Tangent conics and tangent quadrics | Differential Geometry 5 | NJ Wildberger

In this video we further develop and extend Lagrange's algebraic approach to the differential calculus. We show how to associate to a polynomial function y=p(x) at a point x=r not just a tangent line, but also a tangent conic, a tangent cubic and so on. Only elementary high school manipulations are needed--no limits or real numbers-- and we efficiently obtain a hierarchy of approximations to a polynomial at a given point. Instead of derivatives, closely related quantities called sub-derivatives grab the spotlight. They are often simpler and more general quantities! The quadratic approximation, given by the tangent conic, will be crucially important for us in our development of differential geometry: the key point is that the subject is largely what we get when we look at curves and surfaces quadratically! Tangent conics (and higher approximations) of polynomial curves is a potentially rich theory that deserves a lot more attention. We highlight a beautiful observation of E. Ghys: that for a cubic polynomial, the various tangent conics are disjoint (this is in my opinion the loveliest theorem in calculus). Should not all undergraduates be exposed to such natural geometric applications in their calculus courses?? The power of this point of view is shown clearly by the ease with which we can extend it to the multivariable situation. A function of two variables z=p(x,y) defines a surface, which may be studied at a point [x,y]=[r,s] in an analogous way, yielding at each point a tangent plane, a tangent quadric, a tangent cubic surface etc. We explicitly look at the surface associated to the Folium of Descartes, namely z=x^3+y^3+3xy and try to visualize it. This is a powerful but elementary alternative to the usual way of thinking about functions! *** Research Gate page: https://www.researchgate.net/profile/... Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon:   / njwildberger   Your support would be much appreciated. Insights into Mathematics YT channel:    / njwildberger   Insights into Mathematics Playlists:    • The Algebra of Boole, Logic and Circuit An...   (31 videos)    • Box Arithmetic: a new framework for Mathem...   (18 videos)    • Hypergroups and Diffusion Symmetry: an int...   (6 videos)    • Rational Trigonometry for maths, physics a...   (4 videos)    • Sociology and Pure Maths   (44 videos)    • Old Babylonian mathematics and Plimpton 322   (8 videos)    • Math Foundations   (226 videos)    • Math Seminars N J Wildberger   (26 videos)    • Math History (ancient to modern)   (45 videos)    • Geometric Linear Algebra   (43 videos)    • Algebraic Topology   (40 videos)    • Universal Hyperbolic Geometry   (55 videos)    • Differential Geometry   (34 videos)    • Elementary Probability and Statistics   (8 videos)    • Math Terminology for Incoming Uni Students   (9 videos)    • Famous Math Problems   ( 46 videos) Wild Egg Maths Playlists:    • Box Arithmetic Book: A finite multiset fou...      • Intro to Algebraic Calculus with Box Arith...   (4 videos)    • Classical to Quantum (for Members)   (64 videos)    • Solving Polynomial Equations and the Geode...   (45 videos)    • De Casteljau Bezier curves and associated ...   (20 videos)    • Exploring q-series (for Members)   (8 videos)    • Six: A mathematical exploration   (9 videos)    • Algebraic Calculus One: a new foundation f...   (52 videos)    • Advice to prospective research mathematici...   (9 videos)    • The Hexagrammum Mysticum: a Geometric Gem ...   (14 videos)    • Algebraic Calculus Two   (8 videos)    • Special (Polynomial) Functions and Maxel N...   (25 videos)    • Dynamics on Graphs: The ADE Phenomenon & B...   (30 videos)