Calculus

Where Does Calculus Come From?

So, where does calculus come from? How did early mathematicians come up with these complex ideas?

To get an idea of how you could invent calculus, let's start with a seemingly simple problem: to find the area of a circle. Now, we know the formula for the area of a circle:

Graph of a circle

But why is this the case? What kind of thought process leads to this observation? Well, say we don't know this formula. How can we try to find the area of a circle without it? To start, let's try breaking the circle into shapes whose areas are more simple to calculate.

Finding the area of a circle using shapes we know

And after trying to get more and more shapes so that less and less of the circle is left over, let's try a different idea: break the circle up into concentric rings.

Graph of concentric circles

That's great, but now what? Now, let's take just one of these rings, which has a smaller radius, that we will call , that is between 0 and 5.

Graph of concentric circles with one ring highlighted

From here, let's straighten out this ring.

Straightened out ring from concentric circles

With the ring straightened out, now we have a shape whose area is easier to find. But, what shape has an even easier area to find? A rectangle. For simplicity, we can actually approximate the shape of the straightened-out ring as a rectangle.

Approximating the Straightened Out Ring as a Rectangle

This rectangle has a width that is equal to the circumference of the ring, or , and a height of whatever smaller radius of that you chose earlier. Let's rename to , to represent a small difference in radius from one ring of the circle to the next one. So, what do we have now? We have a bunch of rings of the circle approximated as rectangles whose areas we know how to find! And, for smaller and smaller choices of (or breaking the circle into smaller and smaller rings), our approximation of the area of the ring becomes more and more accurate.

Let's go a step further and straighten out all the rings of the circle into rectangles and line them up side by side. Then, placing these rectangles on a graph of the line , we can see that each rectangle extends to the point where it just touches the line.

Rings of concentric circles placed on a graph of the line: y = 2πR

And for smaller and smaller choices of , we can see that the approximation of the total area of the circle becomes more accurate.

Rings of concentric circles placed on a graph of the line: y=2πR with a smaller choice for dr

Now you might notice that as gets smaller, the number of rectangles gets quite large, and won't it be time-consuming to add all their areas together? Take another look at the graph, and you will also notice that the total areas of the rectangles actually look like the area underneath the line, which is a triangle!

Total areas of concentric circles represented as the area under the graph

We know the formula for the area of a triangle:

Which in this case would be:

Which is the formula for the area of a circle!

But wait, how did we get here? Let's take a step back and think about it. We had a problem that could be solved by approximating it with the sum of many smaller numbers, each of which looked like for values of R from 0 to 5. And that small number was our choice of thickness for each ring of the circle. There are two important things to take note of here:

• Not only does play a role in the areas of the rectangles we are adding up, it also represents the spacing between the different values of R.

• The smaller the choice for , the better the approximation. In other words, the smaller we make , the more accurate the answer will be.

These are some pretty interesting ideas, right? So now you might be wondering, why go through this effort for something as simple as finding the area of a circle? Well, let's think for a moment... Since we were able to find the area of a circle by reframing the question as finding the area under a graph, could we not also apply that to other, more complex graphs? The answer is, yes, we can! Say, for example, we take the graph of , a parabola.

Graph of a parabola

How could we possibly find the area under a graph like this, say between the values of 0 and 5? This is a much more difficult problem, isn't it? And let's reframe this problem slightly: let's fix the left endpoint at 0 and let the right endpoint vary. Now the question is, can we find a function, let's call it , that gives us the area under the parabola between the left endpoint of 0 and the right endpoint of x?

Area under a parabola

This brings us to the first big topic of calculus: integrals. To use calculus vocabulary, the function we called is known as the integral of the function of the graph. In our case, would be the integral of . Or in a more mathematical notation:

As we progress through calculus, we will discover the tools that will help us find , but for now, what function represents is still a mystery. What we do know is that gives us the area under the parabola from a fixed left endpoint and a variable right endpoint. Now take a moment and think of what else we know about the relationship between and the graph, .

When we increase x by just a tiny bit, say by an amount we will call , we see a resulting change in the area under the graph, which we will call . This tiny difference in area, , can be approximated as a rectangle, just as we were able to approximate as a rectangle in our circle example. The rectangle approximation for , however, has a height of and a width of . And for smaller and smaller choices of , the approximation of the area under the graph becomes more and more accurate, just as with the circle example.

