程序代写案例-ENG2086

ENG2086 - Engineering Mathematics 2
Solutions
Q1






Question 3

a)


[3 marks]



[3 marks]

-0.5
0
0.5
1
1.5
2
2.5
-8 -6 -4 -2 0 2 4 6 8
Even extension
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-8 -6 -4 -2 0 2 4 6 8
Odd extension

b)
 The period is T=4sthe frequencyisHzandthe angular frequency
isr/s
[2]

c)
Fourier coefficients for the even extension:

The function is even so bn=0
[1 mark]

଴ =
2

න () =
4

න 2ଶ =
்/ଶ

்/ଶ
ି்/ଶ
4
4
න 2ଶ =


න 2ଶ =
2ଷ
3



=
2
3



[3 marks]



௡ =
2

න () cos = න 2ଶ cos =




ିଶ

= 2 ቐଶ
sin
ฬ଴


2

න sin


ቑ =
= 2 ൞ଶ
sin
ฬ଴


2


cos
ฬ଴

+
1

න cos


ቑൢ =
= 2 ቐଶ
sin
ฬ଴


2


cos
ฬ଴

+
1

sin
ฬ଴

ቋቑ =
= 2 ቊ
sin


2


cos

+
1

sin
ൠቋ
=

=
4

൞sin

2

4

− cos

2
+
2sin 2
ቑൢ

[9 marks]


d)
1 = ସ

൜1 − ସ

ቄଶ

ቅൠ=ସ

ቄ1 − ଼
గమ

2 = ସ
ଶగ
ቄ− ସ
ଶగ
ቅ=− ସ
గమ

3 =
4
3
ቊ−1 −
4
3 ൜
−2
3ൠቋ
=
4
3
൜−1 +
8
9ଶൠ

[3 marks]


() =

2
+ ෍ ௡ cos

௡ୀଵ
=
=
1
6
+
4

൜1 −
8
ଶൠ
cos

2

4

cos +
4
3
൜−1 +
8
9ଶൠ
cos3

2
+ ⋯

[1 mark]

















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