Rolle

1
Rolles Theorem and the Mean Value Theorem

• Guaranteeing Extrema

2
Rolles Theorem

• Guarantees the existence of an extreme value in
  the interior of a closed interval

3
Rolles Theorem

• Let f be a continuous function on the closed
  interval a, b and differentiable on the open
  interval (a, b). If
• f(a) f(b)
• then there is at least one number c in (a, b)
  such that f(c) 0.

4
Rolles Theorem

• Three things that must be true for the theorem to
  hold
• (a) the function must be continuous
• (b) the function must be differentiable
• (c) f(a) must equal f(b)

5
Rolles Theorem Examples

• Book
• On-line

6
Mean Value Theorem

• If f is continuous on the closed interval
• a, b and differentiable on the open interval
  (a, b), then there exists a number c in (a, b)
  such that

7
Mean Value Theorem

• In other words, the derivative equals the slope
  of the line.

8
Mean Value Theorem

• Tutorial
• Examples
• More Examples