The paradox of the derivative | Chapter 2, Essence of calculus

Explaining the concept of a derivative and highlighting the subtleties and potential paradoxes associated with it.

Discussing the concept of measuring instantaneous change between separate points in time and the importance of understanding phrases like 'instantaneous rate of change'.

Using the motion of a car over 10 seconds as a central example to explain the relationship between distance, velocity, and time on a graph.

Deepening the understanding of velocity by exploring the relationship between distance traveled per unit time and the concept of velocity at specific points in time.

Delving into the paradox of the derivative by examining the concept of speed at a single point in time and the approach to rates of change as time intervals shrink.

Providing a visual understanding of the derivative by examining the slope of a tangent line, rise-over-run slope, and the concept of approaching zero for time intervals.

Explaining the calculation of velocity using derivatives with a specific example of a distance vs. time function and simplifying the expression as time intervals approach zero.

Discussing the derivative as a constant approximation for rate of change and understanding the concept of instantaneous rate of change in different contexts.