The Mean Value Theorem
Somewhere in between
You drive 120 km in 1.5 hours, an average of 80 km/h. Must your speedometer have read exactly 80 at some moment? Could you have avoided it?
The ochre line joins the curve's endpoints, so its slope is the average rate of change. Slide the indigo point along the curve until its tangent line is parallel to the ochre line. Is there always such a point?
Drag the ochre endpoints or the indigo point.
The Mean Value Theorem
If f is continuous on [a,b] and differentiable on (a,b), then there is at least one c in (a,b) with
In the driving example, position is continuous and differentiable, so at some moment the instantaneous speed equals the average, 80 km/h. Rolle's Theorem is the special case f(a)=f(b): then some c has f'(c)=0.
Both hypotheses are needed. Choose |x| or \sqrt[3]{x^2} on an interval containing 0. Which hypothesis fails, and does any c still work?
The theorem has a big consequence: if f'(x)=0 everywhere on an interval, then f is constant there. That is why two antiderivatives of the same function differ only by a constant.
Solve it step by step
Check the hypotheses, compute the average rate of change, and find c. Decimals to two places are fine. The score counts steps you answer correctly on the first try.
Common mistakes
- Forgetting to check the hypotheses
A corner or a break anywhere in the interval can make the conclusion fail.
- Giving a c outside (a, b)
Solving f'(c) = slope can produce values outside the interval. Only those strictly between a and b count.
- Expecting the theorem to say where c is
It guarantees that c exists. Finding it is a separate calculation, and sometimes there are several.