Calculus Foundations: Taylor Series and Integrals

This lecture covers the foundational concepts of calculus, focusing on Taylor series and integrals. The instructor begins by discussing the importance of limits and the challenges of determining the degree of Taylor expansions needed to resolve indeterminate forms. The lecture emphasizes the significance of understanding the convergence of series and the relationship between functions and their Taylor series representations. An example of a function that is infinitely differentiable but cannot be represented by its Taylor series is presented, illustrating the subtleties of calculus. The discussion then transitions to integrals, explaining the historical context of integral calculus and its applications in calculating areas under curves. The instructor introduces the concept of Riemann sums and the process of approximating areas using rectangles, leading to the formal definition of integrals. The lecture concludes with a theorem regarding the integrability of continuous functions, setting the stage for further exploration of integration techniques in subsequent sessions.