General Cubic Function: Derivatives, Integrals, and Osculating Circles

This interactive GeoGebra example dynamically illustrates a general cubic function f(x)=ax³+bx²+cx+d using sliders and draggable points. It demonstrates the graph of the cubic, its derivative, extrema, inflection point, definite integral area, curvature vectors, and osculating circles. Core mathematical concepts include polynomial functions, derivatives, turning points, inflection points, definite integrals, curvature, and osculating circles. The construction uses GeoGebra commands such as TurningPoint, Extremum, InflexionPoint, Point, If, Integral, CurvatureVector, and Circle. Sliders a, b, c, d adjust the coefficients; green points set the integration limits; a yellow point moves along the curve to explore different osculating circles. Suitable for high school and university students learning calculus, function analysis, and differential geometry, as well as for classroom demonstrations and self-study.