at once arbitrary yet specific and particular 1 life without variables is verboseat once arbitrary...
TRANSCRIPT
![Page 1: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/1.jpg)
At once arbitrary yet specific and particular
1
Life without variables is verbose At once arbitrary yet specific and particular
Functions Imaginary square root of -1
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s ) ๐ก=
๐ฅ๐ฃ
(๐ )2=โ1
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Life without variables is verbose
4
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
STOP
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s )
![Page 3: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/3.jpg)
Life without variables is verbose
5
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s )
Example duration
Example distance
Example speed
![Page 4: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/4.jpg)
Life without variables is verbose
6
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s )
Example duration
Example distance
Example speed
![Page 5: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/5.jpg)
Life without variables is verbose
7
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s )
Example duration
Example distance
Example speed
![Page 6: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/6.jpg)
At once arbitrary yet specific and particular
8
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s )
Example duration
Example distance
Example speed
t0 1 2 3 4 5 6 7 . . .-2 -1
# of seconds
x. . .0 1 2 3 4 5 6 7-2 -1
# of meters
v0 1 2 3 4 5 6 7 . . .-2 -1
# of meters per second
๐ก=๐ฅ๐ฃ
? ? ? ??
? ? ? ??
? ? ? ??
![Page 7: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/7.jpg)
At once arbitrary yet specific and particular
9
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
Example duration
Example distance
Example speed
t0 1 2 3 4 5 6 7 . . .-2 -1
# of seconds
x. . .0 1 2 3 4 5 6 7-2 -1
# of meters
v0 1 2 3 4 5 6 7 . . .-2 -1
# of meters per second
0 1 . . .-1t
0 1 . . .-1x
0 1 . . .-1v
= ๐ก=๐ฅ๐ฃ
![Page 8: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/8.jpg)
At once arbitrary yet specific and particular
10
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
t, an arbitrary yet specific and particular example of a duration measured in seconds whose number value is chosen from the highlighted domain below
x, an arbitrary yet specific and particular example of a distance measured in meters whose number value is chosen from the highlighted domain below
v, an arbitrary yet specific and particular example of a speed measured in meters per second whose number value is chosen from the highlighted domain below
๐ก=๐ฅ๐ฃ
0 1 . . .-1
0 1 . . .-1
0 1 . . .-1
=
0 1 . . .-1
0 1 . . .-1
0 1 . . .-1
t
x
v=
Obvious now, but easy to forget when doing โcalculus of variations,โ (i.e. optimization problems)
?? ?
? ??
? ??
![Page 9: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/9.jpg)
At once arbitrary yet specific and particular
11
Main street
1 N
2 N
3 N
4 N
5 North
1 S
2 S
3 S
4 S
5 South
STOP
t, an arbitrary yet specific and particular example of a duration measured in seconds whose number value is chosen from the highlighted domain below
x, an arbitrary yet specific and particular example of a distance measured in meters whose number value is chosen from the highlighted domain below
v, an arbitrary yet specific and particular example of a speed measured in meters per second whose number value is chosen from the highlighted domain below
0 1 . . .-1
0 1 . . .-1
=
0 1 . . .-1
๐ก=๐ฅ๐ฃ
Obvious now, but easy to forget when doing โcalculus of variations,โ (i.e. optimization problems)
![Page 10: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/10.jpg)
At once arbitrary yet specific and particular
12
Life without variables is verbose At once arbitrary yet specific and particular
Functions Imaginary square root of -1
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s ) ๐ก=
๐ฅ๐ฃ
(๐ )2=โ1
![Page 11: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/11.jpg)
an ordered pair
Functions
13
an arbitrary yet specific and particular object from collection X
the resulting object in collection Y
The function f
Domain X Codomain YGraph F
Essential stipulation: Each maps to precisely one .
