infi-07b-sol11_0 - 2007 11" 20106 11" 1(1.2 x 2 x 1...

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Unformatted text preview: 2007 11 " 20106 11 " 1 (1 ) .2 - x 2 x - 1 x 1 , x 2 1 x - 1 0 2 x2 - 1 . x - 1 2 - x - 2-x- x -1 2 (2 ) x 2 2 - x 0 ,- ( ) .(3 ) (2 - x ) x - 1 x - 4 x + 4 x - 1 4 x 5 x 2 2 2 2 5 4 . ( - , - 1] [1, 5 ] : (3) - (2) ,(1) 4 . - 1 x 2 - 3 x + 1 1 , [ - 1,1] Arcsin x -1 x2 - 3x + 1 0 x2 - 3x + 2 . ,2 - 1 0 = x 2 - 3 x + 2 . ( -,1] [2, ) x 2 - 3 x + 1 1 x ( x - 3) 0 ,3 - 0 0 = x ( x - 3) . [0, 3] . ( ( -,1] [2, ) ) [0, 3] = [0,1] [2, 3] : - 2 (*) ( 2 x + x ) (1 - x ) > 0 . 3 x (1 - x ) > 0 : x = x x 0 . . x 0 . 0 < x < 1 . x (1 - x ) > 0 : x = - x x < 0 . 1 - x < 0 x < 0 . 0 < x < 1 : . tan x = tan x tan x 0 [0, ) - [ - , - ) . 2 2 . tan x 2 sin x : . - , sin x < 0 ( - , - ) - 2 1 2007 11 " 20106 . - , tan( - ) = sin( - ) = 0 - - cos x > 0 [0, ) - 2 tan x = sin x 2 sin x sin x 2 sin x cos x cos x - - sin x > 0 x 0 - , x = 0 - . 0 < x - cos x 1 1 2 cos x 3 2 . [0, ] {- } : 3 . tan x = - tan x tan x < 0 ( , ) - ( - , 0) 2 2 . - tan x 2 sin x : . - , sin x < 0 ( - , 0) - 2 cos x < 0 ( , ) - 2 - tan x = - sin x >0 sin x 2 sin x - sin x 2 sin x cos x - 1 2 cos x - 1 cos x 2 cos x . [ 23 , ) - . [0, ] [ 23 , ) {- } [ - , ] - : 3 3 b-a = b- a - a = b- a + -a 2 2 2 ( 2) (1) (2) (3) a a b- a + -a = b- a + a < + = a 2 2 2 2 2 2 - (1) - x = x ( 2) a a . a = = , (3) 2 2 2 4 (*) , x = x - x - .- 1 1 1 < x -1 < x - 1 < (i ) . 2 2 2 1 3 < x< (*) 1 2 2 . ,- 1 < x 2 9 9 7 9 (*) 4 < x + 3 < , < x + 3 < 2 2 2 2 2 2007 11 " 20106 . x+3 < 9 2 : x 0 (ii ) (*) 2 2 2 3- x 3 - x - 2 x (1) x + 2 x - 3 ( x - 1)( x + 3) (2) 1 1 x -1 x + 3 -1 = = = = 2 x 2x 2x 2x 2x - x = x - , ( - 1) - (1) . ab = a b (2) . 0 < < 1 2 , 1 x < 2 x > 1 (i ) x - 1 < 1 x - 1 < 2 2 . x+3 < 9 2 : .(**) .< 2 90 (*) 3- x 1 9 -1 = 1 x -1 x + 3 1 2 9 = 2 x 2 2 2x 2 2 9 1 = 1 , < : , 1 - 45 10 2 10 : ,( 0 < < 1 ) = 2 1 100 , , . , 2 (1) 3- x 9 9 1 x - 1 < -1 = < 2x 2 200 10 . x - 1 < 1 - (**)- (1) 2 . x 2 + sin x (*) x 2 - 1 x2 - 1 1 x2 - 1 x - 1 > = = 2 x + cos x 2 x + 1 2 x + 2 2 x + 1 2 : x > 2 . . , (*) : x > M , , M = 3000 . , x 2 + sin x x - 1 2000 > > = 1000 2 x + cos x 2 2 ,- M- : x 2 + sin x 2 x + cos x . (*)- 5 ("" or) . '- ' 3 2007 11 " 20106 . [ x ]2 = 4 [ x] = 2 2 x<3 or - 2 x < - 1 . [ x2 ] = 4 4 x2 < 5 2 x< 5 or - 5 < x -2 . . . 0 2 r < 2 - 2 x = 2[ x ] + 2 r . 0 r < 1 , x = [ x ] + r x . [2 x] = 2r < 1 2 [ x] 2 [ x ] + 1 2 r 1 . 0 r < 1 - n x = n + r 2 6 : , . . x R f ( x ) = 0 f f ( x ) = f ( f ( x )) = f (0) = 0 = f ( x ) : x R . R f , f (1) = 0 . . . f ( x) = f f ( x ) = f ( f ( x )) = f ( y ) : . f ( x ) = y x R =y . x = f ( x ) , x = y - ,-- f . , x R x = f ( x ) - : . f ( y ) = x - y R , f . x R . . . x = f ( y) = f f ( y ) = f ( f ( y )) = f ( x ) . , f ( x ) = x x R . f ( x ) < f ( y ) g ( x ) < g ( y ) : . x < y - x, y R . . , f ( x ) + g ( x ) < f ( y ) + g ( y ) - . ( f + g )( x ) < ( f + g )( y ) : . . . f ( x) = x g ( x) 0 , ( f + g )( x ) = f ( x ) .( 0 = g (0) < g (1) = 0 0 < 1 ) g 4 ...
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