Although it may appear daunting, calculus is a neat field of math if you’re looking to do more complicated problems. That is what we’re covering in this post! One of the most daunting things in calculus is doing something that requires a lot of repeated steps that are simple, but it’s easy to get lost. So we’re getting rid of all the fancy math for now and just looking at the basic concepts.

Limits

Limits, the fundamental building block of calculus, are not that bad without worrying about how to calculate them. Let’s see a typical example:

The basic idea of a limit to infinity is: what value does the function approach as we get farther and farther to the right? Here, the function is the thick black line and the zero line (the x-axis) is the thin one. We see that the function gets closer and closer to that line as we move farther to the right. Because the line is thick, it appears to reach zero, but it actually doesn’t!

Now, the limit as we get to infinity is not what value the function has at infinity, because infinity is not a number. It’s a concept! So we can say that as x (how far to the left or right we are) tends toward infinity, our function gets closer and closer to zero, but never reaches it! That’s the basic idea of limits.

Derivatives

How steep is this hill?

It’s a straight line, so what we mathematicians like to do is move one unit to the right and see how far it goes up or down, and that is the steepness of the hill, or its slope.

Okay then, how steep is this hill?

If we move a unit in one place, we go up a certain amount of units. But if we move somewhere else, it might be steeper or less steep! What is math to do?

Well, calculus has a solution! If we take a constant slope (the first hill), it looks like a triangle. So let’s say we want to find the slope at a certain point in the curvy hill. We do the slope calculation, and pretend that it’s a straight line. Then we make how much we move (the “one unit” for the slope) smaller!

When we zoom in, it looks like a straight line for a small part of it! So we take this section and calculate the slope of it! This is an easy task for math, but instead we use the limit (see previous section) of the slope as the area gets smaller and smaller, or more zoomed in.

Now, this is the derivative of a function (in this case, the curvy line) at a point. However, we can take the entire derivative and get an entire function that tells us how steep it is at all points! This gets us something useful!

How do we apply this to the real world? Well, we just make everything real! Let’s say we have a function that for a time t, gives us the amount of water in a tub. If we plot the line and it isn’t straight, the derivative can give us information as to how fast the tub is emptying/filling up. The farther from zero the derivative is, in either direction, the steeper the slope and the faster it is emptying (negative numbers) or filling up (positive numbers). This leaves us one more main idea!

Integration

Integration is doing the reverse. We have how steep the hill is at all points, but we don’t have the hill!

Here’s what an integration problem might look like. We have an equation where we put in any point and get how steep the hill is. All we need is the hill! Using integration, we turn something like the above picture into one hill. You might ask: Why all the lines?

Let’s say you take an entire hill, or rather a random curve in the plane, and move it up or down. That doesn’t change how steep it is! So the equation for how steep it is has nothing to tell you where it is on the up-and-down direction. The slope field shows us a bunch of possibilities, and it doesn’t matter exactly how high or low it is! We choose one solution that matches the problem and go with it, like the example above.

Now, in some cases, it does matter on the up-or-down direction, but those equations are more complicated and this still conveys the basic idea.

Summary

Here we’ve covered the three basic ideas of (single-variable, or 2D) calculus: Limits, Derivatives, and Integration.

• Limits: What does the function get close to towards infinity? This also works for “holes” that occur at a specific x-value in the function, where there’s division by zero that has no effect on the rest of the function (or a myriad of other possibilities).
• Derivatives: How steep is the function at a certain point? If you take a function, or in our example a hill, and differentiate (take the derivative of) all of it at once, we get a graph that shows us how steep it is at all points! This does not work if there’s a sharp corner though, because the math breaks down a little.
• Integration: What hill is this steep? If you get a formula that shows the steepness of something at all points, you can turn it into the original graph/hill.

These also have some fun applications. Integration specifically will give you the area under a curve (like the sine at the top of the article) or the length of a curve, both which require a little more math. However, they’re still neat tools that someone who’s looking for more math in their life should learn how to use! Also, you get fancy symbols from them that make you look smart.

That’s it for this post! I hope you learned something! And if you didn’t, well, here’s a fact that you might not have known (to increase the I-learned-something percentage!): Although FDR and Theodore Roosevelt were not closely related, they’re fifth cousins. Their closest tie is the fact that Franklin Roosevelt’s wife was Theodore Roosevelt’s niece.

Goodbye!