri J. Boyd (Rollins College)Joyati Debnath (Winona State University)
Michelle Merriweather (Bronxville High School)
Joseph Lakey (New Mexico State University)
Elaine A. Terry (Saint Joseph's University)
David Smith (University of the Virgin Islands)
Nicoleta Virginia Bila (Fayetteville State University)
Valeree Falduto (Palm Beach State College)
Kirsten R. Messer (Colorado State University-Pueblo)
William Radulovich (Formerly at Florida State College at Jacksonville)
Erica M. Rutter (Arizona State University)
David Torain (Hampton University)
Catherine Abbott (Keuka College)
Julie Levandosky (Framingham State University)
David McCune (William Jewell College)

## E-Textbooks

The American Yawp: A Massively Collaborative Open U.S. History Textbook by Joseph L. Locke The American Yawp is a free, online, collaboratively built American history textbook. Over 300 historians joined together to create the book they wanted for their own students—an accessible, synthetic narrative that reflects the best of recent historical scholarship and provides a jumping-off point for discussions in the U.S. history classroom and beyond. Computer Networks: A Systems Approach by Larry Peterson, Bruce Davie Suppose you want to build a computer network, one that has the potential to grow to global proportions and to support applications as diverse as teleconferencing, video on demand, electronic commerce, distributed computing, and digital libraries. What available technologies would serve as the underlying building blocks, and what kind of software architecture would you design to integrate these building blocks into an effective communication service? Answering this question is the overriding goal of this book—to describe the available building materials and then to show how they can be used to construct a network from the ground up.

Fundamentals of Business by Stephen J. Skripak (Adapted by) This text is an open education resource intended to serve as a no-cost, faculty customizable primary text for one-semester undergraduate introductory business courses.

Also available: testbank (by request) and interactive self-quizzing (pressbooks version)

## Openstax Solution Manual

Chapter 21: Nuclear Chemistry. Ancillaries. OpenStax projects offer an array of ancillaries for students and instructors. The following resources are available. PowerPoint Slides. Instructor's Solution Manual. Our resources are continually expanding, so please visit http://openstaxcollege.org to view an up-to-date list of the.

### Student Solutions Manual

OpenStax Statistics. Instructor Answer and Solution Guide. Chapter 1: Sampling and Data. CHAPTER 1: SAMPLING AND DATA. Exercise 1. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given .

### Introductory Statistics

May 11, 2013 . OpenStax Tutor name, Openstax Tutor logo, Connexions name, Connexions logo , Rice University name, and. Rice University . OPENSTAX. OpenStax provides free, peer-reviewed, openly licensed textbooks for introductory college and Advanced Placement® . Instructor's Solutions Manual. • Webassign .

### Calculus Volume 1

solution manual, and PowerPoint slides. Instructor resources require a verified instructor account, which can be requested on your openstax.org log-in. Take advantage of these resources to supplement your OpenStax book. Partner Resources. OpenStax Partners are our allies in the mission to make high-quality learning .

### Instructor Solutions Manual for Physics by Halliday, Resnick, and

the back of the book. Exercises and Problems which are enclosed in a box also appear in the Student's Solution Manual with considerably more detail and, when appropriate, include discussion on any physical implications of the answer. These student solutions carefully discuss the steps required for solving problems, .

### Access. The Future of Education. OpenStax College

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### Math 1111 Objectives OpenStax Algebra & Trigonometry CHAPTER

OpenStax Algebra & Trigonometry. CHAPTER 6: EXPONENTIAL AND LOGARITHMIC FUNCTIONS. Section 6.1: Exponential Functions. D Evaluate . 570: Chapter 6 Review Exercises (see solutions manual) p. 573: Chapter 6 Practice Test (see solutions manual). Please note that we are NOT covering Section 6.8, so any .

### Calculus Volume 2

OpenStax projects offer an array of ancillaries for students and instructors. The following resources are available. PowerPoint Slides. Instructor's Answers and Solutions Guide. Student Answers and Solutions Guide. Our resources are continually expanding, so please visit http://openstaxcollege.org to view an up-to- date list .

