The classical definiton of average velocity is the distance traveled divided by time taken to do so. This doesn't make sense if we ask what velocity means at a single moment in time. Here we explore this deep question.
To make sense of velocity at a single moment we had to think about how functions behaved as the input approaches (but does not equal) a specific value. Here we formalize this idea with the concept of a limit, the central technical tool in all of calculus.
The informal approach to limits is all well and good, but if we are to study functions in more depth we need a much more precise defintion. Here we introduce the rigorous, so called epsilon-delta defintion of a limit.
If I travel 100 miles in one hour, even if my speed is fluctuating, there has to be a moment when I'm travelling at exactly 100mph. This is the mean value theorem. Making this precise will give us deeper understanding of a differentiable functions.
This gives us a useful tool for computing certain types of limit. (Interesting Fact: This result is actually due to Johann Bernoulli, who was paid by l'Hospital to give him his mathematical discoveries.)
We calculate the volumes of solid shapes by integrating cross-sectional area functions.
When we think about numbers, we think about the number line. We call these the real numbers. Real numbers come with two fundamental operations: addition and multiplication. If we sit the number line in a plane, is there some natural way to extend these operations, preserving all the usual properties of arithmetic?
A complex number is a point in the Cartesian plane. We justify the term number because we can naturally add and multiply points in the plane in a way that satisfies all the usual properties of arithemtic. Once we transition from the line to the plane, what new properties emerge? Can we make sense of familiar concepts like exponentials?