Tangent to a Curve

The graph of a function where each output is the square of the input creates a smooth curve that bends upward, becoming steeper as one moves further from the center. At any chosen position along this curve, the curve reaches a certain height depending on the input value. This position can be a reference for analyzing how the curve behaves in its immediate vicinity.

To understand the change in the curve near a particular position, imagine selecting another point slightly ahead along the curve. Connecting these two points with a straight line forms a secant line. The secant line cuts across the curve and provides an overall sense of how the curve rises or falls between the two selected points. However, because it stretches over a distance, it only gives an average description of the curve's behavior over that interval.

As the second point gradually moves closer to the first, the secant line adjusts continuously. Its steepness shifts, offering a more precise representation of how the curve behaves over increasingly smaller distances. This adjustment reflects the curve's smooth and continuous nature. As the two points approach each other, the slope of the secant line tends toward a specific value that depends on the location chosen on the curve.

When the two points finally coincide, the secant line ceases to span between two distinct locations and becomes a tangent line. The tangent line touches the curve at exactly one point without crossing it. The specific position and steepness of the tangent line uniquely describe how the curve behaves at that single location. In this way, the tangent line captures the idea of an instantaneous rate of change, offering a precise view of the curve's behavior at an exact point.