ΤΕΛΕΥΤΑΙΑ ΕΠΑΝΑΛΗΨΗ ΜΑΘ ΚΑΤΕΥΘ ΓΛ

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1 - --- 2012 2 ENOTHTA 0 3-4 1 ( ) 5-7 2 ( ) 8-17 3 ( ) 18-46 4 ( )47-72 - . . mini-Book .. . . , 50-60 - - - ..vaggelisnikolakakis@hotmail.com .6937020032 -- ( ) - () - -TO 40() - () - - - - - -MATHEMATICA - () -. () -SPIVAK - Calculus (1980) -SPIVAK ( ) -APOSTOL - Calculus - (1969) 3 0 4 5 10 . ) f (x) =R _e 0 1 < o =. f R f (x) ln ' = o) f, g f'(x)=g'(x) x . c f(x)=g(x)+c xe. ) . f xo . . . ) f , . ) f xo f(xo)>0, f(x)>0 x xo. ) f [,] xoe(,) f'(xo)>0. ) f . f''(x)>0 xe. ) f [,] f(x) >0 ( ) f xdx|o}>0, xoe[,] f(xo)>0. ) f [,] ( ) f xdx|o}=0 f .

20 .. f (x) ln x=R-_e . f R-_e 1f (x)x' = . : . -f() = , R (fof)(2t) : .: 1,.: 0 ,.:1 , .: 1 -f() = , R f(0) = 1 : .: f() = +1 , .: f() = , .: f() = + 1 .: f() = - +1 - f() = , R i (f(4t) + if(4t))2004 : .: -1 , .: 1 ,.: i ,.:-i B. | ) 9 + , 0 : f 0 x) ( > + =xe x x x f q1 6 . : 22 ) (/ xxe x x x x f + + = ouv q. 201 ) (limxx fx+ 30 . z;( ). z1 , z2. : 2 1 2 1z z z z = . . .. zz1= z, z , z i , - z .. ) ( ) ( lim0ox xx f x f = f ox.. 1 ) ( lim =+ x fx f x=1.. f , . f ( x) = 0 x, f .. f , f ( ) =0, , : }=xadt t f x f ) ( ) ( x. 40 A. f ., f , f (x) 0 ' = , f .. y=x+ f +. . . ) z1,z2 z z z z z z 2 1 2 1 2 1+ s + s .) f 1-1 x1,x2 A e x1=x2 f(x1)=f(x2).) f, g ( ) () ( ) ( ) ( )00 0x xlim f x g x f x g x' ' ' ' = ) f [,] xoe(a,) f(xo)=0 f()f()0 i) z z . ii) z-iw. ) f f(1)=0 Re(zw)=0f(x)> 0 | | 1,1 e 12 60 . 1( ) 1,xf x xe x+= + eR z x i = + 1 xw e i+= . . f . xeR z w . . 3 222( )1x xf xxo | + =+. fC + : (2 ) 4 y x c o | = + , , * o | eR . . . f R 1 1 1'( ) ( 1)x x xf x e xe x e+ + += + = + . x -1 + ' f- 0+ f f ( , 1] [ 1, ) + 1 x = ( 1) 0 f = . ( )11 11lim ( ) lim( 1) lim( ) 1 lim( ) 1 11 1xx x x xx xxf x xee e++ + += + = + = + = 1lim ( ) lim( 1)xx xf x xe ++ += + = +. . 1 1 1 2 1 1 1 1( )( ) 1 1 ( )x x x x x x xz w x i e i xe xi i e i xe xi i e xe i e x+ + + + + + + = + = + = + + = + + 11 0 ( ) 0 1xxe f x x+ + = = = . . fC + : (2 ) 4 y x c o | = + : ( )lim 2xf xax+= lim[ ( ) (2 ) ] 4xf x a x |+ = .

