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In-Depth Information
Z
1
t
2
C
10
cos.t /
dt
D
1
10
sin.t /
1
0
1
1
p.t/
D
˛
.5:109/
3
t
3
C
D
0
1
3
1
10
sin.1/
0:417:
D
C
Below, we will also compute linear and quadratic approximations of the functions
in examples
5.1
-
5.3
. We will graph the functions and their approximations there (see
pages
184
-
185
).
5.3.2
Approximation Using Linear Functions
Again we consider a function y
D
y.t/ defined on an interval Œa; b.Ouraimisto
compute a linear approximation
p.t/
D
˛
C
ˇt
(5.111)
of y using the principle of least squares.
Define
F.˛;ˇ/
D
Z
b
a
.˛
C
ˇt
y.t//
2
dt
:
(5.112)
A minimum of F is obtained by finding ˛ and ˇ such that
@F
@˛
@F
@ˇ
D
D
0:
(5.113)
Since
D
2
Z
b
a
@F
@˛
.˛
C
ˇt
y.t//
dt
;
(5.114)
and
D
2
Z
b
a
@F
@ˇ
.˛
C
ˇt
y.t //t
dt
;
(5.115)
the requirements given by (
5.113
) now read
