How this calculator works
The overall rate uses only the first and last observations. Consecutive rates show how the change differs across smaller intervals; their unweighted average need not equal the overall rate. Each rate has units of function value per x unit. Endpoint modes rearrange the same secant relation and reject a missing x when no unique answer exists.
Formula
r = (f(x2) - f(x1))/(x2 - x1); f(x2) = f(x1) + r(x2 - x1).
Worked example
For x = 0, 1, 3 and f(x) = 0, 1, 9, the interval rates are 1 and 4. The overall rate is 9/3 = 3, rather than their simple average 2.5.
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Frequently asked questions
Is this a derivative?
No. It measures change over a finite interval. A derivative describes the limiting rate at one point.
Why must x increase?
A consistent order makes each interval clear and avoids a zero denominator. Reorder both lists together.
Is a graph included?
This tool returns numeric rates and labelled intervals. It does not draw a curve between observations.