Embed Size (px)
DESCRIPTION
At once arbitrary yet specific and particular. Life without variables is verbose. At once arbitrary yet specific and particular. Functions. Imaginary square root of -1. Life without variables is verbose. 5 North. 5 North. 5 North. 4 N. 4 N. 4 N. 3 N. 3 N. 3 N. 2 N. 2 N. 2 N. 1 N. - PowerPoint PPT Presentation
Citation preview
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 ) 𝑡=
𝑥𝑣
(𝑖 )2=−1
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 )
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
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
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
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
𝑡=𝑥𝑣
? ? ? ??
? ? ? ??
? ? ? ??
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
= 𝑡=𝑥𝑣
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
𝑡=𝑥𝑣
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 “calculus of variations,” (i.e. optimization problems)
?? ?
? ??
? ??
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
𝑡=𝑥𝑣
Obvious now, but easy to forget when doing “calculus of variations,” (i.e. optimization problems)
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 ) 𝑡=
𝑥𝑣
(𝑖 )2=−1
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
14
(𝑥 , 𝑓 (𝑥 ) )𝑥
𝑦= 𝑓 (𝑥 )
The function fDomain X Codomain YGraph F
The “squaring” 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 )(−2 , (−2 )2=4 ) 𝑥
0 1 2-2 -1
𝑓 (𝑥 )
1
2
3
4
Graph F
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )=𝑥2Association rule
Functions
(𝑥 , 𝑓 (𝑥 ) )𝑥
𝑦= 𝑓 (𝑥 )
The function fDomain X Codomain YGraph F
( 𝑦 ,𝑔 (𝑦 ) )𝑦 𝑧=𝑔 ( 𝑦 )
The function gCodomain ZDomain Y Graph G
Composition of functions
15
(𝑥 , 𝑓 (𝑥 ) )𝑥
𝑦= 𝑓 (𝑥 )
The function fDomain X Codomain YGraph F
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
( 𝑦 ,𝑔 (𝑦 ) )𝑦 𝑧=𝑔 ( 𝑦 )
The function gCodomain ZDomain Y Graph G
Composition of functions
16
𝑥
Domain X
𝑧=𝑔 ( 𝑓 (𝑥 ) )
Codomain Z
(𝑥 , 𝑓 (𝑥 ) )𝑥
𝑦= 𝑓 (𝑥 )
The function fDomain X Codomain YGraph F
( 𝑦 ,𝑔 (𝑦 ) )𝑦 𝑧=𝑔 ( 𝑦 )
The function gCodomain ZDomain Y Graph G
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
𝑥0 1 2-2 -1
𝑔 ( 𝑓 (𝑥 ) )
1
2
3
4
5
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
Composition of functions
17
𝑥
Domain X
𝑧=𝑔 ( 𝑓 (𝑥 ) )
Codomain Z
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
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
𝑓 (𝑥 )=𝑥2Graph F
0 1 2 3 4 5-2 -1-5-4 -3Codomain Z
𝑔 ( 𝑦 )=𝑦+1Graph G
Inverses of functions
18
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
𝑥
Domain X
𝑧=𝑔 ( 𝑓 (𝑥 ) )
Codomain Z
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
𝑥2 3 40 1
𝑔 ( 𝑓 (𝑥 ) )
1
2
3
4
5
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
Inverses of functions
19
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
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
𝑥
Domain X
𝑥=𝑔 ( 𝑓 (𝑥 ) )
Codomain X
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
𝑓 (𝑥 )2 3 40 1
5
𝑔 ( 𝑓 (𝑥 ) )1234 STOP
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
𝑥
Domain X
𝑥=𝑔 ( 𝑓 (𝑥 ) )
Codomain X
𝑥2 3 40 1
5
𝑓 (𝑥 )1234
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 ) 𝑡=
𝑥𝑣
(𝑖 )2=−1
or
1
2
3
4
or 0 1 2-2 -1 3 4
Square-root “function” 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
𝑓 (𝑥 )=𝑥2Graph F
0 1 2 3 4 . . .-2 -1. . .-4 -3Codomain X
𝑔 ( 𝑦 )=undosquaring (𝑦 )Graph G
(𝑥 , 𝑓 (𝑥 ) )𝑓 (𝑥 )
Co/domain YGraph F
( 𝑓 (𝑥 ) ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G
(𝑥 ,𝑔 ( 𝑓 (𝑥 ) ) )
Graph G FThe function g f
𝑥
Domain X
𝑥=𝑔 ( 𝑓 (𝑥 ) )
Codomain X
Can’t tell which one value to return
? ?
Square-root “function” and
23
𝑦1 2 3 4
-2
-1
𝑔 ( 𝑦 )
1
2
-1-2 0
Square-root “function” and
24
𝑦1 2 3 4
𝑔 ( 𝑦 )
-1-2
-2
-1
1
2
0
1.41 𝑖1.41 𝑖1.41 𝑖
1.411.41
11
000-1
Square-root “function” and
25
𝑦3 4
ℜ [𝑔 ( 𝑦 ) ]
2
-2
-2 0 1 2𝑖
(ℜ [𝑔 (𝑦 ) ]+ 𝑖ℑ [𝑔 (𝑦 ) ] )2
(𝑖 )2=−1
𝑖𝑖 ∙𝑖=−1𝑖𝑖 0 ∙0=0
11 ∙1=1 1.41
1.41 ∙1.41≅ 2
-2
-1
1
2
0
(1.41 𝑖 ) ∙ (1.41 𝑖 )
(1.41 ∙1.41 ) ∙ (𝑖 ∙ 𝑖 )≅−2
𝑔 ( 𝑦 )
𝑖ℑ [𝑔 (𝑦 ) ]
