Rolles Theorem

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Chapter 3Section 2

• Rolles Theorem Mean Value Theorem

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Rolle's Theorem

• For f continuous on a,b and differentiable on
  (a,b), if f(a) f(b) then there is at least one
  number c in (a,b) such that f(c) 0.

3

• Example 1
• Find the two x-intercepts of
• and show the f(x) 0 at some point between
  these two intercepts.

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• Example 2
• Let

Find all values of c in (-2,2) such that f(c)
0.
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The Mean Value Theorem

• If f is continuous on a,b and differentiable
  on (a,b), then there is a number c such that

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• Example 3
• Given

find all values of c in (1,4) such that
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• Example 4
• Two stationary patrol cars equipped with radar
  are five miles apart on a highway, as shown
  below. As a truck passes the first patrol car,
  its speed is clocked at 55 mph. Four minutes
  later, when the truck passes the second patrol
  car, its speed is clocked at 50 mph. Prove that
  the truck must have exceeded the 55 mph speed
  limit at some time in the four minute interval.

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Homework

• 3.2
• p. 167
• 1,2,8,11,29,33,36,40,44