This article serves as the beginning of propositional logic. First, I discuss logical connectives, then constructing truth tables. I also explain tautologies, contradictions, and contingencies. Towards the end, and I consider modus ponens and substitution. This article is a must-read for any wanting to write mathematical proofs.
So what exactly is a tangent line? Is it a line that only once crosses the graph of a function? No! The tangent line concept is more subtle than that. In this article, I discuss the average rate of change, the instantaneous rate of change, what exactly is a tangent line. I also cover the relative rate of change.
Continuity is one of the essential concepts in all of the calculus; indeed, all of mathematics. So in this article, I discuss continuous functions and the cases where a discontinuity may occur. I also present one-sided continuity and the removability of a continuity. In other words, how to extend a function so that it has no discontinuities.
There are different ways of understanding limits: the intuitive approach with tables and graphs and the rigorous methods using epsilon-delta arguments. However, in this article, for continuous functions on their domain, such as algebraic functions, trigonometric functions, and exponential functions, we can use more straightforward techniques to find a limit’s value.
What is an epsilon-delta argument, and why do we need this? Why do we need a precise definition of a limit when I can just use a table of values or a graph to find a limit? When you want a rigorous proof of a limit or to write a thorough proof of a limit theorem, you can use an epsilon-delta argument. I discuss what these are and show you how to use it.
Limits are used to study the behavior of quantities under a process of change. For example, limits can be used to describe the behavior of a function on its domain. Here we study one-sided limits and two-sided limits with emphasis on graphs. We discuss unbounded behavior and oscillating behavior with many examples given. An Intuitive … Read more
Calculus is a broad, well-developed subject, so where do we start? With limits! Calculus begins with limits and continuity of functions. In this article, I take the reader on a tour of the Calculus. Through derivatives and integrals, the story continues. Many applications of these concepts are also detailed. Calculus is an exciting journey, so let’s get started.