Part 2/6:
Graphically, if a tangent line is drawn at a point on a curve, it will exhibit the same slope at that very point. Therefore, a tangent line serves as the instantaneous slope of the curve at that point, which is fundamentally what a derivative represents.
Secant Lines: Connecting Points on a Curve
In contrast, a secant line is a line that intersects a curve at two or more points. The slope of the secant line provides an average rate of change between those two points on the curve. To describe it formulaically, if the curve has points at ( (a, f(a)) ) and ( (b, f(b)) ), the formula for the slope of the secant line, denoted as (M_s), is given by:
[
M_s = \frac{f(b) - f(a)}{b - a}
]