When I first taught myself calculus prior to more formal learning, I already had an idea of the concept of differentiation as a tool to help us measure the rate of change of a curve. I also already had some idea of higher differentials, a vague sense of integration, and some basic suspicions with regards to the meaning of the fundamental theorem of calculus, as I picked up on these concepts from various books on physics and higher mathematics. Similarly, my early interests in physics led to an awareness that there must be some way to measuring things like acceleration when it is not constant, and thus, graphically, not linear.
So there was always some idea of calculus and its connection, but how to prove it and come about truly understanding it?
When trying to understand some of its core concepts, I didn’t have rigorous formal proofs or derivations at hand. I did have a bit of guidance, but mostly I wanted to think through why calculus works and also why some of its deeper connections and applications make sense. I enjoy thinking from first principles, and I believe that mathematics is much more than ‘plug-and’chug’ formulas. To really appreciate and understand mathematical concepts and applications, first principle exercises are important. And so, in that a lot of my early points of entry were based on intuitive reasoning, the following video series shows how to build on knowledge of things like linear graphs and how the area under a line represents the distance traveled. The series reflects on some of my early thoughts on calculus, or at least some of the intuitive reasoning I used. It was fun to do, even for nostalgic purposes. But it may also prove useful for others learning the basics of differentiation and integration.
In future videos I will discuss more rigorous and formal proofs – there are a few that I know, and a couple of them are quite beautiful. In the future, I would also like to make a similar introductory series on multivariable calculus and maybe also another separate series on linear algebra.