In this article, we explore patterns of R-2 across data sets to reveal a fundamental equivalence. If any set of bivariate data is rotated, the R-2 value changes and defines a curve as a function of the rotation angle theta. We show that for any data set whose rotated R-2 curve has the same maximum value, the entire curve is identical up to a shift in the rotation angle. We also find that a recently introduced measure of linearity, Q(2), provides a definitive relation between these data sets, with equal Q(2) values resulting in identically rotated R-2 curves. Finally, we also show that we can consider R-2 as the combination of two components, the linearity of the data and a rotation angle. While rotating data or axes for non spatial data loses the specific meaning of the variables, these results may provide additional understanding of the relationships within the data and can reveal similarities between data sets, the role of sensitivity in correlation, and the nature of correlation itself.

This book provides readers with techniques to apply the power of calculus in order to analyze real problems from data, rather than formulas.

Employing the Remainder Theorem, analytic and graphical tools, and dynamic applets, this paper investigates when the graphs of rational functions intersect horizontal, oblique, and curved asymptotes. Additionally, we consider how the respective remainder function perturbs the asymptotic function to producethe rational function. Dynamic applets assist in the development of the ideas herein. Finally, this paper endswith a significant number of student investigations.

For high school and college instructors and students, this paper connects number systems, field axioms, and polynomials. It also considers other properties such as cardinality, density, subset, and superset relationships. Additional aspects of this paper include gains and losses through sequences of number systems. The paper ends with a great number of activities for classroom use.

What is numerical integration? Numerical integration or (numerical) quadrature is the calculation of the value of a definite integral using numerical formulas, not the fundamental theorem with antiderivatives.

Most any students can explain the meaning of ab, for a∈R and for b∈Z. And some students may be able to explain the meaning of (a+bi)c, for a,b∈R and for c∈Z. However, few may be able to explain adi or a(c+di) let alone (a+bi)(c+di) for a,b,c,d∈R. Through both mathematical explanation and reader investigations via a dynamic graphing applet, we will investigate the meaning of (a+bi)(c+di). Additionally, we provide some course-appropriate investigations which can be used by instructors in various grades 11–16 courses.

Since there are no general solutions to polynomials of degree higher than four, high school and college students only infrequently investigate quintic polynomials. Additionally, although students commonly investigate real roots of polynomials, only infrequently are complex roots - and, more particularly, the location of complex roots - investigated. This paper considers features of graphs of quintic polynomials and uses analytic constructions to locate the functions' complex roots. Throughout, hyperlinked dynamic applets are provided for the student reader to experientially participate in the paper. This paper is an extension to other investigations regarding locating complex roots (Bauldry, Bosse, & Otey, 2017).

The standard technique taught in calculus courses for partial fraction expansions uses undetermined coefficients to generate a system of linear equations; we present a derivative-based technique that calculus and differential equations instructors can use to reinforce connections to calculus. Simple algebra shows that we can use the derivative to calculate coefficients, reinforcing calculus connections. For irreducible quadratic factors, the simple derivative formula with complex arithmetic gives the coefficients, helping students gain facility with complex numbers. For higher multiplicity factors, we can use higher derivative formulas taken from the Laurent expansion. We conclude with examples of using the technique in different courses, highlighting precalculus and calculus classes.

Avoiding complex geometric and analytic constructions, this paper considers techniques for visualizing the location of complex roots of quadratic, cubic, and quartic real polynomial functions. This provides teachers and students of mathematics with a better understanding of the nature of these functions and their respective real and complex roots.

This chapter contains sections titled: Lebesgue Measure and Measurable Sets The Lebesgue Integral Measure, Integral, and Convergence Littlewood's Three Principles

This chapter contains sections titled: Historical Writing Projects Induction Proofs: Summations, Inequalities, and Divisibility Series Rearrangements Newton and the Binomial Theorem Symmetric Sums of Logarithms Logical Equivalence: Completeness of the Real Numbers Vitali's Nonmeasurable Set Sources for Real Analysis Projects Sources for Projects for Calculus Students

