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• Teaching meaningful mathematics with the Computer Algebra System MAXIMA using the example of inequalities

  53-65

  Karl Josef Fuchs

  Views:

  29

  The paper was originally motivated by the request to accentuate the meaningful contribution of inequalities in Mathematics Education. Additionally nationwide approved competences such as estimating come to the fore when organizing mathematical contents along some central Big Ideas. Not least the integration of computers enriches the reasonable discussion of inequalities by modern well accepted methodological principles. The freeware MAXIMA is used as Computer Algebra System (CAS) representatively.

• Infimum problems derived from the proofs of some generalized Schwarz inequalities

  41-57

  Zoltán Boros

  Árpád Száz

  Views:

  63

  We define f_(a;b)(r) = ar + b/r for all a, b, r Є R with r > 0. And, for some subsets A of R, we determine F_{A_+} (a; b) = inf (r Є A_+) f_(a,b) (r) ; where A_+ ={r Є A : r > 0}. The above in ma are mainly motivated by the proofs of some recent generalized Schwarz inequalities established by the present authors.

  Subject Classification: I35

• An idea which yields a lot of elementary inequalities

  61-72

  István Szalay

  Views:

  9

  The aim of the article is to show how studies in higher mathematics can be applied in everyday teaching practice to construct new problems for their pupils. In higher mathematics it is known that the set of real numbers with the addition and multiplication (shortly: (R,+,x)) is an ordered field. Considering a strictly monotonic increasing and continuous function σ with domain ...
  By this idea, using different kinds of functions σ we show a lot of different elementary inequalities.

• Integral part problems derived from a solution of an in mum problem

  43-53

  Paweł Pasteczka

  Árpád Száz

  Views:

  17

  In this paper, we solve the following two integral part problems:
  Find all r ϵ R satisfying r^2 = [r]*([r]+1), resp. r^2≤[r]*([r]+1).
  These problems have been mainly motivated by a solution of an infimum problem of Z. Boros and Á. Száz.

• Maximum and minimum problems in secondary school education

  81-98

  Zsolt Fülöp

  Views:

  31

  The aim of this paper is to offer some possible ways of solving extreme value problems by elementary methods with which the generally available method of differential calculus can be avoided. We line up some problems which can be solved by the usage of these elementary methods in secondary school education. The importance of the extremum problems is ignored in the regular curriculum; however they are in the main stream of competition problems – therefore they are useful tools in the selection and development of talented students. The extremum problem-solving by elementary methods means the replacement of the methods of differential calculus (which are quite stereotyped) by the elementary methods collected from different fields of Mathematics, such as elementary inequalities between geometric, arithmetic and square means, the codomain of the quadratic and trigonometric functions, etc. In the first part we show some patterns that students can imitate in solving similar problems. These patterns could also provide some ideas for Hungarian teachers on how to introduce this topic in their practice. In the second part we discuss the results of a survey carried out in two secondary schools and we formulate our conclusion concerning the improvement of students' performance in solving these kind of problems.

• Powers which commute or associate as solutions of ODEs

  241-254

  Alexander Gofen

  Views:

  16

  This paper is dedicated to the two classical transcendental functions: The locus of points for which powers commute, and the locus of points for which powers associate. These classical functions however are considered in a new perspective: as holomorphic solutions of ODEs passing over the points of singularity of these ODEs.
  Generally, solution functions which are holomorphic at singular points of the phase space of ODEs were studied in [2,3], and it was shown in [3], that certain holomorphic functions may satisfy only singular rational ODEs. This is the frame in which the function of commuting or associating powers are considered in this paper.
  First we obtain several types of ODEs satisfied by these functions. The obtained ODEs happen to have singular points, yet the solutions are proved to be holomorphic at these points, and their Taylor expansions are obtained. However it is not yet known whether these two transcendental functions can satisfy a regular rational ODE at the respective special points. The article also poses an open question about remarkable inequalities related to the commuting powers.

• Gaussian iteration of mean values and the existence of 2^(1/2)

  35-42

  Zoltán Daróczy

  Views:

  34

  We propose a method for proving the existence of √2 and finding its approximate value in secondary education.