Cauchy Mean Value Theorem
Hypotheses
•
f
and
g
are continuous on
[
a, b
]
and di
ff
erentiable on
(
a, b
)
.
•
g
(
x
) = 0
for all
x
in
(
a, b
)
.
Conclusion
•
There is a
c
in
(
a, b
)
such that
f
(
c
)
g
(
c
)
=
f
(
b
)

f
(
a
)
g
(
b
)

g
(
a
)
.
Interpretation of Theorem
This roughly says that at at least one time
c
,
speed of
f
speed of
g
=
change in
f
change in
g
, which is the key idea to
proving L’Hospital’s Rule (when
f
(
a
) =
g
(
a
) = 0
).
Idea of Proof
We are trying to show there is a solution to
f
(
x
)
g
(
x
)
=
f
(
b
)

f
(
a
)
g
(
b
)

g
(
a
)
, where
x
is the variable and
a
and
b
are constants. We would like to apply Rolle’s Theorem, but to what function? Notice
f
(
x
)
g
(
x
)
is
not
the derivative of anything nice.
Let’s get rid of fractions and rearrange to get
(
g
(
b
)

g
(
a
))
·
f
(
x
)

(
f
(
b
)

f
(
a
))
·
g
(
x
) = 0
,
which is of the form
constant
·
f
(
x
)

constant
·
g
(
x
) = 0
. The lefthandside is just the
derivative of
(
g
(
b
)

g
(
a
))
·
f
(
x
)

(
f
(
b
)

f
(
a
))
·
g
(
x
)
, which luckily turns out to have all
the properties we will need (see the proof).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 WILKENING
 Rolle’s Theorem, Rolle

Click to edit the document details