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## Sec 4.2: The Mean Value Theorem

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**Sec 4.2: The Mean Value Theorem**Beijing celebrates the wonder of the amazing Mean Value Theorem the mean value theorem of calculus decorate a pedestrian bridge across Zhushikou Dong Dajie in Beijing The picture is taken a few blocks south of Tiananmen Square (between Qianmen and the Temple of the Heaven, in Chongwen District of the city**Sec 4.2: The Mean Value Theorem**Beijing celebrates the wonder of the amazing Mean Value Theorem**Sec 4.2: The Mean Value Theorem**ROLLE’S THEOREM 1 there is at least one number c in (a, b) f(x) is continuous on [a, b] 2 f(x) is differentiable on (a, b) 3**Sec 4.2: The Mean Value Theorem**Example for 3 ROLLE’S THEOREM 1 there is at least one number c in (a, b) f(x) is continuous on [a, b] 2 f(x) is differentiable on (a, b) 3**Sec 4.2: The Mean Value Theorem**MEAN VALUE THEOREM there is at least one number c in (a, b) 1 f(x) is continuous on [a, b] 2 f(x) is differentiable on (a, b)**Sec 4.2: The Mean Value Theorem**MEAN VALUE THEOREM there is at least one number c in (a, b) 1 f(x) is continuous on [a, b] 2 f(x) is differentiable on (a, b)**Sec 4.2: The Mean Value Theorem**Example from inverse trig MEAN VALUE THEOREM there is at least one number c in (a, b) 1 f(x) is continuous on [a, b] 2 f(x) is differentiable on (a, b) Mathematical Consequences COROLLARY 1 COROLLARY 2