On tangents

The importance of tangents in calculus is emphasized by the relation between tangents and derivatives. The graph of any function F possesses a tangent at the location which has the coordinates (c, F(c)), if the derivative F(c) is present. The reverse of this does not exist. Tangents are also related to the derivatives through tangent vectors. The tangent of a simple arc has a line at P, when the simple arc possesses a non-zero tangent vector at the location P. The reverse generally does not exist, but under certain conditions, when P is a cusp, it may exist.