(553) PRACTICAL TASKS FOR DETERMINING THE EXTREME VALUES - FINDING SOLUTIONS WITH DERIVATIVES AND VISUALISATION WITH GEOGEBRA

摘要

Many important applied problems involve finding the best way to accomplish some task. Often this involves finding the maximum or minimum value of some function: the minimum time to make a certain journey, the minimum cost for doing a task, the maximum power that can be generated by a device, and so on. Many of these problems can be solved by finding the appropriate function and then using techniques of calculus to find the maximum or the minimum value required. In this paper we will examine several specific tasks in which it is required to determine the conditions under which a value reaches a maximum or minimum through visualization and interaction using the free dynamic mathematical software GeoGebra. Visualisation of 2D and 3D objects and changing the dimensions of the objects, allow us to connect the algebraic ideas with dynamic visual representation. "Hand" developed solutions can be compared with the results gained with the Geogebra constructions and formulas. By using Geogebra tool “slider” students can change the value of the length of a side or the size of an angle and thus change the perimeter or the area of 2D figure or volume of 3D body. Thus, for defined range of input values, set of different result can be gained. What can students analyse and examine? Is the gained solution with derivatives the only solution, are there more combinations for the defined input values when the maximum or minimum value of the area, the perimeter or the volume is obtained, how the perimeter, the area or the volume are changing when the value of one slider is decreasing and at the same time the value of other slider is increasing e.t.c. What can we conclude analyzing the function of the formula for calculation of volume for example? By changing the values of the sliders students can see a point on the graph of the function that is moving, where x coordinate is the value of the length of a side or the size of an angle and y coordinates the value of the volume of the body and thus see when the function reaches it's maximum or minimum. Furthermore, they can analyse the change of the values of the moving point on the graph of the function together with the changing dimensions of constructed 2D figure or 3D body from the formulas entered in the input bar of the Algebra window and at the same time have visual presentation of the Geogebra construction. For each of the problems a solution is prepared with derivatives, also GeoGebra applet to explore and make conclusions, GeoGebra applet to examine the graph of the function and directions for interactivity on how to use the applets. The interactive materials are placed on our wikispaces wiki and therefore are accessible to all students worldwide not just from school lan's but also from home at any time. The idea of implementation of new computer technology in the math education is to contribute to the process of developing student-centered instead of teacher-centered learning environment by using appropriate techniques and strategies that promote and enhance the critical, creative, and evaluative thinking capabilities of students and at the same time encourage students self motivation and interest in learning of mathematics.