### Math 1111 Objectives OpenStax Algebra & Trigonometry CHAPTER

OpenStax Algebra & Trigonometry. CHAPTER 5: POLYNOMIAL AND RATIONAL FUNCTIONS. Section 5.1: Quadratic Functions. D Recognize . 458: Chapter 5 Review Exercises (see solutions manual) p. 461: Chapter 5 Practice Test (see solutions manual). Please note that we are NOT covering Section 5.8, so any .

### College Algebra

Chapter Review Exercises, and Practice Test. Ancillaries. OpenStax projects offer an array of ancillaries for students and instructors. Currently the following resources are available. Instructor's Solutions Manual. Student's Solutions Manual. PowerPoint Slides. 1. 4,649 total exercises. Includes Chapter Reviews and Practice .

### Beginning and Intermediate Algebra Student Solutions Manual

Aug 8, 2010 . Solutions manual has not been cross checked for accuracy. If you disagree with this solutions manual you should check with your instructor. Should you find an error, please E-mail [email protected] so it can be corrected. Thank you! Beginning Algebra Student Solutions Manual by Tyler Wallace is .

### Lecture Notes for OpenStax Introductory Statistics

population. To summarize: the process of inferring information about a population based on a representative sample is the practice of inferential statistics. 1These lecture notes are intended to be used with the open source textbook “Introductory . Statistics” by Barbara Illowsky and Susan Dean (OpenStax College, 2013). 1 .

### AICASA Booklist

Required. 1st 9780321910417. OpenStax. 2. Calculus. Calculus for Biology and Medicine. Neuhauser. Required 2nd 9780321644688. Pearson Prentice Hall. 2. Calculus. Student Solutions Manual for Calculus for Biology and. Medicine 2e. Neuhauser. Required 2nd 9780321644923. Pearson Prentice Hall. 3. Cell Biology.

### OpenStax Psychology

OpenStax Psychology. Correlation to APA Guidelines for the Undergraduate Psychology . major. 5.1E Adapt information literacy skills obtained in the psychology major to investigating solutions to a variety of problem solutions. 5.1f Describe how ethical principles of psychology have relevance to nonpsychology settings.

### Precalculus

Aug 21, 2014 . OpenStax projects offer an array of ancillaries for students and instructors. Currently the following resources are available. Instructor's Solutions Manual. Student's Solutions Manual. PowerPoint Slides. Please visit http:// openstaxcollege.org to view an up-to-date list of the Learning Resources for this title and .

### College Physics

College Physics by OpenStax College is licensed under a Creative Commons . community by creating a high quality online homework solution. Page 2. for selected open source textbooks, available at an affordable price to students. • Student Solution Manual The Student Solution Manual is a free download for students.

### Introductory Statistics

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### College Physics

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### Introduction to Sociology

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### College Physics (Kindle)

included in this edition. Answers to odd-numbered multiple-choice questions are included in the Answers section at the end of the book, and answers to all questions are found in the Instructor's Solutions Manual and on the instructor's. PowerLecture CD-ROM. □. Enhanced Content problems require symbolic or conceptual .

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## Book: Calculus (OpenStax)

Standard calculus textbooks

These books are those designed to be the required textbook for a standard high school or university calculus course. Many of them have had a large number of revisions, which makes it likely that you can find slightly older but still perfectly useful copies on the used market. They try to have everything -- proofs, intuitive explanations, illustrations, problems -- but they aren't necessarily great at anything. These books are widely derided by math majors, though I suspect this to be in part motivated by elitism and by grad students' dislike of teaching lower-level math courses. In any case, these are useful even if you just want a cheap source of problems to practice with.

The OpenStax series follows a very common sequencing of topics, which I list here as a representative of what you can expect to find in the books in this section. OpenStax splits these into three separate books, but most series combine them all into one huge book of about 1000 pages. It's also common to see 1 & 2 combined, and a separate book for 3.