A change in area under a parabola

This gives us a new way to think about how is related to . Changing the output of by is about equal to , where is whatever you choose, times . This relationship can be rearranged to:

And, of course, this relationship becomes more and more accurate as we choose smaller and smaller values for . While the function is still a mystery to us, this relationship is key and, in fact, holds true for much more than just the graph of .

This brings us to the next big topic of calculus: derivatives. The relationship between , , and the function of the graph, , written as the ratio of divided by is equal to , is called the derivative of A. In mathematical notation:

Now, you may have noticed that the general formulas we've written for the derivative and integral look like they relate to each other. That's because they do! Derivatives and integrals are actually inverses of each other. In other words, a derivative can be used to find an integral and vice versa. The back-and-forth between integrals and derivatives where the derivative of a function for the area under a graph gives the function defining the graph itself is called the Fundamental Theorem of Calculus.

Let's summarize a bit. Generally speaking, a derivative is a measure of how sensitive a function is to small changes in its input, while an integral is a measure of some area under a graph. The Fundamental Theorem of Calculus links the two together and shows how they are inverses of each other.

Now that we have a solid idea of what calculus is and where it comes from let's dig a little deeper. We can gather from our examples in the previous section that there are some main concepts of calculus:

• Calculus is all about approximation or becoming more accurate as some value approaches another value

• There are two types of calculus:

  • Calculus that deals with derivatives or differential calculus

  • Calculus that deals with integrals, or integral calculus

• There is a fundamental theorem of calculus, and it links differential and integral calculus together

The Idea of a Limit in Calculus

Before we delve into the types of calculus, let's take a look at what sets calculus apart from other types of math: the idea of a limit. Remember in the previous section when we talked about choosing smaller and smaller values for either or ? When we consider these smaller and smaller values, we are improving the accuracy of our approximations by having or approach zero. Why not just use zero directly? Remember, the formula for the derivative of A is the ratio of divided by , and dividing by zero is impossible! This is where the limit comes in. The limit essentially allows us to see what the answer to a problem (for example, the area under a graph) should be as we get closer and closer to whatever value the limit is. In the case of our examples in the "Where Does Calculus Come From?" section, the limit was zero.

Differential Calculus

Differential calculus is the branch of calculus that deals with the rate of change of one quantity with respect to another quantity. In this branch, we divide things into smaller and smaller sections and study how they change from moment to moment.

Integral Calculus

When we know the rate of change of a function, integral calculus can be used to find a quantity. In this branch, we sum small sections of things together to discover their overall behavior.

The Fundamental Theorem of Calculus

The fundamental theorem of calculus links differential and integral calculus by stating that differentiation and integration are inverses of each other and is divided into two parts:

• Part 1 – shows the relationship between derivatives and integrals

• Part 2 – uses the relationship established in part 1 to show how to calculate an integral on a specific range

The definitions for the fundamental theorem of calculus are as follows:

Practical Applications of Calculus

Calculus has a wide variety, and a long history, of useful applications. In general, calculus is used in STEM (Science Technology Engineering Math) applications as well as in medicine, economics, and construction, just to name a few. A form of calculus was used back in ancient Egypt to build the pyramids! But the calculus we are learning today is the calculus that Sir Isaac Newton and Gottfried Leibniz developed in the seventeenth century.

AP Calculus: AB and BC

AP Calculus is broken down into two courses, AP Calculus AB and AP Calculus BC. The difference between these two courses is that AP Calculus BC covers everything that AP Calculus AB covers, plus a couple of extra topics. Please have a look at our articles on each topic for a full study of AP Calculus!

AP Calculus AB

The AP Calculus AB course covers many topics of calculus. A brief overview of them is listed below:

• Functions
• Limits and Continuity
• Derivatives
• Applications of Derivatives
• Integrals
• Applications of Integrals
• Differential Equations

AP Calculus BC

The AP Calculus BC course covers everything that AP Calculus AB does, plus these extra topics:

• Sequences and Series
• Parametric Equations
• Polar Coordinates
• Vectors