Essence of calculus
12 episodes · 2017
The goal here is to make calculus feel like something that you yourself could have discovered.
1Essence of calculus
In this first video of the series, we see how unraveling the nuances of a simple geometry question can lead to integrals, derivatives, and the fundamental theorem of calculus.
2The paradox of the derivative
Derivatives center on the idea of change in an instant, but change happens across time while an instant consists of just one moment. How does that work?
3Derivative formulas through geometry
A few derivative formulas, such as the power rule and the derivative of sine, demonstrated with geometric intuition.
4Visualizing the chain rule and product rule
A visual explanation of what the chain rule and product rule are, and why they are true.
5What's so special about Euler's number e?
What is e? And why are exponentials proportional to their own derivatives?
6Implicit differentiation, what's going on here?
Implicit differentiation can feel weird, but what's going on makes much more sense once you view each side of the equation as a two-variable function, f(x, y).
7Limits, L'Hopital's rule, and epsilon delta definitions
Formal derivatives, the epsilon-delta definition, and why L'Hôpital's rule works.
8Integration and the fundamental theorem of calculus
What is an integral? How do you think about it?
9What does area have to do with slope?
Integrals are used to find the average of a continuous variable, and this can offer a perspective on why integrals and derivatives are inverses, distinct from the one shown in the last video.
10Higher order derivatives
A very quick primer on the second derivative, third derivative, etc.
11Taylor series
Taylor polynomials are incredibly powerful for approximations, and Taylor series can give new ways to express functions.
12What they won't teach you in calculusfinale
A visual for derivatives which generalizes more nicely to topics beyond calculus.
