Average Rate of Change Calculator

The average rate of change of a function f(x) over an interval [a, b] quantifies how the output shifts per unit of input. It is numerically identical to the slope of the secant line connecting the points (a, f(a)) and (b, f(b)). Confusing this with instantaneous rate (the derivative) is a common error in coursework and engineering estimates. A wrong interval or a misread sign can cascade into incorrect velocity readings, flawed cost projections, or misjudged growth trends. This calculator parses your function symbolically and evaluates the difference quotient without floating-point shortcuts.

The tool supports polynomial, trigonometric, exponential, and logarithmic expressions. It plots the curve with the secant line for visual verification. Note: the formula assumes f is defined and finite at both endpoints. Discontinuities or division by zero at a or b will produce an error. For piecewise functions, evaluate each piece separately across its valid sub-interval.

The average rate of change of f(x) on the interval [a, b] is defined as the difference quotient:

This is geometrically equivalent to the slope of the secant line through the two points (a, f(a)) and (b, f(b)). The secant line equation is:

Where f(x) is the function expression, a is the left endpoint of the interval, b is the right endpoint of the interval (b ≠ a), f(a) is the function value at a, and f(b) is the function value at b. The constraint b ≠ a is mandatory to avoid division by zero. As b → a, the average rate of change approaches the instantaneous rate (derivative) f′(a).