First-Order Approximation Zoom: Local Linearization Explorer

① This GeoGebra example visualizes how the first-order Taylor approximation (tangent line) of a function matches the original curve near a selected point, with interactive zoom controls to explore the concept of local linearity. ② Core mathematical concepts include derivatives, tangent lines, first-order Taylor polynomials, local linear approximation, and the behavior of functions in small neighborhoods. ③ Construction highlights: using commands like Point, Polygon, TaylorSeries, and Textfield, the applet builds function f, point A, zoom object, auxiliary points C1-C4, polygon poly1, tangent line TanLine, secant toggle togSecant, input boxes for parameters, and sliders to adjust the viewing window (xMin/xMax/yMin/yMax). ④ This is ideal for high school or introductory college calculus courses to demonstrate the geometric meaning of derivatives and the idea of linearization, and can be used for interactive student exploration.