Taylor Polynomials: Calculating Limits and Derivatives

This lecture focuses on the calculation of Taylor polynomials and their applications in determining limits and derivatives of functions defined from R² to R. The instructor begins by reviewing examples of Taylor expansions, specifically around the point (1,1) for various functions. The lecture emphasizes the importance of understanding the structure of polynomials and how to manipulate them to find their Taylor series. The instructor introduces two methods for calculating Taylor expansions: the direct formula and a variant known as the 'end of the wolf' technique, which allows for centering the expansion around a specific point. The discussion includes practical examples, such as the exponential function and sine function, demonstrating how to derive their Taylor series. The lecture concludes with a generalization of Taylor polynomials to higher dimensions, highlighting the relevance of these concepts in multivariable calculus and their applications in real-world problems.