DOCSLIB.ORG

College Students' Intuitive Understanding of the Concept Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
College Students' Intuitive Understanding of the Concept Of COLLEGE STUDENTS’ INTUITIVE UNDERSTANDING OF THE CONCEPT OF LIMIT AND THEIR LEVEL OF REVERSE THINKING DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University By Kyeong Hah Roh, Ph. D. The Ohio State University 2005 Dissertation Committee: Approved by Professor Sigrid Wagner, Advisor _________________________________ Advisor Professor Douglas Owens School of Teaching and Learning Professor Jeffery Connor Professor Hea-Jin Lee Copyright by Kyeong Hah Roh, 2005 ABSTRACT This thesis addresses the relationship of students’ intuitive understanding of the limit of a sequence to their reversibility, which is an ability to reverse the order between ε and N as required in the rigorous definition of limit. The subjects of this research were students who had not had any experience with rigorous proofs using the ε − N definition of a limit. Eleven students completed a series of 1-hour semi-structured, task- based interviews once a week for 5 weeks. Monotone bounded, unbounded, constant, oscillating convergent, or oscillating divergent sequences were tested. Students represented each sequence numerically as well as graphically in determining convergence of the sequence. Students also used tools, called ε − strips, specially developed for this study to explore the ε − N relation in defining limits through hands-on activities with the ε − strips. Finally, students were presented the following ε − strip definitions, and were asked to evaluate the propriety of the definitions as statements of the limit of a sequence. ε − Strip Definition A: A certain value L is a limit of a sequence when infinitely many points on the graph of the sequence are covered by any epsilon strip as long as the epsilon strip covers L. ε − Strip Definition B: A certain value L is a limit of a sequence when only finitely many points on the graph of the sequence are NOT covered by any epsilon strip as long as the epsilon strip covers L. ii It was found that students’ understanding of the definition of the limit of a sequence was associated with not only their conception of limit but also their level of reversibility. In addition, there was improvement in students’ reversibility and/or their conception of limit through the ε − strip activity, even though there was no procedure for indicating students’ errors, correcting students’ misconceptions about limit, or confirming the propriety of the ε − strip definitions to students during the interviews. This study implies that theε − strip activity, combined with the various types of sequences used in this teaching experiment, is one instructional method for helping students develop and accommodate their conception of the limit of a sequence. iii ACKNOWLEDGMENTS I would like to thank a large number of people who provided interests, encouragement, and support to my study thus made this dissertation possible. First of all, I would like to gratefully acknowledge the enthusiastic supervision of my advisor, Dr. Sigrid Wagner. Always, she listened to my ideas attentively and provided sound advice to my research through numerous stimulating discussions. In particular, Sigrid's optimistic attitude and strong faith in my project gave me big courage to endure hard time in my research. My sincere and warm thanks also go to my dissertation committee members, Dr. Douglas Owens, Dr. Jeffery Connor, and Dr. Hea-Jin Lee. I would like to thank Dr. Owens always his time for fruitful discussions, general advice, and his thoughtful concern about my career from the beginning of my study at the Ohio State University. Numerous have been those stimulating discussions and general advice for both my teaching and my research in which I have been helped by Dr. Connor. Dr. Lee is thanked for managing to read my dissertation so thoroughly, and for her helpful advice. I would like to express my sincere appreciation for support and hospitality of School of Teaching and Learning at the Ohio State University. I would also like to thank all faculty and staffs, particularly Dr. Patricia Brosnan who sincerely encouraged me to acclimate in new circumstance during the first year of my study in the Ohio State University. I would like to thank Dr. David Haury, Robert Day, and Claudette Price in iv ERIC Clearinghouse for