Hello Blog Readers,
As promised in my previous blog entry, I’m going to start posting a few entries about calculus as I gear up to teach the subject in the fall.
The key to studying calculus is to make sure you have a solid understanding of what is being accomplished in the subject area before tackling the subject area. So what is being accomplished? What are the goals of calculus? Well, any elementary course in calculus must include in the least two things: first, a thorough study of the methods for calculating the slope of the tangent line to any curve at a given point; and second, a thorough study of the methods for calculating the area beneath a curve on a given domain. As you study these topics, you come to learn that these two seemingly unrelated calculations are in fact intimately related. Indeed, when appropriately defined, one calculation turns out to be the inverse operation of the other. It is this revelation, uncovered in the late 1600s and early 1700s, that ended up laying the groundwork towards unlocking many of the laws of science, economics, and engineering.
We’re going to focus on the first goal of calculus. For those who do not remember, a tangent line is a line that touches a curve exactly once. To gain a “babystep” understanding for what this concept looks like, let’s take a look at the most perfect curve I know: the circle.
It turns out that with a bit of background knowledge, calculating the slope of a tangent line on a circle is one of the easiest calculations you can make in calculus. No complicated looking math, equations, or formula are needed. The only tools you need, you likely learned as a freshman or sophomore in high school.
To calculate the tangent line, first, you need to remember that a circle is defined by a center and radius (looking at our picture above, the center of our circle is point O). Second, you need to pick the point on the circle that the tangent line is going through (point P in our reference picture). Third, you need to remember that the tangent line of a circle at point P is perpendicular to the radius going through point P (the “right angle” square in our reference picture).
Only the third fact requires a formal proof. If you had an elementary math education through high school, I can almost guarantee that you have already seen this proof. You may not have understood it at the time you saw it, you may not remember it now, but if you wish to further examine it, feel free to Google.
The rest of the calculation is Algebra 1 level mathematics. First, calculate the slope of the line going through points O and P. Call this slope S. Next, recall that perpendicular lines have slopes that are negative reciprocals of one another. Then, the slope of the tangent line at point P must be -1/S. There you have it. The calculation is that simple! We call the slope -1/S the derivative of the circle at point P.
If this was all of the study of calculus, then we probably wouldn’t hold off on teaching the subject until students reach the end of high school/beginning of college (and even then, it seems only a small proportion of students tackle calculus. I read somewhere that only about 10% of the US population has studied calculus. If you start looking at math beyond multi-variable calculus, then less than 1% of the population is going to know what you’re talking about. Crazy!). Yet, it turns out that the simplicity of the circle calculation only works because circles have what is called “perfect curvature.” The rest of the study of calculating tangent lines to curves (formally called calculating “derivatives”) consists of figuring out techniques to do what we just did here, but with curves that are not “perfect,” i.e. for curves that are wrapping and changing, that are “more curvy” or “less curvy” than a circle, or that have other special features. It is this challenge and ingenuity that gives the study of calculus its rigor.
So there you have it. Explaining one of the most fundamental aspects of calculus using as simple of math as possible. Hopefully you enjoyed this ‘babystep’ into the world of calculus. I’ll try to get some more entries up on the subject over the coming weeks. See? You don’t need an advanced math degree to understand all of the math on this blog. Some of it is quite simple.