The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis. This book proposes that an effective way to motivate these definitions is to tell one of the stories (there are many) of the historical development of the subject, from its intuitive beginnings to modern rigor. The definitions and techniques are motivated by the actual difficulties encountered by the intuitive approach and are presented in their historical context However, this is not a history of analysis book. It is an introductory analysis textbook, presented through the lens of history. As such, it does not simply insert historical snippets to supplement the material. The history is an integral part of the topic, and students are asked to solve problems that occur as they arise in their historical context. This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. Written with the student in mind, the book provides guidance for transforming an intuitive understanding into rigorous mathematical arguments. For example, in addition to more traditional problems, major theorems are often stated and a proof is outlined. The student is then asked to fill in the missing details as a homework problem.
In Active Calculus, we endeavor to actively engage students in learning the subject through an activity-driven approach in which the vast majority of the examples are completed by students. Where many texts present a general theory of calculus followed by substantial collections of worked examples, we instead pose problems or situations, consider possibilities, and then ask students to investigate and explore. Following key activities or examples, the presentation normally includes some overall perspective and a brief synopsis of general trends or properties, followed by formal statements of rules or theorems. While we often offer plausibility arguments for such results, rarely do we include formal proofs. It is not the intent of this text for the instructor or author to demonstrate to students that the ideas of calculus are coherent and true, but rather for students to encounter these ideas in a supportive, leading manner that enables them to begin to understand for themselves why calculus is both coherent and true. This approach is consistent with the growing body of scholarship that calls for students to be interactively engaged in class.
Matthew Boelkins, David Austin, Steven Schlicker
Department of Mathematics Grand Valley State University
Advances in Plant Reproductive Ecology and Conservation Biology
New advances and research in plant reproductive ecology continue to provide key insights into common and rare plant species. Similarly, the emerging technologies, techniques, and tools in data collection and analyses continue to advance the fields of plant conservation biology and plant reproductive ecology, allowing for discoveries and interpretations that were not feasible at the beginning of the century.
Brenda Molano-Flores (editor) James Cohen (editor)
MDPI - Multidisciplinary Digital Publishing Institute
This text was written as a prequel to the APEXCalculus series, a three–volume series on Calculus. This text is not intended to fully prepare students with all of the mathematical knowledge they need to tackle Calculus, rather it is de-signed to review mathematical concepts that are o[en stumbling blocks in the Calculus sequence. It starts basic and builds to more complex topics. This text is written so that each secࢢon and topic largely stands on its own, making it a good resource for students in Calculus who are struggling with the supporting mathemathics found in Calculus courses. The topics were chosen based on experience; several instructors in the Applied Mathemathics Department at the Virginia Military Insࢢtute (VMI) compiled a list of topics that Calculus students commonly struggle with, giving the focus of this text.
Amy Chapman, Meagan Herald, Jessica Libertini
Calculus was first developed more than three hundred years ago by Sir Isaac Newton and Gottfried Leibniz to help them describe and understand the rules overning the motion of planets and moons. Since then, thousands of other men and women have refined the basic ideas of calculus, developed new techniques to make the calculations easier, and found ways to apply calculus to problems besides planetary motion. Perhaps most importantly, they have used calculus to help understand a wide variety of physical, biological, economic and social phenomena and to describe and solve problems in those areas. Part of the beauty of calculus is that it is based on a few very simple ideas
Shana Calaway, Dale Hoffman, David Lippman
Basic Analysis: Introduction to Real Analysis
A first course in mathematical analysis. Covers the real number system, sequences and series, continuous functions, the derivative, the Riemann integral, sequences of functions, and metric spaces.
Big Data for Epidemiology: Applied Data Analysis Using National Health Surveys
National data sets provide an avenue for students to practice data analytic skills while also answering meaningful research questions. This open education resource was developed to train future public health professionals how to conduct secondary data analysis of national health surveys using SAS statistical software. SAS software was selected because it is one of the most commonly used software programs used among public health departments and academia.
Gilbert Strang's clear, direct style and detailed, intensive explanations make this textbook ideal as both a course companion and for self-study. Single variable and multivariable calculus are covered in depth. Key examples of the application of calculus to areas such as physics, engineering and economics are included in order to enhance students' understanding. New to the second edition is a chapter on the 'Highlights of Calculus', which accompanies the popular video lectures by the author on MIT's OpenCourseWare.
We believe that calculus can be for students what it was for Euler and the Bernoullis: a language and a tool for exploring the whole fabric of science. We also believe that much of the mathematical depth and vitality of calculus lies in connections to other sciences. The mathematical questions that arise are compelling in part because the answers matter to other disciplines. We began our work with a "clean slate," not by asking what parts of the traditional course to include or discard. Our starting points are thus our summary of what calculus is really about. Our curricular goals are what we aim to convey about the subject in the course.
