Asymptotes, Limits, and Graphing Techniques Summary: In this class, we address the concepts of asymptotes and dominant terms in the analysis of functions. We explore horizontal asymptotes, which describe the behavior of a function as tends to infinity; vertical asymptotes, which indicate infinite limits when approaches certain values; and oblique...
Limit and Continuity Summary: This class addresses the relationship between the limit and the continuity of a function, starting with an intuitive and formal explanation of the term. Continuity at a point and in a set is explored, detailing the necessary conditions for a function to be continuous. The algebraic...
The Derivative as the Limit of a Function Abstract: In this lesson, we will explore the concept of the derivative as the mathematical tool to analyze changes in functions. Starting from the slope of a secant line, and taking the limit as the points get closer, we will define the...
Derivatives of Polynomials, Trigonometric and Logarithmic Functions The derivative is a central tool in differential calculus, with fundamental applications in science, engineering, and economics. This article offers a progressive guide to mastering the differentiation of functions, from polynomials to trigonometric and logarithmic functions. Through demonstrations and concrete examples, it aims...
Chain Rule for the Derivative of the Composition of Functions With what we have seen so far, we already have all the basics to compute almost any derivative. However, we must distinguish between the possibility of computing a derivative and the effort we invest in carrying out such calculations, and...
Weierstrass Extreme Value Theorem Why is it that in so many optimization problems it is almost taken for granted that “the maximum exists” or that “there is always a minimum” on a given interval, when in reality nothing forces that to be the case? The Weierstrass Theorem is the missing...
Maximum and Minimum Values of a Function Where is the “best” point of a function: the maximum you want to achieve or the minimum you need to avoid? That question, which arises in optimization, physics, economics, and engineering, is one of the main applications of differential calculus. And here is...
A Simple Market Model:Basic Concepts and Assumptions Abstract: This lesson introduces the "Simple Market Model," an approach that facilitates the learning of key investment concepts, combining risk-free assets (bonds, with known return) and risky assets (stocks, with uncertain return). We will see how these assets can be combined in a...
Things You Should Know About C++, Its History, and Evolution Did you know that one of the most influential languages in the history of programming was born from the quest to combine efficiency and creativity? Since its beginnings in 1979, C++ has evolved into an indispensable tool in the world...
The Principle of No-Arbitrage Summary: In this class, we will address the Principle of No-Arbitrage, an essential concept in financial theory that underpins the stability and consistency of markets. This principle not only forms the basis of mathematical models for asset valuation but also plays a crucial role in understanding...
