Application of Rates of Change

The first derivative of a position function represents the instantaneous rate of change of position, which is the car's velocity. This derivative equals the slope of the tangent line on a position-time graph at any given point. Velocity indicates both the car's speed and direction of travel along the highway.

When velocity equals zero, the car is momentarily stopped at that instant. This occurs at specific moments in time where the first derivative of the position function equals zero. The car transitions between moving forward and moving backward at these zero-velocity points.

A car speeds up when velocity and acceleration have the same sign, meaning both are positive or both are negative. Conversely, the car slows down when velocity and acceleration have opposite signs. This relationship reveals the exact state of motion beyond simple observation of position changes.

Acceleration measures how velocity changes over time and is found by taking the derivative of the velocity function. This represents the second derivative of the position function with respect to time. Acceleration can be evaluated as an instantaneous value by reducing the time interval to approach zero.

On a position-time graph, the slope of a tangent line shows instantaneous velocity, while a secant line's slope represents average velocity. On a velocity-time graph, the tangent line's slope indicates acceleration. These graphical methods enable interpretation and prediction of real-world motion scenarios in physics and engineering applications.

Average velocity is calculated by dividing the change in position by the corresponding change in time over a specific interval. Instantaneous velocity is found by progressively reducing the time interval to approach zero, corresponding to the derivative of the position function at a single moment.

A positive velocity shows the car is moving forward along the highway, while a negative velocity indicates the car is moving backward toward its initial position. The sign of the velocity, determined by the first derivative of the position function, directly reveals the direction of travel at any given moment.