Mean value theorem
MathDefinition
If a function f is continuous on [a, b] and differentiable on (a, b), then there exists at least one point c in (a, b) where f′(c) equals the average rate of change over the interval. In other words, the instantaneous slope equals the slope of the secant line at some point.
Examples
- If a car travels 150 miles in 3 hours, MVT guarantees the speedometer read exactly 50 mph at some moment during the trip
- For f(x) = x² on [1, 3], the average rate is (9−1)/(3−1) = 4, and f′(c) = 2c = 4 gives c = 2
Key Fact
f′(c) = [f(b) − f(a)] / (b − a) for some c in (a, b)
Study This Concept
Practice Mean Value Theorem with free review games in these units: