As early as 2400 years ago, Greek philosophers were coming up with paradoxes that seemingly had no solution.
Early mathematicians came up with problems that seemed impossible to solve.
It wasn’t until the 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.
I have a wide range of people who listen to this podcast.
There are some of you who consider yourself 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 is even a thing.
I have students who listen to this podcast. If you are 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 is 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 solving problems and never took a few minutes to step back and address why we were taking this course and what the entire branch of mathematics called calculus was about.
So, with that…..
Most ancient mathematics was static. Mathematicians were trying to solve a problem. What is the area of a triangle, or what is the solution to an equation?
For example, what number, when added to two, will equal four? In algebra, we state this as X + 2 = 4. What is X?
In this case, X obviously is 2.
2 + 2 = 4.
That is a very simple equation.
But what if we change it just a little bit? Instead of asking, “What plus 2 equals four?” we ask, “What plus 2 equals something else?”
We would write this out: “x + 2 = y”.
With that small change, we’ve 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=2, then y=4. If x=100, then y=102.
This is no longer an equation. It is 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, and y to be the dependent variable, what you get out.
Functions are extremely powerful. When I was in high school, my school didn’t offer a calculus course. They had to offer a course called “functions”. When I asked why they didn’t just 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 are also handy ways of thinking when you aren’t 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, but the end result would be one value: height.
Now, let’s assume a function that is a bit more complicated. x2 = y. We take the square of the number we input. If x = 2, then y = 4. If x = 4, then y = 16.
If you graph out this function, you will get a parabola. Unlike my previous example, the rate of increase in y keeps getting bigger and bigger as x gets bigger.
The question is, what is the rate of change? In the case of a parabola, the rate of change would be the slope of a line at any point along the parabola that touches the parabola at just that 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 try 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 are 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 have 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 at zero, and your accelerometer is at zero as well.
Now, let’s assume you are driving down a straight road going at a speed of ten (miles or kilometers per hour doesn’t matter). Your odometer would be moving at a constant rate, your speedometer would be pegged to ten, 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. Your speedometer is now slowing going up, and your accelerometer is now not at zero but some number above zero.
Each of these things, distance, velocity, and acceleration, are all linked together with 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.
I want to put a pin in this idea of 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 these shapes.
But what if the area you want to calculate isn’t a regular shape? What if it is just some random shape, or even it is the area under a line defined by a function?
That is 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. You kept on doing this over and over, drawing smaller and smaller squares. You would eventually approach the area of the object, no matter how odd 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 was actually figured out by the Greek mathematician Archimedes over 2000 years ago. His methods were crude, but it was the same fundamental technique that is used in integral calculus today.
You might be wondering, if you filled in the space with squares, there will always be a little bit left over. You can’t have an infinite number of squares.
You are 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 the limit.
Let’s say you added up an infinite series of numbers starting with one, then one half, then one-fourth, then one-eighth, then one-sixteenth, etc. Every number is one-half the value of what came before it.
If you add them all up, what do you get? You might say that there is no way to calculate this because it is 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 these numbers. No matter how small of a number you pick, one billionth, of one trillionth, of one gazillionth, if you keep adding up those numbers, you will get closer to two than that number is from it.
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, and 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 the side of the tank. How fast will the water drain out?
You might think that this is a straightforward question, but it isn’t. That is because the rate at which the water drains out of the hole depends on the amount of water in the tank. When there is a lot of water, the water pressure in the tank will cause water to flow out of the hole faster.
As the tank empties, there is 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 which can only be solved by 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 that fuel then requires more fuel, and so on and so on.
The solution to the problem 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 dermine how the glacier moved, 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 shouldn’t come as a surprise that the explosion of scientific discoveries in the 18th and 19th centuries occurred after the development of calculus.
So, just to give a brief summary,
Functions are mathematical expressions 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 in 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 is does.