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 our ‘math vocabulary’. The next thing we will do – and this is quite important – is try to reflect on these concepts and ask the kinds of questions mathematicians would ask about these concepts with the goal of making additional discoveries.
What We’ve Done So Far
We began with the idea of the limit. This idea allowed us to talk about trends as we approach some critical threshold that we care about. Limits enable us to talk about things like tangent lines, long-term trends, and continuity. We have since focused mostly on the tangent line direction. Tangent lines tell us about how things are changing over time. We gave a special name to the slope of these tangent lines – we call them derivatives. We should think about derivatives as telling us how rapidly something is changing over time. We spent a little bit of time talking about how derivatives are calculated in practice. We delved deeper into this concept to show how derivatives can tell us whether quantities are growing, shrinking, or flat at a particular moment, and we applied those ideas to show how you can find peaks and valleys in a graph. This application is particularly helpful, since very often in real-world situations we want to know how to something optimally – or “the best possible way” – and derivatives help us find what that best possible way is. We also talked about the second derivative – which is the derivative of a derivative – and we showed how that gives us information about acceleration and gravity, as well as more useful ways to solve optimization-type problems.
It is important to reflect not so much on the details of calculating derivatives, but on what sorts of things we’ve been able to do with them. Notice the italicized words in the previous paragraph. I selected those words to put a focus on the sorts of concepts we have in mind when we do derivatives and the sorts of problems we are able to solve with them. By thinking more on these concepts, as mathematicians, we should be trying to come up with new questions about these concepts and new approaches.
When we learn about addition, we next learn about subtraction. When we learn about multiplication, division comes next. This makes sense, because these things reverse each other. It makes sense to learn about them at the same time, because they are closely intertwined with one another.
Since the idea of reversing rules shows up all over mathematics, one of the most important questions we should be asking is
Can we reverse the derivative? If so, what does that tell us?
This is a good question. The second major chunk of our series on calculus will handle exactly that question. The answer is that, you can basically reverse derivatives. The fact that two functions like and have the exact same derivative will be a bit of a hiccup – but when we spend a bit more time working on it we an iron out that difficulty. Coming up with new math isn’t easy, so that difficulty shouldn’t discourage us too much.
Once we do construct this reverse derivative – which mathematicians normally call an antiderivative – we will want to ask questions about what sorts of information it can tell us. If derivatives have built in geometric information about slopes of tangent lines, then maybe antiderivatives tell us some kind of geometry too? Maybe they have some kind of direct or indirect connection with the interpretation of the derivative as describing change over time? It is difficult to tell until we actually get our hands dirty, but these are all good questions to ponder.
Here, we reflect on the geometry and optimization we learned about using derivatives. The ideas of sloping, peaks, and valleys are notions that make sense in three-dimensions as well as in two. Really, if you phrase them the right way, they make sense in any number of dimensions (but our visual intuition wouldn’t work any more above three). Because of the difficulty in imagining high numbers of dimensions, let’s just focus on three. Derivatives told us about slopes on two-dimensional graphs. Well, is there something that can tell us about slopes on three-dimensional graphs? Once we build that something, can we use it to find peaks and valleys of three-dimensional graphs? The answer to both of these is yes. It turns out that defining these new three-dimensional derivatives isn’t actually very hard – but putting them to good use can be a bit tricky. I hope to eventually get into these details, but doing any more than giving bare definitions would actually require a fair amount of extra work that goes outside of calculus into something called linear algebra and matrices. I will very likely do a series explaining these at some point in the future of the blog. For now, we will leave this question unexplored. But rest assured that this is an application we know how to solve – and a scientist in any field at all probably uses these methods explicitly or implicitly in nearly all of their work.
Solving “Differential Equations”?
When we talked about gravity, something odd came up. We had an equation like for a constant number . This is another direction we might explore. How do you solve equations like this?
These are called differential equations. With normal equations, the unknowns you want to solve for are variables (like ) and the operations going on usually just involve multiplication, addition, subtraction, and division. Differential equations are different. There, the unknown you want to discover is a function (like ) and in addition to the normal rules, you can also take derivatives. To give a few examples, you might want to know how to solve equations like
It isn’t exactly obvious how to solve things like this. As it turns out, the key idea we need to rely on is that of the antiderivative – or reverse derivative – to help us. There are several other important ideas that can help us solve equations like these. It can be done – and is done all the time by scientists in fact. My main point here is that, conceptually, we know how important solving equations is to all of the rest of mathematics, so we should expect that solving these so-called differential equations should also be important too.
We’ve talked about derivatives and second derivatives. I mentioned that, in principle, nothing is really stopping you from taking derivatives until you pass out from exhaustion. Then you can wake up and take more derivatives. At least, at face value, that seems to be true. We might ask ourselves questions like
Is it really true that you can always take derivatives to your hearts content, as long as you want to?
The answer to that is no. It is actually fairly difficult to find examples where you can do one derivative but not two. But these exist. Once you’ve found one, it is fairly easy to find examples where you can do two derivatives but not three, and so on in that pattern. These examples are not “ordinary” functions like you’d learn about in school. They are a bit awkward and contrived. If you want to see extremely awkward and contrived functions, look up the Weierstrass function. It is about as bad as they come. This graph, somehow, has the weird property that every single infinitely small point on the graph is a sharp corner.
Mathematicians usually call things like this “pathological functions”. You can see why. They behave very differently from how you’d expect things to behave. This point actually leads us to another question:
When things actually do behave the way we expect them to, is there additional information we can learn about the “nice” functions?
This answer turns out to be a resounding yes. In fact, it is a truly shocking amount of “yes”. As it turns out, if you figure out how derivatives work with complex numbers instead of merely the normal day-to-day real numbers, things start falling into place in a remarkable way. Most people who take advanced courses where they learn about this find themselves in complete disbelief at least five or six times. Statements that look almost insane at face value turn out to be completely true and extremely useful. This is the world of analytic functions, as they are called. Every single type of function we’ve dealt with in this series is analytic. It turns out that analytic functions – like the trigonometric function and exponential function – are basically “infinite polynomials”. We will actually be able to get into this without going into complex numbers, but once you understand the world of complex numbers, the task of studying the “nice” functions becomes a lot easier.
Conclusion: This is How Math Works
This should leave my reader with a lot of questions. That is good. You should have a lot of questions. This is what being a mathematician is like. This is what learning math is like. Unanswered questions are what inspire us – looking for answers is thrilling. I would compare it to exploring an unexplored island, or even an unexplored planet or unobserved galaxy. In mathematics, you have the opportunity to discover new worlds you never knew existed, and if you are fortunate, you will be the first person in all human history to explore that new world. True, it is a hard mountain to climb, but there isn’t anything quite like the view from the peak once you’ve conquered that mountain.
I hope you feel like you’ve conquered a mountain learning about calculus. Because it truly is a mountain worth conquering. And as we continue to conquer that mountain, the view will only get better and better. We shall now move onward to the next main topic in any exploration of calculus – antiderivatives.