Derivatives and Critical Points: Tangent, Slope, and Extrema Visualization

This GeoGebra example dynamically illustrates the relationship between a function's derivative and its critical points. It shows how the slope of the tangent line at a moving point on the graph changes, and uses the condition derivative equals zero to locate critical points (local extrema). Core mathematical concepts include the geometric meaning of the derivative (tangent slope), the definition of critical points (where derivative is zero or undefined), and the first-derivative test for monotonicity and extrema. Construction highlights: using GeoGebra commands Point to place a movable point A on function f, Tangent to create the tangent line g at A, Slope to compute its slope m, OrthogonalLine to construct perpendiculars, Intersect to find intersection points, and f' to represent the derivative function graph. Text objects text1 and text2 display numerical values, while auxiliary points B, C, and other objects complete the exploration environment. Suitable for high school calculus introduction or college-level basic calculus teaching, this example helps students understand how sign changes of the derivative relate to extrema, and is ideal for classroom demonstration or self-guided exploration.