What is Calculus? (Encore)
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The following is an Encore presentation of Everything Everywhere Daily. As early as 2400 years ago, Greek philosophers were coming up with paradoxes that seemingly had no solution, and early mathematicians had come up with problems that seemed impossible to solve. It wasn't until the late 17th century that the techniques were finally developed to solve these problems and unlock new fields of science and mathematics.

Learn more about calculus, what it is, and what it attempts to do on this episode of Everything Everywhere Daily. This episode is sponsored by NerdWallet. When it comes to general knowledge and history, you know I've got you covered. But who do you turn to when you need smart financial decisions? If your answer is NerdWallet, then you're absolutely right. And if it's not, let me change your mind.

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I have a very wide range of people who listen to this podcast. There are some of you who consider yourselves bad at math and would never consider taking a course in calculus. For you, this episode will simply try to explain what in the world calculus is and why it's even a thing. I have students who listen to this podcast. If you're considering taking a calculus course at some time in the future, for you, I hope this episode will give you an idea of what you will be getting out of the course and why it's worth learning.

The remaining group are those like me who have taken a calculus course. This episode is simply what I wished my professors had done on the first day of class. We jumped right into problem solving and never took a few minutes to just step back and address why we were taking this course in the first place and what this entire branch of mathematics called calculus was about. So, with that...

Most ancient mathematics was static. Mathematicians were trying to solve a problem. What's the area of a triangle or what is the solution to an equation? For example, what number when added to 2 will equal 4? In algebra we state this as x + 2 = 4. What we want to know is what is x? In this case, x is obviously 2. 2 + 2 = 4. That's a very simple equation.

But what if we change it just a little bit? Instead of asking what plus two equals four, we instead ask the question, what plus two equals something else? We could write this out as x plus two equals y. With that small change, we have now created something very different. There is no one answer to the question. Instead, we get a completely different answer depending on what we put in.

If x equals 2, then y would be 4. But if x equals 100, then y equals 102. This is no longer an equation. It's called a function. You put something in and you get something out. By convention, mathematicians have defined x to be the independent variable, what you put in. You can pick whatever you want. And y is the dependent variable of what you get out, which is determined by the function.

Functions are extremely powerful. When I was in high school, my high school didn't offer a calculus course. They had to offer a course called functions. When I asked why they just didn't offer calculus or pre-calculus, I was told by my teacher that it had to do with licensing and they got around the rules by just calling the course functions. Functions are everywhere in mathematics and science. They're also handy ways of thinking even if you're not using numbers. Let's say you wanted to create a basic model for how tall someone is.

You could make a function that would list inputs to determine the output of height. Some of the inputs in the height function would be genetics, age, and diet. They would all have different weights, obviously, but the end result would be one value, height. Now let's assume a function that's a bit more complicated. x squared equals y. We take the square of the number we input. So if x equals 2, then y equals 4. And if x equals 4, then y equals 16.

If you make a graph of this function with all the values of x you put in, you will wind up with a parabola. Unlike my previous example, the rate of increase in y keeps getting bigger and bigger as x gets bigger. So the question is, what is the rate of change? In the case of the parabola, the rate of change would be the slope of any line at any point along the parabola that just touches the parabola at one point.

This idea of change and the rate of change is fundamental to what is known as differential calculus, one of the two types of calculus. To clarify this idea of change, I want to use an example that everyone, especially the great students at Truck Driver University, would be familiar with. Let's say you're driving down a road at a constant speed. You have three different instruments on your dashboard. You have an odometer that measures how far you've traveled.

You have a speedometer, which measures how fast you are traveling. And you have an accelerometer, which measures your acceleration. When your vehicle isn't moving, your odometer isn't moving, which means your speedometer is also not moving and your accelerometer is also not moving. But now let's assume you're driving down a straight road going at the speed of 10. And that can be kilometers or miles per hour. It doesn't matter. You're going 10. Your odometer would be moving at a constant rate. Your speedometer would be pegged to 10.

and your accelerometer would still be at zero because your speed is constant. Now let's assume you step on the gas. Your odometer is now moving even faster than it was before and it keeps moving even faster every single instant. Your speedometer is now moving slowly going up and your accelerometer is now not at zero but some number above zero.

Each of these things, position, velocity, and acceleration are all linked together with differential calculus. Velocity measures the rate of change of your distance or position. Acceleration measures the rate of change in your velocity. In calculus, we would say that velocity is the derivative of position as velocity measures the rate of change of your position.

Acceleration is the derivative of velocity, as it measures the rate of change of your velocity. Acceleration would be the second derivative of position. Now I want to put a pin in this idea about the rate of change for just a minute to focus on another problem that seems unrelated, but as you will see is actually closely related. How to define the area of a shape.

