Inscribed Rectangle under a Parabola

This example demonstrates how to construct a rectangle inscribed under a parabola and dynamically adjust its vertices to observe the area change, ultimately finding the position that maximizes the area. The core mathematical concepts include quadratic functions, rectangle area formulas, differential calculus for finding extrema, and the combination of function graphs with geometric shapes. The construction uses GeoGebra's parabola tool to draw a standard parabola, then creates perpendicular and horizontal lines to determine the four vertices of the rectangle, with a slider or draggable point to vary the rectangle's width while displaying the area value. This example is suitable for high school or early college students, serving as an intuitive introduction to derivatives in optimization problems and as a dynamic demonstration in calculus teaching.