Philosophiae Doctor - PhD / The purpose of this study was to explore errors that are displayed by students when learning derivatives of trigonometric functions in an extended curriculum programme. The first aim was to identify errors that are displayed by students in their solutions through the lens of the APOS theory. The second aim was to address students' errors by using the two principles of Vygotsky's socio-cultural theory of learning, namely the zone of proximal development and more knowledgeable others. The research presented in this thesis is a case study located in the interpretive paradigm of qualitative research. The participants in this study comprised a group of students who registered for mathematics in the ECP at Cape Peninsula University of Technology, Cape Town, South Africa. The study was piloted in 2008 with a group of twenty students who registered for mathematics in the ECP for Chemical Engineering. In 2009 thirty students from the ECP registered for mathematics in Chemical Engineering were selected to participate in the main study. This study was conducted over a period of four and a half years. Data collection was done through students' written tasks; classroom audio and video recordings and indepth interviews. Data were analysed through categorising errors from students' written work, and finding common themes and patterns in audio and video recordings and from the in-depth interviews. The findings of this study revealed that students committed interpretation, arbitrary, procedural, linear extrapolation and conceptual errors. Interpretation errors arise when students fail to interpret the nature of the problem correctly owing to over-generalisation of certain mathematical rules. Arbitrary errors arise when students behave arbitrarily and fail to take account of the constraints laid down in what is given. Procedural errors occur when students fail to carry out manipulations or algorithms although they understand concepts in problem. Linear extrapolation errors happen through an overgeneralisation of the property f (a + b) = f (a) + f (b) , which applies only when f is a linear function Conceptual errors occur owing to failure to grasp the concepts involved in the problem or failure to appreciate the relationships involved in the problem. The findings were consistent with literature indicated that errors are based on students’ prior knowledge, as they over-generalise certain mathematical procedures, algorithms and rules of differentiation in their solutions. The use of learning activities in the form of written tasks; as well as classroom audio and video recordings assisted the lecturer to identify and address errors that were displayed by students when they learned derivatives of trigonometric functions. The students claimed in their interviews that they benefited from class discussions as they obtained immediate feedback from their fellow students and the lecturer. They also claimed that their performances improved as they continued to practice with the assistance of more knowledgeable students, as well as the lecturer. This study supports the view from the literature that identification of errors has immense potential to address students’ poor understanding of derivatives of trigonometric functions. This thesis recommends further research on errors in various sections of Differential Calculus, which is studied in an extended curriculum programme at Universities of Technology in South Africa.

This thesis is a study of the analytical development of the trigonometric functions.

This thesis proposes an optimized Co-ordinate Rotation Digital Computer (CORDIC) algorithm in the rotation and extended vectoring mode of the circular co-ordinate system. The CORDIC algorithm computes the values of trigonometric functions and their inverses. The proposed algorithm provides the result with a lower overall latency than existing systems. This is done by using redundant representations and approximations of the required direction and angle of each rotation. The algorithm has been designed to provide the result in a fixed number of iterations $n$ for the rotation mode and $3\lceil n/2 \rceil + \lfloor n/2 \rfloor$ for the extended vectoring mode; where, $n$ is a design parameter. In each iteration, the algorithm performs between 0 and $p/n$ parallel rotations, where, $p$ is the number of precision bits and $n$ is the selected number of iterations. A technique to handle the scaling factor compensation for such an algorithm is proposed. The results of the functional verification for different values of $n$ and an estimation of the overall latency are presented. Based on the results, guidelines to choosing a value of $n$ to meet the required performance have also been presented.

