What is a secant line?

A secant line is a line that intersects a curve at at least two distinct points. It is a fundamental concept in <a href="https://www.wikiwhat.page/kavramlar/calculus">calculus</a>, particularly in understanding derivatives and rates of change.

Key aspects of a secant line:

  • Intersection: A secant line must intersect the curve (e.g., a function's graph) at two or more points. This is what distinguishes it from a <a href="https://www.wikiwhat.page/kavramlar/tangent%20line">tangent line</a>, which touches the curve at only one point (or, more precisely, at a single point of tangency).

  • Slope: The slope of a secant line represents the average rate of change of the function between the two points where the secant line intersects the curve. Given two points, (x<sub>1</sub>, f(x<sub>1</sub>)) and (x<sub>2</sub>, f(x<sub>2</sub>)), the slope m is calculated as:

    • m = (f(x<sub>2</sub>) - f(x<sub>1</sub>)) / (x<sub>2</sub> - x<sub>1</sub>)
  • Approximation: As the two points of intersection get closer and closer together, the secant line approaches the <a href="https://www.wikiwhat.page/kavramlar/tangent%20line">tangent line</a> at a single point. This limit of the secant line's slope is the derivative of the function at that point, which represents the instantaneous rate of change.

  • Applications: Secant lines are used to approximate the slope of a curve at a specific point and are important in numerical methods for approximating derivatives. They also play a key role in understanding the definition of a derivative.