|
|
Integrals and differential equations in apllied problems - a digest of solved examples.
HOLUB, Miroslav
The main topic of the Thesis is the creation of a collection of selected solved interesting tasks intended for expanding the teaching of mathematics seminars at secondary schools, or as supplementary literature for students of various fields of university. The first part deals with the possibilities of applications of integral calculus in problems from common practice. Specific topics of some tasks can also be used in areas of professional circles. The second part of the Thesis focuses on mathematically solved problems, based on exceptional circumstances from everyday life. The goal of each example is to compile an ODE, determine the initial conditions, solve the equation and confront the task.
|
|
|
Teaching basic calculus at the secondary school, case study
Hamšík, Karel ; Vondrová, Naďa (advisor) ; Kvasz, Ladislav (referee)
This thesis deals with teaching the basics of calculus in secondary school. These are specifcally those areas of mathematics (functions, derivatives, integrals) that are subsequently discussed in more depth at universities of science, technology, economics, and also other felds. Most university students will encounter these math problems at least to a small extent. Therefore, I am interested in the extent to which pupils are prepared to solve these problems after leaving high school. The second topic that I am interested in is how much the pupils encountered the topics and terms of calculus during their secondary school studies, how they understand them and whether or not they can use them. The aim of this thesis is to describe the methods used in education of elementary school calculus in a specifc seminar where this material is discussed and subsequently verify the ability of the pupils to independently solve problems from this feld of mathematics by a set of didactic tests and their subsequent analysis. The analysis showed that the subject of calculus is for the most part well understood by the pupils. The seminar therefore fulflls the prerequisites for a successful introduction to this subject, which can be successfully expanded on in further higher education. Key words: teaching calculus,...
|
|
|
Basic notions of the calculus by Newton, Berkely and their followers
Mixa, Lukáš ; Kvasz, Ladislav (advisor) ; Dvořák, Petr (referee)
Seventeenth century is important not only for mathematics but for European social development in general. This thesis offers an overview about development of mathematics in the seventeenth century England. I present only those mathematical discoveries, which were relevantfor the work of Isaac Newton. In the first part I show the construction of logarithms by John Napier, Henry Briggs and Gregory Saint-Vincent. The second part is dedicated to methods of tangents and quadrature. I describe works of Pierre Fermat, John Wallis and Isaac Barrow. In the third part is shown how Isaac Newton used the mentioned findings for the development of the calculus. I use this example to demonstrate, that historical approach offers an illustrative connection between geometry, algebra and mathematical analysis and can be used in teaching. Keywords: Logarithm, tangent, quadrature, fluxion, fluent, calculus
|
|
|
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
|
|
|
Euler's number in calculus
RÁLKOVÁ, Lucie
The main aim of my thesis on the topic of "Euler's number in mathematical analysis" is to create an overview of the Euler numbers in calculus. This essay in the first part deals with the rise of the number e, in other parts of the current use of calculus. Purpose of this work is the insight students of secondary schools and universities to problems Euler numbers and to better understand the importance of e not only in mathematics.
|
|
|
Cavalieri's principle
Kreslová, Iva ; Halas, Zdeněk (advisor) ; Štěpánová, Martina (referee)
The Bachelor thesis deals with the development of key ideas important for formal formulating of Cavalieri's principle, its proof in general form and using Cavalieri's principle in determining the area of plane figures and volumes of solids. Determining of area and volumes using Cavalieri's principle is associated with the derivation of the well-known formulae for calculating area and volume.
|
|
|
Basic notions of the calculus by Newton, Berkely and their followers
Mixa, Lukáš ; Kvasz, Ladislav (advisor) ; Dvořák, Petr (referee)
Seventeenth century is important not only for mathematics but for European social development in general. This thesis offers an overview about development of mathematics in the seventeenth century England. I present only those mathematical discoveries, which were relevantfor the work of Isaac Newton. In the first part I show the construction of logarithms by John Napier, Henry Briggs and Gregory Saint-Vincent. The second part is dedicated to methods of tangents and quadrature. I describe works of Pierre Fermat, John Wallis and Isaac Barrow. In the third part is shown how Isaac Newton used the mentioned findings for the development of the calculus. I use this example to demonstrate, that historical approach offers an illustrative connection between geometry, algebra and mathematical analysis and can be used in teaching. Keywords: Logarithm, tangent, quadrature, fluxion, fluent, calculus
|
|
|
Alternative mathematical notation and its applications in calculus
Marian, Jakub ; Pick, Luboš (advisor) ; Zahradník, Miloš (referee)
We explore the possibility of formalizing classical notions in calculus without using the notion of variable. We provide a new mathematical 'language' capable of performing all classical computations (namely computing limits, finite differences, one-dimensional derivatives, and indefinite and definite integrals) without any need to introduce a variable. Equations written using our notation contain only func- tion symbols (and as such are completely rigorous and don't leave any room for vague interpretations). They also tend to be much shorter and more mathemati- cally transparent than their traditional counterparts (for example, there is no need for introduction of new symbols in integration, and definite integration is formalized in such a way that all rules (including 'substitution' rules) for indefinite integration translate directly to definite integration). We also fully formalize the Landau little-o notation in a way that makes computation of limits using it fully rigorous. 1
|
|
|
Mathematical apparatus of thermodynamics
TESAŘOVÁ, Jaroslava
This Thesis deals with the use of the mathematical apparatus in thermodynamics, specifically the use of Calculus of functions of several variables. The main emphasis is placed on creating a mathematical derivation of the theoretical foundations of thermodynamics, for example Maxwell relations. The following thing explains the linear differential forms with the help of them the laws of thermodynamics are defined. In the Thesis there are examples with solution to illustrate mathematical procedures. To understand this Thesis the knowledge of Calculus is required.
|
|
| |
guest ::