Range of f
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14
(๐ฅ , ๐ (๐ฅ ) )๐ฅ
๐ฆ= ๐ (๐ฅ )
The function fDomain X Codomain YGraph F
The โsquaringโ function f
Domain X
Codomain Y
0 1 2 3 4 . . .-2 -1. . . -4 -3
0 1 2 3 4 . . .-2 -1. . . -4 -3
(0 ,02=0 )(1 ,12=1 )(2 ,22=4 )(โ2 , (โ2 )2=4 ) ๐ฅ
0 1 2-2 -1
๐ (๐ฅ )
1
2
3
4
Graph F
(๐ฅ , ๐ (๐ฅ ) )๐ (๐ฅ )=๐ฅ2Association rule
Functions
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(๐ฅ , ๐ (๐ฅ ) )๐ฅ
๐ฆ= ๐ (๐ฅ )
The function fDomain X Codomain YGraph F
( ๐ฆ ,๐ (๐ฆ ) )๐ฆ ๐ง=๐ ( ๐ฆ )
The function gCodomain ZDomain Y Graph G
Composition of functions
15
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(๐ฅ , ๐ (๐ฅ ) )๐ฅ
๐ฆ= ๐ (๐ฅ )
The function fDomain X Codomain YGraph F
(๐ฅ , ๐ (๐ฅ ) )๐ (๐ฅ )
Co/domain YGraph F
( ๐ (๐ฅ ) ,๐ ( ๐ (๐ฅ ) ) )
Graph G
( ๐ฆ ,๐ (๐ฆ ) )๐ฆ ๐ง=๐ ( ๐ฆ )
The function gCodomain ZDomain Y Graph G
Composition of functions
16
๐ฅ
Domain X
๐ง=๐ ( ๐ (๐ฅ ) )
Codomain Z
(๐ฅ , ๐ (๐ฅ ) )๐ฅ
๐ฆ= ๐ (๐ฅ )
The function fDomain X Codomain YGraph F
( ๐ฆ ,๐ (๐ฆ ) )๐ฆ ๐ง=๐ ( ๐ฆ )
The function gCodomain ZDomain Y Graph G
(๐ฅ ,๐ ( ๐ (๐ฅ ) ) )
Graph G FThe function g f
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๐ฅ0 1 2-2 -1
๐ ( ๐ (๐ฅ ) )
1
2
3
4
5
(๐ฅ , ๐ (๐ฅ ) )๐ (๐ฅ )
Co/domain YGraph F
( ๐ (๐ฅ ) ,๐ ( ๐ (๐ฅ ) ) )
Graph G
Composition of functions
17
๐ฅ
Domain X
๐ง=๐ ( ๐ (๐ฅ ) )
Codomain Z
(๐ฅ ,๐ ( ๐ (๐ฅ ) ) )
Graph G FThe function g f
Domain X
Co/domain Y
0 1 2 3 4 5-2 -1-5-4 -3
0 1 2 3 4 5-2 -1-5-4 -3
๐ (๐ฅ )=๐ฅ2Graph F
0 1 2 3 4 5-2 -1-5-4 -3Codomain Z
๐ ( ๐ฆ )=๐ฆ+1Graph G
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Inverses of functions
18
(๐ฅ , ๐ (๐ฅ ) )๐ (๐ฅ )
Co/domain YGraph F
( ๐ (๐ฅ ) ,๐ ( ๐ (๐ฅ ) ) )
Graph G
๐ฅ
Domain X
๐ง=๐ ( ๐ (๐ฅ ) )
Codomain Z
(๐ฅ ,๐ ( ๐ (๐ฅ ) ) )
Graph G FThe function g f
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๐ฅ2 3 40 1
๐ ( ๐ (๐ฅ ) )
1
2
3
4
5
(๐ฅ , ๐ (๐ฅ ) )๐ (๐ฅ )
Co/domain YGraph F
( ๐ (๐ฅ ) ,๐ ( ๐ (๐ฅ ) ) )
Graph G
Inverses of functions
19
(๐ฅ ,๐ ( ๐ (๐ฅ ) ) )
Graph G FThe function g f
Domain X
Co/domain Y
0 1 2 3 4 . . .-2 -1. . .-4 -3
0 1 2 3 4 . . .-2 -1. . .-4 -3
somethingGraph F
