# Calculus AB: Applications of the Derivative

### The Mean Value Theorem

Connect any two points, (a, f (a)) and (b, f (b)) , on a differentiable function f to form a line:

Figure %: Connecting two points on a continuous function

Intuitively, it should be clear that we can find a point c between a and b where the tangent line is parallel to the secant drawn between a and b . In other words, it should be possible to find a point such that the slope of the tangent at c is the same as the slope of the secant line drawn from a to b .

Figure %: Demonstration of the Mean Value Theorem

This intuitive idea is stated as the Mean Value Theorem, which states that if f is continuous on [a, b] and differentiable on (a, b) , then there exists a point c on [a, b] for which

 f'(c) =

#### Rolle's Theorem

Rolle's theorem is a special case of the mean value theorem in which f (a) = f (b) . It says: if f is continuous on [a,b] and differentiable on (a,b), and f (a) = f (b) , then there is a c on (a, b) where f'(c) = 0 .

The figure below should make clear that this is just a special case of the mean value theorem:

Figure %: Rolle's theorem as a case of the Mean Value Theorem

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