Mathematics Commons^™

Minimizing Travel Time Through Multiple Media With Various Borders, Tonja Miick

Masters Theses & Specialist Projects

This thesis consists of two main chapters along with an introduction and
conclusion. In the introduction, we address the inspiration for the thesis, which
originates in a common calculus problem wherein travel time is minimized across two media separated by a single, straight boundary line. We then discuss the correlation of this problem with physics via Snells Law. The first core chapter takes this idea and develops it to include the concept of two media with a circular border. To make the problem easier to discuss, we talk about it in terms of running and swimming speeds. We first address …

Comparing The Impact Of Traditional And Modeling College Algebra Courses On Student Performance In Survey Of Calculus, Jerry West

Graduate Theses and Dissertations

Students in higher education deserve opportunities to succeed and learning environments which maximize success. Mathematics courses can create a barrier for success for some students. College algebra is a course that serves as a gateway to required courses in many bachelor's degree programs. The content in college algebra should serve to maximize students' potential in utilizing mathematics and gaining skills required in subsequent math-based courses when necessary. The Committee for Undergraduate Programs in Mathematics has gone through extensive work to help mathematics departments reform their college algebra courses in order to help students gain interest in the utilization of mathematics …

Zero-Bounded Limits As A Special Case Of The Squeeze Theorem For Evaluating Single-Variable And Multivariable Limits, Eleftherios Gkioulekas

School of Mathematical & Statistical Sciences Faculty Publications

Many limits, typically taught as examples of applying the ‘squeeze’ theorem, can be evaluated more easily using the proposed zero-bounded limit theorem. The theorem applies to functions defined as a product of a factor going to zero and a factor that remains bounded in some neighborhood of the limit. This technique is immensely useful for both single-variable limits and multidimensional limits. A comprehensive treatment of multidimensional limits and continuity is also outlined.

On Equivalent Characterizations Of Convexity Of Functions, Eleftherios Gkioulekas

School of Mathematical & Statistical Sciences Faculty Publications

A detailed development of the theory of convex functions, not often found in complete form in most textbooks, is given. We adopt the strict secant line definition as the definitive definition of convexity. We then show that for differentiable functions, this definition becomes logically equivalent with the first derivative monotonicity definition and the tangent line definition. Consequently, for differentiable functions, all three characterizations are logically equivalent.

Validation Study: A Case Study Of Calculus 1 (Math 1210), Abibat Adebisi Lasisi

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

This report discusses the process of assessing the validity of a test. We provide an analysis of the validation study of a test administered to the first semester calculus students at Utah State University in the Fall 2011 semester. The results of our analysis suggest that the test has some measurement relevance. This implies that the subject contents of the test correlate with the syllabus of the course. Thus, the measurement has some learning level relevance. Our analysis also revealed that the measurement has a high reliability coefficient which signifies some internal consistency of the mea- surement. The relevancy and …

Motivation For Achievement And Attitudes Toward Mathematics Instruction In A Required Calculus Course At The Norwegian University Of Science And Technology, Donna Sundre, Carol Barry, Vidar Gynnild, Erin Tangen Ostgard

Numeracy

This study from the Norwegian University of Science and Technology (NTNU) examines students’ learning goals and attitudes toward mathematics in a first-year calculus course in undergraduate engineering education. Achievement motivation research using the Achievement Goal Questionnaire (AGQ) is advanced from current literature with two additions: (1) a course specific context using introductory college calculus students, and (2) participation of Norwegian students.

Pre- and posttest measures of attitudes indicate that students do change learning goals over time, unfortunately opposite to the instructors’ aspirations. A significant increase in “Mastery Avoidance” and “Work Avoidance” was accompanied with a drop in “Mastery Approach” and …

Quantum Stochastic Flows: Trotter Product Formula, Dilations And Quantum Brownian Motion., Biswarup Das Dr.

Doctoral Theses

Motivated by the major role played by probabilistic models in many areas of science, several quantum (i.e. non-commutative) generalizations of classical probability have been attempted by a number of mathematicians. The pioneering works of K.R. Parthasarathy, L. Accardi, R.L. Hudson, P.A. Meyer and others led to the development of one such non-commutative model called ‘quantum probability’ which has a very rich theory of quantum stochastic calculus a la Hudson and Parthasarathy. Within the framework of quantum stochastic calculus, the ‘grand design’ that engages us is the canonical construction and study of ∗-homomorphic flows (jt)t≥0 on a given C ∗ or …

Go Figure: Calculus Students' Use Of Figures And Graphs In Technical Report Writing, Thomas J. Pfaff, Michael Rogers, Ali Erkan, Jason G. Hamilton

Numeracy

Understanding how to read and use graphs to communicate scientific and mathematical information is critical for STEM majors, as well as an important part of quantitative literacy. Our study suggests that first-semester calculus students do not know how to use graphs in a technical report without explicit instruction. Although not a surprising result, it leaves us wondering about when such skills are developed, and if calculus I is a place to start. Our work is now exploring the potential benefit on students' use of graphs by having them formally evaluate other students' reports.

