Intermediate Value Theorem

1
Intermediate Value Theorem
2
Objectives

• Students will be able to
• Determine if the Intermediate Value Theorem
  Applies to a particular function
• Use the Intermediate Value Theorem to determine
  if a function has a root on a given interval

3
Intermediate Value Theorem

• Let f(x) be continuous on a closed interval
  (one-sided continuity at the
  endpoints), and
• (we can say this without loss of generality).
  Then for any value d such that
  , there exists a value c such that
  and

4
Example 1

• Use the Intermediate Value Theorem to determine
  if the equation below has a solution in the
  interval (0, 1).

5
Example 2

• Use the Intermediate Value Theorem to determine
  if the function below has at least one root in
  the interval (-1, 1).

6
Example 3

• The price of a gallon of gas at a particular
  QuickTrip was 2.61 on a Monday. The following
  Friday, the price of a gallon of gas was 2.75.
  Therefore, some time between that Monday and that
  Friday, there was a point where the piece of a
  gallon of gas was 2.70.
• Is the previous conclusion valid?