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Weakening the Hypotheses of the Mean Value Theorem
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by Jim Hartman
College of Wooster
Wooster, Ohio
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and Larry Riddle
professor of mathematics
Agnes Scott College
Decatur, Georgia
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Theorem 1: Suppose
is differentiable on an open interval containing
. If both
and
exist, then the two limits are equal, and the common
value is
Proof: Let 
and
. By the Mean Value Theorem, for every positive h sufficiently small, there exists
satisfying
such that:
.
Then:
.
Similarly, for every positive h sufficiently small, there exists
satisfying
such that:
.
Then:
.
This shows that
.
Note: The same proof can be modified to show that if
is continuous at
and differentiable on both sides of
, and if
, then
is differentiable at
with
.
Theorem 2: Suppose p and q are defined on an open interval containing
, and each are differentiable at
Let:
Then f is differentiable at
if and only if 
and 
.
Proof: We know that 
exists if and only if 
.
We have that:
.
Also:
if and only if 
. So f will be differentiable at
if and only if 
and 
.
2003 AB6, part (c)
Suppose the function
is defined by:
where
and
are constants. If
is differentiable at 
what are the values of
and
?
Method 1: We are told that
is differentiable at
, and so
is certainly differentiable on the open interval (0,5).
and 
So the two limits both exist and by Theorem 1 must be equal. Hence 
.
Since
is continuous at
,
. This gives the two equations to solve for
and
.
Method 2: Let
and
. Both are differentiable at
. If
is differentiable at
, then Theorem 2 implies that 
and 
. This yields the two same two equations as Method 1.
Either the note after Theorem 1 or Theorem 2 can be used to show that if we choose
and
, then we can prove that
is differentiable at
.
Jim Hartman teaches at the College of Wooster in Wooster, Ohio. He has been involved with
AP Calculus since 1991 as a Table Leader at the AP Reading and through workshops and AP Summer Institutes.
Larry Riddle is a professor of mathematics at Agnes Scott College in Decatur, Georgia.
He started grading AP Calculus exams shortly after beginning his teaching career, and nineteen years later is now in his fourth year as the Chief Reader for AP Calculus.
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