Lecture 04: The Fundamental Theorem of Calculus

This paragraph introduces the two core concepts of calculus: the derivative and the integral. The derivative is used to find the instantaneous velocity from a position function, while the integral calculates the distance traveled given the velocity at every moment. Both processes involve infinite operations, such as dividing time into smaller intervals for better approximations. The paragraph also highlights the connection between these concepts, suggesting they reflect a common underlying reality, which is the basis of the fundamental theorem of calculus. The scenario of a car moving down a straight road is used to illustrate both derivative and integral processes, emphasizing that knowing the position or velocity at every moment allows for the calculation of velocity or distance, respectively.

The speaker delves into the fundamental theorem of calculus, which connects the derivative and the integral, using the example of a car's motion. Two methods for calculating distance traveled are presented: the integral method, which involves summing up infinitesimally small distances over time, and the simpler method of using the position function to find the net distance by subtracting the starting point from the ending point. The paragraph clarifies that the fundamental theorem of calculus provides an alternative, often simpler, way to compute values that would be difficult or tedious to calculate using the integral method alone.

The paragraph explores the process of finding a car's distance traveled using velocity functions. It contrasts the integral process, which involves dividing time into small intervals and summing the distances covered in each, with the derivative process, which involves finding the position function from which the velocity function is derived. The fundamental theorem of calculus is highlighted as it allows for the computation of distance traveled by simply using the position function's final and initial values, thus avoiding the laborious process of infinite addition.

This section discusses the historical development of calculus, acknowledging the contributions of ancient mathematicians like Eudoxus and Archimedes, and the incremental advancements made by others such as Isaac Barrow. The paragraph focuses on the significant roles played by Isaac Newton and Gottfried Wilhelm Leibniz in systematizing calculus and revealing its fundamental theorem. It also touches on the controversy surrounding the credit for calculus' invention and the poem by Alexander Pope that celebrates Newton's contributions, as well as Leibniz's acknowledgment of Newton's work.

The paragraph delves into the controversy over who should be credited with the invention of calculus, highlighting the reluctance of Isaac Newton to publish his findings, which led to a dispute with Leibniz. Newton developed the concepts in the 1660s but did not publish them until much later, while Leibniz published his work in 1684. The dispute led to a division between British and continental mathematicians, hindering the development of calculus in Britain. The paragraph also discusses Newton's attempt to establish his priority through an anagram and the accusations that Leibniz had plagiarized Newton's work.

This paragraph addresses the conceptual challenges faced by early calculus developers, particularly the lack of a clear understanding of limits and real numbers. It discusses the criticism Newton and Leibniz received for their use of infinitesimals and the lack of clarity in their explanations. The paragraph also mentions George Berkeley's satirical work 'The Analyst,' which questioned the logical foundations of calculus. The resolution of the concept of limits is noted to have occurred much later, in the 19th century, which underscores the significant hurdles in the early understanding and acceptance of calculus.