mathematiko typologio
TRANSCRIPT
- 30 - 20 -10 0 10 20 30 -20-1001020 (Y) : 2003 . 1, , 30020 , , , . ( ) ( ) - - . , . , , , : [email protected] [email protected] " $ , : (, ) , , (,) [, ] , [, ] [, ) [, ), [, ) (, ] (, ], (, ] 4 - 4 . 1 1.1 , , . . x x. x x $ y : y e . . . , . : (1) : = {x / x x} (2) : = {x / x x} (3) : A = {x / x} Boole (1) ()=() (2) ()=() (3) = (4) = (5) ()=()() (6) ()=()() (7) A B A B = De Morgan (8) A B A B = (9) AA=A (10) AA=A 10 . 210 - . 2. ( ) . (Cantor) ( 1-1 ) . - 1-1 . . 0 . ( , ). 0 - 5 . 5 - c kai . To c . Cantor 02=c, . Cantor , - 1, 2, 3 Cantor: c=02=1 H , 5 . : (Natural) {0, 1, 2, 3, } : {, -2, -1, 0, +1, +2, } : (Rational) { / , e } : (Irrational) {x e / x , e } 2 , 3 , e=2,718, =3,1415927 ( x e ) : (Real) {x e / xe xe} : (Complex) {z e / z=x+iy x,y e i = - 1} c c c c c 1.2 . . - 1.3 (Primenumbers) . .. 1, 2, 3, 7, 13, , . . 1.4 AB=A =A . =5 12+ : =1,6180339887498948482 - Fibonacci: ](1)=1, ](2)=1, ](3)=](1)+](2) ]()=](-2)+](-1) 1.5 () () () (), 0 s < , =+ 6 - , 6 . =0 1.6 , 2 =2. ={0, 2, 4, 6} 2, 1. =2+1. - ={1, 3, 5} 1.7 2 : (), 0, 2, 4, 6, 8 3 : 3 4 : 00 4 5 : 0 5 6 : 2 3 7 : - 7 8 : 8 9 : 9 10 : 0 11 : + 11 ( ) 1.8 : -. : ( ) . . 3 . . . 28 14 7 1 2 2 7 28 = 227 = 272001005025512 2 2 5 5 200 = 25 1.9 ... . . : 60 30 15 5 1 2 2 5 5 60=25 18090459312 2 5 3 3 180=253 15007503757515312 2 5 5 5 31500=2 53 ... (60, 180, 1500) = 25 = 45 = 20 (,) : , >. : =1+1 - 7 . 7 =12+2 1=23+3 . -2=-1+ =0, 1, 2, -1, ( ) 1.10 ... : . - : ... (60, 180, 1500) = 2 53 = 41259 = 4500 [,] ... 1 1.11 . :. > = = : : : = : 2 22 1 23 355 3 5 151= = = 77 7 5 3512 22 1 25 5= = = , ..: 1123+ 1.12 : ) ) . 92, 254 = 103, 3333333... 3, 33 = = 71, 1666666... 1, 166 = = 201, 571428571428... 1, 5714287 = = ( ) - . (simple) , 2,66666 (mixed) , 2,3473737373 (73) (34). , : 8 - 8 . 31, 97464646 x = 31, 9746 104x = 319746, 46 102x = 3197, 46 : (104-102)x = 316549 9900 x = 316549 x = 3165499900 : ) - : 23122, 312100= ) , ( ) , 9 : 212 2 2102, 1299 99= = 3 0 30, 39 9= = ) 9 0 . 2512 25 24872, 512990 990= = 47231 472 467590, 4723199000 99000= = 1.13 . . 28=1+2+4+7+14. 27 . 1.14 () () . ()= . 1.15 . =() =() 220 : (220) = {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110} (220) = 284 284 : (284) = {1, 2, 4, 71, 142} (284) = 220 1184 1210, 17296 18416 1.16 (, , ) : =+ ...(, , )=1 (, , ) (, , ) . (, , ) =2, =- =+ (2+1, 2+2, 2+2+1), , () - 9 . 9 1.17 . 2 . : (Lambert 1770). O e Euler, . . . , , 0z2 z1, z2 e) : >, =, 0 >0 +>0 > > > >, >0 > >, >0 > > > ( ,,, ) ! 2+1>2+1 > 2>2 ||>|| >>0 > >>0 1 1 < x>y>0 >1 x>y x>y>0 0 e : > > >>0 n : n>n -n0 0 s0 =0 >0 >0 0 ++0 > > >0 >0 ++=0 ===0 ++0 0 0 0 > > >0 >0 > 0 (+)(+)(+)8, ,,e+ 2(++)(+)+(+)+(+), ,,e+ 2(+)(+) 3(++)(++) 3 + + > , ,,e+ 3 + + > , ,,e+ e (1+2++)(1+2++) (x+y)s (+)(x+y) (++)1 1 1 | |+ + |\ .9 , , 0 : ++3 ++33 ,, : + + + ArithmeticGeometricalHarmonic A-G-H (11+22++)s(2 2 21 2 ... + + + )( 2 2 21 2 ... + + + ) 2 2 2i i i ii 1 i 1 i 1 = = =| | s |\ . 1 21 2 ... = = =(1,2,,) =(1,2,,) , || || Buniakoski-Cauchy-Schwartz : || s |||| 14 - - 14 . p, q>1 1 1p q+ =1 i1 1n n np qqpi i ii 1 i 1 i 1 = = =| | | |s | |\ . \ . |1|p-1/|1|=|2|p-1/|2|= p>1 i, i0 1/1=2/2==n/n 12n 12n : 1 2 n 1 2 n 1 1 2 2 n n ... ... ... n n n+ + + + + + + + +| || | s | |\ .\ . ( )( )1 2 n 1 2 n 1 1 2 2 n n ... ... n( ... ) + + + + + + s + + + 1=2==n 1=2==n 1, 2, 3, , 0 (1+1)(1+2)(1+) 1+(1+2++) , 0 1+2++(-1) < 1 1++-1 x0 : (1+x) > 1+x >1 + | | |\ . \ . \ . - 15 . 15 2.5 = ... _, =1, 1= =+ = ()= | |= |\ . -= 1 | | | |= | |\ .\ . ( ) = 2.6 : x2= x= , =12 , 0 0 = , 1 1 = = = ( )2 = 2 = 4 = ( ) = 12 = 2.7 : x= x= , =1 , = = = + = = = ( ) = = 2.8 x= x=log ex= x=ln 10x= x=log e=2,71828 Euler, >0, 1, >0 loga1=0 logaa=1 log aa = log()=log+log log()=log-log log=log log =1log log= log log ln1=0 lne=1 ln(xy)=lnx+lny ln(xy)=lnx lny ln=ln lnex=x elnx=x ln= log log e 2.9 =1+(-1), =+1- , , .. = 2+ S=1 2+ =1-1, = 1+ , , .. = S=11-1 , S=111 ||1 . : >1 - ( ), (, ) x+=0 0 x= =0 0 =0 x+x+=0 =-4 >0 =0 0 x1 x2 =0 x 0 ( 1 2) ](x) = (x-1)(x-2) =0 ( , ) ](x) = (x-) = 2x2| |+ |\ . 3 : >0 0 (2) D=0 (3) D0 x 0 y xx=-p/2(p/2, 0) p 2px1 y1 = 2px1 y1 < 2px1 y=2px (x1, y1) yy1 = p(x + x1) : . y = 2px (x1, y1) oo oyy y (x x )p = 4.8.7, y=x+ y=2px. p 2 = 0 y=2px (x1, y1). (y y1)=(x x1) y = x + (y1 - x1). 4.9.8 p 2(y1 x1) =0 . y x (x1, y1) yx - 29 . 29 . Z - x y -4 -2 0 2 6 8 10-2 0 2 4 6 8 S L : = 2 2 22 2 S S 4L S 16LS 16L ln2 8L S| |+ + |\ . : = 23SL 30 - 30 . 4.10 ( ) . -. : + = 2 2. ( >). 2 =- 2, 2. (-, 0) (, 0) 2 22 2x y1 + = , =- ( - ) (x, y) : 2 2o o2 2(x x ) ( y y )1 + = - , 2 222 2 2 2 =+ 2(1 )1 = =2 2 = y x y ( ) 2 ( ) =2 2 = >0. - . - ( ) (, 0) (-, 0) : 2 22 2x y1 = , = - (x0, y0) x y : 2 20 02 2( x x ) ( y y )1 = (rectangular) = : x - y = 2 2 . . . 2 22 2x y1 = : y x y x = = -, : 2 2 1 += = > 2 22 2x y1 = 2 22 2y x1 = 2 22 2x y1 = x1, y1 : 1 12 2xx yy1 = xy 2 2 xy 34 - 34 . 2 22 2x y1 = (1) y = x + (2) (1) (2) - - = 0 2 22 2x y1 = (x1, y1). y y1 = (x x1). 4.11.7 =y1 x1 - - (y1 x1) = 0 xx - : 2 222 2 2 2 = x : 2( 1)1 = - Z - xy - 35 . 35 4.12 ( ) , ( , ). . . . . (, , ) , (focus), , (directrix), . (,) 1 1 = = = = = yx : 1 36 - 36 . 4.13 2 ](x,y) = x+xy+y+2x+2y+ = 0 2 . xy xy - 2 x+xy+y+2x+2y+ = 0 x+xy+y+2x+2y+ = 0. : A B A B D B B D ' ' '' ' ' ' = = =' ' ' A B A Bd d ' = = = S = A + = + = S Q=- : / d D DS Q 1 () - - - - = =0 2 () >0 0 0 - - =0 - - - 3 () 0 x x2 2( A ) ( ) 42 + += y y=-1/x 2 2DA x y 0d' ' ' ' + + = z-Sz+d = 0 : 2 2A ( ) 4A2+ + +' = 2 2A ( ) 42+ +' = y=2x 2 2 S A B' =+ (x,y) : o oo oASx BSy ( A ) 0( S )x (ES AE )y ZS 0+ + + = `+ + + + = ) | |= |\ . 38 - 38 . 5 5.1 3 , x, y, z . (x, y, z) x, y, z xy, (0s0, ] [,], e(,) : (x)g(x)dx () g(x)dx () g(x)dx = +} } } ] [,] : (x)g(x)dx () g(x)dx =} } ] [,] : (x)g(x)dx () g(x)dx =} } ] g ] g [,] (x)g (x)dx (x)g(x) g(x) (x)dx' ' = } } : (x) () () = (x)dx} x t x=(t). (t) a b x . a=() b=() (t) [,] ][(t)] [, ] : ( ) b a(x)dx (t ) (t )dt ' =} } x=-t : 0 0 0(x)dx ( t )dt ( t )dt = = } } } x= 2-t : /2 /20 0(x, x)dx (t, t )dt =} } ], : T 0 (x)dx (x)dx+=} }, x=t+ - 87 . 