The function $sin x$ as one of the six trigonometric functions is fundamental in nearly every branch of mathematics, and its applications. In this thesis, we study an integral equation related to that of $sin x$: $mbox{~for~}xin[-frac{hat{pi}_{p}}{2},~frac{hat{pi}_{p}}{2}] mbox{~and~} p>1$ $$x=int_0^{S_{p}(x)}(1-|t|^{p})^{-frac{1}{p}}dt.$$ Here $hat{pi}_{p}=frac{2pi}{psin(frac{pi}{p})}=2int_0^1(1-t^{p})^{-frac{1}{p}}dt.$ We find that the function $S_{p}(x)$ is well defined. Its properties are also similar to those of $sin x$ : differentiation, identities, periodicity, asymptotic expansions, $cdots$, etc. For example, we have $$|S_{p}(x)|^{p}+|S'_{p}(x)|^{p}=1mbox{~~and~~}frac{d}{dx}(|S'_{p}(x)|^{p-2}S'_{p}(x))=-(p-1)|S_{p}(x)|^{p-2}S_{p}(x).$$ We call $S_{p}(x)$ the generalized sine function. Similarly, we define the generalized cosine function $C_{p}(x)$ by $|x|=int_{C_{p}(x)}^{1}(1- t^{p})^{-frac{1}{p}}dt$ for $xin[-frac{hat{pi}_{p}}{2}$,~$frac{hat{pi}_{p}}{2}]$ and derive its properties. Thus we obtain two sets of trigonometric functions: egin{itemize} item[(i)]$~S_{p}(x),~ S'_{p}(x),~ T_{p}(x)=frac{S_{p}(x)}{S'_{p}(x)},~RT_{p}(x)=frac{S'_{p}(x)}{S_{p}(x)},~ SE_{p}(x)=frac{1}{S'_{p}(x)},~ RS_{p}(x)=frac{1}{S_{p}(x)}~;$ item[(ii)]$~C_{p}(x),~ C'_{p}(x),~RCT_{p}(x)=-frac{C'_{p}(x)}{C_{p}(x)},~ CT_{p}(x)=-frac{C_{p}(x)}{C'_{p}(x)},~RC_{p}(x)=frac{1}{C_{p}(x)},~ CS_{p}(x)=-frac{1}{C'_{p}(x)}mbox{~¡C~}$ end{itemize}These two sets of functions have similar differentiation formulas, identities and periodic properties as the classical trigonometric functions. They coincide when $p=2$. Their graphs and asymptotic expansions are also interesting. Through this study, we understand more about the theoretical framework of trigonometric functions.

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.

Made available in DSpace on 2015-03-03T15:36:11Z (GMT). No. of bitstreams: 1 JoaildoM_DISSERT.pdf: 2805469 bytes, checksum: c64255f0e7f702b14602a15e05f2ea1f (MD5) Previous issue date: 2013-04-12 / Universidade Federal do Rio Grande do Norte / In this study, we sought to address the weaknesses faced by most students when they were studying trigonometric functions sine and cosine. For this, we proposed the use of software Geogebra in performing a sequence of activities about the content covered. The research was a qualitative approach based on observations of the activities performed by the students of 2nd year of high school IFRN - Campus Caicfio. The activities enabled check some diculties encountered by students, well as the interaction between them during the tasks. The results were satisfactory, since they indicate that the use of software contributed to a better understanding of these mathematical concepts studied / Nesse trabalho, buscou-se suprir as defi ci?ncias enfrentadas por boa parte dos alunos quando eram estudadas as funfi??es trigonomfietricas seno e co-seno. Para isso, prop?s-se a utilizafi??o do software Geogebra na realizafi??o de uma sequ?ncia de atividades sobre o contefiudo abordado. A pesquisa teve uma abordagem qualitativa, baseada em observa fi??es das atividades realizadas pelos alunos do 2o Ano do Ensino Mfi?dio do IFRN { Campus Caicfi?. As atividades possibilitaram verificar algumas dificuldades encontradas pelos alunos, assim como a interafi??o existente entre eles durante a realizafi??o das tarefas. Os resultados obtidos foram satisfatfi?rios, pois indicam que a utilizafi??o do software contribuiu para uma melhor compreens?o desses conceitos matemfi?ticos estudados

