Final answer:
The slope of a secant line becomes the slope of a tangent line by considering the average rate of change between two points and then taking the limit as the interval approaches zero, resulting in the instantaneous rate of change or instantaneous velocity at a single point.
Step-by-step explanation:
The concept of how the slope of a secant line turns into the slope of a tangent line involves understanding two key ideas of calculus: average rate of change and instantaneous rate. For a function that represents position versus time (x vs. t), the slope of the function at any point signifies the velocity. When we refer to the average rate of change of a function between two points, we essentially measure the slope of the secant line connecting those points. This secant line gives us the average velocity over that interval. Transitioning from the secant line to the slope of a tangent line means focusing on a single point. As the interval between the two points on the curve gets smaller and approaches zero, the secant line becomes increasingly similar to the tangent line at a point. Eventually, when the interval is infinitesimally small, the slope of the secant line represents the instantaneous rate of change, or instantaneous velocity, at that point. Mathematically, this is the limit of the average rate of change as the interval approaches zero, or the derivative of the position with respect to time at a specific instant.