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
)]
.
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 Spring '08
 ALL
 Calculus, Derivative, Mean Value Theorem, Continuous function, General Antiderivative, 145.3 liters

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