Day 1 started with a review of most of the previous content from the course, including functions, derivatives and some basic applications since we went two weeks without any lessons. Afterwards we began to work with the idea of an anti-derivative by showing some examples and deriving useful techniques like u-substitution or integration by parts. We ended class by tying together the idea of an area under a curve and the anti-derivative with the fundamental theorem of calculus. Overall, the class went fairly well and most became comfortable with the idea of an integral by the end.
We kept on reviewing the basic integration concepts from yesterday with most students and helped those who weren’t there the day before get caught up. Now that we had covered the fundamental idea of the integral, we demonstrated some uses in example problems related to physics and showed a couple of very basic differential equations (a topic I might elaborate on in a future blog post). Finally, we tied in the ideas from the course into robotics with a brief lesson on PID control. With the remaining time, we covered some miscellaneous topics like Taylor series and various proofs of e^ix = cos(x) + i*sin(x). By the end, everyone should be ready to apply the concepts of calculus.