Introduction The other day, while doom-scrolling youtube, I came across a video from Michael Penn about finding an extremely high-order derivative using Linear Algebra to avoid the bulk of what the work would otherwise entail. The problem being considered in the video was: Given $f(x) = e^{2x}\sin{3x}$ Find $f^{(3030)}(0)$ Solving the problem The approach taken to solve this problem is given as follows. Consider the following first few derivatives of f:…

Introduction When studying math, I always try to find good geometric intuitions for whatever concept I’m trying to learn. It’s not always possible, especially when dealing with higher-dimensional spaces, but when you can create simple, concrete examples in easier-to-visualize spaces, the intuition you gain makes the generalized form much more clear. In this post I want to discuss some tools I use to play around with the concepts from my lectures and visualize what’s happening.…

Introduction When I was first introduced to the idea of injectivity, the definition felt rather arbitrary and unmotivated. Conceptually, when a function is injective it means that it is one-to-one, or that it maps every element in its domain to exactly one element in its codomain. This is (and was) a clear statement, but my confusion stemmed from the way that we formalize this concept in the language of mathematics.…