• Volume 1: functions, limits, derivatives, integration.
• Volume 2: [more] integration, differential equations, sequences and series, parametric equations, polar coordinates.
• Volume 3: parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, second-order differential equations.

Here are some books in this category, including the most popular ones that I know of. I've tried to link mainly to the best-priced editions so you can get a book for $5-10 (though prices change). • OpenStax: Strang, Herman et al. Calculus. (FREE ONLINE: Vol 1 Vol 2 Vol 3) • Strang, Calculus (FREE ONLINE: Website, Direct link to PDF) • Simmons. Calculus with Analytic Geometry (2e) • Stewart. Calculus: Early Transcendentals (5e) -- Has huge adoption in both high schools and universities. • Leithold. (TC7 TCWAG 6) -- I've seen Louis Leithold credited as the creator of the modern "standard" calculus textbook. • Larson, Hostetler, Edwards. Calculus (8e 7e 6e) • Thomas et al. Thomas' Calculus (12e Upd 10e 9e) -- Thomas retired in 1978, and editions since then have different co-authors, principally Ross Finney. • Lial, Greenwell et al. Calculus with Applications (9e) • Anton, Bivens and Davis. Calculus (7e 7e Student Resource Manual 9e multivariable) • Marsden and Weinstein. Calculus (Vol I: 2e Vol II: 2e Vol III: 2e) • Rogawski. Calculus: Early Transcendentals (2e) • Briggs and Cochran. Calculus: Early Transcendentals (1e) • Bittinger and Ellenbogen. Calculus and Its Applications (9e) These are books that math majors get enthusiastic about. They are typically used in elite universities and honors courses. They emphasize theory from the ground up beginning with the real number system, but they have fewer calculation problems and applications problems (no applications, in some cases). Note that many, maybe most students who read these books do not actually start calculus with them: often they've seen calculus before, maybe they've even had a course or two. Spivak is the most revered of this class of textbooks. It's also the most current: its fourth edition was published in 2008. Spivak has a chatty, conversational style. The book begins by establishing the properties of the real number system, and covers limits, differential and integral calculus, and infinite sequences. Unlike the books that follow, Spivak does not deal with calculus of several variables, so you need another book for that. (He has written a book on that subject, Calculus on Manifolds, but it's not considered an introductory book.) Spivak's book has been called an introduction to analysis you might also call it "calculus for mathematicians". Apostol is, according to my own rather cursory research, the most widely used of this class of textbooks in college classrooms. Volume I in particular is excellent. The language is much more typical "math-speak" than Spivak's, but the approaches to the material are unusual. Apostol begins with integration before covering limits and differential calculus. This approach works, but it is incovenient if you're trying to use it in combination with another textbook or an online course, since almost all other courses follow the order limits-derivatives-integrals. Apostol's problem sets are excellent, perhaps the best of the bunch. Chapter 1 in particular is a crash course in proof by induction that will very quickly get you comfortable with manipulating sums. Volume II introduces linear algebra and calculus of several variables. It seems to be regarded as not as good as Volume I, particularly because its presentation of linear alegebra is dated. However, it does cover some interesting topics not often seen in a calculus textbook: systems of differential equations, probability, and numerical analysis (i.e. approximation methods). The standard edition of these books is, unfortunately, exhorbitantly expensive. However, at least the physical quality of the books is extremely good (the one I have is, at least). Courant, Differential and Integral Calculus, volumes I and II • Courant (Wiley 2nd edition). Vol IVol II • Courant and John (Springer). Vol I, Vol II/1, Vol II/2 • Courant (original). Vol II • Courant (1937/1936 Blackie edition, printed 1961). At Archive.org: Vol I, Vol II Richard Courant, who died in 1972, was a highly respected German-American mathematician. With