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Functions in examples and counterexamples
Janda, David ; Pilous, Derek (advisor) ; Zhouf, Jaroslav (referee)
The aim of my Bachelors thesis is to explicate students coming to the uni- versity the key problems in fundamentals of mathematical analysis. I focus on the most notable terms of continuity and limit, which these secondary students were acquainted with. However, majority of them just intuitively and informaly. I am trying to point out the fact, that the knowledge of many students is distortid and uncomplete. As a result it is necessary to practise and clarify this knowledge so that the intuitive imagination of these terms corresponds to the formal definition. I am trying to get this point by brea- king of intuitive imaginations of students by counterexamples. Important is a chapter named The Construction of Functions, which contains instructi- ons leading to the finding functions with specific features. Not only these features, described in this thesis, but also more complex such as derivation, primitive function or uniform convergence. It is a consequence of the fact, that the principle of examples to practise these terms is in many sights similar and repetitious. In chapters named Continuity and Limit, I am interpreting these terms using the special examples, which are in my opinion optimal for rehearsing. My intention is to help illustrate selected problematical sections of mathematical analysis.
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Limits and L'Hospital's rule
Ranšová, Kateřina ; Staněk, Jakub (advisor) ; Halas, Zdeněk (referee)
Title: Limits and l'Hospital's rule Author: Kateřina Ranšová Department: Department of Mathematics Education Supervisor: RNDr. Jakub Staněk, Ph.D., Department of Mathematics Education Abstract: The aim of this BA thesis is to introduce to the reader the concept of the limit of function and the means of its solution. The main impact of the thesis lies in a didactic approach and in the connection of a limit theory with its graphic representation and different methods of algebraic calculation. The text consists of eight chapters which can be divided into two parts according to their content. The first part explores the term "the limit of the function". Individual types of limits are then defined. To facilitate understanding, most of the definitions are accompanied by a particular example and a graphic representation . The first part is concluded by a unified definition which by means of the term "vicinity" summarizes all preceding types of limits. The other part deals with some basic methods of limits' calculation. Other topics include Taylor Series, l'Hospital's Rule and their applications to the limits. The core of the thesis is a comparison of calculation by means of l'Hospital's Rule and Taylor Series. The conclusion of the thesis presents some advantages and disadvantages of applying Taylor Series and...
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Problem posing for limits of functions
Prskavcová, Pavla ; Pilous, Derek (advisor) ; Kvasz, Ladislav (referee)
Title: Problem posing for limits of functions Author: Pavla Prskavcová Department: Department of Mathematics and Mathematical Education Supervisor: Mgr. Derek Pilous, Ph.D. The aim of the thesis is to describe methods of limit problems solving and of posing new problems that are solvable by those methods. The first part of the thesis contains the necessary theory (definitions and theorems). In the second part of the thesis there are described the standard methods used for evaluating containing limits. The third part suggest ways of posing problems solvable by using elementary methods, based on the analysis of methods describes in the se- cond part of the thesis (without usage of the differential calculus). Keywords: limit, problem solving, problem posing, elementary function.
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Reeducating university students' mechanical knowledge in mathematical analysis
Šmídová, Kristýna ; Vondrová, Naďa (advisor) ; Kvasz, Ladislav (referee)
The topic of this thesis is the didactics of mathematical analysis. The thesis describes selected observations from the reeducation in an individual tutoring environment of for- mal knowledge of university students in the field of calculus. The aim of the thesis is to describe what formal knowledge appeared, to describe and evaluate selected reeducation interventions and on this basis formulate appropriate methodological recommendation. In the first chapter we deal with the contradiction between definition and concept concept of students, we outline how to convey to students the purpose of definitions and we suggest how to teach students to work with definitions properly, including understanding quan- tified propositions. In the second chapter we present the theory of process and concept together with the generic model theory. In the third chapter we explain the methods of work with students and the methods of the analysis of videos from tutoring. In the fourth chapter we analyze cognitive processes of the concept of sequence limits. KEYWORDS reeducation, individual tutoring, mechanical knowledge, calculus, definitions, quantified proposition, infinity, sequence, limit 1
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Limits and L'Hospital's rule
Ranšová, Kateřina ; Staněk, Jakub (advisor) ; Halas, Zdeněk (referee)
Title: Limits and l'Hospital's rule Author: Kateřina Ranšová Department: Department of Mathematics Education Supervisor: RNDr. Jakub Staněk, Ph.D., Department of Mathematics Education Abstract: The aim of this BA thesis is to introduce to the reader the concept of the limit of function and the means of its solution. The main impact of the thesis lies in a didactic approach and in the connection of a limit theory with its graphic representation and different methods of algebraic calculation. The text consists of eight chapters which can be divided into two parts according to their content. The first part explores the term "the limit of the function". Individual types of limits are then defined. To facilitate understanding, most of the definitions are accompanied by a particular example and a graphic representation . The first part is concluded by a unified definition which by means of the term "vicinity" summarizes all preceding types of limits. The other part deals with some basic methods of limits' calculation. Other topics include Taylor Series, l'Hospital's Rule and their applications to the limits. The core of the thesis is a comparison of calculation by means of l'Hospital's Rule and Taylor Series. The conclusion of the thesis presents some advantages and disadvantages of applying Taylor Series and...
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Problem posing for limits of functions
Prskavcová, Pavla ; Pilous, Derek (advisor) ; Kvasz, Ladislav (referee)
Title: Problem posing for limits of functions Author: Pavla Prskavcová Department: Department of Mathematics and Mathematical Education Supervisor: Mgr. Derek Pilous, Ph.D. The aim of the thesis is to describe methods of limit problems solving and of posing new problems that are solvable by those methods. The first part of the thesis contains the necessary theory (definitions and theorems). In the second part of the thesis there are described the standard methods used for evaluating containing limits. The third part suggest ways of posing problems solvable by using elementary methods, based on the analysis of methods describes in the se- cond part of the thesis (without usage of the differential calculus). Keywords: limit, problem solving, problem posing, elementary function.
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Functions in examples and counterexamples
Janda, David ; Pilous, Derek (advisor) ; Zhouf, Jaroslav (referee)
The aim of my Bachelors thesis is to explicate students coming to the uni- versity the key problems in fundamentals of mathematical analysis. I focus on the most notable terms of continuity and limit, which these secondary students were acquainted with. However, majority of them just intuitively and informaly. I am trying to point out the fact, that the knowledge of many students is distortid and uncomplete. As a result it is necessary to practise and clarify this knowledge so that the intuitive imagination of these terms corresponds to the formal definition. I am trying to get this point by brea- king of intuitive imaginations of students by counterexamples. Important is a chapter named The Construction of Functions, which contains instructi- ons leading to the finding functions with specific features. Not only these features, described in this thesis, but also more complex such as derivation, primitive function or uniform convergence. It is a consequence of the fact, that the principle of examples to practise these terms is in many sights similar and repetitious. In chapters named Continuity and Limit, I am interpreting these terms using the special examples, which are in my opinion optimal for rehearsing. My intention is to help illustrate selected problematical sections of mathematical analysis.
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