3 2 33 3( ) 2lim lim limx x xf x ax axx|_o_ _ _+ + ++ = = =+ 2 1 o o o = = 13 3 223 2 2 2 22 2 2 22lim[ ( ) (2 ) ] lim[ ( ) ] lim( )12 ( 1) 2lim[ ] lim( ) lim1 1 1x x xx x xx xf x a x f x xxx x x x x x xx x x x|_| | ||+ + ++ + ++ = = =++ + = = = =+ + + 4 2 | | | = = 70 iz 2 4if (z) , z iz i+= = 1) M(z) , Imf (z) 0 = 2) u z i, w f (z) i = = uw 3) M(z) C K(i) , N(f (z)) C . z x yi, x, y = + e z i (x, y) (0,1) = =, : ix y 2 4i ( y 2 i(4 x))(x i(y 1))f (z)x i(y 1) (x i(y 1))(x i(y 1)) + + + = = =+ + 2 2 2 2x(y 2) (4 x)(y 1) x(x 4) (y 2)(y 1)ix (y 1) x (y 1) + + + + + + = ++ + . 1) (x, y) (0,1) = :2 2Im(f (z)) 0 x(x 4) (y 2)(y 1) 0 x y 4x y 2 0 = + + + = + + + = , 2 24 1 4( 2) 25 0 + = > . 1K 2,2| | |\ . 5R2= . 2) z i = : iz 2 4iuw (z i)(f (z) i) (z i) iz i + | |= = = |\ . 2iz 2 4i zi i(z i) 2 4i 1 3 4iz i+ += = + = +, 2 2| uw| ( 3) 4 9 16 25 5 = + = + = = . 3) | z i | 0 = > , (2) :5| f (z) i | 5 | f (z) i | = =, f (z) K(i) 5. 255 5 = = = 14 80 : 2z 2z 0 + +o = oe 1 2z , z 1 2z , z e . i) . ii) k k1 2z z + e k k1 2z z eI *ke . iii) A 1| z | 2 = , :) . ) . ) :n n *1 2(w z ) (w z ) 0,n = e. i) : ii. ( )k k k k k1 2 1 1 1z z z z 2Re z R + = + = e( )k k k k k1 2 1 1 1z z z z 2Im z I = = eiii. ) 21 2 1 1 1z z z z z 4= = o = o o =o ) : 4 16 12 A= = ( ) ( )11,22n n n n1 2 1 2z 1 3i2 2 3iz2z 1 3iw z w z 0 w z w z )= + = = = = ( )A 1, 3 ( )B 1, 3 . , y 0 = 90 f(z)= z4+z1+z +z4 , zeC , z = 0.) f(z) 9 e... z 9 e* z =1. ) z=x f (4, 12) f(z)= (1,2). ) f(z)=0 z =1 ... w= z5+z2+1 (0,0) =1. ) f(z) 9 e f(z)= ) z ( f z4+z1+z+z4 =z4+z1+z + z4 z1+z=z1+z z1-z1+z- z =0z zz z -( z -z)=0 ( z -z)(1-z z )=0 (z=z 2z =1) z 9 e* z = 0 z =1 15 ) z=x 9 e* 4 ( f(x) ( 12a ) x ( f = 4 ( ( 12 f(z)=f(x)=x4+x1+x+x4=2x4+x1+x f(x)=2x4+x1+x=2x5+x2-x+1=0 h(x)=2x5+x2-x+1 h [1,2] h(1)=4- ( 0 h(2)=69-2) 0 4( ( 12 8 ( 2( 24 -24 ( -2 ( -8 69-24 ( 69-2 ( 69-8 45( h(2) ( 61 : h(1)h(2) ( 0B u. x0e(1,2) h(x0)=0 : h'(x)=10x4+2x- 1 ( x ( 21 ( x4( 1610 ( 10x4( 160 i) 2( 2x( 4ii) -12 ( - ( -4 iii) i),ii),iii) + 0( h'(x) ( 160 : h| (1,2) ) f(z)=0 z4+z1+z+ z4=0 z5+1+z2+z z4=0 z5+z2+1=-z z z3z5+z2+1= - z31 z z2 5+ + =3z =1 100 16 110 17 18 10 z i = + , R e: , >1 f f(x) xz i x 1 ,x 0 = + >. x 0 >2x z xz i + > + : ) f(x) 0 = =0 ) 1 21z zz iz= =

2 1 2 1z z i z z i z z +> ) ( ) f ( )1 2z , z 0z 20 0z z z z i + > + ) = (0) ... 0 f = = ( ) ( )22 2f(x) x i i x 1 ... x x 1 x 1 = o +|+ = = o + | + x 0 >f ( ) 0,+ ( )( )( )( )2 2 222 22 2 2 2xx x 1 z x xz if (x) 1 1 0xz ix x 1 x x 1o +| +|o +| | + +| +' = = = >+o + | + o + | +

