Hmm… what could Noah write that could be so scary? Well, I will draw one picture, and it will have the point of the post in a nutshell. But, then I will use words to explain what is going on. The hint of the post: The ordinal number has something to do with this post (although it’s a little more basic than that)!
Okay, so the image shown here is indeed a calculus image. It illustrates the Three Musketeers of calculus: limits (L), derivatives (a), and integrals (the green lanes). I would like to motivate each of these using everyday examples.
The limit is the basis behind everything else in calculus. Basically, what it tries to find is what happens as you approach a certain point, whether this be in one, two, or more dimensions. Calculus is the study of limiting processes, of which derivatives and integrals both are.
However, for many nice things, the limit is exactly what you might expect. A whimsical example of a limit is given here, where the function is not given explicitly with a formula, but instead just by its variables and parameters:
Let F(g,t) be a function that gives the coordinates of your car at a given time t, where g is the amount of gas in the tank. In that case, we would probably say that the limit of F as g goes to zero should be some gas station, and hopefully not Holiday, at least if you are my Dad who had a bad experience with their gasoline on multiple occasions. This is not explicitly illustrated in the picture above. How convenient that the fuel gauge knows how to compute a limit of coordinates in terms of its parameter! :p
In algebra, one of the common exercises is to compute the slope of a line. You can consider a line between any two points of a non-straight line to be a secant line. If you move the two points on the curve closer together, they will converge to a LIMIT, and the limit of that slope is called the derivative.
In the picture at the top of the post, the red line labelled “a” is the derivative of the purple curve at point L. It is called “a” because it corresponds to the acceleration of the brown point (car?) at the point L, when the purple curve corresponds to velocity. This implies that acceleration is the derivative of velocity. And I think that many people use both of these in everyday life!
In the picture that I have included at the top, you might notice the “lanes” of green that are separated by blue lines. You will also notice the black line along the bottom of the figure. That is the time axis. The area under each curve corresponds to the distance travelled in the given time interval. In this sense, it means that an integral is nothing more than a weighted sum! And I hope that we know how to add numbers together!
All three of these concepts I recall were nicely compiled into a single lab exercise when I was in eleventh-grade physics. Although I was taking calculus at the time as well, the experiment really made it connect for me, and hopefully it does for you too.
In this experiment, we attached a ribbon to a toy car that we could rev up and let run down the hallway. During the experiment, a machine put marks on the ribbon at the starting point of the car, and marked the ribbon at intervals of deciseconds (tenths of a second).
Our analysis was to investigate the distance travelled by the car (easy enough), as well as the velocities at given points, and also its acceleration, by cutting the strips appropriately at intervals of half-seconds. In this way, we essentially constructed a diagram similar to what I have at the top of this post. So, it’s a very intuitive way to describe the first semester of calculus conceptually in a single picture!
I’ll also give the Reader’s Digest version:
LIMIT: A place that you are trying to end up as some resource reaches a critical value.
DERIVATIVE: The number that is given on your speedometer.
INTEGRAL: If you know how far you have travelled and how long, you can find your distance by adding!
Many of the applications of calculus in everyday life refer to physics–after all, calculus was discovered as a way to help solve physics problems! The key thing: even if the computations are difficult, understanding the concepts and intuition is something that anyone should be able to do!
Oh, and one last thing. 234 was the Vector Calculus course for which I have been a TA umpteen times during my time at Northwestern. But I figured that I would just do a Calculus I post instead. 🙂
Today is the two-hundred and thirty-fourth day of M.M.X.I.V. That makes thirty-three weeks and three days.
Today is the thirty-ninth day of the Character Building Trial. That makes five weeks and four days.