Unveiling Unique Mathematical Patterns Through Imagery
Explore a series of captivating mathematical patterns depicted through a collection of images. From intricate equations to visually stunning representations, dive into the intriguing world of mathematical artistry. Each slide unveils a new perspective, inviting you to contemplate the beauty of mathematics in a visual context.
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y ( ) f x = + '( )( f x ) y x x 0 0 0 ( ) f x ( ) f x = 0 x = x x 0 0
( ) ( ) ( ) x M ( ) ( ) ( ) p = f x p x 0 0 p x = f x 0 0 x = f 0 0 ( )( ) ( )( ) n n = f x p x 0 0
(x ) f n n a ( ) n ( ' ' ) ( ) f a f a + + + + 2 n ( ) ( ' )( ) ( ) ... ( ) f a f a x a x a x a ! 2 ! n
( 3 ) ( ' ' ) ( ) f a f a + + + + 2 3 ( ) ( ' )( ) ( ) ( ) ... f a f a x a x a x a ! 2 ! 3
( ) n ( ' ' ) ( ) f a f a + + + + 2 n ( ) ( ' )( ) ( ) ... ( ) f a f a x a x a x a ! 2 ! n ( 3 ) ( ' ' ) ( ) f a f a + + + + 2 3 ( ) ( ' )( ) ( ) ( ) ... f a f a x a x a x a ! 2 ! 3
+ 2 3 1 n n x x x x = + + + + + + + x 1 ... ... e x + ! 2 ! 3 ! ( 1 )! n n
2 . 1e 2 3 2 . 1 + 2 . 1 + = 2 . 1 + + 2 . 1 e 1 .......... ......... ! 2 ! 3 E 2 . 1e % a a
( ) 2! f a ( )( 2 ( ) f x = ( ) f a + - ) + ( - ) f a x a x a ) 3 ( ( ) f a + + 3 ( ) ... x a ! 3 n ) ( ( ) f a + + n ( ) x a R n ! n x n ( - ) x t = ( + 1) n ( ) t dt R f n ! n a
x a "( ) 2! f f x 2 ( ) = ( ) f x + '( )( x - ) + ( - ) i f x f x x x x + 1 + 1 + 1 i i i i i i i (3) ( ) n ( ) 3! ( ) ! n f x x 3 n + ( - ) + ... + ( - ) + i i x x x x R + 1 + 1 i i i i n
( 3 ) " ( ) ( ) f x ! f x = + + + 2 3 ( ) ( ) ( ' ) i i f x f x f x h h h + 1 i i i 2 ! 3 ( ) n ( ) f x + + + n ... i h R n ! n ) 1 + ( n ( ) f + = 1 n R h n + ( 1 )! n
(3) ( ) n ( ) 2! ( ) 3! ( ) ! n f x f x f x ( ) x h 2 3 n ( ) = ( ) f x + + + + ... + + i i i f x f h x h h R + 1 i i i n ( + 1) n ( ) 1)! f x + 1 n = R h n ( + n x
2 4 6 x x x = + + cos( ) 1 x ! 2 ! 4 ! 6 3 5 7 x x x = + + sin( ) x x ! 3 ! 5 ! 7 2 3 4 5 x x x x = + + + + + + ex 1 x ! 2 ! 3 ! 4 ! 5
( ) nP x ( ) x R n
( ) 2! ( ) 3! f x f x 2 3 ( ) ( ) f x ( ) x h + = + + + + L f x h f h h
2 3 4 5 h h h h ( ) ( ) f x ( ) x h ( ) x ( ) x ( ) x ( ) x + = + + + + + + L f x h f f f f f 2! 3! 4 5 2 3 4 5 h h h h ( ) ( ) 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0 ( ) 0 0 + = + + + + + + L f h f f h f f f f 2! 3! 4 5
3 5 7 x x x sin x x = - + - + 3! 5! 7! + 2 1 n x n n n = ( 1) - forall x (2 + 1)! = 0
= ( ) T x x 1 3 x = ( ) T x x 3 3! 3 5 x x = + T x 5 3! 5!
