Introduction to Math-itis
Calculus, and its concepts, are perceived to be unattainable by the general public. In addition, the basic ideas of Calculus, the derivative and the integral, are perceived to not be understandable or straightforward and with good reason. Calculus appears to be painfully difficult. But the basic ideas of calculus are fairly straight forward. And they do not require a classroom or symbols. They are simply ideas that are used everyday, implicitly and intuitively, by all 7 billion inhabitants of this planet (see Go Tell It to the Mathematicians for an indepth explanation of intuition and mathematics).
In short, I am speaking to the general American who has not had years of mathematical training. I am speaking to those Americans who may have an appreciation for math but never studied it beyond high school and those citizens who have diagnosed themselves with the contagious phobia, math-itis. I do not want to leave any of you at the train station. I have written this for you.
In this explanation of calculus, I will use the Minneapolis Light-rail station on 38th St. in South Minneapolis, laxatives, and a Go-Kart. In addition, I have provided two photo sequences: one inbound to the 38th St. station and one outbound from the 38th St. station. Feel free to refer back to either of these two slide shows at any point while reading this article.
Perspective #1 – Inbound Train
First, take the time to view the first slide show one or two times before reading further. Are you familiar with the picture sequence in the slide show? Good. As you can see, the train arrives to the station in stages. In other words, each picture illustrates the train getting closer and closer to you. It’s speed decreases as it comes to a complete stop. You know this from experience. If you remember back to grade school math or physics class, I know it’s painful, you may recall that distance is measured by two points. In this case, you use the point of the train in the first picture and the point of the train in the last picture. This is called the integral in calculus. That is it. The integral is simply the distance between one point and another point. If you need do not quite get this idea, and that is okay, view the slide show as many times as you like.
Again, we can see that the train gets closer and closer with each picture. The idea of the integral allows us to compute, intuitively, that with each picture, our distance decreases with each picture. Thus, we see that each integral, distance, decreases as the train comes towards us.
Perspective #2 – Outbound Train
Before we discuss this other basic idea of calculus, the derivative, please take the time to view the second slide show one or two times. As you noticed, the train is departing from the station. In addition, you noticed the distance, integral, increasing with each slide. This is where the second idea of calculus occurs. This other basic idea is called the derivative. To help make this second idea personal and salient, we are going to take a slight detour.
The derivative computes an instantaneous point in time. It also computes an instantaneous rate of speed. For example, you are sitting at your desk right now, reading this article, and your speed is zero, unless your desk is on a Go-Kart. Then, cool!
Anyway, zero is a speed, right? 0 mph? However, if you get up to go to the bathroom, each step you take is an instantaneous point. Each step you take illustrates your rate of speed. If you really have to go to the bathroom, say someone put a laxative in your coffee, your instantaneous speed, while your running, at each step will be higher. Now lets return to the train.
As you watched the train move away from the station at each point, you not only noticed the distance, integral, increase, but you also recognize that each point in time illustrates a rate of speed. See! Laxatives do come in handy. This is the perfect example of the derivative. Each picture shows the train at an instantaneous point and you know from experience that a train increases speed when it departs a train station. Thus, each point also illustrates a rate of speed. This is the derivative.
Final Thoughts on Your Former Phobia
If you understand these two basic ideas, then you have a basic understanding of calculus. But most importantly, it should be salient to you that you use these two ideas every day, and throughout the day, in your life. When you are driving you slow down if the car in front of you slows down or when coming to a red light or a stop sign. This is your brain, intuitively, computing the integral for you. Your distance is decreasing correct?
When you are driving on the city streets or on the freeway, you are keeping an eye on your speedometer. If you are on Highway 62 in South Minneapolis (Crosstown), you keep your rate of speed at the speed limit, 55 mph (speeding tickets are bad). You understand now that when you look down and view your speedometer, your speed is instantaneous at that point in time. You look down a few seconds later, your speed is what you see at that instantaneous point in time. This is the derivative. However, there is something else going on with these two ideas.
Did you notice that they are happening at the same time in each perspective; that is, both the inbound and outbound trains?