θεωρια μαθηματικων κατευθυνσης

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  • 1. http://lisari.blogspot.com : / - . 1. 0 ; i i , . : i i . 0 0 0i , i 0 0 0 . 2. ; i i : ( ) ( ) ( ) ( ) i i i . i i , i i, : ( i) ( i) ( i) ( i) ( ) ( )i . ( ) ( ) ( ) ( ) i i i . i i : 2 ( i)( i) ( i) i( i) i i ( i)( i) i i i i i ( ) ( )i . : ( )( ) ( ) ( ) i i i . , i i , i 0 , i, : 2 2 2 2 2 2 i ( i)( i) ( ) ( )i i i ( i)( i) . , 2 2 2 2 i i i . 3. z i ; z i z i . z i . : i i . i i , i , i . 4. 1 2 1 2 z z z z . M( , ) M ( , ) z i z i . z i z i , , : 2z z 2z z i . 1 z i 2 z i , : x )(zM M(z) y 4

2. http://lisari.blogspot.com : / 1. 1 2 1 2 z z z z 2. 1 2 1 2 z z z z 3. 1 2 1 2 z z z z 4. 1 1 2 2 z z z z . 5. 1 2 1 2 z z z z z z 6. 1 2 1 2 ... z z z z z ... z 5. : 1 2 1 2 z z z z 1 2 1 2 z z z z : 1 z i 2 z i , : 1 2 z z ( i) ( i) ( ) ( )i 1 2 ( ) ( )i ( i) ( i) z z 1 2 ( ) ( )i ( i) ( i) z z 6. i ; i , 4 , 4, : 4 4 4 1 , 0 i , 1 i i i i (i ) i 1 i i -1 , 2 i , 3 7. 2 z z 0 , , R 0 . 2 z z 0 , , , R 0 . , , : 2 2 z 2 4 , 2 4 . , : 0 , : 1,2 z 2 0 , : z 2 0 , , 2 2 2 2 2 2 ( 1)( ) i ( ) i 24 4 (2 ) , : 22 i z 2 2 . : 1,2 i z 2 , . : z : z , z z z , z z . 3. http://lisari.blogspot.com : / . 8. ; M(x,y) z x yi . z M O , 2 2 |z| |OM| x y 9. . : |z| |z | | z| 2 |z| z z 1 2 1 2 |z z | |z | |z | 1 1 2 2 z z z z | z | | z | . 1 2 1 2 1 2 |z | |z | |z z | |z | |z || | 10. : 1 2 1 2 |z z | |z | |z | : 2 2 2 1 2 1 2 1 2 1 2 |z z | |z | |z | |z z | |z | |z | 1 2 1 2 1 1 2 2 (z z )(z z ) z z z z 1 2 1 2 1 1 2 2 z z z z z z z z , . 11.) , 1 2 |z z | ) 0 |z z | , 0; ) 1 2 |z z | |z z |; ) . : 1 2 1 2 (M M ) |z z | ) 0 |z z | , 0 O . ) 1 2 |z z | |z z | O O . . 1. f A; f f x A . : f(A) {y|y f(x) x A} . f A ( )f A . x M(x,y) |z| a y 5 4. http://lisari.blogspot.com : / 2. f A ; f M(x,y) y f(x) , M(x,f(x)) , x A . - f f C . - , , y f(x) f C . , y f(x) f. - f C f, : ) f f C . ) f f(A) f C . ) f 0 x A 0 x x f C (. 8). Cf O y x () Cf O y x () f() Cf O x=x0 A(x0,f(x0)) x0 y x () f(x0) 8 - f C , f , , f | f|. ) f , x x , f, M (x, f(x)) M(x,f(x)) , x x . (. 9). ) | |f f C x x , x x , f C . (. 10). 3. ) ( )f x x ) 2 ( )f x x , 0 ) 3 ( )f x x , 0 ) ( ) f x x , 0 ) ( )f x x , ( )g x | x | . : ) ( )f x x O y x 9 (x, f(x)) y=f(x) y= f(x) (x,f(x)) O y x 10 y=f(x)y=| f(x)| 5. http://lisari.blogspot.com : / 11 a>0 O x y a0 xO y 0 O x y 0 O x y 1 O x y ()00 f>0 a x0 x 35 f 0 x 0 ( ,x ] 0 [x , ) . , 1 0 2 x x x 1 0 2 f(x ) f(x ) f(x ) . 0 f(x ) f. , , f ( , ) . , 1 2 x ,x ( , ) 1 2 x x . 1 2 0 x ,x ( ,x ] , f 0 ( ,x ] , 1 2 f(x ) f(x ) . 1 2 0 x ,x [x , ) , f 0 [x , ) , 1 2 f(x ) f(x ) . , 1 0 2 x x x , 1 0 2 f(x ) f(x ) f(x ) . , 1 2 f(x ) f(x ) , f ( , ) . , f (x) 0 0 0 x ( ,x ) (x , ) . 21. f ; f f . f , f . 22. f 23. http://lisari.blogspot.com : / : f . f (x) 0 x , f . f (x) 0 x , f . 