87 ] ](x)0 [,] (x)dx} C] x x= x=. ](x)s0 : =-(x)dx} xy 0 2 4 6 8-4 -2 0 2 4 ] ](x)>0 [,] C] x 2V (x)dx = } n x=, x1, x2, , xn= y=](x), y1=](x1), , yn=](xn), h=(-)/n. : o 1 n 1(x)dx h( y y ... y )~ + + +} o 1 2 n 1 nh(x)dx ( y 2y 2y ... 2y y )2 ~ + + + + +} o 1 2 3 n 1 nh(x)dx ( y 4y 2y 4y ... 4y y )3 ~ + + + + + +} Simpson () 88 - 88 . 9.5 (Scalar product) ,=(1, 2, 3) ,=(1, 2, 3) - , , (e ) , ,=, 0 s s , ,=11+22+33 , ,= 0 ,, , ,= ,||, 9.6 (Vector product) ,=(1, 2, 3) ,=(1, 2, 3) ,, : ,,= n, n, (n=1) , , - ,, , n, ( , , n, ) : ,,=1 2 31 2 3i j k , , , ,,= (23-32) i,+(13-31) j,+( 12-21) k, ,,= - ,, ,,= 0, ,||, ,,=n, ,, , : - 89 . 89 10 , : , .. , 1993 Luis Brand, , ... The pocket professor, 2500 Math Formulas , Murray R. Spiegel, John Liu, , ( Schaum) , . , . , . , , . , http://www-groups.dcs.st-and.ac.uk/~history/index.html http://mathforum.org/ , , . Drexel http://mathworld.wolfram.com/ . , , , 90 , : - 90 . C Cantor, 4 Cauchy, 10, 13, 56 Cauchy Euler, 10 D De Moivre, 10, 19 F Fibonacci, 5 L Lagrange, 10 N Newton, 10 P Pascal, 10 , 16 , 6 A-G-H, 13 Bernoulli, 14 Buniakoski-Cauchy-Schwartz, 13 Chebyshev, 14 Hlder, 14 Minkowski, 14 Weierstrass, 14 , 14 , 5 , 5 , 5 , 5 , 5 Euler, 15 , 16 , 5 , 5 , 10 , 10 , 4 , 19 , 6 , 5 De Moivre, 19 , 6 , 6 , 6 , 6 , 6 , 21 , 5 , 6 , 18 , 18 , 18 , 18 , 18 , 10 , 7 , 7 , 20 , 18 , 19 , , 15 , 16 , 4 , 6 , 8 , 4 , 4 , 4 , 4 -, 4 , 4 , 4 , 4 , : - 91 . 91 Pascal, 10 Euler, 19 , 5 92 , : - 92 . 1 ..................................................................................................................... 4 1.1 ......................................................................................................................................... 4 1.1.1 ............................................................................................................................. 4 1.1.1.1 Boole .................................................................................... 4 1.1.2 ....................................................................................................................... 4 1.1.3 .......................................................................................................................... 4 1.1.4 ........................................................................................................ 5 1.2 .................................................................................................... 5 1.3 (Prime numbers) ........................................................................................... 5 1.4 .......................................................................................................................... 5 1.5 ................................................................................................................... 5 1.6 , ............................................................................................................ 6 1.7 ............................................................................................................................... 6 1.8 .............................................................................. 6 1.9 ....................................................................................................... 6 1.10 ................................................................................................. 7 1.11 .................................................................................................................................... 7 1.12 ...................................................................................................................... 7 1.13 .......................................................................................................................... 8 1.14 ..................................................................................................... 8 1.15 ........................................................................................................................... 8 1.16 .................................................................................................................. 8 1.17 ......................................................................................................................... 9 1.18 .................................................................................................................... 9 1.19 .................................................................................................................. 9 2 .................................................................................................................. 10 2.1 ................................................................................................................. 10 2.2 .......................................................................................................................... 10 2.3 ................................................................................................................................ 11 2.4 - ............................................................................................................... 12 2.4.1 : ................................................................................................ 12 2.4.2 ........................................ 13 2.4.3 Buniakoski-Cauchy-Schwartz (B-C-S) ......................................................... 13 2.4.4 Hlder ............................................................................................................... 14 2.4.5 Minkowski. ....................................................................................................... 14 2.4.6 Chebyshev ........................................................................................................ 