Mean Value Theorem for Derivatives (4) – Tangent, Secant, and Dynamic Visualizat

This GeoGebra example dynamically illustrates the geometric meaning of the Mean Value Theorem (MVT) for derivatives, showing that for a continuous and differentiable function on an interval, there exists a point where the tangent slope equals the secant slope. Core mathematical concepts include derivatives, tangent lines, secant lines, slope, and the conditions and conclusion of the MVT. Construction highlights: using commands such as Line, OrthogonalLine, Intersect, Segment, Tangent, and Textfield to build the function curve, secant line, tangent line, and intersection points; InputBox1 and Textfield allow parameter input, while slidingLine and sliders control a moving point; button1 and button2 enable interactive toggling; list1 and text1 display numerical values and explanations. Suitable for senior high school or introductory college calculus, this applet helps students intuitively grasp the geometric essence of Rolle's theorem and Lagrange's MVT, and observe the critical point where slopes become equal during dynamic changes.