secant line (2.1)
a line crosses a curve at two points
A secant line is a straight line that intersects a curve at two points. In calculus, we use secant lines to approximate the behavior of a function between two given points. For instance, if we have a function f(x) with two points x1 and x2, we can draw a straight line that passes through these two points to estimate the slope of the curve between them. This approximation is known as the secant line, and it provides a way to calculate rates of change and slopes of curves.
To find the equation of the secant line passing through two points (x1, f(x1)) and (x2, f(x2)), we use the formula:
y – f(x1) = (f(x2) – f(x1)) / (x2 – x1) * (x – x1)
where x represents a variable point on the secant line at distance from x1. This formula gives us a linear equation that describes the secant line passing between the two points on the curve.
The secant line can be used to estimate the slope of a curve at a particular point by making x2 approach x1, which will cause the secant line to approach the tangent line at that point. This method is known as the method of secant lines and is an essential tool for approximating derivatives in calculus.
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