James Callahan, Northampton, Massachusetts David Cox, Amherst, Massachusetts Kenneth Hoffman, Amherst, Massachusetts
Smith College Open Educational Resources: Textbooks
Contemporary Calculus I . For the students
This is a textbook for differential calculus with explanations, examples, worked solutions, problem sets and answers. It has been reviewed by calculus instructors and class-tested by them and the author. Topics are typically introduced by way of applications, and the text contains the usual theorems and techniques of a first course in calculus. Besides technique practice and applications of the techniques, the examples and problem sets are also designed to help students develop a visual and conceptual understanding of the main ideas of differential calculus.
Control Problems for Conservation Laws with Traffic Applications
Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered.
Bayen, Alexandre, Delle Monache, Maria Laura , Garavello, Mauro., et al.
Data Analysis in the Psychological Sciences: A Practical, Applied, Multimedia Approach
This open resources textbook contains 10 Units that describe and explain the main concepts in statistical analysis of psychological data. In addition to conceptual descriptions and explanations of the basic analyses for descriptive statistics, this textbook also explains how to conduct those analyses with common statistical software (Excel) and open-source free software (R).
Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. Use of this book will help the student develop the insight and intuition necessary to master algebraic techniques and manipulative skills. Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and techniques.
This textbook covers calculus of a single variable, suitable for a year-long (or two-semester) course. Chapters 1-5 cover Calculus I, while Chapters 6-9 cover Calculus II. The book is designed for students who have completed courses in high-school algebra, geometry, and trigonometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but the old idea of an infinitesimal is resurrected, owing to its usefulness (especially in the sciences).
Elementary Differential Equations by William F. Trench Andrew G. Cowles Distinguished Professor Emeritus, Department of Mathematics, Trinity University, San Antonio, Texas, USA,Previously published by Brooks/Cole Thomson Learning, 2000. This book has been judged to meet the evaluation criteria set by the Editorial Board of the American Institute of Mathematics in connection with the Institute’s Open Textbook Initiative.
Informal Calculus with Applications to Biological and Environmental Sciences
This book is an approachable introduction to calculus with applications to biology and environmental science. For example, one application in the book is determining the volume of earth moved in the 1959 earthquake that created Quake Lake. Another application uses differential equations to model various biological examples, including moose and wolf populations at Isle Royale National Park, ranavirus in amphibians, and competing species of protozoa. The text focuses on intuitive understanding of concepts, but still covers most of the algebra and calculations common in a survey of calculus course.
Much of the material of Chapters 2-6 and 8 has been adapted from thewidely used textbook “Elementary differential equations and boundary value problems” by Boyce & DiPrima (John Wiley & Sons, Inc., Seventh Edition, ○c 2001). Many of the examples presented in these notes may be found in this book. The material of Chapter 7 is adapted from the textbook “Nonlinear dynamics and chaos” by Steven H. Strogatz (Perseus Publishing, ○c 1994)
The Hong Kong University of Science and Technology
The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. The lecture notes also contain many well-selected exercises of various levels. Although these topics are written in a more abstract way compared with those available in some textbooks, teachers can choose to simplify them depending on the background of the students. For instance, rather than introducing the topology of the real line to students, related topological concepts can be replaced by more familiar concepts such as open and closed intervals. Some other topics such as lower and upper semicontinuity, differentiation of convex functions, and generalized differentiation of non-differentiable convex functions can be used as optional mathematical projects. In this way, the lecture notes are suitable for teaching students of different backgrounds.
Portland State University Library Portland,
What follows are my lecture notes for Math 3311: Introduction to Numerical Methods, taught at the Hong Kong University of Science and Technology. Math 3311, with two lecture hours per week, is primarily for non-mathematics majors and is required by several engineering departments. All web surfers are welcome to download these notes at http://www.math.ust.hk/~machas/numerical-methods.pdf and to use the notes freely for teaching and learning. I welcome any comments, suggestions or corrections sent by email to jeffrey.chasnov@ust.hk.
This is a text for a two-term course in introductory real analysis for junior or senior mathematics majors and science students with a serious interest in mathematics. Prospective educators or mathematically gifted high school students can also benefit from the mathematical maturity that can be gained from an introductory real analysis course. The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics. The standard elementary calculus sequence is the only specific prerequisite for Chapters 1–5, which deal with real-valued functions. (However, other analysis oriented courses, such as elementary differential equation, also provide useful preparatory experience.) Chapters 6 and 7 require a working knowledge of determinants, matrices and linear transformations, typically available from a first course in linear algebra. Chapter 8 is accessible after completion of Chapters 1–5.
