Tangent lines are really important when we look at how fast something is moving at a specific moment.

When we want to figure out the speed of an object, we look at the slope of the tangent line on a graph that shows its position over time.

Instantaneous Rate of Change

• What does it mean?: The instantaneous rate of change at a certain point is how fast something is changing right at that moment. We find this by looking at the average rate of change and making the time we’re looking at super small—close to zero.

• A Simple Formula: If we say $s(t)$ is where the object is at time $t$, then the speed (which we call velocity) is found using something called a derivative, written as $s'(t)$.

Example

Picture a car driving down a straight road. If we say that the car's position at time $t$ is given by the formula $s(t) = t^2 + 2t$, here’s what that means:

• Finding $s'(t)$: The derivative $s'(t) = 2t + 2$ shows us the car's speed at any time $t$.

• Tangent Line: At the moment $t = 1$, we can find the slope of the tangent line (which tells us the speed) by calculating $s'(1) = 4$. This means the car is moving at a speed of 4 units for every unit of time.

Understanding this connection helps us see how things move in real life!