All right. If you’ve understood all the previous Chapter 1 stuff, you are ready to dip your toe into the ocean of glory that is Calculus.
First, let’s go over what you’ve learned:
The essence of this chapter is that there are many types of functions with many different qualities. That’s a little vague, I know, but it’s important. If you’re trying to understand something in reality, whether it’s a math theorem or a physical phenomenon or a simulation you’re designing, you need to have this library of functions in your head. You need to know how a function behaves over time and how operations on it result in changes.
I will probably say this many times as we branch out into different areas, but it’s worth noting – you have GOT to work problems. If you don’t, you have no sense of what skill level you possess. You may very well understand a concept completely on the first read, but you can’t be sure till you try out a few problems. It is very often the case that you have some small gap in your understanding that will only be brought to light when you fail to answer some question correctly.
So, whenever you’re new at something, AT LEAST work the odd problems. If you’re still feeling fuzzy, work the evens. Remember, mathematics, more than any other field, is a tower of knowledge. If you have a shoddy layer anywhere, everything above it can fall apart.
That’s a cliche metaphor, but let me give you the less obvious aspect of it. If you build on a shitty foundation, the house might not fall over with the first or second or third story. But, eventually it will. How well you cement these basics will determine how high you can go later.
We’ve all heard the story of the awesome high school mathematician who gets knocked on her ass the moment she bumps into differential equations or linear algebra. It’s not because she’s stupid or because there’s a cap on her abilities – it’s because she didn’t understand the fundamentals. With a rough understanding, you really can coast through a lot of calculus without understanding what you’re doing or why it works. With enough rote learning, you can even solve tricky problems. But, if you don’t really know what you’re doing, sooner or later, you’re going to hit a wall.
Now then, onward to the good stuff. Before we go, I give you a quote from Winston Churchill, on his math studies as a young man:
Further dim chambers lighted by sullen, sulphurous fires were reputed to contain a dragon called the’Differential Calculus.’ But this monster was beyond the bounds appointed by the Civil Service Commissioners who regulated this stage of Pilgrim’s heavy journey. We turned aside…
Well, loyal readers, we will not be turning aside. We will meet the dragon, crush his will, and… make him solve equations for us.