What is a secant line?

A secant line is a line that intersects a curve at at least two distinct points. Unlike a tangent line, which touches the curve at only one point (locally), a secant line passes through multiple points on the curve.

Here's a breakdown of key information:

• Intersection Points: The defining characteristic is its intersection with the curve at two or more points. These points are used to calculate the slope of the secant line.

• Slope: The slope of the secant line represents the average rate of change of the function between the two points of intersection. If the points are (x₁ , y₁) and (x₂, y₂), the slope (m) is calculated as:

  m = (y₂ - y₁) / (x₂ - x₁)

• Relationship to Tangent Line: As the two points on the curve where the secant line intersects get closer and closer together, the secant line approaches the tangent line at that point. In calculus, this concept is fundamental to understanding the derivative, which is the slope of the tangent line.

• Applications: Secant lines are frequently used in numerical methods, particularly in finding roots of equations using techniques like the secant method. They also provide a visual and numerical representation of the average rate of change of a function over an interval.

• Visualization: Imagine a curve drawn on a graph. Draw a straight line that crosses the curve at two (or more) distinct points. That line is a secant line.

In short, the secant line provides a simple yet powerful way to approximate the instantaneous rate of change (represented by the tangent line) using readily calculable average rates of change.