14 2.4.7 Weierstrass ....................................................................................................... 14 2.4.8 .......................................................................................................... 14 2.4.9 Bernoulli ........................................................................................................... 14 2.5 .................................................................................................................................... 15 2.6 ................................................................................................................... 15 2.7 .............................................................................................................................. 15 2.8 ................................................................................................................................ 15 2.9 .................................................................................................................................... 15 2.10 ................................................................................................................................... 16 2.10.1 ....................................................................................................... 16 2.10.2 ..................................................................................................... 16 2.10.2.1 .................................................................... 16 2.10.2.2 ..................................................................................................... 16 2.10.2.3 ........................................................................................................ 17 , : - 93 . 93 2.10.3 ......................................................................................................... 17 2.10.4 >3 ..................................................................................... 17 2.10.5 .......................................................................................................... 17 2.10.6 .................................................................................................. 18 2.10.7 5 ............................................................................ 19 3 ................................................................................................... 20 3.1 ..................................................................................................................................... 20 3.2 .................................................................................................................... 20 3.3 .................................................................................................... 20 3.4 .................................................................................................... 20 3.5 ..................................................................................................................... 20 3.6 ........................................................................................................ 20 3.7 - ..................................................................... 20 3.8 ........................................................................................................... 21 3.8.1 ............................................................................................. 21 3.9 ...................................................................................... 21 3.10 ............................................................................................................. 21 3.11 ............................................................................ 21 4 ........................................................................... 22 4.1 ...................................................................................................... 22 4.1.1 ................................................................................ 22 4.1.2 ....................................................................................................... 22 4.2 .......................................................................................... 22 4.2.1 ............................................................................. 22 4.2.2 ............................................................................. 22 4.2.3 ........................................................................... 23 4.3 ................................................................................................................... 23 4.3.1 ........................................................................................... 23 4.3.2 ................................................................................................... 23 4.4 . .................................................................... 23 4.5 ..................................................................................................................... 23 4.6 .................................................................................................................. 23 4.7 ........................................................................................................................................ 24 4.7.1 .................................................................. 24 4.7.2 .............................................................. 24 4.7.3 .............................................................. 24 4.7.4 ............................................ 24 4.7.5 ....................................................................................................... 24 4.7.6 ...................................................................................................... 24 4.7.7 ..................................................................................................................... 