X2 t04 08 odd & even functions (2013)
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Transcript of X2 t04 08 odd & even functions (2013)
![Page 1: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/1.jpg)
Odd & Even Functions xfxf (1) Even
![Page 2: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/2.jpg)
Odd & Even Functions xfxf
aa
a
dxxfdxxf0
2
(1) Even
![Page 3: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/3.jpg)
Odd & Even Functions xfxf
aa
a
dxxfdxxf0
2
(1) Even
NOTE: horizontal shift
ac
c
ac
ac
dxcxfdxcxf 2
![Page 4: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/4.jpg)
Odd & Even Functions xfxf
aa
a
dxxfdxxf0
2
(1) Even
NOTE: horizontal shift
ac
c
ac
ac
dxcxfdxcxf 2
xfxf (2) Odd
![Page 5: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/5.jpg)
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
![Page 6: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/6.jpg)
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
![Page 7: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/7.jpg)
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
![Page 8: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/8.jpg)
a
dxxaf0
Proof:
![Page 9: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/9.jpg)
a
dxxaf0 dxdu
xauProof:
![Page 10: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/10.jpg)
a
dxxaf0 dxdu
xau
0 ,,0
uaxaux
Proof:
![Page 11: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/11.jpg)
a
dxxaf0 dxdu
xau
0 ,,0
uaxaux
Proof:
0
a
duuf
![Page 12: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/12.jpg)
a
dxxaf0 dxdu
xau
0 ,,0
uaxaux
Proof:
0
a
duuf
a
duuf0
![Page 13: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/13.jpg)
a
dxxaf0 dxdu
xau
0 ,,0
uaxaux
Proof:
0
a
duuf
a
duuf0
a
dxxf0
![Page 14: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/14.jpg)
a
dxxaf0 dxdu
xau
0 ,,0
uaxaux
Proof:
0
a
duuf
a
duuf0
a
dxxf0
eveneven even odd even oddevenoddodd
![Page 15: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/15.jpg)
1
1
3sin e.g. xdxi
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1
1
3sin e.g. xdxi function odd function odd 3
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1
1
3sin e.g. xdxi 0 function odd function odd 3
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1
1
3sin e.g. xdxi 0 function odd function odd 3
1
0
2 1 dxxxii
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1
1
3sin e.g. xdxi 0 function odd function odd 3
1
0
2 1 dxxxii 1
0
21 dxxx
![Page 20: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/20.jpg)
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
![Page 21: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/21.jpg)
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
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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
![Page 23: X2 t04 08 odd & even functions (2013)](https://reader033.fdocuments.net/reader033/viewer/2022051513/547c9b1ab479597f508b46e9/html5/thumbnails/23.jpg)
Exercise 2I; 1 bdf, 2 ace, 3
Exercise 2J; 42, 44
The 100 (not 78)