Tangent Line and Slope Change: Limits to Derivatives

This GeoGebra example lets you drag a point along a given curve and observe how the tangent line’s steepness (slope) changes at different locations. The activity is designed to build intuition for the idea that the slope of a tangent line corresponds to a rate of change captured in the limit process. The main math concepts are function graphs, tangent lines, local linear approximation, and the relationship between limits and derivatives. Construction-wise, the setup uses the function object f to represent the original curve, a as the moving point (or chosen tangency point), A as a key auxiliary object tied to the construction, and g as the tangent line (its equation or geometric representation). As a is dragged, g updates accordingly, making the connection between limit-based slope and derivative visually immediate. Suitable for high-school advanced learners and introductory calculus students, especially for teaching and learning limits leading to derivatives via graphing and interactive geometry.