University of Florida/Egm6341/s11.team1.Gong/Mtg20
<
University of Florida
<
Egm6341
< s11.team1.Gong
Mtg 20: Wed, 16 Feb 11
page20-1
Pf of SSET p.19-3 cont'd
(
1
)
p
.19
−
3
R
o
l
l
e
′
s
t
h
m
⇒
∃
ς
1
∈
]
0
,
1
[
s
t
G
(
1
)
(
ς
1
)
=
(
1
)
0
{\displaystyle {\color {red}(1)}{\color {blue}p.19-3\ Rolle's\ thm}\ \Rightarrow \ \exists \ \varsigma _{1}\in \ ]\ 0,1\ [\ st\ G^{\color {blue}(1)(\varsigma _{1})}{\overset {\color {red}(1)}{=}}0}
N
o
w
G
(
1
)
(
0
)
=
(
2
)
0
w
h
y
?
{\displaystyle Now\ G^{(1)}(0)\ {\overset {\color {red}(2)}{=}}0\ {\color {red}why?}}
(
1
)
p
.19
−
1
:
G
(
1
)
(
t
)
=
(
3
)
e
(
1
)
(
t
)
−
5
t
4
e
(
1
)
{\displaystyle {\color {red}(1)}\ {\color {blue}p.19-1:}\ G^{(1)}(t){\overset {\color {red}(3)}{=}}e^{(1)}(t)-5t^{4}e(1)}
(
5
)
p
.18
−
3
:
e
(
t
)
=
A
(
t
)
−
A
2
L
(
t
)
(
4
)
{\displaystyle {\color {red}(5)}{\color {blue}p.18-3:}\ e(t)=A(t)-{A}_{2}^{L}(t)\ {\color {red}(4)}}
e
(
1
)
(
t
)
=
A
(
1
)
−
A
2
L
(
1
)
(
t
)
(
5
)
{\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e^{(1)}(t)=A^{(1)}-{A}_{2}^{L{\color {red}(1)}}(t)\ {\color {red}(5)}}
page20-2
A
(
t
)
=
∫
+
t
−
t
−
=
∫
k
−
t
−
+
∫
t
k
−
{\displaystyle A(t)=\int _{+t}^{-t}-=\int _{k}^{-t}-+\int _{t}^{k}-}
A
(
1
)
=
(
1
)
F
(
−
t
)
+
F
(
t
)
{\displaystyle A^{(1)}{\overset {\color {red}(1)}{=}}F(-t)+F(t)}
k
∈
]
−
t
,
t
[
(
k
i
s
c
o
n
s
t
a
n
t
)
{\displaystyle k\in {\color {red}]}-t,t{\color {red}[}\ ({\color {red}kisconstant})}
(4)
p.20-1
&
(5)
p.18-3:
A
2
L
(
1
)
(
t
)
=
(
2
)
1
3
[
F
(
−
t
)
+
4
F
(
0
)
+
F
(
t
)
]
+
t
3
[
F
(
1
)
(
−
t
)
+
F
(
1
)
(
t
)
{\displaystyle A_{2}^{L{\color {red}(1)}}(t){\overset {\color {red}(2)}{=}}{\frac {1}{3}}[F(-t)+4F(0)+F(t)]+{\frac {t}{3}}[F^{(1)}(-t)+F^{(1)}(t)}
(5)
p.20-1:
e
(
1
)
(
0
)
=
A
(
1
)
(
0
)
−
A
2
L
(
1
)
(
0
)
=
2
F
(
0
)
−
[
2
F
(
0
)
+
0
]
=
0
(
3
)
{\displaystyle e^{(1)}(0)=A^{(1)}(0)-A_{2}^{L(1)}(0)=2F(0)-[2F(0)+0]=0\ {\color {red}(3)}}
(
3
)
a
n
d
(
3
)
p
.20
−
1
⇒
(
2
)
p
.20
−
1
{\displaystyle {\color {red}(3)\ and\ (3)}{\color {blue}p.20-1}\Rightarrow {\color {red}(2)}{\color {blue}p.20-1}}
Recall:
G
(
