Why Integrals?
Cat's Out of Box
by MaybeILikeIt
4y ago
We’ve looked at how limits explain derivatives in an unambiguous and somewhat elegant manner. Unsurprisingly, limits seems to be the foundation for integrals as well. It might, however, be a longer pathway, not just because of the proof but also because of the definition of the integral. Defining the Integral I find that the definition of the integral (textbook or intuitive) is much more elusive than that of the derivative. After scouring the internet, I was only slightly satisfied with Wolfram Alpha’s definition: a mathematical object that can be interpreted as an area or a generalizat ..read more
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Explaining Derivatives
Cat's Out of Box
by MaybeILikeIt
5y ago
Purpose Some people told me that I should explain derivatives in my Mean Value Theorem post if I wanted it to be accessible to younger readers. Because that post was already long, I decided to make a separate post for it. This took a while because I wanted to find an intuitive explanation for derivatives, but the most I could come up with was explaining the limit definition, which is basically every high school classroom’s way of explaining it. I finally got an explanation while discussing uniform motion with my cousin. Well, here it is. Enjoy! Building up the Idea It’s no news that ..read more
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Modular Arithmetic and Large Exponents
Cat's Out of Box
by MaybeILikeIt
5y ago
What are the last two digits of 20192019? These types of questions are popular in tests like the AMC or MMPC, and often seem daunting, but modular arithmetic simplifies the problem down easily. Modular arithmetic can be described as using the remainder of a number when it’s divided by another number. Before we solve the problem, let’s look at a simpler question: what is the last digit of 20192019? Because we’re only concerned with the units digit, we only have to worry about 92019. If a value were squared, then the last value would be b^2. By the same logic, last digit of ..read more
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LR 1: Mean Value Theorem
Cat's Out of Box
by MaybeILikeIt
5y ago
This post begins a multi-part build up to proving L’Hôpital’s rule. In order to build ground up, however, some theorems must be understood, so I’m going to start off with understanding the mean value theorem. One last thing: the mean value theorem is not really a part of understanding L’Hôpital’s rule, but Cauchy’s mean value theorem is. As Cauchy’s is an extension and generalization of the mean value theorem, I think that it’s important that readers have a good idea of what it is. THE CONCEPT (finally) As you’re driving through a highway with a speed limit of 75 mph, you notice tha ..read more
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Gabriel’s Horn
Cat's Out of Box
by MaybeILikeIt
5y ago
How can something have a finite volume and infinite surface area? In the physical world, it can’t. Simply because infinite anything isn’t possible as far as we know. But logically, it’s completely possible. In fact, that’s what Gabriel’s Horn is. Gabriel’s horn is a hypothetical object that supposedly has infinite surface area and finite volume. With calculus, it’s easy to create this object. https://www.geogebra.org/resource/pbnv5GfG/nzHRo8dT0MwBuSCK/material-pbnv5GfG.png If you think about it, the horn is simply the graph of the function rotated over the x-axis. Created using ..read more
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Cat’s Out of Box
Cat's Out of Box
by MaybeILikeIt
5y ago
Hi there! If you’re here, thank you! In this blog, I really just want to look at science/math phenomena and try to break them down. Basically, I want to take the mystery out of things that are hard to wrap our heads around. If you’re wondering, the blog’s name alludes to Erwin Schrödinger’s thought experiment (Schrödinger’s cat) that suggests that if a cat is in a box and has a 50% chance of being dead, until you open the box, the cat is both dead and alive: essentially, you don’t know. With this blog, I want to open the hypothetical box, the cat’s status being the topics I want to ..read more
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