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Many experimental problems in the natural sciences result in data which can best be represented by linear combinations of exponentials of the form f(t) = ∑[with p above and k=1 below] α_k e^-λ_kt. Among such problems are those dealing with growth, decay, ion concentration, and survival and mortality. Also, in general, the solution to any problem which may be represented by linear differential equations with constant coefficients is a linear combination of exponentials. In most problems like those which have been mentioned, the parameters α_k and λ_k have biological or physical significance. Therefore, in fitting the. function f(t) to the data it is not only necessary that the function approximate the data closely, but it is also necessary that the parameters α_k and λ_k be accurately estimated. Furthermore, a measure of the accuracy of the estimation of the parameters is required. A new estimation procedure for linear combinations of exponentials is developed in this paper. Unlike the iterative maximum likelihood and least-squares methods for estimating the parameters for such a model, the new procedure is noniterative and can be easily applied. Also, in contrast to other non-iterative methods, error estimates are available for the parameter estimates yielded by the new procedure. In the model for the new procedure the points t_i at which observations are taken are assumed to be equally spaced and the number of such points is specified to be an integral multiple of the number of parameters to be estimated. Moreover, each observation is specified to have expectation f (t_i), where f is the function mentioned earlier. The coefficients α_k are assumed to be non-zero and the exponents λ_k are assumed to be distinct and positive. Then in the derivation of new procedure, the observations are reduced to as many sums as there are parameters to be estimated. Each of these sums is equated to its expected value and the resultant equations are solved for estimators of the parameters. The estimators from the new procedure are shown to be asymptotically normally distributed as either the number of points at which observations are taken or the number of observations made at each such point approaches infinity. The asymptotic variances obtained are used to form approximate confidence limits for the α_k and λ_k. The statistical properties of the estimators are also studied. It is found that they are consistent, but not in general unbiased or efficient. Asymptotic efficiencies are calculated tor a few sets of parameter values and a bias approximation is obtained for two special cases. The new method is also shown to be optimum relative to certain similar methods and necessary conditions for the new procedure to lead to admissible estimates are studied. In the last portion of the thesis a sampling study is reported for observations generated with a model containing only one exponential term and with errors which are normally distributed. The small sample biases and variances for the estimates computed from these observations are given and the effects of changes in the parameters in the model are investigated. Then some actual experimental data are fitted using both the new procedure and some alternative methods. The final chapter in the body of the thesis contains a critical evaluation of the new procedure relative to other estimation methods. / Ph. D.

The purpose of this paper is to define and develop a set of functions of an area in such a manner as to be analogous to the trigonometric and the hyperbolic functions.

This thesis explores how to best choose data when curve fitting using power exponential functions. The power exponential functions used are μ(b; x)=(xe1-x)b and Φ(ρ; x)=((1-x)ex)ρ . We use a number of designs such as the equidistant design, the Chebyshev design and the the D-optimal design to compare which design gives the best fit. A few examples including the logistic and the heidler function are looked at during the comparison. The measurement of the errors were made based on the sum of least squares errors in the first part and the maximum error in the second part. MATLAB was used in this comparison.

abstract: This dissertation report follows a three-paper format, with each paper having a different but related focus. In Paper 1 I discuss conceptual analysis of mathematical ideas relative to its place within cognitive learning theories and research studies. In particular, I highlight specific ways mathematics education research uses conceptual analysis and discuss the implications of these uses for interpreting and leveraging results to produce empirically tested learning trajectories. From my summary and analysis I develop two recommendations for the cognitive researchers developing empirically supported learning trajectories. (1) A researcher should frame his/her work, and analyze others’ work, within the researcher’s image of a broadly coherent trajectory for student learning and (2) that the field should work towards a common understanding for the meaning of a hypothetical learning trajectory. In Paper 2 I argue that prior research in online learning has tested the impact of online courses on measures such as student retention rates, satisfaction scores, and GPA but that research is needed to describe the meanings students construct for mathematical ideas researchers have identified as critical to their success in future math courses and other STEM fields. This paper discusses the need for a new focus in studying online mathematics learning and calls for cognitive researchers to begin developing a productive methodology for examining the meanings students construct while engaged in online lessons. Paper 3 describes the online Precalculus course intervention we designed around measurement imagery and quantitative reasoning as themes that unite topics across units. I report results relative to the meanings students developed for exponential functions and related ideas (such as percent change and growth factors) while working through lessons in the intervention. I provide a conceptual analysis guiding its design and discuss pre-test and pre-interview results, post-test and post-interview results, and observations from student behaviors while interacting with lessons. I demonstrate that the targeted meanings can be productive for students, show common unproductive meanings students possess as they enter Precalculus, highlight challenges and opportunities in teaching and learning in the online environment, and discuss needed adaptations to the intervention and future research opportunities informed by my results. / Dissertation/Thesis / Doctoral Dissertation Mathematics Education 2018