Calculus, Biology And Medicine: A Case Study In Quantitative Literacy For Science Students, Kim Rheinlander, Dorothy Wallace

Numeracy

This paper describes a course designed to enhance the numeracy of biology and pre-medical students. The course introduces students with the background of one semester of calculus to systems of nonlinear ordinary differential equations as they appear in the mathematical biology literature. Evaluation of the course showed increased enjoyment and confidence in doing mathematics, and an increased appreciation of the utility of mathematics to science. Students who complete this course are better able to read the research literature in mathematical biology and carry out research problems of their own.

Teaching Calculus With Wolfram Alpha, Andrew Lang

College of Science and Engineering Faculty Research and Scholarship

This article describes the benefits and drawbacks of using Wolfram|Alpha as the platform for teaching calculus concepts in the lab setting. It is a result of our experiences designing and creating an entirely new set of labs using Wolfram|Alpha. We present the reasoning behind our transition from using a standard computer algebra system (CAS) to Wolfram|Alpha in our differential and integral calculus labs, together with the positive results from our experience. We also discuss the current limitations of Wolfram|Alpha, including a discussion on why we still use a CAS for our multivariate calculus labs.

Measuring Resource Inequality: The Gini Coefficient, Michael T. Catalano, Tanya L. Leise, Thomas J. Pfaff

Numeracy

This paper stems from work done by the authors at the Mathematics for Social Justice Workshop held in June of 2007 at Middlebury College. We provide a description of the Gini coefficient and some discussion of how it can be used to promote quantitative literacy skills in mathematics courses. The Gini Coefficient was introduced in 1921 by Italian statistician Corrado Gini as a measure of inequality. It is defined as twice the area between two curves. One, the Lorenz curve for a given population with respect to a given resource, represents the cumulative percentage of the resource as a function …

Biology In Mathematics At The University Of Richmond, Lester Caudill

Department of Math & Statistics Faculty Publications

In an effort to meet the needs of science students for modeling skills, three new courses have been created at the University of Richmond: Scientific Calculus I and II, and Mathematical Models in Biology and Medicine. The courses are described, and lessons learned and future directions are discussed.

Students' Reasoning About The Concept Of Limit In The Context Of Reinventing The Formal Definition, Craig Alan Swinyard

Dissertations and Theses

Many researchers (Artigue, 2000; Bezuidenhout, 2001; Cornu, 1991; Dorier, 1995) have noted the vital role limit plays as a foundational concept in analysis. The vast majority of topics encountered in calculus and undergraduate analysis are built upon understanding the concept of limit and being able to work flexibly with its formal definition (Bezuidenhout, 2001). The purpose of this study was to: (1) Develop insight into students' reasoning about limit in relation to their engagement in instruction designed to support their reinventing the formal definition of limit, and; (2) Inform the design of principled instruction that might support students' attempts to …

A Closer Look At The Crease Length Problem, Sean F. Ellermeyer

Faculty Articles

An optimization problem that appears as an exercise in most modern calculus textbooks is the crease length problem. Here, Ellermeyer provides a solution of the general crease length problem in which all possible foldings of a corner to the opposite edge are taken into account. One of his findings will be that the minimum crease length is never produced by a Case 2 fold and hence that the general crease length problem always yields a different minimum than the constrained problem that is treated in the textbooks. He discovers a criterion that determines which foldings must be performed in order …

Quantum Stochastic Dilation Of A Class Of Quantum Dynamical Semigroups And Quantum Random Walks., Lingaraj Sahu Dr.

Doctoral Theses

No abstract provided.

Use Of Pen-Based Technology In Calculus Courses, John R. Hubbard

Department of Math & Statistics Faculty Publications

The author and his students used Tablet computers in Calculus I and Calculus II classes, providing students with dynamic digital transcripts that they could replay at their convenience. He and his students agreed that these graphic replays provide an effective alternative to the static explanations found in textbooks and in traditional course notes. Two specific examples are given in this paper.

Mathematics Placement Test: Helping Students Succeed, Norma Rueda, Carole Sokolowski

Mathematics Faculty Publications

A study was conducted at Merrimack College in Massachusetts to compare the grades of students who took the recommended course as determined by their mathematics placement exam score and those who did not follow this recommendation. The goal was to decide whether the mathematics placement exam used at Merrimack College was effective in placing students in the appropriate mathematics class. During five years, first-year students who took a mathematics course in the fall semester were categorized into four groups: those who took the recommended course, those who took an easier course than recommended, those who took a course more difficult …

Spectral Triples And Metric Aspects Of Geometry On Some Noncommutative Spaces., Partha Sarathi Chakraborty Dr.

Doctoral Theses

Quantization of mathematical theories is now more than half a century old idea in mathe- matics. It goes back to Gelfand-Naimarks seminal paper [37] in 1943. As the name suggests noncommutative geometry is the quantization" of differential geometry. It is the study of noncommutative algebras as if they were algebras of functions on spaces like the commuta- tive algebras associated to affine algebraic varieties, smooth manifolds, topological spaces. One can trace its roots in the Gelfand-Naimark theorems (1943, 37]). In modern terminol- ogy their theorem says there is an antiequivalence between the category of (locally) compact Hausdorff spaces and (proper, …

Nsu Innovative Teaching In Math 2002, Nova Southeastern University

Abraham S. Fischler College of Education and School of Criminal Justice College Archive

No abstract provided.

