Supporting hyperplanes

This lecture introduces supporting hyperplanes for differentiable functions in R to the N, aiming to approximate the function's graph by hyperplanes. Starting with one-dimensional theory, the instructor explains the concept of tangent lines and extends it to higher dimensions, emphasizing the difference between lines and hyperplanes. The lecture covers the generalization of tangent lines to hyperplanes in higher dimensions, providing equations and examples for hyperplanes in two and three dimensions. The instructor also discusses linear approximations and supporting hyperplanes, demonstrating how to find the equation of a supporting hyperplane and its normal vector. Through detailed explanations and examples, the lecture illustrates the process of determining supporting hyperplanes and linear approximations for functions in different dimensions.