Science, Mathematics, and Environmental Education for their hospitality to support helpful equipments. I wish to express my deep appreciation to Department of Mathematics at Ohio University, for their interests, support, and help with setting up equipment for my research. Especially, I would like to say a big thank you to all students and instructors who participated in my research for this dissertation. I am grateful to all my friends, Eun Jung Part, Jennifer Kaminski, and other my colleague for their constant interest and helpful discussion.. Finally, I am forever indebted to my parents for their constant love, endless patience and encouragement which enable me to stand all difficulties. v VITA September 1, 1970 . .Born - Kwangju, Korea. 1993. B.S. Mathematics Education, Ewha Womans University, Seoul, Korea. 1995 . M.S. Mathematics, Seoul National University, Seoul, Korea. 2000 . .Ph.D. Mathematics, Seoul National University, Seoul, Korea. 2000 – 2001 . .Senior Researcher Ewha Educational Research Institute, Ewha Womans University, Seoul, Korea. 2002 – 2003. Graduate Teaching and Research Associate, ERIC Clearinghouse for Science, Mathematics, and Education, The Ohio State University. 2003 – 2005. Visiting Assistant Professor, Department of Mathematics, Ohio University. PUBLICATIONS 1. Roh, K. (2004). The role of reverse thinking in the contexts of limit. In D. E. McDoull & J. A. Ross(Eds.), Proceedings of the twenty-sixth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 191-192). Ontario, ON; University of Toronto. 2. Roh, K. et al. (2003). Curriculum development for mathematics and information science in a knowledge based society, Seoul, Korea; Ewha Womans University Publisher. vi 3. Roh, K. (2003). Problem based learning in mathematics education. ERIC Digest. EDO-SE-03-07. Columbus, OH; ERIC Clearinghouse for Science, Mathematics, and Environmental Education. 4. Roh, K. (2002). Inquiry by mathematics. In L. A. Milbourne (Ed.), The connected teacher’s companion to inquiry in science and mathematics (pp. 25-30). SE067123. Columbus, OH; ERIC Clearinghouse for Science, Mathematics, and Environmental Education. 5. Haury, D. L., Lee, H., Lorson, M., Roh, K., & Sonmez, D. (2002). The connected teacher’s companion to problem solving in science and mathematics. SE067122. Columbus, OH; ERIC Clearinghouse for Science, Mathematics, and Environmental Education. 6. Choi, H. Han, C. Moon, H., Roh, K., & Wee. N. (1999). Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves, Computer-Aided Design, 31, 59-72. 7. Choi, H. Han, C. Moon, H., Roh, K., & Wee. N. (1998). A new algorithm for offset of plain domain via MAT, Differential/topological techniques in geometric modeling and processing ’98, 161-185. FIELDS OF STUDY Major Field: Education vii TABLE OF CONTENTS Abstract…………………………………………………………………………… ii Acknowledgments………………………………………………………………… iv Vita……………………………………………………………………………….. vi List of Figure…………………………………………………………………….... x List of Table……………………………………………………………………….. xiii Chapter 1 STATEMENT OF THE PROBLEM……………………………… 1 The Problem………………………………………………………. 1 Research Questions……………………………………………….. 11 Theoretical Framework…………………………………………… 12 Chapter 2 LITERATURE REVIEW…………………………………………. 24 Readiness for Instruction on Limits………………………………. 24 Misconceptions About Limits…………………………………….. 29 Cognitive Obstacles Influencing Misconceptions About Limits…. 35 Instructional Approaches…………………………………………. 42 Conclusion………………………………………………………... 47 Chapter 3 RESEARCH METHODOLOGY…………………………………. 49 Preparation of the Research………………………………………. 49 Data Collection Methods…………………………………………. 54 Data Analysis……………………………………………………... 72 viii Chapter 4 RESULTS…………………………………………………………. 79 Summary of Results by Interview Task........................................... 79 Students’ Conception of the Convergence of a Sequence………… 91 Students’ Conception of the Limit Value of a Sequence………….. 120 Relationship Between Concept of Limit and Definition of Convergence………………………………………………………. 134 Students’ Reversibility in the Context of the Limit of a Sequence.. 136 Chapter 5 CONCLUDING REMARKS AND DISCUSSION...…………….. 183 Results in Terms of Research Questions………………………….. 185 Implications of the Research for Teaching and Learning ………... 201 Future Research……………………………………………………206 Conclusion………………………………………………………... 210 References …………………………………………………………………….. 211 Appendices …………………………………………………………………….. 218 Appendix A. Description of Research Project….