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MAD 4401
Quiz 11
1.
Let
f
(
x
)
be continuous and suppose that the sequence
x
0
,
f
(
x
0
),
f
(
f
(
x
0
)),
f
(
f
(
f
(
x
0
))),
…
converges to
z
.
Show that
f
(
z
)
=
z
.
This
sequence can also be defined as
x
n
{ }
n
=
0
∞
where
x
0
is arbitrary and
x
n
+
1
=
f
(
x
n
)
for
n
=
0,1,2,3,
…
.
2.
State the Mean Value Theorem.
3.
Let
f
:
R
→
R
be a differentiable function.
Suppose that
f
(
z
)
=
z
is a fixed
point for
f
.
Suppose that for some
ε
>
0
for all
c
∈
[
z
−
,
z
+
]
it is true that
′
f
(
c
)
<
1
.
Show that for any
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Unformatted text preview: x not equal to z that f ( x ) − z < x − z . 4. Consider the equation f ( x ) = and suppose that z is a solution. Show that z is a fixed point of the function g ( x ) = x − f ( x ) ′ f ( x ) . 5. Compute the derivative of g ( x ) = x − f ( x ) ′ f ( x ) . What is the value of ′ g ( z ) ?...
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This note was uploaded on 11/12/2011 for the course STA 3032 taught by Professor Kyung during the Summer '08 term at University of Florida.
 Summer '08
 Kyung
 Statistics

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