Explore the historical development and fundamental concepts of calculus in this comprehensive lecture. Trace the origins of calculus from ancient Greek mathematicians to its expansion in the 17th century, examining the contributions of key figures like Fermat, Newton, and Leibniz. Delve into the two main branches of calculus—differential and integral—and their applications in solving problems related to tangents, maxima and minima, areas, and volumes. Discover how power series, infinitesimals, and differentials shaped the modern understanding of calculus, and gain insights into the notations and approaches that revolutionized mathematical thinking.