Herbert Robbins, he also wrote What is Mathematics?, a remarkable work that attempts to explain a large swathe of mathematics to a popular audience. This calculus textbook is particularly noted for its wealth of physics applications, making it by far the most "applied" of this group. However, its problems are also notoriously difficult, so much so that they're not really appropriate for a learner at this level, at least not without a supplement of easier problems to practice with. Since Courant has been dead for several decades, a newer edition was published (by Springer) with Fritz John as co-author. That edition is also highly respected. The most notable difference is that Courant and John splits Volume II into two parts and adds several sections of new material. Shahriari, Approximately Calculus (1e) A more recent book of this style. It's intended as a second book on calculus. It's full of difficult problems that push your theoretical understanding of calculus, approaching the topic in unusual ways. These books don't fit cleanly into the categories that I described above. These take the attitude that emphasis on rigorous proofs gets in the way of understanding calculus. Ash and Ash, The Calculus Tutoring Book (1e paperback 1e hardcover) This book just doesn't bother with theorems or proofs unless they are necessary. It focuses on giving you an intuitive grasp of calculus. I like it a lot. I wouldn't advocate it as your only book on calculus, but I think it would make a very good first book. It covers the standard calculus curriculum as I decribed in the "Standard calculus textbooks" section above. Although it focuses heavily on pedagogy, I think that this may not be a good book "for dummies", so to speak. It may be that stronger math students are the ones who benefit the most from this approach, because they are used to learning by understanding rather than by memorizing rules. Though I mention that proofs are largely dispensed with here, they are not entirely absent: I found that, when they were included, it was most often in support of a statement that I wasn't sure if I believed, which seems the perfect balance. Doesn't cover a few standard topics, such as differential equations or parametric equations. The paperback seems a bit flimsy, so if you're planning to use this you might want to get the hardcover edition. This book is out of print, so get a used copy while you still can! Thompson and Gardner, Calculus Made Easy (Revised) An intuitive introduction to calculus. This isn't thorough enough to be a course textbook, but it makes a good companion to a textbook. It's for someone who's just beginning calculus. The original version (by just Thompson) is in the public domain. It's shorter and some people feel that it does a better job of keeping things simple. Keep in mind that it's old! It was published in 1914. Its casual style emphasizes its old-fashionedness. Some might find this charming, others annoying. You can get it online here: https://archive.org/details/CalculusMadeEasy Kline, Calculus: An Intuitive and Physical Approach (Dover) The title pretty much says what this is. It's older (originally published in 1967, updated in 1977), and was written by Morris Kline, a prolific writer on mathematics (and critic of math education) who authored several books for a popular audience. Focuses on supporting calculus with geometric and physical explanations, and motivating it with the historical problems that motivated the development of calculus. Problems focus on understanding. Early emphasis on differential equations. Zeldovich and Yaglom. Higher Math for Beginners (Archive.org) Thomas, Calculus and Analytic Geometry (c) 3rd Alternate This entry refers to the original series of textbooks written by George B. Thomas himself, up to the Fourth Edition (1969). As of 2015 they can still be bought used (quite cheap!), and thanks to the durable hardback bindings they used in those days, you can probably still find them in good condition. (Just hope some jerk didn't scribble all over your copy.) Depending on who you ask, the pinnacle of the series was either the 4th Edition or the 3rd Alternate Edition. As advertised, there is a strong emphasis here on analytic geometry and applications. Thomas taught at MIT, and this is clearly a "calculus for engineers" approach. Serge Lang is famous for his difficult graduate-level texts in a dozen different subjects, but he has also put his hand to writing a handful of textbooks for lower levels of math. Of those, probably the most successful is this, his calculus book. This book is really distinct from all the others here. It is relatively short. It ignores applications to focus on equations, but it is not really a rigorous textbook either. It has proofs, but only the ones that Lang thought were essential. It simply covers the topics of calculus in a clear and straightforward manner. There have been five editions, and each added more material (getting longer), until Lang actually brought back the original edition under the title Short Calculus. I have the Fourth Edition. Because of its focus on the standard topics without a bunch of fuss over applications, this is currently my preferred reference for anything that it covers. Körner, Calculus for the Ambitious (1e) CUP's promotion says, "In a lively and easy-to-read style, Professor Körner uses approximation and estimates in a way that will easily merge into the standard development of analysis. By using Taylor's theorem with error bounds he is able to discuss topics that are rarely covered at this introductory level." I haven't seen this book in the flesh yet, but it's inexpensive. Stein. Calculus in the First Three Dimensions (Dover Solutions manual (PDF)) Kiesler. Elementary Calculus: An Infinitesimal Approach (FREE ONLINE Dover) Lax and Terrell. Calculus With Applications Online calculus resources MIT OCW has a series of online lectures for calculus. The lecturing isn't outstanding, but the material covered is good for the most part. The lectures can be watched on YouTube here: Other online lecture offerings include ones from Ohio State and Khan Academy. Both seemed a bit slow and basic for my taste, whereas MIT's was more my speed, but then again I'd already taken some calculus in high school so I wasn't starting from scratch. The University of Wisconsin-Madison also has free calculus texts that you can get online they don't look at all adequate as stand-alone texts for self study, but they do have problems with some solutions included. Whitman College also has free calculus texts. For multivariate calculus, there seems to be a general consensus that the texts that teach "all" of calculus don't do a great job with it. (Leithold, for example, stops giving so many proofs at that point in the curriculum and states that they belong in a book dedicated to the subject.) The following texts are dedicated to this part of the calculus curriculum in particular (arranged in descending order according to my impression of their usefulness): • Hubbard and Hubbard, Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach (5e at Matrix Editions, 5e at Amazon, 4e at Amazon) • Schey, Div, Grad, Curl and All That (4e, 3e, 2e, 1e) • Baxandall and Liebeck, Vector Calculus (Dover) • Friedman, Advanced Calculus (Dover) • C. E. Edwards. Advanced Calculus of Several Variables (1e) • H. M. Edwards. Advanced Calculus: A Differential Forms Approach (3e, 2014 reissue, 3e, 1994) • Shifrin, Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds (1e) • Corral, Vector Calculus (FREE ONLINE) • Marsden and Tromba. Vector Calculus (5e, 4e) • Colley. Vector Calculus (2e, 1e, 2e) • Matthews. Vector Calculus (1e) • Bressoud. Second Year Calculus: From Celestial Mechanics to Special Relativity (1e) • Flanigan and Kazdan. Calculus Two: Linear and Nonlinear Functions (2e) • Bray. Multivariable Calculus (1e at CreateSpace, 1e) Author's page: https://services.math.duke.edu/ The Feynman Lectures also discuss vector calculus in the context of electromagnetism, which is supposedly a great way to tie it in with your understanding of the physical, if you already have some background in physics. See Physics. It may be that the problems in whatever textbook you're using don't give you as much practice as you want. One option is to get another textbook and do the problems from that, too. However, there are a number of books dedicated to providing problems. One advantage to these is that they generally tend to give you explanations of how to solve them as well. • Demidovich, Problems in mathematical analysis [can be found