Active Prelude to Calculus

Active Prelude to Calculus is designed for college students who aspire to take calculus and who either need to take a course to prepare them for calculus or want to do some additional self-study. Many of the core topics of the course will be familiar to students who have completed high school. At the same time, we take a perspective on every topic that emphasizes how it is important in calculus. This text is written in the spirit of Active Calculus and is especially ideal for students who will eventually study calculus from that text. The reader will find that the text requires them to engage actively with the material, to view topics from multiple perspectives, and to develop deep conceptual understanding of ideas. Many courses at the high school and college level with titles such as “college algebra”, “precalculus”, and “trigonometry” serve other disciplines and courses other than calculus. As such, these prerequisite classes frequently contain wide-ranging material that, while mathematically interesting and important, isn\u27t necessary for calculus. Perhaps because of these additional topics, certain ideas that are essential in calculus are often under-emphasized or ignored. In Active Prelude to Calculus, one of our top goals is to keep the focus narrow on the following most important ideas. Those most important ideas include: functions as processes; average rate of change; a library of basic functions; families of functions that model important phenomena; the sine and cosine are circular functions; inverses of functions; exact values versus approximate ones; and long-term trends, unbounded behavior, and limits of functions. See more in the preface of the text at https://activecalculus.org/prelude/preface-our-goals.html. The text is available in three different formats: HTML, PDF, and print, each of which is available via links on the landing page at https://activecalculus.org/. The first two formats are free.https://scholarworks.gvsu.edu/books/1020/thumbnail.jp

Active Calculus Multivariable

Active Calculus Multivariable is the continuation of Active Calculus to multivariable functions. The Active Calculus texts are different from most existing calculus texts in at least the following ways: the texts are free for download by students and instructors in .pdf format; in the electronic format, graphics are in full color and there are live html links to java applets; the texts are open source, and interested instructors can gain access to the original source files upon request; the style of the texts requires students to be active learners — there are very few worked examples in the texts, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems, and developing understanding of key calculus concepts; each section begins with motivating questions, a brief introduction, and a preview activity, all of which are designed to be read and completed prior to class; the exercises are few in number and challenging in nature.https://scholarworks.gvsu.edu/books/1014/thumbnail.jp

Active Calculus 2.1

Active Calculus is different from most existing calculus texts in at least the following ways: the text is freely readable online in HTML format and is also available for in PDF; in the electronic format, graphics are in full color and there are live links to java applets; there are live WeBWorK exercises in each chapter, which are fully interactive in the HTML format and included in print in the PDF; the text is open source, and interested users can gain access to the original source files on GitHub; the style of the text requires students to be active learners — there are very few worked examples in the text, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems, and developing understanding of key calculus concepts; each section begins with motivating questions, a brief introduction, and a preview activity, all of which are designed to be read and completed prior to class; following the WeBWorK exercises in each section, there are several challenging problems that require students to connect key ideas and write to communicate their understanding. For more information, see the author\u27s website and blog.https://scholarworks.gvsu.edu/books/1018/thumbnail.jp

Active Calculus Multivariable: 2018 Edition

Active Calculus Multivariable is the continuation of Active Calculus to multivariable functions. The Active Calculus texts are different from most existing calculus texts in at least the following ways: the texts are freely readable online in HTML format (new in this version of Active Calculus Multivariable) and are also available for in PDF; in the electronic format, graphics are in full color; the texts are open source, and interested instructors can gain access to the original source files on GitHub; the style of the texts requires students to be active learners — there are very few worked examples in the texts, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems, and developing understanding of key calculus concepts; each section begins with motivating questions, a brief introduction, and a preview activity, all of which are designed to be read and completed prior to class; each section contains a collection of WeBWorK exercises (with solutions available in the HTML version, new in this version) followed by several challenging problems that require students to connect key ideas and write to communicate their understanding.https://scholarworks.gvsu.edu/books/1019/thumbnail.jp

Active Calculus 1.0

Active Calculus is different from most existing calculus texts in at least the following ways: the text is free for download by students and instructors in .pdf format; in the electronic format, graphics are in full color and there are live html links to java applets; the text is open source, and interested instructors can gain access to the original source files upon request; the style of the text requires students to be active learners — there are very few worked examples in the text, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems, and developing understanding of key calculus concepts; each section begins with motivating questions, a brief introduction, and a preview activity, all of which are designed to be read and completed prior to class; the exercises are few in number and challenging in nature.https://scholarworks.gvsu.edu/books/1010/thumbnail.jp

On the spectral radius of a positive operator

This dissertation surveys results on the spectral radius, r(A), of an operator A on a Banach space, under the hypothesis that A is positive. A primary goal of the work is to present an accessible introduction and overview to the spectral properties of a positive operator to readers from outside the field. An historical overview is presented first, taking the reader from Perron\u27s Theorem for positive matrices to the Krei n-Rutman Theorem for compact, strongly positive operators on a partially ordered Banach space, then moving on to a preview of related results on r(A) which followed. Chapter 2 is devoted to a careful examination of ordered Banach spaces, and synthesizes a variety of properties of cones, positive functionals, and positive operators from across the research literature. Next follows a development of the existence theory of positive eigenvalues and positive eigenvectors of positive operators. These theorems are presented in order of increasing stringency of hypotheses, reflect recent and simpler proofs, and are used to prove the culminating result of Chapter 3, the Krei n-Rutman Theorem. The remaining two chapters focus on finding bounds and numerical approximations for r(A). Chapter 4 includes propositions on pointwise bounds for r(A), an alternate characterization of r(A) due to Karlin, the monontonicity results of Marek, Schaefer\u27s work on positive invertibility and its links to r(A), and more. In the final chapter, a generalized power method is introduced and shown to converge, extending a result of Krasnoselskii. Furthermore, a specific iterative technique for approximating the spectral radius of an integral operator is developed and analyzed in light of results of Kershaw and Bauer. Finally, a numerical technique of Cryer is presented, and the text closes with a selection of numerical examples

Differential Equations with Linear Algebra

Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach,