For the function defined by the graph below, find the
half-range FCS. [JNTU 2003 S]
SolutionThe graph is composed of two line seg-
ments OP and PB in intervals
0,,
22
pp
p
⎛⎞⎛⎞
⎜⎟⎜⎟
⎝⎠⎝⎠
and
,
respecti
v
el
y
, their slopes being +1 and −1.
Equation of OP : y=x
0,
2
p
⎛⎞
⎜⎟
⎝⎠
Equation of PB : y=−(x−p)
,
2
p
p
⎛⎞
⎜⎟
⎝⎠
0
y
p
P
BA
y
=
p
−
2
p
−
2
x
p p
,
⎞
⎠
⎛
⎝
22
So, the function f (x) is defined by
,0
()
,
x
fx
x
p/
pp/p
<<2
⎧⎫
=
⎨⎬
−2
<
⎩⎭
x
x<
(1)
The half-range FCS is
0
1
1
()cos
2
n
n
fxaanx
=
=+
∑
(2)
where
/2
0
00/2
222
()()afxdxxdxxdx
ppp
p
p
ppp
==+−
∫∫∫
Example 10.28
Find the half-range cosine series for f ( x) =x(2 −x)
in 0 ...
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