24 4.7.7.1 .................................................... 24 4.7.7.2 1 3 ............................................................................. 24 4.7.7.3 2 4 ............................................................................. 24 4.7.7.4 x .................................................................................. 24 4.7.7.5 y .................................................................................. 24 4.7.7.6 x ............................................................................................................ 24 4.7.7.7 y ............................................................................................................ 25 4.7.8 .......................................................................................... 25 4.7.9 1 2 ...................................................................... 25 4.7.10 ............................................................................. 25 4.8 ....................................................................................................................................... 26 4.8.1 ............................................................................................................................... 26 4.8.2 .................................................................................................................. 26 4.8.3 ...................................................................... 26 94 , : - 94 . 4.8.4 ........................................................................................ 26 4.8.5 3 .................................................................. 26 4.8.6 (x1, y1) ........................................................... 26 4.8.7 ................................................... 26 4.8.8 ............................................................ 26 4.9 .............................................................................................................................. 27 4.9.1 ............................................................................................................................... 27 4.9.2 ........................................................................................................... 27 4.9.3 ................................................................................................ 27 4.9.4 ............................................................................................... 28 4.9.5 ........................................................................... 28 4.9.6 ................................................................................... 28 4.9.7 ........................................................................................... 28 4.9.8 ............................................ 28 4.9.9 ...................................................................... 28 4.9.10 ..................................................................................... 28 4.9.11 .................................................................. 29 4.10 ...................................................................................................................................... 30 4.10.1 ............................................................................................................................... 30 4.10.2 ................................................................................................................ 30 4.10.3 ............................................................................................. 30 4.10.4 ................................................................................................ 30 4.10.5 ................................................. 31 4.10.6 ............................................................... 31 4.10.7 .......................................................................... 31 4.10.8 .............................................................................................. 31 4.10.9 ................................................................................... 31 4.10.10 .................................................................................... 31 4.11 ................................................................................................................................... 33 4.11.1 ................................................................................................................................ 33 4.11.2 ...................................................................................................... 33 4.11.3 ............................................................................................... 33 4.11.4 ................................................................................................... 33 4.11.5 ............................................................................................................. 33 4.11.6 ..................................................................................... 33 4.11.7 ................................................ 34 4.11.8 ...................................................................... 34 4.11.9 ............................................................................................... 34 4.11.10 ............................................................................... 34 4.12 .................................................................................................... 35 4.13 .................................................................................................... 36 4.13.1 ............................................................................................................................... 36 4.13.2 ........................................................................................................................... 36 4.13.3 ......................................................................................... 