…………….…... 219 Appendix B. Student Consent Form…………………………….... 221 Appendix C. Survey ……….……………………………………... 222 Appendix D. Pretest ………..…………………………………….. 228 Appendix E. A List of Task Activities & Interview Protocols……. 230 Appendix F. Posttest ……………………………………………… 254 ix LIST OF FIGURES Figure Page 1.1 Formal cognition versus intuitive cognition…………………………...…... 15 1.2 Formal cognition versus two kinds of intuition: Primary intuition and 19 secondary intuition………………………………………………………… 3.1 Procedure of the teaching experiment and data analysis for they study….... 53 3.2 Mirror activity…………………………………………………………….. 59 3.3 An example of a symbolic representation of a sequence:
Load more

Recommended publications
  • Math 137 Calculus 1 for Honours Mathematics Course Notes
    Math 137 Calculus 1 for Honours Mathematics Course Notes Barbara A. Forrest and Brian E. Forrest Version 1.61 Copyright c Barbara A. Forrest and Brian E. Forrest. All rights reserved. August 1, 2021 All rights, including copyright and images in the content of these course notes, are owned by the course authors Barbara Forrest and Brian Forrest. By accessing these course notes, you agree that you may only use the content for your own personal, non-commercial use. You are not permitted to copy, transmit, adapt, or change in any way the content of these course notes for any other purpose whatsoever without the prior written permission of the course authors. Author Contact Information: Barbara Forrest ([email protected]) Brian Forrest ([email protected]) i QUICK REFERENCE PAGE 1 Right Angle Trigonometry opposite ad jacent opposite sin θ = hypotenuse cos θ = hypotenuse tan θ = ad jacent 1 1 1 csc θ = sin θ sec θ = cos θ cot θ = tan θ Radians Definition of Sine and Cosine The angle θ in For any θ, cos θ and sin θ are radians equals the defined to be the x− and y− length of the directed coordinates of the point P on the arc BP, taken positive unit circle such that the radius counter-clockwise and OP makes an angle of θ radians negative clockwise. with the positive x− axis. Thus Thus, π radians = 180◦ sin θ = AP, and cos θ = OA. 180 or 1 rad = π . The Unit Circle ii QUICK REFERENCE PAGE 2 Trigonometric Identities Pythagorean cos2 θ + sin2 θ = 1 Identity Range −1 ≤ cos θ ≤ 1 −1 ≤ sin θ ≤ 1 Periodicity cos(θ ± 2π) = cos θ sin(θ ± 2π) = sin
    [Show full text]
  • G-Convergence and Homogenization of Some Sequences of Monotone Differential Operators
    Thesis for the degree of Doctor of Philosophy Östersund 2009 G-CONVERGENCE AND HOMOGENIZATION OF SOME SEQUENCES OF MONOTONE DIFFERENTIAL OPERATORS Liselott Flodén Supervisors: Associate Professor Anders Holmbom, Mid Sweden University Professor Nils Svanstedt, Göteborg University Professor Mårten Gulliksson, Mid Sweden University Department of Engineering and Sustainable Development Mid Sweden University, SE‐831 25 Östersund, Sweden ISSN 1652‐893X, Mid Sweden University Doctoral Thesis 70 ISBN 978‐91‐86073‐36‐7 i Akademisk avhandling som med tillstånd av Mittuniversitetet framläggs till offentlig granskning för avläggande av filosofie doktorsexamen onsdagen den 3 juni 2009, klockan 10.00 i sal Q221, Mittuniversitetet, Östersund. Seminariet kommer att hållas på svenska. G-CONVERGENCE AND HOMOGENIZATION OF SOME SEQUENCES OF MONOTONE DIFFERENTIAL OPERATORS Liselott Flodén © Liselott Flodén, 2009 Department of Engineering and Sustainable Development Mid Sweden University, SE‐831 25 Östersund Sweden Telephone: +46 (0)771‐97 50 00 Printed by Kopieringen Mittuniversitetet, Sundsvall, Sweden, 2009 ii Tothememoryofmyfather G-convergence and Homogenization of some Sequences of Monotone Differential Operators Liselott Flodén Department of Engineering and Sustainable Development Mid Sweden University, SE-831 25 Östersund, Sweden Abstract This thesis mainly deals with questions concerning the convergence of some sequences of elliptic and parabolic linear and non-linear oper- ators by means of G-convergence and homogenization. In particular, we study operators with oscillations in several spatial and temporal scales. Our main tools are multiscale techniques, developed from the method of two-scale convergence and adapted to the problems stud- ied. For certain classes of parabolic equations we distinguish different cases of homogenization for different relations between the frequen- cies of oscillations in space and time by means of different sets of local problems.
