Application of Rates of Change

Consider a car moving along a straight highway. Its position over time is modeled by a function whose graph is a smooth curve.

This function tracks the car's displacement relative to a fixed origin point over time.

The instantaneous rate of change of position of the car is measured by its velocity, showing its speed and direction. It is found by taking the first derivative of the position function.

The car is momentarily stopped when its velocity equals zero. For the given position function, this zero-velocity condition is met at specific moments.

A positive velocity shows that the car is moving forward, while a negative velocity means the car is moving backward toward its initial position.

Similarly, acceleration measures how the velocity changes. It is found by taking the derivative of the velocity function.

The car speeds up when velocity and acceleration have the same sign, and it slows down when they have opposite signs

This shows how the rate of change predicts the exact state of motion of the car at any given time beyond simple observation.