nÃo hà / Discutir o uso pedagÃgico de recursos tecnolÃgicos nas aulas de MatemÃtica tem sido foco de muitos estudos, uma vez que por si sà eles nÃo garantem um novo modelo educacional. Neste sentido, esse trabalho tem o objetivo de explorar o software GeoGebra como recurso pedagÃgico no ensino das funÃÃes trigonomÃtricas para alunos da 2 sÃrie do Ensino MÃdio, de uma escola da rede particular no estado do CearÃ, atravÃs da visualizaÃÃo das funÃÃes trigonomÃtricas no cÃrculo trigonomÃtrico, suas representaÃÃes grÃficas e o papel dos parÃmetros na construÃÃo dos grÃficos dessas funÃÃes, oportunizando o contato com recursos diferentes dos presentes em sala de aula. Esse trabalho mostra que o uso da simulaÃÃo computacional pode ser um meio eficiente de promover o aprendizado de forma significativa, facilitando assim o desenvolvimento de habilidades na compreensÃo por parte dos discentes dos conteÃdos relacionados Ãs funÃÃes trigonomÃtricas, num ambiente onde os conteÃdos abordados sÃo trabalhados utilizando o amplo campo de informaÃÃes e interaÃÃes do computador. / Discuss the pedagogical use of technological resources in mathematics classes has been the focus of many studies, since they alone do not guarantee a new educational model. In this sense, this work aims to explore the GeoGebra software as a teaching resource in the teaching of the trigonometric functions for students of 2nd year of high school, a school of private schools in the state of CearÃ, through visualization of the trigonometric functions in the unit circle , their graphical representations and the role of parameters in the construction of the graphs of these functions, providing the opportunity for contact with the different features present in the classroom. This work shows that the use of computer simulation can be an effective means of promoting learning significantly, thus facilitating the development of skills in understanding by students of the content related to trigonometric functions, in an environment where the subjects covered are worked using the broad field of information and computer interactions.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2014-10-31T11:26:53Z No. of bitstreams: 2 Dissertação - Emílio Curi Neto - 2014.pdf: 3380066 bytes, checksum: d90415ab2be40c912a8f3437adf514bb (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2014-10-31T13:45:48Z (GMT) No. of bitstreams: 2 Dissertação - Emílio Curi Neto - 2014.pdf: 3380066 bytes, checksum: d90415ab2be40c912a8f3437adf514bb (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-10-31T13:45:48Z (GMT). No. of bitstreams: 2 Dissertação - Emílio Curi Neto - 2014.pdf: 3380066 bytes, checksum: d90415ab2be40c912a8f3437adf514bb (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-07-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work the main goal is focused on applying the theory of Taylor polynomial approximations applied on the trigonometric function defined by f : [0; 2 ] 􀀀! R, where f(x) = sin(x). To achieve this goal, eight sections were developed, in which initially a reflection on the problem and the need to obtain the values in this respect in that it is wide angle measure x is presented. Is presented and subsequently treated a problem involving the movement of a pendulum, which uses the approximation sin(x) x where x belongs to a certain range. In the sections that follow a literature review of the theories of differential and integral calculus is presented, and the related theory of Taylor approximation of functions by polynomials. Later we used these theories to analyze and determine polynomials approximating the function f(x) = sin(x) in a neighborhood of the point x = 0, and estimate the error when we applied these approaches. At this time the error occurred due to the approach used in the pendulum problem was also analyzed. Finally a hint of practice to be held in the classroom using the theories treated here as well as the study of the problem of heat transfer in a bar through the theory of Fourier activity is presented. / Neste trabalho o objetivo principal está focado em aplicar a teoria de Taylor relativa à aproximações polinomiais aplicadas à função trigonométrica definida por f : [0; 2 ] 􀀀! R, onde f(x) = sen(x). Para alcançar esse objetivo, foram desenvolvidas oito seções, nas quais inicialmente é apresentada uma reflexão sobre a necessidade e a problemática de obtêr-se os valores desta relação a medida em que varia-se a medida do ângulo x. Posteriormente é apresentado e tratado um problema envolvendo o movimento de um pêndulo, o qual utiliza a aproximação sen(x) x onde x pertence o um certo intervalo. Nas seções que seguem é apresentada uma revisão bibliográfica das Teorias do Cálculo Diferencial e Integral, assim como da Teoria de Taylor relacionada à aproximação de funções através de polinômios. Posteriormente utilizou-se estas teorias para analisar e determinar polinômios que aproximam a função sen(x) em uma vizinhança do ponto x = 0, assim como estimar o erro gerado ao utilizar-se estas aproximações. Nesse momento também foi analisado o erro ocorrido devido à aproximação utilizada no problema do pêndulo. Por fim é apresentada uma sugestão de atividade prática a ser realizada em sala de aula utilizando as teorias aqui tratadas, assim como o estudo do problema de transferência de calor em uma barra através da teoria de Fourier.

