A contribution to the development of functional thinking related to convexity and one-dimensional motion

Miodrag Mateljević, Marek Svetlik

Mathematical concepts are defined precisely using the language of the branch of mathematics to which they belong. But their meaning can be enriched through different interpretations and those of them belonging to the real world situations, we call "vivid" mathematics. In contacts with Professor M. Marjanović, we investigated a case of "vivid" mathematics in some earlier papers and we continue to do so in this paper. Suppose that a liquid (water) flow has a constant inflow rate and that a vessel has the form of a surface of revolution, and suppose that this process begins at moment $t=0$ and ends at moment $t=T$. We study the dependence of the height $h(t)$ of the liquid level at the time $t$, which will be called the {t height filling function}. It is convex or concave depending on the way how the level of the liquid changes. This vivid interpretation holds in general, namely we prove that given a strictly increasing convex (concave) continuous function on $[0,T]$ satisfying certain conditions, there exists a vessel such that its height filling function is equal to the given function. This is a fact that seems to be new and we continue paying attention to it. In this way, we hope that we are providing a matter that can serve as a motivation and an illustration for a deeper understanding of basic concepts and ideas of the differential and integral calculus. It can also serve for a further development of functional thinking in teaching mathematics. We also consider a more general concept of one-dimensional motion, including changes in direction of motion and the difference between velocity and acceleration defined by the position and the path as functions of time. We indicate how one can apply this for studying the height filling function of a liquid flow, which can be considered now as a one-dimensional motion of a liquid along the axis of rotation of the vessel.