Newton's Method: Geometric Visualization of Root Finding

This GeoGebra example geometrically illustrates Newton's Method (Newton-Raphson method) for finding roots of a real-valued function. It visually demonstrates how successive approximations converge to the root of a function $f(x)$ by drawing tangent lines at successive points. The core mathematical concepts involved are derivatives, tangent lines, limits of sequences, and iterative numerical root-finding. The construction utilizes key GeoGebra commands such as `Tangent` to construct the tangent line to the curve $f$, `Intersect` to find the intersection of the tangent with the x-axis (generating the next iterate $x_{n+1}$), `Segment` to draw the steps of iteration, and `Textfield` to display the numerical values of the sequence. It is designed for calculus students and educators in high school or university levels to build an intuitive, geometric understanding of convergence and derivative applications.