x2 t04 08 odd & even functions (2012)
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Odd & Even Functions xfxf (1) Even
Odd & Even Functions xfxf
aa
a
dxxfdxxf0
2
(1) Even
Odd & Even Functions xfxf
aa
a
dxxfdxxf0
2
(1) Even
NOTE: horizontal shift
ac
c
ac
ac
dxcxfdxcxf 2
Odd & Even Functions xfxf
aa
a
dxxfdxxf0
2
(1) Even
NOTE: horizontal shift
ac
c
ac
ac
dxcxfdxcxf 2
xfxf (2) Odd
Odd & Even Functions xfxf
aa
a
dxxfdxxf0
2
(1) Even
NOTE: horizontal shift
ac
c
ac
ac
dxcxfdxcxf 2
xfxf
0
a
a
dxxf
(2) Odd
Odd & Even Functions xfxf
aa
a
dxxfdxxf0
2
(1) Even
NOTE: horizontal shift
ac
c
ac
ac
dxcxfdxcxf 2
xfxf
0
a
a
dxxf
(2) Odd
NOTE: horizontal shift 0
ac
ac
dxcxf
Odd & Even Functions xfxf
aa
a
dxxfdxxf0
2
(1) Even
NOTE: horizontal shift
ac
c
ac
ac
dxcxfdxcxf 2
xfxf
0
a
a
dxxf
(2) Odd
NOTE: horizontal shift 0
ac
ac
dxcxf
(3) aa
dxxafdxxf00
a
dxxaf0
Proof:
a
dxxaf0 dxdu
xauProof:
a
dxxaf0 dxdu
xau
0 ,,0
uaxaux
Proof:
a
dxxaf0 dxdu
xau
0 ,,0
uaxaux
Proof:
0
a
duuf
a
dxxaf0 dxdu
xau
0 ,,0
uaxaux
Proof:
0
a
duuf
a
duuf0
a
dxxaf0 dxdu
xau
0 ,,0
uaxaux
Proof:
0
a
duuf
a
duuf0
a
dxxf0
a
dxxaf0 dxdu
xau
0 ,,0
uaxaux
Proof:
0
a
duuf
a
duuf0
a
dxxf0
eveneven even odd even oddevenoddodd
1
1
3sin e.g. xdxi
1
1
3sin e.g. xdxi function odd function odd 3
1
1
3sin e.g. xdxi 0 function odd function odd 3
1
1
3sin e.g. xdxi 0 function odd function odd 3
1
0
2 1 dxxxii
1
1
3sin e.g. xdxi 0 function odd function odd 3
1
0
2 1 dxxxii 1
0
21 dxxx
1
1
3sin e.g. xdxi 0 function odd function odd 3
1
0
2 1 dxxxii 1
0
21 dxxx
1
0
25
23
21
2 dxxxx
1
1
3sin e.g. xdxi 0 function odd function odd 3
1
0
2 1 dxxxii 1
0
21 dxxx
1
0
27
25
23
72
54
32
xxx
1
0
25
23
21
2 dxxxx
1
1
3sin e.g. xdxi 0 function odd function odd 3
1
0
2 1 dxxxii 1
0
21 dxxx
1
0
27
25
23
72
54
32
xxx
10516
072
54
32
1
0
25
23
21
2 dxxxx
Exercise 2I; 1 bdf, 2 ace, 3
Exercise 2J; 42, 44
The 100 (not 78)
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