I really enjoy learning from 3Blue1Brown math courses. In the calculus essence playlist, I jot down the notes of half of them:
Limits, L’Hopital’s rule, and espilon delta definitions
Integration and the fundamental theorem of calculus
What does area have to do with slope?
Higher order derivatives
What they won’t teach you in calculus
Limits, L’Hopital’s rule, and espilon delta definitions: the accurate math definition of derivative is as follows, note the notion of limit, or dx approaching to zero is emphasized here.
Integration and the fundamental theorem of calculus: it seems so obvious yet is the fundamental theorem of calculus:
What does area have to do with slope? Grant takes the meditation of fundamental theorem of calculus further to find the average height of an area. The question he asks and relates is the slope. Intuitively, average slope of a curve (sinx here) is equal to the slope between the beginning and ending point of the curve. Generally, any curve/area, the average slope of f(x)/derivative is the slope decided by the starting point and end point of its anti-derivative curve.
Higher order derivatives and Taylor series, Prof.Shankar has an online class (fundamental physics 16), a graph below by Grant illustrates:
What they won’t teach you in calculus, an example Grant showed is amazing and so inspiring in learning math and appreciate the fun of it.