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(a) Use (
5.142
)and(
5.144
) to show that
Z
b
F.˛;t/
dt
D
Z
b
a
d
d˛
@
@˛
F.˛;t/
dt
:
(5.145)
a
(b) Strictly speaking, the derivation above is only valid when (
5.142
) holds, and
it holds for sufficiently smooth functions and we do not want to be precise on
that issue. But the relation (
5.141
) may also be derived by a direct calculation.
Define
Z
b
d
d˛
.˛
p.t//
2
dt
L.˛/
D
(5.146)
a
and
R.˛/
D
Z
b
a
@
@˛
.˛
p.t//
2
dt
:
(5.147)
Show, by direct computations, that
L.˛/
D
2˛.b
a/
2
Z
b
a
p.t/
dt
and
R.˛/
D
2˛.b
a/
2
Z
b
a
p.t/
dt
:
Conclude that (
5.141
) thus holds.
˘
Exercise 5.6.
Compute a constant approximation, using the least squares method,
for the following functions:
(a) y.t/
D
1
C
1
100
sin.t /;
0
t
:
(b) y.t/
D
e
t
;
0
t
1=e:
(c) y.t/
D
p
t;
0
t
:
˘
Exercise 5.7.
Compute a linear least squares approximation for the following func-
tions:
(a) y.t/
D
e
t
;
0
t
1:
(b) y.t/
D
e
t
;
0
t
1:
(c) y.t/
D
1=t;
1
t
2:
˘