= 1 . 0 . 0 5 . 0 . 0 2 . 1 + 4 3 2 ( ) 15 25 f x x x x x
= 1 . 0 . 0 5 . 0 . 0 2 . 1 + 4 3 2 ( ) 15 25 f x x x x x ( ) @ ( ) f x = 1.2 - 0.25 = 0.95 if = 1 f x h h + 1 i i
2 ( ) @ 1.2 - 0.25 - 0.5 = 0.45 if = 1 f x h h h + 1 i
Errors Reduced step size f(x) = 0.1x4 - 0.15x3 - 0.5x2 - 0.25x + 1.2
( ) 6 ( ) 4 = ( ) 4 f f = 74, 125 , f ( ) 4 ( ) 4 = x f f = ( ) x 30, = 4 6 f 2 3 h h ( ) ( ) f x ( ) x h ( ) x ( ) x + = + + + + L f x h f f f 2! 3! = , 4 x = = 6 4 2 h
2 3 2 2! 2 3! ( ) ( ) 4 ( ) 4 2 ( ) 4 ( ) 4 4 + 2 = + + + f f f f f 2 3 2 2! 2 3! ( ) 6 ( ) = 125 + 74 2 + 30 + 6 f = = + + + 125 341 148 60 8 ( ) 6 f = 4 x
1 = = expansion series Taylor Obtain of at 1 f(x) a x 1 x = = ( ) ) 1 ( 1 f x f x 1 2 = = ( ' ) ) 1 ( ' 1 f x f 2 x ) 2 ( ) 2 ( = = ( ) ) 1 ( 2 f x f 3 x 6 4 ) 3 ( ) 3 ( = = ( ) ) 1 ( 6 f x f 2 3 1 : ) 1 = ) 1 + ) 1 ) 1 + Taylor Series Expansion ( ( ( ( ... a x x x 37
= ( ) f x sin( ) x
p p n f (n)(x) f (n)( ) ( ) 4 an= f (n)( 1 1 a = )/n! 4 1 2 4 1 p sin = = 0 sin( ) x 2 0 0! 2 1 1 2 1 1 2 1 1 2 1 1 2 1 ( ) 1 p a = = cos( ) x cos = 1 4 1 2 1! 2 ( ) ( ) 1 p sin( ) x a = - sin = - 2 4 2 2 2! 1 p a = cos( ) x - cos = - 3 3 4 3! 2 ( ) 1 p a = sin( ) x sin = 4 4 4 2 4! 1 1 1 1 2 1 1 2 1 1 2 1 1 2 2 3 4 5 ( ) ( ) ( ) ( ) ( ) p p p p p + - - - - - + - + - - K x x x x x 4 4 4 4 4 2 3! 4! 5! 2 2
= ( ) cos( 2 ) f x x p = x 4
( ) f x ( ( ) ( )( ) x f x f x x x + - 0 - 0 0 ( 2 3 ) - ) x x x + ( ) + ( ) 0 0 f x f x 0 0 2! 3!
p p ( ) f x cos(2 ) cos 0 x f = = = 4 2 p p ( ) x 2sin(2 ) 2sin 2 f x f = - = - = - 4 2 p p ( ) x 4cos(2 ) 4cos 0 f x f = - = - = 4 2 p p ( ) x 8sin(2 ) 8sin 8 f x f = = = 4 2
( ) f x ( ) ( )( ) f x f x x x + - 0 - 0 0 ( 2 3 ( ) - ) x x x x + ( ) + ( ) 0 0 f x f x 0 0 2! 3! p ( ) f x 0 2 x - - 4 2 3 p p - - x x 4 4 0 8 + + 2! 3! 3 4 3 p p ( ) f x 2 x x - - + - 4 4
The exponentia function l computed is by 2 3 4 n x x x x = + + + + + + + x 1 ... ... e x 2 ! 3 ! 4 ! n x = 0.5
|a| < s s
= + ( ) sin( ) cos( 2 ) f x x x
2 x (0) ( ) f x (0) (0)2 f f x f + + ( ) ( ) ( ) sin( ) cos( ) sin( ) = - cos(2 ) 2sin(2 ) 4cos(2 ) - (0) (0) (0) 1 1 f x f f x x f = = + = = = - x x x x f f - 4 x x
2 ( ) f x 1 2 x x + - 2 (1.5) 1 1.5 + 2 (1.5) 2 f - = -
( )( x ( ) f x ( ) f x ) f x x + - 0 - 0 0 2 ( ) x x + ( ) 0 f x 0 2 p = = 1.5708... x 0 2
