Section 5.2

1
Section 5.2 Rolles Theorem and the Mean Value
Theorem

• Objective
• Use Rolles Theorem.
• Use the Mean Value Theorem .
• Warm up
• Four equal squares are arranged to form one
  larger square. If the perimeter of each of the
  smaller squares is 36, what is the area of the
  larger square?
• 72
• 81
• 144
• 324
• 1296

2
Rolles Theorem

• Let f be continuous 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.
• This theorem guarantees extrema in the interior
  of a, b.
• c the horizontal tangent line

3
Example

• Find the 2 x intercepts of
• Show that f(x) 0 at some point between the 2
  intercepts.
• Set f(x) 0
• Solve for x
• Check to see if the function is continuous on the
  interval.
• Differentiate
• Solve for x.
• Pick value that lies in between the two
  intercepts.

4
Mean Value Theorem

• If f is continuous on a, b and differentiable
  on (a, b) then there exists a number c in (a, b)
  such that
• Refers to a value where the average rate of
  change is equal to the instantaneous rate of
  change in a, b.

5
Examples

• Use the Mean Value Theorem to find c in -2, 1
  of the function
• Plug into MVT formula.
• Differentiate function.
• Set derivative equal to the value found with the
  MVT formula.
• Solve for c.

6
Class work and Homework

• Class work is p. 326 8, 14, 22, 24, 26, 28
• Homework is p. 326 7, 13, 21, 23, 25