    [Show full text]
  • Generalizations of the Riemann Integral: an Investigation of the Henstock Integral
    Generalizations of the Riemann Integral: An Investigation of the Henstock Integral Jonathan Wells May 15, 2011 Abstract The Henstock integral, a generalization of the Riemann integral that makes use of the δ-fine tagged partition, is studied. We first consider Lebesgue’s Criterion for Riemann Integrability, which states that a func- tion is Riemann integrable if and only if it is bounded and continuous almost everywhere, before investigating several theoretical shortcomings of the Riemann integral. Despite the inverse relationship between integra- tion and differentiation given by the Fundamental Theorem of Calculus, we find that not every derivative is Riemann integrable. We also find that the strong condition of uniform convergence must be applied to guarantee that the limit of a sequence of Riemann integrable functions remains in- tegrable. However, by slightly altering the way that tagged partitions are formed, we are able to construct a definition for the integral that allows for the integration of a much wider class of functions. We investigate sev- eral properties of this generalized Riemann integral. We also demonstrate that every derivative is Henstock integrable, and that the much looser requirements of the Monotone Convergence Theorem guarantee that the limit of a sequence of Henstock integrable functions is integrable. This paper is written without the use of Lebesgue measure theory. Acknowledgements I would like to thank Professor Patrick Keef and Professor Russell Gordon for their advice and guidance through this project. I would also like to acknowledge Kathryn Barich and Kailey Bolles for their assistance in the editing process. Introduction As the workhorse of modern analysis, the integral is without question one of the most familiar pieces of the calculus sequence.
    [Show full text]
  • Sequences, Series and Taylor Approximation (Ma2712b, MA2730)
    Sequences, Series and Taylor Approximation (MA2712b, MA2730) Level 2 Teaching Team Current curator: Simon Shaw November 20, 2015 Contents 0 Introduction, Overview 6 1 Taylor Polynomials 10 1.1 Lecture 1: Taylor Polynomials, Definition . .. 10 1.1.1 Reminder from Level 1 about Differentiable Functions . .. 11 1.1.2 Definition of Taylor Polynomials . 11 1.2 Lectures 2 and 3: Taylor Polynomials, Examples . ... 13 x 1.2.1 Example: Compute and plot Tnf for f(x) = e ............ 13 1.2.2 Example: Find the Maclaurin polynomials of f(x) = sin x ...... 14 2 1.2.3 Find the Maclaurin polynomial T11f for f(x) = sin(x ) ....... 15 1.2.4 QuestionsforChapter6: ErrorEstimates . 15 1.3 Lecture 4 and 5: Calculus of Taylor Polynomials . .. 17 1.3.1 GeneralResults............................... 17 1.4 Lecture 6: Various Applications of Taylor Polynomials . ... 22 1.4.1 RelativeExtrema .............................. 22 1.4.2 Limits .................................... 24 1.4.3 How to Calculate Complicated Taylor Polynomials? . 26 1.5 ExerciseSheet1................................... 29 1.5.1 ExerciseSheet1a .............................. 29 1.5.2 FeedbackforSheet1a ........................... 33 2 Real Sequences 40 2.1 Lecture 7: Definitions, Limit of a Sequence . ... 40 2.1.1 DefinitionofaSequence .......................... 40 2.1.2 LimitofaSequence............................. 41 2.1.3 Graphic Representations of Sequences . .. 43 2.2 Lecture 8: Algebra of Limits, Special Sequences . ..... 44 2.2.1 InfiniteLimits................................ 44 1 2.2.2 AlgebraofLimits.............................. 44 2.2.3 Some Standard Convergent Sequences . .. 46 2.3 Lecture 9: Bounded and Monotone Sequences . ..... 48 2.3.1 BoundedSequences............................. 48 2.3.2 Convergent Sequences and Closed Bounded Intervals . .... 48 2.4 Lecture10:MonotoneSequences .
