4'2 The Mean Value Theorem

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4.2 - The Mean Value Theorem
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Theorems
If the conditions (hypotheses) of a theorem are
satisfied, the conclusion is known to be true.
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Rolles Theorem

• Let f be a function that satisfies the following
  three hypotheses
• f is continuous on the closed interval a, b.
• f is differentiable on the open interval (a, b).
• f (a) f (b)
• Then there is a number c in (a, b) such that
• f '(c) 0.

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Example Rolles Theorem
Verify that the function satisfies the three
hypotheses of Rolles Theorem on the given
interval. Then find all numbers c that satisfy
the conclusion of Rolles Theorem.
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The Mean Value Theorem

• Let f be a function that satisfies the following
  two hypotheses
• f is continuous on the closed interval a, b.
• f is differentiable on the open interval (a, b).
• Then there is a number c in (a, b) such that

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Example Mean Value Theorem
Verify that the function satisfies the two
hypotheses of Mean Value Theorem on the given
interval. Then find all numbers c that satisfy
the conclusion of Mean Value Theorem.
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Theorem
If f '(x) 0 for all x in an interval (a, b),
then f is constant on (a, b).
Corollary If f '(x) g'(x) for all x in an
interval (a, b), then f g is constant on (a,
b) that is, f(x) g(x) c where c is a
constant.
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Proof By Contradiction

• Assume that something is true.
• Show that under your assumptions, the conditions
  of a known theorem are satisfied. This guarantees
  the conclusion of that theorem.
• Show that the conclusion of the theorem is, in
  fact, not true under the assumptions.
• Since the conclusion of the theorem must be true
  if the assumptions were satisfied, the only
  conclusion left is that the assumptions must be
  incorrect.

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Example
Show that the equation 2x 1 sin x 0 has
exactly one real root.