Tangents to Parametric Curves (New) | Algebraic Calculus One | Wild Egg

Tangents are an essential part of the differential calculus. Here we introduce these important lines which approximate curves at points in an algebraic fashion -- finessing the need for infinite processes to support takings of limits. We begin by a discussion of tangents historically, especially for conics, touching base with the polarity of Apollonius and the key theorem of Pascal on hexagons on cubics. Then we introduce an explicit technology for obtaining tangents for parametric curves by specializing formulas for chords or secants. This is an important replacement to the usual "tangents via limits of chords" approach found in traditional courses. Using projective coordinates makes this especially elegant, and we discover pleasant formulas for a wide variety of important curves. Note: this is a modification of a previous video, where now we use projective coordinates in the following way: the affine point [x,y] is represented by [1:x:y], while the line a+bx+cy=0 is represented by (a:b:c). This is more in keeping with the general principle that simpler things should come first! Video Contents: 0:00 Intro 0:13 Tangent line 1:38 Descartes, Fermat 2:52 Constructing tangents to a conic 9:02 Tangents to a conic via Pascal's theorem 13:26 Tangents as "limits of chords" 15:54 Chords and tangents to a parabola 18:21 Chord 22:03 Strategy for finding/defining tangent 24:25 Curve, Parametrization, Tangent ****** 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   Wild Egg Maths Playlists:    • 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) 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)    • Elementary Mathematics Explained (K-6)   (40 videos)    • Ancient Mathematics and insights of Howard...   (7 videos)    • Wild West Banking: A mathematician goes We...   (7 videos)    • Playing Go: the ancient oriental board game   (19 videos)    • Maths and Music   (21 videos)    • Year 9 Mathematics (review fractions, deci...   (10 videos)    • Wild Trig: An introduction to Rational Tri...   (94 videos)