    [Show full text]
  • Euler's Calculation of the Sum of the Reciprocals of the Squares
    Euler's Calculation of the Sum of the Reciprocals of the Squares Kenneth M. Monks ∗ August 5, 2019 A central theme of most second-semester calculus courses is that of infinite series. Simply put, to study infinite series is to attempt to answer the following question: What happens if you add up infinitely many numbers? How much sense humankind made of this question at different points throughout history depended enormously on what exactly those numbers being summed were. As far back as the third century bce, Greek mathematician and engineer Archimedes (287 bce{212 bce) used his method of exhaustion to carry out computations equivalent to the evaluation of an infinite geometric series in his work Quadrature of the Parabola [Archimedes, 1897]. Far more difficult than geometric series are p-series: series of the form 1 X 1 1 1 1 = 1 + + + + ··· np 2p 3p 4p n=1 for a real number p. Here we show the history of just two such series. In Section 1, we warm up with Nicole Oresme's treatment of the harmonic series, the p = 1 case.1 This will lessen the likelihood that we pull a muscle during our more intense Section 3 excursion: Euler's incredibly clever method for evaluating the p = 2 case. 1 Oresme and the Harmonic Series In roughly the year 1350 ce, a University of Paris scholar named Nicole Oresme2 (1323 ce{1382 ce) proved that the harmonic series does not sum to any finite value [Oresme, 1961]. Today, we would say the series diverges and that 1 1 1 1 + + + + ··· = 1: 2 3 4 His argument was brief and beautiful! Let us admire it below.
    [Show full text]
  • (Sn) Converges to a Real Number S If ∀Ε > 0, ∃Ns.T
    32 MATH 3333{INTERMEDIATE ANALYSIS{BLECHER NOTES 4. Sequences 4.1. Convergent sequences. • A sequence (sn) converges to a real number s if 8 > 0, 9Ns:t: jsn − sj < 8n ≥ N. Saying that jsn−sj < is the same as saying that s− < sn < s+. • If (sn) converges to s then we say that s is the limit of (sn) and write s = limn sn, or s = limn!1 sn, or sn ! s as n ! 1, or simply sn ! s. • If (sn) does not converge to any real number then we say that it diverges. • A sequence (sn) is called bounded if the set fsn : n 2 Ng is a bounded set. That is, there are numbers m and M such that m ≤ sn ≤ M for all n 2 N. This is the same as saying that fsn : n 2 Ng ⊂ [m; M]. It is easy to see that this is equivalent to: there exists a number K ≥ 0 such that jsnj ≤ K for all n 2 N. (See the first lines of the last Section.) Fact 1. Any convergent sequence is bounded. Proof: Suppose that sn ! s as n ! 1. Taking = 1 in the definition of convergence gives that there exists a number N 2 N such that jsn −sj < 1 whenever n ≥ N. Thus jsnj = jsn − s + sj ≤ jsn − sj + jsj < 1 + jsj whenever n ≥ N. Now let M = maxfjs1j; js2j; ··· ; jsN j; 1 + jsjg. We have jsnj ≤ M if n = 1; 2; ··· ;N, and jsnj ≤ M if n ≥ N. So (sn) is bounded. • A sequence (an) is called nonnegative if an ≥ 0 for all n 2 N.
    [Show full text]
  • BAIRE ONE FUNCTIONS 1. History Baire One Functions Are Named in Honor of René-Louis Baire (1874- 1932), a French Mathematician
    BAIRE ONE FUNCTIONS JOHNNY HU Abstract. This paper gives a general overview of Baire one func- tions, including examples as well as several interesting properties involving bounds, uniform convergence, continuity, and Fσ sets. We conclude with a result on a characterization of Baire one func- tions in terms of the notion of first return recoverability, which is a topic of current research in analysis [6]. 1. History Baire one functions are named in honor of Ren´e-Louis Baire (1874- 1932), a French mathematician who had research interests in continuity of functions and the idea of limits [1]. The problem of classifying the class of functions that are convergent sequences of continuous functions was first explored in 1897. According to F.A. Medvedev in his book Scenes from the History of Real Functions, Baire was interested in the relationship between the continuity of a function of two variables in each argument separately and its joint continuity in the two variables [2]. Baire showed that every function of two variables that is continuous in each argument separately is not pointwise continuous with respect to the two variables jointly [2]. For instance, consider the function xy f(x; y) = : x2 + y2 Observe that for all points (x; y) =6 0, f is continuous and so, of course, f is separately continuous. Next, we see that, xy xy lim = lim = 0 (x;0)!(0;0) x2 + y2 (0;y)!(0;0) x2 + y2 as (x; y) approaches (0; 0) along the x-axis and y-axis respectively. However, as (x; y) approaches (0; 0) along the line y = x, we have xy 1 lim = : (x;x)!(0;0) x2 + y2 2 Hence, f is continuous along the lines y = 0 and x = 0, but is discon- tinuous overall at (0; 0).
