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Calculus using proximities: a mathematical approach in which students can actually prove theorems

Richard O’Donovan (2017)

Open Mathematics

Teaching and learning calculus are notoriously difficult and the didactic solutions may involve resorting to intuitive but vague definitions or informal gestures offered as proofs. The teaching literature is rife with examples of metaphors, adverb manipulations and descriptions of what happens “just before” the limit. It is then difficult to leave the domain of the mental image, thus losing the training in rigour. The author (with Karel Hrbacek and Olivier Lessmann) has endeavoured a radically different...

Equation f ( p ( x ) ) = q ( f ( x ) ) for given real functions p , q

Oldřich Kopeček (2012)

Czechoslovak Mathematical Journal

We investigate functional equations f ( p ( x ) ) = q ( f ( x ) ) where p and q are given real functions defined on the set of all real numbers. For these investigations, we can use methods for constructions of homomorphisms of mono-unary algebras. Our considerations will be confined to functions p , q which are strictly increasing and continuous on . In this case, there is a simple characterization for the existence of a solution of the above equation. First, we give such a characterization. Further, we present a construction...

Geometrické důkazy v matematické analýze

Antonín Slavík (2019)

Pokroky matematiky, fyziky a astronomie

Článek představuje alternativní pohled na některé známé výsledky z matematické analýzy. Ukazuje jejich geometrickou interpretaci, případně využívá vhodných ilustrací ke zdůvodnění platnosti příslušných tvrzení. Uvedené postupy nejsou vždy zcela precizní, na rozdíl od formálně přesných důkazů však umožňují hlubší porozumění problematice.

Grafy funkcí z pohledu žáků a studentů základních, středních a vysokých škol

David Janda, Derek Pilous (2017)

Učitel matematiky

The article focuses on students' understanding of graphs of functions. Interviews were conducted with 22 students in who solved some tasks on classifications of graphs of functions. For the analysis of data, we used the theory of prototypes and the theory of exemplification and the framework of hypothetical learning trajectories. Some extracts from the interviews are given to illustrate main results. For instance, we observed that students focus on various aspects of the graph (linearity, passing...

Každý má svou posloupnost

Vlastimil Dlab (2013)

Učitel matematiky

The article concerns a certain type of recursive sequence whose subtype is Fibonacci sequence. Its goal is to show the mathematical core of the general sequence and deduce a general formula for the n -th term by the method which comes from Abraham de Moivre. An application of the above is given, for tasks of the kind “mixing two solutions consisting of certain percentage of certain liquid”.

Limita posloupnosti a věta o sevření

Emil Calda (2012)

Učitel matematiky

Článek pojednává o větě o sevření, známé také jako věta o dvou policajtech. Je zde uveden její důkaz a příklady, jak se dá použít ke zjišťování limit posloupností.

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