Differentiability and Tangent Planes in Multivariable Functions

This lecture covers the concept of differentiability in multivariable functions, focusing on the geometric interpretation of tangent planes. The instructor begins by discussing administrative updates regarding the course and exam format, emphasizing the importance of understanding the material. The lecture then delves into the definition of differentiability, explaining how a function is differentiable at a point if it can be approximated by a linear function near that point. The instructor illustrates this with examples, showing how to derive the equation of the tangent plane at a given point. The relationship between differentiability and the existence of partial derivatives is also highlighted, along with the conditions under which a function is continuous. The lecture concludes with a discussion on the implications of differentiability in higher dimensions, including the concept of hyperplanes and the conditions for differentiability in multiple variables. Visual aids and examples are used throughout to enhance understanding of these complex concepts.