Abstract not available

The purpose of this report is to describe the course, Analysis of the Real Line, taught at The University of Texas at Austin. Course materials are presented using the inquiry based learning method. Students work a series of warm up problems before being presented rigorous problems in calculus, including topics on integration, exponential functions, and real number line analysis. Additionally, students consider aspects of these problems that could be incorporated into a high school curriculum. Typical problems in several major areas are summarized along with warm up problems that introduce or extend the topics. / text

Submitted by Luciana Alcantara De Toledo (luatoledo@yahoo.com) on 2018-09-28T18:33:17Z No. of bitstreams: 1 DISSERTAÇÃO - ENSINO DA FUNÇÃO EXPONENCIAL - ANÁLISE DE RESULTADOS - LUCIANA ALCANTARA DE TOLEDO - VERSÃO FINAL.pdf: 8323445 bytes, checksum: f3f79aaa7aa209ffe5cfa3b2b1f02a29 (MD5) / Rejected by Elza Mitiko Sato null (elzasato@ibilce.unesp.br), reason: Solicitamos que realize correções na submissão seguindo as orientações abaixo: Problema 01) A paginação deve ser sequencial, iniciando a contagem na folha de rosto e mostrando o número a partir da introdução, a ficha catalográfica ficará após a folha de rosto e não deverá ser contada. Problema 02) Solicito que corrija a descrição na natureza da pesquisa(Folha de rosto e de aprovação): Dissertação apresentada como parte dos requisitos para obtenção do título de Mestre em Matemática, junto ao Programa de Pós-Graduação Mestrado Profissional em Matemática em Rede Nacional – PROFMAT, do Instituto de Biociências, Letras e Ciências Exatas da Universidade Estadual Paulista “Júlio de Mesquita Filho”, Câmpus de São José do Rio Preto. Problema 03) No arquivo a data da defesa está 17 de agosto de 2018, porém nos metadados você preencheu 18 de agosto de 2018, qual a data correta? Lembramos que o arquivo depositado no repositório deve ser igual ao impresso, o rigor com o padrão da Universidade se deve ao fato de que o seu trabalho passará a ser visível mundialmente. Agradecemos a compreensão on 2018-09-28T19:03:49Z (GMT) / Submitted by Luciana Alcantara De Toledo (luatoledo@yahoo.com) on 2018-10-04T19:16:14Z No. of bitstreams: 1 DISSERTAÇÃO - ENSINO DA FUNÇÃO EXPONENCIAL - ANÁLISE DE RESULTADOS - LUCIANA ALCANTARA DE TOLEDO - VERSÃO FINAL.pdf: 8323445 bytes, checksum: f3f79aaa7aa209ffe5cfa3b2b1f02a29 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-10-04T19:38:25Z (GMT) No. of bitstreams: 1 toledo_la_me_sjrp.pdf: 8323445 bytes, checksum: f3f79aaa7aa209ffe5cfa3b2b1f02a29 (MD5) / Made available in DSpace on 2018-10-04T19:38:25Z (GMT). No. of bitstreams: 1 toledo_la_me_sjrp.pdf: 8323445 bytes, checksum: f3f79aaa7aa209ffe5cfa3b2b1f02a29 (MD5) Previous issue date: 2018-08-17 / Este trabalho tem como objetivo apresentar as análises dos resultados referentes ao ensino da Função Exponencial com o uso da metodologia de Resolução de Problemas para os alunos do Ensino Médio. Baseando-se na teoria proposta por Onuchic (1999), foram elaboradas quatro atividades envolvendo problemas para introduzir os conceitos referentes a Função Exponencial no primeiro ano e verificar os conhecimentos já adquiridos pelos alunos do segundo e terceiro anos. Os problemas envolvem o crescimento populacional de uma bactéria e a meia-vida de um fármaco. Os alunos do primeiro ano apresentaram melhor desempenho com esta nova abordagem de ensino baseada em soluções de problemas, e no segundo e terceiro anos apresentaram dificuldades com relação ao tópico de funções. / This study evaluated the impact teaching of exponential functions through a problem-solving approach to students in secondary education. Based on the theory proposed by Onuchic (1999), we created four activities involving problems to introduce the concepts related to exponential functions to tenthgrade students and to verify the acquired knowledge of eleventh-grade and twelfth-grade students. The problems involve the population growth of bacteria and the half-life of a drug. All the students welcomed the new didactic proposal. The tenth-grade students were able to understand the new concepts based on problem-solving teaching and eleventh-grade and twelfth-grade students presented difficulties regarding the topic of functions.

