Connecting Derivatives and Integrals: From Slope to Area

This interactive GeoGebra example visually demonstrates the inverse relationship between derivatives and integrals. By selecting points on the graph of a function f, tangent slopes are computed to construct the derivative function; simultaneously, the original function is reconstructed by accumulating small rectangle areas, illustrating integration as a process of area summation. Core mathematical concepts include the geometric meaning of the derivative (tangent slope), the geometric meaning of the definite integral (area under a curve), the Fundamental Theorem of Calculus, and the correspondence between a function, its derivative, and its antiderivative. The construction uses GeoGebra commands such as Point, Sequence, and Textfield, along with objects like slider inc to control the increment, points A and topend to mark key positions, function f with display toggles chkShowF and chkShowFprime, multiple line segments (dlines1, dlines2, flines, fsegs, dersegs) for drawing tangents and partition rectangles, and intgraph, intgraph2 with area toggle chkShowArea to display the integral area. This resource is suitable for high school to early college calculus instruction, particularly for helping students grasp the fundamental theorem of calculus, and can be used for classroom demonstrations or self-directed exploration.