636 Higher Engineering Mathematics
3. Determine the half-range Fourier sine series
for the function defined by:
f (t ) =
t, 0 < t < 1
(2 − t ), 1 < t < 2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
f (t ) =
8
π 2
sin
πt
2
−
1
3 2 sin
3πt
2
+
1
5 2 sin
5πt
2
− · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
4. Show that the half-range Fourier cosine series
for the function f (θ) = θ
2 in the range 0 to 4
is given by:
f (θ) =
16
3
−
64
π 2
cos
πθ
4
−
1
2 2 cos
2πθ
4
+
1
3 2 cos
3πθ
4
− · · ·
Sketch the function within and outside of the
given range.