What Calculus is.
Calculus has traditionally been viewed as the pinnacle of high school mathematics. It is a relatively young field of mathematical study, and does indeed present some difficult concepts that are completely different from the courses the usually precede it. However, Calculus is not the bogey monster it's reputation suggests.
Calculus is a study of change. It was developed to explain and model the physics being observed by Isaac Newton and his contemporaries. At it's core, there are only three big ideas - the derivative, the integral, and the limit. In Advanced Placement Calculus - where students are expected to take the AP exam and possible earn college credit - a great deal of emphasis is placed on technical skill working with difficult and sophisticated models and equations. We aren't determined by the exam, though, and in our class we will focus on the ideas, primarily. We will spend a lot of time developing our understanding of derivatives from a variety of perspectives, connect that to the related idea of the integral and underpin it all with the limit. The second quarter will focus more on interesting applications, with the final exam being a project designed to consolidate the work done over the course of the semester.
This is a course about ideas, not equations or solutions, necessarily. Expect to be asked to think, and expect to be challenged to communicate what you uncover.
Calculus is the study of change, and nearly all of it is based on three concepts - the derivative, the integral, both of which rely on the limit.
We’ll start by examining the integral as it appears in the movement of a ball rolling down a ramp. The concept and definition of the derivative is rich, so we’ll spend some time thoroughly understanding what it is and what it tells us before moving on to the varied ways of finding out the actual value of the derivative.
Next, we’ll examine integrals, which are closely related to derivatives. Same structure - we need to understand what it is we are finding, so some weeks are spent on the definition and interpretation of integrals before we spend a lot of time finding the value of integrals.
Along the way, we will have had to consider Limits, at least informally. The concept of the limit is what defines both the derivative and the integral. This topic gets a lot more attention in AP Calculus, but without the test to worry about, we use limits to get what we need from derivatives and integrals. They are interesting in their own right, though, so we will examine them in more detail later in the course.
As we close out the course, we will take a tour through various applications of these concepts and techniques. Calculus being the study of change, and things being as they are - always changing - it turns out calculus is going on all the time around us. Classic applications we will explore include Newton’s Method, Optimization, Related Rates, and computing the volumes of special solids.
Where to go for help.
In class, I will often first direct you to consult with your classmates - I will be a last resort for answers. However, should you need more help with the material, do come after school and that is when I can be more direct with the instructional style. I am usually available for at least 30 minutes every day, but I am hosting "office hours" on Tuesdays and Thursdays. The best policy is to let me know you're planning on coming and I can not only be prepared for you, we can immediately clear up any conflicts in each of our schedules.
If any issues come up for you at all - be they confusion on an assignment or just to let me know of some other difficultly, please send me an email. I am often able to respond and that alone is often helpful. If you know ahead of time you are going to have trouble completing an assignment, let me know that by email too.