We have now discussed an extremely important tenet in the development of calculus, which is called the first part of the Fundamental Theorem of Calculus. We briefly recall what we deduced in the last post: Fundamental Theorem of Calculus, Part 1: If is a function that tells us the area underneath the graph of fromContinue reading “Fundamental Theorem of Calculus, Part 2 (Explaining Calculus #17)”

# Category Archives: Mathematics

## Why Prove a Theorem Twice?

I have written before on the problem and solution of the problem of “Pythagorean Triples”. The problem, based on the Pythagorean theorem for right triangles, asks for all possible solutions to this equation which have all of whole numbers. Not just any right triangle works – for instance if then , which is definitely notContinue reading “Why Prove a Theorem Twice?”

## The Realm of Integers (Types of Numbers #3)

To kick off the tower of numbers, so to speak, the last article in this series discussed the “basic” numbers – mainly focusing on the positive whole numbers but also including zero. Now, we haven’t quite explored every aspect of the number zero yet, and we ran into problems with subtraction within the so-called “naturalContinue reading “The Realm of Integers (Types of Numbers #3)”

## Fundamental Theorem of Calculus, Part 1 (Explaining Calculus #16)

We have recently talked about two concepts that appear rather disparate. We have discussed antiderivatives – which is the idea of reversing all the rules for taking derivatives, and Riemann sums, which are used to calculate the areas underneath complicated shapes. What do these have to do with each other? It does not look likeContinue reading “Fundamental Theorem of Calculus, Part 1 (Explaining Calculus #16)”

## The Natural Numbers and Zero (Types of Numbers #2)

I now begin the escapade into different sorts of numbers. This may seem rather a strange adventure to anyone who hasn’t listened to too many professional mathematicians talks, or anyone who is too far removed from high school education. I would guess that most people who have read this have some particular picture of numberContinue reading “The Natural Numbers and Zero (Types of Numbers #2)”

## Riemann Sums and Areas (Explaining Calculus #14)

At this point in calculus, we are taking what appears to be a sharp turn in a direction completely away from what we’ve been doing so far. We now turn to talking about areas. The Problem Geometry is one of the most important branches of all mathematics, both theoretical mathematics and applied mathematics. When middleContinue reading “Riemann Sums and Areas (Explaining Calculus #14)”

## Why Ask Questions About Numbers? (Types of Numbers #1)

This is a first post in a series in which I’d like to discuss various different types of numbers that mathematicians study. This may sound strange – and that is fair. After all, aren’t all numbers just… numbers? Why would some numbers be so different from other numbers? What sense does that make? I’m honestlyContinue reading “Why Ask Questions About Numbers? (Types of Numbers #1)”

## The Importance of Derivatives and What Comes Next (Explaining Calculus #13)

This post doesn’t exactly introduce anything new about calculus. But, we must remember, it is important to reflect whenever we learn. Even in mathematics. In this post, I plan on achieving two main goals. One, we will reflect on what we’ve done so far by introducing the new ideas of limits and derivatives to ourContinue reading “The Importance of Derivatives and What Comes Next (Explaining Calculus #13)”

## Higher Order Derivatives and Their Applications (Explaining Calculus #12)

Up to this point, I’ve focused my efforts on derivatives of functions and what those derivatives mean. In particular, derivatives tell us about how things change over time – with derivatives, we can measure quantities like speed and growth. But derivatives are also functions, which means they have their own derivatives. Can we learn anythingContinue reading “Higher Order Derivatives and Their Applications (Explaining Calculus #12)”

## Exponential Models (Explaining Calculus #11)

We’ve used calculus for a couple different applications at this point. I’d like to tack on another application to our list, perhaps the most important one – at least from the perspective of a society very reliant on technology, engineering, and science. The application I have in mind goes under the very broad title ofContinue reading “Exponential Models (Explaining Calculus #11)”