Suppose \(f\) is continous on the closed interval \([a,b]\) and differentiable on \((a,b)\). Then there is a number \(c\) such that \(a < c < b\) and

\(\dfrac{f(b)-f(a)}{b-a} = f'(c)\)

• Lamar's notes

Suppose that \(f\) is continous on \([u,v]\). Then there are numbers \(u'\) and \(v'\) such that \(u < u' < v' < v, v'-u' = (u-v)/3\), and

\(\dfrac{f(v')-f(u')}{v'-u'} = \dfrac{f(v)-f(u)}{v-u}\)

Suppose that \(f\) is differentiable at \(c\), and \(\{u_n\}_{n=1}^{\infty}\) and \(\{v_n\}_{n=1}^{\infty}\) are sequences such that as \(n \to \infty\), \(u_n \to c^{-}\) and \(v_n \to c^{+}\). Then

\(\lim_{n \to \infty} \dfrac{f(v_n) - f(u_n)}{v_n - u_n} = f'(c)\)