ARC Definition

ARC an abbreviation of average rate of change is the change in the value of a quantity divided by the elapsed time. For a function, this is the change in the y-value (Δy) divided by the change in the x-value (Δx) for two distinct points on the graph. It should be noted that average rate of change is not the same thing as the slope of the secant line of a curve. There are several formulas that can be used to calculate average rate of change. They include: average rate of change = ^Δy⁄_Δx = ^{y₂ - y₁}⁄_{x₂ - x₁} = ^{f(x₂) – f(x₁)}⁄_{x₂ - x₁} = ^{f(x + h) – f(x)}⁄_h.

Given a function f(x) plotted in the cartesian plane above as y = f(x), the average rate of change (or average rate of change function) of f from x to a is given by: A(x, a) = ^{f(x) – f(a)}⁄_{x - a}. This corresponds the slope of the secant line connecting the points (x, f(x)) and (a, f(a)). The limiting value f’(x) = lim_a->x ^{f(x) – f(a)}⁄_{x - a} as the point a approaches x gives the instantaneous slope of the tangent line to f(x) at each point x, which is a quantity known as the derivative of f(x), denoted f’(x) or d f / dx.