Submitted by Nayara de Novaes Rezende Villani null (na_villani@hotmail.com) on 2017-11-29T19:22:14Z No. of bitstreams: 1 DissertacaoPROFMAT_Nayara_2.pdf: 1915936 bytes, checksum: 027573c203bbb23b0c54dc6b4cfe75c5 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2017-11-30T18:31:00Z (GMT) No. of bitstreams: 1 villani_nnr_me_sjrp.pdf: 1915936 bytes, checksum: 027573c203bbb23b0c54dc6b4cfe75c5 (MD5) / Made available in DSpace on 2017-11-30T18:31:00Z (GMT). No. of bitstreams: 1 villani_nnr_me_sjrp.pdf: 1915936 bytes, checksum: 027573c203bbb23b0c54dc6b4cfe75c5 (MD5) Previous issue date: 2017-09-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho são apresentadas propostas de atividades para abordar o número de Euler no Ensino Médio, uma vez que existem muitas situações cotidianas em que a única maneira de descrevê-las convenientemente por meio de modelos matemáticos é utilizando a função exponencial com a base sendo o número de Euler. Por exemplo, o decaimento radioativo, a lei do resfriamento de Newton, o estudo de uma certa epidemia numa população e investimento de capital. Inicialmente, são apontados fatos históricos desde o surgimento do número de Euler até o momento de sua notação. Em seguida, são apresentadas uma abordagem sobre a função exponencial e propostas de atividades envolvendo as situações cotidianas mencionadas. / In this work we present proposals of activities to approach the Euler number in High School, since there are many daily situations in which the only way to describe them conveniently by means of mathematical models is to use the exponential function with the base being the Euler number. For example, radioactive decay, Newton's law of cooling, the study of a certain epidemic in a population and capital investment. Initially, historical facts are pointed out from the appearance of the Euler number until the moment of its notation. Next, we present an approach on the exponential function and proposals of activities involving the daily situations mentioned.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Neste trabalho vamos fazer uma abordagem elementar sobre a funÃÃo exponencial visando entender o significado de potÃncias com expoente natural, inteiro, racional e irracional bem como as propriedades que fazem dela uma das funÃÃes mais importantes da MatemÃtica. Paralelamente, serà feita outra abordagem mostrando essas propriedades usando uma ferramenta poderosa da MatemÃtica: o cÃlculo diferencial e integral. TambÃm vamos tratar da funÃÃo logarÃtmica por ela ser a inversa da funÃÃo exponencial e por ser tÃo importante quanto esta. / In this work we make an elementary approach to the exponential function in order to understand the meaning of powers with natural exponent, integer, rational and irrational as well as the properties that make it one of the most important functions of mathematics. In parallel, another approach will be showing these properties using a powerful tool of mathematics: differential and integral calculus. We will also discuss the logarithmic function because it is the inverse of the exponential function and being as important as this.