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In-Depth Information
So it is fair to conclude that (
5.109
) is a natural continuous version of the discrete
average given in (
5.110
).
(b) Did you notice that we have used
Z
b
.˛
y.t//
2
dt
D
Z
b
a
d
d˛
@
@˛
.˛
y.t//
2
dt
‹
a
Is that a legal operation? We discuss this in Exercise
5.5
.
(c) Is ˛ given by (
5.109
) really a minimum? Yes it is, since
F
00
.˛/
D
2.b
a/ > 0:
Example 5.1.
Consider
y.t/
D
sin.t /
defined on 0
t
=2. A constant approximation of y is given by
Z
=2
2
2
p.t/
D
˛
.5:109/
Œcos.t /
=2
0
D
sin.t /
dt
D
0
2
2
:
D
.0
1/
D
Example 5.2.
Consider
y.t/
D
e
t
e
t
defined on 0
t
1. A constant approximation of y is given by
Z
1
e
t
e
t
dt
D
e
t
C
e
t
1
0
p.t/
D
˛
.5:109/
D
e
C
e
1
.1
C
1/
D
0
e
C
e
1
2
1:086:
D
Example 5.3.
Consider
1
10
cos.t /
y.t/
D
t
2
C
defined on 0
t
1. A constant approximation of y is given by