1
)
(
ξ
1
)
=
0
(
1
)
p
.20
−
1
{\displaystyle G^{(1)}(\xi _{1})=0\ {\color {red}(1)}{\color {blue}p.20-1}}
G
(
1
)
(
0
)
=
0
(
2
)
p
.20
−
1
{\displaystyle G^{(1)}(0)=0\ {\color {red}(2)}{\color {blue}p.20-1}}
page20-3
R
o
l
l
e
′
s
t
h
e
o
r
e
m
⇒
∀
ξ
2
∈
]
0
,
ξ
1
[
s
t
G
(
2
)
(
ξ
2
)
=
0
(
1
)
{\displaystyle Rolle'stheorem\Rightarrow \ \forall \xi _{2}\in ]0,\xi _{1}[st\ \ G^{(2)}(\xi _{2})=0\ {\color {red}(1)}}
A
g
a
i
n
,
G
(
2
)
(
0
)
=
(
2
)
0
H
W
∗
4.3
{\displaystyle Again,\ G^{(2)}(0){\overset {\color {red}(2)}{=}}0\ {\color {blue}HW^{*}4.3}}
(
1
)
a
n
d
(
2
)
R
o
l
l
e
′
s
t
h
e
o
r
e
m
⇒
∀
ξ
3
∈
]
0
,
ξ
2
[
G
(
2
)
(
ξ
2
)
=
0
s
t
G
(
3
)
(
ξ
3
)
=
(
2
)
0
{\displaystyle {\color {red}(1)\ and\ (2)}\ Rolle'stheorem\ \Rightarrow \ \forall \xi _{3}\in ]0,\xi _{2}[\ G^{(2)}(\xi _{2})=0\ st\ \ G^{(3)}(\xi _{3}){\overset {\color {red}(2)}{=}}0}
(
1
)
p
.19
−
1
:
G
(
3
)
(
t
)
e
(
4
)
(
3
)
(
t
)
−
t
2
⏟
(
ξ
)
(
4
)
(
3
)
e
(
1
)
{\displaystyle {\color {red}(1)}{\color {blue}p.19-1:}\ G^{(3)}(t){\overset {\color {red}(4)}{e}}^{(3)}(t)-{\color {red}{\underset {(\xi )(4)(3)}{\underbrace {\color {black}t^{2}} }}}e(1)}
e
(
3
)
(
t
)
=
H
W
∗
4.4
−
t
3
[
F
(
3
)
(
t
)
−
F
(
3
)
(
−
t
)
]
(
5
)
{\displaystyle e^{(3)}(t){\underset {\color {blue}HW^{*}4.4}{=}}-{\frac {t}{3}}[F^{(3)}(t)-F^{(3)}(-t)]\ {\color {red}(5)}}
G
(
3
)
(
ξ
3
)
=
−
ξ
3
3
[
F
(
3
)
(
ξ
3
)
−
F
(
3
)
(
−
ξ
3
)
]
⏞
A
p
p
l
y
D
M
V
T
−
60
(
ξ
3
)
2
e
(
1
)
=
(
6
)
f
r
o
m
(
3
)
0
{\displaystyle G^{(3)}(\xi _{3})=-{\frac {\xi _{3}}{3}}{\color {green}{\overset {Apply\ DMVT}{\overbrace {\color {black}[F^{(3)}(\xi _{3})-F^{(3)}(-\xi _{3})]} }}}-60(\xi _{3})^{2}e(1){\underset {{\color {blue}from}{\color {red}(3)}}{\overset {\color {red}(6)}{=}}}0}
=
D
M
V
T
(
f
)
−
ξ
3
3
[
2
ξ
3
⏟
ξ
3
−
(
−
ξ
3
)
F
(
4
)
(
ξ
4
)
]
−
60
(
ξ
3
)
2
e
(
1
)
ξ
4
∈
]
−
ξ
3
,
ξ
3
[
{\displaystyle {\underset {\color {red}(f)}{\overset {\color {blue}DMVT}{=}}}-{\frac {\xi _{3}}{3}}[{\color {blue}{\underset {\xi _{3}-(-\xi _{3})}{\underbrace {\color {black}2\xi _{3}} }}}F^{(4)}(\xi _{4})]-60(\xi _{3})^{2}e(1)\ \xi _{4}\in \ ]-\xi _{3},\xi _{3}[}