36 4.13.3.1 , , : ..................................................................................... 36 4.13.3.2 ................................................................................................................. 37 5 ............................................................................. 38 5.1 ...................................................................................................... 38 5.1.1 ................................................................................ 38 5.1.2 ................................................................................. 38 5.1.3 .................................................................................... 38 5.1.4 ......................................................... 38 5.2 ............................................................................................................ 39 , : - 95 . 95 5.3 ...................................................................................... 39 5.4 ..................................................................................................................... 39 5.5 .................................................................................................................... 39 5.6 ..................................................................................................................................... 40 5.6.1 .................................................................................................... 40 5.6.2 ..................................................................................................................... 40 5.6.3 ........................................................................ 40 5.6.4 3 ............................................................... 40 5.6.5 ............................................................................................................. 40 5.6.6 ............................................................................... 41 5.6.7 ........................................................................................ 41 5.6.8 ........................................................................................ 41 5.6.9 ......................................................................................... 41 5.7 .................................................................................................................... 42 5.7.1 ............................................................................................... 42 5.7.2 ..................................................................................................................... 42 5.7.3 ............................................................................................................... 42 5.7.4 ......................................................................................................... 43 5.7.5 ......................................................................................................... 43 5.8 .................................................................................................................. 44 5.8.1 .................................................................................................................................. 44 5.8.2 ........................................................................................................................ 44 5.8.3 ..................................................................................................................... 44 5.8.4 .................................................................................................................... 45 5.8.4.1 ............................................................................................ 45 5.8.4.2 .......................................................................................... 45 6 .................................................................................................. 47 6.1 ...................................................................................................................................... 47 6.1.1 ................................................................................................................................ 47 6.1.2 .................................................................................................................. 47 6.2 .................................................................................................................................. 48 6.2.1 ..................................................................................................... 48 6.2.2 ......................................................................................... 49 6.2.3 ............................................................................................................ 49 6.3 ............................................................................................................... 49 6.3.1 Cramer ......................................................................................................... 49 6.3.2 Gauss ............................................................................................................ 50 7 ........................................................................................................ 51 7.1 ..................................................................................................................................... 51 7.2 ................................................................................................................... 52 7.3 ....................................................................... 52 7.4 ...................................................................................................... 52 7.5 , , , ............................................................................ 53 7.6 ......................................................................................................... 