Definition: The mean value theorem says that for a function continuous on [a, b] and differentiable inside, some point has a tangent slope equal to the average slope.
What does the theorem state?
If f is continuous on [a, b] and differentiable on (a, b), then there is a number c in (a, b) where f'(c) = (f(b) – f(a))/(b – a). Somewhere the instantaneous rate matches the average rate.
How do you find c?
Worked example: f(x) = x3 on [0, 3]. The average rate is (27 – 0)/3 = 9. Solve f'(c) = 3c2 = 9, so c2 = 3 and c = √3, about 1.73, which lies inside (0, 3).
Real-life use: if a car travels 90 miles in 1.5 hours, its average speed is 60 mph, so at some moment the speedometer read exactly 60.
What mistakes occur?
Applying it without checking the conditions. The function f(x) = |x| on [-1, 2] is not differentiable at 0, so the theorem does not apply. Mathbot verifies continuity and differentiability first.
Where to learn it
- Using Derivatives to Analyze Graphs and Optimize – lesson
- Motion, Related Rates and Linearization – lesson