Tangent conic sections to the graph of a function and their derivatives

Abstract

The geometric problem of the tangent line to the graph of a function at a point P is studied within the derivative concept. The geometric problem of the tangent can be extended to other tangent curves and not only to the simplest curve (the tangent straight line). The tangent line is a specific case of a tangent conic section since a line is a degenerate conic section. Therefore a tangent conic section is a more general solution as this contains also the geometric problem of the tangent line. Here, the tangent conic sections to the graph of a function are determined. The tangent conic section contains the point P as its vertex whose tangent line is equal to the tangent line of the function at that point P. The second-degree equation of a tangent conic section is an implicitly defined function. So, the parametric equations are used to obtain the graphs and the derivatives easily. Tangent conic sections to a given point P of a function are calculated in illustrative examples.