This preview shows pages 1–2. Sign up to view the full content.
Project 23
MAC 2311
1. In Lecture 23 we saw that Mean Value Theorem can be used to prove the following: If
f
and
g
are two functions such that
f
0
(
x
) =
g
0
(
x
)
on an interval
(
a;b
)
, then there is a constant
C
such that
f
(
x
) =
g
(
x
) +
C
for all
x
in
(
a;b
)
. How does this explain our formula for the
general antiderivative of a function (that is, if F is an antiderivative of
f
, why must all others
be F plus a constant)?
You can use this property to prove interesting identities. For example.
..
Look at the functions
y
= tan
2
(
x
)
and
y
= sec
2
(
x
)
. Show that they have the same derivative.
Use this relationship to prove the identity:
tan
2
(
x
) + 1 = sec
2
(
x
)
on
(
±
±
2
;
±
2
)
.
Show that the function
y
=
x
1
ln(
x
)
is constant on its domain. (Do this by calculating the
derivative using logarithmic differentiation.) What is the constant?
By taking the derivative of each side, prove that for positive functions
f
and
g
, we have
ln[
f
(
x
)
g
(
x
)] = ln[
f
(
x
)] + ln[
g
(
x
)]
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2. What is the derivative of
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 ALL
 Mean Value Theorem

Click to edit the document details