Free Access Appendix B: A Brief Calculus Chronology William C. Bauldry, William C. BauldrySearch for more papers by this author Book Author(s):William C. Bauldry, William C. BauldrySearch for more papers by this author First published: 25 June 2009 https://doi.org/10.1002/9781118164419.app2 AboutPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShareShare a linkShare onFacebookTwitterLinked InRedditWechat Introduction to Real Analysis: An Educational Approach RelatedInformation

Preface. Acknowledgments. 1 Elementary Calculus. 1.1 Preliminary Concepts. 1.2 Limits and Continuity. 1.3 Differentiation. 1.4 Integration. 1.5 Sequences and Series of Constants. 1.6 Power Series and Taylor Series. Summary. Exercises. Interlude: Fermat, Descartes, and theTangent Problem. 2 Introduction to Real Analysis. 2.1 Basic Topology of the Real Numbers. 2.2 Limits and Continuity. 2.3 Differentiation. 2.4 Riemann and Riemann-Stieltjes Integration. 2.5 Sequences, Series, and Convergence Tests. 2.6 Pointwise and Uniform Convergence. Summary. Exercises. Interlude: Euler and the "Basel Problem". 3 A Brief Introduction to Lebesgue Theory. 3.1 Lebesgue Measure and Measurable Sets. 3.2 The Lebesgue Integral. 3.3 Measure, Integral, and Convergence. 3.4 Littlewood's Three Principles. Summary. Exercises. Interlude: The Set of Rational Numbers isVery Large andVery Small. 4 Special Topics. 4.1 Modeling with Logistic Functions-Numerical Derivatives. 4.2 Numerical Quadrature. 4.3 Fourier Series. 4.4 Special Functions-The Gamma Function. 4.5 Calculus Without Limits: Differential Algebra. Summary. Exercises. Appendix A: Definitions and Theorems of Elementary Real Analysis. A.1 Limits. A.2 Continuity. A.3 The Derivative. A.4 Riemann Integration. A.5 Riemann-Stieltjes Integration. A.6 Sequences and Series of Constants. A.7 Sequences and Series of Functions. Appendix B: A Very Brief Calculus Chronology. Appendix C: Projects in Real Analysis. C.1 Historical Writing Projects. C.2 Induction Proofs: Summations, Inequalities, and Divisibility. C.3 Series Rearrangements. C.4 Newton and the Binomial Theorem. C.5 Symmetric Sums of Logarithms. C.6 Logical Equivalence: Completeness of the Real Numbers. C.7 Vitali's Nonmeasurable Set. C.8 Sources for Real Analysis Projects. C.9 Sources for Projects for Calculus Students. Bibliography. Index.

From the Publisher:In this easy-to-understand manual, you'll learn to use the Maple computer algebra system as a problem-solving tool in exploring such topics as slopes of functions, graphs of polynomials, rational functions, Riemann sums, Taylor polynomials and convergence, polar coordinates, splines, and much more. In each project, you'll find a statement of requisite Maple commands, a tutorial section, a description of the investigation, a statement that characterizes the "minimal project report" required, and project Extensions that are more challenging and require more thought and more time to complete.

At the end of a chapter on Newton's Law of Cooling, Nagle and Saff's Fundamentals of Differential Equations and Boundary Value Problems (NS, pg 107) poses the following problem: Two friends sit down to talk and enjoy a cup of coffee. When the coffee is served, the impatient friend immediately adds a teaspoon of cream to his coffee. The relaxed friend waits 5 minutes before adding a teaspoon of cream (which has been kept at a constant temperature). The two now begin to drink their coffee. Who has the hotter coffee? Assume that the cream is cooler than the air and use Newton's Law of Cooling. We will demonstrate a classroom experiment of this problem using a TI-CBL TM unit, hand-held technology that comes with temperature and other probes. We will explain how to import the data into Maple TM , and look at an interactive Maple demo created to complement the experiment in which students must answer questions to progress. We will then discuss problems that arose with using the CBL Unit in this experiment, such as the fact that in the first experimental run, the impatient friend had cooler coffee, while in the second experimental run the relaxed friend had cooler coffee. Using results obtained from the simplified experiment, we will look at an improved attempt at a classroom experiment. For Maple Demos and other information, see http://www.mathsci.appstate.edu/∼sjg/ICTCM13/