Whose Limit Is It Anyway?, Joseph E. Borzellino

Mathematics

In a tongue-in-cheek manner, we investigate the notion of limit. We illustrate some of its shortcomings and show that addressing these shortcomings can often lead to unexpected consequences.

The Fermat Point And The Steiner Problem, J. N. Boyd, P. N. Raychowdhury

Virginia Journal of Science

We revisit the convex coordinates of the Fermat point of a triangle. We have already computed these convex coordinates in a general setting. In this note, we obtain the coordinates in the context of the Steiner problem. Thereafter, we pursue calculations suggested by the problem.

Quantum Stochastic Dilation Of Completely Positive Semigroups And Flows., Debashish Goswami Dr.

Doctoral Theses

The central theme of the present thesia is quantum stochastic dilation af semigroupe of completely panitive mapa on operator algebran. It is the sim of all mathemati- cal, or even all scientific theorics, to understand a given class of objects through a tanonical and simpler subclass of it. For example, abstract C"-algebras are studied through their conerete realisation as elgebra of operators, contractions on a Hilbert space by unitaries. Hilbert modules by the factorissble ones, to mention anly a few. In most af these caes, a general object of the relavant class is sociated with a canonical candidate of the …

A Tour Of Mistakes, Paul H. Edelman

Vanderbilt Law School Faculty Publications

In these pages, Steven Lubet recently reviewed A Tour of the Calculus, by David Berlinski. Inspired by both the beauty of calculus and Berlinski's description of it, Lubet waxes poetic on the many parallels between the law and calculus. It is completely understandable--even admirable that one might be led to ruminations on the relationship between calculus and one's own discipline. There is little doubt that the subject of calculus stands as one of the great intellectual feats of Western thought. It has had profound implications for physics, engineering, economics and many other disciplines-so why not law? Alas, these philosophical musings …

Was Newton's Calculus A Dead End? The Continental Influence Of Maclaurin's Treatise Of Fluxions, Judith V. Grabiner

Pitzer Faculty Publications and Research

We will show that Maclaurin's Treatise of Fluxions did develop important ideas and techniques and that it did influence the mainstream of mathematics. The Newtonian tradition in calculus did not come to an end in Maclaurin's Britain. Instead, Maclaurin's Treatise served to transmit Newtonian ideas in calculus, improved and expanded, to the Continent. We will look at what these ideas were, what Maclaurin did with them, and what happened to this work afterwards. Then, we will ask what by then should be an interesting question: why has Maclaurin's role been so consistently underrated? Thse questions will involve general matters of …

Wheels On Wheels On Wheels-Surprising Symmetry, Frank A. Farris

Mathematics and Computer Science

While designing a computer laboratory exercise for my calculus students, I happened to sketch the curve defined by this vector equation: (x, y) = (cos(t), sin(t)) + 1/2(cos(7t), sin(7t)) + 1/3(sin(17t), cos(17t)). I was thinking of the curve traced by a particle on a wheel mounted on a wheel mounted on a wheel, each turning at a different rate. The first term represents the largest wheel, of radius 1, turning counter-clockwise at one radian per second. The second term represents a smaller wheel centered at the edge of the first, turning 7 times as fast. The third term is for …

Some Problems In Joint Spectral Theory., Tirthankar Bhattacharya Dr.

Doctoral Theses

No abstract provided.

Metode De Calcul În Analiza Matematică, Florentin Smarandache, C. Dumitrescu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

An Algebraic Approach To Derivatives, Julia Lee Roman

Theses & Dissertations

The purpose of this thesis is to relate a unique method of dealing with the concept of the derivative of a function. The traditional approach to teaching calculus introduces the idea of the limit of a function early in the course as evidenced in the Essential Elements of the State Board of Education of Texas which places the limit concept as the second essential element of calculus in Texas high schools (Texas Education Agency (TEA) Essential Elements 1991). The "concepts and skills associated with the derivative" (TEA Essential Elements 1991) follow immediately after the introduction of the limit of a …

Nonstandard Analysis Based Calculus, Kathleen Renae Gibson

Theses Digitization Project

In the first part of the project the elementary development of an extended number system called Hyperreals is discussed. The second half of this project develops the basics of Nonstandard Analysis, including the theory of ultrafilters, and the formal construction of the Hyperreals.

What They Didn’T Tell Me About Calculus And The Computer, Melvin Henriksen

All HMC Faculty Publications and Research

My first attempt at using computers as an aid to teaching calculus began in 1966 and culminated with the publication in 1971 of Single Variahle Calculus, written jointly with Milton Lees. Everyone used a mainframe and punched cards and few knew how to program. Computer use was limited to supplementary exercises that could be done today with a hand-held programmable calculator. A constructive sequential approach to limits and elementary numerical analysis were emphasized. The absence of programming displeased computer scientists, but it was too avant garde for all but a tiny minority of teachers of calculus. Attempts by others …