For simple shapes like triangles, squares, and circles, ancient mathematicians figured out equations to calculate the area of those shapes. But what if the area you wanted to calculate isn't a regular shape? What if it's just some random shape, or even if it's the area under a line defined by a function? Well, that's a problem. There is no simple equation for figuring out the area under such a curve. One way you could do it is to fill it with shapes that you know how to calculate.

Let's say you drew a square as big as you could inside the shape. Then you drew more squares in the leftover space, and you just kept on doing this over and over and over drawing smaller and smaller squares. You could eventually approach the area of the object no matter how oddly it was drawn. Determining the area under something, or the volume as the case may be, is known as an integral, and this branch of calculus is known as integral calculus.

This approach to determining the area of something by adding up an infinite number of objects was actually figured out by the Greek mathematician Archimedes over 2,000 years ago. His methods were crude, but it was the same fundamental technique that's used in integral calculus today. You might be wondering, if you filled in the spaces with squares, there will always be a little bit left over. You can't have an infinite number of squares. And you're technically correct.

One of the big intellectual developments that led to the creation of calculus was the idea of a limit. The idea of a limit gets around the problems with infinity. If you remember back to my episode on Zeno and his paradoxes, these were finally resolved with the technique of using a limit. Let's say you added up an infinite series of numbers starting with 1 and then 1 half, 1 fourth, 1 eighth, 1 sixteenth, etc. Every number is 1 half the value which came before it.

If you add up all of them, an infinite amount, what do you get? You might say that there's no way to calculate this because it's an infinite number of numbers. However, the concept of a limit gets around the problem. You can't add up every number, but a mathematician would say that the limit of that sum is two. That means you can get arbitrarily close to the number two by adding up those numbers. No matter how small of a number you pick, a billionth of a trillionth of a gazillionth,

If you keep adding those numbers up, you will get closer to two than that tiny number you picked is from it, no matter what number you pick. Now, what does finding the area of something have to do with finding the rate of change of something? It turns out that these two things, derivatives and differential calculus, integrals and integral calculus, are related. In fact, they are sort of the opposites of each other in the same way that subtraction is the opposite of addition and division is the opposite of multiplication.

These techniques of finding the derivative or integral of a function are the basis of all calculus. Acceleration is the derivative of velocity and velocity is the integral of acceleration. What are some examples of problems that calculus can solve? Let's assume you have a tank of water with a hole on one side of the tank. How fast will the water drain out?

You might think that this is a straightforward question, but it's not. That's because the rate at which the water drains out of the hole depends on the amount of water in the tank. When there's a lot of water in the tank, the water pressure in the tank will cause water to flow out the hole faster. As the tank empties, there's less water pressing down, slowing the flow of water out of the hole. The flow of water when the tank is full will be different than when the tank is near empty. This is a problem that can only be solved with calculus.

Another problem is the rocket equation, which I talked about in a previous episode. To launch something into space requires a certain amount of fuel. But that fuel now requires more fuel to be launched, and then that fuel requires fuel, and so on and so on. The solution to that problem also requires calculus. When I took a class on fluid dynamics, we had to determine how a glacier moved. The problem is that the speed of a glacier at the top is faster than it is at the bottom.

To determine how the glacier moves, you had to use a whole lot of calculus. Pretty much every discipline that uses mathematics has problems that can only be solved with calculus. Engineering, chemistry, physics, economics, biology, and astronomy all require the tools of calculus. In a previous episode, I covered the breakthroughs made by Isaac Newton and Gottfried Leibniz, who developed the techniques of integral and differential calculus.

It should come as no surprise that the explosion of scientific discoveries in the 18th and 19th centuries that I have covered in many different episodes of this podcast all occurred after the development of calculus. So, once again, for all of those who say you're bad at math, functions are a mathematical expression where you have one or more inputs and get a single output. You can find the derivative of a function which determines the rate of change at any point along the function.

Or you can integrate a function, which can be used to find the area under a function if you're using two dimensions. There is obviously a whole lot more to it than what I've just described, but that wasn't the purpose of this episode. But now, even if you don't know how to do calculus, at least you know why it exists and what it does. The executive producer of Everything Everywhere Daily is Charles Daniel. The associate producers are Benji Long and Cameron Kiefer.

I want to give a big shout out to everyone who supports the show over on Patreon, including the show's producers. Your support helps me put out a show every single day. And also, Patreon is currently the only place where Everything Everywhere daily merchandise is available to the top tier of supporters. If you'd like to talk to other listeners of the show and members of the Completionist Club, you can join the Everything Everywhere daily Facebook group or Discord server. Links to everything are in the show notes.