online] • Maron, Problems in Calculus of One Variable (c) • Mendelson, Schaum's 3,000 Solved Problems in Calculus (c) [reportedly has some errors] • Kelley, The Humongous Book of Calculus Problems (c) • Bluman, Problem Book for First Year Calculus • Jones, Calculus: 1,001 Practice Problems For Dummies (1e) -- By PatrickJMT of YouTube fame. • REA. The Calculus Problem Solver (Unspecified edition) John Erdman of Portland State University has posted a number of free problem books as PDFs on his website (answers to odd-numbered exercises given): http://web.pdx.edu/ This site has links to lots of other sites with problem sets and practice exams: http://www.calculus.org Simmons. Differential Equations with Applications and Historical Notes (3e, 2e) Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems (9e, 8e) Edwards and Penney. Differential Equations and Linear Algebra (2e) Edwards, Penney, Calvis. Differential Equations and Boundary Value Problems: Computing and Modeling (5e, 4e (Edwards/Penney)) Piskunov, Differential and Integral Calculus (Vol I: Mir/CBS Vol II: Mir/CBS) From what I gather, this pair of volumes is legendary for its quality (though problems are too easy), but the physical quality of this edition is terrible. • Tenenbaum, Ordinary Differential Equations (Dover) • Hurewicz, Lectures on Ordinary Differential Equations (Dover) • Coddington, An Introduction to Ordinary Differential Equations (Dover) • Brauer and Nohel. The Qualitative Theory of Ordinary Differential Equations: An Introduction (Dover) • Arnold. Ordinary Differential Equations (MIT Press, tr. Silverman, Springer, tr. Cooke) - Springer is a later version. • Coddington and Levinson, Theory of Ordinary Differential Equations (Krieger 1984) • Imhoff, Differential Equations in 24 Hours: with Solutions and Historical Notes (1ed, 2015) • Logan. A First Course in Differential Equations • Strauss. Partial Differential Equations: An Introduction (1e, 2e) • Farlow. Partial Differential Equations for Scientists and Engineers (Dover) • Logan. Applied Partial Differential Equations (3e, 2e) • John. Partial Differential Equations (4e) • Bleecker and Csordas. Basic Partial Differential Equations (1e) • Zachmanoglou and Thoe. Introduction to Partial Differential Equations with Applications (Dover) • Asmar. Partial Differential Equations with Fourier Series and Boundary Value Problems (3e Dover) • Hillen, Leonard, van Roessel. Partial Differential Equations: Theory and Completely Solved Problems (1e) • Olver. Introduction to Partial Differential Equations • Logan. Applied Partial Differential Equations Advanced (more prequisites) • Vasy. *Partial Differential Equations: An Accessible Route Through Theory and Applications( (1e) • Gustafson. Introduction to Partial Differential Equations and Hilbert Space Methods (Dover) • Taylor. Partial Differential Equations (Vol I: Basic Theory2e Vol II: Qualitative Studies of Linear Equations2e Vol III: Nonlinear Equations2e) • Evans. Partial Differential Equations: Second Edition (2e) • Folland. Introduction to Partial Differential Equations (2e) • McOwen. Partial Differential Equations: Methods and Applications (2e) The term "advanced calculus" seems to be somewhat vague. Sometimes it means calculus of several variables, sometimes it means analysis, sometimes it means applications that become possible after you have a couple of years of calculus under your belt. The ones I list here straddle the line between calculus and analysis. ## ►TBQ, “College Physics” Vol 2 of 3 (Chapters 13 – 24) 2013 “College Physics” Vol 2 of 3 (Chapters 13 – 24) 2012 (download free pdf), 498 pages, 40 Mb). See volume chapter headings below. An open license textbook originally published by Openstax College of Rice University. ### Volume 2 of 3: Chapter Headings 13 Temperature, Kinetic Theory, and the Gas Laws 14 Heat and Heat Transfer Methods 15 Thermodynamics 16 Oscillatory Motion and Waves 17 Physics of Hearing 18 Electric Charge and Electric Field 19 Electric Potential and Electric Field 20 Electric Current, Resistance, and Ohm’s Law 21 Circuits, Bioelectricity, and DC Instruments 22 Magnetism 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies 24 Electromagnetic Waves Appendices A Atomic Masses B Selected Radioactive Isotopes C Useful Information D Glossary