Family Of Polynomial Functions Research Articles

Minimizing inside a triangle with GeoGebra

In this article, we consider certain minimization problems. If and are the distances of a boundary or inner point to the sides of a given triangle, find the point which minimizes for positive integer n. These problems can be afforded easily with GeoGebra. We consider two examples, the first concerns an isosceles triangle and the second a scalene triangle. In both cases, we divide the triangle horizontally into line segments parallel to the base and look at a family of polynomial functions. Using GeoGebra, we observe in the case of isosceles triangle that the minimum point of each member of the family lies on the y-axis. Tracking these points vertically we discover the critical point which minimizes , In particular, we show that the sequence of these critical points converges to the incenter of the triangle. In the case of a scalene, we observe that the minima points of the polynomials lie on a curve, the minimum of which can be traced with GeoGebra and computed with basic calculus. Finally, we consider a discussion with some references concerning general solutions of ‘minimal sums of distances’ and ‘minimal sums of squared distances’.

Becoming Aware Of Mathematical Gaps In New Curricular Materials: A Resource-Based Analysis Of Teaching Practice

The study featured in this article, with its central focus on resources-in-use, draws upon salient aspects of the documentational approach of didactics. It includes an a priori analysis of the curricular resources being used by a teacher for the first time, followed by detailed in situ observations of the unfolding of her teaching practice involving these resources. The central mathematical problem of the lesson being analyzed deals with families of polynomial functions. The analysis highlights the teacher's growing awareness of the mathematical gaps in the resources she is using, which we conjecture to be a first step for her in the evolutionary transformation of resource to document, as well as an essential constituent of her ongoing professional development.

Apparently first closed‐form solutions for inhomogeneous vibrating beams under axial loading

Closed‐form solutions are presented for fundamental natural frequencies of inhomogeneous vibrating beams under axially distributed loading. The mode shape is postulated as coinciding with the static deflection of the associated homogeneous beam without distributed axial loading. Then the inverse problem of determining the stiffness and mass density distributions, producing the above mode shape, is solved. To describe these variations, the family of polynomial functions is used. Several sets of boundary conditions are considered. It is shown that the natural frequency vanishes when the intensity of the axially distributed loading equals the critical buckling value. A linear relationship is established between the square of the natural frequency and the load ratio for all reported sets of boundary conditions, in contrast to uniform beams where the exact linear relationship holds for columns with simply supported and/or sliding ends.