0 1 2 3 4 . . .-2 -1. . .-4 -3Codomain X
undo somethingGraph G
๐ฅ
Domain X
๐ฅ=๐ ( ๐ (๐ฅ ) )
Codomain X
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Inverses of functions
20
Domain X
Co/domain Y
0 1 2 3 4 . . .-2 -1. . .-4 -3
0 1 2 3 4 . . .-2 -1. . .-4 -3
somethingGraph F
0 1 2 3 4 . . .-2 -1. . .-4 -3Codomain X
undo somethingGraph G
๐ (๐ฅ )2 3 40 1
5
๐ ( ๐ (๐ฅ ) )1234 STOP
(๐ฅ , ๐ (๐ฅ ) )๐ (๐ฅ )
Co/domain YGraph F
( ๐ (๐ฅ ) ,๐ ( ๐ (๐ฅ ) ) )
Graph G
(๐ฅ ,๐ ( ๐ (๐ฅ ) ) )
Graph G FThe function g f
๐ฅ
Domain X
๐ฅ=๐ ( ๐ (๐ฅ ) )
Codomain X
๐ฅ2 3 40 1
5
๐ (๐ฅ )1234
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At once arbitrary yet specific and particular
21
Life without variables is verbose At once arbitrary yet specific and particular
Functions Imaginary square root of -1
2s=20m
(10 m / s )5s=
50m(10 m / s ) 7s=
70m(10 m / s ) ๐ก=
๐ฅ๐ฃ
(๐ )2=โ1
![Page 20: At once arbitrary yet specific and particular 1 Life without variables is verboseAt once arbitrary yet specific and particular FunctionsImaginary square](https://reader038.vdocument.in/reader038/viewer/2022110208/56649dde5503460f94ad7cac/html5/thumbnails/20.jpg)
or
1
2
3
4
or 0 1 2-2 -1 3 4
Square-root โfunctionโ and
22
Domain X
Co/domain Y
0 1 2 3 4 . . .-2 -1. . .-4 -3
0 1 2 3 4 . . .-2 -1. . .-4 -3
๐ (๐ฅ )=๐ฅ2Graph F
0 1 2 3 4 . . .-2 -1. . .-4 -3Codomain X
๐ ( ๐ฆ )=undosquaring (๐ฆ )Graph G
(๐ฅ , ๐ (๐ฅ ) )๐ (๐ฅ )
Co/domain YGraph F
( ๐ (๐ฅ ) ,๐ ( ๐ (๐ฅ ) ) )
Graph G
(๐ฅ ,๐ ( ๐ (๐ฅ ) ) )
Graph G FThe function g f
๐ฅ
Domain X
๐ฅ=๐ ( ๐ (๐ฅ ) )
Codomain X
Canโt tell which one value to return
? ?
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Square-root โfunctionโ and
23
๐ฆ1 2 3 4
-2
-1
๐ ( ๐ฆ )
1
2
-1-2 0
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Square-root โfunctionโ and
24
๐ฆ1 2 3 4
๐ ( ๐ฆ )
-1-2
-2
-1
1
2
0
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1.41 ๐1.41 ๐1.41 ๐
1.411.41
11
000-1
Square-root โfunctionโ and
25
๐ฆ3 4
โ [๐ ( ๐ฆ ) ]
2
-2
-2 0 1 2๐
(โ [๐ (๐ฆ ) ]+ ๐โ [๐ (๐ฆ ) ] )2
(๐ )2=โ1
๐๐ โ๐=โ1๐๐ 0 โ0=0
11 โ1=1 1.41
1.41 โ1.41โ 2
-2
-1
1
2
0
(1.41 ๐ ) โ (1.41 ๐ )
(1.41 โ1.41 ) โ (๐ โ ๐ )โ โ2
๐ ( ๐ฆ )
๐โ [๐ (๐ฆ ) ]