Apply the average rate of change formula to any function on an interval. See the rise over run, the secant line drawn through both endpoints, and a shrinking-interval table of average rates of change.
The Mean Value Theorem guarantees some c where the instantaneous rate equals this average rate — verify the hypotheses with the MVT/Rolle/IVT panel .
The average rate of change formula gives (f(b) - f(a)) / (b - a) — the rise over the run of a function across an interval, and the slope of the secant line joining (a, f(a)) to (b, f(b)).
Pick a function, set the interval endpoints, and this tool draws the curve, the secant line, and the dashed rise/run right triangle on the same axes, then narrates every step. A shrinking-interval table recomputes the average rate of change over ever-smaller intervals anchored at one endpoint, showing what happens as the interval closes in.
The average rate of change of f on [a, b] is (f(b) - f(a)) / (b - a): the rise divided by the run. It equals the slope of the secant line through (a, f(a)) and (b, f(b)).
Read f(a) and f(b) straight out of the table rows for your two input values, subtract to get the rise, subtract the inputs to get the run, then divide. You never need a formula for f.
Yes — they are two names for the same number. The average rate of change over an interval is exactly the slope of the secant line through the interval endpoints.
The difference quotient is (f(x + h) - f(x)) / h, the average rate of change over an interval of width h. Shrinking h is what the trend table on this page shows.
Yes. It is negative when f finishes lower than it started on the interval, and zero when f(a) = f(b), which makes the secant line horizontal.
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