Faniya Ahmetova

The paper considers the technique of symbolic and numerical differentiation in the MathCAD; shows the perspective of the use of the application package in the learning process. All actions performed under differentiation in MathCAD are illustrated with specific examples. Of course, first-year students first have to learn the technique of differentiation without the software. However, MathCAD package can be used as a mean of control and self-control in solving the tasks on differentiation. After solving a particular problem analytically, correctness of the answer can be checked by using MathCAD. Thus, MathCAD is a perfect tool to help students in their independent work.

The paper is devoted to teaching theory of normal systems solutions in the course “Differential Equations” and to the problems that arise in the presentation of the material. In studying the discipline, students face difficulties in finding the general solution of the normal systems of ordinary differential equations (ODE). One of the methods for solving systems is the method of allocation of integrable combinations, that is getting out of the system such equations that can be integrated and thus obtain the first integrals. Their aggregate determines the general decision or the general integral of the system. In this regard, the article describes the normal system of ODE and their symmetrical forms of recording. The basic concepts are interpreted: Cauchy problem, theorem of existence and uniqueness of normal ODE systems solutions, first integrals for systems. It illustrates in detail the method of finding integrable combinations, first integrals and general solutions of various systems on a wide range of tasks. The paper will be useful for students of technical universities and lecturers during seminars on the subject “Differential equations”.

The paper is devoted to the historical development of geometry. The purpose of the research is to show that apart from the geometry, which is taught in schools and universities, there is another geometry, called Lobachevskian geometry. It significantly differs from the Euclidean geometry. Coming from delivered purposes, the following major objectives are identified in this work: the main provisions of Euclidean geometry and the foundations of the Lobachevskian geometry, the consistency of Lobachevskian geometry. The paper will be useful to students of physical and mathematical departments of universities and pedagogic institutions of higher education. It can be used by lecturers and students in classes with profound study of mathematics.

The paper is devoted to teaching theory of limits in the course of mathematical analysis and the problems that arise in presenting educational material. To master the recording skills of the Cauchy function limit determination in the language «epsilon-delta», the authors propose a table that addressed all possible argument aspirations, written through the surroundings and intervals; provide definitions of the function limit for all cases reviewed in the table and explain in detail the content and meaning of these basic definitions on the examples of tasks. The authors offer the table that summarizes the uncertainties and shows the ways of their elimination. Calculation method of all kinds of limits on a wide range of tasks is also illustrated. The paper can be useful to lecturers and first-year students.