    [Show full text]
  • Sequences and Their Limits
    Sequences and their limits c Frank Zorzitto, Faculty of Mathematics University of Waterloo The limit idea For the purposes of calculus, a sequence is simply a list of numbers x1; x2; x3; : : : ; xn;::: that goes on indefinitely. The numbers in the sequence are usually called terms, so that x1 is the first term, x2 is the second term, and the entry xn in the general nth position is the nth term, naturally. The subscript n = 1; 2; 3;::: that marks the position of the terms will sometimes be called the index. We shall deal only with real sequences, namely those whose terms are real numbers. Here are some examples of sequences. • the sequence of positive integers: 1; 2; 3; : : : ; n; : : : • the sequence of primes in their natural order: 2; 3; 5; 7; 11; ::: • the decimal sequence that estimates 1=3: :3;:33;:333;:3333;:33333;::: • a binary sequence: 0; 1; 0; 1; 0; 1;::: • the zero sequence: 0; 0; 0; 0;::: • a geometric sequence: 1; r; r2; r3; : : : ; rn;::: 1 −1 1 (−1)n • a sequence that alternates in sign: 2 ; 3 ; 4 ;:::; n ;::: • a constant sequence: −5; −5; −5; −5; −5;::: 1 2 3 4 n • an increasing sequence: 2 ; 3 ; 4 ; 5 :::; n+1 ;::: 1 1 1 1 • a decreasing sequence: 1; 2 ; 3 ; 4 ;:::; n ;::: 3 2 4 3 5 4 n+1 n • a sequence used to estimate e: ( 2 ) ; ( 3 ) ; ( 4 ) :::; ( n ) ::: 1 • a seemingly random sequence: sin 1; sin 2; sin 3;:::; sin n; : : : • the sequence of decimals that approximates π: 3; 3:1; 3:14; 3:141; 3:1415; 3:14159; 3:141592; 3:1415926; 3:14159265;::: • a sequence that lists all fractions between 0 and 1, written in their lowest form, in groups of increasing denominator with increasing numerator in each group: 1 1 2 1 3 1 2 3 4 1 5 1 2 3 4 5 6 1 3 5 7 1 2 4 5 ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;::: 2 3 3 4 4 5 5 5 5 6 6 7 7 7 7 7 7 8 8 8 8 9 9 9 9 It is plain to see that the possibilities for sequences are endless.
    [Show full text]
  • Infinitesimals
    Infinitesimals: History & Application Joel A. Tropp Plan II Honors Program, WCH 4.104, The University of Texas at Austin, Austin, TX 78712 Abstract. An infinitesimal is a number whose magnitude ex- ceeds zero but somehow fails to exceed any finite, positive num- ber. Although logically problematic, infinitesimals are extremely appealing for investigating continuous phenomena. They were used extensively by mathematicians until the late 19th century, at which point they were purged because they lacked a rigorous founda- tion. In 1960, the logician Abraham Robinson revived them by constructing a number system, the hyperreals, which contains in- finitesimals and infinitely large quantities. This thesis introduces Nonstandard Analysis (NSA), the set of techniques which Robinson invented. It contains a rigorous de- velopment of the hyperreals and shows how they can be used to prove the fundamental theorems of real analysis in a direct, natural way. (Incredibly, a great deal of the presentation echoes the work of Leibniz, which was performed in the 17th century.) NSA has also extended mathematics in directions which exceed the scope of this thesis. These investigations may eventually result in fruitful discoveries. Contents Introduction: Why Infinitesimals? vi Chapter 1. Historical Background 1 1.1. Overview 1 1.2. Origins 1 1.3. Continuity 3 1.4. Eudoxus and Archimedes 5 1.5. Apply when Necessary 7 1.6. Banished 10 1.7. Regained 12 1.8. The Future 13 Chapter 2. Rigorous Infinitesimals 15 2.1. Developing Nonstandard Analysis 15 2.2. Direct Ultrapower Construction of ∗R 17 2.3. Principles of NSA 28 2.4. Working with Hyperreals 32 Chapter 3.