53 7.7 ................................................................................................................... 53 7.8 ..................................................................................... 53 7.9 (/2) .................................................................................... 54 7.10 ........................................................................................ 54 7.11 ..................................................................... 54 7.12 , ............................................................................................ 54 7.13 ............................................................... 55 7.14 .............................................................................................................. 55 7.15 .................................................................................................................... 56 7.15.1 ............................................................................................................. 56 96 , : - 96 . 7.15.2 ....................................................................................................... 56 7.15.3 ...................................................................................................... 56 7.15.4 ........................................................................................................... 56 7.15.5 Brigg ................................................................................................................... 56 7.15.6 ................................................................................................................ 56 8 ............................................................................................................ 58 8.1 ..................................................................................................................................... 58 8.2 : ..................................................................................................................... 58 8.3 : ................................................................................................................. 58 8.4 1-1: .................................................................................................. 58 8.5 ............................................................................................................ 59 8.6 .............................................................................................................. 59 8.7 ........................................................................................................... 59 8.8 .............................................................................................................. 59 8.9 ............................................................................................................. 59 8.9.1 : ............................................................................................................................... 59 8.9.2 : ............................................................................................................................. 59 8.10 ................................................................................................ 60 8.11 ....................................................... 60 8.12 y = ](x) = x+ ............................................................................. 61 8.13 y=](x)=x+x+ ..................................................................... 61 8.14 y=](x)=xn ....................................................................................................... 61 8.15 x .............................................................................................. 62 8.16 ............................................................................................................. 62 8.17 ...................................................................................................... 62 8.18 ....................................................................... 63 8.18.1 ........................................ 64 8.18.2 ........................................................................ 64 8.18.3 ..................................... 64 8.18.4 ............................ 64 8.19 ......................................................................................................... 65 8.19.1 ............................................................................................................................... 65 8.19.2 ......................................................................... 65 8.19.3 ...................................................................... 65 8.19.4 ........................................ 65 8.19.5 ............................................................................................................. 65 8.19.6 ............................................................................................................ 66 8.19.7 .................................................................................................................... 66 8.19.8 .............................................................................................................. 66 8.19.9 ................................................................................. 66 8.19.10 ................................... 66 8.19.11 ................................................................... 66 8.19.12 .......................................................... 67 8.20 ................................................................................... 69 8.20.1 ............................................... 69 8.20.2 ............................ 69 8.20.3 ................................................. 70 9 .................................................................................................................. 