MATHEMATICAL ANALYSIS 08
Mean Value Theorem
At some point, the instantaneous rate of change equals the average rate of change across the whole interval.
The theorem
If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there exists \(c\in(a,b)\) such that
Geometric meaning
The secant through the endpoints has slope \([f(b)-f(a)]/(b-a)\). The theorem guarantees a tangent inside the interval with exactly the same slope.
Example
For \(f(x)=x^2\) on \([1,3]\), the average slope is \((9-1)/(3-1)=4\). Since \(f'(x)=2x\), the required point satisfies \(2c=4\), so \(c=2\).
Why the assumptions matter
Continuity is required on the closed interval and differentiability in its interior. Without them, the conclusion can fail.
Connection to Rolle
Subtract the secant line from \(f\). The resulting function has equal endpoint values, so Rolle's theorem gives a point where its derivative is zero.