Slope of the Tangent Line

Definition

The slope of the tangent line at a specific point on a curve represents the instantaneous rate of change or the steepness of the curve at that point.

Analogy

Think of driving a car on a curvy road. The slope of the tangent line is like your speedometer, showing how fast you're going at any given moment. Just as your speed can change rapidly on a curvy road, the slope of the tangent line can vary greatly along a curve.

Related terms

Derivative: The derivative is a function that gives you the slope of the tangent line at any point on a curve.

Average Rate of Change (Secant Line): The average rate of change between two points on a curve is represented by the slope of the secant line connecting those two points.

Instantaneous Rate of Change: Similar to slope of the tangent line, it refers to how quickly something is changing at an exact moment in time.