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2014-12-09T20:54:30Z No. of bitstreams: 1 Dissertação-Wellington Silva Tavares-2013.pdf: 5557876 bytes, checksum: 305728e8a14e05901ef3038bdb447272 (MD5) / Rejected by Jaqueline Silva (jtas29@gmail.com), reason: on 2014-12-09T20:54:39Z (GMT) / Submitted by Jaqueline Silva (jtas29@gmail.com) on 2014-12-11T18:42:24Z No. of bitstreams: 1 Dissertação-Wellington Silva Tavares-2013.pdf: 5557876 bytes, checksum: 305728e8a14e05901ef3038bdb447272 (MD5) / Rejected by Jaqueline Silva (jtas29@gmail.com), reason: on 2014-12-11T19:14:55Z (GMT) / Submitted by Jaqueline Silva (jtas29@gmail.com) on 2014-12-12T19:37:24Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação-Wellington Silva Tavares-2013.pdf: 5557876 bytes, checksum: 305728e8a14e05901ef3038bdb447272 (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2014-12-16T09:24:16Z (GMT) No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação-Wellington Silva Tavares-2013.pdf: 5557876 bytes, checksum: 305728e8a14e05901ef3038bdb447272 (MD5) / Made available in DSpace on 2014-12-16T09:24:16Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertação-Wellington Silva Tavares-2013.pdf: 5557876 bytes, checksum: 305728e8a14e05901ef3038bdb447272 (MD5) Previous issue date: 2013-10-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / One of the main causes of the great di culty of learning mathematics is the lack of the interest from the students. Aiming to seek an improvement in teaching in order to make students more motivated, this project was developed, which proposes a di erent methodology in the teaching of trigonometric functions with the utilization of mathematic software Winplot. Activities were developed for application of Winplot which might provide to the teacher a more e cient method in the teaching of trigonometric functions. The use of computers in mathematics education aims to make young people be motivated to learn this discipline, changing the routine of class and arousing the interest of the student involved. The Mathematics has undergone signi cant changes in their teaching methods, thus develop logical reasoning, encourage independent thinking, creativity and problem-solving skills became essential to all citizen who wishes to have an active and critical participation in society. Therefore, the utilization of informatics in teaching mathematics o ers these characteristics in a simple and fun without leaving to encourage the student in their learning. / Uma das principais causas da grande di culdade de aprender Matemática é a falta de interesse dos alunos. Visando buscar uma melhoraria para o ensino, a m de tornar os alunos mais motivados, foi desenvolvido este projeto, que propõe uma metodologia diferenciada no ensino das funções trigonométricas com a utilização do software matemático Winplot. Foram desenvolvidas atividades para aplicação do Winplot que poderão proporcionar ao professor um método mais e ciente no ensino das funções trigonom étricas. O uso da informática no ensino da matemática tem o objetivo de fazer com que os jovens se motivem a aprender esta disciplina, mudando a rotina da classe e despertando o interesse do aluno envolvido. A matemática vem sofrendo mudanças signi cativas em seus métodos de ensino, assim desenvolver o raciocínio lógico, estimular o pensamento independente, a criatividade e a capacidade de resolver problemas tornou-se essencial a todo cidadão que deseja ter uma participação ativa e crítica na sociedade. Logo, a utilização da informática no ensino da matemática oferece essas características de uma maneira simples e divertida sem deixar de estimular o aluno em sua aprendizagem.

O objetivo principal deste trabalho é o estudo das funções univalentes e de suas propriedades. Este estudo é direcionado principalmente aos polinômios univalentes e à investigação de prolemas extremos envolvendo seus coeficientes, seus zeros e suas propriedades geométricas. Encontramos uma relação interessante entre os polinômios univalentes e os polinômios univalentes definida por Suffridge. Geramos várias funções univalentes estreladas e convexas através de suas propriedades geométricas e da localização de seus zeros. / The main aim of this work is the study of univalent functions and their properties. This study is focused speciall on the univalent polynomials and theinvestigation of extremal problems involving their coefficients, zeros and geometric properties. We find a very ineresting relation between univalent polynomials through a class of univalent starlike and convex functions involving their geometric properties and the location of their zeros.