The slope of a secant line between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a function $f(x)$ is given by the difference quotient: $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$.
As the two points on the secant line get closer together, the secant line approaches the tangent line at that point.
Secant lines are fundamental in understanding limits and derivatives as they provide an initial approximation before taking the limit.
In calculus, secant lines help illustrate how average rate of change transitions into instantaneous rate of change (derivative).
Secant lines can be used to estimate values for functions when exact computation is complex or impractical.
Review Questions
What is the formula for finding the slope of a secant line between two points on a function?
How does a secant line relate to a tangent line as the interval between two points decreases?
Why are secant lines important in understanding limits and derivatives?
Related terms
Tangent Line: A straight line that touches a curve at exactly one point without crossing it. It represents the instantaneous rate of change at that point.
Difference Quotient: $\frac{f(x+h) - f(x)}{h}$ provides an average rate of change over an interval and is used to find derivatives.
Derivative: The limit of the difference quotient as $h$ approaches zero; it represents an instantaneous rate of change of a function.