A Word About Mathematical Models
Throughout this book we've made use of mathematical models. Indeed, this book started and ended with a look a mathematical model of a projectile. Throughout this book, I hope you've come to get a very deep understanding of this particular model, and how calculus allows us to extract so much more information than you might have guessed at the outset. This model sparked the curiosity of Newton and Liebniz, and I hope you enjoyed it.
We also saw numerous other mathematical models throughout this book including a model of the length of the day in Johnson Vermont, the traffic on Route 15 in Johnson Vermont, and the number of users of the popular app Instagram. I hope these examples helped you to grasp the concepts of calculus more deeply.
One question you may have asked yourself as we explored these models however is: what exactly is a mathematical model?
It's a good question. Back at the start of this book when we used Geogebra to build a quadratic model of a projectile, you might have just assumed that the mathematical model Geogebra produced is magic. The ease with which Geogebra produced that model and others certainly does appear magical.
Needless to say mathematical models are not magic.
Definition: A mathematical model is a sketch of the natural world that use the tools and objects of mathematics as its medium.
That's it. In the same way that the Mona Lisa is a sketch of a human being that uses oil paint on a poplar panel as its medium, so too is a mathematical model a sketch. In the same way that Ansel Adams' photograph of the Grand Teton is a sketch of the famous mountain that uses film as its medium, so too is a mathematical model a sketch. In the same way that Shakespeare's sentence "All the world's a stage and all the men and women merely players" is a sketch of a fundamental truth about humanity, so too is a mathematical model a sketch.
These sketches can be tremendously valuable. For instance our sketch of the height of a projectile helped us understand not only the peak height of the projectile, but also the time it would land (this was 
At first you may be confused by his bold statement that "Ceci n'est pas une pipe" ("This is not a pipe"). But of course it's not a pipe. It's a collection of pixels that are being reproduced on your computer screen and which look like a little bit like a pipe. But it is absolutely positively not a pipe. It's an image.
The point of this interlude is to help you realize that images and sketches, and so also mathematical models, are not reality. They are reflections of reality; they are sketches of reality. In these reflections we can likely learn a great deal about the reality they reflect. But never make the mistake of fully conflating an image or a mathematical model with the part of the world it's sketching, or else you have fallen prey to the Treachery of Images.
This might seem outlandish at first, but in practice this way of thinking about mathematical models clarifies a tremendous amount of confusion you might have had as you read this book. Perhaps, for instance, when you saw your first function, maybe you were concerned about why that function dipped below the x-axis, or extended left of the y-axis. After all, dipping below the x-axis means the projectile goes subterranean, and that makes no sense. Also, the extension of the function left of the y-axis is "negative time", and that makes no sense either. But when you take my advice, and see
SPLAT! at the end of Falling Stuff on Earth). Similarly our sketch of the rate of traffic on Route 15 helped us not only make estimates about the busiest and quietest time of the day, but also helped us make estimates about the total traffic on the road throughout the day. Also, the sketch of Mona Lisa helps us understand beauty and technical perfection. The sketch of the Grand Teton helps us understand the orogenic processes that led to its formation, the natural splendor that surrounds it, and the importance of environmental conservation. Shakespeare's sketch of humanity helps us understand a little bit of why people do what they do in life, good and bad. I can't list all the things you can learn from sketches here because it would take too long.
But these sketches are only sketches. They are not reality.
To help you understand what this means, I encourage you to consider the famous work of René Magritte, The Treachery of Images.
At first you may be confused by his bold statement that "Ceci n'est pas une pipe" ("This is not a pipe"). But of course it's not a pipe. It's a collection of pixels that are being reproduced on your computer screen and which look like a little bit like a pipe. But it is absolutely positively not a pipe. It's an image.
The point of this interlude is to help you realize that images and sketches, and so also mathematical models, are not reality. They are reflections of reality; they are sketches of reality. In these reflections we can likely learn a great deal about the reality they reflect. But never make the mistake of fully conflating an image or a mathematical model with the part of the world it's sketching, or else you have fallen prey to the Treachery of Images.
This might seem outlandish at first, but in practice this way of thinking about mathematical models clarifies a tremendous amount of confusion you might have had as you read this book. Perhaps, for instance, when you saw your first function, maybe you were concerned about why that function dipped below the x-axis, or extended left of the y-axis. After all, dipping below the x-axis means the projectile goes subterranean, and that makes no sense. Also, the extension of the function left of the y-axis is "negative time", and that makes no sense either. But when you take my advice, and see g(x) as simply a sketch of the natural world, the resolution becomes apparent: this sketch is just not that good. The sketch really should stop at the y-axis, and not extend leftward, and it should also stop at the x-axis, and not extend downward. But that would have complicated things tremendously when we were just starting out, so there was no point in getting into that nuance at that point.
In the next activity we'll learn about the domain of a function which allows us to "crop" functions to improve them as sketches, but even after we do so, you can't think that you've now obtained reality. Even after adding more strokes to your sketch, you can never get around the fundamental fact that your model is a sketch. It will never be reality no matter how many strokes you add or improve.
Some people spend a lot of time worrying about this, but that is wasted time and energy. No matter how much sophisticated mathematics or physics or statistics you learn, you will always only be able to sketch the world you live in. You never become god with math or science, creating reality out of sheer will; you only ever become a better sketch artist.
The sooner you get used to this fact, the sooner you'll transcend to being more proficient with the tools of mathematics. Indeed, I don't know about you, but for me it's far easier to enjoy and play and use mathematics when I think about myself as a sketch artist than as a god.