This chapter is all about one of the central concepts in Calculus: Limits. Let’s first build some intuition about why we might care limits (and derivatives to come) in the following videos by investigating the Tangent Problem.
Video1.0.1.The Velocity Problem Numerically.
If I know my location at two different times, what is my velocity? There are two different notions to consider: average velocity - over an interval - and instantaneous velocity - at a point.
Post-video reflection.
What is the difference between instantaneous velocity and average velocity over a time interval?
Video1.0.2.Video 2: The Velocity Problem: Graphically.
We numerically and graphically investigate what happens as we approach the interesting point. Through this, we define the notion of a limit.
Given a distance vs time plot, the average velocity over an interval is the same as the slope of the secant line between those two points. But taking smaller and smaller intervals near a point, the slopes of these secant lines approach the instantaneous velocity.
Post-video reflection.
Draw a different curve than I drew, and sketch your own visual of the limit of secant lines approaching a tangent line at a specific point.
