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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Rolles Theorem
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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.
f(x) (x - 3)(x 1)2 on -1,3
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Example Rolles Theorem
f(x) (x - 3)(x 1)2 on -1,3

• Check to see if Rolles Theorem applies
• f(x) is continuous on -1,3
• f(x) is differentiable on (-1,3)
• f(-1)0 AND f(3)0

 
 
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• Check to see if Rolles Theorem applies
• f(x) is continuous on 0,3
• f(x) is differentiable on (0,3)
• f(0)2 AND f(3)2

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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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Tangent parallel to chord.
Slope of tangent
Slope of chord
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• Apply the MVT to on
  -1,4.

Heres the idea behind the MVT There must be
some value of c in the interval -1,4 where the
slope of the tangent line at c is the same as the
slope of the line connecting the endpoints (i.e.
the slope of the secant line or the AVERAGE rate
of change).
 
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• 1. Apply the MVT to
  on -1,4.

f(x) is continuous on -1,4.
MVT applies!
f(x) is differentiable on -1,4.
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• 2. Apply the MVT to on -1,2.

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• 2. Apply the MVT to on -1,2.

f(x) is continuous on -1,2.
f(x) is not differentiable at x 0.
MVT does not apply!
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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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Application Mean Value Theorem
You are driving on I-95 at 55 mph when you pass a
police car with radar. Five minutes later, 6
miles down the road, you pass another police car
with radar and you are still going 55 mph. She
pulls you over and gives you a ticket for
speeding citing the mean value theorem as proof.
WHY ?
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Application Mean Value Theorem
You are driving on I-95 at 55 mph when you pass a
police car with radar. Five minutes later, 6
miles down the road you pass another police car
with radar and you are still going 55mph. He
pulls you over and gives you a ticket for
speeding citing the mean value theorem as proof.
Let t 0 be the time you pass PC1. Let s
distance traveled. Five minutes later is 5/60
hour 1/12 hr. and 6 mi later, you pass PC2.
There is some point in time c where your average
velocity is defined by
72 mph
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AP QUESTION
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