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Mathematical Proofs (Using Various Methods)

Are you someone who relies on logic and evidence for solving problems? Mathematical proofs will help you refine and take advantage of this valuable way of thinking as it applies to mathematics and potentially other areas such as philosophy and computer science.

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Quantifiers and Predicate Logic

Okay, so you know about propositional logic; now comes quantifiers and predicate logic. First, I discuss the universal and existential quantifiers. Then I explain the uniqueness quantifier and negating quantifiers. Towards the end, and I consider counterexamples and combining quantifiers.

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Propositional Logic (Truth Tables and Their Usage)

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.

Rate of Change and Tangent Lines example

Rate of Change and Tangent Lines

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.

Horizontal Asymptotes and Limits at Infinity

Horizontal Asymptotes and Vertical Asymptotes

What exactly are asymptotes? They are often mentioned in precalculus. But without a rigorous definition, you may have been left wondering. In this article, I go through, rigorously, exactly what horizontal asymptotes and vertical asymptotes are. I also illustrate them using graphs of functions.

Continuous Its Meaning and Applications

Continuous (It’s Meaning and Applications)

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.

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Find the Limit (Techniques for Finding Limits)

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.

Limit Definition Precise Definition of Limit epsilon delta argument

Limit Definition (Precise Definition of Limit)

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.

Evaluating Limits Analytically Using Limit Theorems

Limits (Calculus Starts with Limits)

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

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What is Calculus? (An Introduction)

Maybe you’re asking: what is the purpose of calculus? Well, you’re in the right place. This article discusses what calculus is, what it has been, why it’s still essential today, and will be for the foreseeable future.

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Calculus (Start Here) – Enter the World of Calculus

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.

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