    [Show full text]
  • A Compact Introduction to a Generalized Extreme Value Theorem
    Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Topology Mathematics via Primary Historical Sources (TRIUMPHS) Summer 2017 A Compact Introduction to a Generalized Extreme Value Theorem Nicholas A. Scoville Ursinus College, [email protected] Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_topology Part of the Curriculum and Instruction Commons, Educational Methods Commons, Geometry and Topology Commons, Higher Education Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits ou.y Recommended Citation Scoville, Nicholas A., "A Compact Introduction to a Generalized Extreme Value Theorem" (2017). Topology. 5. https://digitalcommons.ursinus.edu/triumphs_topology/5 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Topology by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. A Compact Introduction to a Generalized Extreme Value Theorem Nicholas A. Scoville∗ April 19, 2021 1 Introduction Compactness is a concept that is often introduced in a first course in analysis or topology, but one which students in those courses often find to be considerably difficult. Not only can it be challenging to figure out what the standard definition is saying, it can be even more of a mystery tofigure out what possessed someone to write down such a definition in the first place. In this project, we look back to when compactness was first defined (albeit slightly differently than it is today) tosee what use it had then and, more importantly, the role it continues to play in mathematics today.
    [Show full text]
  • The Strange Properties of the Infinite Power Tower Arxiv:1908.05559V1
    The strange properties of the infinite power tower An \investigative math" approach for young students Luca Moroni∗ (August 2019) Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. Paul Lockhart { \A Mathematician's Lament" Abstract In this article we investigate some "unexpected" properties of the \Infinite Power Tower 1" function (or \Tetration with infinite height"): . .. xx y = f(x) = xx where the \tower" of exponentiations has an infinite height. Apart from following an initial personal curiosity, the material collected here is also intended as a potential guide for teachers of high-school/undergraduate students interested in planning an activity of \investigative mathematics in the classroom", where the knowledge is gained through the active, creative and cooperative use of diversified mathematical tools (and some ingenuity). The activity should possibly be carried on with a laboratorial style, with no preclusions on the paths chosen and undertaken by the students and with little or no information imparted from the teacher's desk. The teacher should then act just as a guide and a facilitator. The infinite power tower proves to be particularly well suited to this kind of learning activity, as the student will have to face a challenging function defined through a rather uncommon infinite recursive process. They'll then have to find the right strategies to get around the trickiness of this function and achieve some concrete results, without the help of pre-defined procedures. The mathematical requisites to follow this path are: functions, properties of exponentials and logarithms, sequences, limits and derivatives.
    [Show full text]
  • Chapter 2 Limits of Sequences
    Chapter 2 Limits of Sequences Calculus Student: lim sn = 0 means the sn are getting closer and closer to zero but n!1 never gets there. Instructor: ARGHHHHH! Exercise 2.1 Think of a better response for the instructor. In particular, provide a counterexample: find a sequence of numbers that 'are getting closer and closer to zero' but aren't really getting close at all. What about the 'never gets there' part? Should it be necessary that sequence values are never equal to its limit? 2.1 Definition and examples We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero Then we define what it means for sequence to converge to an arbitrary real number. Finally, we discuss the various ways a sequence may diverge (not converge). In between we will apply what we learn to further our understanding of real numbers and to develop tools that are useful for proving the important theorems of Calculus. Recall that a sequence is a function whose domain is Z+ or Z≥. A sequence is most usually denoted with subscript notation rather than standard function notation, that is we write sn rather than s(n): See Section 0.3.2 for more about definitions and notations used in describing sequences. 43 44 CHAPTER 2. LIMITS OF SEQUENCES 1 Figure 2.1: s = : n n 2 1 0 0 5 10 15 20 2.1.1 Sequences converging to zero. Definition We say that the sequence sn converges to 0 whenever the following hold: For all > 0, there exists a real number, N, such that n > N =) jsnj < .
    [Show full text]