Project 29
MAC 2311
YOU MUST SHOW YOUR WORK TO RECEIVE FULL CREDIT!!
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
(
x
) =
g
(
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.
What then must be the relationship between the two
functions?
Use this relationship to prove the identity tan
2
(
x
) + 1 = sec
2
(
x
) .
Look at the function
y
=
x
1
ln(
x
)
. Calculate the derivative using logarithmic
differentiation. (Does the answer surprise you?) What other functions have
this derivative?
Use this relationship to evaluate
x
1
ln(
x
)
for all
x
in its domain.
(Hint: the
value
e
simplifies nicely when you plug it in.)
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2. What is the derivative of sin(2
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 Fall '08
 ALL
 Calculus, Derivative, Mean Value Theorem, Continuous function

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