Explore an advanced lecture on differential geometry that delves into Lagrange's algebraic approach to calculus. Learn how to associate polynomial functions with tangent lines, conics, and higher-order approximations using only elementary high school math. Discover the concept of sub-derivatives and their importance in differential geometry. Examine the quadratic approximation and tangent conics, including a beautiful theorem about cubic polynomials. Extend these concepts to multivariable functions and surfaces, using the Folium of Descartes as an example. Gain insights into a powerful alternative approach to understanding functions and their geometric properties.