Average Rate of Change Calculator

The average rate of change tells you how fast a function rises or falls across an interval, and this calculator works it out for any function between any two x-values. Enter a function of x, the start and end points of the interval, and it returns the average rate of change along with the two function values and the changes in x and y, updating as you type. The idea is the foundation of calculus and one of the most useful in all of mathematics, because it answers a very natural question: over this stretch, how much does the output change for each unit of input? The answer is simply the change in y divided by the change in x, which is exactly the slope of the straight line, called the secant, that joins the two points on the graph. That is the same arithmetic as working out an average speed from distance and time, or an average growth rate from a starting and ending value, which is why the concept shows up far beyond the classroom. It is also the stepping stone to the derivative: shrink the interval until the two points almost touch, and the average rate of change becomes the instantaneous rate, the slope at a single point. The calculator accepts everyday function notation, with powers written using the caret and the usual functions like sin, cos, sqrt, exp and ln, so you can test polynomials, roots, exponentials and more. That makes it genuinely useful for students learning calculus and pre-calculus, for checking homework on slopes and secants, and for anyone needing an average rate from a formula. Because the result updates live, you can shorten the interval and watch the average rate approach the slope at a point. The formula and a worked example are explained clearly below.

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Use x, ^ for powers, and functions like sin, cos, tan, sqrt, exp, ln, abs.