![{\displaystyle {\color {red}(1)}{\color {blue}p.19-3\ Rolle's\ thm}\ \Rightarrow \ \exists \ \varsigma _{1}\in \ ]\ 0,1\ [\ st\ G^{\color {blue}(1)(\varsigma _{1})}{\overset {\color {red}(1)}{=}}0}](../../../320b0a14b843c9963bec3852c1b9e9dc9ff2bb46.svg)
page20-2
![{\displaystyle k\in {\color {red}]}-t,t{\color {red}[}\ ({\color {red}kisconstant})}](../../../a63ebf807459841b14d25d819824c578d7376333.svg)
(4)p.20-1& (5)p.18-3:
![{\displaystyle A_{2}^{L{\color {red}(1)}}(t){\overset {\color {red}(2)}{=}}{\frac {1}{3}}[F(-t)+4F(0)+F(t)]+{\frac {t}{3}}[F^{(1)}(-t)+F^{(1)}(t)}](../../../2ebb20455c1febb5f2c7d16880efa65ecf7ef4ed.svg)
(5) p.20-1:
![{\displaystyle e^{(1)}(0)=A^{(1)}(0)-A_{2}^{L(1)}(0)=2F(0)-[2F(0)+0]=0\ {\color {red}(3)}}](../../../eaffcb493c57acd601feeb608ff11961b276989e.svg)
Recall:
![{\displaystyle Rolle'stheorem\Rightarrow \ \forall \xi _{2}\in ]0,\xi _{1}[st\ \ G^{(2)}(\xi _{2})=0\ {\color {red}(1)}}](../../../70a16829e66f81333255f1b48fbc635df78bc475.svg)
![{\displaystyle {\color {red}(1)\ and\ (2)}\ Rolle'stheorem\ \Rightarrow \ \forall \xi _{3}\in ]0,\xi _{2}[\ G^{(2)}(\xi _{2})=0\ st\ \ G^{(3)}(\xi _{3}){\overset {\color {red}(2)}{=}}0}](../../../e258b398945df71a670ae3474a33ca02baf3e9bf.svg)
![{\displaystyle e^{(3)}(t){\underset {\color {blue}HW^{*}4.4}{=}}-{\frac {t}{3}}[F^{(3)}(t)-F^{(3)}(-t)]\ {\color {red}(5)}}](../../../9b331208fccb768062144a0b8122ac90cf7244e7.svg)
![{\displaystyle G^{(3)}(\xi _{3})=-{\frac {\xi _{3}}{3}}{\color {green}{\overset {Apply\ DMVT}{\overbrace {\color {black}[F^{(3)}(\xi _{3})-F^{(3)}(-\xi _{3})]} }}}-60(\xi _{3})^{2}e(1){\underset {{\color {blue}from}{\color {red}(3)}}{\overset {\color {red}(6)}{=}}}0}](../../../c269ba45d12e663ae28c8b6e8c07dc2d539ff7e2.svg)
![{\displaystyle {\underset {\color {red}(f)}{\overset {\color {blue}DMVT}{=}}}-{\frac {\xi _{3}}{3}}[{\color {blue}{\underset {\xi _{3}-(-\xi _{3})}{\underbrace {\color {black}2\xi _{3}} }}}F^{(4)}(\xi _{4})]-60(\xi _{3})^{2}e(1)\ \xi _{4}\in \ ]-\xi _{3},\xi _{3}[}](../../../5143313f8539ebd5ce71079ce1360ab99952b6a0.svg)
