MA005: Calculus I

We used a similar processes to "solve" both the tangent line problem and the area problem. First, we found a way to get an approximate solution, and then we found a way to improve our approximation. Finally, we asked what would happen if we continued improving our approximations "forever", that is, we "took a limit". For the tangent line problem, we let the point $Q$ get closer and closer and closer to $P$, the limit as $b$ approached $a$. In the area problem, we let the widths of the rectangles get smaller and smaller, the limit as $w$ approached 0. Limiting processes underlie derivatives, integrals, and several other fundamental topics in calculus, and we will examine limits and their properties in Chapter 1.