|Volume XVI , issue 1 ( 2013 )
|Generalizations of some remarkable inequalities
In this paper, we present a generalization of Mitrinović's inequality for polygons, and triangles, a generalization of J. Radon's inequality and a generalization of Nesbitt's inequality. The main tool in the proofs is the inequality of Jensen.
Keywords: Jensen inequality; Mitrinović inequality; J. Radon inequality; Bergström inequality; Nesbitt inequality.
|On computing the derivative of a function
This note discusses a fact and its application on examining the existence of a derivative of a function at a point. The application provides a relatively easier method while avoiding laborious computations when standard computing is used. This may be introduced as a part of applications of the derivative in a Calculus course.
Keywords: functions; derivatives; mean value theorem.
|From matchstick puzzles to isoperimetric problems
In this paper, we are going to present a way from a simple matchstick puzzle to isoperimetric problems. This is a much more interesting, simple and instructive way to learn the basics of isoperimetry. The elements of mathematical history used in the paper are meant to make the topics described more colorful and interesting. One of the important features of the paper is that all the issues are discussed using elementary mathematics, so that it would be easily accessible to anyone.
Keywords: matchstick; puzzle; isoperimetric; perimeter; area; polygon.
|A somewhat unexpected concavity
This classroom note considers the slightly counterintuitive concavity of a rational function.
Keywords: rational function; concavity.
|Polynomial division and Gröbner bases
Division in the ring of multivariate polynomials is usually not a part of the standard university math curriculum. However, the algorithm is elementary and it has very important consequences for algebraic computations. In this paper, the algorithm is explained and illustrated with some examples, and the importance of the choice of monomial ordering is stressed. The notion of Gröbner basis is introduced and explained on examples. The paper can be used by math students and teachers as a brief description of this very important topic and introduction for reading more detailed textbooks.
Keywords: multivariate polynomial division; Gröbner basis.
|Definition of the definite integral
In this paper we suggest a new approach for a definition of definite integral of a real function in the first course in Mathematical Analysis. The definite integral exists if for any sequence of partitions, the upper sum and the lower sum of Darboux have the same limit. If the definite integral of a real function exists, then we can simply compute it, as a limit of a sequence of integral sums of Riemann.
Keywords: Definite integral; upper and lower sum of Darboux; Riemann sum.
|A geometric maximization problem
We consider an area maximization problem to illustrate the importance of analytical work before solving an equation numerically.
Keywords: maximization; area; numerical method.
|A survey on use of computers in mathematical education in Serbia
This paper examines the issues involved in using computer facilities and educational software GeoGebra in teaching mathematics. It documents them in the light of results of the surveys conducted during a six months period. We have interviewed 43 Serbian teachers during three different seminars about mathematical education. The results suggest that even though computers are present in Serbian schools, the use of computers and mainly the educational software GeoGebra are still not on a satisfying level.
Keywords: Serbian teachers; teaching mathematics; computers; GeoGebra.