71 9.1 ...................................................................................................................... 71 9.1.1 ..................................................................................................................... 71 9.1.2 .................................................................................................................. 71 9.1.3 ......................................................................................................................... 71 , : - 97 . 97 9.1.4 .............................................................................................. 71 9.1.5 , .................................................................................... 71 9.1.6 ........................................................................................................ 72 9.1.7 ...................................................................................................... 72 9.2 (continuity) .......................................................................................... 72 9.2.1 x .................................................................................................................. 72 9.2.2 .................................................................................................................. 72 9.2.3 .......................................................................................... 72 9.2.4 ............................................................................................................. 73 9.2.5 ........................................................................................... 73 9.3 .......................................................................................................... 74 9.3.1 y=](x) x .......................................... 74 9.3.2 ......................................................................... 74 9.3.3 .................................................................................................... 74 9.3.4 ..................................................................................... 74 9.3.5 ...................................................................... 75 9.3.6 .......................................................................................... 75 9.3.7 ..................................................................... 75 9.3.8 ...................................................... 75 9.3.9 - ............................. 75 9.3.10 ............................................................................................... 76 9.3.10.1 Leibnitz .......................................... 76 9.3.10.2 .................................................................. 76 9.3.11 ................................................................................................. 77 9.3.12 Rolle ............................................................................................................ 77 9.3.13 Darboux ...................................................................................................... 77 9.3.14 ( ) ................................................................ 77 9.3.15 (Cauchy) ......................................................................... 77 9.3.16 De L Hospital ............................................................................................. 78 9.3.16.1 ..................................................................................... 78 9.3.17 Taylor ............................................................................................................... 78 9.3.18 ............................................................................. 79 9.3.18.1 ( ) .................................................................. 79 9.3.18.2 ( ) ..................................................................... 79 9.3.18.3 ( ) ................................. 79 9.4 ......................................................................................................................... 81 9.4.1 ....................................................................................................... 81 9.4.1.1 .................................................................................... 81 9.4.1.2 .................................................... 82 9.4.1.3 (partial fractions) ............................... 83 9.4.2 ................................................................................................. 83 9.4.2.1 ........................................................................................ 84 9.4.2.2 ............................................................................................. 84 9.4.2.3 .......................................................................... 84 9.4.2.4 Leibnitz ............................................ 85 9.4.2.5 ....................................................... 85 9.4.2.6 ............................................ 85 9.4.2.7 ............................................................................................... 85 9.4.2.8 .................................................................................... 86 9.4.2.9 ....................................................................................................... 86 9.4.2.10 .................. 86 9.4.2.11 ......................................................................... 87 9.4.2.12 ................................................................................... 87 9.4.2.13 .................................................................. 87 98 , : - 98 . 9.5 .......................................................................................... 88 9.6 ........................................................................................... 88 10 , : ............................................................................................... 89