What a great start to the year! Calculus is my favorite branch of mathematics, and I have the fortunate opportunity to teach two calculus classes this year. What makes calculus intellectually rich is that it is based on the concept of infinity, a somewhat abstract concept, but is also highly practical in its applications.
The biggest concept we’ve explored in Calculus AB so far is the concept of a limit. We spend a lot of time on limits because they are foundational to calculus. In order to think about limits, imagine you are ten feet away from the finish line of a race. In order to complete the race, you would have to run five feet, then two and a half feet, then one and a quarter feet, and so on. Could you ever cross the finish line, or are you stuck continually cutting the remaining distance in half forever? Of course, common sense tells you can cross the finish line, but the thought of having to walk an infinite number of increasingly tiny steps can be confusing to think about. This thought experiment is said to be have been first considered by the Greek philosopher Zeno, circa 450 BC. One way the confusion is resolved is by understanding that this infinite sum of distances is actually a finite number; as you continue to halve the distance between you and the finish line, you are approaching the finish line, so the sum of all of these increasingly smaller distances is ten feet. The concept of seeing what something approaches (in the previous case, the sum of all of the steps that are becoming smaller and smaller) is called finding the limit, and just like in this example, it can be used to solve problems that are out of reach when only thinking about the finite, rather than the infinite.
In Calculus BC, we’ve also explored the idea of a derivative, along with its applications. Derivatives are instantaneous rates of change. The speed at which you are driving at any given time is a derivative. You can also think about derivatives when thinking about rings on a tree. If a tree gets the same amount of nutrients, on average, from year to year, the rings become smaller and smaller from year to year, and you can calculate how much so using derivatives. We have also explored the concept of a marginal profit, which is an economics term for the profit obtained from producing and selling one more item of a product in a business. You can also use derivatives to find the maximum or minimum value of a quantity. The most classic example of this is the area enclosed by a fence. The maximum area enclosed by a rectangular fence occurs when the rectangle is a square, and it turns out that you can prove this using derivatives. These are examples of derivatives that I find fascinating, and they are just the beginning of the applications that we will look at this year.
While it is still early in the year, it is never too early to start thinking about the AP tests in May. AP test style questions are unique, so becoming familiar with the types of questions on the AP test and how to answer them in the way that the test demands is important, especially for knowing how to earn partial credit on the AP tests. The true or false questions in the book require students to justify their answers using words, not just math, so those problems have been a favorite of mine to ask students to do in class. As the year progresses, we will continue to go and over more and more AP style questions in both classes.
-Gage Edgar, Math Instructor