of Key Symbols and Notation Index (Vol 1 – 3) ### Volume 3 of 3 Chapter Headings 25 Geometric Optics 26 Vision and Optical Instruments 27 Wave Optics 28 Special Relativity 29 Introduction to Quantum Physics 30 Atomic Physics 31 Radioactivity and Nuclear Physics 32 Medical Applications of Nuclear Physics 33 Particle Physics 34 Frontiers of Physics Appendices A Atomic Masses B Selected Radioactive Isotopes C Useful Information D Glossary of Key Symbols and Notation Index (Vol 1 – 3) ### Volume 1 of 3 Chapter Headings 1 Introduction: The Nature of Science and Physics 2 Kinematics 3 Two-Dimensional Kinematics 4 Dynamics: Force and Newton’s Laws of Motion 5 Further Applications of Newton’s Laws: Friction, Drag, and Elasticity 6 Uniform Circular Motion and Gravitation 7 Work, Energy, and Energy Resources 8 Linear Momentum and Collisions 9 Statics and Torque 10 Rotational Motion and Angular Momentum 11 Fluid Statics 12 Fluid Dynamics and Its Biological and Medical Applications Appendices A Atomic Masses B Selected Radioactive Isotopes C Useful Information D Glossary of Key Symbols and Notation Index (Vol 1 – 3) ## Kinematic Equations from Integral Calculus Let&rsquos begin with a particle with an acceleration a(t) is a known function of time. Since the time derivative of the velocity function is acceleration, we can take the indefinite integral of both sides, finding [int frac v(t) dt = int a(t) dt + C_<1>,] where C1 is a constant of integration. Since (int frac v(t) dt = v(t)), the velocity is given by [v(t) = int a(t) dt + C_ <1>ldotp label<3.18>] Similarly, the time derivative of the position function is the velocity function, Thus, we can use the same mathematical manipulations we just used and find where C2 is a second constant of integration. We can derive the kinematic equations for a constant acceleration using these integrals. With a(t) = a, a constant, and doing the integration in Equation ef<3.18>, we find [v(t) = int a dt + C_ <1>= at + C_ <1>ldotp] If the initial velocity is v(0) = v0, then which is Equation 3.5.12. Substituting this expression into Equation ef <3.19>gives [x(t) = int (v_ <0>+ at) dt + C_ <2>ldotp] Doing the integration, we find [x(t) = v_ <0>t + frac<1> <2>at^ <2>+ C_ <2>ldotp] so, C2 = x0. Substituting back into the equation for x(t), we finally have [x(t) = x_ <0>+ v_ <0>t + frac<1> <2>at^ <2>ldotp] Example 3.17: Motion of a Motorboat A motorboat is traveling at a constant velocity of 5.0 m/s when it starts to decelerate to arrive at the dock. Its acceleration is a(t) = (-frac<1><4>) t m/s 2 . (a) What is the velocity function of the motorboat? (b) At what time does the velocity reach zero? (c) What is the position function of the motorboat? (d) What is the displacement of the motorboat from the time it begins to decelerate to when the velocity is zero? (e) Graph the velocity and position functions. (a) To get the velocity function we must integrate and use initial conditions to find the constant of integration. (b) We set the velocity function equal to zero and solve for t. (c) Similarly, we must integrate to find the position function and use initial conditions to find the constant of integration. (d) Since the initial position is taken to be zero, we only have to evaluate the position function at t = 0 . We take t = 0 to be the time when the boat starts to decelerate. 1. From the functional form of the acceleration we can solve Equation ef <3.18>to get v(t):$v(t) = int a(t) dt + C_ <1>= int - frac<1><4>tdt + C_ <1>= - frac<1><8>t^ <2>+ C_ <1>ldotp$At t = 0 we have v(0) = 5.0 m/s = 0 + C1, so C1 = 5.0 m/s or v(t) = 5.0 m/s &minus (frac<1><8>) t 2 . 2. v(t) = 0 = 5.0 m/s &minus (frac<1><8>) t 2 (Rightarrow) t = 6.3 s 3. Solve Equation ef<3.19>:$x(t) = int v(t) dt + C_ <2>= int (5.0 - frac<1><8>t^<2>) dt + C_ <2>= 5.0t - frac<1><24>t^ <3>+ C_ <2>ldotp$At t = 0, we set x(0) = 0 = x0, since we are only interested in the displacement from when the boat starts to decelerate. We have$x(0) = 0 = C_ <2>ldotp$Therefore, the equation for the position is$x(t) = 5.0t - frac<1><24>t^ <3>ldotp$4. Since the initial position is taken to be zero, we only have to evaluate x(t) when the velocity is zero. This occurs at t = 6.3 s. Therefore, the displacement is$x(6.3) = 5.0(6.3) &minus frac<1><24>(6.3)^ <3>= 21.1 m ldotp\$