zx i 0 + > 2x z xz i + > + f ( ) 0,+ =0 ) 11z z = 11z z ( >1) = oo

2z iz =2z iz =i z = z= 1z z ) f( )22 2f(x) x x 1 x 1 = o + | + ( ) 0,+ ( ) ( )1 2z , z . f | ) 0,+ ( )1 2z , z . ( )1 2z , z 0zC e( )0 1 2z z , z e ( )( ) ( )2 102 1f z f zf zz z' = () 2 2 20 000 0z x z z z zf (x) 1 0 f ( z ) 1 0 1xz i z z i z z i+| +| +|' ' = > = > >+ + +.... 20 0z z z z i + > +3 19 20 1 2z, z , 1 2 1 2z +z z -z >. ) i 1 2 W = , W. ) i i ()1z 1 g i = + +() 2z 1 gi = + 0 > ( ) ig R () 0 0 g =( ) 0 0 g' = , = e . ) i ii () ( ) Im f w = f .Rolle | | 0 , 1 ( ) ( )( )1 11 1g ge e + = 30 20 A) ( )( )( )( )2 21 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2z z z z z z z z z z z z z z z z + s + s + + s ( )( ) ( )( )1 2 1 2 1 2 1 2 1 1 1 2 2 1 2 2 1 1 1 2 2 1 2 2z z z z z z z z z z z z z z z z z z z z z z z z + + s + + + s +( )( )1 2 1 2 1 2 1 2 1 22z z 2z z 0 2 z z z z 0 4Re z z 0 + s + s s x 0 s ) ( ) f x1z 1 i = + o ( )2z 1 f x i = + + ( )1 2Re z z 0 s . ( )( )( ) ( ) ( )( ) ( )( )( ) f x f x f x f x1 2z z 1 i 1 f x i 1 f x i i i f x = + o + + = + + + o + o o. ( ) ( )( ) f x1 2Re z z 0 1 f x 0 s + o s . . . . , . Fermat , . . ) :. Rolle , . 40 21 22 50 23 60 2 f(x)=eInx

, x>0 i) g(x)=) x ( f) x ( xf ) x ( f x' ' ' 2o + , xe(0,1) (1,+) . ii) i) () f(x) g(x) : x=2, x= >2 iii) + lim E ) ( i) 0s > = = } } 0 x 4 1 0 4x1dx 4x1dx ) x ( g ) x ( f22222 1-4x21x2141x 1 x 4 02 2 2s s s s > x 2 > 2 2 2x14 4x10 4x1 = < ()=} } } = + =((

+ = + = 2 2 222 221 18 4x1) 2 ( 4 dx )x1( dx 4 dx )x14 (=(4+ )217 1.. iii) + = E217 14) (2 4 0 0 4) (lim = + = E+ 70 f : R R , R () ()3f x 2f x 0 + =(1) x R e . i) f , ii) iii) f 1f iv) ( )( )12f xg xx= gC+ gC x 1 = x e = 25 26 80 f ( ) 4 , : ( )( ) ( ) x ' ' f x ' f 3 ex f= (1) x = ) 4 , ( x e , 0 ) x ( ' ' f > f ( ) 4 , . ) (3) 0 ) x ( f ) x ( ' f x0 0 0= + . ) x ( xf ) x ( g = . g [0,1] f(x) (), x g (0,1) ). x ( ' xf ) x ( f ) x ( ' g + = 0 ) 0 ( f 0 ) 0 ( g = = 0 ) 1 ( f 1 ) 1 ( g = = . Rolle 0 ) x ( f ) x ( ' f x 0 ) x ( ' g : ) 1 , 0 ( x0 0 0 0 0= + = e) (2) 12c3x) x ( ' f1x31) x ( ' f131)) x ( ' f () x ( ' ' f+ = '|.|

\|='||.|

\| = x=1 : 34c c31111 1 = + = 4 x ,x 4 3) x ( ' f3 4 x) x ( ' f1343x) x ( ' f1 < < , . ) ( ) ( ) ( )x 01f (0, e] ( limf x , f e ] ( , ]e+= = . ( )x x xln x 1lim f x lim lim 0x x+ + += = = , . ( ) ( ) ( )f1 1 1f D f (0, e] f [e, ) ( , ] (0, ] ( , ]e e e=+ = = ) . , , ) 69 . x 0,2t | |e |\ . , , . ( ) ( ) f x f x x x q = ouv q =ouv. x4t= ) . . . . . , , , ) . . , , . ) 70 150 71 72 160(to day) f , R , ( ) ( )x1g x f t dt =} ( ) ( )x 21 1z f t dt i f t dt = +} } x R e . Cf (1,2)(2,4) A. A ( )( )( )f 21f 1f x dx 2012=} , :

( ) i ( ) ( )21g 2 f t dt =}

() ii z . . ( ) ( )x 1xh x f t dt+= } ( ) i Ch . () ii Ch0=1 . . ( )( )2 2x2x xx1lim e e f t dt+ } . ( ) 2-3 -2 73 !!. - : , .!!!! -: .- , . - .- .- .- .- , . - . .- . - , . - ; . !!! ... . lzano ( ) ( ) g x f x 1 = Rolle ( ) G x . f , Rolle ( ) ( )xG x f t dt xo= }.. - , . - . .

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