Linear Regression Using R: An Introduction to Data Modeling
Linear Regression Using R: An Introduction to Data Modeling presents one of the fundamental data modeling techniques in an informal tutorial style. Learn how to predict system outputs from measured data using a detailed step-by-step process to develop, train, and test reliable regression models. Key modeling and programming concepts are intuitively described using the R programming language. All of the necessary resources are freely available online.
https://conservancy.umn.edu/bitstream/handle/11299/189222/LinearRegression_fulltext.pdf? sequence=5&isAllowed=y
University of Minnesota Libraries Publishing
This text is an outgrowth of lectures given at the University of Windsor, Canada. One of our main objectives is updating the undergraduate analysis as a rigorous postcalculus course. While such excellent books as Dieudonn´e’s Foundations of Modern Analysis are addressed mainly to graduate students, we try to simplify the modern Bourbaki approach to make it accessible to sufficiently advanced undergraduates.
This book covers the standard material for a one-semester course in multivariable calculus. The topics include curves, differentiability and partial derivatives, multiple integrals, vector fields, line and surface integrals, and the theorems of Green, Stokes, and Gauss. Roughly speaking the book is organized into three main parts corresponding to the type of function being studied: vector-valued functions of one variable, real-valued functions of many variables, and finally the general case of vector-valued functions of many variables. As is always the case, the most productive way for students to learn is by doing problems, and the book is written to get to the exercises as quickly as possible.
Notes on Diffy Qs: Differential Equations for Engineers
This book originated from my class notes for teaching Math 286 at the University of Illinois at Urbana-Champaign in Fall 2008 and Spring 2009. It is a ﬁrst course on differential equations for engineers. I also taught Math 285 at UIUC and Math 20D at UCSD using this book. The standard book at UIUC was Edwards and Penney, Differential Equations and Boundary Value Problems: Computing and Modeling [EP], fourth edition. The standard book at UCSD is Boyce and DiPrima’s Elementary Differential Equations and Boundary Value Problems [BD]. As the syllabus at UIUC was based on [EP], the early chapters in the book have some resemblance to [EP] in choice of material and its sequence, and examples used. Among other books I used as sources of information and inspiration are E.L. Ince’s classic (and inexpensive) Ordinary Differential Equations [I], Stanley Farlow’s Differential Equations and Their Applications [F], now available from Dover, Berg and McGregor’s Elementary Partial Differential Equations [BM], and William Trench’s free book Elementary Differential Equations with Boundary Value Problems [T]. See the Further Reading chapter at the end of the book.
Precalculus is adaptable and designed to fit the needs of a variety of precalculus courses. It is a comprehensive text that covers more ground than a typical one- or two-semester college-level precalculus course. The content is organized by clearly-defined learning objectives and includes worked examples that demonstrate problem-solving approaches in an accessible way.
From the preface, "These are notes for a course in precalculus, as it is taught at New York City College of Technology - CUNY (where it is offered under the course number MAT 1375). Our approach is calculator based. For this, we will use the currently standard TI-84 calculator, and in particular, many of the examples will be explained and solved with it. However, we want to point out that there are also many other calculators that are suitable for the purpose of this course and many of these alternatives have similar functionalities as the calculator that we have chosen to use. An introduction to the TI-84 calculator together with the most common applications needed for this course is provided in appendix A. In the future we may expand on this by providing introductions to other calculators or computer algebra systems."
This Statistics resource was developed under the guidance and support of experienced high school teachers and subject matter experts. It is presented here in multiple formats: PDF, online, and low-cost print. Statistics offers instruction in grade-level appropriate concepts and skills in a logical, engaging progression that begins with sampling and data and covers topics such as probability, random variables, the normal distribution, and hypothesis testing. This content was developed with students in mind, incorporating statistics labs, worked exercises, and additional opportunities for assessment that incorporate real-world statistical applications. For instructors, resources are available to support the implementation of the Statistics textbook, including a Getting Started Guide, direct instruction presentations, and a solutions manual.
Numerical methods are designed to approximate one thing or another. Sometimes roots, sometimes derivatives or definite integrals, or curves, or solutions of differential equations. As numerical methods produce only approximations to these things, it is important to have some idea how accurate they are. Sometimes accuracy comes down to careful algebraic analysis—sometimes careful analysis of the calculus, and often careful analysis of Taylor polynomials. But before we can tackle those details, we should discuss just how error and, therefore, accuracy are measured. There are two basic measurements of accuracy: absolute error and relative error. Suppose that p is the value we are approximating, and p˜ is an approximation of p. Then p˜ misses the mark by exactly the quantity p˜− p, the so-called error. Of course, p˜ − p will be negative when p˜ misses low. That is, when the approximation p˜ is less than the exact value p. On the other hand, p˜ − p will be positive when p˜ misses high. But generally, we are not concerned with whether our approximation is too high or too low. We just want to know how far off it is. Thus, we most often talk about the absolute error, |p˜ − p|. You might recognize the expression |p˜ − p| as the distance between p˜ and p, and that’s not a bad way to think about absolute error.