Significance

The acceleration function is linear in time so the integration involves simple polynomials. In Figure (PageIndex<1>), we see that if we extend the solution beyond the point when the velocity is zero, the velocity becomes negative and the boat reverses direction. This tells us that solutions can give us information outside our immediate interest and we should be careful when interpreting them.

A particle starts from rest and has an acceleration function (a(t)=left(5-left(10 frac<1> ight) t ight) frac>). (a) What is the velocity function? (b) What is the position function? (c) When is the velocity zero?

## Book: Calculus (OpenStax)

The Mathematics Department uses the following required textbooks in these core undergraduate courses:

• M 302: My Math Lab by Miller, 12th ed., Prentice Hall or Heart of Mathematics by Starbird, 3rd ed. Wiley--OR--the ebook of the same title.
• M 305G: Algebra and Trigonometry by Abramson
Precalculus by Abramson
(from openstax)
• M 408C: Calculus: Early Transcendentals by Stewart, 8th edition, Cengage--OR--the ebook of the same title. The custom textbook sold in 2011 by the Co-op can also be used.
• M 408D: Calculus: Early Transcendentals by Stewart, 8th edition, Cengage--OR--the ebook of the same title. The custom textbook sold in 2011 by the Co-op can also be used.
• M 408K: Calculus: Early Transcendentals by Stewart, 8th edition, Cengage--OR--the ebook of the same title. The custom textbook sold in 2011 by the Co-op can also be used.
• M 408L: Calculus: Early Transcendentals by Stewart, 8th edition, Cengage--OR--the ebook of the same title.The custom textbook sold in 2011 by the Co-op can also be used.
• M 408M: Calculus: Early Transcendentals by Stewart, 8th edition, Cengage--OR--the ebook of the same title.The custom textbook sold in 2011 by the Co-op can also be used.
• M 408N: Calculus: Early Transcendentals by Stewart, 8th edition, Cengage-OR--the ebook of the same title. The custom textbook sold in 2011 by the Co-op can also be used.
• M 408S: Calculus: Early Transcendentals by Stewart, 8th edition, Cengage-OR--the ebook of the same title. The custom textbook sold in 2011 by the Co-op can also be used.
• M 316: StatsPortal, Freeman Publishers (online)
• M 316K/L: Reconceptualizing Mathematics for Elementary School Teachers by Sowder, Sowder, and Nickerson 3rd edition
• M 325K: Discrete Mathematics--An Introduction to Mathematical Reasoning by Epp, Brief edition, Thomson -OR--the ebook of the same title.
• M 328K: Elementary Number Theory and its Applications by Rosen, 6th edition, Addison-Wesley -OR Number Theory Through Inquiry by Marshall/Odell/Starbird, 1st ed. MAA
• M 427J: Differential Equations and Their Applications by Martin Braun, 4th ed., Springer-OR--the ebook of the same title.
• M 427K: Elementary Diff. Equations and Boundary Value Problems by Boyce/Diprima, 9th ed., Wiley-OR--the ebook of the same title.
• M 427L: Vector Calculus by Marsden & Tromba, 6th edition, Freeman -OR--the ebook of the same title. (new edition)
• M 340L: Linear Algebra and its Applications by Lay, 4th edition, Addison-Wesley -OR--the ebook of the same title.
• M 341: Elementary Linear Algebra by Andrilli & Hecker, 5th edition, Harcourt -OR--the ebook of the same title.
• M 362K: A First Course in Probability by Ross, 8th edition, Prentice Hall --OR--the ebook of the same title.-OR--Probability by Pitman, 1st ed., Springer

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