23. 0 0 A(x ,f(x )) f ; 0 0 A(x ,f(x )) f , :. f 0 ( ,x ) 0 (x , ) , , f C 0 0 A(x ,f(x )) . 0 0 A(x ,f(x )) f C , f 0 x 0 x . 24. f ; : 0 0 A(x ,f(x )) f f , 0 f (x ) 0 : f ; f : i) f . ii) f . : f o ( , ) 0 x ( , ) . f 0 x f C 0 0 A(x ,f(x )) , 0 0 A(x ,f(x )) f C . 25. 0 x x f C ; 0 x x f. , 0x x lim f(x) , 0x x lim f(x) 26. y f ( ). y f ( ), x lim f(x) ( x lim f(x) ) . 24. http://lisari.blogspot.com : / 27. y x f , ; y x f , , x lim [f(x) ( x )] 0 , x lim [f(x) ( x )] 0 . 28. () y x ; : y x f , , x f(x) lim R x x lim [f(x) x] R , : x f(x) lim R x x lim [f(x) x] R . 1. : 2 . P(x) Q(x) , P(x) , . 2. , f : f . , f . , , ( , ) , ( , ) . 29. de LHospital. 1o 0x x lim f(x) 0 , 0x x lim g(x) 0 , 0 x R { , } , g'(x) 0 o x o x 0x x f (x) lim g (x) ( ), : 0 0x x x x f(x) f (x) lim g(x) g (x) lim . 2o 0x x lim f(x) , 0x x lim g(x) , 0 x R { , } , g'(x) 0 o x o x 0x x f (x) g (x) lim ( ), : 0 0x x x x f(x) f (x) lim lim g(x) g (x) . : ) . . ) g'(x) 0 o x , o x . ) . . 1. f ; 25. http://lisari.blogspot.com : / f F :F'(x) f(x) , x . 2. f . F f , : G(x) F(x) c, c R , f . G f G(x) F(x) c, c R . G(x) F(x) c, c R , f , G'(x) (F(x) c)' F'(x) f(x), x . G f . , x F (x) f(x) G (x) f(x), : G'(x) F'(x) , x . c , G(x) F(x) c, x . 3*. f [ , ] . f [ , ] . 0 1 2 x x x ... x [ , ] x . 1 [x ,x ] , {1,2,..., } , 1 2S f( ) x f( ) x f( ) x f( ) x , , : 1 S f( ) x . S , 1 lim ( ) f x R . f , f(x)dx f . , 1 f(x)dx lim f( ) x 4. f(x)dx . ) : f(x)dx f(x)dx f(x)dx 0 f(x) 0 x [ , ] , f(x)dx 0 . ) f,g [ , ] , R . : f(x)dx f(x)dx [f(x) g(x)]dx f(x)dx g(x)dx [ f(x) g(x)]dx f(x)dx g(x)dx xv-1 v y=f(x) k 21 x x2x1 xv=a=x0O y 10 26. http://lisari.blogspot.com : / ) f , , , f(x)dx f(x)dx f(x)dx ) f [ , ] . f(x) 0 x [ , ] f , f(x)dx 0 . 5. x F(x) f(x)dx ,x , f . : x a F'(x) f(t)dt f(x) , x . ) : f , x F(x) f(t)dt , x , f . : x a f(t)dt f(x) , x . ) : g(x) a f(t)dt f'(g(x) g'(x), . 5. f [ , ] . G f [ , ] , : f(t)dt G( ) G( ) , x F(x) f(t)dt f [ , ] . G f [ , ] , c , G(x) F(x) c. (1) (1), x , G( ) F( ) c f(t)dt c c, c G( ) . , G(x) F(x) G( ) , , x , G( ) F( ) G( ) f(t)dt G( ) f(t)dt G( ) G( ) . 6. . ) : f(x)g (x)dx [f(x)g(x)] f(x)g(x)dx, f ,g [ , ] . ) : 2 1 u u f(g(x))g (x)dx f(u)du , f,g , u g(x) , du g (x)dx 1 u g( ) , 2 u g( ). 7.) f , x , x x x , f(x) 0 x [ , ] f . 27. http://lisari.blogspot.com : / ) f,g f(x) g(x) x [ , ] , f,g x , x : E( ) (f(x) g(x))dx ) f [ , ] f(x) 0 x [ , ] , f , x , x x x E( ) f(x)dx ) f,g [ , ] , c R , f(x) c g(x) c 0 , x [ , ] . (. 20) . () O x y y=g (x) y=f (x) () O x y y=f (x)+c y=g (x)+c 20 , (1), : ( ) ( ) [(f(x) c) (g(x) c)]dx (f(x) g(x))dx . E( ) (f(x) g(x))dx ) f(x) g(x) [ , ] , f,g x x E( ) |f(x) g(x)|dx ) x x , g, g(x) 0 x [ , ] x x : E( ) g(x)dx , x x f(x) 0 , E( ) (f(x) g(x))dx [ g(x)]dx g(x)dx . , g g(x) 0 x [ , ] , : E( ) g(x)dx O x y=g(x) y 21